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DEFECT-INDUCED ELECTRICAL/OPTICAL PROPERTIES OF SrTiO3-X (001) BY PHOTO-ASSISTED TUNNELING SPECTROSCOPY Asa Frye A Dissertation in Materials Science and Engineering Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 1999 Dissertation Supervisor Graduate Group Chairperson ABSTRACT DEFECT-INDUCED ELECTRICAL/OPTICAL PROPERTIES OF SrTiO3-X (001) BY PHOTO-ASSISTED TUNNELING SPECTROSCOPY Asa Frye Dawn A. Bonnell The (001) surface of monocrystalline strontium titanate is used in a variety of commercial applications as an active substrate or electrode and therefore represents an important technological material. Its functionality is enabled by defect states energetically located in the forbidden gap which are introduced upon removal of oxygen from the lattice. Studies by conventional surface analysis techniques as well as first principles calculations have not yet resulted in agreement regarding the microscopic origin of the acceptor type surface states. The combined techniques of optical spectroscopy and scanning tunneling spectroscopy, however, afford a unique opportunity to probe the local origins of deep level surface states by direct modification of the surface charge density through optical excitation. The technique of photo-assisted tunneling spectroscopy (PATS) using continuous illumination of mono-energetic light was applied to study the optical responsivity of a series of samples with increasing degrees of reduction. The surface structures were characterized by conventional scanning tunneling microscopy (STM), photo-assisted tunneling microscopy (PATM), and low energy electron diffraction (LEED). A theoretical model was developed to iv generate tunneling spectra and facilitate interpretation of the experimental results. This work reports the first STM images and STS spectra obtained on undoped and transparent single crystalline SrTiO3-x. The surface structure and optical responsivity was found to strongly depend on the processing conditions, where the latter increased with increasing degree of reduction. The results were explained in terms of variations of the local surface potential induced by local charge transfer mechanisms, where evidence of both increasing and decreasing surface charge was observed depending on the incident photon energy. It has been determined that oxygen vacancy association is necessary to introduce a deep level gap state centered at 1.77 eV below the conduction band edge and that this state is localized on surface terrace sites. This work represents the first successful demonstration of spectroscopic PATS, combined with theoretical modeling, as a strong metrological tool to study the local electrical/optical properties of wide band v gap semiconducting oxide materials. To my mother, the love that put me on the right track; and to my wife, the love that keeps me from derailing. ii ACKNOWLEDGMENTS I’d like to thank GOD for giving me the gift of imagination and the courage to use it wisely. I’d also like to thank my thesis advisor, Prof. Dawn Bonnell, who provided me the opportunity to work on a challenging and unique research problem. Special thanks and appreciation is owed to my thesis committee members — Prof. Peter Davies, Prof. Takeshi Egami, Prof. Jack Fischer, and Prof. Roger French — all of whom have offered valuable insight and guidance towards the success of my studies and accomplishments, as well as my development as a scientist. I am indebted to my wife Solita Moran-Frye and my son Atiba Rivera who have endured six years of sacrifice while remaining both supportive and encouraging. Deep appreciation is extended to my office mates, Kelly Brown, Bryan Huey, Sergei Kalinin, James Kiely, Marilyn Nowakowski, Jack Smith and Paul Thibado who have shared and contributed in special ways to make my experience at Penn both enjoyable and rewarding. Additional deep appreciation is extended to: Dr. Fred Allen for his friendship and genuine interest in my success as a graduate student; Prof. John DiNardo, Prof. B. Graham, Prof. C. Graham, Prof. John Vohs, Prof. Alan T. “Charlie” Johnson, Dr. Xiaomei Li, Dr. Xue-Feng Lin, and Dr. Dave Carroll who have all generously shared equipment and/or time and expertise that helped to broaden my technical skills; Prof. L. A. Girifalco for insightful scientific discussions and helping me to see the value in “sticking to my guns”; and Prof. David Luzzi for recognizing both my strengths and weaknesses and, most importantly, telling me about them. iii And last, but not least, a thousand thanks are extended to Irene Clements, Pat Overend, Donna Samuel, Donna Hampton, and Cora Ingrum, all of whom have always been there to offer meaningful words of support, encouragement iv and understanding. Table of Contents Abstract ............................................................................................................................. iv List of Tables..................................................................................................................... ix List of Figures .................................................................................................................... x Chapter 1 Introduction and Background .................................................................... 1 1.1: Motivation for study..................................................................................................... 1 1.1.1 SrTiO3: a critical technological material.............................................................. 1 1.1.2 Photo-assisted tunneling microscopy and spectroscopy........................................ 2 1.2: Background on SrTiO3 ................................................................................................. 3 1.2.1 Bulk structure and properties................................................................................. 3 1.2.2 Surface structure and properties.......................................................................... 16 1.3: Photo-assisted tunneling spectroscopy....................................................................... 25 1.3.1 Introduction to PATS............................................................................................ 25 1.3.2 Three basic photon absorption mechanisms ........................................................ 27 1.3.3 Limitations of PATS ............................................................................................. 30 1.4: Thesis objectives ........................................................................................................ 35 References ......................................................................................................................... 36 Chapter 2 Experimental .............................................................................................. 42 2.1: Photo-assisted tunneling spectroscopy....................................................................... 42 2.1.1 Experimental arrangement................................................................................... 42 2.1.2 Experimental method............................................................................................ 47 2.1.3 Experimental noise............................................................................................... 49 vi 2.2: Sample preparation and characterization ................................................................... 51 2.2.1 Sample processing history.................................................................................... 51 2.2.2 Methods of characterization ................................................................................ 56 References ......................................................................................................................... 57 Chapter 3 Tunneling Spectroscopy ............................................................................ 58 3.1: Introduction ................................................................................................................ 58 3.1.1 Quantum mechanical tunneling and the WKB approximation............................. 58 3.1.2 The purpose of modeling tunneling spectra ......................................................... 61 3.2: The tunneling model .................................................................................................. 63 3.2.1 One-dimensional quantum transmission.............................................................. 63 3.2.2 Effects of specular transmission........................................................................... 68 3.2.3 The potential distribution functions ..................................................................... 73 3.2.4 The potential barrier functions ............................................................................ 79 3.2.5 Determination of the defect-induced current ....................................................... 90 3.3: Sample of calculation................................................................................................. 93 3.3.1 Simulated vs experimental spectra....................................................................... 93 3.3.2 Parametric study of tunneling model ................................................................... 97 3.3.3 Discussion .......................................................................................................... 103 References ....................................................................................................................... 106 Chapter 4 Characterization of The Bulk ................................................................. 108 4.1: Bulk properties of reduced SrTiO3 ........................................................................... 108 4.1.1 Hall/resistivity measurements ............................................................................ 108 4.1.2 Optical measurements ........................................................................................ 112 vii 4.1.3 Discussion .......................................................................................................... 120 4.1.4 Conclusions ........................................................................................................ 123 References ....................................................................................................................... 125 Chapter 5 Characterization of Vicinal SrTiO3 (001) .............................................. 126 5.1: Structure and chemistry of reduced SrTiO3 (001).................................................... 126 5.1.1 LEED/Auger observations.................................................................................. 126 5.2: Morphological structure by STM............................................................................. 133 5.2.1 Surface morphology of V–930............................................................................ 133 5.2.2 Surface morphology of V–1100.......................................................................... 138 5.2.3 Surface morphologies of the H series ................................................................ 144 5.2.4 Surface morphology of V–930Nb ....................................................................... 148 5.2.5 Summary of observed morphologies .................................................................. 151 5.3: Surface electronic properties by STS and PATS ..................................................... 153 5.3.1 Terrace and step edge electronic properties by STS.......................................... 153 5.3.2 Terrace optical responsivity by PATS................................................................ 158 5.3.3 Summary of observed optical responsivity......................................................... 180 References ....................................................................................................................... 181 Chapter 6 Discussion and Conclusions .................................................................... 182 6.1: Discussion of results ................................................................................................ 182 6.1.1 Photo-assisted tunneling microscopy and spectroscopy.................................... 182 6.1.2 Surface structures and morphologies................................................................. 183 6.1.3 Defect-induced electronic properties ................................................................. 186 6.1.4 Conclusions ........................................................................................................ 191 viii References ....................................................................................................................... 193 Chapter 7 Summary of Dissertation......................................................................... 194 Appendix A: Franck-Condon principle and the spectroscopic resolution .............. 196 Appendix B: Semiconductor defect statistics ............................................................. 201 Appendix C: Mathematica code for modeled tunneling spectra .............................. 208 ix List of Figures Figure 1.1 Coordinated octahedra structure as adopted by strontium titanate. ............... 5 Figure 1.2 Top: Calculated bulk electronic band structure of SrTiO3............................. 6 Figure 1.3 Ordering of oxygen vacancy point defects in nonstoichiometric cubic perovskite ...................................................................................................................... 12 Figure 1.4 Sphere packing model showing the ideal (001) termination of a ABO3 perovskite surface.......................................................................................................... 16 Figure 1.5 TiO2 termination of (001) SrTiO3 showing titanium adatoms at a terrace site and at a step edge. ......................................................................................................... 23 Figure 1.6 Surface photovoltage effect upon illuminating a n-type depletion semiconductor with energies equal to or greater than the band gap energy, Eg............ 27 Figure 1.7 Other photoabsorption mechanisms............................................................. 29 Figure 1.8 The laser induced thermovoltage versus irradiance..................................... 31 Figure 1.9 Electric field intensity between tip and sample versus irradiance ............... 33 Figure 2.1 Experimental arrangement for photo-assisted tunneling spectroscopy. ...... 43 Figure 2.2 Spectral response of experimental optics..................................................... 45 Figure 2.3 Quantification of current variance ............................................................... 50 Figure 2.4 Laue back-reflection photograph showing 〈001〉 orientation....................... 52 Figure 2.5 AFM image showing the stepped surface of SrTiO3 ................................... 53 Figure 2.6 STM images showing a stepped surface...................................................... 53 Figure 2.7 A 500 nm × 500 nm AFM image showing a stepped surface...................... 55 Figure 2.8 AFM images of heavily reduced SrTiO3 (001)............................................ 55 x Figure 3.1 A particle wave of energy E propagating within a piecewise constant potential or within a continuous potential function....................................................... 60 Figure 3.2 Schematic representation of the energy band structure of a metal with respect to a semiconductor in a non-equilibrium (Va ≠ 0) configuration...................... 69 Figure 3.3 Equivalent circuit for a metal-vacuum-semiconductor tunnel junction at forward bias................................................................................................................... 73 Figure 3.4 Calculated potential across the sample, Vs, as a function of the total applied bias, Va. ......................................................................................................................... 77 Figure 3.5 An equilibrium (Va = 0) configuration for a metal-vacuum-semiconductor tunnel junction separated by a gap of width s. .............................................................. 78 Figure 3.6 Calculated spatial and voltage dependent vacuum potential barrier............ 81 Figure 3.7 Calculated surface potential (i.e., band bending) as a function of the voltage component across the sample........................................................................................ 87 Figure 3.8 Comparison of equilibrium band bending for monovalent and divalent donors in a semiconductor with band gap energy Eg = 3.2 eV. .................................... 89 Figure 3.9 Calculated tunneling spectrum using the parameters in Table 3.1. ............. 94 Figure 3.10 Comparison of calculated and experimental spectra. ................................ 95 Figure 3.11a,b a) Increasing carrier density: 1, 5, and 10 × 1019 cm-3. b) Increasing surface potential: 0.25, 0.30, and 0.35 eV..................................................................... 98 Figure 3.11c,d c) Increasing static dielectric constant: 100, 210, and 300. d) Increasing effective mass: 5, 12, and 50. .................................................................. 99 Figure 3.11e,f e) Increasing tunneling gap: 8, 9, and 10 Å. f) Increasing electron affinity: 2.6, 3.0, and 3.4 eV. ...................................................................................... 100 xi Figure 3.12 The predicted effect of increasing surface charge density in steps of ∼1.40×10-7 coulombs per cm2. .................................................................................... 102 Figure 4.1 Resistivity and carrier density of undoped single crystal SrTiO3 .............. 110 Figure 4.2 The dispersion curves for the optical constants (n and k) of SrTiO3......... 114 Figure 4.3 The dispersion curves for the absorption coefficient of SrTiO3. ............... 117 Figure 4.4 Dielectric function of SrTiO3 below anomalous dispersion. ..................... 119 Figure 5.1 Chart recorder traces showing AES spectra of vacuum reduced SrTiO3 (001) surface............................................................................................................... 127 Figure 5.2 Two distinct LEED patterns from SrTiO3-x (001) vicinal surfaces............ 128 Figure 5.3 AES spectra of sample STO–8 heat treated successively at: 500, 1000, 1100, and 1200 °C for 5 minutes each. ....................................................................... 130 Figure 5.4 The 2 × 2R45o superstructure corresponding to the LEED pattern of Figure 5.2b. ................................................................................................................. 132 Figure 5.5 Multiple unit cell high step edges observed on V–930.............................. 134 Figure 5.6 Comparison of dark and illuminated surfaces, with incident photon energy of 3.6 eV...................................................................................................................... 134 Figure 5.7 Comparison of dark and illuminated surfaces, with incident photon energy of 1.9 eV...................................................................................................................... 135 Figure 5.8 Comparison of dark versus illuminated sections of a single surface. ........ 136 Figure 5.9 Step with apparent holes along the edge and at the kink. .......................... 137 Figure 5.10 Surface hole formed by local chemical attack. ........................................ 138 Figure 5.11 Surface morphology of V–1100 showing wandering step edges. ........... 139 Figure 5.12 Apparent cluster-free surface with concave and convex step edges........ 140 xii Figure 5.13 Series of convex step edges separated by ∼20 Å...................................... 141 Figure 5.14 Terrace cluster structure of heavily reduced SrTiO3 (001)...................... 142 Figure 5.15 Comparison of dark and illuminated surfaces, with incident photon energy of 2.95 eV. ....................................................................................................... 143 Figure 5.16 Terrace and step edge morphology of H–700.......................................... 145 Figure 5.17 Terrace and step edge morphology of H–1000........................................ 146 Figure 5.18 Local terrace cluster structure.................................................................. 147 Figure 5.19 Comparison of dark and illuminated surfaces, with incident photon energy of 3.8 eV. ......................................................................................................... 147 Figure 5.20 Terrace and step edge morphology of V–930Nb..................................... 148 Figure 5.21 Terrace and step edge morphology of V–930Nb..................................... 150 Figure 5.22 Local terrace cluster structure of V–930Nb............................................. 150 Figure 5.23 Linear and semi-log plots comparing the terrace electronic properties in the H series.............................................................................................................. 154 Figure 5.24a Step edge on V–930Nb where local electronic structure is observed to vary as shown by the tunneling spectra in Figure 5.24b. ............................................ 155 Figure 5.24b,c Terrace versus step edge electronic behavior.. ................................... 156 Figure 5.25 Dark versus light spectra for H–700 illuminated with 3.8 eV light......... 160 Figure 5.26 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.040 eV; ∆β = 3.5 × 10-4 C/V; ∆χ = 0.80 eV. .................................................. 162 Figure 5.27 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.016 eV; ∆β = 1.5 × 10-4 C/V; ∆χ = 0.50 eV. .................................................. 163 xiii Figure 5.28 Surface photo-effect matched with the following parameter variations: ∆ψ = - 0.023 eV; ∆β = - 3.3 × 10-4 C/V; ∆χ = - 0.65 eV. ........................................... 164 Figure 5.29 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.035 eV; ∆β = 0 C/V; ∆χ = 0.40 eV. ............................................................... 165 Figure 5.30 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.145 eV; ∆β = 1.0 × 10-4 C/V; ∆χ = 0.70 eV. .................................................. 166 Figure 5.31 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.040 eV; ∆β = - 6.0 × 10-4 C/V; ∆χ = 0.20 eV................................................. 167 Figure 5.32 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.150 eV; ∆β = - 6.3 × 10-4 C/V; ∆χ = 0.40 eV................................................. 168 Figure 5.33 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.170 eV; ∆β = - 6.3 × 10-4 C/V; ∆χ = 0.40 eV................................................. 169 Figure 5.34 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.054 eV; ∆β = 0 C/V; ∆χ = 0.30 eV. ............................................................... 170 Figure 5.35 Surface photo-effect matched with the following parameter variations: ∆ψ = - 0.090 eV; ∆β = 0 C/V; ∆χ = 0.10 eV. ............................................................. 171 Figure 5.36 Surface photo-effect matched with the following parameter variations: ∆ψ = - 0.140 eV; ∆β = 3.0 × 10-4 C/V; ∆χ = - 0.70 eV. ............................................. 172 Figure 5.37 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.018 eV; ∆β = 0 C/V; ∆χ = 0.26 eV. ............................................................... 173 Figure 5.38 Surface photo-effect matched with the following parameter variations: ∆ψ = - 0.043 eV; ∆β = - 1.5 × 10-4 C/V; ∆χ = - 0.50 eV. ........................................... 174 xiv Figure 5.39 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.032 eV; ∆β = 1.5 × 10-4 C/V; ∆χ = 0.50 eV. .................................................. 175 Figure 5.40 Surface photo-effect matched with the following parameter variations: ∆ψ = 0.014 eV; ∆β = 9.6 × 10-4 C/V; ∆χ = 0.40 eV. .................................................. 176 Figure 5.41 The photo-induced change in surface charge determined by modeling the observed changes in the tunneling spectra. ........................................................... 177 Figure A.1 Illustration of molecular energy as a function of internuclear distance.... 197 Figure A.2 Scheme to depict defect thermal ionization energies; scheme to depict the same defect spectroscopic energies............................................................................. 198 Figure B.1 a) A defect-related state occupied by two electrons of opposite spin; b) the same defect-related state with one electron removed to the conduction band. ........... 203 xv List of Tables Table 1.1: The irreps of high symmetry points and directions in a simple cubic lattice.... 7 Table 1.2: The ten irreps and corresponding term symbols for point group Oh................. 7 Table 1.3: Schottky and “Schottky-like” defect reactions in cubic SrTiO3 ..................... 10 Table 1.4: Theoretical and observed defect-induced ionization energies in SrTiO3-x...... 24 Table 3.1: Parameters used to calculate the tunneling spectrum in Figure 3.9. ............... 93 Table 4.1: Thermal history and Hall/resistivity measurements...................................... 109 Table 4.2: Optical transition energies deduced from Figure 4.3a. ................................. 118 ix Chapter 1: Introduction and Background Recent unresolved issues regarding the structure and properties of oxygen deficient SrTiO3 (001) are presented in this chapter. Following a general description of the current understanding of bulk and surface structures, the method of photon-assisted tunneling spectroscopy is described and the objectives of the present study are proposed. 1.1 MOTIVATION FOR STUDY 1.1.1 SrTiO3: a critical technological material Transition metal oxides (TMO) are an important class of technological materials that have received a great deal of attention in recent years. The variability in oxidation state of the transition-metal cation largely accounts for the observation of various stable bulk and surface structures as well as versatility in physical properties. Considerable progress has occurred over the past several decades in the development of experimental tools and techniques aimed towards elucidating the fundamental nature of a variety of processes that occur on surfaces [1]. The application of many of these methods to the study of TMOs is well-represented in the literature [2]. Despite these gains, the details of the microscopic mechanisms of processes on oxide surfaces remain largely undetermined. Particular knowledge of the effect of local surface geometric and electronic structure, for example, in determining the efficiency of surface reactions is of vital use to several industries. Strontium titanate is an excellent example of a model TMO that has found widespread application in various technologies. It has been identified as a good substrate candidate for use in photoelectrochromic devices where a charge transfer (redox) reaction is facilitated by a defect state energetically located in the band gap of the oxide [3]. Similar surface defect-related 10 properties have made SrTiO3 the focus of research in the fields of photocatalysis and solar energy conversion [4,5]. Advancements in other technologies where SrTiO3 has been identified as a critical or potentially critical material, such as gas sensors [6], superconducting thin film growth [7], and memory storage devices [8], depend on increased understanding of corrosion mechanisms, high temperature reconstructions, defect interactions, etc., at the surfaces and grain boundaries. Much of the photo- and chemical-reactivity of the (001) surface have been linked to extrinsic states in the forbidden energy gap. The pursuit to determine the microscopic origin of these states or the microscopic mechanisms of these reactions, however, has given rise to controversies [2]. 1.1.2 Photo-assisted Tunneling Microscopy and Spectroscopy Recent developments in scanning probe techniques have permitted the surface science community to witness the marriage of scanning tunneling microscopy and optical spectroscopy. In principle, the high energy resolution of optical transition processes [9] combined with the high spatial resolution of the STM presents a unique opportunity for the characterization of adsorption modes and photocatalytic (or other charge-transfer) reactions. The former was demonstrated recently in a study of water adsorption on RuS2 and TiO2 electrodes [10]. Increasing humidity was observed to substantially increase photocurrent efficiencies. This was explained in terms of a molecular adsorbate-induced channel for hole annihilation at the surface of the electrode via recombination with tunneling electrons. It was thus suggested that the enhanced photocurrent can be used to distinguish between molecular and dissociative adsorption. Other charge-transfer mechanisms have been observed by photo-assisted tunneling spectroscopy (PATS) as will be discussed further in section 1.3. 11 There are three primary ways (depending on the energy of the incident light) in which photoabsorption may be detected in tunneling spectra using the method described in Chapter 2. Comparison of dark and illuminated spectra can give (1) a surface photovoltage, (2) a direct photocurrent, or (3) modified band bending. The last effect suggests the potential to correlate local trapped surface charge with associated surface defect structure. 1.2 BACKGROUND ON SrTiO3 The physical properties of strontium titanate have been studied extensively for well over thirty years. These research efforts have accumulated a profusion of facts leading to deeper understanding of some phenomena and greater bewilderment of others (compare, for example, [11] with [12] and [13] with [14]). This section reviews the current level of knowledge of bulk and surface structure and properties of monocrystalline SrTiO3. 1.2.1 Bulk structure and properties The cations Sr and Ti, in terms of the ideal ionic model, assume their group oxidation states. It is thus clear from the chemical formula SrTiO3 that all ions have closed-shell electronic configurations. The attractive component of the lattice energy is dominated by electrostatic interactions and may be estimated using the formal charges Z by an equation of the form [15] E coh 1 = 2Nc Z iZ je 2 ∑i ∑ rij j ≠i [1.1] where the indices i and j vary independently over all ions in the crystal, r is the separation between two ions i and j, and Nc is the number of formula units (or unit cells) in the crystal. This energy (also known as the Madelung potential) is the most significant contribution to the cohesion of the solid. A recent calculation of the cohesive energy gives a value of approximately 12 149 eV per formula unit [16]. This energy may be compared to the lattice energies for other oxides such as TiO2 (126 eV), Al2O3 (165 eV), ZrO2 (116 eV) or SrO (33.4 eV) [17]. At room temperature SrTiO3 adopts the ideal cubic perovskite structure which may be described as a close packing of Sr+2 and O-2 ions with Ti+4 occupying one quarter of the octahedral interstices. Alternatively, one may consider the structure as a network of polyhedra, as illustrated in Figure 1.1, from which its simple cubic symmetry (crystallographic space group Pm3m) is readily apparent. The basic structural unit is the Ti+4-O6-2 octahedron and the crystal consists of corner shared octahedra with Sr+2 occupying the icosahedral interstices. Each oxygen is coordinated to two Ti ions (linearly) and to four Sr ions, where the Ti-O bond length is smaller than the Sr-O bond length. There are eight other oxygen ions surrounding each oxygen with an O-O bond length equivalent to the Sr-O bond length. The unit cell edge is 3.905Å [18] so that the bond lengths are approximately 1.95Å and 2.76Å for the Ti-O bond and Sr-O (O-O) bond, respectively. Early determinations of the electronic structure of strontium titanate [19–21] utilized the LCAO (linear combination of atomic orbitals) or tight binding approach of Slater and Koster [22] or molecular-orbital (MO) methods based on a local density approximation for electron correlation. The limitations of both methods are well-known — the former overestimates the band gap energy while it is underestimated by the latter. These calculations, however, afford significant insight into the electrical and optical properties of SrTiO3 and several predictions based on these results are consistent with experimental observations [23–25]. 13 Figure 1.1 Coordinated octahedra structure as adopted by strontium titanate. The small black spheres are Ti ions; the large sphere is the Sr ion. 14 Figure 1.2 Top: Calculated bulk electronic band structure of SrTiO3. The vertical energy scale is measured in Rydbergs (≈ 13.6 eV) [ref. 20]. Bottom: The Broullouin zone for a simple cubic lattice, where the symmetry points and directions are labeled in terms of the BouckaertSmoluchowski-Wigner symbols [ref. 26]. 15 Table 1.1: The irreps of high symmetry points and directions in a simple cubic lattice. Symbol Symmetry Γ (zone center) Oh ∆ ([001]) C4v Λ ([111]) C3v Σ ([110]) C2v M,X D4h R Oh T C4v Z,S C2v Table 1.2: The ten irreps and corresponding term symbols for point group Oh. BSW symbol Term symbol Γ15 Γ25 ′ Γ25 Γ1 Γ1′ Γ2 Γ2′ Γ12 Γ12′ Γ15′ A1 A1 A2 A2 Eg Eu T1g T1u T2g T2u The band structure calculation of L. F. Mattheiss [20] and the Broullouin zone corresponding to a direct lattice with cubic symmetry are shown in Figure 1.2 where the representations of the points and directions of high symmetry are given by the conventional BSW symbols [27] as outlined in Table 1.1. Of particular interest to the present study is the behavior of the electronic structure along the ∆ direction from the center (Γ) to the edge (Χ) of the zone. The ten irreducible representations (or irreps) for the cubic point group Oh and their corresponding term symbols are listed in Table 1.2. The terms highlighted in bold represent the symmetry classes to which the oxygen s and p Bloch sums are adapted in order to interact with the metal d orbitals. This interaction gives the “molecular orbitals” which overlap and form the energy bands of the solid [28]. For [001] propagation, the five Ti d states reduce to the following 16 classes of the ∆ group: ∆1, ∆2, ∆2', and ∆5. The s and p Bloch sums propagating along [001] may be classified into all but the ∆2' class. Consequently, the t2g state with ∆2' symmetry does not “mix” with the oxygen states giving a flat lowest energy conduction band. All other d states (two t2g states with ∆5 symmetry and two eg states with ∆1 and ∆2 symmetry, respectively) mix with the appropriate symmetry adapted oxygen s and p states to form the higher energy conduction bands. The lower band states consist of non-bonding (Γ15 and Γ25) as well as bonding (Γ15) oxygen Bloch sums which predominately constitute the valence band. Several experimental investigations have produced results supporting as well as disputing the qualitative features of the band structure calculations of Mattheiss and others [19]. For example, Cardona [24] studied the electronic structure using reflectivity spectra which contained features consistent with the calculated splitting of the valence bands at zone center. Perkins and Winter [29] later demonstrated the correlation between Cardona’s results and the calculated joint density of states based on theoretical band structure. The early transport measurements of Frederikse and co-workers [23] reported a mobility effective mass of approximately 16me (where me is the free electron mass) in close agreement with that predicted by the band structure calculations of Kahn and Leyendecker [19]. Although some groups attributed the fundamental absorption edge to a direct transition at zone center (Γ15→Γ25) in agreement with Mattheiss’ results, it has been demonstrated [30] that this excitation is in fact indirect (Γ15→Χ3) and assisted by an optical phonon mode. It is well established that the energy of this transition (the optical band gap) lies near 3.21 eV at room temperature. It should be noted, however, that a recent approximate calculation based on electron and hole formation energies suggests a thermal band gap energy of 4.35 eV 17 [16]. Moreover, Reihl et. al. [31] found it necessary to assume a gap energy of 4.5 eV in order to achieve the best fit between theoretical density of states (DOS) and experimental DOS obtained from photoemission studies. It is not unusual that the band gap energy is an adjustable parameter for theoretical band structure calculations. The Frank-Condon principle explains why the thermal excitation energy for a particular ionization process is expected to differ from the optical excitation energy for the same process. Based on this principle, however, and in contrast to what is suggested in the discussion above, the optical excitation energy is expected to exceed the thermal ionization energy. (Appendix A contains a more thorough discussion on optical excitation processes which are accompanied by large vibrational coupling.) Nominally pure SrTiO3 is an electronic insulator at room temperature. Incorporation of point defects into the lattice can generate free charge carriers or charged ionic species. In the latter case, the defects may be associated or unassociated. In semiconducting SrTiO3 at room temperature the charge carriers are predominately electrons introduced by donor impurity doping or heating in a reducing atmosphere. The latter treatment introduces an approximately equivalent density of oxygen vacancies which are known to exhibit a significantly large lattice mobility, particularly at elevated temperatures [32]. SrTiO3 is thus considered a mixed electronic-ionic conductor. Oxygen vacancies may also form as a mode for incorporation of acceptor-type impurities or intrinsic acceptor defects. Acceptor-type impurities, such as Al or Fe, were once believed to be present in oxides at levels typically no less than 10–100 ppma (parts per million atomic) [33]. In a given sample of SrTiO3 this suggests an accidental impurity density of the order 1017–1018 cm-3. Intrinsic acceptor defects — i.e., strontium vacancies — are formed during high temperature processes such as sintering or annealing. Note that the dominant ionic disorder in 18 this close-packed lattice has been shown to be Schottky and “Schottky-like” [16]. Table 1.3 lists the reactions that have the lowest defect formation energies. Highly charged defects such as VTi−4 are energetically unfavorable in ionic structures. It has thus been suggested that a deficiency in Sr should be observed for samples processed at sufficiently high temperatures. The stability of strontium vacancies in the structure is further supported by the observation of a finite solubility (up to 1000 ppma) of excess TiO2 in SrTiO3 and an associated modification of the equilibrium conductivity as expected from the following incorporation scheme [14]: TiO 2 → TiTi + 2OO + VSr−2 + VO+2 . [1.2] In the above reaction, the ionic defects are assumed unassociated. It should be noted that an earlier study [13] suggested excess TiO2 compensation by neutral (i.e., associated) vacancy pairs (V −2 Sr +2 ,VO ) since the observed conductivity behavior remained unaffected. Excess SrO, resulting from reaction I in Table 1.3, may be easily consumed by the formation of Ruddlesden-Popper phases [34], and thus also leave the equilibrium conductivity behavior unaffected. Table 1.3: Schottky and “Schottky-like” defect reactions in cubic SrTiO3 Formation energy (eV)* Reaction −2 Sr +2 O I SrTiO3 → V + V + SrO 1.53 II SrTiO3 → VTi−4 + 2VO+2 + TiO 2 2.48 III SrTiO3 → VSr−2 + VTi−4 + 3VO+2 + SrTiO3 * see reference 16 19 1.61 The structural accommodation of oxygen vacancies in perovskite oxides may be illustrated as shown schematically in Figure 1.3. The far left image shows stoichiometric AMO3. Removal of one oxygen from the lattice reduces two M+4 to M+3 and replaces two octahedra with two square-pyramids, as shown by the center image. The limiting structure consisting of all M+4 in the lattice being reduced to M+3 with all octahedra being replaced with square-pyramids is shown by the far right image. The perovskite structure cannot support further removal of oxygen from the lattice. Note, however, that this limiting phase has a reduced lattice point symmetry (i.e., C4v) which suggests the lifting of energy band degeneracies. Evidence of this type of oxygen vacancy ordering in perovskite oxides has been observed in heavily reduced CaMnO3 [35,36]. Grossly oxygen deficient strontium titanate, in the limit corresponding to the formula SrTiO2.5, will contain an oxygen vacancy density of the order 1021 cm-3. It should be noted, however, that under extreme reducing conditions ( PO 2 ≈ 10-13 Torr) the oxygen vacancy density has been reported to only be in the range 2.0–7.6 × 1019 cm-3 [13]. This corresponds to one oxygen vacancy per 512 unit cells, or chemical formula SrTiO2.998. 20 Figure 1.3 Ordering of oxygen vacancy point defects in nonstoichiometric cubic perovskite [ref. 36]. The introduction of band gap states, due to removal of oxygen from the lattice, has been investigated theoretically using a tight-binding model including consideration for the effects of lattice relaxation [37]. The results for an isolated vacancy, obtained from a second-neighbor approximation, suggest that a doubly occupied (i.e., charge neutral) state with Eg symmetry lies 0.116 eV below the conduction band edge. The singly ionized state was found to have an energy and symmetry that depended on the strength of the attractive potential assumed for the defect site. Less attractive potentials give 0.27 eV, where the state has Eg symmetry. More attractive potentials give 0.41 eV, where the state has Eu symmetry. These energies may be compared to those determined from resistivity and Hall data which were found to be in the range 0.07–0.14 eV for carrier densities in the range 3×1018–8×1016 cm-3, respectively [38]. The discrepancy between these values may be due to the neglect of mutual screening effects from neighboring oxygen vacancies (see below). Relaxation of the titanium ions towards the vacancy increases all defect binding energies, while the latter decrease when the titaniums are relaxed away from the 21 vacancy site. It is concluded that the oxygen vacancy may be characterized as an ionic species — i.e., the localized gap states result only from the potential due to the charge of the defect. Similar conclusions have been drawn from recent ab initio supercell band structure calculations [39]. The incorporation of two or more oxygen vacancies in a supercell containing eight unit cells gives rise to a variety of unique density of state functions depending on the distribution of defects in the supercell. The defect densities considered (one to three defects per supercell) are already of the order 1021 cm-3. Metallic behavior is thus predicted in every case. The significant finding is that vacancy clustering is necessary in order to form well-defined band gap trap states. In the most heavily reduced case, corresponding to three homogeneously distributed vacancies per unit cell (i.e., SrTiO2.625), conduction band states were found to extend to 1.5 eV below the Fermi level. When all three defects reside on the same octahedra, a more narrow band of states result at 0.5 eV below the conduction band edge. In light of the previous discussions, one might expect singly-ionized oxygen vacancies to predominate at room temperature for heavily reduced samples, while doubly-ionized oxygen vacancies are expected to predominate for moderately reduced samples. Furthermore, for lightly reduced samples, it has been proposed that transport occurs via the conduction band at room temperature whereas at sufficiently low temperatures (< 50 K) transport occurs via an oxygen vacancy induced defect band [38]. As the oxygen vacancy density increases, it is assumed that the width of this band also increases, while the ionization energies decrease. In heavily reduced SrTiO3 (as in Nb-doped crystals) it is often assumed that the defect state may be described by the hydrogenic model according to which the binding energy of an electron to a singly-ionized defect is given by [28] 22 Eb = RH (m* m e ) κ 2st , [1.3] where RH is the Rydberg constant (13.6 eV), m * is the density of states effective mass, and κ st is the static dielectric constant of the undoped/stoichiometric crystal. Clearly, the large dielectric constant of SrTiO3 (~ 300) ensures that all defect states will be ionized down to very low temperatures, as observed by Yamada and Miller [18]. Assuming an electron effective mass of 12, [1.3] gives Eb ≈ 1.8 meV. Therefore, in heavily reduced and Nb-doped SrTiO3, transport occurs exclusively via the conduction band. On the other hand, if a significant degree of defect interaction occurs in the oxygen deficient case, a localized defect band is expected to trap electrons at room temperature such that the conduction band carrier density will exactly equal the density of oxygen vacancies in the lattice. Much of the uncertainty regarding the nature of the oxygen vacancy defect and its occupancy may be due to the variety of observed optical activity which is strongly correlated with the degree of sample reduction (see Table 1.4, page 24). Optical transmission studies have established a window of transparency for pure stoichiometric SrTiO3 in the energy range 0.25– 3.1 eV [40,41]. The lower energy is determined by the onset of optical phonon excitation while the upper energy is determined by the onset of charge transitions between the valence and conduction bands. Additional absorption bands are observed in reduced SrTiO3 and BaTiO3 single crystals. In heavily reduced SrTiO3, free carrier absorption gives rise to a 0.9 eV peak. There is also a 2.9 eV peak that is observed to saturate with increased reduction [42,43], and a 2.14 eV peak introduced in strongly reduced SrTiO3 which is speculated to be due to divacancies (i.e., two associated singly-ionized oxygen vacancies) [44]. Both reduced and Nb-doped samples have been reported to exhibit absorption at 2.4 eV which was assumed to be due to a charge 23 transition from the lowest conduction band to a higher conduction band of mixed Ti-4p and O-3p nature [43,45]. For moderately reduced samples, absorption at 1.77 eV is observed for nominally pure samples but not for Nb-doped samples [43,44]. It is believed that this is due to excitation of a single charge from a singly-ionized oxygen vacancy, or vacancy-related defect, to the conduction band. Both the 1.77 eV and 2.4 eV energies were also reported to explain transient absorption (i.e., photochromic) behavior in reduced SrTiO3 in terms of charge transitions from band gap states to the conduction band [46]. A similar effect was observed in reduced BaTiO3 where the energies were reported to be 1.8 eV and 2.6 eV [47]. It can be concluded that the exact nature of the oxygen vacancy defect, in particular the mechanism of optical absorption in the bulk, remains largely unresolved. A similar uncertainty holds for the nature of the oxygen vacancy and associated defects on the (001)-terminated surface. Indeed, there is doubt as to whether the assumption of surface band gap states is necessary to explain the often observed photoelectrochemical activity of reduced SrTiO3 since the same behavior can be explained in terms of a bulk response [48]. The following section discusses the presently understood properties of the (001)-terminated surface of strontium titanate. The format begins with a description of the structure of the ideal bulktruncated lattice. Next, the mechanisms of relaxation and restructuring are discussed for both the stoichiometric and non-stoichiometric surfaces. Finally, the effects of the above geometrical considerations on the electronic structure at the surface are described and discussed in light of recent experimental observations. 1.2.2 Surface structure and properties Truncation of the bulk lattice along 〈001〉 can result in one of two possible terminations: one with stoichiometric composition SrO or one with stoichiometric composition TiO2. This is illustrated 24 in Figure 1.4 where the large black spheres represent Sr ions, the small black spheres represent Ti ions and the white spheres are the oxygen ions. The oxygen coordination of Sr decreases from the bulk value of 12 to a surface value of 8. Similarly, the oxygen coordination of Ti decreases from the bulk value of 6 to a surface value of 5. In order to understand the relative stability of these two possible terminations, one must consider the manner in which charge is redistributed at the surface and how this gives rise to shifts in atomic positions. Figure 1.4 Sphere packing model showing the ideal (001) termination of a ABO3 perovskite surface. The large black spheres are A ions and the small black spheres are B ions. This model also shows the geometry of a surface oxygen vacancy on the BO2 plane [ref. 2]. The mechanism of charge redistribution depends on the surface type. Based on electrostatic criteria, P. W. Tasker [49] established a classification of surface types. A type I surface requires that the charge in each plane parallel to the surface is distributed such that these 25 planes are charge neutral. If these planes are not charge neutral but are stacked to give a repeat unit (or bi-layer) with a net zero dipole moment, this is called a type II surface. A type III surface is a polar surface — i.e., each layer parallel to the surface has a finite charge σ and the stacking supports a net dipole between each layer throughout the crystal. Type III surfaces are associated with a high surface energy and the instability of such surfaces has recently been explained in terms of a simple electrostatic effect as discussed below. The (001) surfaces of SrTiO3 are generally considered type I (assuming formal charges on the ions). As pointed out by Noguera and co-workers, however, the real charges on the ions are not likely to be the formal charges so that perovskite (001) surfaces are likely to have type III properties [50]. They proposed to describe the (001) termination as “weakly polar” to distinguish it from standard polar surfaces such as ZnO terminated at the basal planes. Assuming that surface relaxation is negligible, Noguera and co-workers have developed a criteria for the stability of polar and “weakly polar” surfaces. It is based on the fact that each bilayer contains a constant electrostatic field such that the electrostatic potential increases monotonically with the thickness of the crystal. The total dipole moment is directly proportional to the number of bi-layers so that an infinite energy is associated with macroscopic crystals and accounts for the instability of the surface. This macroscopic dipole moment, however, is found to be completely removed if the magnitudes of the surface σs and subsurface σss charges are modified to fulfill the following condition: σ ss − σ s = σb , 2 [1.4] where σb is the magnitude of the charge on the bulk (001) planes. This is referred to as the electrostatic condition for the stability of a polar surface. 26 It is straightforward to show that in the absence of a modification of surface/ subsurface intraplane covalency (i.e., σss = σb), the stability condition is met for the (001) termination of SrTiO3 simply as a result of the bond breaking mechanism in the formation of the SrO and TiO2 surfaces [50]. Therefore, significant modification in the electronic structure is not expected. The reduction of the surface ionic charges with respect to the bulk has been verified by X-ray photoemission studies [4,51]. The latter has also given evidence, however, in support of enhanced covalency of the (001) surface due to a decrease in the Madelung potentials as compared with the bulk. When this decrease is significantly large, relaxation will be observed. Relaxation can be anticipated from the results of the ionic model. The general approach requires a minimization of the full expression for the lattice energy (i.e., [1.1] plus a short-range repulsive term) to determine the equilibrium inter-atomic separation, rijeq . The well-known result shows that rijeq is an increasing function of the coordination number and inversely proportional to the Madelung constant (where the Madelung constant varies more slowly than the coordination number). Since both the SrO and TiO2 terminations necessarily contain under coordinated atoms, one might expect a contraction for the outer layers of both surface terminations. Detailed theoretical calculations [52] and a detailed LEED study [11] both confirm an inward relaxation of the SrO-terminated plane. The former study also predicted an inward relaxation of the TiO2 plane, while the latter study observed an outward relaxation of this plane (at T = 120K). Yet another group reported an outward relaxation on both terminations [12]. Whatever the actual case may be, it is clear that the equilibrium structure is influenced by a balance between shortrange (intra-atomic) and long-range (inter-atomic) Coulombic forces. The former favors increased ionicity while the latter favors increased covalency. This competition is found to explain and predict a variety of surface effects in SrTiO3 including a planar ferroelectric domain 27 structure on the SrO face [52], as well as the surface electronic structure for the stoichiometric and oxygen deficient TiO2-terminated face. Other structural modifications at the (001) surface may be attributed to differences in the properties between the surface ions. The large polarizability of the anions as compared to the cations, for example, is effective in shielding the often significantly large field associated with the surface dipole moment. This field exerts a greater force on the cations than on the anions giving rise to surface rumpling. This effect can be thought of as an inhomogeneous inward relaxation of the surface where the cations are relaxed more strongly than the anions. Evidence of rumpling was observed in the LEED investigations by Bickel et al. [11]. Understanding the crystallography and morphology of the real SrTiO3-x (001) surface is critical to the interpretation of the observed electrical/optical properties and the associated chemical reactivity. It can never be too overstated that the observed structure and properties are strongly dependent on the thermal history of the sample. It has become clear that the resulting chemistry and morphology vary considerably with temperature, oxygen partial pressure, annealing time, as well as cooling rate [53–62]. Many of the conflicting results reported to date have been argued to arise due to varying sample preparation recipes from one research group to the next. Often at the heart of debate is whether or not the observed structure represents an equilibrium structure. The most comprehensive study should combine the available tools of modern surface science to examine chemistry, crystallography, morphology, and electronic structure. Moreover, comparing the analysis of the various data must draw consistent conclusions that are well-substantiated on theoretical grounds. Below are brief discussions of two views currently proposed to describe the real surface of (001) SrTiO3-x. The first argues in 28 favor of a phase separation or demixing phenomenon, while the second argues in favor of defect ordering. The observation of a variation in cation:cation stoichiometry, facilitated by modification of the surface Sr content, has led to the idea that surface restructuring occurs by the formation of new oxides [53,54]. The mechanism of this restructuring in some cases is attributed to migration of Sr between the surface and the bulk, and its affect has been reported to extend up to 200 [54] to 1000 [53] atomic planes beneath the surface. The two most comprehensive studies of this phenomenon, however, do not agree on the details of the resulting new surface oxide. In one case it was determined that upon heating in a reducing atmosphere the surface restructures to form various orders (n) of the sub-oxide Srn+1TinO3n+1 known as Ruddlesden-Popper (R-P) phases, where n = 1 or 2 depending on the extent of local reduction [53]. In the other case, it was determined that reducing the surface results in the formation of TiO-rich phases, known as Magneli phases, with R-P phases forming in the sub-surface region [54]. The latter study also reported that the order of this bi-layer structure reversed upon high temperature equilibration in an oxidizing atmosphere. One might expect that the formation of new surface oxides cannot proceed without a simultaneous modification of the electronic structure and associated electrical/optical properties. The three to five times increase in the unit cell upon the formation of R-P phases (for example) not only reduces the size of the Broullouin zone, but the reduction in lattice symmetry should modify the band structure in terms of lifting energy degeneracies in the Bloch functions. No study, theoretical or experimental, has yet reported a modification in electronic structure with the formation of these new surface phases. 29 The other view of surface restructuring explains the mechanism in terms of oxygen vacancy ordering. Again, a simple electrostatic argument is sufficient for a qualitative description of this effect. The introduction of a single oxygen vacancy on the TiO2 termination redistributes charge q such that a quadrupole (co-planar with the surface) forms with -q on the two linearly coordinated Ti ions and +2q on the vacancy site. Numerical studies suggest that aligned quadrupoles on the TiO2 surface have a large attractive interaction while the interaction is repulsive if two quadrupoles are normal to each other [50]. It can therefore be expected that large densities of Ti-Vo-Ti quadrupoles will preferentially order to form a parallel row structure. This structure maximizes the attractive energy while minimizing the repulsive energy. It should be noted, however, that this simple explanation ignores relaxation effects. Indeed the literature reports the formation of a variety of superstructures observed with LEED, RHEED and STM including 2 × 1, 2 × 2 , c( 4 × 2 ) , c( 6 × 2) and 5 × 5 − R26.6 o [53–61]. Interestingly, atomic scale STM images showing features 8.7Å apart and aligned in parallel rows have been ascribed to oxygen vacancies on the TiO2-terminated surface [57]. These images were acquired with a tunneling microscope reported to be biased such that the features reflect the occupied density of states in the surface band gap which are attributed to oxygen vacancies or oxygen vacancyrelated defects. A similar structure was reported to be observed on the reduced (001) surface of BaTiO3 [63]. Oxygen vacancies may also be considered to form on the SrO-terminated surface. Following a similar electrostatic argument, however, it is clear that this does not occur without tending to destabilize the surface. The charge redistribution (neglecting relaxation effects) for a single oxygen vacancy shifts electron density towards the Ti located in the subsurface plane introducing a dipole oriented normal to the surface. If planar SrO1-x surfaces were able to form, it 30 can be expected that, for sufficiently large x, a superstructure will develop due to repulsion from adjacent (parallel) dipoles. As previously discussed, however, surfaces sustaining a net dipole moment are associated with large energies. In fact, the oxygen defect formation energy on a SrO surface is larger than that on the TiO2 surface. This is supported by the absence of surface states on SrO-terminated samples annealed under UHV conditions [59]. It was previously mentioned that the interplay between inter- and intra-atomic Coulombic forces influence many of the observed surface properties including the electronic structure. Early theoretical determinations of the surface electronic structure for the perfect (001) TiO2terminated plane predicted the existence of band gap states of pure d-orbital character extending from the center of the gap to the conduction band edge with a density of about 1015 electrons per cm2 [64,65]. As such states were not observed experimentally, the theory was refined to include electron correlation effects which pushed the previously predicted states up towards the conduction band. The enhancement of covalency at the surface, due to a decrease in the Madelung potential of the ions, transfers charge from the oxygens to the titaniums. This lowers the energies of the surface d-bands into the gap. The effect of the intra-atomic Coulombic force, however, tends to shift these states up in energy so that surface resonances in the conduction band are expected. Such resonances are said to be difficult to distinguish experimentally from bulk conduction band states [31]. The introduction of oxygen vacancy defects is a mechanism by which the inter-atomic force can dominate the intra-atomic force, as a result of a further reduction of the titanium Madelung potentials, and thus shift the d-states into the band gap region [66]. 31 A Titanium Oxygen B Figure 1.5 TiO2 termination of (001) SrTiO3 showing titanium adatoms at (A) a terrace site and at (B) a step edge. The B adatom is symmetrical with the corner Ti as defined by the shaded mirror plane. Recent advanced calculations based on tight binding [67] and first-principles pseudopotential methods [68] qualitatively predict similar surface electronic structures. It is worth noting that in the former study [67] no gap states were derived upon the introduction of oxygen vacancies until relaxation of the titaniums towards the center of the defect was considered. It was found, however, that no degree of relaxation was sufficient to explain the experimentally observed deep defect levels (see discussion below). It was therefore assumed that upon significant removal of oxygen ions, titanium adatoms may be present on TiO2 terraces or adsorbed on step sites, as schematically shown in Figure 1.5. Theoretical treatment of these configurations indeed derived band gap defect states in reasonable agreement with experiment. In the case of a terrace adatom (A) two levels were derived at 2.44 eV and 3.01 eV above the top 32 of the valence band edge. When cation adatoms appear at step sites (B) a state is found to lie 1.3 eV above the valence band edge. This deeper lying state is a direct consequence of the greater Ti–Ti interaction at corner sites as opposed to terrace sites. Indeed, step sites on vicinal (001) SrTiO3 are believed to be active in the dissociative adsorption of H2O, whereas the latter adsorbs molecularly upon terraces [4]. Table 1.4: Theoretical and observed defect-induced ionization energies in SrTiO3-x. Energy (eV) Proposed origin Reference 1.30 Argon ion sputtering 56 1.77 single electron excitation to CB 18,43,44 1.90 Ti adatom at step site 67 2.14 possible divacancies 44 2.40 interband excitation (CB to CB) 18,43,45 2.90 saturable upon reduction 18,42,43 The surface electronic structure has also been studied experimentally using photoelectron spectroscopy (PES), resonant photoemission (RESPE), and inverse photoemission spectroscopy (IPES) for surfaces prepared by several methods such as vacuum fracturing, Ar-ionbombardment, and polishing followed by vacuum annealing [31,55,56,69]. The general observations may be summarized as follows: a) vacuum fractured surfaces do not show band gap emission; b) fractured (or Ar-bombarded) and annealed surfaces show band gap emission of ~1013 electrons per cm2 centered at E F − 0.7 eV and E F − 1.3eV attributable to extrinsic states, where the former is associated with a bulk state and both defect states show strong Ti-3d character; c) surface enhanced covalency is confirmed by observed Ti resonance over the entire width of the valence band; and d) the introduction of oxygen vacancy defects reduces the density 33 of states near the top of the valence band, as might be expected since charge is transferred to the defect states. Table 1.4 summarizes defect ionization energies believed associated with oxygen non-stoichiometry as observed by bulk or surface probes or as determined by first principles calculations. 1.3 PHOTO-ASSISTED TUNNELING SPECTROSCOPY 1.3.1 Introduction to PATS The effects of coupling light to the STM junction have been investigated in many experimental and theoretical studies over the past decade [70–84]. There are various methods of coupling light to the junction as well as a variety of signals generated at the junction. The first, and perhaps simplest, application of photo-STM utilized the generation of a local surface photovoltage (LSPV) to increase the carrier density in semi-insulating GaAs to facilitate image acquisition [70]. The bulk of the literature to date report using the LSPV effect to either image topography [71] or to generate LSPV maps [72–76]. The latter gives information regarding spatial variations in materials properties such as band structure, doping density, and surface defect structure (see section 1.3.2). The LSPV effect in tunneling spectra has been observed in a number of investigations on both narrow- and wide-band gap semiconductors [77–80]. The photo-induced tunneling current at constant voltage has also been measured as a function of photon energy (from below to above the fundamental absorption edge) to characterize the efficiency of photocell material [81]. In addition to photovoltage or photocurrent measurements, the signal generated at the junction may arise from: a) thermal expansion of the tip and sample due to local heating; b) a thermoelectric potential due to differential heating of the tip — i.e., the Thompson effect; c) excitation of tip-induced plasmons on a metallic surface; or d) enhancement of the incident 34 electric field vector due to the geometry of the tunneling junction (antenna effect). Depending on the details of the experimental method chosen for photo-STM, one or more of these effects may have significant influence on the observed data and may obscure the particular phenomenon under investigation. Care must be taken to minimize unwanted contributions. In some more elaborate experimental schemes these effects may be exploited to generate photo-induced signals. Nonlinearities in the thermal expansion (or surface polarization), for example, have been used to generate harmonics in the tunneling current (or a displacement current) due to the difference-frequency signal (kilohertz to megahertz range) generated by illuminating the junction with two monochromatic lasers of different wavelengths [82]. When the difference frequency generated is in the gigahertz range, the inherent nonlinearity of the tunneling junction currentvoltage characteristic results in a rectification of the tunneling current proportional to the square root of the product of the incident laser powers [83]. The former effect has been utilized for imaging and local conductivity characterization [84]. Ebi eVB' < eVB eVB Ec Ef -------- Eg +++ ++++ +++ depletion width Figure 1.6 Ev Ec Ef -------++ +++ ++ Ev depletion width Surface photovoltage effect upon illuminating a n-type depletion semiconductor with energies equal to or greater than the band gap energy, Eg. The left sketch shows the band 35 structure at equilibrium (dark); the right sketch shows the band structure upon injection of excess carriers. Note the decrease in both the surface potential and width of the depletion region. 1.3.2 Three basic photon absorption mechanisms The experimental geometry and method chosen for the application of PATS is described in Chapter 2. This section describes the three primary charge excitation mechanisms that may be detected in the tunneling spectra using such a method. These include: 1) generation of a surface photovoltage; 2) decrease of surface charge via surface and/or bulk photoabsorption; and 3) increase of the surface charge via bulk photoabsorption. In principle, one or more of these mechanisms may be activated (if allowed) depending on the energy and intensity of the incident light. Figure 1.6 illustrates the generation of a surface photovoltage for a n-type semiconductor in depletion. At equilibrium the Fermi energy is assumed to be determined by the energy of the acceptor-type surface states. The negative charge on the surface is compensated by a positive space charge consisting of a uniform distribution of ionized donor defects. The latter is referred to as the depletion region and its width is defined by the point in the bulk where the sample is charge neutral. The displacement of charge at the surface of the semiconductor is compensated by a built-in electric field, Ebi, that opposes further drift of electrons to the surface states. If the sample absorbs light energy equal to or greater than the band gap, excess carriers are introduced in the form of mobile electrons in the conduction band and holes (with considerably less mobility) in the valence band. If the excitation occurs within the depletion width, the electrons are swept into the bulk by the built-in field and the holes are swept to the surface where they recombine with the charge in occupied surface states. At steady state, a constant supply of holes to the surface may be facilitated by electron flow from the metal tip (not shown in the figure). 36 Therefore, at zero external bias, a net current is induced by the absorption of band gap light. The decrease in the depletion width is a direct consequence of the decrease in the surface charge and an induced field within the tip-sample gap accompanies the displacement of the Fermi levels. The resulting steady state LSPV is thus the manifestation of a balance between minority carrier injection at the surface and recombination via surface states. eVB' < eVB eVB' > eVB Ec Ef -------- Ev ++ +++ ++ depletion width Ec Ef -------- Ev ++ +++ ++ depletion width b a Figure 1.7 Other photoabsorption mechanisms that (if allowed) may accompany illumination of a n-type depletion semiconductor with photon energies less than the band gap energy. The surface potential and depletion width decrease in (a); the surface potential is expected to increase in (b) with little or no change in the depletion width. Sub-band gap light may be absorbed via excitation of charge from occupied surface or bulk states to the conduction band or via excitation of charge from the valance band to acceptortype states within the depletion region. These mechanisms are sketched in Figure 1.7. Similar to the LSPV effect, both the surface charge and depletion width are reduced as shown in the case of Figure 1.7a. In contrast to the LSPV effect, a zero bias current is not sustained; only a redistribution of charge to accommodate a reduced surface potential. This 37 effect has profound influence on the characteristics of tunneling spectra (see section 3.3). A different absorption mechanism, illustrated in Figure 1.7b, will result in an increase in the surface potential due to an increase in the positive space charge. In oxygen deficient SrTiO3, where the oxygen vacancies are singly ionized at room temperature, such a mechanism may occur if sufficiently energetic light induces excitation from deep lying oxygen vacancy or vacancy-related trap states in the depletion region. In reality it is not too unreasonable to expect several surface and/or bulk absorption mechanisms to occur simultaneously under the appropriate conditions. 1.3.3 Limitations of PATS A few comments are in order regarding the potential contribution of additional signals generated at the tunneling junction, as discussed in section 1.3.1. In the present application of PATS, the most probable origins of extraneous signals include tip (or sample) expansion due to local heating, thermovoltages due to differential tip and/or junction heating, and rectification due to field enhancement at the tip. These effects have been studied in detail by various groups [85–92] and the general findings are summarized below. Tip and sample heating due to junction illumination is simply unavoidable. Some researchers have resorted to back-illumination [78] or laser modulation [85] techniques to minimize this effect. In the latter case, it has been shown that there exists a cutoff frequency, fc, dependent on the thermal properties of the sample and tip, above which the effect is rectified [86,87]. The high frequency roll-off of current pre-amplifiers, however, may present difficulties in obtaining desired photo-induced signals using modulation techniques. Constant illumination is applied in the present implementation of PATS. If the thermal conductivity of the metal tip is much larger than that of the material under investigation, then the initial response upon 38 illumination will be dominated by the response of the tip. For a tungsten tip, the tunneling signal increases to a maximum within 18µs after a 20ns laser pulse (~6.4µJ) due to tip expansion [89]. One might conclude that an alternative method for minimizing expansion effects is to use low incident power and acquire the tunneling signal rapidly (in ≤ 18µs). This, however, must be balanced against the time to average multiple spectra (typically 5 to 10 per acquisition) and coupling enough power to the junction to generate a response above the noise floor of the electronics. Figure 1.8 The laser induced thermovoltage versus irradiance for two different cone angles resulting from surface heating only [ref. 89]. 39 Thermovoltages are induced within a metal rod as a result of a temperature gradient along the rod (the Thomson effect). Additionally, a heated metal/semiconductor junction may be cooled by the induction of a current flow from semiconductor to metal (the Peltier effect). In a STM junction, the latter effect has been found to be negligible compared to the former [89]. Figure 1.8 shows a theoretical determination of induced thermovoltage due to surface heating as a function of incident power density. Thermovoltages on the order of a few tens of millivolts are predicted for sharper tips (half-cone angle = 15º) and a few millivolts for blunter tips (half-cone angle = 30º) when illuminated with irradiance in the range 1–2 MW/cm2. For tunneling spectroscopy on wide band gap systems, which typically requires a voltage sweep over several volts, this effect may be lost within the noise floor of the electronics. Under suitable circumstances, however, thermovoltage effects can be important. Indeed several studies have used this effect for chemical differentiation in STM images, to study the effects of surface defect structure on the behavior of a two-dimensional electron gas, as well as to obtain average thermoelectric coefficients for surface structures with better than 1nm lateral resolution [90]. 40 Figure 1.9 Calculated electric field intensity between tip and sample versus irradiance for illumination of λ = 10 µm light. The half-cone angle used was 15° and the field was evaluated at a distance 2000 Å from the tip apex [ref. 91]. Finally, the junction comprised of a sharp tip and conductive sample is similar to the well-studied metal-metal point contact diode. Rigorous calculations have been performed to describe the nonlinear I-V characteristics of the latter [91]. It has been established that the geometry of the junction is such that when incident radiation has a wavelength larger than the dimensions of the tip apex (i.e., infrared radiation), oscillating currents are induced on the tip surface as if it were a wire antenna. These oscillating currents induce charges at the tip apex and mirror charges on the conducting sample giving rise to an oscillating voltage at the junction. The resulting electric fields in the junction have been shown to be up to a thousand times larger than that in the incident radiation. As shown in Figure 1.9, the magnitudes of these fields can be of the same order as those induced by the applied bias and in some cases sufficient to induce field 41 emission (rather than tunneling) of electrons from the metal surface. These effects are not expected to be significant for light in the visible region. On the other hand, results from recent theoretical calculations for 488 nm and 850 nm light suggest that field enhancement is greatest when the angle of incidence (measured with respect to the surface normal) is less than 10–20 degrees and it is larger for sharper tips [92] owing to an inverse relationship between the optical electric field at the tip apex and the radius of curvature of the tip. Furthermore, the field enhancement is greatest when the electric field vector lies in the plane of incidence (the plane containing the k vector of the incident wave and the axis of the tip). A non-zero field enhancement for E-fields polarized normal to the plane of incidence may arise due to irregularities in the tip shape (i.e., deviation from ideal conical symmetry), a condition more likely close to reality. It is concluded that the aforementioned light-induced effects may be minimized to varying degrees of success by proper experimental design. Specifically, one should consider illumination power and angle of incidence, tunneling tip material and shape, and finally junction illumination duration versus data acquisition mode/rate. Note that the bias applied at the tunneling junction, with a typical gap width of 5–10 Å, induces fields of the order 107 volts per cm. The experimental flux data (see Figure 2.2) indicates that the irradiance of the light source used is of the order 1.64 mW per cm2. These powers are off to the left of Figure 1.9 so that the expected electric field induced by the light source in the present application is at least three orders of magnitude less than the field induced by the applied bias and thus should not present an experimental difficulty. 42 1.4 THESIS OBJECTIVES The previous sections have presented an overview of the progress made in increasing the understanding of bulk and surface structure and properties of electron-doped strontium titanate. The various, and at times conflicting, observations of the surface structure highlights the need for continued and focused research on the subject. Specific unresolved issues regard the microscopic nature of the reactivity or optical response of the defective (001) surface. Is it surface or subsurface mediated? Are acceptor-type surface states associated with oxygen vacancies, titanium adatoms, or other undetermined defects such as surface dislocations or a new oxide? One objective of this thesis is to contribute an unambiguous piece to the growing puzzle of information, and/or to establish a relationship amongst the existing pieces of the puzzle, through a study of the optical activity of (001) surfaces of electron-doped strontium titanate. This is accomplished by another objective of this thesis which is to demonstrate the utility of the technique of photo-assisted tunneling microscopy and spectroscopy for identification and characterization of local charge transfer mechanisms at surfaces. The investigation seeks to identify the local origins of surface (or subsurface) optical responsivity associated with previously identified energies as outlined in Table 1.4. The quantitative assessment of the electronic behavior is derived through PATS combined with theoretical modeling. The details of the experimental design are described in Chapter 2 and the theoretical background of tunneling spectroscopic analysis is presented in Chapter 3. REFERENCES 1. D. P. Woodruff and T. A. Delchar Modern Techniques of Surface Science 2nd ed. Cambridge University Press, Cambridge (1994). 43 2. V. E. Henrich and P. A. Cox The Surface Science of Metal Oxides Cambridge University Press, Cambridge (1994). 3. J. P. Zielger, E. K. 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Möller, A. Rettenberger, K. Läuger Phys. Rev. B 52 [19] (1995) 13,796 91. T. E. Sullivan, P. H. Cutler and A. A. Lucas Surface Science 62 (1977) 455 49 92. M. J. Hagmann J. Vac. Sci. Technol. B 15 [3] (1997) 597 50 Chapter 2: Experimental Method The details of the experimental instrumentation are described and sample preparation techniques are discussed. A brief introduction to tunneling microscopy is also presented with distinctions identified for the approach to PATS. 2.1 PHOTO-ASSISTED TUNNELING SPECTROSCOPY 2.1.1 Experimental arrangement A schematic diagram of the experimental arrangement appears in Figure 2.1. Unpolarized white light from an Oriel 400 W xenon arc lamp was focused onto the 2 mm width entrance slit of a Fastie-Ebert-type grating monochromator. After passing through a 0.25 mm width exit slit and a 25 mm diameter, 150 mm focal length (F.L.) spherical lens, the monochromatic light was focused onto the surface of the sample by a 68 mm F.L. cylindrical mirror mounted above the tunneling microscope inside a vacuum chamber. This configuration was chosen to focus the image of the exit slit onto the STM junction (magnification = −0.6) without excessive throughput loss due to beam reflection and divergence. The monochromator contained two reflection gratings blazed for 600 nm (low blaze) and 1200 nm (high blaze), respectively. The former grating is most efficient for wavelengths in the range 400–900 nm, while the latter is most efficient in the range 800–1800 nm. The linear dispersion for the low blaze grating is 3.3 nm per mm. Therefore, an exit slit width of 0.25 mm combined with a counter accuracy of ±1 nm gives a total monochromator-limited wavelength uncertainty of ±1.83 nm (i.e., an energy uncertainty of ±14 meV). Finally, a filter was used at the exit slit of the monochromator in order to minimize unwanted energies contributed from higher order diffraction. It should be noted, however, that the monochromator design maximizes first order diffraction from the low blaze 42 grating. This grating has a large groove density (1180 g/mm) which gives it a relatively small angular dispersion. Overlap between first and second order diffraction is therefore assumed to be negligible. incident light sample holder θ ≈ 42° sample vacuum chamber mirror lens 400W Xe arc lamp monochromator ion pump sample computer Figure 2.1 STM stage STM electronics Experimental arrangement for photo-assisted tunneling spectroscopy. Since the incident light necessarily interacted with several optical components (including passage through the vacuum chamber window), a characterization of the spectral response of the optical set-up was performed by placing a Si pn-type photodiode at the position of the tip apex and measuring the total flux reflected from the mirror. Using a pre-amp and the photo-sensitivity data furnished by Hamamatsu Photonics, the current from the photodiode was converted to total incidence flux as shown in Figure 2.2. It can be seen that the behavior reflects the efficiency expected of the reflection gratings — i.e., the low blaze grating is more efficient in the near UV 43 range, while the high blaze grating is more efficient in the near IR range. It should be noted that the particular photodiode used for these measurements had a suppressed sensitivity for wavelengths greater than 750 nm; therefore, the actual flux in this region is expected to be greater than that shown in Figure 2.2. The average flux is of the order 1013 photons per second. It is assumed that this photon density is sufficient to produce a measurable photo-response, if the cross section for charge carrier excitation is close to unity. The active area for the photodiode was 0.012 cm2. One watt of incident monochromatic power is equivalent to λ (5.03 × 1015 ) photons per second, where λ is in nanometers, such that the irradiance of the light source on the junction was of the order of a few mW per square centimeter. As illustrated in Figure 2.1, the angle of incidence with respect to the axis of the tip was approximately 42 degrees. These last two points, as well as the fact that the experiment was confined to a bandwidth of 628 nm from 327 to 955 nm, suggest that the data should not be significantly skewed by antenna effects, as discussed in section 1.3.3. 44 1.2E+14 1E+14 high blaze 8E+13 low blaze 6E+13 4E+13 2E+13 960 920 880 840 800 760 720 680 640 600 560 520 480 440 400 360 320 280 240 200 0 Wavelength (nm) Figure 2.2 Spectral response of experimental optics measured by a Si photodiode. The STM used was built in-house based on a concentric piezoelectric tube design [1] where the inner tube served to provide linear motion of the tip in three mutually orthogonal directions over the surface being imaged. The outer tube served to dynamically translate the sample and to stabilize the STM against thermal drift due to thermal expansion of the piezo tubes [2]. All movement and data acquisition were controlled and monitored by computer via the STM electronics. The latter consisted of a differential feedback unit and a high voltage triple amplifier. The STM was housed inside a vacuum chamber where all measurements were performed at base pressures in the low 10-10 Torr (ultra-high vacuum, UHV) range. The chamber plus ion pump were mounted on a T-shaped steel bracket which was suspended upon three pneumatic legs providing isolation from building (acoustic) vibrations. Additional mechanical stability was provided by mounting the STM upon a stack of viton-spaced steel plates, with springs placed between the bottom two plates to improve vibration damping. 45 Perhaps the most significant contribution to the mechanical stability of the STM is attributed to materials selection. All components were constructed from Macor® glass ceramic except for the piezo tubes (which were PZT), the rails upon which the sample holder was translated (tungstencarbide), and parts of the sample holder itself (stainless steel). These metals (as well as a few nuts and bolts) served primarily to provide electrical connections to carry the tunneling signal. The rigidity of the glass ceramic as well as PZT tubes ensures a high resonance frequency of the STM assembly, well above that which may be excited by the scanning mechanism during operation of the microscope [3]. A low-pass filter is often used to remove high frequency (electronic) noise from images. This was observed to be problematical when attempting to rapidly acquire spectra since it contributed additional capacitance to the circuit. A filter was thus not used. Other than the STM, the chamber was equipped only with an ion gauge and two linear feedthroughs. One feedthrough served to translate the sample to and from the chamber. The other served to translate the sample to and from the STM as well as providing the electrical connection for in-situ cleaning of the sample surface by joule heating of a tantalum foil. The absence of several analytical peripherals on the chamber was beneficial to the extent that additional acoustical coupling was absent. The piezoelectric constant of the tube scanner was measured by the double piezoelectric response technique [4] to be 44.7 Å/V. The maximum voltage applied to the electrodes was ±100 V, giving a maximum scanning area of 8940 × 8940 Å2. The edge of such a scan is less than one hundredths the width of a fine human hair (which is about 100 microns in diameter). Hence this device was designed for high resolution imaging. 46 2.1.2 Experimental method All tunneling images acquired in this thesis represent constant current images. These are generally obtained by specifying a current/voltage combination (i.e., a set point) in the control electronics. As the tip is scanned across the sample, a differential op-amp provides a voltage to the z piezo that is proportional to the difference between the set point and the monitored tunneling current. Variation in the opacity of the tunneling barrier gives rise to variations in the tunneling current. The op-amp output voltage serves to restore the tunneling current to the set point value. Knowing the scanning piezo constant, a map of the restoring voltage as a function of tip position constructs the STM image. Often the latter is interpreted as a map of constant surface state density. On the other hand, a local variation in work function (perhaps as a result of local variation in surface charge) will vary the opacity of the tunneling barrier and also give rise to image features. Therefore, STM images obtained in this way are also often interpreted as maps of constant surface charge density. In general, contrast in STM images may result from variations in surface state density, surface charge, morphology, or any combination thereof. To appreciate image interpretation, it is important to recall that the feedback circuit functions to maintain a constant current. If it may be assumed that the active area of the tunneling tip does not change in the course of the measurement, the STM produces images of constant current density. Therefore, one should have a sufficient understanding of the charge transport mechanism near the surface of the material under investigation. Near surface transport for metals is different than that for traditional semiconductors which can be different than that for low mobility semiconducting oxides. In the presence of a Schottky barrier, the dominant mechanism can vary depending on the position of the equilibrium Fermi energy, EF (see section 3.2.5). A degenerate n-type semiconductor (i.e., 47 such that EF lies above the conduction band minimum) at forward bias will generate images of constant local state density modified by topography. A non-degenerate semiconductor (EF lies below the conduction band minimum), however, will generate images of constant surface charge modified by topography. The difference is due only to the dominant transport mechanism in the depletion region of the semiconductor. In the former case tunneling dominates; in the latter case emission dominates. The usual sample-and-hold scheme for measuring tunneling spectra was employed with a slight modification for PATS. Upon identifying a surface feature of interest, the feedback unit was disengaged momentarily while the bias applied to the sample was varied over a fixed voltage range and the variation in current was recorded. Ten such acquisitions were averaged together for a given spectrum. During a conventional measurement, the feedback unit would be re-engaged briefly between each of the ten acquisitions in order to minimize thermal drift effects. For the PATS application, however, re-engaging the feedback can possibly introduce an unwanted variation in the tip-sample distance, s, if a photocurrent is induced at the set point voltage. Such a variation in s may be misinterpreted as a charge transition photoresponse. To avoid this problem the feedback was not re-engaged between each of the ten acquisitions of a given spectrum. Thermal drift effects were minimized by acquiring the spectra rapidly — a typical spectrum was acquired on the order of a few tenths of a second. The rate varied depending on voltage sweep range and voltage step size (together determining the number of data points) and the delay between each voltage increment. Proper selection of a probe tip is critical to successful STM and STS measurements. Both materials selection and tip geometry can have a significant impact on the quality of the images and the interpretation of the spectra. Mechanically clipped Pt-Ir wire is most often used 48 as a probe because the material is inert in an ambient environment and often the mechanical forming results in a small cluster of atoms near the apex that can facilitate atomic resolution imaging. Unfortunately, these tips often do not appear microscopically sharp and thus can be limited to surfaces that are “smooth” on the scale of a few tenths of a micron squared. Many of the surfaces studied in this thesis may be characterized as microscopically rough (i.e., see Figure 2.8 on page 55) and hence required the use of tips prepared by chemical etching. Tungsten tips were thus prepared by the dc drop off method [4] using 5M KOH solution and 12 volts between anode (tip) and cathode. The tips were rinsed with ethanol and deionized water followed by annealing in a reducing environment at ≥ 725 ºC overnight to remove residual oxides formed during the etching process. 2.1.3 Experimental noise In addition to the potential artifacts associated with the optical measurement as discussed in section 1.3.3, there are certain aspects of conventional STM experiments that may contribute noise to the phenomena being observed. The most significant source of electronic noise results from stray capacitance throughout the STM circuitry that precedes the current preamp. Indeed the decision to apply Pt electrodes to the samples was primarily to minimize the capacitance between the sample and the clips of the sample holder. A typical characterization of the experimental noise inherent to the measurements presented in this thesis is shown in Figure 2.3 as a plot of the sample standard deviation in the tunneling current versus sample bias. This plot was generated by acquiring ten separate tunneling spectra and calculating the variance for each I/V coordinate from this set. The result suggests that the accuracy of the measurement decreases for larger absolute values of the applied bias. The rapid decrease in variance observed beyond roughly ±3 volts is a consequence of the 49 saturation of the current preamp and should not be interpreted as an improvement in reproducibility. Since the set point for the spectra obtained to generate Figure 2.3 was 500 pA at -3 V, it is rather interesting that the variance near -3V is about five times the set point current itself! This behavior should be considered when analyzing experimental tunneling spectra such that one might be cautious about interpreting spectral features far from the equilibrium Fermi level. It should also be kept in mind, however, that Figure 2.3 represents a worst-case scenario given the manner in which the data was acquired. In practice, the spectra are acquired in rapid succession such that phenomena like thermal drifting has negligible effects on a given spectrum. 3 2.5 2 1.5 1 0.5 4.01 3.68 3.35 3.02 2.69 2.35 2.02 1.69 1.36 1.03 0.69 0.36 0.02 -0.31 -0.64 -0.97 -1.30 -1.63 -1.97 -2.30 -2.63 -2.96 -3.29 -3.63 -3.93 0 Sample Bias (V) Figure 2.3 Quantification of current variance inherent to the tunneling measurements obtained in this study. 50 2.2 SAMPLE PREPARATION AND CHARACTERIZATION 2.2.1 Sample processing history Initial attempts to prepare clean and “flat” (001) surfaces of single crystal SrTiO3 did not proceed without considerable difficulty. The process began by orienting a single crystal boule (obtained from Atomergic Chemetals Corp.) such that the 〈001〉 direction was aligned normal to the surface. An observed 4-fold rotational symmetry by back-reflection Laue verified this orientation, as shown in Figure 2.4. The boule was then wafered using a 0.015" diameter diamond wire blade. The wafers were polished using diamond impregnated lapping films down to 0.1µm and finally to 0.05µm using alumina paste. Note that these last two steps could not be controlled to absolute precision such that preservation of the original orientation could not always be assumed. Variation in surface misorientation will result in a variation in step size and density. For this study, it was only important that steps be present and accessible within the STM maximum scanning range. The crystals were then heat treated in air by a slow temperature ramp (3 degrees per minute) to 1000 °C, held for 10 hours, and subsequently furnace cooled (5 degrees per minute). Platinum electrodes were added to the samples using Engelhard #6926 unfritted paste. An additional vacuum heating was necessary to reduce the undoped crystals and to activate the Nbdoped crystal (see discussion in section 4.1.1). The air anneal resulted in flat surfaces characterized by plateaus separated by straight-edged (and kinked) multiple unit cell steps, as observed by AFM (Figure 2.5). After vacuum reducing, however, no surface yielded a LEED pattern until further heat treatment in the analysis (high vacuum) chamber at ≥1000 °C for 3–5 minutes. The LEED patterns on these surfaces contained relatively broad spots and significant background intensity. These surfaces were also extremely difficult to image with STM. Figure 51 2.6 is a representative STM image obtained from one of these crystals. It can be seen from the left image that the surface was not uniformly clean and flat. A magnification of the region circled in the left image appears in the right image. This stepped region reveals the nascent ordering that probably accounted for the observed (albeit poor quality) 1×1 LEED patterns. Figure 2.4 Laue back-reflection photograph showing 〈001〉 orientation — i.e., a 4-fold rotation axis is centered in and normal to the plane of the photograph. 52 Figure 2.5 Atomic force microscope (AFM) image showing the stepped surface of SrTiO3 after polishing and air annealing for up to 10 hours at 1000 °C. Figure 2.6 Scanning tunneling microscope (STM) images showing a stepped surface of SrTiO3 after a similar preparation as the sample in Figure 2.5 with an additional vacuum anneal at 1000 °C for 30 minutes. The image to the right is a magnification of the circled region appearing in the image to the left. 53 It was subsequently determined that the reduction anneal deposited carbon upon the surface, most likely due to contamination in the vacuum furnace. This required extended heat treatments in the analysis chamber for surface cleaning which resulted in heavily reduced samples. Auger electron spectroscopy (AES) measurements also revealed a growing sulfur peak with extended vacuum annealing. The initial vacuum treatment could not be eliminated because it was necessary for the formation of good Pt electrodes. To solve this problem, chemical etching and an additional air anneal were introduced into the preparation procedure. After polishing, all samples were etched using a buffered NH4–HF (BHF) solution for 1.25 minutes. The pH was determined to be in the range 4.5-4.7 using colorpHast® indicator strips. This solution is expected to preferentially attack the more basic (i.e., the SrO) planes leaving flat surfaces predominately TiO2-terminated [5]. After the initial vacuum annealing, the samples were briefly re-annealed in air to remove carbon contamination and restore them to a common insulating state. The resulting surfaces appear quite similar to those without etching except for the occasional addition of deep trenches and square etch pits due to chemical attack upon residual polishing damage. Figure 2.7 shows an AFM image of a typical sample prepared with the BHF solution before additional vacuum annealing. Figure 2.8 shows AFM images of a typical sample similarly prepared after being heavily reduced by vacuum annealing. Preparing the samples in this way resulted in clean (i.e., carbon free) surfaces that could be reduced in a controlled manner to obtain samples with varying conductivities and surface morphologies. 54 Figure 2.7 A 500 nm × 500 nm AFM image showing a stepped surface of SrTiO3 after a similar preparation as the sample in Figure 2.5 with an additional BHF etching and re-annealing in air. Figure 2.8 AFM images of heavily reduced SrTiO3 (001) showing morphological development as compared to the air-annealed sample in Figure 2.7. Left: 3 µm × 3µm scan. Right: 1 µm × 1µm scan of lower right edge of the image to the left. 55 2.2.2 Methods of characterization In addition to SPM (scanning probe microscopy), other methods were used to characterize the developing structure, chemistry and physical properties of the samples. The cleanliness of the surfaces was verified by Auger electron spectroscopy (AES) and surface ordering was verified using low energy electron diffraction (LEED). The bulk optical constants (index of refraction and extinction coefficient) were characterized using ellipsometric and transmission spectroscopies, and bulk majority carrier density/mobility was determined by Hall measurements. REFERENCES 1. J. W. Lyding, S. Skala, J. S. Hubacek, R. Brockenbrough and G. Gammie Rev. Sci. Instrum. 59 [9] (1988) 1897 2. D. W. Pohl Rev. Sci. Instrum. 58 [1] (1987) 54 3. D. A. Bonnell in Scanning Tunneling Microscopy and Spectroscopy: Theory, Techniques and Applications, D.A. Bonnell, ed. VCH Publishers Inc., New York (1993) 4. J. C. Chen Introduction to Scanning Tunneling Microscopy Oxford University Press, Oxford (1993) 5. M. Kawasaki, K. Takahashi, T. Maeda, R. Tsuchiya, M. Shinohara, O. Ishiyama, T. Yonezawa, M. Yoshimoto and H. Koinuma Science 266 (1994) 1540 56 Chapter 3: Tunneling Spectroscopy Theory Tunneling spectra are simulated for the purpose of interpreting the changes observed in experimental data. The approach to the simulations follows from the contributions of various authors over the past ten years or so. A brief background in the concepts of 1D tunneling is presented, followed by the details of the mathematical formulas used to generate the simulated points. Additional details are found in Appendix B. 3.1 INTRODUCTION 3.1.1 Quantum mechanical tunneling and the WKB approximation Before discussing the theory developed for interpretation of experimental tunneling spectra, a brief review of quantum mechanical tunneling is presented in this section. Detailed treatment of particle scattering by various one-dimensional potentials may be found in any textbook on quantum physics. Only the salient features of quantum transmission through a potential barrier are discussed here. The problem presented is sketched in Figure 3.1. A particle of energy E, propagating in the positive x direction, is expected to be totally reflected by the potential barrier by the laws of classical physics. The principles of quantum physics, however, permit the incident particle to be treated as an incident wave described by a suitable wave function ψ which exactly solves the time-independent wave equation — i.e., Schrödinger’s eigenvalue equation (S-E) given by d 2 ψ (x) − κ2 ψ (x ) = 0 , 2 dx where κ= 2m (E − V(x)) . h2 58 The most general solution to the S-E for a piecewise constant potential function (Figure 3.1a) consists of a linear sum of plane waves, ψ = Ceiκx , propagating in both the +x and x directions, where C is the amplitude of the wave. The central problem amounts to a determination of the ratio of the transmitted amplitude (at x > +a) to the incident amplitude (at x < -a). In other words, one is interested in determining the transmission probability, |T|2, of the potential barrier. For a particularly opaque barrier (i.e., where κa is large) the well-known solution is of the form 2 T ∝e −4 κa . The transmission probability is thus finite and strongly dependent on the width of the barrier as well as the magnitude of the kinetic energy associated with the particle (which is negative in the classically forbidden region, |x| < a). Note that this extreme sensitivity to κa is responsible for the intrinsically high vertical spatial resolution of scanning tunneling microscopy. (The lateral resolution depends on the experimental conditions and may be limited by the physical properties of the electrodes.) 59 V(x) E -a +a x a V(x) E b b′ x b Figure 3.1 A particle wave of energy E propagating (a) within a piecewise constant potential or (b) within a continuous potential function. Real barriers are often not as simple as depicted in Figure 3.1a, so that one must find suitable solutions to the S.E. and determine the transmission probability for a potential function of arbitrary shape as shown in Figure 3.1b. This problem has not yet been treated with exact solutions. Instead, an approach known as the Wentzel-Kramers-Brillouin (WKB) method is usually taken. It is assumed that the eigenfunctions for the potential in Figure 3.1b will be very nearly in form to those for the potential in Figure 3.1a if the variation in V(x) within one de Broglie wavelength is small compared to the kinetic energy of the particle. This condition is easily satisfied for typical barrier functions except near the boundaries labeled b and b′ in Figure 60 3.1b. These are known as the classical turning points and the WKB method continues with the derivation of a connection rule that establishes continuity between the appropriate WKB wave functions far from, and on both sides of, these points. The derivation of connection rules requires an approximation regarding the nature of the potential in the vicinity of the turning points [1]. Simply stated, the kinetic energy is assumed to vary linearly over the distance where the WKB functions are invalid, and the S-E is solved exactly over this region. Two conditions are identified where the WKB method contributes greater error to the determination of |T|2 for real tunneling barriers. The first is when the kinetic energy approaches zero such that the incident energy is near the top of the tunneling barrier. In this case, the connection rules are invalid because the slope of the potential varies rapidly and a linear approximation to the behavior of the kinetic energy is not possible. The second condition is when the barrier is very narrow such that the classical turning points approach each other. In this case, a greater portion of the total potential function is treated by the (artificial) straight-line approximation. 3.1.2 The purpose of modeling tunneling spectra The system defined by a metal tip, a vacuum gap, and a semiconductor may be viewed as a MIS (metal-insulator-semiconductor) junction. The theories concerning the behavior of planar MIS junctions and the associated Schottky barrier are well-developed [2–4]. Similarly, the study of MIM (metal-insulator-metal) as well as MIS tunnel junctions has resulted in spectroscopic methods for the determination of surface and interface electronic structure that corresponds well with experimental data [5–10]. It is common practice to model the effects of the junction dynamics on experimentally obtained tunneling spectra using the tools of MIS and tunneling 61 theories [11–13]. This is necessary since the electric field at the junction is known to perturb the electronic structure of both the tip and the material under investigation [9,14]. Ultimately, one must be able to distinguish between the electrical characteristics of the tip-sample tunnel junction as a MIS diode (i.e., device behavior) and the effects due purely to surface states — the two are not independent. Therefore, this chapter discusses a method by which the tunneling current between a metal probe and a semiconducting sample (separated by free space) may be calculated as a function of the potential difference applied across the junction. The discussion begins with the development of Harrison’s formula [15] for one dimensional tunneling across a planar barrier which, when applied to experimental tunneling spectroscopy (TS), correctly describes the transport of charge across both the tunneling gap and (given the appropriate conditions) the semiconductor depletion region (section 3.2.1). Following will be a discussion of the treatment of dynamic band bending (section 3.2.4), and finally the relative importance of tunneling versus thermionic emission as a means by which charge carriers negotiate the semiconductor diffusion barrier (section 3.2.5). The focus throughout the discussion is on the application of this method as an aid in the study of Nb-doped or reduced strontium titanate (SrTiO3) (001) surfaces. Therefore, the Fermi statistics appropriate for monovalent as well as divalent defect centers is considered (see Appendix B). A parametric study of the model will be discussed for the purposes of establishing a fitting procedure and a context in which to interpret experimental data (section 3.3.2). 62 3.2 THE TUNNELING MODEL 3.2.1 One-dimensional quantum transmission Walter A. Harrison [15] developed an expression for one-dimensional tunneling of an independent electron where the initial and final states were considered to be single particle solutions to the Schrödinger equation for two distinct Hamiltonians separated by a planar junction. This approach is similar to the many-body description of tunneling based on an extension of first-order time-dependent perturbation theory [16], where Bardeen [17] demonstrated their equivalence given the appropriate evaluation of the tunneling matrix element, Mab. The model therefore begins with a fundamental result of time-dependent perturbation theory known as the Golden Rule which gives the probability per unit time that an electron makes a transition from state a to state b as 2 2π Pab = M ab ρb , h [3.1a] where |Mab| is the transition matrix element (analogous to the transmission probability |T| of section 3.1.1), ρb is the density of final states per unit energy, and O is the reduced Planck constant. Including the density of initial states per unit energy as well as the probability functions, f(E), for the occupancy of the initial and final states obtains 2 2π Pab = M ab ρa ρb fa (1− fb ) . h [3.1b] Two restrictions imposed on the states defining the system are that: 1) the initial and final state energies must be equal, and 2) the transverse wave vector, kt, of the tunneling electron must be conserved. In other words, the model is restricted to elastic tunneling (1) obtained by specular transmission (2) through the energy barrier. 63 A summation is performed over the allowed energy range (as determined by the applied bias) for fixed kt, as well as over the appropriate range of allowed kt (see section 3.2.2). Adding a factor of 2 for spin degeneracy and e for the fundamental unit of charge, the total current density from electrode a to electrode b is 2 4πe Mab ρaρb fa (1− fb )dE k t . j a→b = ∑ ∫ h kt [3.2a] The current density from electrode b to electrode a is similarly given by 2 4πe M j b→a = ρaρb fb (1− fa )dE k t , ∑ ab ∫ hk [3.2b] t so that the net one-dimensional current density is given by the difference 2 4πe j z = j a→b − jb→ a = Mab ρ a ρb (fa − fb )dE k t . ∑ ∫ h kt [3.2c] Following Bardeen [17], as long as the system has the property of separability (i.e., the Hamiltonians for each electrode do not overlap in configuration space) [18], the tunneling matrix element may be evaluated as h * ∂ψ b ∂ψ a * M ab = −ihˆJab = − ih ψ − ψ . b 2mi a ∂z ∂z [3.3] It is assumed that the potential varies slowly in the barrier region permitting use of the WKB method to find the initial (ψa) and final (ψb) state solutions to the Schrödinger equation. The details of obtaining such solutions are beyond the scope of this text and will not be included here; however, the results of the solution [15] gives M ab 2 2 zb h2 m m ∂E a ∂E b = exp −2 k dz ∫ z , 2m h2 L a h2 L b ∂k z ∂k z za where the 1D density of states, ρ, for a particle in a “box” of length L is given by 64 [3.4] 1 ∂E π . ≡ ρ ∂k z L Using [3.4] and [3.5], and replacing ∑ kt by ∫ 1 4π2 [3.5] d 2 k t [3.2c] becomes e 1 j z = ∫ dE(fa − fb )∫ d 2 k t 2 exp(−η) 4π hπ [3.6a] zb η = 2 ∫ k z dz [3.6b] za The wave vector of the tunneling electron is given by k z = 2 me ϕ h2 , me is the free electron mass, and ϕ is the average tunneling barrier height. Equation [3.6] is the formula derived by Harrison for one-dimensional single particle tunneling through a planar junction. The limits of the integral in [3.6b] are the classical turning points which must be determined as a function of both the electron energy and the applied bias. They are simply given by the roots of the equation ϕ (z, V, E) = 0. The limits of the total energy integral in [3.6a] are determined by the width of the energy window for elastic tunneling as also determined by the applied bias. Lastly, the limits of the wave vector integral in [3.6a] are determined by the overlap of the projections of the constant energy surfaces of both electrodes onto the plane of the tunneling barrier. It is important to note that it is by this latter restriction that some vestige of the dependence on the density of states is recovered since an explicit dependence vanishes upon making the necessary replacements to derive [3.6]. A more direct dependence on the band structure of either electrode is represented by the density of states (DOS) effective mass for the tunneling electron which is introduced by transforming the wave vector integral into one over the longitudinal (normal) energy [19]. 65 This may be done simply by a Jacobian transform to polar coordinates, so that [3.6] may be re-written as 4πem * dEΘ ± [E − E i ]∫ dE L exp(− η), jz = h3 ∫ [3.7] where m* is the DOS effective mass of the tunneling electron, and Ei is the band edge for band i (i.e., i = CB or VB). The positive sign corresponds to the conduction band (CB), whereas the negative sign corresponds to the valence band (VB). It has been assumed that the shape of the energy bands are approximately parabolic as well as isotropic. The latter means that the constant energy surfaces are assumed to be concentric spheres in k space over the energy range where the assumption of an isotropic band structure is valid. The value for the effective mass is determined by the electrode with the larger E(k) curvature (i.e., smaller sphere) for a given total energy since this electrode will also determine the limits of the normal energy integral in [3.7]. In a MIS structure, where S represents a n-type semiconductor in depletion, if it may be assumed that the electron effective mass in the metal is approximately equal to the free electron mass, then (in absence of other limiting factors) the tunneling current at small forward and moderate reverse bias will be limited predominately by the electronic structure of the semiconductor. With a sufficient magnitude of the applied voltage, the tunneling current will tend to be limited by the metal, depending on the curvature of the energy bands. (This point will be restated and demonstrated schematically in section 3.2.2.) This does not inhibit the semiconductor band structure from further influencing the features of the tunneling spectrum. For now, however, it should be appreciated that the tunneling spectrum is generally a complex convolution of the electronic properties of both the metal and the semiconductor [7,8]. 66 In writing [3.7] the Fermi distribution functions were assumed to take their absolute zero forms so that (fa - fb) was replaced by the step function Θ (also known as the Heaviside function) defined by 0, z ≤ 0 . Θ(z ) = 1, z > 0 [3.8] This replacement is justified given that the energy resolution of experimental TS is not expected to be much better than several times kT [20]. It should be observed that Θ , as defined in this way, has the effect of completely ignoring tunneling events between the metal and energy states within the band gap of the semiconductor. The task ahead consists of the following: a) determining the limits of integration in [3.7], and b) determining the average tunneling barrier height as a function of the applied bias. There are two barrier functions which must be considered — the insulator (i.e., vacuum gap) barrier and the semiconductor Schottky-type barrier. The determination of both of these functions will be discussed following a consideration of the correct distribution of the applied bias, Va, across the MIS structure. For convenience, the equilibrium semiconductor Fermi level is taken as the reference energy, E SC F = 0 . Furthermore, the semiconductor is held at ground potential so that the bias is applied to the tip. The bulk conduction and valence band edges are thus simply given by B E Bcb − E SC F = E cb = ξ E Bvb = E cbB − E g = ξ − E g where Eg is the semiconductor band gap and ξ is a function of the temperature and/or the properties of the material. The functional form of ξ depends on whether the majority carrier 67 density (n) is exceeded by or exceeds the majority carrier band effective density of states ( N ) [21,22]. The former case refers to a non-degenerate semiconductor and ξ is given by kTln n , N [3.9a] while the latter case refers to a degenerate semiconductor and ξ is given by 3n π 2 3 h2 ∗ . 8m [3.9b] 3.2.2 Effects of specular transmission Figure 3.2 is an example of a simple construction that may be used to illustrate the effects of specular transmission on the limits of integration in [3.7]. To this end, it also serves to qualitatively demonstrate the effects of the electrode band structures on the characteristics of the calculated tunneling spectra. Shown plotted on an energy versus wave vector scale are the conduction band of the metal displaced by −eVa with respect to the valence band of the semiconductor. In this non-equilibrium configuration, a current is expected to result due to electrons transporting from the occupied states of the valence band to the unoccupied states of the metal conduction band. The bold line highlights the energy window over which transport occurs elastically. Therefore, it can be immediately seen that the limits of the total energy integral will be given by the energy of the valence band edge E V and the Fermi energy of the metal, E M F . Since Θ ignores band gap energies, these limits may simply be chosen as the Fermi energies of the two electrodes. In the present example 68 E SC EF EV eVa E′ M EF Eo SC kt,E′ Figure 3.2 M k kt,E′ Schematic representation of the energy band structure of a metal with respect to a semiconductor in a non-equilibrium (Va ≠ 0) configuration. For a tunneling electron with total energy E′, the transmission factor is integrated from the zone center to E′, where the effective mass in [3.7] is given by the curvature of the valence band. EM F E SC F −eVa ∫ dE → ∫ E SC F E SC F − eVa dE → ∫ dEΘ − [E − E ]. B vb [3.10] 0 For a given total energy, shown as E′ in Figure 3.2, the limits of the normal energy integral reflect the limits of the integral over d2kt in [3.6a]. Indicated along the k axis are the values of the maximum transverse wave vectors for an electron with energy E′ in the metal ( kM t ,E ′ ) and in SC M the semiconductor ( kSC t ,E ′ ). Since kt ,E ′ < k t, E ′ , the current is limited by the semiconductor at this 69 energy and the integration over the normal energy extends from the center of the zone to E′. This gives E′ ∫ dE E (k =0) E′ L → ∫ dE L exp(−η), [3.11] E Bvb and the effective mass in [3.7] will be given by the curvature of the valence band. A similar construction for the conduction band gives identical results except for a sign change in the argument for the Heaviside function and E Bvb being replaced by E Bcb in [3.10] and [3.11]. Note that the selection of these limits does not depend on the nature of band bending which contributes only to the depletion region transmission factor integral, ηsc. It is instructive to examine the overlap of the band structures of the metal and the semiconductor in a manner similar to Figure 3.2 for both valence and conduction bands and for bands with varying curvature. The following conclusions regarding the effects of specular transmission into the conduction band may be drawn from such an exercise: 1) When the curvature of the semiconductor conduction band (CB) is identical to the curvature of the metal CB, the current will be limited by the semiconductor until the bottom of the metal conduction band increases above the bottom of the semiconductor CB. At this point the current will be limited entirely by the metal. 2) When the curvature of the semiconductor CB is smaller than that of the metal CB, there will be an energy, Eo, at which the two dispersion curves intersect for a given applied bias. When Eo lies above the metal Fermi level, the current is always limited by the semiconductor. (Note that this is opposite to the case for tunneling from the valence band as can be seen in Figure 3.2.) When the bias reaches some critical value, Va′, so that Eo < E M F , the current will be partially limited by the semiconductor and partially limited by the metal. As the bias continues 70 to exceed Va′, the metal assumes an increasing dominance over the current until, as before, the current is limited entirely by the metal. 3) When the curvature of the semiconductor CB is larger than that of the metal CB, the current is always limited by the semiconductor until the bottom of the metal CB increases above the semiconductor CB edge. At this point the current is partially limited by the metal which assumes an increasing dominance with voltage (as before) until the current is entirely metallimited. A similar exercise with the valence band draws similar conclusions except that the current is never limited completely by the metal. When the curvature of the valence band and the metal CB are identical, the current is semiconductor limited until a critical value of the bias Va′ is reached (where Eo > E M F ) and the current becomes partially limited by the metal. When the valence band curvature is smaller than the curvature of the metal CB, Va′ is reduced in magnitude. The reverse is true when the valence band curvature is larger. Since the Fermi energy in a metal is determined primarily by its atomic density, the greater the density of the metal, the less it will influence the current for a given bias. Most metals have quite large densities (owing to their efficient packing) so that as a general rule of thumb, the smaller the curvature of the semiconductor bands, the greater the influence of the metal on limiting the tunneling current. Recall that these effects arise as a consequence of the restriction to specular transmission; a condition that is not guaranteed to exist in every experimental situation. however, that the assumption of specular transmission It is believed, is reasonably valid for surface roughness comparable to the wavelength of the incident electrons [3]. Recall 71 the de Broglie relation λ = 150.4 E , where E is in eV, and λ is in Å. For electrons with low energy, such as those tunneling at low biases, λ is ∼100Å. At ±2V, the high-energy electrons will have a wavelength of 8.7Å. At ±4V, λ is 6.1Å. Therefore, higher applied biases require successively smoother surfaces for the assumption of specular transmission to be valid. One might consider, however, the meaning of “roughness” on the scale of the tunneling junction cross-section (usually assumed to be ∼25Å2). A roughness corresponding to atomic corrugations (i.e., ≤1.5Å) extends the valid voltage range out to ±67V. A roughness on the order of a unit cell step edge on (001) SrTiO3 (i.e., ∼4Å) extends the valid voltage range to ±9.4V, well beyond typical experimental voltage sweeps. 72 Vi Va RM≈0 Vs RG≈∞ a e(∆-Vi) χ (ψs,o-Vs) φM EC SC B b s Figure 3.3 ξ eVa M EF +++ ++++ +++ EF EV ω a) Equivalent circuit for b) a metal-vacuum-semiconductor tunnel junction at forward bias. (See text for symbol definitions.) 3.2.3 The potential distribution functions Equation [3.7] correctly describes the transport of charge across a classically impenetrable barrier. The barriers in a STM may be represented by a resistor in series with a diode, as shown schematically in Figure 3.3a. Any applied bias, Va, will be divided between the resistor RG (free space gap) and the diode (semiconductor depletion region). These components of Va will be referred to as Vi and Vs, respectively. Figure 3.3b schematically shows the spatial behavior of the energy bands for a forward biased junction. The symbols of the figure have the following meanings: 73 φM work function of the metal χ electron affinity of the semiconductor ∆ equilibrium (“contact”) potential across the insulating region ψs,o equilibrium surface potential ξB bulk value of the conduction band edge with respect to E SC F s metal-to-semiconductor separation ω width of the depletion region From inspection of Figure 3.3b one can immediately write φ M = eVa + ξ + e(ψ s,o − Vs )+ χ + e(∆ − Vi ). B [3.12] At thermal equilibrium, it is assumed that there is an initial negative surface charge density of magnitude QSS, which is compensated by a positive charge density of magnitude QSC in the depletion region and a positive charge density of magnitude QM induced on the metal, so that QM = −(−QSS + QSC ). [3.13] Gauss’ law gives ∆=− sQ M s (−QSS + QSC ) , = εi εi [3.14] where ε i is the permittivity of the insulator (or free space in this case). Also from Gauss’ law, the field at the surface of the semiconductor at equilibrium and non-equilibrium are respectively given by Fs,o = − Qsc,o εs and Fs,a = − Q sc,a , εs [3.15] where the difference between Fs,o and Fs,a is proportional to the component of the applied bias that falls across the semiconductor. Using [3.14] and [3.15] one may write 74 ∆o = s(−Qss,o − εs Fs,o ) εi and ∆a = s (−Q ss,a − ε sFs,a ) εi so that the component of the applied bias that drops across the insulator is Vi = ∆ a − ∆ o = s(−Qss,a − ε s Fs,a ) s(−Qss, o − ε sFs,o ) s(ε sFs, a − εs Fs, o ) s(Qss,a − Qss,o ) − =− − εi εi εi εi or Vi = − s(∆Qsc + ∆Q ss ) . εi [3.16] Therefore, the development of the component of the external field which drops across the insulator is accompanied by a change in both the space charge and the surface charge. Assuming an abrupt junction approximation, from semiconductor theory [2] Qsc, o = 2eε sN D ψ s,o 1 kT 2 − e [3.17] and 1 Qsc,a kT 2 = 2eε sN D ψ s,o − − Vs , e [3.18] where ND is the density of ionized single electron donor defects. The equilibrium band bending ψ s, o may be selected as an initial condition. This value will be close to φ M − χ in the absence of a large density of surface states; otherwise, ψ s, o will be independent of φ M − χ . From [3.17] and [3.18] it is clear that ∆Qsc is determined by Vs as stated earlier. Since ∆Qss is not known a priori, an assumption is made regarding its behavior. From [3.12] and Figure 3.3b it is clear that 75 both the “contact” potential and the surface band bending change linearly with applied bias so that it seems reasonable to assume that Q ss (which constitutes the discontinuity of the field across the insulator-semiconductor junction) also varies linearly with applied bias. This permits writing ∆Qss (Va ) = −βVa , [3.19] where β is an adjustable parameter and the minus sign indicates that a positive bias decreases the depletion region charge density and is thus expected to decrease the surface charge density. This parameter shall be refered to as the capacitance constant; it has units of C/V cm2. Combining [3.16]–[3.19], and using the identity Va = Vs + Vi , gives 1 1 −1 2 2 sβ s kT kT − Vs − 2eεs ND ψ s,o − . [3.20] Va = 1− Vs − 2eε sN D ψ s,o − εi εi e e By varying Va and numerically solving for Vs, and using Va = Vs + Vi to find Vi, the potential distribution functions Vs(Va) and Vi(Va) are completely determined to be used further to determine the bias-dependent behavior of the average tunneling barrier heights. The equilibrium band bending is determined by the equilibrium surface charge. It is also assumed that most of the compensation for surface charge occurs in the depletion region of the semiconductor (i.e., changes in surface charge on the metal are assumed to be negligible) so that from [3.17] 1 ′ − Q sc,o ∆Qss, o = Qsc,o 1 kT 2 kT 2 = 2eε sND ψ ′s, o − − 2eε N ψ − s D s,o e , [3.21] e where ψ s,′ o and ψ s, o are the required fitting parameters. 76 2 2 a 1.5 1 Sample Bias, Vs (V) 1 Sample Bias, Vs (V) b 1.5 0.5 0 -0.5 -1 -1.5 0.5 0 -0.5 -1 -1.5 -2 -2 -3 Figure 3.4 -2 -1 0 1 Tip Bias, Va (V) 2 3 4 -3 -2 -1 0 1 Tip Bias, Va (V) 2 3 4 Calculated potential across the sample, Vs, as a function of the total applied bias, Va. The effect of variation of donor density, ND, is demonstrated: solid line =1018; dashed line = 1019; dot-dashed line = 1020 cm-3; a) β = 0; b) β = 0.002. Figure 3.4 shows the results of using the above procedure to calculate the voltage drop across the sample, Vs, as a function of the bias applied to the tip, Va, for a metal tip–vacuum gap–semiconductor junction. The effect of increasing the donor density is to decrease |Vs| for all non-zero values of Va. This effect can be expected since increasing the free carrier density increases the screening strength in the semiconductor and thus a smaller potential is expected to drop across the diode in Figure 3.3a. It should be noted that increasing the static dielectric constant, which also increases screening, has a similar effect. A comparison of Figure 3.4a and 3.4b demonstrates the effect of a bias-dependent variation of the surface charge — i.e., β ≠ 0 in 77 Figure 3.4b. For a given tip bias, |Vs| decreases with decreasing Qss. It can also be seen that this effect is greater at larger biases. The main point of Figure 3.4 is to illustrate how the potential distribution curves depend on the experimental variables and is most useful when fitting experimental data. Values which give unreasonable potential distribution and/or surface potential curves are assumed unphysical. φM-φS-eVi χ φM φS EC M F SC E E′ EF s Figure 3.5 +++ ++++ +++ EV ω An equilibrium (Va = 0) configuration for a metal-vacuum-semiconductor tunnel junction separated by a gap of width s. 78 3.2.4 The potential barrier functions An arbitrary electron at the surface of the metal must overcome a barrier determined by the sum of the surface work function, φm, and the difference in energy between the metal Fermi level and the energy of the electron, E′. This is shown schematically on the left hand side (z=s) of the barrier in Figure 3.5. The barrier is thus given by φ m + (E F − E ′) = (E F − E ′)+ (E cb − E F M SC S SC )+ χ + (φ m − φ s − eVi ), [3.22] where E Scb is the value of the conduction band edge at the surface of the semiconductor due to band banding. On the semiconductor side (z=0) of the junction, an electron at the same energy must overcome a barrier given by χ + (E cb − E F S SC )+ (E SC F − E ′). [3.23] Notice that χ + (E Scb − E SC F ) may be replaced by the semiconductor work function φ s so that both [3.22] and [3.23] may be written as SC E F + φm − eVi − E ′ (z = s) E SC F + φs − E ′ (z = 0) from which a trapezoidal function is constructed to describe the vacuum barrier. Recalling that SC EF ≡ 0 , z z ϕ trap (z,Vi , E′ ) = (φ m − eVi − E ′) + (φ s − E ′ ) 1 − . s s [3.24] Following Simmons [23], a term is included to account for the image force lowering of the barrier given by ϕ image ( z) = − 0.4e 2 s , 8πκ op ε o z(s − z ) 79 [3.25] where κ op is the optical dielectric constant of the barrier region (assumed to be unity for a vacuum barrier). The total vacuum barrier is the sum of [3.24] and [3.25]. Figure 3.6a shows the calculated barrier “seen” by an electron at the Fermi energy for a junction at equilibrium for several values of the tip-sample separation, s. It can be seen that the average barrier height decreases and becomes more symmetrical as s decreases. Figure 3.6b shows the effect of a junction under non-equilibrium conditions for a constant tip-sample separation of 9 Å. The bold curve corresponds to equilibrium — i.e., Vi = 0. Increasing in the forward bias (Vi > 0) direction decreases the average barrier height and there is a substantial decrease in the width of the barrier. Increasing in the reverse bias (Vi < 0) direction, however, increases the average barrier height and the barrier width increases only slightly. 80 Potential (eV) 3.5 a 3 2.5 2 1.5 1 0.5 0 0 2 4 6 8 Gap width (Å) Potential (eV) 6 b 5 4 3 2 1 0 0 2 4 6 8 Gap width (Å) Figure 3.6 Calculated spatial (a) and voltage (b) dependent vacuum potential barrier (in eV). The abscissa is measured with respect to the electrode with the smaller work function. In (a) the arrow indicates the direction of decreasing tip-sample separation, s. In (b) the top and bottom arrows indicate the direction of increasing reverse and forward bias, respectively, in voltage steps of 0.9 eV. 81 A second potential barrier often exists in the form of band bending within the depletion region of the semiconductor (see Figure 3.3 and Figure 3.5). Feenstra and Stroscio [11] demonstrated the importance of including voltage-dependent (i.e., dynamic) band bending to improve agreement between experimental and theoretical tunneling spectra. Their approach consisted of calculating the equilibrium surface potential φs of the semiconductor as a function of the applied bias. This potential is assumed to decrease parabolically from the surface (z = 0) towards the bulk, reaching a value defined as the equilibrium bulk potential φb at the depletion edge (z = w). The resulting energy profile φbb is added to both the valence and conduction band edges to model the band bending. The parabolic potential profile in the depletion region can be written as φ bb (z, Vs ) = (ω − z)2 ω 2 φsbb (Vs ), [3.26] where ω is the width of the depletion region given by ω= 2εs φsbb (Vs ) . e2 ND [3.27] Therefore, the potential barrier associated with band bending in the semiconductor depletion region (for an electron with energy E ′ ) is given by ϕ sc (z,Vs, E′ ) = φ bb (z,Vs ) + E cb − E ′ . [3.28] A similar approach is employed here to determine φ sbb (Vs ) , which begins with a single integration of the one-dimensional form of Poisson’s equation (in MKS units): ∇ ⋅ D z = ρz , [3.29] where D z = ε s Fz is the z component of the electric displacement vector. Below are defined the dimensionless parameters that are usually introduced for convenience [24,25]. Since the band 82 bending at the surface is given by the difference between the potential at the surface and the potential in the bulk, φ sbb = (φ b − φ s ) [3.30] where φz ≡ 1 (E F − E I, z ). e [3.31] In [3.31] E I, z is called the intrinsic Fermi level which maintains a constant relation with respect to the conduction and valence band edges, and is given by 3 m *vb 4 1 E I = (E vb + E cb ) + kT ln * . 2 m cb [3.32] Therefore, [3.31] may be referred to as the depletion region potential function and with [3.30] gives the surface band bending (φsbb) when z = 0. ( uz = By writing [3.31] in a reduced form eφ z dφ ), and noting that Fz = − z , Poisson’s equation may be written as: kT dz eρ z d 2 uz . 2 = − dz κεokT [3.33] Here κ is the static dielectric constant of the semiconductor. The equilibrium charge distribution function ρ z = ρ D + ρ A + e (p z − n z ) is found employing Fermi statistics. The free carrier densities, in the degenerate limit, are given by the usual relations [26] 3 m * kT 2 E − E cb = 2 cb 2 n z = Ncb F 1 F F 1 (uz − wcb,I ) 2 2πh 2 kT 3 2 m kT E vb − E F F1 (w − uz ) = 2 2 2πh2 2 vb,I kT * vb pz = Nvb F 1 83 [3.34] where w1,2 = E1 − E 2 for any two energies, Ni is the effective density of states for energy band i, kT and F12 is the Fermi-Dirac (F-D) integral of order 1 2 . The latter is one of a family of dimensionless integrals [21] defined by ∞ ε jdε 1 F j (η/ ) = . Γ ( j + 1) ∫0 1 + exp (ε − η/ ) [3.35] These integrals cannot be solved analytically. Extensive tabulations exist (see for example Appendix B of reference 21) for F-D integrals of various orders and for both positive and negative values of the argument, η / . In the non-degenerate limit, these integrals may be replaced in [3.34] by the “classical” approximation Fj (η/ ) ≈ exp(η/ ). The ionized donor ρ D and acceptor ρ A charge densities are also determined by equilibrium Fermi statistics. (In the present formalism it is assumed that ρ A = 0.) For ionized single donor defect centers (i.e., Nb•Ti or VO• ), the contribution to the space charge density is given by (see Appendix B) 1 ρ D = eND . 1 + 2 exp[uz − wD1, I ] [B.5] Assuming donor defects that may accommodate two electrons, the charge density due to ionized donor sites (i.e., both VO• and VO• • ) can be shown to be given by f −f f ρ D = eND E ′ E ′ D1 1 − fD1 + fE ′fD1 [B.22] where the functions fE ′ and fD1 are similar to the bracketed term in [B.5] and are derived in Appendix B. Poisson’s equation can be rewritten using [B.5] or [B.22] combined with [3.34]. 84 As an example, consider the case of a degenerate semiconductor with divalent donor defects. Equation [3.33] thus takes the form d 2 uz Ncb 1 fE ′ − fE ′fD1 Nvb + = − F w − u − F u − w 1 ( ) 1 ( ) z cb,I dz 2 L2D 1− fD1 + fE ′ fD1 N D 2 vb,I ND 2 z [3.36] where the extrinsic Debeye length LD is defined by LD ≡ εs kT . e2N D It is convenient to make a change of variables [2,3] by writing F/ = − that F/ dF/ = [3.37] du z , from which it follows dz d 2 uz du . Integrating from the bulk to the surface, observing the boundary condition dz2 z du z = 0 (i.e., zero electric field in the charge neutral region) gives dz b F/ s 1 ∫ F/ dF/ = 2 F/ 2 s F/b =0 1 f − f f E ′ E ′ D1 + Nvb F 1 (wvb,I − uz )− Ncb F 1 (uz − wcb,I ) duz [3.38] − 2 ∫ LD 1− fD1 + fE ′ fD1 N D 2 ND 2 ub us = The first part of the integrand on the RHS of [3.38] integrates as exp(ub ) + exp(wE ′, I ) 1 2exp w arctan ( E ′,I ) 2 LD exp(wE ′,I + wD1,I )− exp(2wE ′, I ) exp(us ) + exp(wE ′,I ) − arctan exp (wE ′,I + wD1,I )− exp (2wE ′, I ) 85 1 × . + w − exp 2w exp w ( E ′,I D1,I ) ( E ′,I ) The F-D integrals are integrated using the formula [27] dF j (η) = jF j −1 (η), dη so that the remainder of the integrand on the RHS of [3.38] integrates as 2 1 Ncb 2 F F 3 (u − wcb,I ) 3 (us − wcb, I ) − 2 3 2 b LD ND 3 2 − 2 Nvb 2 F F 3 (w − ub ) . 3 (wvb, I − us )− 3 2 vb ,I ND 3 2 The latter two solutions (sans the factor L−2 D ) are referred to below as sol1(us) and sol2(us), respectively, so that for the electric field at the surface of the semiconductor Fs = 1 kT kT duz kT 2 2 F/ s = − = ± sol1(u ) + sol2(u ) [ s s ] . eLD e e dz s [3.39] Recalling [3.15], the net space charge per unit surface area, Qsc,a, as a function of the reduced surface band bending, us, is written as 1 Qsc,a = eND LD 2 [sol1(us ) + sol2(us )]2 . [3.40] Equations [3.18] and [3.40] together give a means by which to model the band bending behavior. Varying us and numerically solving for Vs obtains a sample bias-dependent surface band bending function, φsbb(Vs). 86 3.5 a 3 Surface Potential (eV) Surface Potential (eV) 3 3.5 2.5 2 1.5 1 0.5 b 2.5 2 1.5 1 0.5 0 0 -0.5 -0.5 -3 Figure 3.7 -2 -1 0 1 2 Sample voltage,Vs (V) 3 -3 4 -2 -1 0 1 2 3 4 Sample voltage,Vs (V) Calculated surface potential (i.e., band bending) as a function of the voltage component across the sample. a) Variation of defect density ND: solid = 1018 cm-3; dashed = 1019 cm-3; dot-dashed = 1020 cm-3. b) Variation of initial band bending ψ s, o : solid = 0.3eV; dashed = 1.5eV; dot-dashed = 2.4eV Figure 3.7 shows calculated surface potential curves for the monovalent donor case and their dependence on ND and ψ s, o . As can be anticipated, decreasing ψ s, o shifts the curves uniformly towards more positive sample biases (Figure 3.7b). It is interesting to observe that increasing ND increases the range of band bending at positive sample bias (Figure 3.7a). There are three important points to consider on these plots. 1) At zero applied bias (Vs=0) they show the equilibrium surface band bending, or Schottky barrier height, associated with the “contact” potential. 2) At forward bias (Vs<0) ∆Qss is negative (see [3.19]) and the majority carrier density increases at the surface of the semiconductor thus decreasing both φsbb and ω. When sufficient bias is applied, the surface potential and depletion width fall to zero. The sample voltage at 87 which this occurs is called the flat band voltage and it marks the onset of majority carrier accumulation at the surface of the semiconductor. Figure 3.7 shows this to be given by the voltage at which the surface potential reaches the onset of the lower plateau. 3) In the reverse bias (Vs>0) case, the surface potential and depletion width both increase. This moves the sample Fermi level towards the valence band edge in the near surface region. The bias at which the Fermi energy (EF) falls below the intrinsic Fermi energy (EI) at the surface defines the onset of inversion [2]. This is given by the voltage at which the surface potential reaches the onset of the upper plateau. One may also compare calculated band bending for monovalent and divalent donor defect semiconductors. In the latter case an additional occupied donor level resides a few eV below the conduction band edge. At sufficient reverse bias, the Fermi energy will approach this level near the surface and thus the state is expected to become unoccupied, assuming there exists a mechanism for this process to occur. Applying the present formalism, the effect on band bending appears to shift the onset of inversion to a lower applied bias, depending on the energy level of the deep donor state, as shown in Figure 3.8. Note that the rather abrupt transition to inversion as shown is believed to be an artifact resulting from use of the abrupt junction approximation. Hence Figure 3.8 demonstrates a possible limitation of simplified semiconductor device theory when applied to semiconductors with multivalent defect centers. The true behavior is likely to lie somewhere between the solid curve and the broken curves — i.e., a gradual increase in slope giving rise to an earlier onset of inversion. 88 3.5 Surface Potential (eV) 3 2.5 2 1.5 1 0.5 0 -0.5 -3 -2 -1 0 1 2 3 4 Sample voltage,Vs (V) Figure 3.8 Comparison of equilibrium band bending for monovalent (solid) and divalent donors in a semiconductor with band gap energy Eg = 3.2 eV. The defects lie at 1.7 (dashed), 1.3 (dot-dashed), and 0.9 eV (dotted) below the conduction band minimum. Note that the curves in Figures 3.7 and 3.8 are calculated in what is referred to as a quasiequilibrium formalism, where the surface of the semiconductor is always assumed to be in thermal equilibrium with the bulk. One consequence of this assumption is that the difference between the surface potential energies corresponding to the upper and lower plateaus is given by the semiconductor band gap. There is both experimental [11] and theoretical [28] evidence which support the view that non-equilibrium conditions via minority carrier extraction prevail in 89 practice due to a sufficiently large tunneling current. The absence of spectral features corresponding to the onset of inversion has been interpreted in terms of non-equilibrium conditions which pin the majority carrier quasi-Fermi level to the Fermi level of the metal [28]. Consequently, band bending is no longer limited to the width of the forbidden energy gap [13]. Considering that oxides tend to have large band gap energies, the intrinsic carrier density is negligible. Hence, non-equilibrium via minority carrier extraction for oxides may be assumed a non-measurable effect. This justifies adopting the quasi-equilibrium approach described above. An absence of inversion for wide band gap (i.e., large dielectric constant) systems may be expected by considering the potential distribution curves in Figure 3.4. Depending on the experimental conditions, a relatively small fraction of the applied voltage may fall across the semiconductor depletion region due to screening effects. This suggests that, for wide band gap semiconductors, experimental band bending will be confined to a narrow voltage range about the origin in Figures 3.7 and 3.8. 3.2.5 Determination of the defect-induced current In defect semiconductors, there is a current density that is induced by the contribution of free carriers from the defect states. There are three ways in which this current density may be treated in the model. At the appropriate density of defects, there may be a defect band introduced into the band gap of the solid. This band can be treated in the same manner as the valence and conduction bands — i.e., sum the tunneling probability over the energy window corresponding to the width of the band. If the conditions are such that the semiconductor is degenerate, then the Fermi level will lie a few meV above the conduction band edge. The current at small forward bias will generally include two components due to the (defect-induced) carriers in the conduction 90 band — a tunneling component and an emission component. The tunneling component is automatically accounted for by the limits of the integral in [3.10]. The standard Richardson-Schottky (R-S) equation is derived based on the assumption that the electrons in the conduction band behave approximately as an ideal gas. Consequently, the electron energy is assumed to be completely kinetic in nature and the emission current is given as eφ j DI = A* T 2 exp − b exp (β E 0 ), kT where E0 is the electric field at the semiconductor surface, [3.41] 4πem* k 2 A = h3 * and 1 e 2 e . In semiconducting insulators, however, the behavior of the conduction β = kT 4πε o κ op band electrons cannot always be described by free electron theory. This is true a fortiori for oxides wherein strong electron-lattice coupling is believed to determine the mechanism of charge transport at low carrier densities. Relaxation of the ideal gas assumption results in a modified RS equation as 3 m* 2 eφ j DI = αT E 0 µ exp − b exp(β E 0 ), kT me 3 2 where α = 3 × 10 −4 As cm3 K 3 2 [3.42] and µ is the electron mobility in the semiconductor. This relation was first derived by Simmons [29] and recently shown to demonstrate improved consistency with experimental mobility data for Ba1-xSrxTiO3 (BST) thin films [30]. In the present application, the forward-biased defect current is assumed to be dominated by the emission process (i.e., depletion region diffusion currents are neglected) and emission from metal to 91 semiconductor is ignored. The net current is thus assumed to be given by [3.42] modified by a vacuum barrier transmission factor and including the term 1 4 e 3 N kT D ψs − ∆ φ′o = 2 2 e 8π ε sε op [3.43] for image force lowering of the diffusion barrier. Most insulating semiconductors are non-degenerate at low to moderate defect densities. The Fermi level thus will lie below the conduction band edge and the defect-induced current density may be assumed to be given entirely by [3.41] or [3.42]. The vacuum barrier transmission factor is calculated exactly as before (equation [3.6b]) assuming all of the electrons “see” the barrier as measured from the bottom of the conduction band at the surface. In the degenerate semiconductor case, the relative importance of tunneling versus thermionic emission may be considered in terms of the ratio [3] Ξ= kT , eE 00 [3.44] where 1 E 00 and α = m ∗ me 1 eh N 2 N 2 = *D = 18.5 × 10 −15 D , 2 m εs ακ [3.45] . As a rough guide, tunneling is expected to dominate when Ξ is much less than unity, thermionic-field emission is expected for Ξ close to unity, and thermionic emission dominates when Ξ is much greater than unity. It can be seen from [3.45] that a larger carrier density favors tunneling, whereas a larger effective mass and dielectric constant favor thermionic emission as a means to negotiate the semiconductor diffusion barrier. Assuming α = 12 and κ = 300, for tunneling to be considered dominant, [3.44] and [3.45] require N D >> 6.57 × 1021 cm-3, 92 or approximate stoichiometry SrTiO2.625 for a reduced crystal. For carrier densities much less than this value, it is reasonable to use [3.41] or [3.42] in all current calculations. 3.3 SAMPLE CALCULATIONS 3.3.1 Simulated vs experimental spectra The current density due to tunneling between the conduction band (jCB) and the valence band (jVB) states are calculated independently using [3.7] and together with [3.42] the total current is given by I = j total A = (jCB + jVB + jDI )π r2 , [3.46] where r is the radius of curvature of the metal tip. A sample calculation was performed using the parameters in Table 3.1. Figure 3.9a shows the spectrum on a linear scale, while Figure 3.9b is a semi-log plot of the same result. Table 3.1: Parameters used to calculate the tunneling spectrum in Figure 3.9. tip-to-sample separation radius of curvature of tip conduction band effective mass valence band effective mass static dielectric constant optical dielectric constant equilibrium surface potential donor defect density electron mobility capacitance constant band gap energy metal work function sample electron affinity s r 9Å 5Å 12 1 210 5 m *cb m *vb κ sc κ op ψ s, o ND µe 0.30 eV 5 × 1019 cm−3 2 30cm Vs 0.0001 3.2 eV 5.65 eV 3.0 eV β Eg φm χ 93 10 7.5 5 a -4 -3 2.5 -2 -1 -2.5 1 2 3 4 -5 -7.5 -10 Sample voltage,Va (V) 2 1.5 b Log|I| (nA) 1 0.5 0 -0.5 -1 -1.5 -2 -3 -2 -1 0 1 2 3 4 Sample voltage,Va (V) Figure 3.9 Calculated tunneling spectrum using the parameters in Table 3.1. 94 Experimental limitations may truncate the measured spectra. The upper dotted line in Figure 3.9b marks the point at which the current pre-amp voltage saturates, while the lower dotted line marks a typical upper limit to the noise floor of the electronics. The spectrum in Figure 3.9b is replotted in Figure 3.10 and compared to an experimentally obtained spectrum. No effort was made to “fit” the calculated curve to the experimental one; however, the agreement between the qualitative features of the curves quite remarkable. 1 Log|I| (nA) 0.5 0 -0.5 -1 -3 -2 -1 0 1 2 3 4 Sample voltage,Va (V) Figure 3.10 Comparison of calculated (solid) and experimental (dashed) spectra. 95 is The modeled spectrum may be characterized by an asymmetry in the slope of the current as a function of applied bias. The current at small forward bias rises more slowly compared to reverse bias. This current is due to the finite density of extrinsic carriers in the conduction band which must overcome a decreasing surface potential as well as a similarly evolving vacuum barrier (Figure 3.6b). The rate at which this current rises is strongly dependent on the strength of dynamic band bending, which itself depends strongly on the potential distribution functions. That is, a larger potential drop in the semiconductor gives a greater degree of dynamic band bending. A rapid drop in the surface potential corresponds to a rapid increase in surface electron density and hence a steep slope in the curve at negative sample bias. Hence, the relative slope in the spectra at forward bias is a measure of the relative degree of dynamic band bending. At reverse bias, however, there is a much faster rise in current at the onset voltage. In this case, the current is determined by the density of states of the electrodes and a large density of electrons from the metal tunnel into a large “volume” of momentum space provided by the unoccupied states in the oxide conduction band. In contrast to negative biases, the current slope at positive bias is less dependent on dynamic band bending. The dominant portion of the tunneling current involves exchange between states near the top of the diffusion barrier where the width varies most slowly. At sufficient negative sample bias, the metal Fermi level crosses the valence band edge so that a large increase in current due to electrons tunneling from the valence band is expected. This may be observed as a kink or slope change. Since the valence band current is superimposed on the defect-induced current, the occurrence of such feature depends on the initial surface potential as well as the strength of dynamic band bending. 96 In most cases for insulating semiconductors, as in Figure 3.9, the current at forward bias is dominated by the defect-induced current density. 3.3.2 Parametric study of tunneling model The behavior of the calculated curves may be studied as a function of several experimental and material parameters. The goals of such a parametric study are 1) to establish a fitting procedure, and 2) to facilitate identification of experimental artifacts due to changes in tip-sample separation or sample permittivity, for example. Figure 3.11 shows the results of varying the specified parameters about their initial values which appear in Table 3.1. The magnitude of the specified parameters increases from the solid curve to the dot-dashed curve. The dashed curve in all of the plots of Figure 3.11 is identical to the calculated curve appearing in Figures 3.9 and 3.10. It may be immediately observed that variations in carrier density, surface potential, and sample permittivity all strongly affect the behavior at forward bias. Only a very weak dependence at reverse bias is predicted. Varying the effective mass is predicted to have a weak effect on the curve at all applied biases. And finally, varying the tip-sample separation or the electron affinity is predicted to strongly affect the curve at reverse bias while only weakly affecting it at forward bias. The dependence upon each of these parameters is discussed separately. Carrier density: A larger carrier density increases the image force lowering term in the R-S emission relation [3.43], thus increasing the carrier density at the semiconductor surface for a given applied bias. The screening efficiency also increases with increasing free carrier density such that less voltage is expected to drop within the semiconductor. These two effects are opposing and when the latter outweighs the former, the calculated current decreases with increasing carrier density. 97 Surface potential: Modifying the surface potential is a more direct means to modify the surface carrier density. The latter constitutes the fraction of mobile electrons with sufficient kinetic energy in the direction of the surface to overcome the diffusion potential. The surface carrier density thus naturally falls 98 with increasing surface potential. 1 Log|I| (nA) 0.5 0 -0.5 -1 a -3 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 1 Log|I| (nA) 0.5 0 -0.5 -1 b -3 Sample voltage,Va (V) Figure 3.11a,b a) Increasing carrier density: 1, 5, and 10 × 1019 cm-3. b) Increasing surface potential: 0.25, 0.30, and 0.35 eV. 99 1 Log|I| (nA) 0.5 0 -0.5 -1 c -3 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 1 Log|I| (nA) 0.5 0 -0.5 -1 d -3 Sample voltage,Va (V) Figure 3.11c,d c) Increasing static dielectric d) Increasing effective mass: 5, 12, and 50. 100 constant: 100, 210, and 300. 1 Log|I| (nA) 0.5 0 -0.5 -1 e -3 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 1 Log|I| (nA) 0.5 0 -0.5 -1 f -3 Sample voltage,Va (V) Figure 3.11e,f e) Increasing tunneling gap: 8, 9, and 10 Å. f) Increasing electron affinity: 2.6, 3.0, and 3.4 eV. 101 This effect is due solely to emission over the Schottky barrier and is not observed at reverse bias where the transport mechanism is described completely by tunneling. Only a weak shift in the current onset is observed, attributable to the dependence of the depletion width on the surface potential. As previously discussed, the dominant portion of the tunneling current at reverse bias occurs near the top of the diffusion barrier where the width varies most slowly, hence the weak dependence. Dielectric constant: The total polarizability of a medium is represented by the complex dielectric constant. A larger value thus corresponds to greater screening strength and thus (similar to decreasing resistivity) reduces dynamic band bending. Effective mass: The absence of a significant dependence on the DOS effective mass is not surprising considering that it only enters as a term in the pre-exponential factors of [3.7] and [3.42]. When the semiconductor is degenerate, however, the Fermi energy, measured with respect to the conduction band edge, has an inverse dependence on the effective mass. Vacuum gap: An exponential dependence on the vacuum gap is expected from [3.6]. The asymmetry in Figure 3.11e thus cannot be explained in a straighforward manner. Perhaps it may be understood by considering the development of the vacuum barrier as shown in Figure 3.6b. At forward bias the barrier “seen” by electrons at the surface of the semiconductor decreases significantly in width as well as in height. Furthermore, the majority of the “hot” electrons are near the top of the barrier where the WKB treatment begins to break down (see section 3.1.1). To a first approximation, it may be concluded that the voltage-induced narrowing of the barrier at forward bias very nearly compensates for the gap increase. Electron affinity: The vacuum barrier is determined by the average value of the work functions of the two electrodes. Figure 3.11f is simply a reflection of an increasing vacuum 102 barrier potential. The weaker dependence at forward bias may be explained by the same argument used to explain the weak dependence on the vacuum gap. Figures 3.11b and 3.11f are a bit unrealistic to the extent that surface potential and electron affinity are both directly dependent on the surface charge density. Thus the variations in these figures may be associated with an increase in surface charge such that the true variation in the spectra might appear as shown in Figure 3.12. It may be concluded that a variation in surface charge is predicted to have an asymmetric effect on the tunneling spectra, where the effect is stronger at forward bias than at reverse bias. 1 Log|I| (nA) 0.5 0 -0.5 -1 -3 Figure 3.12 -2 -1 0 1 2 3 Sample voltage,Va (V) 4 The predicted effect of increasing surface charge density in steps of ∼1.40×10-7 coulombs per cm2. 103 3.3.3 Discussion Figure 3.10 must be regarded with caution because the physical properties of the sample were not verified to match those used in Table 3.1. The values chosen, however, are average values for semiconducting SrTiO3 and, except for the electron mobility, they are not expected to deviate by orders of magnitude from the actual values of the sample in question. In practice, materials parameters measured by the methods described in section 2.2.2 give average macroscopic values; scanning tunneling spectroscopy measures local surface properties which may deviate significantly from their average values. Other experimental parameters, such as the tip-sample distance, are impossible to know precisely. Figure 3.10 thus serves as a reasonable test case for the reliability of the model. The greatest error is introduced by parameters that give rise to big effects for small to moderate variations in their values. The predictions of any theoretical model strongly depend on the how the model is constructed. Several assumptions, approximations, and restrictions have been made throughout this chapter which potentially may separately or collectively steer the predictions away from the true behavior of the system. In some cases the assumptions are justified for the right experimental conditions or rather limit interpretation to a specified experimental window. This, for example, is true for the conditions of elastic/specular transmission as well as assuming the abrupt junction approximation. In other cases, however, the assumptions are a more significant deviation from reality and their use is justified only by the resulting simplicity that they facilitate in the computations. An example of this is the assumption of an isotropic band structure for the semiconducting electrode. Inspection of Figure 1.2 immediately demonstrates that this assumption is significantly challenged. The band structure influences the calculated results through the DOS effective mass and through the transverse wave vector integral. As Figure 104 3.11d shows, error in the effective mass has negligible effects on the results. The isotropic assumption, on the other hand, can possibly introduce more profound error through the limits of the integral in [3.6a] since the shapes of the Broullion zone constant energy surfaces projected onto the tunneling junction are not ideally circular. It has been assumed that the materials parameters do not change during the course of a measurement. It has already been argued, however, that the surface charge varies with applied bias and that the semiconductor electron affinity varies with surface charge. One might therefore expect a voltage-dependent electron affinity to appear in [3.24], and its value should increase with increasing reverse bias. Qualitatively, this will cause the current to rise more slowly at larger reverse bias and improve the agreement between experiment and theory in Figure 3.10. It has also been recently observed that the dielectric constant decreases for applied electric fields greater than 300 kV/cm [32]. The average depletion width in semiconducting SrTiO3 is of the order 102 Å. For biases of the order of a volt, this gives electric fields across the depletion width of the order 103 kV/cm. For fields of this magnitude the dielectric constant decreases by nearly a factor of two. This decrease in screening strength will result in greater dynamic band bending, and it can be anticipated from Figure 3.4 that this effect will be most apparent at forward sample bias. Qualitatively, this will cause the current to rise more quickly at larger forward bias, worsening the agreement between experiment and theory in Figure 3.10. It should be noted, however, that the value of the electron mobility was chosen arbitrarily and can be up to an order of magnitude smaller in reality. To improve the fit between experiment and theory, it may be necessary to adjust parameters such as the electron mobility or the capacitance constant β which decreases band bending with increasing magnitude (see Figure 3.4). 105 Finally, it has been implicitly assumed that electrons with positive kinetic energy (i.e., those with energies greater than the potential barriers) will have a transmission probability of unity. The phenomenon of quantum mechanical reflection (QMR) has not yet been treated rigorously in this context. Crowell and Sze used a numerical approach to determine QMR of electrons at metal-semiconductor Schottky barriers [33] and their results give some insight into what one might expect in the case of a MIS junction. Although their treatment is only approximate, it is consistent with the present model to the extent that it is a one-dimensional isotropic effective mass approach. The general results predict that QMR can be up to 60%–70% when the incident energy is equal to the barrier height for “abrupt” potentials, and decreases slowly with energy in excess of the barrier maximum. Moreover, QMR increases a) with the electric field at the surface of the semiconductor, and b) when the semiconductor effective mass is larger than that of the metal. Both points a and b are significant to the present work. In practice, the electric field across the depletion region is typically of the order 106 V/cm, and the effective mass of SrTiO3 is several times larger than that of the metal electrode. Despite these challenges to the accuracy of the model, the agreement between experiment and theory represented in Figure 3.10 is sufficient for the present application for the following reason. As discussed in section 1.4.2, the photoinduced effects are expected to involve modification of the surface charge. Therefore the effect will be mainly observed as a modification of band bending. Exact matching of the experimental current densities over the entire width of the voltage sweep is not necessary for a quantitative assessment of the change in surface charge. A “good fit” might be characterized by a matching of the apparent surface gap as well as the slope of the curves at the current onset voltages. A suitable fitting procedure might therefore consist of adjusting the unknown parameters until a good fit is obtained for the “dark” 106 spectra, then adjusting the surface potential (and electron affinity) until a good fit is obtained for the “illuminated” spectra. The photo-induced change in the surface charge is then given by [3.21]. This approach will be employed to interpret the data presented in Chapter 5. REFERENCES 1. D. Bohm Quantum Theory Prentice-Hall, New York (1951) 2. S. M. Sze Physics of Semiconductor Devices 2nd ed. John Wiley & Sons, New York (1981) 3. E. H. Rhoderick and R. H. Williams Metal-Semiconductor Contacts Clarendon Press, Oxford (1988) 4. H. K. Henisch Semiconductor Contacts Clarendon Press, Oxford (1984) 5. J. Shewchun, A. Waxman and G. Warfield Solid-State Electronics 10 (1967) 1165 6. E. Wolf Electron Tunneling Spectroscopy Oxford University Press, Oxford (1986) 7. J. A. Stroscio and W. J. Kaiser, Eds. Scanning Tunneling Microscopy Academic Press, Inc., San Diego (1993) 8. D. A. Bonnell, Ed. Scanning Tunneling Microscopy and Spectroscopy: Theory, Techniques and Applications VCH Publishers, Inc., New York (1993) 9. W. J. Kaiser, L. D. Bell, M. H. Hecht and F. J. Grunthaner J. Vac. Sci. Technol. A 6 [2] (1988) 519 10. E. Burstein and S. Lundqvist, Eds. Tunneling Phenomena in Solids Plenum Press, New York (1969) 11. R. M. Feenstra and J. A. Stroscio J. Vac. Sci. Technol. B 5 [4] (1987) 923 12. R. M. Silver, J. A. Dagata, and W. Tseng J. Appl. Phys. 76 [9] (1994) 5122 107 13. Ch. Sommerhalter, Th. W. Matthes, J. Boneberg, P. Leiderer, and M. Ch. Lux-Steiner J. Vac. Sci. Technol. B 15 [6] (1997) 1876 14. M. McEllistrem, G. Haase, D. Chen, and R. J. Hamers Phys. Rev. Lett. 70 [16] (1993) 2471 15. W. A. Harrison Phys. Rev. 123 [1] (1961) 85 16. J. R. Oppenheimer Phys. Rev. 31 (1928) 66 17. J. Bardeen Phys. Rev. Lett. 6 [2] (1961) 57 18. C. B. Duke Tunneling In Solids Academic Press, New York 1969 19. J. Bono and R. H. Good, Jr. Surface Science 175 (1986) 415 20. P. Hansma Tunneling Spectroscopy: Capabilities, Applications, and New Techniques Plenum Press, New York (1982) 21. J. S. Blakemore Semiconductor Statistics Pergamon Press, New York (1962) 22. E. Spenke Electronic Semiconductors McGraw-Hill, New York (1958) 23. J. G. Simmons J. Appl. Phys. 34 [9] (1963) 2581 24. R. H. Kingston and S. F. Neustadler J. Appl. Phys. 26 [6] (1955) 718 25. R. Seiwatz and M. Green J. Appl. Phys. 29 [7] (1958) 1034 26. K. Seeger Semiconductor Physics 4th ed. Springer-Verlag, New York (1989) 27. J. McDougall and E. C. Stoner Trans. Roy. Soc. London A 237 (1938) 67 28. M. A. Green, F. D. King and J. Shewchun Solid-State Electronics 17 (1974) 551 29. J. G. Simmons Phys. Rev. Lett. 15 [25] (1965) 967 30. S. Zafar, R. E. Jones, B. Jiang, B. White, V. Kaushik, and S. Gillespie Appl. Phys. Lett. 73 [24] (1998) 3533 108 31. Wolfram Research, Inc., Mathematica, Version 3.0, Champaign, IL (1996) 32. R. A. van der Berg, P. W. M. Blom, J. F. M. Cillessen, and R. M. Wolf Appl. Phys. Lett. 66 [6] (1995) 697 33. C. R. Crowell and S. M. Sze J. Appl. Phys. 37 [7] (1966) 2683 109 Chapter 4: Characterization of The Bulk All ellipsometry and transmission measurements were performed and analyzed with technical assistance at the DuPont experimental station in Wilmington Delaware. Hall measurements were performed in the standard configuration with the exception that a bipolar power source, modulated at 1kHz, supplied an AC current to the sample and a lock-in amplifier was used to detect the Hall voltage. The magnetic field was 9.5 kGauss and the currents used were between 35 and 85 mA. Materials parameters (i.e., the free carrier density and optical dielectric constant) obtained from these measurements were used to model the tunneling spectra as discussed in Chapter 3. 4.1 BULK PROPERTIES OF REDUCED SrTiO3 4.1.1 Hall /resistivity measurements The samples were heat treated by one of two methods: 1) vacuum annealing by joule heating; or 2) furnace annealing in a hydrogen atmosphere. In the former case, the temperature was monitored by an optical pyrometer. Upon cooling, the temperature was observed to drop below 800–850 ºC (the “freeze-in” point for oxygen vacancies) within 50 to 100 seconds, corresponding to an initial quench rate of approximately 150 º/min. In the latter case, the cooling rate was not easily verified since the samples were simply extracted from the hot zone of the furnace to initiate cooling. The quench rate in this case is thus expected to have been much slower. All samples appeared uniform in color indicating a uniform distribution of point defects. Table 4.1 summarizes the results of the Hall and resistivity measurements performed on four out of the six samples studied. The results for the hydrogen reduced samples (the H series) are also presented graphically in Figure 4.1. The geometries of samples V–930 and V–1100 (the 108 V series) were not compatible with the geometry of the apparatus designed to make these measurements as a result of cracking during thermal treatment. Therefore, all conclusions regarding the oxygen vacancy density dependence are restricted to the H series. Table 4.1: Thermal history and Hall/resistivity measurements. Sample ID Thermal history VH (mV) ρ (Ω-cm) n (cm-3) V–930 930 °C, 10 min / vacuum — — — V–1100 1100 °C, 12 min / vacuum — — — V–930Nb 930 °C, 10 min / vacuum 2.1 0.042 ± 0.001 8.8 × 10+17 H–700 700 °C, 2 hrs / H2 furnace 2.0 0.430 ± 0.008 3.3 ± 0.7 × 10+17 H–850 850 °C, 2 hrs / H2 furnace 2.4 0.069 ± 0.002 3.2 ± 1.1 × 10+17 H–1000 1000 °C, 2 hrs / H2 furnace 3.2 0.026 ± 0.003 1.3 ± 0.5 × 10+17 109 4 0.45 Resistivity (ohm-cm) 3.5 0.35 3 0.3 2.5 0.25 2 0.2 1.5 0.15 1 0.1 0.5 0.05 Carrier Density x 10+17 (cm-3) 0.4 0 0 700 850 1000 Temperature (°C) Figure 4.1 Resistivity and carrier density of undoped single crystal SrTiO3 as a function of annealing temperature. These samples show a decrease in resistivity with increasing degree of reduction (i.e., increasing annealing temperature). The measured carrier density, however, apparently also decreases with reduction. Note the larger error in measurement of the carrier density for the H850 sample. The lower end of the variance (2.16 × 10+17 cm-3) lies mid-way between H–700 and H–1000 to give a reasonable trend, although not exactly in concert with the observed resistivity trend as shown in Figure 4.1. The simple theory developed to relate the Hall voltage (VH) to the carrier density, (n), necessarily assumes that the transport properties of the material may be adequately described by the Drude model or the semiclassical model — i.e., charge carriers behave as a free electron gas. The carrier density is simply related to the Hall coefficient (RH) and can be written in terms of measurable parameters as 110 n= IH z , VHe t [4.1] where I is the measured current, t is the thickness of the sample, and Hz is the applied magnetic field. Furthermore, for n-type extrinsic materials, the Hall mobility µH is given by µH = RH 1 = , ρ en ρ [4.2] and is very nearly equivalent to the true carrier drift mobility when acoustic phonon scattering is the predominant scattering mechanism [1]. Since electron-lattice coupling is often strong in transmission metal oxides (giving rise to small or large polaron formation), it is not obvious that [4.1] and [4.2] will give accurate results. It was anticipated that sample V–930Nb would be a good case to test the relative accuracy of the Hall measurement since this sample was commercially prepared with a specified doping of 0.17 weight percent Nb2O5. This means that in one gram of doped crystal one may expect 1.28 × 10-5 moles of added impurity atoms. Niobium substitutes on the Ti sublattice and is known to introduce shallow (i.e., hydrogenic) donor states. If the crystal were stoichiometric, then the formula SrTi1-xNbxO3 describes the crystal where x ≈ 0.00235. This corresponds to a carrier density of 3.95 × 1019 cm-3, much larger than that determined by the Hall measurement. On the other hand, this estimated value for n predicts a Hall voltage of 0.047 mV whereas the observed experimental noise was 0.1 mV. The observed Hall voltage was well reproduced and the behavior was not consistent with a magnetoresistance effect. Using [4.2] and assuming a 1019 carrier density gives a mobility of 3.73 cm2/Vs, whereas the measured carrier density gives 168.34 cm2/Vs. The former estimate is close to tabulated 111 values for “small” x in SrTiO3-x [2]. The latter value, however, is not too unreasonable and is still an order of magnitude less than the electron mobilities in Si (1500 cm2/Vs), Ge (3900 cm2/Vs), or GaAs (8500 cm2/Vs). The reported purity of this sample was 10 ppma (i.e, an accidental impurity density of 1017 cm-3). Therefore, accidental acceptor impurity compensation for the donor impurities is not an acceptable explanation for the discrepancy in n. It is possible that the sample was unintentionally under doped. An alternative explanation for such observations was reported by Perluzzo and Destry [3]. These authors also observed discrepancies in the ratio of estimated carrier densities to measured carrier densities for Nb-doped single crystals until a thermal “activation” process restored the ratio to approximate unity. It should be noted that, as described in Chapter 2, all samples were processed with similar thermal histories prior to vacuum or hydrogen annealing. The doped crystal was observed to become insulating (although it remained blue in color) after the air anneal and therefore required further processing to “activate” the carriers. Since it was desirable to yield a surface structure comparable to the reduced crystals, the activation procedure recommended by Perluzzo and Destry could not be applied. Thus, a possible explanation of the low observed carrier density could be that a small fraction of the defects were re-activated by the thermal treatment described in Table 4.1. It will be shown in the following chapter that the assumed carrier density does not present difficulty in obtaining good agreement between the experimental and calculated tunneling spectra. 4.1.2 Optical measurements It was mentioned earlier that all samples appeared uniform in color. Both V–930 and H–700 appeared clear, both V–930Nb and H–850 transmitted in the blue region of the visible spectrum, and both V–1100 and H–1000 appeared black. All samples can be expected to 112 strongly absorb light with energies above approximately 3.2 eV where electronic interband transitions are excited between the highest occupied valence band and the lowest unoccupied conduction band. The changing color of the crystals with increasing defect density suggests increases in absorption at energies throughout the infrared and visible regions of the spectrum. This interpretation is confirmed by the optical measurements presented below. Spectroscopic ellipsometry is a technique that yields the optical constants of a material from measurements of light reflected from its surface. The technique generally consists of illuminating the surface with circularly- or elliptically-polarized monochromated light and detecting changes in amplitude and phase upon reflection. One component of the incident light is linearly polarized in the plane of incidence (the p wave) and another component is linearly polarized normal to the plane of incidence (the s wave). The actual parameters measured by an ellipsometer are the angles Ψ (Psi) and ∆ (Del), where the former is a measure of the ellipticity introduced to the reflected wave due to a relative change in amplitude between the p and s waves, and the latter is a measure of the change in phase difference between the p and s waves upon reflection. Subsequently, an appropriate model must be assumed for the reflecting system upon solving an equation of the form [4] i∆ tan Ψe = r p (nˆ ,φ) , s r (nˆ ,φ ) [4.3] where r p and r s are called the Fresnel reflection coefficients. They are functions of known experimental parameters (such as the angle of incidence, φ , and the refractive index of air) and the (unknown) complex index of refraction, nˆ , of the material under investigation. The appropriate forms of r p and r s depend on the assumed model of the reflecting system. In the present case, a simple model with two reflecting surfaces (the top and bottom of a single crystal) 113 was sufficient to fit the experimental data. Optical transmission measurements were performed to obtain better accuracy in the optical constants in spectral regions where absorption is very small and thus the ellipsometer greater error. 2.5 a 0.01 Extinction coefficient 0.008 2 0.006 0.004 1.5 0.002 0 1.52 1.54 1.56 1.58 1.6 1.62 1 0.5 0 2 2.5 3 3.5 4 4.5 5 Energy (eV) 4 b Palik oxidized H-700 H-850 H-1000 Refraction index 3.5 3 2.5 2 1.5 2 2.5 3 3.5 4 4.5 Energy (eV) 114 5 produced data with Figure 4.2 (previous page) The dispersion curves for the optical constants (n and k) of SrTiO3. The extinction coefficient (a) and index of refraction (b) are shown for oxidized and reduced single crystals as indicated by the legend. These are also compared to literature values measured over the same spectral range (solid curve). The inset in (a) shows an absorption tail for H-850 near the “red” region of the visible spectrum. Figure 4.2 shows the measured dispersion of the extinction coefficient (k) and the refractive index (n) for each sample in the H series. Also included are results of measurements from an oxidized sample of the same boule as well as the tabulated values for undoped SrTiO3 at room temperature [5]. All curves in Figure 4.2b show an increase in refractive index with increasing photon energy up to some energy where n reaches a maximum. This is called normal dispersion. At energies above 3.8 eV to 4.1 eV n decreases with increasing photon energy. This is called anomalous dispersion. A shoulder appears above the peak energy for all samples. This is indicative of two overlapping dissipating processes. Except for a finite spectral range from 2.0 to 3.0 eV, all tabulated values appearing in Figure 4.2b are compiled from a single reference. The variation (i.e., ∆n) in the tabulated values from 2.0 to 3.0 eV are in the range 0.043 to 0.125; variations between the Palik values and those from the oxidized sample are in the range 0.12 to 0.24 over the same spectral window. (At higher energies |∆n| is as large as 0.4 near 3.5 eV and 4.8 eV.) Therefore, the deviation between the oxidized crystal data and the tabulated values is not too unreasonable. The results of Figure 4.2a, however, suggest a significant difference in the band structure which is observed to evolve upon reduction. To see this, the data in Figure 4.2a are used to calculate the absorption coefficient (α) which is related to the extinction coefficient by 115 α= 4πk , λ [4.4] where λ is the wavelength of the light. The results are shown in Figure 4.3a. Within the independent electron approximation, the absorption coefficient near the fundamental absorption edge is assumed to be approximated by [6] a 0.4 (αEph)2 1 0.3 0.25 0.2 0.15 0.1 6 Absorption coeff. x 10 (1/cm) 0.35 0.8 0.05 0 3.2 3.4 3.6 3.8 4 4.2 0.4 (αEph)0.5 0.35 0.6 0.3 0.25 0.2 0.15 0.1 0.4 0.05 0 3.2 3.4 3.6 3.8 4 4.2 0.2 0 2 2.5 3 3.5 4 4.5 5 Energy (eV) 400 b Absorption depth (nm) 350 300 250 200 150 100 Palik oxidized H-700 H-850 H-1000 50 0 2 2.5 3 3.5 Energy (eV) 116 4 4.5 5 Figure 4.3 (a) The dispersion curves for the absorption coefficient of SrTiO3. The insets demonstrate that this measurement gives evidence of both direct and indirect transition mechanisms. The transition energies are deduced by linear fitting near the absorption edge. (b) The inverse of the curves in (a). Table 4.2: Optical transition energies deduced from Figure 4.3a. Sample ID Direct Gap Energy (eV) Indirect Gap Energy (eV) Palik 3.52 3.28 oxidized 3.79 3.59 H–700 3.88 3.77 H–850 3.90 3.68 H–1000 3.58 3.00 (hω − E ) , α~ /n g hω [4.5] where hω = Eph is the energy of the incident photon, and the value of n/ depends on the transition mechanism. For indirect transitions n/ = 2; for direct transitions n/ = 0.5. 1 Plotting (αE ph )n/ versus Eph near the absorption edge obtains a linear fit when the appropriate transition mechanism is assumed, and the intercept with the abscissa gives the energy of the transition. Table 4.2 lists the absorption edge transition energies deduced in this manner from the insets in Figure 4.3a. Both direct and indirect transition energies are found. In addition, the energies for the oxidized sample are greater than the Palik values. The trend amongst the second to fourth entries in the table show an increase in the direct gap energy. The heavily reduced sample, however, shows a substantial decrease in this transition energy which agrees better with the Palik value. The lowest energy transitions are indirect and follow a similar 117 trend with reduction as observed for the direct gap energies except that the energy decrease has already begun with H–850. Figures 4.2a and 4.3a also show a general trend in the H series of decreasing absorption with reduction in the UV region (i.e., Eph > 4.5 eV). There is, however, an initial increase in absorption from the oxidized sample to H–700. These trends are reversed for lower photon energies (i.e., Eph < 4.0 eV). H–1000 is observed to cross H–850 near 4.5 eV; it crosses H–700 near 4.0 eV. Below the latter energy the absorption increases with reduction, with an initial decrease from the oxidized sample to H–700. The strongest absorption in the visible region is observed for sample H–1000 as shown in Figure 4.3b. This is a plot of the inverse of the absorption coefficient and thus determines the depth to which the transmitted light intensity falls below approximately 37% of its incident value. It can be seen that the bulk of the intensity drops within approximately 50 to 375 nm of the crystal surface for energies well below the onset of interband transitions. All other crystals remain transparent in the visible, becoming opaque only in the UV where the bulk of the transmitted intensity drops within 10 nm of the crystal surface. 118 6 band gap (eV) oxidized H-700 H-850 H-1000 Dielectric constant 5.8 3.79 3.88 3.90 3.58 5.6 5.4 5.2 5 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Energy (eV) Figure 4.4 Dielectric function, εˆ = n2 − k 2 , of SrTiO3 below anomalous dispersion. 4.1.3 Discussion The data presented in this chapter was acquired for two primary reasons. To determine the values of physical parameters to be used in tunneling calculations (such as the carrier density and optical dielectric constant) and to characterize changes in the physical properties of the bulk upon reduction to facilitate interpretation of observed surface properties which strongly depend on bulk properties. Examples of the latter are the surface potential and electrostatic energy which both have an inverse dependence on the density of ionized defects. If the structure at the surface does not strongly disrupt the Ti-O coordination, then the surface electronic structure may not deviate strongly from the bulk electronic structure. Thus the optical data is a good starting point to characterize changes in the band structure induced by reduction annealing. The allowed direct transitions in cubic SrTiO3 were determined by Casella 119 [7], based on band structure calculations and the dipole selection rule, to be: Γ15 → Γ2 5′ ; ∆ 5 → ∆ 2 ′ ; ∆ 5 → ∆ 5 ; Χ5 ′ → Χ3 ; Χ4 ′ → Χ5 ; and Χ5 ′ → Χ 5 , where the last two transitions were assumed possible in the presence of a [001] uniaxial stress. Based on the observed splitting (0.86 eV) of the primary absorption peak from reflectivity data, however, Cardona concluded that both Χ4 ′ → Χ5 and Χ5 ′ → Χ 5 transitions accounted for the observed dispersion in the refractive index [8]. A similar splitting of 0.8 eV is observed at the onset of anomalous dispersion for the oxidized curve in Figure 4.2b. The decreasing intensity in both n and k among the H series in this spectral region can thus be interpreted as a decrease in the joint density of states for these transitions. It has been established by photoemission studies that this is caused by a decrease in the density of states of the upper valence band upon reduction [9]. There is an apparent increase in the joint density of states for these transitions (accompanied by a slight decrease in the splitting to ∼ 0.6 eV) within the early stages of reduction that, to the author’s knowledge, has not been previously reported. The H series differs from the oxidized crystal only by the additional heating in 100% flowing hydrogen. It will be shown in the following chapter that extended vacuum annealing resulted in the appearance of a sulfur peak in Auger spectra. Galbraith analysis determined a bulk sulfur concentration of less than 0.08 ppm by weight (or ≈ 0.46 ppm based on anion sites). This is equivalent to ∼ 7.7 × 1015 cm-3 sulfur impurities. If it is assumed that sulfur diffuses to the surface with vacuum heat treatment, such a low density could hardly account for a large change in optical properties upon removal from the lattice. Moreover, it is implausible that introducing a small amount of sulfur to the oxygen sublattice depletes states either from the top of the valence band or from the bottom 120 of the conduction band in a manner reminiscent of strong nonstoichiometry. The origin of the initial increase thus remains undetermined. Consequently, the classical Lorentz oscillator model perhaps does not appropriately describe the behavior of SrTiO3 at the early stages of nonstoichiometry. This model predicts that the dielectric function εˆ = n2 − k 2 should decrease with increasing band gap if the dispersion may be approximately described as the response of a single harmonic oscillator [10]. Figure 4.4 shows that this behavior is observed only between the H–700 and H–850 samples. The observed decrease in εˆ with decreasing band gap from H–850 to H–1000 is perhaps a consequence of the fact that a single oscillator cannot be assumed for the heavily reduced sample such that the band gap is rather ill-defined. The electronic structure of heavily reduced SrTiO3 thus can not be interpreted in terms of a rigid band model. Indeed, the broad absorption throughout the visible suggests a broadening in the static dielectric constant. Local variations in nonstoichiometry might give rise to local variations in κst and thus a spread in hydrogenic binding energies. The following discussion of this last point leads to a possible interpretation of the resistivity and carrier density results. It is perhaps customary to interpret nonstoichiometric crystal properties in terms of a random distribution of point defects. From a thermodynamic point of view, however, if the enthalpy of formation of associated defects (i.e., oxygen vacancy pairs) is large and negative, then the free energy of the crystal will be minimized by the formation of defect clusters. Recent theoretical work suggests that oxygen vacancy clustering is required in order to form localized states in the band gap of SrTiO3 [11]. Such states will have the potential to depopulate the conduction band by free carrier trapping. It is reasonable to expect defect clustering in crystals with larger nonstoichiometry. It should be emphasized, however, that within the context of the 121 theory, a larger overall defect density is neither a necessary nor sufficient condition for the formation of band gap states. Rather than a random distribution, it may be more appropriate to think in terms of a statistical distribution of point defects with local density variations. The static dielectric constant tends to decrease with increasing nonstoichiometry [12] such that if the defect state energies are assumed to be approximated by [1.3], then a distribution of binding energies will accompany a distribution of defect densities. This can lead to optical absorption over a broad spectral range in the visible (assuming defect to conduction band transitions are symmetry allowed) as observed in Figure 4.3b. The above interpretation involves the removal of free carriers from the conduction band as the oxygen vacancy density (and especially clustering) increases. Hence, a decrease in carrier density as measured by a Hall probe would be expected. This was indeed observed as shown in Figure 4.1. The foregoing qualitative argument is supported only with data obtained by a method which has been shown to inadequately describe temperature-dependent transport properties in semiconducting transition metal oxides [13]. The values for n in Table 4.1 are, however, within the range observed in previous studies on reduced single crystal SrTiO3 [14,15]. The predicted Hall mobilities for H–700, H–850, and H–1000 are 44, 281, and 1,945 cm2/V s, respectively. Except for the last value, these are reasonably smaller than carrier mobilities in traditional semiconductors as discussed above, although much larger than values expected for systems with a tendency for large polaron formation. The average value obtained by previous authors is 6.07 cm2/V s [15]. If the carrier mobility is assumed to be independent of carrier density and clustering, and given by the latter value, one can work backwards using sample resistivities in Table 4.1 to predict the carrier densities 122 to be 2.45, 0.24, 1.5, and 3.96 × 1019 cm-3, for V–930Nb, H–700, H–850, and H–1000, respectively. Now the carrier density for the doped crystal agrees better with the estimated value based on the doping specification. It should be noted, however, that a carrier density of 2.4 × 1018 cm-3 for sample H700 is difficult to reconcile with the fact that the sample appeared transparent and colorless! 4.1.4 Conclusions The measured optical and transport properties of single crystal SrTiO3 give results that are in reasonable agreement with tabulated and other reported values. Evidence of decreasing valence to conduction band transitions is provided by decreasing n and k values in the UV region supporting the view that oxygen nonstoichiometry depletes densities of states in the upper valence band in agreement with previous photoemission studies. Both direct and indirect band gap energies were determined, the values of which apparently depend upon the degree of reduction. Although the resistivities of the samples decrease with reduction as expected, the carrier density also decreases suggesting a significant increase in electron mobility. Given this result, there may be some skepticism in the accuracy of the Hall measurements; however, alternative explanations are not consistent with experimental observations. The results are thus assumed acceptable. The validity of this assumption is supported by the successful matching of experimental tunneling spectra as presented in Chapter 5. REFERENCES 1. J. P. McKelvey Solid State and Semiconductor Physics Krieger Pub. Co., Florida (1982) 2. P. A. Cox Transition Metal Oxides: An Introduction to Their Electronic Structure and Properties Oxford University Press, New York (1992) 3. G. Perluzzo and J. Destry Can. J. Phys. 56 (1978) 453 123 4. H. G. Tompkins A User’s Gride to Ellipsometry Academic Press, Inc., Boston (1993) 5. Handbook of optical constants of solids II Edward D. Palik, Ed., Academic Press, Inc., San Diego, CA (1991) 6. S. M. Sze Physics of Semiconductor Devices 2nd ed. John Wiley & Sons, New York (1981) 7. R. C. Casella Phys. Rev 154 [3] (1967) 743 8. M. Cardona Phys. Rev. 140 [2A] (1965) A651 9. V. E. Henrich, G. Dresselhaus and H. J. Zeiger Phys. Rev. B 17 [12] (1978) 4908 10. F. Wooten Optical Properties of Solids Academic Press, San Diego, CA (1972) 11. N. Shanthi and D. D. Sarma Phys. Rev. B 57 [4] (1998) 2153 12. H. B. Lal Indian J. Pure Appl. Phys. 8 (1970) 81 13. S. Fu Field Effect Study of the Transport Properties of TiO2, Ph.D. Thesis, University of Pennsylvania (1998) 14. W. S. Baer Phys. Rev. 144 [2] (1966) 734 15. H. Yamada and G. R. Miller J. Solid State Chem. 6 (1973) 169 124 Chapter 5: Properties of Vicinal SrTiO3 (001) The surfaces of several electron doped samples are characterized in terms of their crystallographic structure and morphology using measurements primarily from low energy electron diffraction (LEED) and STM. The local electrical and optical properties are characterized by STS and PATS. 5.1 STRUCTURE AND CHEMISTRY OF REDUCED SrTiO3 (001) 5.1.1 LEED/Auger observations Auger measurements were performed primarily to verify the cleanliness of the surfaces. A separate vacuum chamber equipped with an Omicron retarding field rear view LEED was used and the results displayed on a chart recorder. Prominent peaks for Sr (106 eV), Ti (391 and 420 eV) and O (515 eV) were observed in all spectra. Figure 5.1 shows two typical spectra obtained from a vacuum annealed sample cut from the same boule and processed in the same manner as all samples studied in this thesis. Weak carbon (273 eV) and sulfur (153 eV) peaks were observed when the samples were subjected to low annealing conditions (i.e., 600 °C for a few minutes) Figure 5.1. as shown in the top curve of When subjected to more extensive annealing conditions (i.e., 1000 °C for 6 hours), reduction in the carbon peak and an increase in a sulfur peak (to varying intensities) was often observed, as shown in the bottom curve of Figure 5.1. 126 C Ti O S Sr Figure 5.1 Chart recorder traces showing AES spectra of vacuum reduced SrTiO3 (001) surface. Top: 600 °C for a few minutes; bottom: 1000 °C fro 6 hours. LEED patterns similarly showed evidence of surface crystallographic and morphological development. Two of the most frequently observed patterns are shown in Figure 5.2, where both patterns were obtained with a primary beam energy of 100 eV. Figure 5.2a shows the square symmetry as observed for sample V–930. This is usually interpreted as a 1 × 1 pattern (using the notation of E. A. Wood) suggesting that the surface symmetry is equivalent to that expected from a projection of the bulk lattice onto the surface plane. The pattern shown in Figure 5.2b was observed for sample V–1100 as well as other samples subjected to extended thermal treatment. 127 a b Figure 5.2 Two distinct LEED patterns from SrTiO3-x (001) vicinal surfaces, both obtained with an incident beam energy of 100 eV. Comparison of these two LEED patterns also suggests an increase in the roughness of the surface as indicated by a slight increase in background intensity. This intensity is due to diffuse scattering, rather than diffraction, of the primary beam. STM images confirm an increase in roughness as will be shown in the following section. To better understand the changes in surface structure and chemistry with thermal history and to establish a “recipe” for preparing flat surfaces, one sample (designated STO–8) was cut from the same boule as all other undoped samples and successively heat treated in vacuum at 500, 1000, 1100 and 1200 °C for five minutes each. A qualitative assessment of 128 the AES results, shown in Figure 5.3, suggests that the surface sulfur content did not change significantly as compared to Figure 5.1. A rather strong carbon peak is seen in Figure 5.3 that also did not change significantly with heat treatment. The latter observation can be interpreted in one of two ways: a) The complex geometry of the vacuum chamber precluded a visual inspection of the primary beam incident upon the surface of the sample such that focusing the primary beam was difficult. A sufficiently broad beam can generate Auger signals from other parts of the sample holder which contained carbon contamination (i.e., the Ta peak in Figure 5.3 was generated by the foil used to heat the sample). b) If surface carbides were present as a result of carbon bonding to surface cations (i.e, Ti), removing them would require heating at substantially higher temperatures than was accessible in the present experimental configuration. Also interesting is the former observation of the apparent constant sulfur peak. This also may be interpreted in one of two ways: a) A similar broad beam argument suggests that the signal may have originated from a source other than the surface of the sample. b) If the sulfur indeed originated from the bulk of the sample, then it appears as if some sort of surface saturation occurred which slowed or stopped further accumulation. It should be noted that sample V-930Nb also showed a sulfur peak similar in intensity to those observed in Figure 5.3. Since this crystal did not originate from the same boule as the H series, this raises the prospect that the observed peaks had a common source unrelated to the oxide crystals. Recalling that the Galbraith analysis measured a sulfur content less than 0.08 ppm by weight in the bulk, the surface structures are not expected to be strongly influenced by impurity segregation or precipitation. 129 Ti O C S Ta Sr Figure 5.3 AES spectra of sample STO–8 heat treated successively (from top to bottom) at: 500, 1000, 1100, and 1200 °C for 5 minutes each. The arrows indicate baseline shifting due to instability of the chart recorder. LEED patterns were also acquired for STO–8 with the following results: a) the 500 °C anneal produced the pattern in Figure 5.2a that was observed independent of sample orientation with respect to the incident electron beam; b) the 1000 °C anneal produced the pattern in Figure 5.2b, also independent of orientation; c) both the 1100 and 1200 °C anneals produced both patterns in Figure 5.2 depending on orientation of the sample with respect to the incident electron beam. LEED patterns that varied as the sample was rotated on its axis were associated with samples that showed non-uniformity in color, suggesting non-uniformity in surface defect density and/or surface structure. It should be noted that this sample was left in the vacuum chamber for up to nine days following the 1200 °C thermal anneal. The same LEED 130 pattern as described in (c) was subsequently reproduced, indicating that the surface was very stable in the 10-9 Torr environment for up to nine days. The surface ordering can be determined directly from inspection of Figure 5.2. Since the real space primitive mesh vectors are parallel to the reciprocal space primitive mesh vectors, where the magnitude of the latter is given by b∗ = ∗ 2π a 2π = = , b 2 2a [5.1] the real space surface ordering is readily derived as shown in Figure 5.4. It is determined that Figure 5.2b corresponds to a 2 × 2R45o superstructure. As discussed in Chapter 1, there is some belief that the observed superstructures on reduced SrTiO3 (001) are due to ordering of oxygen vacancies. The latter may be represented as shaded spheres in Figure 5.4. It is common practice to describe this superstructure as c( 2 × 2) in order to use a notation analogous to that used to describe the 1 × 1 structure. Note, however, that a centered square unit mesh (indicated by the dotted line in Figure 5.4) is not one of the five unique two dimensional Bravais lattices. 131 a* b b* [01] (0,0) [10] Figure 5.4 The 2 × 2R45o (sometimes called c( 2 × 2) ) superstructure (left) corresponding to the LEED pattern (right) of Figure 5.2b. The open circles are the anions; the small circles are the cations; the shaded circles indicate anion vacancies. 5.2 MORPHOLOGICAL STRUCTURE BY STM The images that are presented in this section demonstrate the changing morphology of undoped/reduced SrTiO3. For comparison, the surface images of niobium-doped SrTiO3 are also presented. All images were acquired in constant current mode. The set points are indicated in the text of the figures with the bias specified as that applied to the sample with respect to the tip. The results were processed only by a flattening procedure to reduced the tilt that is often present to varying degrees between the tip axis and the surface normal. 5.2.1 Surface morphology of V–930 The surface appears clean and flat with crystallographically aligned step edges as shown in Figure 5.5. It is believed that these steps are aligned along 〈01〉 [1]. All step heights are multiples of the unit cell edge (a = 3.9 Å). In this image the step heights are 4a and 8a. The scan size is roughly one fifth the size of the image which appears in Figure 2.7. At this higher 132 resolution the “texture” of the surface can be resolved. There does not appear to be an ordering of surface features. Therefore, the long range order observed in Figure 5.2a has not been resolved. Analysis on the large terrace gives a z range (i.e., black-to-white scale) of 61 Å and a calculated rms roughness of 6.4 Å. (The rms roughness is defined as the standard deviation of the image z values within the area being analyzed.) The higher intensity near the edge of this terrace is most likely due to a tilt in the surface normal with respect to the axis of the STM tip. In other cases (i.e., heavily reduced samples) there may actually be material build up or a change in electronic properties near these edges (see section 5.2.2). Figure 5.6 compares two images taken with (right) and without (left) illumination of the surface with greater than band gap light. The primary difference between the two is due to a southwest shift as a result of thermal drifting. This is commonplace in STM imaging and the effect can be very small or very large. Other than the latter effect, the images are virtually identical. No photo-induced effects are apparent. In the upper left of both images is resolved a much smaller step edge which was determined to be equivalent to a single unit cell height. This was the only such observation for this surface. All other step edges were multiples of the unit cell edge as in Figure 5.5. 133 Figure 5.5 Multiple unit cell high step edges observed on V–930. The image scan size is 1,084 Å × 1,084 Å taken with a set point of -2.0 V and 0.25 nA. light dark Figure 5.6 Comparison of dark and illuminated surfaces, with incident photon energy of 3.6 eV. The image scan size is 1,084 Å × 1,084 Å taken with a set point of -2.0 V and 0.25 nA. No photo-induced effects are observed. 134 dark Figure 5.7 light Comparison of dark and illuminated surfaces, with incident photon energy of 1.9 eV. The image scan size is 2,167 Å × 2,167 Å taken with a set point of -2.0 V and 0.25 nA. No photo-induced effects are observed. Another comparison is shown in Figure 5.7. The energy of the incident light was 1.9 eV, the energy expected to excite charge from states localized on step edges. Again, no photoinduced effects are evident in the right image as compared to the left. Thermal drifting is in a southeast direction in this case. Figure 5.8 shows an image acquired while illuminating the surface with 2.4 eV light for the first 100 of 200 scan lines. Recall from Table 1.4 that this energy was believed to give rise to interband electronic transitions between lower and upper conduction bands. Assuming that a photo-induced effect would be uniform, and that “hot carriers” significantly increased the surface conductivity, the bottom half of the image would be darker than the top, indicating an inward displacement of the tip towards the surface. No such shift was observed, although it would be difficult to distinguish from a tip expansion due to local heating. Therefore, Figure 5.8 demonstrates at least that a tip expansion did not occur at this 135 photon energy. This was also found to be true, however, for all other photon energies used in this study. light dark Figure 5.8 Comparison of dark (bottom) versus illuminated (top) sections of a single surface. Incident photon energy = 2.4 eV. The image scan size is 2,167 Å × 2,167 Å taken with a set point of -2.0 V and 0.25 nA. 136 Figure 5.9 Step with apparent holes along the edge (20 Å deep) and at the kink (depth undetermined). The image scan size is 1,084 Å × 1,084 Å taken with a set point of -2.0 V and 0.25 nA. Figures 5.6 and 5.8 also show the presence of surface clusters near step edges and upon terraces. These clusters range in height from as small as 7 Å to as high as 35 Å. The diameters of these features (measured at half height) range from 38 Å to 80 Å. In addition to these protruding features, depressions were occasionally observed near step edges or step kinks, as indicated by the arrows in Figure 5.9. The origin of these types of apparent depressions will be discussed in section 5.3.1. They are distinguished, however, from much larger holes that are most likely due to local chemical attack. An example is shown in Figure 5.10 where a ∼300 Å deep pit is observed to have a square-shaped bottom and clear ∼28 Å step edges are resolved proceeding down into the hole. Comparison with images taken by AFM on heavily annealed samples, as in Figure 2.8, suggests that these pits are rounded out as a result of reshaping of step edges. This will be further demonstrated in the following section. 137 Figure 5.10 Surface hole formed by local chemical attack. The depth of the hole is ∼300 Å; the step sizes are ∼28 Å. The image scan size is 1,084 Å × 1,084 Å taken with a set point of -2.5 V and 1.0 nA. 5.2.2 Surface morphology of V–1100 Heavy vacuum annealing results in extensive development of the (001) surface. Figures 5.11 through 5.15 immediately show striking deviations from the structure of the surfaces shown in Figures 5.5 through 5.10. Step edges are no longer crystallographically aligned but appear to wander randomly. Figure 5.11a and 511b show several steps edges with similar curvature — i.e., concave with respect to the terraces. The curvature of the edges can be very large at some points giving step edge bulging along the surface as shown in Figure 5.11b. The step heights are difficult to extract in Figure 5.11a; in Figure 5.11b they are twelve times the unit cell height. 138 a Figure 5.11 b Surface morphology of V–1100 showing wandering step edges. The image scan sizes are (a) 5,418 Å × 5,418 Å and (b) 3,386 Å × 3,386 Å, taken with set points -2 V at (a) 0.5 nA and (b) 1.0 nA. 139 Figure 5.12 Apparent cluster-free surface with concave and convex step edges. Large 22a deep holes suggest possible healing of chemical etch pits. The image scan size is 5,418 Å × 5,418 Å taken with a set point of -2.0 V and 0.5 nA. In some cases both concave and convex step edges were observed as shown in Figure 5.12. The step height in the center of the image measures approximately 6a. The large hole in the upper left has a depth of 22a. This is significantly shallower than the depth of the etch pit measured in Figure 5.10, suggesting that the vacuum anneal may redistribute material in order to close these pits. The step heights are about the same as that observed for sample V–930. This is also observed in Figure 5.13a which shows a series of convex steps separated by ∼20 Å. 140 a b Figure 5.13 (a) Series of convex step edges separated by ∼20 Å. Local ordering of surface features normal to the step edges is apparent. The image scan size is 1,355 Å × 1,355 Å taken with a set point of -2.0 V and 1.0 nA. (b) Local terrace structure showing apparent ordering of clusters. The image scan size is 542 Å × 542 Å taken with a set point of -2.5 V and 0.5 nA. The terraces of these steps appear to have some local alignment of clusters in a direction normal to the step edge. A magnified view of this ordering appears in Figure 5.13b. This image bares some resemblance to previously observed surface ordering; however, the scale of these “row structures” is much larger than those formerly reported [2]. The rows are separated by approximately 76 Å (i.e., ∼ 20a). This type of ordering was frequently observed on this surface, although there were several regions where the local structure was less well-defined, as shown in Figure 5.14. The cluster heights range from 10 Å to 20 Å with an average of 15 Å. The z range in Figure 5.14 is 90 Å and the calculated rms roughness is about 6.8 Å, only slightly larger than that observed for sample V–930. 141 Figure 5.14 Terrace cluster structure of heavily reduced SrTiO3 (001). Local ordering is apparent but less well-defined than in other areas as shown in Figure 5.13. The image scan size is 542 Å × 542 Å taken with a set point of -2.5 V and 0.5 nA. 142 light dark Figure 5.15 eV. Comparison of dark and illuminated surfaces, with incident photon energy of 2.95 The image scan size is 542 Å × 542 Å taken with a set point of -2.0 V and 1.0 nA. Finally, Figure 5.15 demonstrates again that photo-induced effects were not observed in the STM images. The photon energy used in this measurement was 2.95 eV (i.e., λ = 420 nm). This energy is in the region where the measured flux from the light source (see Figure 2.2) was maximized and thus any possible photo-induced effects were expected to also be maximized. The absence of photo effects in the images is surprising given that tunneling spectra show strong effects at certain energies, as will be shown in section 5.3. 5.2.3 Surface morphologies of the H series The images presented in the previous sections completely characterize the surface structures observed on all samples processed under mild and heavy vacuum reducing conditions. A few images are presented in this section for the H series to highlight some differences that may have occurred due to annealing in a hydrogen atmosphere or cooling at a different rate. It should be 143 noted that all of the samples in this series were probed with the same tunneling tip. Continued scanning with one tip tends to blunt the tip end as a result of random tip crashes. Blunt tips often can produce cleaner, well-reproduced spectra but usually the quality of images diminish or the images are completely featureless. This was the case for all images acquired on sample H–850 and all but one acquired on sample H–700. Only discernible images are presented. It is assumed that the surface structures of all the H samples may be described by the images presented in this section, which are not very different than those presented in the last section. As an example, Figure 5.16 is an image acquired from H–700. The terrace contains clusters similar to those observed on V–1100. A possible step edge is apparent towards the upper left corner of the image. Step height analysis suggests that this step is 2 Å (i.e., half the unit cell height); however, this may be inconclusive given the extensive cluster structure which complicated the analysis. The z range is ∼40 Å and calculated rms roughness is 4 Å. 144 Figure 5.16 Terrace and step edge morphology of H–700. The image scan size is 800 Å × 800 Å taken with a set point of -2.0 V and 0.25 nA. Similar structures were observed for H–1000 as shown in Figure 5.17. This figure also shows that the step edges are straighter than those observed for V–1100. It is assumed that these edges are crystallographically aligned similar to those observed for V–930. Two step edges can be resolved in Figure 5.17 with 45 degrees between them. If one edge is assumed to be parallel to 〈01〉, the other is necessarily parallel to 〈11〉. This was not observed for V–930. These step heights are 6 Å and 12 Å, respectively. The z range is 44 Å and the calculated rms roughness is 5 Å. Figures 5.18 and 5.19 are the results of measurements for optical effects. Single and multiple unit cell steps are observed in Figure 5.18, with step sizes not more than 3a. There is no well-defined order to the surface clusters although there may be a slight preference for accumulation at step ledges as seen in Figure 5.18b. The images in Figure 5.18 were acquired with illumination during the bottom halves of the scans with 145 2.4 eV (Figure 5.18a) and 2.14 eV (Figure 5.18b) light. The comparison in Figure 5.19 sought to show the effect using more energetic photons at 3.8 eV. Light-induced effects were not apparent in any of the images. 45° Figure 5.17 Terrace and step edge morphology of H–1000. The bottom half was illuminated with 2.05 eV light. The image scan size is 800 Å × 800 Å taken with a set point of +2.0 V and 0.25 nA. 146 a Figure 5.18 b Local terrace cluster structure. Illuminated with (a) 2.4 and (b) 2.14 eV light during the bottom half of each scan. The image scan sizes are (a) 600 Å × 600 Å and (b) 800 Å × 800 Å. Both images were acquired with a set point of +2.0 V and 0.25 nA. light dark Figure 5.19 Comparison of dark and illuminated surfaces, with incident photon energy of 3.8 eV. The image scan size is 1,000 Å × 1,000 Å taken with a set point of +2.0 V and 0.25 nA. 147 5.2.4 Surface morphology of V-930Nb The surface of the niobium-doped crystal was also distinct from the vacuum-annealed undoped crystals. A large area scan is shown in Figure 5.20. The overall appearance is that of a severely damaged surface as a result of chemical or thermal etching. The former is more probable since this sample was prepared commercially. The rugged step edges in Figure 5.20 are characteristic of SrTiO3 (001) prepared by a standard Syton polish. This sample was further etched with BHF and subsequently air annealed. It is well-documented that annealing single crystal SrTiO3 in an oxygen-rich environment tends to repair the surface and produce clean steps with crystallographically aligned edges [3,4]. A significant density of oxygen vacancies may ensure that the mobility of the anion is large enough that sufficient material redistribution is possible as the crystal is oxidizing. The absence of oxygen vacancies in the doped crystal may therefore explain why the surface for V-930Nb does not resemble V-930, despite the fact that their thermal treatments were identical. 148 Figure 5.20 (previous page) Terrace and step edge morphology of V–930Nb. The image scan size is 2,000 Å × 2,000 Å taken with a set point of +2.0 V and 0.25 nA. Notice the presence of holes near step edges. A particularly large hole is indicated by the arrow. Except for the rugged contours of the step edges, this surface has a closer resemblance to V–1100 as well as H–1000. This can be seen in Figure 5.21. The terraces are extensively covered with clusters. There is an apparent increase in intensity near the step edges and many “holes” can be seen near step corners. Some of the latter are indicated by arrows in Figures 5.20 and 5.21. Step height analysis is difficult when the step edge is not straight; however, the majority of the steps in Figures 5.20 and 5.21 are apparently 4 Å, with very few measuring close to 2 Å. The clusters range in size from 4 Å to 17 Å, where the larger sizes are seen accumulated near the bottom part of the image in Figure 5.21. Consistent with the observations on the undoped samples, photo-induced effects were not observed in images for any of the photon energies used. Examples are shown in Figures 5.21 and 5.22 where the surface was illuminated 2.14 eV light, respectively, during the bottom halves of each scan. with 2.05 eV and No photo-effects are observed on the terrace or near the step edge. The latter measures 4 Å high; the cluster-covered terrace has a z range of 126 Å and a calculated rms roughness of 6 Å. 149 Figure 5.21 Terrace and step edge morphology of V–930Nb. The bottom half was illuminated with 2.05 eV light. The image scan size is 1,000 Å × 1,000 Å taken with a set point of +2.0 V and 0.25 nA. The arrows indicate apparent holes near step edges. a Figure 5.22 dark b light Local terrace cluster structure of V–930Nb. Illuminated with (a) 2.4 eV and (b) 2.14 eV light during bottom half of each scan. The image scan sizes are 800 Å × 800 Å. Both images were acquired with a set point of +2.0 V and 0.25 nA. The arrows locate a resolved unit cell high step edge. 150 5.2.5 Summary of Observed Morphologies The morphological structures of reduced and niobium-doped SrTiO3 surfaces have been characterized on a nanometer scale. This work presents the first surface images of optically transparent (i.e., within the very early stages of oxygen nonstoichiometry) SrTiO3 (001) obtained by STM. The structure appeared similar to that observed by AFM. The terraces were flat with crystallographically aligned step edges. It has been observed that material redistribution proceeded upon vacuum reduction resulting in apparent hole formation, wandering step edges, and/or locally ordered cluster coverage. The “row structures” observed were separated by ∼20a; this represents an ordering on a larger scale than previously reported. By comparison, no “row structures” were observed on the surface of the niobium-doped crystal. Hydrogen annealing also resulted in cluster formation, although ordering was not clearly observed on these samples. The step edges on the hydrogen-annealed samples appeared straighter than the vacuum annealed samples; they appeared more rugged on the niobium-doped sample, due to prior chemical treatment. Dark regions were apparent on the more heavily reduced crystals, as well as the niobium-doped crystal, mostly near step corners and kinks. In some cases hole formation was apparent. Roughness calculations on terraces yielded similar results for all samples imaged — rms values ranged from 4 Å to 6.8 Å. Photo-induced effects were not observed in any of the images acquired. 5.3 SURFACE ELECTRONIC PROPERTIES BY STS AND PATS The local electrical and optical properties are demonstrated by measurements obtained from the samples in the H series as well as the niobium-doped sample. Only these samples were characterized in terms of their bulk properties in Chapter 4. The similarity in STM images between vacuum-annealed and hydrogen-annealed surfaces suggests that the electronic 151 properties are not significantly different. (Recall that a STM image is an image of electronic information as well as structural information.) Therefore, the conclusions drawn from the study of the H series are likely to be applicable to the V series. A notable limitation of the study is that spectra could not be easily correlated with surface features. This has already been made clear by the images presented in the previous section which do not show the photo-induced effects that are readily apparent in the tunneling spectra. A well-known rule of thumb in tunneling microscopy is that sharper tips give improved image quality (particularly for larger area scans on “rough” surfaces), while blunt tips are required for reproducible spectra. This was observed to be true a fortiori for imaging on these oxide surfaces. In most cases when tunneling spectra were well-reproduced, the images appeared featureless. Similarly, the spectra were usually completely random when images revealed sharp features. The series of spectra presented in section 5.3.2, which represent the reproducible photo-induced effects, were all obtained from apparently flat (i.e., terrace) regions of the samples. The electronic structure of a step is quite unique as shown in the following section, and thus could be easily distinguished if thermal drift were to bring the STM tip into its vicinity. Consequently, the results are not likely to reflect the optical activity of surface steps. All spectra are presented without smoothing and only corrected for the junction capacitance. 5.3.1 Terrace and step edge electronic properties by STS A comparison of the tunneling spectra acquired from terrace regions of the samples in the H series appears in Figure 5.23. All spectra show the characteristics expected of a n-type semiconductor in depletion. The observed trend is a decreasing conductance for increasing degree of reduction. This is indicated by an increase in the conductance well, where the latter is defined as the region between the current onsets at opposite sides of the equilibrium Fermi level. 152 A noteworthy result is the “metallic” behavior of the spectrum for sample H–700. Identical set point conditions were used during the establishment of tunneling on all of these samples. If the tip–sample distance, s, is assumed to be determined by the bulk conductivity, then since the latter increases from H–700 to H–1000, s is expected to increase in kind. If, on the other hand, s is assumed to be determined more by surface properties, such as the surface potential barrier, then an increasing surface potential (and hence overall increase in the tunneling barrier opacity) will result in a decreasing s. Both mechanisms cannot simultaneously account for the observed trend in Figure 5.23 since they suggest a change in s with opposite signs. Bulk properties are known to have profound effects on tunneling spectra; however, the differences observed in Figure 5.23 are believed to be due predominately to variations in surface properties (i.e., the surface potential). Note that a larger carrier density increases screening and is expected to reduce dynamic band bending effects. Recall, however, that the Hall measurements suggest a slight decrease in carrier density with vacuum annealing. This can induce greater dynamic band bending and contribute to increasing the width of the conductance well. 153 2 Tunneling Current (nA) 1.5 1 0.5 0 -0.5 -1 -1.5 a -2 -3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 1 Log|I| (nA) 0.5 0 -0.5 -1 b -1.5 -3 Sample voltage,Va (V) Figure 5.23 Linear (a) and semi-log (b) plots comparing the terrace electronic properties in the H series. An increasing conductance well is observed with increasing reduction, consistent with an increasing surface potential. Solid = H–700; large dash = H–850; small dash = H–1000 154 solid spectrum dotted spectrum a Figure 5.24a Step edge on V–930Nb where local electronic structure is observed to vary as shown by the tunneling spectra in Figure 5.24b. A similar “metallic” spectrum was observed upon moving the tunneling tip near a step edge imaged on sample V–930Nb as shown in Figure 5.24a. The comparison between terrace and step edge spectra are shown in Figure 5.24b. Again, the STM feedback maintains the set point such that the tip–sample gap is not expected to be constant between the two spectra. The tip will be displaced due to a change in the local tunneling barrier. Recall the many dark regions near the corners of the steps in the STM images. This suggests that the local barrier increases near step edges resulting in an inward displacement of the tunneling tip. A similar displacement is not observed in images obtained by AFM. The most consistent explanation for these observations is that the increase in conductance observed in the spectra is due predominately to a decrease in s as a result of an increase in the surface potential. 155 2 Tunneling Current (nA) 1.5 b 1 0.5 0 -0.5 -1 -1.5 -2 -3 -2 -1 0 1 0 0.5 2 3 10 c Tunneling Current (nA) 7.5 5 2.5 0 -2.5 -5 -7.5 -10 -1 -0.5 1 1.5 Sample voltage,Va (V) Figure 5.24b,c Terrace versus step edge electronic behavior. (b) Solid curve acquired on the terrace; dashed curve acquired near the step. (c) The dashed curve on a larger xy scale. The NDC feature near +0.7 V is associated with an increase in local state density. 156 Note that this explanation is opposite to the case discussed for Figure 5.23 where s is assumed to have decreased, but the conductance well increased due to the more dominant influence of an increasing surface potential as well as a possible decrease in screening strength. On the other hand, many of the images that contained dark regions in the step corners also showed increased intensity just above on the edge of the step. It is possible that the spectra in Figure 5.24b reflect the properties of this region rather than the dark corners of the step. The bright features at the step edge mean that the tip was displaced away from the surface. The observed increase in conductance can only be explained in terms of a local decrease in the tunneling opacity due to a local decrease in the surface potential barrier. The latter involves the local trapped charge density. The foregoing arguments are based on the strong dependence of the tunneling current on s and the tunneling barrier height. As can be seen from equation [3.7], an exponential prefactor contains contributions from the local density of states. Variation in the latter is expected to have a small influence on the tunneling current. The following observations, however, are offered in support of an additional explanation of the observed changes in the spectrum. An increase in the local density of states will provide a proportionally greater tunneling probability (see equation [3.1b]). The filling of these states at reverse bias is expected to result in two well-known effects. Since these states do not extend into the bulk to form a band in which charge can propagate (i.e., step-related states are localized on the surface), tunneling electrons from the metal tip initially populate these states giving rise to a measurable capacitance current. This is manifested as a finite current over the voltage range where the conductance well is expected. This charging current must be removed when plotting results on a semi-log scale so that the apparent equilibrium Fermi level always appears at zero applied bias. The changes in 157 the current necessary to shift the spectra in Figure 5.24a were 0.005 nA and 0.011 nA for the terrace and step edge spectra, respectively. Therefore, an increase in local capacitance has been observed. The other effect is related to the increased local capacitance. Since the local surface charge increases as these states populate, a Coulomb field is believed to induce a blockade to additional tunneling electrons such that the current is expected to increase more slowly with increasing reverse bias and tend to flatten the current-voltage curves or give decreasing conductance with increasing applied bias. This effect is referred to as negative differential conductance (NDC) since the derivative of the spectra has a negative value in the voltage range of these features. This effect is not strongly observed in Figure 5.24b. The spectra for the step edge is replotted on a larger xy scale in Figure 5.24c where a shoulder in the conductance is apparent near +0.7 V and is likely associated with a NDC effect. These two latter observations are in support of an increase in local density of states near the step edge. It is likely that the differences observed in Figure 5.24b are due to both variations in local density of states and variations in local surface potential. This is reasonable since a larger state density can trap a larger charge density. From the preceding arguments it can be deduced that the “metallic” spectrum in Figure 5.24b reflect the dark regions near the step corners and can thus be used to distinguish this feature from others on surface terraces when the tip conditions are such that images are not particularly coherent. 5.3.2 Terrace optical responsivity by PATS All tunneling spectra were acquired using the modified sample-and-hold technique (as described in Chapter 2) where the feedback was momentarily disengaged, the surface illuminated with mono-energetic light, then the voltage varied from -4 V to +4 V before the feedback was re158 engaged. The measurement was fast so that all of the features in one spectrum correlate with one region on the surface. Thermal drift, however, is always present to varying degrees such that two sequentially acquired spectra can appear completely different if the tunneling tip wanders over a region with spatially varying electronic structures. A more insidious problem can occur due to instabilities in the feedback electronics. During the stage when the feedback is disengaged, even a small instability in the amplifier voltage output (i.e., millivolts) can cause an unwanted tip displacement and hence place the measurement on a new current/voltage curve. Random fluctuations of this kind can account for the error observed in the standard deviation curve plotted in Figure 2.3. In principle, if the tunneling gap narrows significantly, the large electric field at the tip can continuously redistribute atoms near the tip apex during a single voltage sweep [5]. The resulting spectrum can be a composite of several different spectra and thus not be useful for interpreting the constant properties of the surface. All of these mechanisms (thermal drift, circuit instabilities, and tunneling tip instabilities) can separately or collectively result in non-reproducibility in the features of the spectra. This difficulty was encountered frequently and it was often necessary to allow the STM to “settle” before taking additional spectra. Careful documentation of frequently observed characteristics distinguished “good data” from misleading data. It was C. B. Duke that said, “Tunneling is an art, not a science” [6]. It may also be said that scanning tunneling spectroscopy requires certain experimental artistry to recognize the science. Figures 5.25 through 5.40 represent the science observed from the application of PATS to vicinal SrTiO3. It is believed that this data reflects the photo-activity of only the terrace regions. Figure 5.25 shows representative spectra obtained from H–700. The solid curve was obtained 159 without illumination. The dot-dashed curve was obtained while illuminating with 3.8 eV light. This figure is identical to the results observed at all energies used in the study. Thus, no evidence has been found to support photo-activity on the surface of H–700. 160 2 Tunneling Current (nA) 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 1.5 1 Log|I| (nA) 0.5 0 -0.5 -1 -1.5 Sample voltage,Va (V) Figure 5.25 Dark versus light spectra for H–700 illuminated with 3.8 eV light. No photo-induced effects are observed, consistent with the absence of effects in the images. 161 The plots in Figures 5.26 to 5.40 demonstrate optically-induced changes in the electrical behavior and also show the corresponding modeled spectra. The observed noise in the tunneling current signal was typically 25 mV. The data are all truncated in the semilog plots at the same upper and lower limits set at 1 and -1.6, respectively. The dark spectra were modeled initially to match the experimental dark spectra. Three parameters requiring adjustment to match the photoinduced data were the equilibrium surface potential (ψ), the capacitance constant (β), and the electron affinity (χ). All depend on the surface charge, which is the single dominant property believed to vary to induce the observed effect. The required values of ∆ψ, ∆β and ∆χ are noted within the text of each figure. The parameters required to determine the change in surface charge are the values of the surface potential before and after illumination. The changes in surface charge were calculated using [3.21] and the results plotted in Figure 5.41a. It can be expected that a spectral variation in the light intensity, as demonstrated in Figure 2.2, will be reflected in the measured changes in tunneling spectra. The light flux at the tunneling junction was not measured simultaneously with the spectra so that exact quantification of materials properties can not be assumed. Approximate values can be determined, however, using the data in Figure 2.2 assuming that the spectral response of the optics did not change significantly from the original measurement. Dividing the data in Figure 5.41a by the product of the photon flux (from Figure 2.2) and the fundamental unit of charge, and applying the appropriate conversion factors to express the results in electrons per mW, gives the data as shown in Figure 5.41b. The term “responsivity” is used here similarly to its use to describe photonic devices [7]. Processed in this way the data better represents the response of the material to incident light in terms of the number of single electron transitions per unit incident power. 162 log linear 2 H-850 2.90 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 1 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 2 -2 -1 0 1 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 4 -3 Sample voltage,Va (V) Figure 5.26 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.040 eV; ∆β = 3.5 × 10-4 C/V; ∆χ = 0.80 eV. 163 log linear 1 2 H-850 2.05 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 -3 3 -2 -1 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 1.5 0.5 1 Log|I| (nA) Tunneling Current (nA) 0 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 -2 -1 0 1 2 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 Sample voltage,Va (V) Figure 5.27 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.016 eV; ∆β = 1.5 × 10-4 C/V; ∆χ = 0.50 eV. 164 log linear 1 2 H-850 1.77 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 -3 3 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 2 -2 -1 0 1 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 4 -3 Sample voltage,Va (V) Figure 5.28 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = - 0.023 eV; ∆β = - 3.3 × 10-4 C/V; ∆χ = - 0.65 eV. 165 log linear 1 2 H-1000 3.80 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 -3 3 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 -2 -1 0 1 2 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 Sample voltage,Va (V) Figure 5.29 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.035 eV; ∆β = 0 C/V; ∆χ = 0.40 eV. 166 log linear 1 2 H-1000 2.90 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 -3 3 -2 -1 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 1.5 0.5 1 Log|I| (nA) Tunneling Current (nA) 0 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 2 -2 -1 0 1 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 4 -3 Sample voltage,Va (V) Figure 5.30 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.145 eV; ∆β = 1.0 × 10-4 C/V; ∆χ = 0.70 eV. 167 log linear 1 2 H-1000 2.82 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 -2 -1 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 1.5 0.5 1 Log|I| (nA) Tunneling Current (nA) 0 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 -2 -1 0 1 2 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 Sample voltage,Va (V) Figure 5.31 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.040 eV; ∆β = - 6.0 × 10-4 C/V; ∆χ = 0.20 eV. 168 log linear 1 2 H-1000 2.40 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 -3 3 -2 -1 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 1.5 0.5 1 Log|I| (nA) Tunneling Current (nA) 0 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 2 -2 -1 0 1 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 4 -3 Sample voltage,Va (V) Figure 5.32 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.150 eV; ∆β = - 6.3 × 10-4 C/V; ∆χ = 0.40 eV. 169 log linear 1 2 H-1000 2.14 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 -3 3 -2 -1 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 1.5 0.5 1 Log|I| (nA) Tunneling Current (nA) 0 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 -2 -1 0 1 2 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 Sample voltage,Va (V) Figure 5.33 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.170 eV; ∆β = - 6.3 × 10-4 C/V; ∆χ = 0.40 eV. 170 log linear 1 2 H-1000 2.05 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 -3 3 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 2 -2 -1 0 1 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 4 -3 Sample voltage,Va (V) Figure 5.34 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.054 eV; ∆β = 0 C/V; ∆χ = 0.30 eV. 171 log linear 2 H-1000 1.90 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 1 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 -2 -1 0 1 2 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 Sample voltage,Va (V) Figure 5.35 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = - 0.090 eV; ∆β = 0 C/V; ∆χ = 0.10 eV. 172 log linear 2 H-1000 1.77 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 1 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 2 -2 -1 0 1 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 4 -3 Sample voltage,Va (V) Figure 5.36 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = - 0.140 eV; ∆β = 3.0 × 10-4 C/V; ∆χ = - 0.70 eV. 173 log linear 2 H-1000 1.30 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 1 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 -2 -1 0 1 2 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 Sample voltage,Va (V) Figure 5.37 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.018 eV; ∆β = 0 C/V; ∆χ = 0.26 eV. 174 log linear 2 V-930Nb 3.80 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 1 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 2 -2 -1 0 1 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 4 -3 Sample voltage,Va (V) Figure 5.38 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = - 0.043 eV; ∆β = - 1.5 × 10-4 C/V; ∆χ = - 0.50 eV. 175 log linear 2 V-930Nb 2.90 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 1 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 2 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 4 -3 2 -2 -1 0 1 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 4 -3 Sample voltage,Va (V) Figure 5.39 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.032 eV; ∆β = 1.5 × 10-4 C/V; ∆χ = 0.50 eV. 176 log linear 2 V-930Nb 2.40 eV 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 1 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 1 2 3 4 1 2 3 4 1 2 dark dark 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 -2 -1 0 1 2 light light 0.5 1 Log|I| (nA) Tunneling Current (nA) 1.5 0.5 0 -0.5 -1 0 -0.5 -1 -1.5 -1.5 -2 -3 -2 -1 0 1 2 3 -3 4 Sample voltage,Va (V) Figure 5.40 -2 -1 0 Sample voltage,Va (V) Surface photo-effect matched with the following parameter variations: ∆ψ = 0.014 eV; ∆β = 9.6 × 10-4 C/V; ∆χ = 0.40 eV. 177 15 V-930Nb 10 H-850 H-1000 5 0 1.3 1.77 1.9 2.05 2.14 2.4 2.82 2.9 3.8 2.4 2.82 2.9 3.8 -5 -10 -15 a Photon Energy (eV) 10 V-930Nb H-850 H-1000 5 0 1.3 1.77 1.9 2.05 2.14 -5 -10 -15 b Photon Energy (eV) Figure 5.41 (a) The photo-induced change in surface charge determined by modeling the observed changes in the tunneling spectra. (b) The responsivity of the surfaces determined by normalizing for the spectral response of the experimental optics. 178 Except at energies 1.77, 1.90, and 3.8 eV light, most observed photo-induced responses may be described as a decrease in tunneling conductance. Photo-induced effects were observed at only three different energies for the H–850 sample and the V–930Nb sample. The H–1000 sample showed photo-effects at all incident energies, similar to the broad band bulk absorption observed from the ellipsometry measurements. Only the V–930Nb sample showed an increase in conductance at 3.8 eV light. The qualitative behavior of the spectra for V–930Nb is similar to H–850 in terms of the average width of their conductance wells. This implies a similarity in surface electronic properties so that the electronic contribution to terrace images acquired from V–930Nb might well represent the electronic contribution to terrace images acquired from H–850. The modeled spectra were seen to fit the experimental data rather closely within reasonable experimental error (see discussion in section 2.1.3). In some cases the fit was nearly exact. All spectra appear to be interpretable in terms of a change in surface charge. The behavior, however, was not always as predicted in Chapter 3. Recall that modifying surface charge was predicted to modify the width of the conductance well, with the shift at forward bias being stronger than that at reverse bias. In at least one case (Figure 5.29) the opposite was observed. In another case a strong effect was observed at forward bias only (Figure 5.35). The model was observed to systematically underestimate the current at larger forward biases for samples H–850 and V–930Nb; the possible reasons for this were discussed in Chapter 3. On the other hand, the model systematically overestimated the current at larger forward biases for H–1000 which is believed to be due to the effects of band gap states. The model does not treat gap states directly. A particular demonstration of this fact was observed for the H–1000 spectra which showed possible additional photo-induced effects. The semilog plots of Figures 179 5.32 and 5.33 contain a change in the slope of the current near -1.5 eV giving an apparent conductance tail which decreases upon illumination with 2.4 eV and 2.14 eV light. The values of ∆β reported along with the figures are small or zero, consistent with the fact that surface states are not anticipated to depend on the presence, or energy, of incident light. The values of ∆χ are also reported with the figures. The latter are given mainly for the purpose of demonstrating the possible limitations of the model. It should be noted that ∆χ was systematically larger than ∆ψ, often by more than an order of magnitude. Also, the absolute values of χ that were necessary for a “good fit” at reverse bias deviated from the tabulated values by up to ∼ 50% of the latter. This does not negate the validity of the conclusions derived from the results since they are based only on the values of ∆ψ which are believed to accurately describe the variations of the surface charge. The results are summarized in the bar plots of Figure 5.41, where Figure 5.41b is a closer representation of the spectroscopic response of the material to incident light. It is clear that the strongest effect was observed on H–1000 and occurs near 1.77 eV light where the responsivity is more than an order of magnitude larger than at any other energy. This absorption was also observed for H–850 but with much lower response. The second strongest responses from H– 1000 occurred near 1.9 eV and 2.4 eV, where the effects are due to decreasing and increasing electronic charge, respectively. All three samples showed similar responsivity at 2.9 eV, where an increase was observed from H–850 to H–1000, suggesting a correlation to carrier density or trap density. Finally, the weakest response was observed to occur at 2.82 eV, and only for H– 1000. This energy does not appear in Table 1.4; it is included because absorption was consistently observed at this energy. 180 5.3.3 Summary of Observed Optical Responsivity Evidence of optically active on the terrace sites of reduced SrTiO3 was observed spectroscopically as increases and decreases in the width of the conductance well in the tunneling spectra. The largest effect, after normalizing for the contribution of the experimental optics, occured for absorption of 1.77 eV monochromatic light. The smallest effect occurs near 2.8 eV monochromatic light. Broad absorption effects were observed over the entire energy range of interest for the H-1000 sample. The H-700 sample did not display absorption effects at any of the energies, and H-850 showed absorption at only three energies. The effect may be described as an increase in surface electron density or a decrease in surface electron density where the average order of magnitude in responsivity was 1011 electrons/mW. REFERENCES 1. N. Ikemiya, A. Kitamura, and S. Hara J. of Crystal Growth 160 (1996) 104 2. Y. Liang and D. A. Bonnell Surface Science Lett. 285 (1993) L510; Y. Liang and D. Bonnell J. Am. Ceram. Soc. 78 [10] (1995) 2633 3. M. Naito and H. Sato Physica C 229 (1994) 1 4. B. Stäuble-Pümpin, B. Ilge, V. C. Matijasevic, P. M. L. O. Scholte, A. J. Steinfort and F. Tuinstra Surface Science 369 (1996) 313 5. J. C. Chen Introduction to Scanning Tunneling Microscopy Oxford University Press, Oxford (1993) 6. C. B. Duke Tunneling In Solids Academic Press, New York 1969 7. B. E. A. Saleh and M. C. Teich Fundamentals of Photonics John Wiley and Sons, Inc., New York (1991) 181 Chapter 6: Discussion and Conclusions 6.1 DISCUSSION OF RESULTS 6.1.1 Photo-assisted tunneling microscopy and spectroscopy One of the goals of this thesis was to explore the feasibility of enhancing the detectability of defect-induced band gap states by activating charge-transfer transitions near the surface. By modifying local surface charge densities, tunneling spectra are expected to show characteristic variations largely in the form of an increase or decrease in the width of the conductance well. Since images obtained by STM are essentially a map of the surface electronic properties (or electronic structure) modulated by the surface topography, one might expect photo-induced effects to enable spatially resolved imaging of surface defects. In the event that several different transitions are optically active, which may be associated with different types of defects, the prospect of selectively imaging defect structures, to facilitate distinction between the local environments responsible for the resulting defect states, is intriguing. This would be a valuable metrological tool for the study of photocatalytic processes. Evidence of optical activity was not observed in any of the STM images acquired. This is despite the fact that the tunneling spectra showed strong photo-induced effects on three out of the four samples studied. One implication of this observation is that STM images are not influenced by variations in electronic properties and thus show only topography similar to AFM imaging. That this is an unacceptable conclusion has been demonstrated by the difference in step edge structure observed by AFM (i.e., Figure 2.7) as compared to that observed by STM (i.e., Figure 5.5). The latter contains additional intensity on top of step ledges and decreases in intensity near step corners. These regions are where tunneling spectroscopy has indicated a 182 variation in electronic properties, such as the existence of a local density of states associated with the step edge structure. It is possible that the “features” associated with the photo-effects simply have not been resolved. The lateral resolution of the STM is determined by the properties of both electrodes at the junction. The images invariably required sharp tips in order to resolve the details of step edges and terrace clusters. Blunt tips (necessary for reproducibility in measured spectra) do not always generate coherent images. On the other hand, the large dielectric constant of SrTiO3, combined with the condition of low carrier density, results in a relatively large depletion width, usually on the order of several hundred angstroms or larger. Depending on the set point conditions, this large depletion width can “spread out” the contribution of the electronic properties to an image [1]. The half-illuminated images (such as Figure 5.18), however, should still contain at least a uniform displacement due to the extreme sensitivity of the STM to changes in the tunneling barrier function. The images and the spectra are thus in disagreement regarding the electronic properties of the (001) surface of SrTiO3. This apparent paradox remains presently unresolved, a situation that represents a unique opportunity for continued progress. 6.1.2 Surface structures and morphologies The surface structures and morphologies of oxygen deficient SrTiO3 were studied in order to identify a correlation between surface structure and observed optical activity. In the early stages of reduction, STM images of the (001) surface appear similar to the AFM images of fully oxidized crystals. This implies that the electronic properties of this surface are uniform within the (lateral) resolution limit of the tunneling microscope. Mild thermal treatment (such as short time anneals or annealing below the “freeze-in” temperature for oxygen vacancies) apparently does not result in significant 183 surface restructuring and/or that oxygen vacancies are distributed randomly such that sharp 1 × 1 LEED patterns are obtainable. Upon further reduction the surface morphology undergoes extensive restructuring. Roughening of the surface is likely to be due mostly to the formation of holes. This is supported by the fact that the calculated rms roughness was similar for all of the terraces imaged from different samples. It has been suggested that hole formation is evidence of pre-existing line defects (such as dislocations) or microcracks which may form during thermal processing or mechanical polishing. The local stress fields near such defects can cause preferential sublimation during high temperature vacuum annealing [2]. This seems reasonable since hole formation was not observed for high temperature air annealing, where the large abundance of oxygen is likely to accelerate defect repair. In vacuum annealing, the formation of holes is probably driven by the reduction in strain energy associated with mechanically or thermally induced defects. Wandering step edges with local ordering of clusters on terraces are the manifestations of this system attempting to lower the excess free energy of the surface, which for ionic solids has a large electrostatic component. These changes have been documented in terms of long range defect ordering or a demixing phenomena giving rise to the formation of new surface phases [3,4]. The LEED patterns observed for vacuum-reduced samples suggest a tendency to form a 2 × 2R45o superstructure. It is significant that higher temperature annealing sometimes returned the structure to 1 × 1 ordering. Although not shown in Chapter 5, it should be noted that upon one occasion a two domain 2 × 1 superstructure was also observed after UHV heating at 900 °C for a few minutes. A perusal of the literature shows a rich variety of annealing temperatures and annealing times. (Unfortunately, the sample cooling rates were not carefully 184 documented.) There is also a corresponding richness in observed superstructures. These facts imply that the crystallographic structure (and perhaps also morphology) is ultimately controlled by kinetic effects and supports the view that mobile species such as oxygen vacancies are the mechanism by which the surface structure is obtained (see discussion below). This also implies that there should be a direct correlation between the structure and the electrical/optical properties of the surface that depend on the oxygen vacancy density. A look at the structure/property relationship of the niobium-doped sample compared to the vacuum-reduced samples draws implications regarding the roll of oxygen vacancies versus the doping of electrons into the conduction band on determining the electronic properties and perhaps surface structure. The apparent terrace morphology of the vicinal surface appears to have a greater dependence on the type (and density) of defect than on the density of carriers in the conduction band. Recall, however, that the Hall measurements show the free carrier density in V–930Nb to be larger than those for the H series, but still of the same order of magnitude. The vacuum anneal for this sample was identical to that for V–930; however, the LEED pattern indicated that it formed a 2 × 2R45o superstructure similar to V–1100. On the other hand, no row-type structures were observed in the images of V–930Nb. The absence of the latter might very well be correlated with the absence of optical responsivity near 1.77 eV as was observed for the reduced samples in Figure 5.41b. The important point is that it is now more difficult to assign the observed superstructure to the ordering of oxygen vacancies when the latter is not expected to be present in large abundance in the niobium-doped crystal. Furthermore, the observed LEED pattern for V–1100 suggests a surface described by C4 symmetry, while the apparent surface morphology suggests C2 symmetry. These observations indicate that the “structure” observed by a LEED probe is not 185 the same as the “structure” observed by a STM probe. This is not unreasonable since a diffraction probe detects order in the lattice on the scale of the coherence width of the incident electron beam. The STM, on the other hand, generally has a more convoluted signal. In the event that tip morphology is not influencing the scans, and the surface is relatively flat, the STM probe detects order in the electronic behavior or electronic structure. Therefore, even in the event that the surface has a well-defined crystallographic order in its top-most layer, a different order in the local environment of perhaps the subsurface plane means that the surface cations have a different order in their local Madelung potentials. The STM will be sensitive to the variations in the local surface potentials. Therefore, the difference in apparent morphologies between say V–930Nb and V–1100 must be correlated to the difference in electronic properties as demonstrated by the difference in their local surface optical responsivities. 6.1.3 Defect-induced electronic properties The correlation between surface structure and surface photo-activity is inferred from Figure 5.41 as an overall increase in absorption across the experimental spectral window with increasing reduction. A characteristic decrease is observed in the spectral region (i.e., 2.82 eV) corresponding to the “color” of the crystal. These properties should be expected since a similar broad band increase in absorption has been observed by ellipsometry as a bulk response to increased reduction, and transmission measurements similarly showed an increase near 2.82 eV. The bulk measurements are interpreted as an increase in the density of occupied band gap states which supply electronic transitions to unoccupied conduction band states. Reduction also varies the bulk electronic structure in terms of variations in the band gap energy and a decrease in the density of states at the top of the valence band. The variation in the band 186 gap, however, is not expected to have a strong influence on the characteristics of the observed tunneling spectra and thus is not relevant to the optical responsivity observed by PATS. All previous STM studies on oxygen deficient strontium titanate report annealing the crystal in a reducing environment until it became “dark blue” or “black,” at which point it was assumed that the crystal was “sufficiently conductive” for tunneling to be established. This work is the first STM study to report tunneling images and spectra obtained from a reduced crystal that remained colorless and transparent. Therefore, the series of spectra in Figure 5.23 represent a unique look at the changing electronic behavior of the surface from the early stages of reduction to strong reduction. The results indicate an increasing surface potential which, for a n-type semiconducting oxide in depletion, is associated with an increasing surface charge density. This must follow since it is counterintuitive (and contrary to the resistivity measurements) that the “clear” sample is more conductive than the “black” sample. The increase in surface charge density manifests also as an increase in surface optical responsivity. The surface is most responsive at 1.77 eV incident light. The number of electrons excited at this energy grows with increasing reduction and a similar effect is not observed for the niobium-doped crystal. This suggests a correlation with the oxygen vacancy defect. The increase in the response observed indicates either an increase in the density of the defect or a growth in the size of domains in which these defects are ordered. Since atomic scale resolution had not been achieved in these experiments and (more importantly) photo-induced effects were not imaged, it is presently not possible to assign a causation relation between the observed behavior and the structure of the defect sites. Some comments can be made, however, regarding the nature of the defect at this energy. 187 Firstly, the mechanism of the charge transfer can be deduced by considering the fact that the tunneling measurement can only detect photo-responses that result in redistribution of charge between the surface and the bulk. An increase in tunneling conductance corresponds to a decreasing surface potential as a result of electron excitation from surface trap states to the conduction band where they can be swept into the bulk by the built-in electric field. Alternatively, electrons from the valence band might be excited to bulk trap states and the surface charge decreased by recombination of holes with surface electrons (see Figure 1.7a). The former mechanism is more likely since the low mobility of holes make it more probable that the latter mechanism will be suppressed by bulk recombination effects. It is therefore believed that the mechanism is described by the depopulation of surface trap states resulting in the negative value of the surface responsivity in Figure 5.41b. Consequently, the mechanisms associated with the broad absorption at higher energies must involve the transfer of electrons to the surface to increase the surface potential and thus give positive values to the surface responsivity. A plausible mechanism is illustrated in Figure 1.7b, where a bulk transition involves excitation from singly ionized oxygen vacancies, and the electrons charging the surface are supplied by the metal STM tip. Secondly, if the defect state is described by non-itinerant wave functions, then it will not exhibit dispersion when represented in a band diagram — i.e., it will be represented as a straight line across the Broullouin zone. It might therefore be expected that excitations are possible at all energies greater than the threshold energy to induce defect-to-conduction band (CB) transitions. Figure 5.41, however, does not suggest the same type of transition behavior for all energies above 1.77 eV so that the observed behavior might be attributed to the narrow width of the lowest CB. The strong response suggests that the transition is symmetry-allowed, which requires 188 that the defect states have opposite parity to the lower CB states. If transitions are allowed between lower and upper CB states (which is believed to account for absorption at 2.4 eV [5-7]), then these states must also have opposite parity. It is thus clear that the defect states must have the same parity as the upper conduction band states, the latter of which are composed of Ti–4p and O–3p orbitals and thus have odd parity. It is suggested that the defect state may also be described by functions of odd parity. Thirdly, there is a clear correlation between the apparent surface ordering and the increase in band gap absorption at 1.77 eV. The least reduced sample showed sufficient conductivity that is clearly associated with its oxygen vacancy density; however, the absence of defect ordering resulted in a 1 × 1 surface structure and no observed optical responsivity in the tunneling spectra. It follows that the surface defect state is not merely associated with the oxygen vacancy, but rather with the ordered arrangement of oxygen vacancies. Moreover, the ordering of oxygen vacancies responsible for the state may not be confined to the surface. (Recall that the 1.77 eV absorption band was also observed by transmission studies [5,6].) The electrostatic fields of bulk vacancies may modify the surface potentials and give rise to an ordering observed by STM that is distinct from that observed by LEED. Note that a similar argument was recently used to explain the atomically resolved long range order observed on oxygen deficient SrTiO3 by STM [8]. Lastly, the determined responsivity suggests a relatively low occupancy for the surface state. This may actually be a consequence of low incidence flux upon the junction. For example, the calculated change in surface charge for the 1.77 eV peak can be used to approximate the cross section for transitions at this energy. The value of ∆Qss in Figure 5.41a at 1.77 eV corresponds to a change in surface electron density of 0.00076 electrons 189 per unit mesh, where the unit mesh is defined by the area 0.39 nm × 0.39 nm. The average flux from Figure 2.2 was of order 1013 photons per second. Considering the active area of the photodiode and converting to nanometer units, this corresponds to 8.3 × 10-2 photons per second per square nanometer, or 0.013 photons per second per unit mesh. Assuming an absorption cross section of unity, and an experimental measuring time of ∼ 0.1 s (see section 2.1.2), gives an expected response of 0.0013 electrons per unit mesh. Therefore, the actual cross section for excitation is closer to 60%, close enough to unity to describe a surface with strong optical responsivity. This implies that the use of solid state lasers, which can supply up to 100 mW or more in power, might induce a greater response and even drive the surface to flat band conditions by completely depopulating the gap state. This can lead to a direct experimental quantification of the density of states associated with the oxygen vacancy defect. Considering the manner in which the experiment was performed, the coarse view of the results is not surprising. Therefore, it is difficult to assess how the resolution of the monochromator (0.014 eV) may have influenced the results. It is nevertheless interesting that the strongest optical effect at the surface occurred at the same energy as oxygen vacancy-related absorption observed in the bulk [5,6]. This suggests that the mere creation of the surface does not strongly modify the surface electronic structure from that of the bulk, as predicted by recent theoretical models [9]. One final speculation is suggested when considering the difference between the responsivity of the niobium-doped sample and the reduced samples. Figure 1.6 illustrates a mechanism by which surface charge may be reduced when greater than band gap energy is incident upon the surface. Only the niobium-doped sample showed a decrease in surface charge consistent with this type of mechanism. Figure 4.3 clearly shows that the majority of the light is 190 absorbed within 50 nm of the surface plane for energies well above the band gap in all of the hydrogen-reduced samples. This strong absorption is due primarily to interband transitions between the valence band and the conduction band. On the other hand, midgap states are believed to act as efficient recombination centers. These excitations may not be detectable in the reduced crystals due to efficient recombination via the bulk oxygen vacancy states within the depletion region. For the niobium-doped crystal, however, the impurity places a hydrogenic (i.e., shallow) state near the conduction band edge. These are not efficient recombination centers and thus charge redistribution as illustrated in Figure 1.6 may explain the observed decrease in surface potential at 3.8 eV. The origin of the trapped surface charge on V–930Nb is not expected to be the same as that on the reduced crystals. Other than step sites, which were not probed by the PATS study, the results do not offer an origin for acceptor type defect states on the terraces of the niobium doped crystal. 6.1.4 Conclusions This experimental work has demonstrated the successful application of the recently developed technique of photo-assisted tunneling spectroscopy (PATS) to the study of electron-doped SrTiO3 (001) surfaces. The technique combined with theoretical modeling has been shown to characterize surface charge transfer transitions in terms of threshold energy and absorption mechanism. The related technique of photo-assisted tunneling microscopy (PATM) did not identify the origins of surface optical responsivity in STM images. This is an unexpected result given the close relationship between tunneling spectra and tunneling images which identifies a potential opportunity for future progress. The quantification of surface properties, such as the magnitude of the change in the surface charge, relies on the accuracy of the theoretical model used to match the experimental 191 data. The model developed in this thesis produced tunneling spectra that closely matched the experimental data using values for the electron affinity which often deviated from the tabulations, highlighting the weakness of the theory under reverse bias conditions. Under forward bias conditions, however, the model is less sensitive to the absolute value of the electron affinity. The behavior of the surface charge is directly modeled as changes in the surface potential which is treated accurately under forward bias conditions. Strong absorption at 1.77 eV is ascribed to the association of oxygen vacancies as measured on terrace sites under heavy reducing conditions. This association may also determine the row-type surface structure observed by STM and it is likely to involve oxygen vacancy ordering in the depletion region. The scale of the ordering observed in this work is larger than that previously reported. Given the sensitivity of the surface structure to thermal history and quality of the crystal, the reason for this difference would be no more than speculation. It is, however, clear that the observed structure and the observed optical responsivity are correlated. Evidence of optically active step sites was not observed; however, conventional tunneling spectroscopy indicate the presence of an occupied localized density of states at step sites. The existence of optically active step sites is therefore not ruled out. REFERENCES 1. D. A. Bonnell, I. Solomon, G. S. Rohrer and C. Warner Acta. Metall. Mater. 40 Suppl. (1992) S161 2. B. Stäuble-Pümpin, B. Ilge, V. C. Matijasevic, P. M. L. O. Scholte, A. J. Steinfort and F. Tuinstra Surface Science 369 (1996) 313 192 3. Y. Liang and D. A. Bonnell Surface Science Lett. 285 (1993) L510; Y. Liang and D. Bonnell J. Am. Ceram. Soc. 78 [10] (1995) 2633 4. K. Szot, W. Speier, J. Herion and Ch. Freiburg Appl. Phys. A. 64 (1997) 55; K. Szot and W. Speier (unpublished) 5. H. Yamada and G. R. Miller J. Solid State Chem. 6 (1973) 169 6. C. Lee, J. Destry and J. L. Brebner Phys. Rev. B 11 [6] (1975) 2299 7. R. L. Wild, E. M. Rockar and J. C. Smith Phys. Rev. B 8 [8] (1973) 3828 8. Q. D. Jiang and J. Zegenhagen Surface Science 425 (1999) 343 9. J. Goniakowski and C. Noguera Surface Science 365 (1996) L657 193 Chapter 7: Summary of Dissertation The use of the (001) surface of strontium titanate (SrTiO3) as a catalyst, as a substrate for epitaxial growth of oxide superconductors, as well as a photoelectrode for solar energy conversion, demonstrates the versatility of the perovskite oxides in widespread commercial applications today. A deeper understanding of the microscopic (or nanoscopic) structure/property relations is therefore vital to several industries Many of the important applications of SrTiO3 are possible due to occupied defect states energetically located within the band gap. Most bulk and surface sensitive probes have identified this state; however, the details of the structural origin are often left to first principles calculations or even to speculation. In particular, agreement has not yet been reached regarding the physical nature of the oxygen vacancy-related band gap states. The objective of this thesis work was to develop a technique, with theoretical understanding, utilizing the combined methods of tunneling spectroscopy and optical spectroscopy, to enable the identification and characterization of optically active surface defect structure. This technique could then be applied to study the (001) vicinal surface of oxygen deficient SrTiO3 in order to characterize the physical origins of the observed optically active deep level defects. The technique successfully identifies surface and subsurface charge transfer transitions giving both the energy of an optically active surface state and the mechanism by which the state is populated or depopulated. The mechanism is modeled as a change in the surface charge. A model has been developed to simulate tunneling spectra on wide band gap (i.e., large dielectric constant) oxides which has been shown to accurately match the experimental behavior at forward 194 biases. The theory-experiment fit at reverse bias was made possible only by assuming large deviations in the electron affinity from its tabulated values, therefore identifying the need for continued refinements in the model. It has been determined that reduction annealing introduces deep level band gap states in SrTiO3 that are optically active and are correlated with oxygen vacancy association on the terrace sites of the (001) surface. This work represents the first successful application of spectroscopic PATS to the study of a wide band gap oxide material. The results demonstrate the strong potential for its use as a metrological tool to study adsorption modes and other surface mediated processes that occur via surface or subsurface gap states. 195 APPENDIX A: FRANCK-CONDON PRINCIPLE AND THE SPECTROSCOPIC RESOLUTION The strength of optically induced charge transitions in solids, and the characteristics of observed absorption bands, in principle may be limited by the properties of the crystal matrix. The most fundamental restriction on the strength of any transition between some initial state, ψi, to some final state, ψf, may be derived from a consideration of the transition rate. This requires the evaluation of an appropriate transition matrix element such as [1] +∞ M fi = ∫ψ ∗ f (τ)g(τ)ψ i (τ )dτ , [A.1] −∞ where the form of the operator g( τ) depends on the nature of the particular transition process, and the volume element τ includes both electronic and nuclear coordinates. A finite result is obtained from [A.1] only when the integrand is an even function. Therefore, for a given transition process where the parity of g( τ) is fixed, a selection rule is established regarding the parities of ψi and ψf. The matrix element for a radiative transition, when the wavelength of the radiation is r large relative to the size of the absorbing atom/molecule, is given by [A.1] where g( τ) = µ e [2]. The latter is the electric dipole operator — an odd function. The integral may be separated based on the electronic (dτe) and nuclear (dτn) coordinates such that [A.1] becomes [3] +∞ +∞ r e M fi = ∫ θ φ µ eθ iφ idτ e dτ n = M fi ∫ φ∗f φ i dτn , ∗ f ∗ f −∞ [A.2a] −∞ +∞ where r M = ∫ θ ∗f µe θi dτ e . e fi −∞ 196 [A.2b] One can see from [A.2b] that the selection rule for a radiative transition requires θf and θi to have opposite parities. The parities of the state functions in a solid are primarily determined by the lattice symmetry. Therefore, a radiative transition is said to be symmetry-forbidden if the initial and final states are described by functions of like parity. When such a transition is symmetryallowed, the magnitude of Mfi is determined by the second integral in [A.2a]. The latter is also known as the Franck-Condon (F-C) factor, where φi and φf correspond to different vibrational states of the system. Figure A.1 Illustration of molecular energy as a function of internuclear distance R for two different electronic states of a diatomic molecule [from reference 5]. 197 The Franck-Condon principle is based on the fact that the characteristic time for an electronic transition is several orders of magnitude shorter than the period associated with nuclear vibrations [4]. The consequence of this fact is demonstrated in Figure A.1 which illustrates transitions between some ground state and an excited state of a diatomic molecule. (Note that rotational energy levels do not exist in a lattice; however, the molecular vibrational levels are analogous to the normal modes of a lattice.) The transitions are shown as vertical lines in accordance with a rapid absorption or fluorescence process. If the equilibrium geometry of the initial and final states are the same, the F-C factor is maximized when the vibrational momentum is conserved; otherwise the F-C factor will be maximized for a transition involving a finite change in vibrational momentum (as exemplified in Figure A.1). In a lattice, this corresponds to the creation or annihilation of a phonon. ECB EVB ED ED EA EA a Figure A.2 b a) Scheme to depict defect thermal ionization energies; b) scheme to depict the same defect spectroscopic energies. The Gaussian distribution about each energy results from phonon interaction in accordance with the Franck-Condon principle. The usual scheme used to depict donor and acceptor ionization energies is shown in Figure A.2a. These levels are assumed to describe thermal excitation processes. A scheme 198 suggested to be more appropriate to depict spectroscopic energies is shown in Figure A.2b, which acknowledges the superposition of a F-C envelope when charge-transfer excitations are accompanied by phonon interaction [6]. Based on this discussion, it may be appreciated that the nature of optical absorption bands in oxides is determined by the density of initial and final states that satisfy the electric dipole selection rule, as well as the allowed phonon spectrum which gives rise to a F-C spectral distribution. The latter imposes a fundamental limit on the resolution of spectroscopic energies determined experimentally. To the author’s knowledge, there are no reported attempts to deconvolute the F-C envelope from existing absorption spectra for transition metal oxides. It is therefore difficult to estimate the F-C limited spectral resolution for a given material such as SrTiO3. It is interesting to note, however, that measured absorption bands associated with defect-induced absorption in SrTiO3 can have a typical FWHM bandwidth of approximately 1.5eV. REFERENCES 1. J. P. Elliott and P. G. Dawber Symmetry In Physics Oxford University Press, Oxford 1979 2. A. T. Fromhold, Jr. Quantum Mechanics for Applied Physics and Engineering Dover Publications, New York 1991 3. L. S. Forsler in Concepts of Inorganic Photochemistry A. W. Adamson and P. D. Fleischauer, Eds. Robert E. Krieger Publishing, Inc., Florida 1984 4. K. Nassau The Physics and Chemistry of Color Wiley, New York 1983. 5. R. Eisberg and R. Resnick Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles 2nd ed. Wiley, New York 1985 199 6. P. A. Cox Transition Metal Oxides: An Introduction To Their Electronic Structure and Properties Oxford University Press, Oxford 1995 200 APPENDIX B: SEMICONDUCTOR DEFECT STATISTICS One of the problems central to statistical thermodynamics is the determination of the occupation probability of a set of states with energy degeneracy Z in order to predict the observable properties of a material. The fundamental postulate of statistical thermodynamics states that all possible microstates for a closed and isolated assembly of N particles in a given macrostate are equally probable. For the states in the conduction band of a metal or semiconductor, the number of possibilities for the occupation of the ith level is assumed to be independent of the state of occupancy of the (i+1) level; the occupancy of the conduction band for a given macrostate is thus derived from the product ∞ Zi ! , i − N i )! ∏ N !(Z i=1 i [B.1] also known as the thermodynamic probability. The numerator gives the total number of ways to permute the energy states of level i; the denominator gives the total number of ways to permute the energy states that do not result in unique permutations, since the particles being considered are assumed indistinguishable. Maximizing [B.1], subject to the constraints of conserved mass and energy, gives the well-known Fermi function Ni 1 = f(E i ) = . Ei − EF exp ( Zi kT) + 1 [B.2] In single electron donor semiconductors, an expression similar to [B.1] is also written as (2 )N !(NN −! N )! N × D D D× D [B.3] D× where N D × is the density of occupied (and hence charge neutral) defect states. From [B.1] and [B.3] one derives 201 N D× ND = f (E D ) = 1 2 exp( 1 E D −E F kT )+ 1 . [B.4] The pre-exponential factor in the denominator of [B.4] is associated with the factor 2 N D × in [B.3] and is a consequence of electron spin degeneracy. When solving Poisson’s equation, the quantity of interest is actually N D + , the density of ionized defect states. This may be obtained by a bit of manipulation of [B.4] as follows. By first taking the inverse of [B.4] and adding -1 to both sides obtains N D − ND × ND × = 1 E − EF exp D . kT 2 Next, taking the inverse again and adding +1 to both sides obtains N D× + N D+ N D+ = 1 + 2exp The numerator in this expression is simply ND. EF − ED . kT Taking another inverse and expressing the energy terms in a reduced form (see section 3.2.4), the desired expression may be written as N D+ ND = 1 . 1 + 2exp[uz − wD1,I ] [B.5] In double electron donor semiconductors, a complication arises from the fact that the occupancies for the singly-ionized states and for the neutral states are mutually dependent upon each other [1]. The problem now requires the derivation of a Fermi function that not only determines how the existing states will be filled, but also which states exist to be filled. Figure B.1 illustrates this point and serves as the working hypothesis of the following derivation. 202 ECB ED2 ED1 a Figure B.1 b a) A defect-related state occupied by two electrons of opposite spin; b) the same defect-related state with one electron removed to the conduction band. The second electron requires considerably more energy to also make a transition to the conduction band. Figure B.1 represents the conditions presented by the presence of oxygen vacancies in oxides such as SrTiO3 or TiO2. It is assumed that these defect states can accept two electrons to satisfy their valency. The first electron can be accepted as either spin up or spin down into a state described by ionization energy ED1. The second electron can be accepted with only spin up or only spin down into a state described by ionization energy ED2 (< ED1). The total number of defect-associated electrons N are accounted for by the number contained in the conduction band ∞ ∑ N , the number contained in singly-ionized states N i =1 i the number contained in neutral states 2ND × , so that ∞ N = 2N D× + N D + + ∑ Ni . i =1 Another conservation rule is 203 [B.6] D+ , and ∞ U = 2N D× E D2 + ND + E D1 + ∑ Ni E i , [B.7] i =1 and the total number of donor states, ND, consists of the sum of neutral, singly-ionized and doubly-ionized states; i.e., N D = ND × + ND + + ND ++ . [B.8] Now, the total number of different ways to occupy ND states with a single electron is given by (2 )N !(NN −! N )! . N + D D D+ [B.9] D+ D The total number of different ways to occupy N D − ND + states with an electron pair is given by (N − N )! . !(N − N − N )! D N D× D+ D+ D [B.10] D× Therefore, the thermodynamic probability for a double-donor semiconductor is written as D + ( ( ) ND − N D+ ! ∞ Z i! N D! . [B.11] ∏ N D + ! N D − ND + ! ND × ! ND − N D+ − N D× ! i =1 Ni !(Z i − Ni )! ( ) N W= 2 ( ) ) Maximizing [B.11] subject to the constraints of [B.6] and [B.7] yields the desired distribution functions. That is, one must solve ∂H ∂X = 0, where Xi are the variables N i , N D + , N D × , λ, and i µ, and H = ln W . Using Stirling’s formula, ln A!≈ A ln A − A , so that d dA ln A!≈ ln A , one must solve the five separate equations ∂H ∂H ∂H ∂H = ∂H = 0 . ∂Ni = ∂ND + = ∂ND × = ∂λ ∂µ The last two equations just return the conservation rules [B.6] and [B.7]. The first equation gives [B.2]. The second and third equations give ND + N D − ND × = 1 2 exp( 1 E D1 − E F 204 kT )+1 [B.12] and ND × N D − ND + = [ exp 2( 1 E D2 −E F kT )]+ 1 , [B.13] respectively. Using [B.8], the LHS of both [B.12] and [B.13] may be re-written as and ND × N D ++ + ND × ND + N D ++ + ND + , respectively, so that taking the inverse of both gives N D ++ ND + = 1 E − EF exp D1 kT 2 [B.14a] and N D ++ ND × = exp 2E D 2 − 2E F . kT [B.14b] Dividing [B.14b] by [B.14a], adding +1 and taking the inverse gives f (E D2 ) = ND × ND + + N D× = 1 2exp ( 2E D2 − E D1 −E F kT )+ 1 . [B.15] This is the occupation probability of neutral states. The occupation probability of singly-ionized states is given by [B.12] and is re-written below as f (E D1 ) = N D+ ND ++ + ND + = 1 2 1 exp( E D1 − E F kT )+1 . [B.16] The re-casting of [B.12] and [B.13] in the respective forms [B.15] and [B.16] was done for the convenience of application to Poisson’s equation, in which case the quantities of interest are N D + and N D ++ . For convenience let E ′ = 2E D 2 − E D1 . Also let fD1 = N D+ + N D − ND × = 2 exp( 1 E F − E D1 and 205 kT )+1 [B.17] fE ′ = N D+ ND − N D ++ = 1 2 exp( 1 EF − E′ kT )+ 1 , [B.18] where [B.17] and [B.18] are obtained in a straightforward manner from [B.15] and [B.16]. In a similar way, it can be easily shown that N D× = 1 ND + exp (E F −E ′ kT). 2 [B.19] Substituting [B.19] into [B.17] gives a relation that, together with [B.18], provide two independent equations with two unknowns — namely, N D + and N D ++ . The results of the necessary algebraic manipulations give 1 − fD1 N D + = N DfE ′ 1 − fD1 + fE ′fD1 [B.20] fE ′ fD1 N D ++ = ND . 1 − fD1 + fE ′fD1 [B.21] and Equations [B.20] and [B.21] are used in Poisson’s equation for determining the relation between band bending and space charge per unit surface area. In [3.33], the donor charge density is given by ρ D = 2eN D++ + eN D + or f −f f ρ D = eND E ′ E ′ D1 . 1 − fD1 + fE ′fD1 REFERENCES 1. Dr. E. Spenke Elektronische Halbleiter Springer-Verlag, Berlin 1965 206 [B.22] APPENDIX C: MATHEMATICA CODE FOR MODELED TUNNELING SPECTRA The following code was written to generate a simulation of the tunneling current-voltage behavior of a n-type semiconducting material in depletion probed by a metal counter–electrode. It will execute successfully on a PC or workstation running Mathematica version 2.2.2 or 3.0. All modeling of experimental spectra in this thesis were executed on a PC running version 3.0 at 400 MHz. The calculations were completed within 10 minutes to 2 hours, depending on the average magnitude of the tunneling current. Larger currents result in longer calculation times. The output of this code gives: a) the total dynamic band bending function; b) the potential distribution functions; c) the predicted experimental dynamic band bending function; d) a linear current-voltage plot of the result; and e) a semi-log current-voltage plot of the result. (* Output control *) Off[General::spell1]; (* turn off similar spelling warning *) Off[NIntegrate::precw]; (* turn off less than precision warning *) Off[NIntegrate::slwcon]; (* turn off slow convergence warning *) Off[NIntegrate::ncvb]; (* turn off non-convergence warning *) Off[FindRoot::cvnwt]; (* turn off non-convergence warning *) Off[FindRoot::precw]; (* turn off less than precision warning *) Off[InterpolatingFunction::dmwarn]; (* turn off domain size warning *) Off[Plot::plnr]; SetDirectory["user files:Asa:mathematica:c results"]; 208 (* define physical constants *) jeV = 1.60217733 * 10^-19; (* joules to electron volts conversion factor *) k = (1.380658 * 10^-23/jeV); (* Boltzmann's constant; eV/K *) hbar = (1.05457266 10^-34/jeV); (* reduced Planck's constant; eV s *) h = 2 Pi hbar; (* Planck's constant; eV s *) m0 = 9.1093897 * 10^-31/jeV; e = 1.60217733 * 10^-19; (* free electron mass; eV s2/m2*) (* fundamental unit of charge; C *) ep0 = 8.854187817 * 10^-12; (* permittivity of free space; F/m = C/(V m) *) (* define variables *) temp = 300; (* temperature; Kelvins *) s = 9; (* tunneling gap width; angstroms *) r = 5; (* radius of curvature of tip; angstroms *) mceff = 12; (* conduction band relative effective mass *) mveff = 0.99; (* valence band relative effective mass *) mmeff = 0.99; (* metal electrode relative effective mass *) kappasc = 300; (* semiconductor static dielectric constant *) kappaschf = 5; (* semiconductor high frequency dielectric constant *) kappain = 1; kappainhf = 1; nd = 1.0 * 10^19; (* insulator static dielectric constant *) (* insulator high frequency dielectric constant *) (* free carrier density; cm-3 *) initBB = 0.30; beta = 0.002; (* initial surface potenial; eV *) (* adjustable parameter for potential distribution *) 209 eGap = 3.2; (* semiconductor band gap; eV *) scfermi = 0; (* semiconductor Fermi level; eV *) wfM = 4.55; (* tungsten metal work function; eV *) eAff = 3.0; emobil = 30; (* semiconductor electron affinity; eV *) (* semiconductor electron mobility; cm2/V s *) numPoints = 100; (* number of points in IV curve *) vaMin = -4; (* starting applied tip bias; V *) vaMax = 4; (* ending applied tip bias; V *) eDC1 = 0.0012; (* ionization energy of donor defect; eV *) (* derive constants *) ceffm := If[mceff>mmeff, mmeff, mceff]; (* eff. masses for integral limits *) veffm := If[mveff>mmeff, mmeff, mveff]; barNc = 2 * (* conduction band effective dos; m-3 *) ((mceff m0 k temp)/(2 Pi (hbar^2)))^1.5; barNv = 2 * (* valence band effective dos; m-3 *) ((mveff m0 k temp)/(2 Pi (hbar^2)))^1.5; eConB := N[scfermi - ((3 (nd * 10^6))/Pi)^(2/3) * (* conduction band edge; eV *) (h^2/(8 mceff m0))] /; nd >= (barNc * 10^-6); eConB := N[scfermi - (k temp Log[(nd * 10^6)/barNc])] /; nd < (barNc * 10^-6); wfSC = eAff + eConB; (* semiconductor work function; eV *) eValB = eConB - eGap; (* valence band edge; eV *) eIntB := N[0.5(eConB + eValB) + 210 (k temp Log[(mveff/mceff)^(3/4)])]; (* intrinsic Fermi level; eV *) eD1 = eConB - eDC1; (* energy of donor state; eV *) debeye = Sqrt[(kappasc ep0 jeV k temp)/(e^2 (nd * 10^6))]; (* Debeye length; m *) ub = (scfermi - eIntB)/(k temp); hevi[t_] := If[t>0,1,0]; (* Heaviside function *) (* derive variables *) wdep[vs_] := Sqrt[(2 kappasc ep0 jeV Abs[bbFunc[vs]])/(nd * 10^6 e^2)]; (* depletion layer width *) phicSC[z_,vs_,longE_] := N[(((wdep[vs] - z)^2/wdep[vs]^2) * bbFunc[vs]) + eConB - longE]; phivSC[z_,vs_,longE_] := N[-1*((((wdep[vs] - z)^2/wdep[vs]^2) * bbFunc[vs]) + eValB - longE)]; phiVac[z_,vi_,longE_] := N[(((wfM - vi - longE) * (z/(s * 10^-10))) + ((wfSC - longE) * (1-(z/(s 10^-10)))) - (((0.4 e^2)/(8 Pi kappainhf jeV ep0)) * ((s 10^-10)/(z ((s 10^-10) - z)))))]; phiczSC[vs_,longE_] := N[((((wdep[vs] - z)^2/wdep[vs]^2) * bbFunc[vs]) + eConB - longE)]; phivzSC[vs_,longE_] := N[-1*((((wdep[vs] - z)^2/wdep[vs]^2) * bbFunc[vs]) + eValB - longE)]; phizVac[vi_,longE_] := N[(((wfM - vi - longE) * (z/(s * 10^-10))) + ((wfSC - longE) * (1-(z/(s 10^-10)))) 211 - (((0.4 e^2)/(8 Pi kappainhf jeV ep0)) * ((s 10^-10)/(z ((s 10^-10) - z)))))]; (* generate equilibrium band bending function -> bbFunc[] *) w[p_,q_] := (p - q)/(k temp); fD1 = (1 + 2 Exp[x-w[eD1,eIntB]])^-1; donorint := NIntegrate[fD1, {x,ub,us}]; xval = (eValB-energy)/(k temp); fdival[j_,eta_]:= Re[(j!^-1)*NIntegrate[(xval^j)/(Exp[xval-eta]+1), {energy,-30,0}, WorkingPrecision->15, AccuracyGoal->10]]; xcond = (energy-eConB)/(k temp); fdicond[j_,eta_]:= Re[(j!^-1)*NIntegrate[(xcond^j)/(Exp[xcond-eta]+1), {energy,0,30}, WorkingPrecision->15, AccuracyGoal->10]]; elecintdeg := (((2 barNc)/(3 nd * 10^6))* ((fdicond[1.5,us-w[eConB,eIntB]])-(fdicond[1.5,ub-w[eConB,eIntB]]))); holeintdeg := (((2 barNv)/(3 nd * 10^6))* ((fdival[1.5,w[eValB,eIntB]-us])-(fdival[1.5,w[eValB,eIntB]-ub]))); elecintnondeg = N[(barNc/(nd * 10^6)) * (Exp[us-w[eConB,eIntB]]-Exp[ub-w[eConB,eIntB]])]; holeintnondeg = N[(barNv/(nd * 10^6)) * (Exp[w[eValB,eIntB]-us]-Exp[w[eValB,eIntB]-ub])]; elecint := elecintdeg /; nd >= (barNc * 10^-6); elecint := elecintnondeg /; nd < (barNc * 10^-6); holeint := holeintdeg /; nd >= (barNc * 10^-6); 212 holeint := holeintnondeg /; nd < (barNc * 10^-6); tvs := N[initBB - (k temp) - (((e debeye^2 nd 10^6)/(kappasc ep0)) * (-1 donorint - elecint + holeint))]; phibbs := N[(ub - us) * (k temp)]; bbCurve = Table[{-tvs,phibbs}, {us,-73,73,1.0}]; bbFunc = Interpolation[bbCurve,InterpolationOrder->1]; bbPlot = ListPlot[bbCurve, PlotJoined -> True, Frame -> True, FrameLabel -> {" Sample voltage,Vs (V)",None}, AspectRatio -> 1, PlotRange ->{{-vaMax,-vaMin},{(-eGap+initBB),(eGap+0.5)}}, AxesLabel -> {None, "Equilibrium Surface Potential (eV)"}]; Clear[bbCurve,us,tvs]; " = extrinsic Debeye length (meters)" debeye; " = reduced bulk potential" ub " = equilibrium depletion width (meters)" wdep[0] " = depletion width to Debeye length" wdep[0]/debeye (* define limits for barrier integrals *) zcbtemp[vs_,longE_] := NSolve[phiczSC[vs,longE] == 0,z]; deltazcb[vs_,longE_] := z /. zcbtemp[vs,longE][[1]]; zccb[vs_,longE_] := If[Re[deltazcb[vs,longE]] > 0,Re[deltazcb[vs,longE]],0]; zvbtemp[vs_,longE_] := NSolve[phivzSC[vs,longE] == 0,z]; deltazvb[vs_,longE_] := z /. zvbtemp[vs,longE][[1]]; zcvb[vs_,longE_] := If[Re[deltazvb[vs,longE]] > 0,Re[deltazvb[vs,longE]],0]; 213 zvactemp[vi_,longE_] := NSolve[phizVac[vi,longE] == 0,z]; zsoltemp[vi_,longE_] := z /. zvactemp[vi,longE][[1]]; vsol[vi_,longE_] := If[Re[zsoltemp[vi,longE]]<0,2,1]; za[vi_,longE_] := z /. zvactemp[vi,longE][[vsol[vi,longE]]]; zb[vi_,longE_] := z /. zvactemp[vi,longE][[(vsol[vi,longE]+1)]]; (* define transmission factor integrals *) phibarVac[z_,vi_,longE_] := If[0<phiVac[z,vi,longE], phiVac[z,vi,longE], 0]; phibarcSC[z_,vs_,longE_] := If[0<phicSC[z,vs,longE], phicSC[z,vs,longE], 0]; phibarvSC[z_,vs_,longE_] := If[0<phivSC[z,vs,longE], phivSC[z,vs,longE], 0]; etaVac[vi_,longE_] := (2 * Sqrt[(2 m0)/(hbar^2)]) * NIntegrate[Sqrt[phibarVac[z,vi,longE]],{z,Re[za[vi,longE]],Re[zb[vi,longE]]}, WorkingPrecision -> 10, AccuracyGoal -> 8]; etacSC[vs_,longE_] := (2 * Sqrt[(2 m0)/(hbar^2)]) * NIntegrate[Sqrt[phibarcSC[z,vs,longE]],{z,0,zccb[vs,longE]}, WorkingPrecision -> 10, AccuracyGoal -> 8]; etavSC[vs_,longE_] := (2 * Sqrt[(2 m0)/(hbar^2)]) * NIntegrate[Sqrt[phibarvSC[z,vs,longE]],{z,0,zcvb[vs,longE]}, WorkingPrecision -> 10, AccuracyGoal -> 8]; 214 (* define energy band integrals *) dVac[vi_,longE_] := Exp[-1 etaVac[vi,longE]]; dcSC[vs_,longE_] := Exp[-1 etacSC[vs,longE]]; dvSC[vs_,longE_] := Exp[-1 etavSC[vs,longE]]; eValBs[va_] := If[bbFunc[(-1*vsFunc[va])] > 0, bbFunc[(-1*vsFunc[va])] + eValB, eValB]; jCB[va_] := N[(4 Pi e m0 ceffm)/h^3] * NIntegrate[hevi[+1 (totE - eConB)]*dVac[viFunc[va],w] *dcSC[(-1*vsFunc[va]),w], {totE,0,-va}, {w,eConB,totE}, WorkingPrecision -> 10, AccuracyGoal -> 8]; 215 jVB[va_] := N[-(4 Pi e m0 veffm)/h^3] * NIntegrate[hevi[-1 (totE - eValBs[va])]*dVac[viFunc[va],w] *dvSC[(-1*vsFunc[va]),w], {totE,eValBs[va],-va}, {w,eValBs[va],totE}, WorkingPrecision -> 10, AccuracyGoal -> 8]; (* define defect induced current *) alpha = 3 * 10^-4; surfE[va_] := N[(-1*vsFunc[va])/wdep[0]]; sbl[va_] := N[(((e^3 nd 10^6)/(8 Pi^2 ep0^3 kappasc kappaschf^2)) * (bbFunc[(-1*vsFunc[va])] + (k temp)))^0.25]; (* barrier lowering *) dI[va_] := N[1*(alpha * temp^1.5 * surfE[va] * emobil * mceff^1.5 * (etaVac[viFunc[va],bbFunc[(-1*vsFunc[va])]]) * Exp[(-1*(bbFunc[(-1*vsFunc[va])] + Abs[eConB] - sbl[va]))/(k temp)] * Exp[(e/(k temp)) * (e/(4 Pi ep0 kappaschf))^0.5 * ((surfE[va])^0.5)])]; jDI[va_] := If[va>=0,dI[va],0]; (* generate potential distribution functions -> viFunc[] and vsFunc[] *) vaFunc := N[((1 - (s 10^-10 kappain^-1 ep0^-1 beta))^-1) * (vs - ((s 10^-10 kappain^-1 ep0^-1) * (Sqrt[(2 e kappasc ep0 nd 10^6) * (initBB - (k temp) - vs)] 216 Sqrt[(2 e kappasc ep0 nd 10^6) * (initBB - (k temp))])))]; vss[va_] := FindRoot[vaFunc == va, {vs,10^1}]; vs[va_] := Re[vs /. vss[va]]; vi[va_] := va - vs[va]; viCurve = Table[{va,vi[va]}, {va,vaMin,vaMax,0.5}]; viFunc = Interpolation[viCurve,InterpolationOrder -> 3]; viPlot = Plot[viFunc[x], {x,vaMin,vaMax}, Frame -> True, PlotRange -> {{vaMin,vaMax},{vaMin,vaMax}}, FrameLabel -> {"Tip Bias, Va (V)",None}, AspectRatio ->1, AxesLabel -> {None, "Insulator voltage, Vi (V)"}]; vsCurve = Table[{va,vs[va]}, {va,vaMin,vaMax,0.5}]; vsFunc = Interpolation[vsCurve,InterpolationOrder -> 3]; vsPlot = Plot[vsFunc[x], {x,vaMin,vaMax}, Frame -> True, PlotRange -> {{vaMin,vaMax},{vaMin,vaMax}}, FrameLabel -> {"Tip Bias, Va (V)",None}, AspectRatio ->1, AxesLabel -> {None, "Sample voltage, Vs (V)"}]; Clear[vsCurve,viCurve]; Show[vsPlot,viPlot, PlotRange -> {{vaMin,vaMax},{vaMin,vaMax}}, AxesLabel -> {None,None},AspectRatio -> 1]; surfPot = Table[{-va,bbFunc[(-1*vsFunc[va])]}, {va,vaMin,vaMax,0.05}]; surfPotPlot = ListPlot[surfPot, PlotJoined -> True, Frame->True, FrameLabel -> {"Sample Bias, Va (V)", None}, PlotRange -> {{-vaMax,-vaMin},{(-eGap + initBB),(eGap + 0.3)}}, 217 AxesLabel -> {None, "Surface Potential (eV)"},AspectRatio->1]; (* define total current density *) current[va_] := (N[jCB[va]] + N[jVB[va]] + N[jDI[va]]) N[Pi] r^2 10^-20; 218 (* generate iv curve *) i[v_] := Re[current[v]]; ivCurve = Table[{(-1*v),i[v]}, {v,vaMin,vaMax,((vaMax - vaMin)/numPoints)}]; ivCurve >> sro2_22; (* name of file to save results*) Clear[ivCurve]; rawdata = << sro2_22; linearspectra=Interpolation[rawdata]; vmin = (-vaMax); vmax = (-vaMin); Plot[(linearspectra[x]*10^9),{x,vmin,vmax},PlotRange->{{vmin,vmax},{-10,+10}}]; Plot[Log[10,Abs[(linearspectra[x]*10^9)]],{x,vmin,vmax},AxesOrigin->{vmin,-1.6}, PlotRange->{{vmin,vmax},{-1.6,1}},Frame->True,AspectRatio -> 1]; 219