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Thermal and electrical properties of semiconductors measured by means of photopyroelectric and photocarrier radiometry techniques. DISSERTATION zur Erlangung des Grades „Doktor der Naturwissenschaften“ an der Fakultät für Physik und Astronomie der Ruhr-Universität Bochum von Michał Pawlak aus Bydgoszcz Bochum 2009 1. Gutachter: Prof. Dr. Josef Pelzl 2. Gutachter: Prof. Dr. Andreas Wieck Datum der Disputation: 18.01.2010 ii Acknowledgements First and foremost, I would like to thank Prof. Dr. J.Pelzl for his kind supervision, valuable guidance, helpful comments on this research and also for his full support during this two very hard years I have had. From the group of many researchers that I was fortune to work with, I would like to thank prof A. Mandelis who I have participated with in developing of the photocarrier radiometry in laboratories in Bochum as well in Toronto, Canada, as well prof. H.Meczynska for her support and valuable advices. During my visit in Toronto I worked also with Dr. J.Tolev and I want to thank him for his support and a lot of valuable guidance in photocarrier and photothermal radiometry. Also, I would like thank Prof A. Wieck, head of the GRK 348 program, for opportunity to participate in interesting lectures and projects related with micro- and nanoelectronics. Many thanks go to my colleagues from Ruhr University Bochum (Mr B.K.Bein, Dr. D. Dietzel, Dr. R. Meckenstock, Dr D.Spoddig, Mr. J.Gibkes, Mr M. Mueller, Dr S.Chotikaprakhan), for their helpful advices and discussions about Germany culture during my stay in Bochum. Many thanks go to my colleagues from Nicolaus Copernicus University in Torun, Poland. Especially I would like thank prof F.Friszt, prof. S.Legowski for their valuable comments, Dr J.Zakrzewski for familiarizing me with photoacoustic science and Mrs.A.Marasek, Dr J.Szatkowski and Dr K.Strzalkowski for produced the CdMgSe single crystals used in this work. iii Many thanks go to my colleagues from University of Toronto (Dr Tolev, Dr. X.Gou, Dr J.Garcia) and also to Prof. Mihai Chirtoc (from University of Reims, France) for useful advice during the construction of the PPE experimental setup. Also, I would like to thank prof. M.Malinski from Technical University of Koszalin, Poland for his advices and helpful comments. I want to thank my parents and Grandfather for their support throughout all of my life especially during last two years. Finally, I would like to thank Malgosia for her love and encouragement, as well as her patience during the many hours I spent completing this work. Bochum, October 2009 Michal Pawlak iv Abbreviations 1-D one dimensional 3-D three dimensional BNC Bayonet Neill-Concelman CDW carrier density wave CdSe cadmium selenide DSC differential scanning calorimetry IPPE inverse PPE GaAs Gallium arsenide GaN Gallium nitride Ge-Si Germanium silicon alloys Mg Magnesium MgSe Magnesium selenide NIR near infrared PAS photoacoustic spectroscopy PPE photopyroelectric technique PPT photothermal piezoelectric technique PZT lead zirconate titanate SCL space charge layer Si Silicon Si -Si02 silicon-silicon dioxide interface TW thermal wave J0 Bessel function of first kind of order 0 J1(λr) Bessel function of first kind of order 1 PL Photoluminescence v Nomenclature Symbol Unity -2 Name Nt [m ] charged interface state density energy Et. W0 [m] space charge layer width I 0 (β , hν ) [W m-2] optical intensity ∆W [m] effective SCL width W0 [m] dc component of the SCL Wm [m] modulated component of the SCL D* [m2/s] ambipolar diffusion coefficient τ [s] bulk recombination lifetime qψ s 0 [J] interface potential energy Tri(ω) [s] complex interface lifetime τri [s] charged interface recombination lifetime r r F (r , t ) [W/m2] heat flow ρ [kg/m3] mass density C [J/kg·K] specific heat k [W/mK] thermal conductivity [J] heat source [K] temperature distribution (or field) Dt [m2/s] thermal diffusivity T0 [K] ambient temperature [K] steady temperature distribution [K] temporal temperature distribution [m-1] thermal wave number r Q(r , t ) r T (r , t ) r Tdc (r ) r Tac (r , t ) σt vi e [Ws1/2/m2K] thermal effusivity µ th [m] thermal diffusion length Eph [J] photon energy EG [J] energy band gap h [J s] Planck constant v -1 [s ] wave frequency r QIB (r , t , λ ) [J/m3] intraband heat release rate per unit volume ηG [m-3] quantum efficiency for photogenerated carriers α (λ ) [m-1] optical absorption coefficient at the excitation wavelength λ N 0 (t ) Lj photon deposition rate per volume [m] diffusion length, where j=n (electron) or p (hole) η FS front surface non-radiative quantum efficiency η BS rear surface non-radiative quantum efficiency N (0, t , λ ) [m-3] photogenerated electron density distribution in a one dimensional geometry N0 [m-3] equilibrium density SFS [m/s] front recombination velocity SBS [m/s] rear recombination velocity N(r,t) r g (r , t ) [m-3] concentration of the excess electron-hole pairs [m-3] carrier generation rate per unit volume from an external source of excitation µe [m2/Vs] electron mobility µh [m2/Vs] hole mobility σe [m-1] carrier density-wave wave number ηQ quantum yield of the photogenerated carriers t [s] Time ω [s-1] angular frequency f [s-1] Frequency ϕ [rad] Phase σ [J/K] Stefan-Boltzmann constant vii τR [s] radiative recombination lifetime τNR [s] non-radiative recombination lifetimes q [C] Charge B [m3/s] radiative recombination probability W [m] laser beam of a spot size EC [J] energy of the conduction EV [J] energy of the valance band edge p [C/m2K] pyroelectric coefficient of the detector A [m2] transducer area WCdSe [mK/W] thermal resistivities of CdSe WMgSe [mK/W] thermal resistivities of MgSe CCd-Mg [mK/W] nonlinear parameter R Reflectivity c [m/s] speed of light m* [kg] effective mass n reflactive index viii Contents Acknowledgements………………………………………………………………………..…iii Abbreviations and Nomenclature……………………………………………………..……..v Contents…………………………………………………………………………………........ix Chapter 1: Introduction……………………………………………………………………...1 1.1 Motivation and objectives………………………………………………………………….1 1.2 Organization of the thesis………………………………………………………………….2 Chapter 2: Physical Principles of the thermal waves………………………………………3 2.1 Historical …………………………………………………………………………………..3 2.2 Heat conduction equation ……………………………………………………………….…4 2.3 Review of the photothermal methods……………………………………………………...7 Chapter 3: Physical basics of the carrier density waves in semiconductors…………..…11 3.1 De-excitation processes in semiconductors………………................................................11 3.2 Ambipolar diffusion equation…………………………….................................................14 3.3 One dimensional excess CDW field in Cartesian geometries………..…………………...15 3.4 Three dimensional CDW field in cylindrical geometries………………………………...16 3.5 Recombination processes in semiconductors……………………………………………..18 Chapter 4: Experimental methods, signal generation mechanisms and instrumentation of PCR and PPE methods…………………………………………………………………..21 4.1 The Photocarrier Radiometry (PCR) signal generation mechanism and instrumentation..21 4.1.1. Introduction…………………………………………………………………………….21 4.1.2. Contribution to the PCR ……………………………………………………………….22 4.1.3. Instrumentation and normalization of PCR signals……………………………………24 4.1.4. The one dimensional Photocarrier Radiometry Signal ………………………………..25 4.1.5. The three dimensional PCR signal……………………………………………………..30 4.1.6. PCR Dimensionality criterion………………………………………………………….31 4.1.7 Photo-carrier radiometry microscope…………………………………………………...33 4.2 Photopyroelectric effect (PPE)……………………………………………………………36 ix 4.2.1. Experimental setup …………………………………………………………………….36 4.2.2. The PPE signal generation …………………………………………………………….37 4.2.3. Normalization of the PPE signal……………………………………………………….40 Chapter 5: Thermal properties of Cd1-xMgxSe single crystals measured by means of photopyroelectric technique………………………………………………………………..50 5.1. Materials…………………………………………………………………………………50 5.2. Experimental results and computational algorithm……………………………………...52 5.2.1 Thermal diffusivity – PPE phases………………………………………………………52 5.2.2 Thermal conductivity – Normalized PPE amplitude…………………………………...55 5.2.3 Discussion………………………………………………………………………………56 Chapter 6: Influence of the space charge layer (SCL) on the charge carrier transport properties measured by means of the photocarrier radiometry (PCR)…………………61 6.1 Theory of optically modulated p-type SiO2-Si interface energetics in the presence of charged interface states ………………………………………………………………………61 6.2 The expression of the PCR signal including effects due to an existing SCL……………..64 6.3 Numerical simulations of an influence of the existence of the SCL on the electronic transport properties…………………………………………………………………………...65 6.3.1 Numerical simulation of the PCR signal dependence on the electrical transport properties in the presence of SCL……………………………………………………………65 6.3.2 Numerical simulation of the PCR signal dependence on the existence of SCL width…68 6.4 Experimental conditions and materials……………………………………………...……71 6.4.1 Experimental methodology……………………………………………………………..71 6.4.2. Experimental set up…………………………………………………………………….72 6.4.3. Materials………………………………………………………………………………..73 6.5 Experimental results………………………………………………………………………73 6.5.1 Effect of chemical etching on the PCR signal………………………………………….73 6.5.2 The perturbation effects of the primary modulated laser beam on the PCR signal…….75 6.5.3 The effect of polishing on the PCR signal……………………………………………...76 6.6 Determination of carrier transport properties in SCL and the depth profile reconstruction…………………………………………………………………………………77 6.7 Summary………………………………………………………………………………….82 x Chapter 7: Non-linear dependence of photocarrier radiometry signals from p-Si wafers on optical excitation intensity and its effect on charge carrier transport properties…...84 7.1 Introduction……………………………………………………………………………….84 7.2 Experimental methodology and materials………………………………………………...87 7.2.1 Low resolution PCR system…………………………………………………………….87 7.2.2 High resolution PCR system……………………………………………………………89 7.2.3 Materials………………………………………………………………………………..90 7.3. Numerical simulations of the PCR signal as a function of the non-linear coefficient β and photo-injected carriers………………………………………………………………………..90 7.4 Experimental results and discussion……………………………………………………..93 7.4.1 Laser power dependencies ……………………………………………………………..93 7.4.2. Modulation frequency dependence at the low resolution system……………………...99 7.4.3. Modulation frequency dependence at 532 nm…..……………………………………108 7.5 Summary…………………………………………………………………………….…..112 Chapter 8: Conclusion and outlook ………………………………………………………113 Bibliography………………………………………………………………………………..116 Curriculum Vitae and conference contributions………………………………………...121 Appendix A: Current controller…………………………………………………………..128 xi xii Chapter 1: Introduction 1.1 Motivation and objectives Nowadays, the trend is increasing to develop semiconductor nano-devices. To obtain high quality appropriate substrates are required. A good choice of the substrate in electronic or opto-electronic nano-devices is detrimental task of the design process. Hence, monitoring of quality of the substrate during technological process is very important task. Whereas the development of characterization strategies capable of evaluating the effects of the bulk substrate Si properties on the performance of microelectronic devices is an issue of growing importance, as the electronic properties of the bulk can seriously affect the electrical characteristics of the device [Schroder, 1997]. For these reasons on-line techniques able to measure the electrical properties are required. On the other hand, the thermal management is also a very important aspect. For instance, thermal conductivity is an important parameters that determines the maximum power at operation of semiconductor devices. For semiconductors used in thermoelectric energy conversion the thermal conductivity is one of the most important parameters 1 determining the efficiency of the device [Tritt, 2004]. For miniaturized semiconducting devices thermal management of the energy dissipation has become a key problem. In this context, the thermal diffusivity is a very important physical parameter in device modeling. It is a parameter specific for each material, which dependents on the composition and structural characteristic of the sample. 1.2 Organization of the thesis The following chapters in this thesis are organized as follows: Chapter 2 introduces the basic concepts of thermal waves and reviews the different experimental techniques. Chapter 3 describes on the basic concepts of carrier density waves in semiconductors. Different recombination processes in semiconductors are discussed. Chapter 4 reports on the experimental set-ups which are constructed during the work, such as photopyroelectric (PPE) and photocarrier radiometry (PCR). This chapter shows also numerical simulations of the photopyroelectric as well photoradiometry signals. Changes depending on the experimental conditions and the constructed experimental set-ups are discussed. Chapter 5 discusses the results of the frequency-dependent and temperature-dependent measurements on Cd1-xMgxSe single crystals by means of photopyroelectric technique. For this purpose the photopyroelectric cell with Peltier element was constructed. Chapter 6 is devoted to the development of photocarrier radiometry. The two-laser beam PCR system was constructed and the experimental verification of the Mandelis theory [Mandelis, 2005a] is presented. Chapter 7 reports on the photocarrier radiometry experimental results of the frequencydependent and intensity-dependent measurements on silicon wafers. In this chapter the nonlinear parameter is introduced to take into account the nonlinearities phenomena. Chapter 8: summarizes the results of this thesis and presents some propositions for interesting directions of future work. 2 Chapter 2: Physical principles of the thermal waves 2.1 Historical Thermal wave (TW) is an temperature distribution oscillating in time and space representing a continuous energy dissipation [Mandelis, 2001]. Thermal waves were used first by A. Angström in the mid 19th century. In 1861 Ångström [Ångström, 1861] had reported determination of the thermal diffusivity of the long copper bar by means of detection and interpretation of the periodical heating from the investigated material. If the thermal wave is excited by the photons the thermal response is named: photothermal effect. Graham Bell and his co-workers were the first who observed that sunlight modulated by chopper incident on a strongly absorbing substance causes audible sound emitted from the substance [Bell, 1880]. It was almost a century later when Rosencwaig and Gersho explained Bell’s photoacoustic experiment in the frame of the thermal waves [Rosencwaig and Gersho, 1976]. Since then, 3 much more attention to the photothermal effect was attributed and a lot of new experimental techniques based on the effect were developed. 2.2 Heat conduction equation When temperature gradient exists in a material then a heat transfer from places with higher temperature to places with lower temperature is observed. There are three distinct methods of this transfer: conduction, convection and radiation. In most solid state problems conduction is the most important process of the thermal energy transfer. Mathematically the heat conduction process is described by heat diffusion equation, which is simply an expression of the energy conservation principle. In the case of the isotropic homogeneous solid the general form of the heat conduction equation in Cartesian co-ordinate is given by [Carslaw and Jaeger, 1959]: r r r r r ∂T (r , t ) ρC = −∇ ⋅ F (r , t ) + Q(r , t ) , ∂t r r where F (r , t ) is the heat flow which is defined in Fournier’s law: r r r F (r , t ) = − k∇T (r , t ) . (2.1) (2.2) Here, ρ , C, k are mass density, specific heat and thermal conductivity, respectively. The negative sign indicates the direction of heat flow from hot to cold areas. By appropriate r boundary conditions and the strength and localization of the heat source Q(r , t ) , the r temperature distribution (or field) T (r , t ) can be evaluated from solution of the heat diffusion equation. Inserting (2.2) to (2.1) and ordering particular terms equation (2.1) becomes r r r 1 ∂T (r , t ) Q(r , t ) 2 ∇ T (r , t ) − =− , Dt ∂t k (2.3) where Dt is a thermal diffusivity of the solid and is defined by Dt = k . ρC The heat sources caused increase the temperature inside material: r r r T (r , t ) = T0 + Tdc (r ) + Tac (r , t ) , (2.4) (2.5) 4 r r where T0, Tdc (r ) and Tac (r , t ) are the ambient temperature, steady and temporal temperature distribution due to the heat sources in material, respectively. Assuming that the temporal r Fournier transform of T (r , t ) exist one can write [Mandelis, 2001] ∞ θ (r , ω ) = ∫ T (r , t ) ⋅ e −i⋅ω⋅t dt r r (2.6) −∞ and taking the Fournier transform of Eq. (2.3) yields [Mandelis,2001] r ∞ ∞ r2 ∞ r r 1 ∂T (r , t ) −iωt 1 −i ⋅ω ⋅t ∇ ∫ T (r , t ) ⋅ e dt − e dt = − Q(r , t )e −iωt dt ∫ ∫ Dt −∞ ∂t k −∞ −∞ (2.7) leads to the transformed equation r r r Q(r , ω ) 2 ∇ θ (r , ω ) − σ t θ (r , ω ) = − , k (2.8) iω ω = (1 + i ) [m-1] Dt 2 Dt (2.9) 2 where the definition σ t (ω ) = was used. Mandelis [Mandelis, 2001] has proven that in the special case where the heat source is harmonically modulated at angular frequency ω0 the equation is valid ∞ r r r T (r , t ) = ∫ θ (r , t )e iωt dω ≡ T (r , ω 0 ). (2.10) −∞ Therefore the heat diffusion can be re-written as follows: r r r Q(r , ω ) 2 ∇ T (r , ω ) − σ t (ω )T (r , ω ) = − k 2 (2.11) where simple changing the symbol ω0 back to ω. In one-dimensional geometry the heat conduction equation can be written d Q ( x, ω ) T ( x, ω ) − σ t2 (ω )T ( x, ω ) = − . dx k (2.11a) Mandelis [Mandelis, 2001] used the Green function method and homogeneous Neumann boundary condition at x0 = 0 to solve the heat conduction equation (2.11a): Dt T ( x, ω ) = F0 ⋅ k⋅ ω −σ ⋅x +i⋅(ω⋅t ) ⋅e t (2.12) where F0 = η ⋅ I 0 . In fact the thermal wave field is given by the real part of (2.12) Dt T ( x, ω ) = F0 ⋅ k⋅ ω −σ (ω )⋅x π ⋅e t ⋅ cos ω ⋅ t − − σ t (ω ) ⋅ x 4 (1.13) 5 From the structure of the thermal wave formula one can deduce that the physical meaning of the earlier defined parameter σt (the real part of the definition (2.9)) is related with wave-like behavior and can be named the thermal wave number [m-1]. In addition the ratio k is Dt another an important thermal parameter: a thermal effusivity e which is the relevant parameter for time-varying heating or cooling processes of surfaces and heat transport across composite layered bodies and can be also written as e = kρC (2.14) The thermal diffusivity-(2.4)-describes the rate at which heat distributed in a material. High values of the thermal effusivity lead to low surface temperature oscillations while high values of the thermal diffusivity contribute to a relatively deeper penetration of the thermal wave. Main features of the thermal waves can be deducted from (2.13). As compared to normal-wavelike behavior, the thermal waves are very heavily damped with a decay length which is the reciprocal of the real part of the thermal wave-number (2.9) µ th = 2 Dt ω (2.15) µ th is known as thermal diffusion length [m]. The depth to which the thermal waves can penetrate increase with the square root of the thermal diffusivity of a material (if D is high then waves reach deeper region in a material) and with the reciprocal of the square root of the modulation frequency of the heating (if the frequency is low then waves penetrate in deeper region of the material). This profilometric feature gives the thermal waves methodology great attention in science and technology. Quantitatively, along the distance of the thermal diffusion length the thermal wave is damped by 1 / e = 0.386 of its beginning value. This parameter therefore defines the range of effective use of the thermal wave technique [Almond and Patel, 1996]. The phase lag between thermal wave field described by (2.13) and the optical modulation heating is given by ∆ϕ = ω 2 Dt ⋅x+ π 4 (2.16) ∆φ increases linearly with the propagation distance x of the thermal wave. The phase lag shows also that the thermal waves are highly dispersive, because the high frequency thermal waves propagate faster than low frequency thermal waves. 6 2.3 Review of the Photothermal Methods The periodic heating of the sample modifies also other physical properties of the sample. These resulting modifications which oscillate at the same frequency as the heating can be used to detect the thermal wave propagation in the sample. Figure 2.1 illustrates schematically the different physical properties and parameters used to detect the thermal wave response. Figure 2.1 When a modulated laser beam strikes a surface, it generates a thermal wave field, which, in turns, causes a refractive index gradient to appear, IR emission, acoustic wave generation or propagate through the material [Mandelis, 2000] Based on the parameters shown in Fig.2.1 a variety of experimental techniques have been developed to measure the photothermal effect: The most important ones are sketched in Fig.2.2. 7 Figure 2.2 The schematic representation of the different configuration in photothermal techniques [Pelzl and Bein, 1990] The photoacoustic effect relies on measurements of the pressure fluctuation, induced in the gas volume by the heat flux across the solid/gas interface, by means of a microphone mounted inside the cell (Figure 2.2a). The first theoretical explanation of the photoacoustic signal generation in a solid state was given by [Rosencwaig and Gersho, 1976]. They found that photoacoustic signal is proportional to the average of the local modulated temperature rise resulting from optical heating. Based on their work the photoacoustic spectroscopy (PAS) was established and it was found that PAS enables to be used on a broad range of materials such as solids [Murphy and Aamodt, 1977, Pelzl and Bein, 1992], gases [Meyer and Sigrist, 1990, Harren et al., 2000], semiconductors. The limitation of this method lies in problems with enclose the sample in a photoacoustic cell. 8 The photopyroelectric effect is bound up with generate the electrical potential when a material, which has a pyroelectric feature, is heated or cooled. This effect is used in a photopyroelectric technique (PPE) where the sample is heated by absorption of a modulated light. Then a direct thermal contact is performed by placing a pyroelectric sensor at the rear (normal PPE) or at the front (IPPE-inverse PPE) of the sample. The temperature changes from the sample reach the pyroelectric sensor where are converted to current and measurements by means of lock in detection. A theoretical explanation was given by [Mandelis and Zver, 1985] and [Chirtoc and Mihailescu, 1989]. The piezoelectric effect (and detection) relies on generation of a voltage in response of applied mechanical stress. In the photothermal piezoelectric technique (PPT) sensor is connected with a sample by means of a metallic hemi-sphere which can collect stresses generated in a sample [Zakrzewski, 2003]. The photothermal piezoelectric technique was used in investigation of the optical and thermal properties of semiconductor. Infrared emission (photothermal radiometry) relies on the Stefan-Boltzmann law which connects the energy of the emitted radiation E with emitter’s temperature T. Applying this law to the photothermal methodology where only an ac component of the temperature distribution can be monitored the Stefan-Boltzman law becomes dE = 4εσT 3Tac (2.17) where σ is the Stefan-Boltzmann constant, and ε is the emissivity of the material. The photothermal methods used this phenomena is called photothermal radiometry or infrared radiometry was purposed by [Nordal and Kanstad, 1979]. This technique is non-contact and non-destructive hence is applied to broad range of materials. Mirage effect (photothermal beam deflection) is based on the changes of the refractive index of the surrounding gas due to the thermal waves propagating from solid state into the gas. The thermal wave is excited by pump laser within a solid while a second laser beam laser beam probe the gradient of the refractive index perpendicular and parallel to the sample surface. This technique was first proposed by Boccara et al. (1980) who used position sensitive detectors such as quadrant or lateral diodes to measure the deflection angle down to 10-8 radians. 9 Photothermally modulated optical reflection relies on the changes of the optical reflection by the thermal waves. A second laser beam can be used to measure changes of the reflection index of the surface [Rosencwaig, 1985]. The measured signal provides a relationship between the temperature dependence of the optical reflectivity [Gruss et al., 1997, Schaub, 2001] and electrical properties in the case of semiconductors [Fournier, 1992, Kiepert et al. 1999, Dietzel, 2001, Fotsing, 2003, Dietzel et al., 2003a]. This technique is used in industry for inspection of wafers due to the fact that is rapid, non-contact and non-destructive. 10 Chapter 3: Physical basics of the carrier density waves in semiconductors 3.1 De-excitation processes in semiconductors When the energy of incident photons, E ph = hν , is greater than the energy band gap of a semiconductor E ph > EG , then electrons from the valance band to the conduction band are excited (Fig.3.1.) Figure 3.1 n-type semiconductor energy-band gap diagram showing excitation and recombination processes. Energy emission processes include nonradiative intraband and interband decay accompanied by phonon emission, as well as, direct band-to-band recombination radiative emissions of energy hν (λG ) and band – to –defect/impurity-state recombination IR emissions of energy hν IR (λ D ) [Mandelis, 2003]. 11 The energy difference hν − EG is deposed in the kinetics energy of the electrons. This energy is gradually lost on collisions with other carriers and lattice phonons until thermal equilibrium is achieved. This process is named the thermalization (process (a) in Fig 3.1) and as a result creates a heat source and can be accurately described by the optical absorption distribution [Mandelis, 1998]: r r W QIB (r , t , λ ) =η G N 0 (t )(hv − E g ) ⋅ exp(− α (λ ) r ) 3 , m (3.1) r where QIB (r , t , λ ) is the intraband heat release rate per unit volume of the optically excited semiconductor, ηG is the quantum efficiency for photogenerated carriers, α (λ ) is the optical absorption coefficient at the excitation wavelength λ, and N 0 (t ) is the photon deposition rate per volume. Assuming an average intraband relaxation time to be in order of 10-12 [s] [Bandeira et al.,1982], effects due to intraband thermalization may be neglected on the time scale of the conventional frequency domain photothermal response of the semiconductor as is seen in Figure 3.2. Figure 3.2 shows discussed de-excitation processes as a function of time. Figure 3.2 De-exctitation processes in semiconductor as a function of time. After the thermalization electron (hole) to the bottom (the top) of the conduction band (the valance band) electrons and holes create electron-hole pairs. The energy of these pairs can be changed to other existing type of energy (as e.g. heat) on several processes depending on the type of the energy bandgap (direct or indirect), defects and/or concentration of these pairs. Photo-exited carriers can diffuse (process 3 in Figure 3.2) through a distance called a 12 diffusion length L j = D jτ j (where j=n (electron) or p (hole)) and then recombine radiatively or non-radiatively through the energy bandgap (processes b and d in Figure 3.1 and process 4 in Figure 3.2) with respect to theirs lifetime τ which can be written 1 τ = 1 τR + 1 τ NR , (3.2) where τR and τNR are lifetimes related with radiative and non-radiative processes, respectively. In a semiconductor with a direct bandgap (e.g. CdSe) the emission of photons is a result of the radiative recombination of photo-excited pairs (in fact in CdSe electrons and holes create an excition). In case of semiconductors with an indirect energy bandgap (as silicon or germanium) the emission of photons is accompanied with phonons (process d on Figure 3.1). It is also non-zero probability that excess electron-hole pairs can recombine through nonradiative bulk interband transition whence generate a heat source [Quimby et al., 1980]. The heat release rate per unit volume due to non-radiative recombination is given according Bandiera et al. [Bandiera, 1982], Thielemann and Rheinlaender [Thielemann et al., 1985] by: r r W QBB (r , t ; λ ) = η Gη NR N 0 (t )EG exp(− α (λ ) r ) , only relevant when hν ≥ EG 3 m (3.3) where η NR is the non-radiative quantum efficiency. Besides above discussed processes photo-excited carriers can nonradiatively recombine at the semiconductor surface generating another heat source. This surface heat release rates per unit area is given by Flasier and Cahen [Flasier et al, ] and Bandeira [Bandiera et al.,1982] for a front surface: QFS (0, t , λ ) = η FS [N (0, t , λ ) − N 0 ]S FS EG (3.4a) QBS (L, t , λ ) = η BS [N (L, t , λ ) − N 0 ]S BS EG (3.4b) and for a rear surface: where η FS ( η BS ) is the front (rear) surface non-radiative quantum efficiency, N (0, t , λ ) is the photogenerated electron density distribution in a one dimensional geometry, N0 is the equilibrium density, and SFS [m/s] (SBS) is the front (rear) recombination velocity. Both surface recombination velocities are parameters which characterize the density of the surface defect states. When S=0 the surface retains bulk properties. When S>0 the surface acts as a sink for photogenerated carriers. Values of S in the range 0 ≤ S < 100 [cm/s] generally indicate good passivation for silicon surface [Guidotii et al. 1989]. 13 3.2 Ambipolar diffusion equation At the point r in a semiconductor, the time rate of change in the concentration N(r,t) of the excess electron-hole pairs is governed by both diffusion and recombination, and can be described by an ambipolar diffusion equation of the form [Sze, 1981]: r r r r r r r ∂N (r , t ) N (r , t ) * 2 − − BN 2 (r , t ) − γN 3 (r , t ) = g (r , t ) (3.5) D ∇ N (r , t ) − ∂t τ r where g (r , t ) is the carrier generation rate per unit volume from an external source of excitation and D* is an ambipolar diffusion coefficient defined by k T 2 µ e µ h B q D* = (µe + µh ) (3.6) where µe and µh are an electron and a hole mobility, respectively. The others parameters from Eq. (3.5) will be discussed in Section 3.6. The ambipolar diffusion equation can be linearized when the excess carrier density is sufficiently small. For example in silicon with the concentration up to N = 1× 1017 [cm-3] Guidotti et al. [Guidotti et al.,1989] found that BN 2 ~ γN 3 << N τ and eq. (3.5) can be written as r r r2 r r ∂N (r , t ) N (r , t ) − = g (r , t ) D ∇ N (r , t ) − ∂t τ * (3.7) r Assuming that the temporal Fournier transform of N (r , t ) exist one can write r N (r , ω ) = ∞ ∫ N (r , t )⋅ e r −i ⋅ω ⋅t dt (3.8) −∞ and taking the Fournier transform of Eq. (3.7) yields r r ∞ ∞ ∞ r2 * ∞ r r 1 ∂N (r , t ) −iωt N (r , t ) −iωt −i⋅ω ⋅t ∇ D ∫ N (r , t ) ⋅ e dt − ∫ e dt − ∫ e dt = − ∫ g (r , t )e −iωt dt τ −∞ ∂t τ −∞ −∞ −∞ (3.9) Follow by Mandelis [Mandelis, 2001] an integration by parts in the second term on the leftr handside (l.h.s) and using the boundary condition for N (r , t ) at t → ±∞ , results in the transformed equation r r r 1 r ∇ 2 N (r , ω ) − σ e (ω ) ⋅ N (r , ω ) = − * q (r , ω ) D (3.10) 14 where the real part of the σ e = 1 + iωτ is carrier density-wave wave number, [m-1]. The * τD CDW wave number is completely different from the thermal wave one. In the case of the CDW wave number the real and the imaginary parts are unequal. This inequality brings on that CDW arise only when condition ωτ ≥ 1 is fulfilled. 3.3 One dimensional excess CDW field in Cartesian geometries Mandelis [Mandelis, 2001] considered the excess CDW field in a semiconductor of the thickness L. He assumed that the excess carrier distribution is generated according to the Beer-Lambert Law q ( x, ω ) = αη Q I 0 −αx e (1 + e iωt ) , where β is an absorption coefficient, ηQ is 2 hν the quantum yield of the photogenerated carriers, hν is the incident photon energy and I0 is the optical intensity. Figure 3.3 shows such geometry in one dimension. Figure 3.3 Illustration of the concept of one-dimensional carrier density wave. In this one-dimensional geometry, the boundary conditions can be written as D* d N ( x, ω ) x =0 = S1 N (0, ω ), dx (3.11a) 15 − D* d N ( x, ω ) x = L = S1 N (L, ω ), dx (3.11b) where S1 and S2 are the surface recombination velocities on the two bounding surfaces x=0 and x=L, respectively. Mandelis [Mandelis, 2001] used a Green Function Formalism to solve Eq. (3.10). The resulting excess CDW field is N ( x, ω ) = αη Q I 0 ( 2 hν D α − σ e * 2 2 ) Γ2γ 1 − γ 2 Γ1e −(σ e +α )L − 2σ L Γ2 − Γ1e e −σ e x γ 1 − γ 2 e −(α −σ e )L e + − 2σ e L Γ2 − Γ1e −σ e ( 2 L− x ) −αL e −e (3.12) where Γ1 = D ∗σ e − S1 D ∗σ e + S1 (3.13a), Γ2 = D ∗σ e + S 2 D∗σ e − S 2 (3.13b), γ1 = D ∗α + S1 D ∗σ e + S1 (3.13c) and γ 2 = D ∗α − S 2 D ∗σ e − S 2 (3.13d) 3.4 Three dimensional CDW field in cylindrical geometries Mandelis [Mandelis, 2001] deduced that the full three-dimensional cylindrically symmetric photo-excited carrier-density-wave field in a cylindrical domain of infinite lateral dimensions, which is generated by a Gaussian source, such as TEM00 laser beam of spot size W, is given by the solution of a one-dimensional carrier density wave field generated by a uniform source producing the same incident optical flux under the same boundary condition, according to the operational transformation ∞ N 3 D (r , z , ω ) = 2W 2 ∫ N1D [z , σ e → ξ e (λ , ω )]e λW − 2 2 J 0 (λr )λdλ (3.14) 0 where ξ e (λ , ω ) = λ2 + σ e2 . Based on the presented theorem the three dimensional expression of the excess carrier density wave field in an electronic laterally infinite solid of thickness L can be easily obtained by means of Eq. (3.12). The three dimensional geometry is presented in figure 3.4 where a free 16 carrier density flux is generated by a normally incident Gaussian laser beam of a spot size W. The incident photons with energies hν cause the absorption and CDW generation which is αη Q I 0 −Wr e occurred according to the Beer-Lambert Law: Q(r , ω ) = 2 hν 2 2 −αz (1 + e ). i ωt Figure 3.4 Illustration of the concept of three-dimensional carrier density wave. Putting Eq.(3.12) to the Eq.(3.14) one can obtain the expression for three dimensional excess CDW −λ W 4 −ξ e ( 2 L − x ) −αz e e − e 2 2 α − ξ e 2 G g − g 2 G1e − (ξ e +α )L αη Q I 0 ∞ 2 1 N (r , z, ω ) = G 2 − G1e − 2σ e L 2hνD * ∫0 −ξ e z g1 − g 2 e −(α −ξ e )L e + − 2ξ e L g 2 − g1e J 0 (λr )λdλ 2 ( (3.15) where J0(λr) is the Bessel function of first kind of order 0 and G1 = D ∗ξ e − S1 D ∗ξ e + S1 (3.16a), G2 = D ∗ξ e + S 2 D ∗ξ e − S 2 (3.16b), D ∗α + S1 g1 = ∗ D ξ e + S1 (3.16c) and g 2 = D ∗α − S 2 . D ∗ξ e − S 2 (3.16d) 17 ... ) 3.5 Recombination processes in semiconductors Photoexcitated carriers in semiconductors can recombine by one of three mechanisms [W.M.Bullis and H.R.Huff, 1996]: a Shockley-Read-Hall (SRH) recombination which is related to multi-phonon release [W.Shockley and W.Read, 1952 and R.Hall 1952]; photon release (radiative recombination) and Auger recombination, in which the recombination energy is carried away by a third carrier. r Assuming that trapping is negligible, so the number of excess holes P (r , t ) is equal the r number of excess electrons N (r , t ) , the total bulk recombination rate Rtotal, will be the sum of individual rates and the average carrier lifetime can be expressed as τ= ∆N 1 = −1 −1 −1 Rtotal τ SRH + τ rad + τ Auger (3.17) The SRH lifetime can be reproduced from [W.Shockley and W.Read, 1952] for one existing dominant center: τ SRH = τ P 0 ( N 0 + N1 + ∆N ) + τ N 0 (P0 + P1 + ∆P ) N 0 + P0 + ∆N (3.18) where N0 and P0 are the equilibrium electron and hole densities, respectively. Here N1 and P1 are the equilibrium carrier densities related with energy of the defect center ED coincides with the Fermi level and can be written as N1 = N C exp P1 = NV exp − ( EC − ET ) (3.19a) kT − ( ED − EV kT ) (3.19b) where NC and NV are the densities of states in the conduction and valance band, respectively. EC and EV are the energies of the conduction and valance band edge, respectively, k is the Boltzmann constant, and T is the temperature. The time constant for capture of an electron (hole) by an empty (full) defect state given by τ N0 = τ P0 = 1 N Dσ Nυ th 1 N Dσ Pυ th (3.20a) (3.20b) 18 where ND is the density of defect states. σ N and σ P are the capture cross sections for electrons and holes by the defect, and υ th is the thermal velocity of carriers. The thermal velocity of carriers is depend on temperature and is expressed by υ th = 8kT πm (3.21) where m is a mass of an electron. The second mechanism of recombining carriers is related with a radiative recombination lifetimes. In semiconductors with direct band gap transition of excited electrons from minimum of conduction band to maximum of valance band (band-to-band) is more probable than in the case of semiconductor with indirect energy band gap such as silicon. The probability of this transition for both types of the semiconductors, with direct and indirect energy band gap, can be described by the radiative recombination probability B which is related to recombination lifetime in formula [Varshni, 1967] τ rad = 1 B ( N 0 + P0 + ∆N ) (3.22) The coefficient B for semiconductors with indirect energy band gap is in order of 10-15 [Varshni, 1967; Gerlach et al., 1972; Augustine et al. 1992] while for direct energy band gap semiconductor is 10-9 [cm3/s]. The recombination lifetime expression in the case of Auger recombination can be written as [Schroder, 1998] τ Auger = ( C P P + 2 P0 ∆N + ∆N 2 0 2 ) 1 + C N N 02 + 2 N 0 ∆P + ∆P 2 ( ) (3.23) where CP and CN are the Auger coefficient for holes and electrons, respectively. In highly doped silicon those coefficients were found by Dziewoir and Schmid [Dziewoir and Schmid, 1977]: C N = 2.8 × 10 −31 C P = 9.9 × 10 −32 [cm6/s]. For demonstration purposes one can calculate recombination lifetimes for p-type silicon is shown in Figure 3.5 as a function of injection level for assumed defect level in the middle of the band gap and the following parameter values: C N = 2.8 × 10 −31 C P = 9.9 × 10 −32 [cm6/s], B = 4 × 10 −15 [cm3/s], σ P = σ N = 1× 10 −14 [cm2], ν th = 1.07 × 107 [cm/s], N C = 2.86 × 1019 , NV = 3.1× 1019 , NT = 1012 [cm-3], ET = 0.562 [eV]and p A = N A = 1015 [cm-3] which corresponds to a resistivity of approximately 15 Ωcm. 19 10 1 SRH radiative Auger total -1 Recombination Lifetime [s] 10 -3 10 -5 10 -7 10 -9 10 10 -11 10 15 16 17 10 10 18 10 10 19 10 20 -3 Carrier Injection level [cm ] Figure 3.5 Calculated recombination lifetimes in silicon as a function of injection level for a p-type Si wafer with a doping density p A = N A = 1015 [cm-3]. Assumed parameter values are given in the text. It is clear seen that the radiative recombination lifetime is much longer than the SRH or Auger recombination time constant and has negligible effects on the overall recombination rate. For low injection level (of an order 1017 [cm-3]) the recombination processes are dominated by the SRH recombination time constant while at higher injection level by the Auger recombination time constant. 20 Chapter 4: Experimental methods, signal generation mechanisms and instrumentation of PCR and PPE methods 4.1 The Photocarrier Radiometry signal generation mechanism and instrumentation 4.1.1. Introduction Mandelis et al. [Mandelis et al., 2003] proposed a new technique for the measurement of the carrier-density – wave and named it the Photocarrier Radiometry (PCR). A modification of this technique is the room- or high temperature near infrared photoluminescence (NIR-PL). In the past the NIR-PL has been associated with the presence of impurities or defects and band-to-band recombination [King and Hall, 1994, Haynes 1956; Varshni 1967]. 21 4.1.2. Contribution to the PCR signal The PCR is associated with room- or high temperature near infrared photoluminescence. In Figure 4.1 one can see that the photoluminescence spectrum of silicon at room temperature has two peaks [King and Hall, 1994] The first at 1.09 eV (≈ 1.14 µm) is associated with band – to – band transitions [Haynes 1956; Varshni 1967]. The second one is an approximately at 0.73 eV (≈ 1.6 µm) and is observed only for silicon wafers grown by Czochralski method and is associated with oxygen dependent defects complexes [Kitgawara 1992; King and Hall, 1994]. Figure 4.1 Measured photoluminescence spectra at T=30, 130 and 300 K for Czochralski-grown Si annealed at T=450 C. The data were obtained using a Ge photodetector [King and Hall, 1994]. The PCR signal theory was discussed thoroughly by Mandelis et al. [Mandelis et al., 2003]. They authors considered an elementary slice of thickness dz, centered at depth z in a semiconductor slab supported by a backing, but not necessarily in contact with the backing, Fig 4.2. 22 Figure 4.2 Cross-sectional view of contributions to front-surface radiative emissions of IR photons from (a) a semiconductor strip of thickness dz at depth z; (b) reentrant photons from the back surface due to reflection from a backing support material; (c) emissive IR photons from the backing at thermodynamic temperature T. The carrierwave depth profile results in a depth dependent IR absorption/emission coefficient due to free carrier absorption of the infrared photon fields, both ac and dc [Mandelis, 2003]. A modulated laser beam at angular frequency ω=2πf and wavelength λvis impinges on the front surface of the semiconductor. The super-bandgap radiation is absorbed within a (short) distance from the surface and excited carriers are subjected to several de-excitation processes discussed in Chapter 3. At thermal and electronic equilibrium, a detailed consideration of all IR emission, absorption, and reflection processes [Mandelis, 2003] yields an expression for the total IR emissive power at the fundamental modulation frequency across the front surface of the material in the presence of a backing support which acts both as reflector of semiconductor-generated IR radiation with spectrum centered at λ. Instrumental filtering of all thermal infrared emission contributions and bandwith matching to the IR photodetector allows for all Planck-mediated (8-12 µm) terms to be eliminated and the PCR signal can be written follow as.[Mandelis, 2003] λ2 L λ1 0 S PCR (ω ) = ∫ dλ [1 − R1 (λ )][1 + Rb (λ )][1 + R1 (λ )]η RWeR (λ )∫ ε fc ( z , ω , λ )dz (4.1) 23 where R1 is the front surface reflectivity, Rb is the backing support material reflectivity, εfc is the IR emission coefficient due to the free photoexcited carrier wave, WeR(λ) is the spectral power per unit wavelength, the product to the recombination transition rate from band do band or from bandedge to defect or impurity state, as the case may be, multiplied by the energy difference between initial and final states, ηR is the quantum yield for IR emission upon carrier recombination into one of these states. For a semiconductor which is in thermal and electronic equilibrium with its environment Kirchhoff’s theorem is fulfilled: ε fc ( z , ω , λ ) = α fcIR ( z , ω , λ ) , (4.2) where αfc is absorption coefficient due to the free photo-excited carriers. For relatively low carrier densities the absorption coefficient depends on the free carrier density as [Smith, 1978] α fcIR ( z , ω , λ ) = qλ2 ∆N ( z , ω , λ ) . 4π 2ε 0 c 3 nm *2 µ (4.3) Putting (4.3) into Eq. (4.1) one can write the expression for the PCR signal in one dimension L S PCR (ω ) = F (λ1 , λ2 )∫ ∆N ( z , ω )dz , (4.4) 0 where F (λ1 , λ 2 ) = λ2 ∫λ [1 − R (λ )][1 + R (λ )][1 + R (λ )]η 1 b 1 R WeR (λ )C (λ )dλ . 1 4.1.3. Instrumentation and normalization of PCR signals Four photocarrier radiometric systems have been used. The first was related with study of the effect of the space charge layer on the PCR signal system and the results are presented in Chapter 6. The second and third systems were constructed in the Center for Advanced Diffusion Wave Technologies, University of Toronto, Canada [Shaungnessy, 2005] and were used for the study of the influence of the optical excitation intensity on the PCR signal (Chapter 7). The last one is the PCR microscope constructed in order to monitor the ion implanted process in silicon wafers. The common part of all these systems is an InGaAs p-i-n photodetector (Thornlabs model PDA 400) with the following parameters: spectral bandwidth of 700-1800nm; an active area of 0.8 mm2, and an adjustable transimpedance gain; the unit was used at the intermediate gain setting (1.5 ×105 V/A) at which it has a noise equivalent power (NEP) of 3.8 ×10-12 W Hz-1/2 (at 1310 nm) and a frequency bandwidth of 700 kHz) a long-pass filter (Spectrogon model LP-1000: a steep cut-on (5% at 1010 nm, 78% at 1060 24 nm) and a transmission range 1042 – 2198 nm is placed in front of the detector in order to ensure that any diffuse reflections of the excitation source do not contribute to the signal). The spectral responsivity is shown in Fig 4.3. Fig 4.3 Spectral responsivity of the PCR detector. All frequency dependent measurements were normalized by the corresponding widebandwidth instrumental transfer functions. The transfer functions were obtained by measuring the amplitude and phase of modulated laser radiation scattered from a microscopically rough metallic surface positioned at the focal plane of the parabolic mirror, and partly transmitted through the filter. 4.1.4. The one dimensional Photocarrier Radiometry (PCR) Signal The PCR signal in one dimensional geometry (as this from Figure 3.3) can be written with help of Eq. (4.4) and Eq. (3.12) L S1D (ω ) = F (λ1 , λ2 )∫ ∆N ( x, ω )dx = F (λ1 , λ2 )E1D (ω )M 1D (ω ) (4.5) 0 where E1D (ω ) = M 1D (ω ) = I 0ηα (1 − R ) 2hνD * α 2 − σ e2 ( γ 1 (Γ2 + e −σ L ) − γ 2 (Γ1e − (α +σ e Γ2 − Γ1e − 2σ e L ) e )L (4.6) + e −αL ) − σ (1 − e ) α e −αL (4.7). 25 For silicon wafers, this equation has been applied with superband-gap radiation of absorption coefficient α(hν) > 103 cm-1, such that the semiconductor material is entirely opaque to the incident radiation, and thus e-αL ≈ 0. The quantities in (4.6) and (4.7) were defined in Chapter 3 (Eqs.: 3.13 a-d). Using MATLAB program and equation (4.5) simulations of the electronic parameters on the PCR signal were performed. 10 21 0 τn= 5 µ s τn= 5 0 µ s τn= 1 0 0 µ s τn= 5 0 0 µ s τn= 1 0 0 0 µ s -2 0 -4 0 PCR Phase [deg.] 20 PCR Amplitude [a.u.] 10 τ n= 5 µ s τ n= 5 0 µ s 10 τ n= 1 0 0 µ s 19 -6 0 τ n= 5 0 0 µ s τ n= 1 0 0 0 µ s -8 0 10 18 10 1 10 2 10 3 10 4 10 5 10 1 F re q u e n c y [ H z ] 10 2 10 3 10 4 10 5 F re q u e n c y [H z ] Figure 4.4 The PCR Amplitude (a) and Phase (b) of p-type silicon wafer versus modulation frequency with the different values of minority bulk recombination lifetime. Parameter settings: S1=300cm/s, S2=105 cm/s, Dn* = 30 cm2/s, L = 550 µm, α(λ=514nm) = 7.76×103 cm-1. Figure 4.4 shows a behavior of the PCR amplitude (a) and phase (b) for a silicon wafer with the minority recombination lifetime τn in the variations in the range 5 µs ≤ τ n ≤ 1 ms. A decrease in τn diminishes the PCR amplitude and shifts the position of the turning point (“knee”) to higher frequencies as the density of the carrier-wave over one period decreases with decreasing recombination time [Mandelis, 2001]. The PCR phases exhibit zero delay with respect to the modulation source at low frequencies, such that ωτn << 1 but they begin to lag behind the source phase as soon as this condition is not valid. As τn decreases, the foregoing condition becomes violated at progressively higher frequencies, whence the shift of the PCR phases in Fig. 4.4b [Mandelis et al., 2003]. 26 0 20 10 -40 * PCR Phase [deg.] PCR Amplitude [a.u.] -20 2 D n=5 [cm /s] * 2 * 2 D n=5 [cm /s] * 2 D n=10 [cm /s] * 2 D n=20 [cm /s] * 2 D n=30 [cm /s] D n=10 [cm /s] D n=20 [cm /s] D n=30 [cm /s] D n=45 [cm /s] * 2 * 2 * 2 * 2 -60 D n=45 [cm /s] 19 10 -80 1 10 2 10 3 10 Frequency [Hz] 4 10 5 10 1 10 2 3 10 10 4 10 5 10 Frequency [Hz] Figure 4.5 The PCR Amplitude (a) and Phase (b) of p-type silicon wafer versus modulation frequency with the different values of ambipolar diffusivity. Parameter settings: τn = 100 µs, S1 = 300 cm/s, S2 = 105 cm/s, L = 550 µm, α(λ=514nm) = 7.76×103 cm-1 Figure 4.5a shows the change in the PCR amplitudes affected by altering the ambipolar diffusivity Dn*. If Dn* is controlled by the bulk of the semiconductor, then an increase of this quantity will decrease the PCR amplitude. This behavior can be explained by the fact that the CDW “centroid” – center of the charge carriers - shifts away from detection point at surface therefore the contribution of the recombining carrier density wave to the PCR signal generated at the surface (or/and subsurface) is smaller. At low frequency the PCR phase doesn’t show any lag, until ωτ >> 1 is fulfilled, then an onset of the PCR phase lag is observed. The PCR phase lag exhibits a shift to higher frequencies with increasing Dn*. High frequencies can affect the position of the CDW centroid shifting it to smaller depth. This effect is observed on the PCR amplitude and the PCR phase. 27 0 20 10 PCR Phase [deg.] PCR Amplitude [a.u.] -20 -40 S1=0 [cm/s] S1=0 [cm/s] S1=0.1 [cm/s] S1=0.1 [cm/s] S1=1 [cm/s] -60 S1=1 [cm/s] S1=10 [cm/s] S1=10 [cm/s] S1=1000 [cm/s] S1=1000 [cm/s] 19 10 -80 -100 1 10 2 10 3 10 4 10 Frequency [Hz] 5 10 1 10 2 10 3 10 4 10 5 10 Frequency [Hz] Figure 4.6 The PCR Amplitude (a) and Phase (b) of p-type silicon wafer versus modulation frequency with the different values of front recombination velocity. Parameter settings: τn = 100 µs, S2 = 105 cm/s, Dn* = 30 cm2/s, L = 550 µm, α(λ=514nm) = 7.76×103 cm-1. Figure 4.6 shows the effect of changing the recombination velocity S1 on the PCR frequency scans. The PCR amplitudes decrease (fig. 4.6a) as the value of S1 increases. This behavior is quite similar to the case observed for decreasing of the minority recombination lifetime τn, although the “knee” shift to higher frequencies is not as pronounced. A similar behavior is observed for the PCR phases (fig. 4.6b), where the phases lag move to higher frequencies. Additionally, the phase lag shows gradual decrease with increasing S1 due to the sub-surface ac diffusion length (or “centroid”) of the CDW which is no longer controlled by the bulk recombination lifetime τn alone but it becomes controlled by an effective lifetime, τeff, defined as follows [Mandelis 2005]: 28 1 τ eff = 1 τp 1 + (4.8) τs where τs is the interface lifetime related to the interface recombination velocity S1. This time constant begins to influence the effective lifetime (and hence the phase saturation level) at S1 values such as τs ~ τn. Figure 4.7 shows the effect of changing recombination velocity S2 on PCR frequency scans. 0 -20 20 -40 S2=0 [cm/s] PCR Phase [deg.] PCR Amplitude [a.u.] 10 S2=0 [cm/s] S2=0.1 [cm/s] S2=0.1 [cm/s] S2=1 [cm/s] -60 S2=1 [cm/s] S2=10 [cm/s] S2=10 [cm/s] S2=1000 [cm/s] S2=1000 [cm/s] 19 10 -80 1 10 2 10 3 10 Frequency [Hz] 4 10 5 10 -100 1 10 2 10 3 10 4 10 5 10 Frequency [Hz] Figure 4.7 The PCR Amplitude (a) and Phase (b) of p-type silicon wafer versus modulation frequency with the different values of rear recombination velocity. Parameter settings: τn = 100 µs, S1 = 300 cm/s, Dn* = 30cm2/s, L = 550 µm, α(λ=514nm) = 7.76×103 cm-1. 29 4.1.5. The three dimensional PCR signal In Chapter 3 (Section 3.4) it was shown that the photo-excited carrier density wave field in three dimensional cylindrical geometries can be described by: (see Eq. (3.14) ) N 3 D (r , z , ω ) = 2W ∞ 2 ∫ N [z, σ 1D e → ξ e (λ , ω )]e λW − 2 2 J 0 (λr )λdλ (4.9) 0 λW 2 − ~ where N 3 D (λ , z , ω ) = 2W 2 N 1D [z , σ cc → ξ e (λ , ω )]e 2 is the Hankel transform of Eq. (3.14). In order to account for contributions over the thickness of the wafer the Hankel transform of the carrier density field has to be integrated over the depth: L ~ ~ N 3 D (λ , ω ) = ∫ N 3 D (λ , z , ω )dz (4.10) 0 The finite area of the detector must be taken into consideration to account for carrier diffusion out of the field of view of the collection optics/detector assembly [Shaughnessy, 2005]. Assuming a disc of radius a2 and area A as the effective detector size and using the relation [Ikari et al., 1999]: a 1 2~ 1 ~ N 3 D (λ , ω )J 0 (λρ )ρdρ = N 3 D (λ , ω )J 1 (λa 2 ) ∫ A0 πa 2 λ (4.11) where J1is the Bessel function of first kind of order 1. The PCR signal can be expressed in final form as the inverse Hankel transform of (4.9) integrated over the detector area S PCR ,3 D (ω ) = ∞ C ~ N 3 D (λ , ω )J 1 (λa2 )dλ πa2 ∫0 (4.12) where ~ N 3 D (λ , ω ) = E3 D (λ , ω )M 3 D (λ , ω ) E3 D (λ , ω ) = − λ2W 2 αη I 0 (1 − R )e 4 2hνD * (α 2 − ξ e2 ) (4.14) (λ , ω ) = (1 − e ) [C (λ , ω ) + C (λ , ω )e ]− (1 − e ) ξ α (4.15) g − g 2 e − (α −ξe )L C1 (λ , ω ) = G1G2 1 − 2ξ e L G2 − G1e (4.16) −ξ e L M 3D (4.13) −ξ e L 1 −α L 2 e 30 C 2 (λ , ω ) = g1G1 − g 2 G2 e −(α −ξe )L G2 − G1e −2ξe L (4.17) Coefficients in (4.12) - (4.17) were defined in Chapter 3. 4.1.6. The photocarrier radiometry (PCR) dimensionality criterion Figure 4.8 shows a frequency-scan simulations in two sets of linear PCR signals from silicon based on 1-D (full lines) and 3-D (squares and rhombs) theoretical models (Eqs 4.5 and 4.10, respectively) with laser wavelength 830 nm, spotsize 4 mm and recombination lifetimes τ = 20 µs and 800 µs. The optical absorption coefficient was taken to be αP = 635 cm-1. The carrier transport parameters were assumed equal for both sets of curves. The amplitude curves are normalized to unity at f = 10 Hz. Log(Amplitude) [a.u.] 1 1-D Simulation τ - 20 µs 1-D Simulation τ - 800 µs 0.1 0 Phase [degrees] -20 -40 -60 1-D Simulation τ - 20 µs 1-D Simulation τ - 800 µs -80 -100 0.01 0.1 1 10 Log (Frequency [kHz]) 100 Figure 4.8 PCR frequency scan simulations with short and long carrier recombination lifetimes using 1-D and 3-D theoretical model. 3-D simulations with τ = 20 µs (■) and 800 µs (●); 1-D simulations with τ = 20 µs (—) and 800 µs (—) coincide with the corresponding 3-D simulation. Other transport parameters: D = 15 cm2/s, S1 = 200 cm/s, S2 = 105 cm/s, αp = 659 cm-1. Laser beam spotsize: 4 mm. 31 It is clear that the simulations using the 1-D and the 3-D equations coincide, as expected, for the chosen large spotsizes compared to the maximum carrier-wave diffusion length, LD(ω) = | σn(ω) |-1 at the lowest frequency f = 10 Hz where LD(10 Hz) =173.2 µm for τ = 20 µs and 1,095 µm for τ = 800 µs. In practice, the use of 1-D theory to explain PCR data is warranted when a change of the beam spotsize on the semiconductor surface does not produce measurable change in the PCR phase. This is an important dimensionality criterion, therefore, the dependence of the PCR phase on laser spotsize using the 1-D or the 3-D theory with various lifetimes (or diffusion lengths of the photo-excited free carrier density-wave) at two frequencies is presented in Fig. 4.9. 24 µm 387 µm 830 µm -20 PCR Phase [degree] PCR Phase [degree] b -10 -5 -10 -15 0 a 0 387 µm 830 µm 24 µm Lifetimes: τ1 - 1 µs; τ2 - 20 µs; 100 -50 -60 -80 τ4 - 200 µs; 10 -40 -70 τ3 - 50 µs; -20 -30 1000 Log(Laser Spotsize [µm]) 10 100 1000 Log(Laser Spotsize [µm]) Figure 4.9 Laser beam spotsize dependence of PCR phases for frequency 1 kHz (a) and 100 kHz (b) with broad range of lifetimes and otherwise same other transport parameters: D = 10 cm2/s, S1 = 500 cm/s, S2 = 105 cm/s, αp = 659 cm-1. It is clearly seen that, for PCR signals with τ = 1 µs, 1-D theory can be used with spotsizes 2W ≥ 387 µm at 1 kHz and 100 kHz. However, with τ = 20 µs, 50 µs, and 200 µs, the condition 2W ≥ 830 µm is required at 100 kHz. In the latter cases all lower frequency ranges require a 3-D theoretical approach. 32 4.1.7 Photo-carrier radiometry microscope In this section the capability of the PCR technique to monitor a quality of ion implanted wafers is presented. In order to present that PCR signal is sensitive to the change of carrier transport properties the photocarrier radiometry microscope was constructed and it is presented on Figure 4.10. As an excitation source of carrier density waves a 808 nm laser diode (0.5mm beam radius) was used. The power of the laser diode was typically 200 mW. The diode laser beam was focused onto the sample surface using lens. The position of the laser beam is coincident with the focal point of an off-axis paraboloidal mirror that collects a portion of infrared radiation from the samples. The collected light is then focused onto the detector by means of lens. Sample was placed onto aluminum holder (acted as a mechanical support and signal amplifier by redirecting the forward emitted IR photons back toward the detector [Mandelis, 2003]. The x-y position scans were realized by means of homemade x-y motor stage. All instruments, data acquisition, and data storage are controlled by a computer running Pascal program with a graphical user interface and realtime display of experimental data. 33 Figure 4.10. The photocarrier radiometry microscope. Typical result obtained using modulation frequency 10 kHz for an ion implanted wafer is presented at Fig. 4.11. Figure 4.11: The PCR amplitude and phase as a function of x-y-scan of the silicon wafer implemented with 6.3 1016 doses of protons [cm-2]. Sample preparation: The 34 energy of the protons was 1 MeV with an implantation depth of 18 µm, the beam was focused on 5x5 mm2 area on the silicon wafer surface. The squares on Figure 4.11 depict the ion implanted regions. Figure 4.12 shows the PCR phase as a function of coordinate. Figure 4.12: The PCR phase a a function of x-scan of the silicon wafer implemented with 6.3 1016 doses of protons [cm-2] at the difference frequencies. Sample preparation: The energy of the protons was 1 MeV with an implantation depth of 18 µm, the beam was focused on 5x5 mm2 area on the silicon wafer surface. From Fig. 4.12 one can see that for 1 kHz the PCR technique is unable to detect any changes in the PCR phase so in the carrier transport properties. Whereas above 1 kHz changes in the PCR phases are clearly seen. The PCR phase lag was observed in an ion implanted region. This is can be explained because the photo-carrier diffusion length is too 35 large to detect any inhomogeneous in the free carrier density depth in the implantation region. 4.2 The photopyroelectric (PPE) signal generation mechanism and instrumentation 4.2.1. Experimental Setup The PPE measurements were performed in the back detection configuration, where the heat is generated on the front side of the sample and the temperature oscillations are measured with the pyroelectric detector contacted to the back side of the sample. The experimental setup constructed for the back detection configuration is presented in Fig. 4.13. Figure 4.13: PPE experimental set up The thermal wave are excited by an argon ion laser with output power 200 mW and operating wavelength λ = 514 nm. The laser beam of 1.89 mm diameter was intensity modulated by means of an acousto–optical modulator in the frequency range 1 Hz to 10 Hz and focused onto the sample. The front surface of the sample was covered by an optically opaque 20 µm to 30 µm graphite coating. Samples were attached to a pyroelectric detector by means of a grease layer 36 (Apiezon T grease). As the grease layer was very thin, its contribution to the PPE signal could be neglected. A 0.98 mm thick lead zirconate titanate PZT crystal was used as a pyroelectric detector. The PPE signal detection was performed by means of a lock-in amplifier (Stanford 830). The detector was placed on a cooper plate with a drilled hole (inside was air). The sample – detector – copper support assembly was placed in an aluminum chamber. A schematic of the PPE chamber is presented in Fig. 4.14. Fig. 4.14: PPE chamber: 1 Peltier-element, 2 aluminum support, 3 cylindrical cooper support, 4 the PZT detector, 5 sample with optically opaque cover-layer, 6 quartz window, 7. BNC connector for PPE Signal The temperature was varied in the range from 20° C up to 40° C by means of a Peltier element which was driven by a homemade current controller (Appendix A). 4.2.2. The PPE signal generation mechanism. The average temperature oscillation Tp at angular frequency ω0 in a pyroelectric detector leads to variations of the surface charge density Q due to the pyroelectric effect and can be written according to B. R. Holeman [Holeman, 1972] as: 37 Q (ω0 ) = p Tp (ω0 ) , (4.18) where p is the pyroelectric coefficient of the detector. Time-dependent variations of the surface charge causes a current flow through the detector of the thickness Lp [Mandelis and Zver, 1985; Rombouts et al., 2005]; I (ω 0 ) = A d Q(ω0 ) dt = pA d Tp (ω 0 ) dt 1 d = pA ∫ Tp ( x, ω 0 )dx e iω0t = ipAω 0θ p (ω 0 )e iω0t , (4.19) Lp L dt p where A is the transducer area and θ p (ω 0 ) = 1 Lp ∫T p ( x, ω 0 )dx . Tp(ω0,x) is the temperature field in Lp the pyroelectric detector. Figure 4.15 shows a schematic of the sample’s model. Figure 4.15 Schematic of the sample’s model 38 In the used arrangement the front surface was covered by a 20 µm to 30 µm thick graphite layer which prevents exciting light to penetrate into the sample. The rear surface was connected to the detector which monitored the thermal wave transmitted through the sample. The distribution of the thermal wave is the solution of one-dimensional thermal transport equations as a result of heat conduction through the sample. Similar theoretical models were considered by Chirtoc and Mihalescu [Chirtoc and Mihalescu, 1989] and Mandelis and Zver [Mandelis and Zver, 1985]. In both works the influence of the thermal interface between the rear surface of the sample and the detector was neglected. Experimentally a good thermal contact was achieved with a very thin grease layer. As in the presented experiments the thermal waves were generated by surface heating the contribution to the heat transport problem of the thermally thin graphite surface layer can be neglected. Also, the thermal contact of the sample to the detector by the grease layer is considered to be ideal. In some cases, however, the thermal diffusivity Dt of the sample can be underestimated due to the influence of the grease layer as demonstrated by Salazar [Salazar,2003]. He calculated the error of the Dt estimation in the presence of about a 2 µm to 3 µm thick grease layer. He found that the error is large for thin and good thermal conductors at high frequencies and decreases with increasing thickness and decreasing thermal diffusivity of a material and modulation frequencies. Although in our measurements we used a different grease, one can deduce that the investigated samples as well as a glassy carbon are rather poor thermal conductors. Furthermore, this effect is additionally reduced because measurements were performed at modulated low frequencies. When the sample and the detector are both thermally thick and optically opaque the temperature field can be obtained using a formula of Chirtoc and Mihalescu [Chirtoc and Mihalescu, 1989] and Mandelis and Zver [Mandelis and Zver, 1985]: η s Dp Θ p (ω 0 ) = k D p k p 1 + s k p Ds exp − ω 0 L exp − i π + ω0 L , s s 2 2 Ds 2 Ds ω 0 (4.20) where ηs is the nonradiative conversion efficiency of the absorbing layer: The sample is characterized by a thickness Ls, a thermal diffusivity Ds and a thermal conductivity ks. The 39 detector is characterized by a thermal diffusivity Dp and a thermal conductivity kp. The PPE signal is then given by η s Dp ω 0 π ω 0 I (ω 0 ) = ( pI 0 A) Ls exp − i + Ls , (4.21) exp − 2 Ds 2 Ds k p (1 + bsp ) 2 where I0 is the intensity of the optical excitation: The phase of the PPE-signal is given by ϕ=− π 2 − πf Ds Ls = − π 2 −m f . (4.22) The coefficient m can be easily determined from experimental data. Thus, with the known sample thickness the thermal diffusivity can be deduced by the relation Ds = πL2 s m2 . (4.23) The PPE amplitude can be written as L2 s ln (I (ω0 )) = B + − 2 Ds pI 0 Aη s Dp where B = ln ks Dp k p 1 + k p Ds 2 ω0 = B − πL s Ds f = B−m f , (4.24) . 4.2.3. Normalization of the PPE Signal Figure 4.16 shows the PPE amplitude and the PPE phase from the detector alone as a function of the modulation frequency in the temperature range from 26 °C to 36 ºC. Error bars for PPE phases were approximately 1.5°. 40 1.80E-010 96 1.75E-010 PPE amplitude [A] 94 1.70E-010 92 PPE phase [deg] 0 26 C 0 27 C 0 28.5 C 0 31 C 0 33 C 0 36.5 C 1.65E-010 90 1.60E-010 2 4 6 8 10 2 4 6 8 10 Frequency [Hz] Frequency [Hz] Fig. 4.16: PPE amplitudes (a) and phases (b) of the detector alone at different temperatures (in °C). From the experimental data in Fig. 4.16b, one can see that in the investigated range of the temperature the PPE phases remains constant within error bars. This means that the thermal properties of the detector can be assumed constant under our measurement conditions. We had also observed small changes in the PPE amplitudes, Fig. 4.16a, but these can be caused by the temperature-dependence of the pyroelectric coefficient and/or thermal effusivity ep (ep = kp(Dp)1/2 ) of the PZT detector as well as long term fluctuations of the laser intensity. These effects can be minimized by an appropriate normalization procedure. Detenclos et al. [Delenclos et al., 2001] normalized the PPE signal from an investigated material to the one obtained with the detector alone or to the signal obtained with a reference sample. They considered the PPE signal for the sample and the detector both thermally thick and optically opaque and pointed out that the normalized signal is not influenced by the temperature-dependence of the pyroelectric coefficient, hence, only a knowledge of thermal effusivity of the detector is required. In fact, PPE amplitudes were normalized to the reference sample instead of the detector alone as the absorption of laser light at the detector electrode is different from that in the graphite layer [Delenclos et al., 2001]. In addition, it is also possible that the heating spot (laser beam spot) 41 interacts (energy exchange) with silver contacts on the surface of the detector, and this could lead to a worse signal-to-noise ratio (SNR) than in the case of normalization to a reference material. As a reference sample, a 0.98 mm thick piece of a glassy carbon (type G) was used. The specific heat capacity C of the glassy carbon in the temperature range from 26 °C to 80 °C was determined from differential scanning calorimetry (DSC) measurements and these results are presented in Table 4.1. Table 4.1 presents the specific heat capacity of the glassy carbon (thickness L=0,98mm) in different temperatures estimated from the PPE phases and amplitudes and result of the DSC measurements. Temperature [°C] The specific heat capacity C [J/kg°C] 26.85 1054.568 36.85 1064.533 46.85 1151.807 56.85 1268.092 66.85 1397.104 76.85 1510.841 86.85 1612.371 The error limit of the differential scanning calorimetry (DSC) measurements was 3% to 5%. One can see that in the covered temperature range (from 20 to 40° C) the specific heat capacity C is about 1050 J⋅kg-1⋅K-1 within the error limit. Figures 4.17a and 4.17b present the PPE signal phases and amplitudes of the glassy carbon, respectively, at room temperature as a function the square root of the modulation frequency. It is worthwhile to note that the error bars were approximately 0.5°. 42 PPE Phase / deg 0.0 e -15 e -16 ln(Amplitude)/au experimental data the best linear fitting 0.5 -0.5 -1.0 1.5 2.0 2.5 3.0 3.5 experimental data the best linear fitting 1.0 1.5 Sqrt(Frequency) / Hz 2.0 2.5 3.0 3.5 Sqrt (Frequency) /Hz Fig.4.17 The PPE Phase and Amplitude of the glassy carbon (thickness L=0.98µm) as a function of a square frequency, respectively. The linear fitting of the experimental PPE Phase to the Eq. 4.22 gives m=-0,860±0,003, whereas the linear fitting of experimental PPE amplitude to the Eq. 4.24 gives m=0,857±0,001. Using Eqs. (4.22) and (4.23) the thermal diffusivity from the as measured PPE phases of the glassy carbon was estimated to Ds = 4.22x10-6 m2⋅s-1. The same value of the thermal diffusivity was obtained from the measured PPE amplitudes by Eq. (4.24) and Eq.(4.23). Using the literature value of the mass-density ρ = 1.42⋅103 kg⋅m-3 [http://www.htw-gmbh.de/] the thermal conductivity of the glassy carbon type G was calculated to ks = 6.3 W⋅m-1⋅K-1 which is in excellent agreement with the value deduced from the data sheet of the producer of the glassy carbon [http://www.htw-gmbh.de/]. This demonstrates the reliability of the present experimental setup and measurement procedure for the experimental determination of the thermal diffusivity. Table 4.2 presents temperature dependence of the thermal diffusivity of the glassy carbon calculated using Eq. (4.22) - (4.23). 43 Table 4.2 presents the thermal diffusivity of the glassy carbon (thickness L=0,98mm) at different temperatures determined from the PPE phases and amplitudes. Temperature mAmpl mPhase [°C] DAmpl x 10-6 DPhase x 10-6 [m2/s] [m2/s] 22.2 -0.841 -0.845 4.266 4.226 24.7 -0.842 -0.845 4.256 4.226 27.4 -0.843 -0.845 4.246 4.226 29.8 -0.845 -0.847 4.226 4.206 33.1 -0.845 -0.848 4.226 4.196 36.0 -0.847 -0.849 4.206 4.186 39.0 -0.849 -0.852 4.186 4.156 As compared to the black glassy carbon, the investigated semiconductor samples have smaller absorption coefficients β; thus, the light can penetrate deeper into the sample generating heat also in the subsurface regions. For this reason a thin black graphite layer was deposited on the surface. The optically opaque cover layer at the present experimental conditions (514 nm laser) avoids the super-bandgap excitation that creates photocarriers which can act as scattering centers for phonons. The scattering centres alter the thermal transport properties (decreasing k) of investigated semiconductors and complicate the experimental data interpretation. Figure 4.18a shows the frequency dependence of the PPE amplitude and phase in the presence of the graphite cover layer on the glassy carbon sample. 44 0.5 -15 e ln(Amplitude)/au PPE Phase / rad 0.0 -0.5 -1.0 -16 e experimental data the best linear fitting -1.5 1.0 1.5 2.0 2.5 3.0 3.5 1.0 1.5 2.0 2.5 3.0 3.5 Sqrt(Frequency) / Hz Sqrt (Frequency) /Hz Figure 4.18 The PPE phase and amplitude of the glassy carbon (thickness L=0.98mm) with a graphite layer as a function of a square frequency, respectively. The linear fitting of the experimental PPE phase to the Eq. 4.22 gives m=-0,860±0,003, whereas the linear fitting of experimental PPE amplitude to the Eq. 4.24 gives m=0,863±0,003. It is clearly seen that the graphite layer does not affect the PPE phase. The same value of the coefficient m was obtained for PPE amplitudes, Fig. 4.18b. Therefore, the thermal diffusivity can be calculated from the as measured PPE phases and amplitudes in the presence of the graphite coating in the same way as for the glassy carbon sample alone. Formula (4.20) describes the thermal wave field in the sample when the sample and the detector are both thermally thick and optically opaque. Experimentally this condition is fulfilled in a certain frequency range for a given sample and detector geometry (thickness). The lower frequency limit – the threshold frequency –for the applicability of the simplified formulaes such as equ. 4.20. is very difficult to determine in the experiment. In order to illustrate the effect of the threshold frequency one can proceed from equ.3.8, the full-expression for the thermal wave [Mandelis and Zver, 1985] θ p (ω 0 ) = I0 2σ p L p {2(b [[ β sη s k β 2 −σ 2 s s s ( r + 1) − (rs + 1)(bsg + 1)e sg s σ s Ls ) ({[ ] [ ] } σ p Lp −σ L e − 1 (bbp + 1) − 1 − e p p (bbp − 1) × ... + (rs − 1)(bsg − 1)e −σ s L s ]e − β s Ls ]) β pη p e − β s Ls + k β 2 −σ 2 p p p ( ) × 45 {[({[e σ p Lp (b ps [ − 1](bbp + 1) − 1 − e − 1)}(bbp − rp )e −β p Lp −σ p L p ](b bp ) − r1 {(b bp {[ } − 1) (bps rr + 1) + e + 1)(bbs + 1)e σ pLp [ − 1](bps + 1) + 1 − e σ p Lp + (bbp − 1)(bbs − 1)e ](b − 1)}(bps rp − 1) + e −σ p L p ]× ]× −σ p L p }[1 − e −β p Lp p (b sg + 1)eσ s Ls + [1 − e −σ p L p ](b ps [({[e σ pLp ] [ − 1 (bbp + 1) − 1 − e + 1)}(bbp − rp )e −β p Lp −σ p L p bp )− r1 {(b {[ + 1)(b ps − 1)e σ p Lp bp σ pLp ] − 1 (bps − 1) + + (bbp − 1)(bps + 1)e −σ p L p }× p [1 − e (b sg −β p Lp ]](b sg [ [ }} ( − 1)e −σ s Ls ÷ (bsg + 1)(bbp + 1)(bps + 1)e − 1)(bbp + 1)(bps − 1)e σ pLp σ pLp + (bbp − 1)(bps + 1)e −σ p L p + (bbp − 1)(bps − 1)e ]e ) −σ s L s −σ p L p ]e σ s Ls + (4.25) σ j = (1 + i )a j where ω a j = 0 2 Dth bmn = k m am k n an X = exp(σ s Ls ) Y = exp (σ p Ls ) Z = exp (σ p (Ls + L p )) E= F= I 0 β pη p ( 2k p β p2 − σ p2 ( ( ) exp − β s I 0 β sη s 2k s β s2 − σ s2 ( ) − β p )Ls ) rj = βj σj Substituting Eq. (4.25) into Eq.(4.21) one obtains a full expression for the PPE signal. Figure 4.19 presents the frequency dependence of the numerical simulations of the PPE amplitudes with thermal properties of the detector as parameters. The normalized amplitudes and phases of the PPE signal were simulated numerically using MATLAB program and Eq. (4.25). The glassy carbon was used as a reference sample. 46 -7 α p =10 [m /s],k p =0.9[W /m K] -0.2 -7 α p =10 [m /s],k p =0.9[W /m K] -7 ln(Normalized PPE Amplitude[a.u.]) α p =5x10 [m /s],k p =1.13[W /m K] -7 α p =5x10 [m /s],k p =1.13[W /m K] -7 α p =8x10 [m /s],k p =1.6[W /m K] -7 α p =8x10 [m /s],k p =1.6[W /m K] -0.4 -0.6 0 1 F threshold 1/2 2 1/2 Frequency [H z ] Figure 4.19: Numerical simulations based on equ. (4.25) (lines) and equ (4.20)(scatters) of normalized PPE amplitudes as function the square root of the modulation frequency for different thermal properties of PZT detector. The others parameters are: Lp=0.98 mm, Ls=1.325 mm, Lr=0.98 mm, Ds=4.8 10-6 [m/s], Dr=4.2 10-6 [m/s], Dg=2.2 10-5 [m/s],ks=9 [W/mK], kr=6.3 [W/mK], kr=0.026 [W/mK],αs=106[1/m], αr=106 [1/m]. It is clearly seen that the threshold frequency Fthreshold (indicated in Figure 4.19) is independent of the thermal properties of the PZT detector. Changes of the magnitude of the normalized PPE amplitude are unfavorable when the thermal conductivity has to be determined. From Figure 4.19 it is evident that the normalization procedure doesn’t solve this problem in the used range of the thermal properties. In fact, comparing Fig. (4.16) and Fig. (4.19) one can deduce that in limited temperature range around room temperature the thermal properties of the detector don’t change much and so their influence on the normalized PPE amplitudes can be neglected. 47 Marinelli [Marinelli, 1992] found that from the as-measured PPE phase it is possible to estimate an absolute value of the thermal diffusivity. The numerical simulations of the asmeasured PPE phases are presented in figure 4.20. -5 PPE Phase [deg.] -10 -15 0 -5 -10 -15 PPE Phase [deg.] -20 -20 -25 -30 -35 -40 -45 -50 -55 0 1 2 Frequency -25 0 3 1/2 4 1/2 [Hz ] 1 1/2 Frequency 1/2 [Hz ] Fthreshold 2 Figure 4.20 Numerical simulations based on equ. (4.25) (lines) and equ (4.20) (scatters) of PPE phasess as a function of the square root of the modulation frequency for different thermal properties of PZT detector. The others parameters are: Lp=0.98 mm, Ls=1.325 mm, Lr=0.98 mm, Ds=4.8 10-6 [m/s], Dr=4.2 10-6 [m/s], Dg=2.2 10-5 [m/s],ks=9 [W/mK], kr=6.3 [W/mK], kr=0.026 [W/mK], αs=106[1/m], αr=106 [1/m]. It is clearly seen that the as measured PPE Phases are not affected by any changes in thermal properties of the detector in the investigated frequency range when the sample and the detector are both thermally thick and optically opaque. Below the threshold frequency the PPE phases described by Eq. (4.25) deviates from their equivalent curves plotted from Eq. (4.20). For 48 this reason the m coefficient for glassy carbon were determined in the frequency range from 2 to 10 Hz. Comparing Fig. (4.19) and Fig.(4.20) one can conclude that the threshold frequency is more less the same for normalized PPE amplitudes and as measured PPE phases. 49 Chapter 5: Thermal properties of Cd1-xMgxSe single crystals measured by means of photopyroelectric technique. 5.1. Materials: Cd1-xMgxSe single crystals Cd1-xMgxSe single crystals were grown by the high-pressure Bridgman method without seed under an argon overpressure [F. Firszt et al., 1995]. The mixture of CdSe and metallic Mg was put into a graphite crucible. The purity of CdSe and Se reaction components was 6N that of Mg was 99.8 %. The temperature of the heating zone was kept at (1880 ± 0.5) K. The crucible was held at that temperature for 2 h and then moved out from the heating zone with lowering speed 4.2 mm⋅h-1. The obtained crystals were cylinders with raw dimensions 8 mm to 10 mm in diameter and 40 mm to 50 mm in length. The crystals have a longitudinal gradient of magnesium concentration (Mg content increases from the tip to the end of the crystal). For a crystal with x = 0.3, the Mg concentration gradient is about 0.01 cm-1. The crystals exhibit the wurtzite structure. The lattice constant of Cd1-xMgxSe crystals decreases with increasing Mg content [F. Firszt et al. , 1998]. The crystals were cut perpendicular to the growth direction into 0.9 mm to 1.5 mm thick plates. Next the plates were mechanically polished and chemically etched in a mixture of 50 K2Cr2O7, H2SO3, and H2O in the proportion 3:2:1. Then they were treated in CS2 and hot 50 % NaOH solution and finally rinsed in water and ethyl alcohol. Table 5.1: Basic information of Cd1-x MgxSe single crystals. Sample Technological Magnesium concentration, Thickness, L (mm) Number x (mole fraction) 1 A423_10 0.00 1.325 2 A557_III 0.00 0.923 3 A558_2 0.06 1.043 4 A440A_VI 0.14 1.306 5 A559_3 0.15 0.944 6 A433_XIV 0.33 1.174 7 A424A_X 0.36 1.294 51 5.2. Experimental results and computational algorithm 5.2.1 Thermal diffusivity – PPE Phases Figure 5.1 shows experimental phase lags of investigated mixed semiconductors as a function of the square root of the modulation frequency. 0.0 PPE phase, radians -0.5 -1.0 -1.5 Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 -2.0 -2.5 -3.0 -3.5 1.2 1.4 1.6 1.8 2.0 2.2 1/2 2.4 2.6 2.8 3.0 3.2 1/2 Frequency , Hz Fig. 5.1: Experimental PPE phase lags (scatters) of the investigated crystals with linear fittings (lines) vs. square root of the modulation frequency. The fitting parameters are collected in Table 5.2. The thermal diffusivities reported in Table 5.2 were calculated using Eq. (4.23) by fitting the as– measured PPE phases, from Fig. 5.1, with Eq. (4.22). 52 Table 5.2 Results of linear fits to Eq. (4.22) and calculated values of the thermal diffusivities of Cd1-xMgxSe mixed crystals from Eq. (4.23). Sample Magnesium Thickness, concentration, x (mole L (mm) m Thermal diffusivity D (10-6 m2⋅s-1) fraction) 1 0.00 1.325 -1.08 4.73 2 0.00 0.923 -0.77 4.48 3 0.06 1.043 -1.26 2.15 4 0.14 1.306 -1.79 1.67 5 0.15 0.944 -1,34 1.58 6 0.33 1.174 -1.85 1.26 7 0.36 1.294 -2.12 1.17 From Table 5.2 it is clearly seen that with increasing magnesium concentration the thermal diffusivity decreases markedly. Figure 5.2 shows experimental data and theoretical curves of the temperature dependence of thermal diffusivities for the investigated crystals. The fits assume a linear temperature variation Dt(T)=aT. 53 5.0 -6 2 Thermal diffusivity, x 10 m ·s -1 4.5 4.0 3.5 Sample 1,a=-0.04 Sample 2,a=-0.04 Sample 3,a=-0.01 Sample 4,a=-0.006 Sample 5,a=-0.006 Sample 6,a=-0.006 Sample 7,a=-0.006 3.0 2.5 2.0 1.5 1.0 20 22 24 26 28 30 32 34 36 38 40 42 44 Temperature, °C Fig. 5.2: Temperature dependence of the thermal diffusivity for the investigated crystals fitted with a linear temperature dependence Dt(T)=aT with a given in the figure. It is clearly seen that with increasing temperature the thermal diffusivity for all investigated crystals decrease. The steepness of this slope decreases with increasing magnesium concentration. 54 5.2.2 Thermal conductivity – normalized PPE amplitude The obtained values of thermal diffusivities for investigated crystals from raw (asmeasured) PPE phases and amplitudes were substituted in the normalized PPE amplitude An(f) defined by An ( f ) = Amp sample Amp glassycarbon , (ratio of the amplitude of the sample and that of the reference sample ) in order to determine thermal conductivities. The nonlinear data-fitting procedure relied on minimizing the following expression in a least-squares sense: 1 N (F (k s , f i ) − An ( f i ))2 ∑ 2 i =1 (5.1) Only one parameter (thermal conductivity ks) was applied. F(ks,fi) is the theoretical PPE amplitude (and normalized by theoretical response of the PPE amplitude of the glassy carbon) described by Eq. (4.21) and N is the number of experimental points. The nonlinear data-fitting was based on the built-in MATLAB function LSQCURVEFIT. The following parameters were used during the fitting procedure for a pyroelectric detector kp=1.13 W⋅m-1⋅K-1 and Dp=4.95x10-7 m2⋅s-1 and for the glassy carbon ks=6.3 W⋅m-1⋅K-1 and Ds=4.22⋅10-6 m2⋅s-1. It was assumed that the temperature dependence of the thermal conductivity of the glassy carbon results only from that of the thermal diffusivity as determined from the PPE phases. Figure 5.3 shows the best fits to the normalized PPE amplitude for sample 1. 55 0.50 Normalized PPE Amplitude, a.u. o 23 C o -1 -1 23 C, k = 9.94 W·m ·K o 25.1 C o -1 -1 25.1 C, k = 9.6 W·m ·K o 27.7 C o -1 -1 27.7 C, k = 9.28 W·m ·K o 31 C o -1 -1 31 C, k = 9.14 W·m ·K o 33.0 C o -1 -1 33 C, k = 8.94 W·m ·K o 36.3 C o -1 -1 36.3 C, k = 8.39 W·m ·K 0.45 0.40 0.35 0.30 4 6 8 10 Frequency, Hz Fig. 5.3: Best fittings (lines) to the normalized PPE amplitude (scatters) of sample 1 at different temperatures. 5.2.3 Discussion It is clearly seen that with increasing temperature the thermal conductivity for sample 1 decreases. For the temperature 27.7 °C (300.7 K) the thermal conductivity of sample 1 (CdSe) is 9.28 W⋅m-1⋅K-1. This value is in good agreement with the thermal conductivity of CdSe (9 W⋅m1 ⋅K-1) crystal obtained by Slack [Slack, 1972]. Additionally, for single crystals, the thermal conductivity may depend on crystallographic directions as in the case of Zn0 [Slack, 1972]. Although our samples were not oriented, this effect can be neglected because measurements were performed at high temperatures. The frequency-dependent normalized amplitude obtained at different temperatures from the other samples have been fitted in the same way. The best results for all crystals of the fitting procedure are collected in Table 5.3 and Fig. 5.4. Figure 5.4 presents the temperature dependence of the thermal conductivity for Cd1-xMgxSe mixed crystals with different magnesium concentrations. 56 Sample 1 n=-0.32 Sample 3 n=-0.28 Sample 4 n=-0.23 Sample 5 n=-0.29 Sample 6 n=-0.29 Sample 7 n=-0.3 -1 Thermal Conductivity, W·m ·K -1 9 6 3 25 30 35 40 45 50 Temperature, C Fig. 5.4: Temperature dependence of the thermal conductivity (scatters ) for Cd1-xMgxSe mixed crystals with different Mg concentrations and the best fits (lines) to k(T)=aTn on log-log scale. The dynamics of these changes can be expressed by coefficient n extracted from k(T)=aTn on loglog scales. Values of coefficient n are presented in Fig. 5.4. Obviously, n is nearly constant for all investigated crystals within the error limit of the fitting (±0.05). The electron contribution to the thermal conductivity can be calculated for a degenerate semiconductor (because carrier concentrations of investigated crystals are high) from a formula given by [Bhandari and Rowe, 1998], k k e = e 2 σLT , (5.2) where T is temperature, e is the elementary charge, and k is the Boltzmann constant. The electrical conductivity σ for samples 1, 4 and 6 were taken from Hall measurements [Perzynska 57 et al., 2000] and the Lorenz factor was assumed to be π 2 3 . The obtained results are collected in Table 5.3. Table 5.3 Thermal conductivity k, electrical conductivity σ and electron contribution to thermal conductivity ke of Cd1-xMgxSe mixed crystals at room temperature as results of the best fits using MATLAB Sample Magnesium Thickness L concentration, x (mm) ks (W⋅m-1⋅K-1) Electrical ke (W⋅m-1⋅K-1) Conductivity σ (Ω-1 ⋅m-1) (mole fraction) 1 0.00 1.325 9.28 1000 0.007 2 0.00 0.923 - - - 3 0.06 1.043 5.34 - - 4 0.14 1.306 3.82 1200 0.009 5 0.15 0.944 3.93 - - 6 0.33 1.174 3.13 500 0.004 7 0.36 1.294 3.00 - - Since the electron contribution to the thermal conductivity is very small indicating that the heat is mainly carried by phonons. This behavior is expected for CdSe crystals at room temperature [Slack, 1972]. Therefore, the thermal conductivity can be related only to the lattice contribution and the thermal resistivity can be described as the inverse of the (lattice) thermal conductivity. Figure 5.5 presents the thermal resistivity W as a function of magnesium concentration at room temperature. The thermal resistivity W increases with increasing magnesium concentration. This behavior is expected when large numbers of magnesium atoms are added to the host lattice and they act as scattering centers for phonons. A similar behavior of the thermal resistivity was found by Adachi [Adachi, 1983] in III-V semiconductors. He gave the expression of the thermal resistivity as a function of concentration x [Adachi, 1983]: WCdMgSe ( x) = xWCdSe + (1 − x )WMgSe + C Cd -Mg x(1 − x ) , (5.4) where WCdSe and WMgSe are the thermal resistivities of CdSe and MgSe, respectively. The coefficient CCd-Mg is called a nonlinear parameter and it is a contribution arising from the lattice 58 disorder generated in ternary Cd1-xMgxSe system by random distribution of Cd and Mg atoms in one of the two sublattice sites [S. Adachi, 1983]. In Fig. 5.5 the best fitting (line) to Eq. (5.4) is also shown. 0.55 0.50 Thermal resistivity, m·K·W -1 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Mg concentration, x Fig. 5.5: Reciprocal thermal conductivity (thermal resistivity) of the Cd1-x MgxSe mixed crystals as a function of magnesium concentration at room temperature. The nonlinear parameter was found to be 1.59 W⋅m-1⋅K-1 while the thermal conductivity of the MgSe is 7.69 W⋅m-1⋅K-1. Although MgSe does not exist in nature, the value obtained from fitting is rather unexpected. This can be explained because of the Adachi model is a simplification of the Abeles model [Abeles, 1963]. Adachi had taken into account only strain scattering while for GeSi alloys mass-defect scattering could be important. Similarly for the mass difference can be important. Tsen et al. [Tsen et al., 1997] reported that the total electron-longitudinal optical phonon scattering rate in GaN is about one order of magnitude larger than that in GaAs. They attributed this enormous increase in the electron-longitudinal optical phonon scattering rate to the 59 much larger ionicity in GaN. In Cd1-xMgxSe mixed crystals an increasing Mg content in the solid solution will increase the iconicity. Supplementary Hall measurements on the mixed crystals indicate a high carrier concentration; hence, this scattering mechanism cannot be neglected [K. Perzynska et al., 2000]. . 60 Chapter 6: Influence of the space charge layer (SCL) on the charge carrier transport properties measured by means of the photocarrier radiometry (PCR) In Chapter 4 an introduction to the PCR technique was presented. It was shown that the PCR signal is sensitive to the carrier transport properties and its linear function of the optical excitation intensity. In this chapter the influence of the space charge layer on the PCR signal is presented. Part of the following results were published by A.Mandelis, J.Batista, M.Pawlak, J.Gibkes and J.Pelzl in J. Phys. IV France 125 (2005) 565-567 and Journal of Applied Physics 97 083507 (2005). 61 6.1 Theory of optically modulated p-type SiO2-Si interface energetics in the presence of charged interface states [Mandelis, 2005a] The Si02-Si interface energy diagram of p-type Si in the presence of the positively charged interface state density Nt (m-2) acting as traps of free minority carriers (electrons) is presented in Figure 6.1. Figure 6.1 Band-structure energetics at a p-type Si-SiO2 interface with a positively charged interface state (trap) density Nt assumed to be at energy Et. The band bending is modulated by the external optical field. [Mandelis, 2005a]. Other parameters are defined in text. At equilibrium in the dark [Sze, 1981] the energy bands at the interface are bent with a total interface potential energy qψ s 0 , measured with respect to the intrinsic Fermi level, causing the maximal value of the space charge layer W0. Additional W0 serves as the reference value for the non equilibrium configuration under optical incidence of intensity I 0 (α , hν ) (W m-2). The effective SCL width can be defined by ∆W = W0 − Wm and Wm were changed with the incident modulated laser intensity I 0 . W0 and Wm were the dc and modulated components of the SCL: W ( I 0 ) = W0 − W m e i ω t . (6.1) 62 The modulated excitation generates a free-carrier density wave mostly within depth µ = 1 α [m-1] from the surface [Mandelis, 2001] that usually includes the very thin ( ~ 0.5µm in Si) space charge layer. This is a coherent excitation of minority carriers which is characterized by a frequency-dependent (ac) diffusion length Ln (ω ) = Dn* ⋅ τ n , (1 + i ⋅ ω ⋅ τ n ) (6.2) where Dn* is the ambipolar diffusion coefficient and τ n is the bulk lifetime of the CDW. Mandelis [Mandelis, 2005a] proved that these parameters are composite (effective) quantities involving interface as well as bulk values. Additionally, he pointed out that the interface lifetime value is affected by the details of the trapping dynamics. The minority carriers (electrons) within an ac diffusion length from the edge of the SCL can be swept into it and slide down the energyband slope of the SCL edge under depletion or inversion conditions for the majority carriers (hole). This increases the charge density within the SCL, which in turns, affects the occupation of the interface affects the occupation of the interface state Et. Under interface illumination with intensity I 0 (λ )e iωt there is a non-zero probability that a fraction of the occupied interface states, Nt0, will absorb photons from the incident radiation and will eject trapped electrons into the conduction bandedge, which will increase the degree of band-bending [Mandelis 2005a]. The SCL acts as a thin spatial region (~ 0.5 µm) in which recombination is essentially absent due to efficient separation of the local electron-hole pair and the completely ionized impurity states at, or above, room temperature. As a consequence, the recombination lifetime in this region has been set equal to infinity, Fig. 6.2. 63 Figure 6.2 Optical source depth profile and carrier-density-wave transport parameters in p-type semiconductor Si with a transparent surface oxide and interface state density Nt [Mandelis, 2005a] 6.2 The expression of the PCR signal including effects due to an existing SCL The expression of an influence the SCL width under optical modulation on the PCR signal was given by Mandelis [Mandelis, 2005a] η Q [1 − R1 (λα )]I 0α Tri (ω ) S PCR ,SCL ( I 0 , ω ;α ) ≅ C 0 ( R1 , R2 ;α IR ) 1 − (1 + αWm )e −α∆W 2hν α (6.3) γ1 1 + * 2 (1 + αWm ) − [1 + (α − σ e )Wm ]e −σ e L } Γ2 + e −σ e L 2 − 2σ L Dn (α − σ e ) σ e (Γ2 − Γ1e e ) [ { [ ( + ∆W − Wm (1 − α∆W ) Γ2 + e −2σ e L ] ( ) )] ] − 21α [(1 + αW ) − e ]− [∆W − W (1 − α∆W )] }e −αL m −α∆W m 64 where Tri(ω) is a [Mandelis,2005a]) Tri (ω ) ≡ complex interface lifetime defined as (Appendix A in τ ri ,where τri is a charged interface recombination lifetime. In 1 + iωτ ri case of no charged surface states, then ∆W = 0 and W0 = Wm ( I 0 ) = 0 for I 0 > 0 , equation (6.3) reduces to (4.5). It is worthwhile to emphasize that the conventional expression of the PCR signal (4.5) exhibits a linear dependence on I 0 while the structure of equation (6.3) depicts – in case of the presence of a nonzero density of charged interface states – that there will be a nonlinear lowintensity range bound up with SCL oscillation from maximum value of W0 (in the dark) and 0 (full illumination). 6.3 Numerical simulations of an influence of the existence of the SCL on the electronic transport properties 6.3.1 Numerical simulation of the PCR signal dependence on the electrical transport properties in the presence of SCL For the simulation purpose it was assumed that exponential dependence of the SCL width W0 (=1µm) on IAC, ∆W ≡ W0 − Wm = A exp(− BI ac ) , (6.5) where A,B are constants (set 0.1x10-5 and 2x10-4, respectively). The results of numerical simulations for the fixed modulation frequency (200 Hz) of the effects of changing bulk minority recombination lifetime are shown in Fig. 6.3, and those of changing interface recombination velocity are shown in Fig. 6.4. 65 0,1 1 13 10 13 12 10 11 10 PCR amlitude[a.u.] 10 12 10 11 10 τ=1µs linear range 10 10 τ=10µs 10 10 linear range 0,1 1 2 Intensity [W/cm ] 0,0 -0,1 PCR phase [deg.] -0,2 -0,3 τ=1µs τ=10µs -0,4 -0,5 -0,6 -0,7 -0,8 1 2 3 2 Intensity [W/cm ] Figure 6.3: Simulated PCR amplitudes (a) and phases (b) as a function of optical intensity for p-type Si with recombination lifetime as a parameter. Other parameters: Dn* = 30 cm2/s, , S1 = 300 cm/s, S2 = 105 cm/s, L = 550 µm, α(λ=514nm) = 7.76×103 cm-1. 66 PCR amlitude[a.u.] 0,1 1 10 13 10 13 10 12 10 12 10 11 S1=10 cm/s linear range 10 S1=100 cm/s 11 linear range 0,1 1 2 Intensity [W/cm ] -1,3 -1,4 -1,5 PCR phase [deg.] -1,6 -1,7 -1,8 S1=10cm/s -1,9 S1=100cm/s -2,0 -2,1 -2,2 -2,3 -2,4 -2,5 0 1 2 3 2 Intensity [W/cm ] Figure 6.4 Simulated PCR amplitudes (a) and phases (b) as a function of optical intensity for p-type Si with surface recombination as a parameter. Other parameters: Dn* = 30 cm2/s,τ = 100 µs, S2 = 105 cm/s, L = 550 L = 550 µm,α(λ=514nm)=7.76×103 cm-1. 67 In both cases a similar shape of the PCR amplitudes is observed. Additionally, if the SCL exists the PCR signal deviates from the linear dependence on the optical excitation intensity. The straight line in Fig.6.3a and 6.4a follows the linear dependence on the optical excitation intensity above around intensity 1.5 W/cm2 (the intensity for which the PCR is a linear function of the intensity is named Ilinear) where the SCL is completely vanished (calculated value of the effective for 2 W/cm2 is ∆W=2x10-8 m). In both cases the shapes of the amplitude curves, A(Ilinear), are similar. The phases, φ(Ilinear), remain flat, because the interface recombination lifetime was assumed constant. From the PCR Phases one can see that this effect can be more important for wafers with a long lifetime. 6.3.2. Numerical simulation of the PCR Signal dependence on the existence of SCL width Assuming negligible Wm for low intensity of the modulated laser compared to W0 an influence of the dc W0 and modulated components Wm of the SCL width on the PCR signal the numerical simulations using MATLAB are shown in Figure 6.5 and Figure 6.6, respectively. 68 0 20 10 (a) (b) -40 W 0=0µm W 0=0µm W 0=0.2µm 19 W 0=0.2µm W 0=0.4µm 10 W 0=0.4µm W 0=0.6µm W 0=0.6µm W 0=1µm -60 W 0=1µm PCR phase [deg.] PCR amplitude [a.u.] -20 -80 18 -100 10 10 100 1000 10000 Frequency [Hz] 100000 10 100 1000 10000 100000 Frequency[Hz] Fig.6.5: PCR amplitudes (a) and phases (b) vs. modulation frequency simulations for ptype Si with constant transport parameters and with the SCL width as a parameter. Other parameters: Wm = 0, Dn* = 30 cm2/s,τn = 100 µs, S1 = 300 cm/s, S2 = 105 cm/s, L = 550 µm, α(λ=514nm) = 7.76×103 cm-1. Figure 6.5 shows that increasing W0 results in a monotonically decreasing amplitude and essentially no change in phase occurs. This behavior is expected, since an increased degree of band-bending is the result of increased interface charge density in the model and thus higher trap density and loss of free carriers. The process takes place right at the Si-SiO2 interface where free minority carriers de-excite mostly non-radiatively in trap states over the space-charge barrier [Johnson, 1958] and therefore are not available to contribute to the PCR signal through radiative NIR emissions. Furthermore, there is no measurable phase shift because the interface 69 recombination lifetime τs was assumed constant. This simplification turns out not to be true experimentally, but the simulation points out the important fact that it is not the value of W0 itself that causes a phase shift, but rather the change in this value has an effect on the transport properties at the Si-SiO2 interface. (a) 0 (b) 20 10 -40 W m=0µm W m=0µm W m=0.2µm W m=0.2µm W m=0.4µm W m=0.4µm W m=0.6µm W m=0.6µm W m=1µm PCR phase [deg.] PCR amplitude [a.u.] -20 -60 W m=1µm 19 10 -80 -100 1 10 2 10 3 10 4 10 5 10 1 10 Frequency [Hz] 2 10 3 10 4 10 5 10 Frequnecy[Hz] Figure 6.6 PCR amplitudes (a) and phases (b) vs. modulation frequency simulations for p-type Si with constant transport parameters and with the optically modulated SCL width, Wm, as a parameter. W0 = 1 m, Dn* = 30 cm2/s, τ= 100 µs, S1 = 300 cm/s, S2 = 105 cm/s, L = 550 µm, α(λ=514nm) = 7.76×103 cm-1. Figure 6.6 corresponds to the case where the maximum value, W0, of the SCL width is fixed, but is subject to oscillating amplitude, Wm, which increases with, e.g., increasing intensity of the modulated laser source. As expected, the PCR amplitude increases as the modulated band curvature tends to offset the effect of the CDW-inhibiting band bending through more effective 70 neutralization of interface charges over the modulation cycle. As a result a higher density of free minority carriers can survive over one illumination period and contribute radiatively to the increased PCR signal. As in the case of Fig. 6.6b, the phase does not show any change over the entire range of Wm values used in this simulation. When the changes in the SCL widths, Figs. 6.5 and 6.6, and in transport parameters, Figs. 4.4-4.7, are combined in a frequency plot for fixed values of the transport parameters, it is found that the former simply shift the PCR amplitude accordingly, while the phase remains fixed. This conclusion proves to be very helpful in interpreting experimental PCR measurements of the SCL. It is worthwhile to emphasis that changing of the physical parameters responsible for the transport properties of the carriers (electronic parameters) and the space charge layer can shift the PCR amplitude. Unlike to the PCR Phase which is only sensitive for altering electronic parameters. 6.4 Experimental Conditions and Materials 6.4.1 Experimental Methodology In chapter 4 it was shown that the PCR signal is not only sensitive to the electrical and optical properties of a material but also to experimental conditions like the beam size. The effect of the beam size on the PCR signal was discussed in Section 4.1.6. In some cases the optical excitation intensity, another experimental condition, can affect the linear dependence of the PCR signal on the optical excitation intensity. In case of the presence of a nonzero density of charged interface states – that there will be a nonlinear low-intensity range of the PCR signal dependence on the optical excitation intensity bound up with SCL oscillation from maximum value of W0 (in the dark) and 0 (full illumination). For monitoring effects of the SCL on the PCR signal two-laser system was proposed. A fixed (low) intensity modulated (“ac”) laser produces a fixed density of free electron pairs (EHP) carrier waves acting as PCR probe. During that time a coincident unmodulated (“dc”) laser source with variable intensity, I dc , substantially exceeding that of the modulated laser but for the lowest values, can change the occupation of the surface states up to the flatband condition when I dc = I FB and acts as optical bias. In this configuration the dc laser induces a change in degree of 71 steady band bending at the surface from the dark (maximum) value, W0 (I dc = 0 ) = Wmax ,of the SCL width up to flatband (minimum) value, W0 (I FB ) = 0 . In this ideal case, the modulated laser is non-perturbing, acting only as a PCR signal carrier with negligible effect on the SCL width, since Wm(IAC)<< W0(IDC). A variation in dc laser intensity within the range 0 ≤ I DC ≤ I FB is expected to act as a variable optical bias by means of the steady-state excess recombination events of minority carriers into impurity states. The enhanced steady bulk recombination further affects (increases) the steady SCL minority charge density Qsi, driving the SiO2-Si interface into deeper depletion or inversion. This change alters the (thermo)dynamically coupled interface charge state coverage [Mandelis,2005a] causing a concomitant change in the degree of bandbending between maximum and zero (flatbands). Because these interface changes perturb the small CDW generated by the modulated laser, the entire process can be described according to Eq. (6.3), with an effective steady value of W0(IDC) for each IDC value and a fixed Wm(IAC). 6.4.2. Experimental set up Figure 6.7 presents the designed and constructed two-laser PCR system. Figure 6.7 Two laser PCR experimental set up. 72 As an excitation source of carrier density waves a modulated (by means of a chopper) lowpower laser a He-Ne laser (632.8 nm; 0.4 mm beam radius) or a 830 nm GaAlAs laser diode (0.3mm beam radius) was used. The non-modulated optical bias laser was an Ar-ion laser (514 nm; 1.89 mm beam radius). The power of the modulated laser was typically PAC ~ 0.5 - 4 mW, whereas the power of the non-modulated laser was varied up to 350 mW. The laser beams were focused onto the sample surface using lens. The position of the laser beams is coincident with the focal point of an off-axis paraboloidal mirror that collects a portion of infrared radiation from the samples. The collected light is then focused onto the detector by means of lens. Samples were placed onto aluminum holder (acted as a mechanical support and signal amplifier by redirecting the forward emitted IR photons back toward the detector [Mandelis, 2003]. All instruments, data acquisition, and data storage are controlled by a computer running Pascal program with a graphical user interface and real-time display of experimental data. Taking into account the reflectivity of Si at 514 nm at normal incidence (R = 0.38) and the laser beam radius r = 1890 µm, the effective maximum photon flux at 350 mW was Fp,max = 1.103×1018 photons/cm2s. (6.4) 6.4.3. Materials Samples were four and six-inch 5-10 Ω−cm, 550-µm thick, p-type Si wafers which were oxidized with a gate oxide of ca. 1000 Å. In addition, n-type wafers were also oxidized with ca. 5000 Å oxide. 6.5 Experimental results 6.5.1 Effect of chemical etching on the PCR signal Some wafers were exposed to variable optical bias with the laser beam incident on the SiO2 and then were etched to hydrophobia in an aqueous solution of 10% vol. HF in water, indicating that the SiO2 layer was fully removed. The PCR signal amplitudes and phases of a p-Si sample before and after etching are shown in Fig. 6.8. While the oxidized wafer exhibits complete photosaturation at irradiation with approx. 250 mW of continuous laser power, it is clear that the removal of the oxide also removed the interface-charge layer very efficiently while increasing the 73 non-radiative electron trapping efficiency at the etched surface. Accordingly, the PCR amplitude dropped significantly and remained independent of the power of the unmodulated laser. The Arion laser-beam reflectance of the wafer before and after etching was measured to be 0.340 and 0.408, respectively. The reflectance of the primary He-Ne beam was 0.295 and 0.339, respectively. These differences cannot account for the drastic change in the PCR amplitude after etching, giving further support to enhanced non-radiative recombination. The PCR phase, Fig. 6.8(b), shows a very reproducible curvature for the oxidized sample. On the other hand, the phase of the etched sample is independent of PDC and became noisier due to the low signal associated with this sample. Signal-to-noise ratios (SNR) for oxidized wafers were in the 100 – 200 range with error bar sizes similar to the symbol size used in the plots, whereas those for the etched samples were approx. 18 - 25. 160 (b) (a) 140 100 95 90 100 80 85 60 40 20 0 0 p-Si Unetched p-Si Etched 100 200 PDC (mW) PCR phase [deg.] PCR Amplitude(µV) 120 80 300 0 100 200 75 300 PDC (mW) 74 Figure 6.8 (a) Amplitudes and (b) phases of an oxidized p-Si wafer before and after etching the SiO2 away. The modulated beam was provided by a mechanically chopped He-Ne laser. Chopping frequency: 200 Hz. In the case of oxidized wafer the PCR amplitude exhibits complete photosaturation at irradiation approximately 250 mW of laser power. Unlike to oxidized wafer, the PCR amplitude of etched wafer dropped drastically and remained independent of the power of the unmodulated laser. This significant change in PCR amplitudes was due to etching which enhanced nonradiative recombination and so attenuated radiative recombination. In figure it is seen that the PCR phase was reproducible only in case of oxidized wafer signal. The PCR phase of the wafer after etching depends on the non-modulated laser power and becomes noisier due to the low signal associated with this sample. 6.5.2 The perturbation effects of the primary modulated laser beam on the PCR signal The issue of possible perturbation effects of the primary modulated laser beam on the SCL measurements was investigated by changing the power of the He-Ne laser and repeating the experiment of Fig. 6.8 using another oxidized p-Si wafer. Typical results are shown in Fig. 6.9. As expected, the PCR signal amplitude does scale with PAC (or, equivalently, I0), Eq. (6.3), however, the re-scaled amplitude of the measurement with decreased power at 1.4 mW coincides with the 3.5-mW amplitude, when normalized to the highest point of the latter curve. Similarly, the phases coincide with no rescaling within error, with the phase obtained at the lower power exhibiting higher noise. These results demonstrate that, in the range of the reported measurements, the modulated laser was not perturbing the electronic properties of the semiconductors and the observed signal variations with PDC were solely due to the effects of the dc laser on the sample. 75 90 89 140 120 100 80 PAC = 3.5 mW 60 PAC = 1.4 mW 40 PAC = 1.4 mW; normalized PCR Phase (deg) PCR Amplitude (µV) (b) (a) 160 88 PAC = 3.5 mW 87 PAC = 1.4 mW 86 85 20 0 84 0 50 100 150 200 250 300 PDC (mW) 0 50 100 150 200 250 300 PDC (mW) Figure 6.9 a) Amplitudes and (b) phases of an oxidized p-Si wafer using two different power levels of the primary (modulated) laser beam. He-Ne laser chopping frequency: 200 Hz. 6.5.3 The effect of polishing on the PCR signal Figure 6.10 shows that the effects of surface polishing on the PCR signal from the p-type wafer of Fig. 6.9 (amplitudes and phases) are minor. The back matte surface was also oxidized and therefore it was expected that it would exhibit an SCL behavior similar to the front surface. Based on the fits to the theory as described below, these effects can be accounted for by small differences in the respective surface recombination velocity, effective lifetime and SCL depth profile. 76 -5.0 Polished surface Unpolished surface 180 -5.5 160 -6.5 120 -7.0 PCR Amplitude (µV) 140 -7.5 100 -8.0 80 PCR Phase (deg) -6.0 -8.5 60 -9.0 40 -9.5 20 0 50 100 150 200 250 -10.0 300 PDC (mW) Figure 6.10 Effects of surface polishing on the PCR signal amplitude and phase. He-Ne laser chopping frequency: 200 Hz. 6.6 Determination of carrier transport properties in SCL and the depth profile reconstruction The developed PCR theory [Mandelis, 2005a] resulting in Eq. (6.4) was tested through a series of PCR measurements with Si-SiO2 interfaces, which were aimed at reconstructing the depth profile of the SCL from scans of IDC at a fixed frequency. Results on a p-Si sample such as those shown in Figs. 6.10-6.12 were supplemented by frequency scans launched at several values of IDC, as shown in Fig. 6.11. 77 (b) 0 -20 10 Phase (degrees) PCR Amplitude (mV) (a) Pdc = 0 mW Pdc = 10 mW Pdc = 20 mW Pdc = 30 mW Pdc = 50 mW Pdc = 90 mW Pdc = 200 mW Pdc = 350 mW 1 10 1 10 2 10 3 10 PDC = 0 mW PDC = 10 mW -40 PDC = 20 mW PDC = 30 mW -60 PDC = 50 mW PDC = 90 mW -80 PDC = 200 mW PDC = 350 mW 4 Frequency (kHz) 5 10 -100 10 1 2 10 10 3 10 4 10 5 Frequency (kHz) Figure 6.11 Amplitude (a) and phase (b) frequency scans of a p-Si – SiO2 interface from the polished surface of Fig. 6.8, under various dc laser power levels and 830-nm modulated excitation source. Theoretical fits to Eq. 6.6 are indicated by the continuous lines. Unique best fits were determined by the set of parameters W0 , τeff , S1, S2, and Deff yielding the minimum variance in Eq. 6.6. For all fits it was found that 0.86 % < Var < 1.14 %. The individual frequency scans were fitted to Eq. (6.3) by means of FORTRAN program with ∆W = W0 since the IAC intensity was too low compared to IDC to affect the band-bending as demonstrated in Fig. 6.9. The fitted parameters at each value of IDC included τeff, S1, S2, Deff ≈ Dn and W0. During the fitting procedure, the four transport properties τeff , S1, S2, and * Deff were set as free parameters. The best-fitting procedure commenced at IDC = IFB , where W0 = 0. Then, the first non-zero value of W0 was incremented for the next lower IDC and all five parameters were allowed to vary until the absolute minimum of Eq. (6.6) was attained. The values of the five parameters yielding the absolute minimum in Eq. (6.6) were unique in each case within the physically expected value ranges and fitting errors were recorded. The procedure was repeated for all IDC ≥ 0. Figure 6.12 shows the IDC dependence of the front surface (interface) recombination velocity, S1, in the range used in our experiments. 78 700 600 S1 (cm/s) 500 400 300 200 100 0 0 50 100 150 200 250 300 350 PDC (mW) Figure 6.12 Recombination velocity for a p-Si – SiO2 interface as a function of the Ar-ion laser dc power of a p-Si – SiO2 interface. Data obtained from simultaneous best fits to amplitude and phase frequency scans shown in Fig. 6.11. The decrease of this parameter essentially down to zero is consistent with the physical process of optical neutralization of the interface states by photo-excited minority electrons. The results are also consistent with earlier derived dependencies of the front SRV on the excess electron density in optically biased photoconductance-decay experiments [Aberle, 1996; J. Schmidt and A. G. Aberle, 1997], from surface photovoltage measurements using negative corona charging [Schmidt and Aberle,1997], and from basic Shockley-Reed recombination theory [A. G. Aberle et al. 1992, M. Schoefthaler et al.,1994]. Using the dependence τ s = const. × S1 −1 [Mandelis, 2005a; Appendix Eq. (A5)] results in an increase of the interface recombination lifetime with incident dc laser power. Figures 6.13 and 6.14 show the IDC dependencies of the CDW effective diffusion coefficient, Deff, and effective recombination lifetime, τeff . 79 30 28 26 2 Deff (cm /s) 24 22 20 18 16 14 12 10 0 50 100 150 200 250 300 350 PDC (mW) Figure 6.13 Effective diffusion coefficient of the carrier-density-wave as a function of the Ar-ion laser dc power of a p-Si – SiO2 interface. Data obtained from simultaneous best fits to amplitude and phase frequency scans shown in Fig. 6.11. 500 τeff (µs) 475 450 425 400 0 50 100 150 200 250 300 350 PDC (mW) Figure 6.14 Effective lifetime of the carrier-density-wave as a function of the Ar-ion laser dc power of a p-Si – SiO2 interface. Data obtained from simultaneous best fits to amplitude and phase frequency scans shown in Fig. 80 6.11. In the high dc-power range, PDC > 100 mW, Fig. 6.12 shows that S1 ≈ 0 . Therefore, in that range, according to [Mandelis 2005a: Appendix Eq. (A7)] 1 τ eff = 1 τB + 1 τs , τ eff ≈ τ B ≈ 460 µs, a constant value reflecting purely bulk recombination and consistent with Fig. 6.14. Here τB is the bulk recombination lifetime. Nevertheless, for the same PDC range the effective diffusion coefficient of the CDW decreases from an essentially electron minority carrier diffusivity value of 26 cm2/s following a relatively steep increase. The onset of decrease at high optical bias corresponds to photo-excited carrier densities of 1017 cm-3 (calculated from Eq. (6.4) for excitation at 830 nm) and is consistent with the onset of nonnegligible carrier-carrier scattering reported for Si(111) surfaces [Li et al., 1997]. The increase of Deff at low PDC bias power is the result of the ambipolar nature of Deff ≈ Dn [Mandelis * 2005a] as the free-electron CDW increases in the near-interface region with increased optical interface charge neutralization, thus changing the value of the ambipolar diffusivity from the majority Dp ( ~ 12 cm2/s) to the minority Dn (~ 30 cm2/s) range. The minimum in the value of τeff around PDC = 50 mW , Fig. 6.14, is most likely associated with the increasing values of Deff and τs : The IR emitting photo-carrier density-wave shifts to deeper sub-interface locations, in agreement with the increased PCR phase lag in Figs. 6.8b, 6.9b and 6.10, and this results in the disappearance of a number of contributing carriers from the field of view of the IR detector [Ikari et al.,1999].The computational application of Eq. (6.3), interprets this relative scarcity of carriers as a decreased recombination lifetime. At higher PDC the flattening bands bring about an increased free-carrier density-wave in the immediate sub-interface region (decreased phase lag in Figs. 6.8b, 6.9b and 6.10) which restores the CDW infrared photon emission within the range of the InGaAs visibility solid-angle. Figure 6.15 is the reconstructed depth profile of the SCL width from full band-bending to the complete flatband condition associated with photo-saturation of the PCR signal. 81 0.35 0.30 SCL width (µm) 0.25 0.20 0.15 0.10 0.05 0.00 0 50 100 150 200 250 300 350 PDC (mW) Figure 6.15 SCL width of a p-Si – SiO2 interface as a function of the Ar-ion laser dc optical bias. The reconstruction was obtained from simultaneous best fits to amplitude and phase frequency scans shown in Fig. 6.11. 6.7 Summary The interface modulated-charge-density theory developed by Mandelis [Mandelis, 2005a] was used for physical simulations involving PCR signals in order to study the effects of the various SCL optoelectronic transport parameters on the PCR amplitude and phase signal channels. Furthermore, an experimental configuration was used involving n- and pdoped Si – SiO2 interfaces and a low-intensity (non-perturbing) modulated laser source as well as a co-incident dc laser of variable intensity acting as interface-state neutralizing optical bias. The application of the theory to the experiments yielded the various transport parameters of the samples as well as the depth profile of the SCL. It was shown that PCR can monitor the complete flattening of the energy bands at the interface of p-Si – SiO2 with dc optical powers up to a 300 mW. The uncompensated charge density at the interface was also calculated from the theory. The SCL profile for the n-type Si wafers was also investigated. Typical results are shown in Fig. 6.16. 82 110 40 unetched etched unetched etched 105 100 30 95 25 90 20 85 15 10 0 PCR Phase (deg) PCR Amplitude (µV) 35 80 50 100 150 200 250 75 300 PDC (mW) Figure 6.16 Amplitudes and phases of an oxidized n-Si wafer before and after etching the SiO2 away. The modulated beam was provided by a mechanically chopped HeNe laser chopping frequency: 200 Hz. Both lasers were incident on the exposed SiO2 layer. In all cases, large differences were observed in the PCR amplitudes between intact and etched samples. Nevertheless, there was no indication of flatband saturation within the range of IDC intensities available and phases were always essentially independent of the dc laser power. A comparison among the amplitude shapes and absolute signal levels in Figs. 6.8 and 6.16 shows that it is much harder for the optical bias to induce complete flattening of the bands in n-Si than in p-Si. This observation indicates the relatively low efficiency of this methodology for driving n-Si into the flatband state. The independence of PCR phases from IDC is an additional indicator that the optical bias does not affect the transport properties of the SiO2 - n-Si interface. This behavior can be explained by the much lower (ca. 100 times [Aberle, 1992]) minority carrier capture cross-section of the n-type interface. This fact leads to much shorter interface lifetimes, τs [Mandelis, 2005a], making it harder for an optical source to build up the neutralized interface charge coverage observed with p-Si – SiO2 interfaces, Fig. 6.15. The same effect makes the interface recombination velocity essentially independent of excess carrier density at the n-Si – SiO2 interface [Aberle, 1992] which results in the independence of the PCR phase from PDC, Fig. 6.16. 83 Chapter 7: Non-linear dependence of Photocarrier Radiometry signals from p-Si wafers on optical excitation intensity and its effect on charge carrier transport properties In Chapter 4 an introduction to the PCR technique was presented. It was shown that the PCR signal is sensitive to the carrier transport properties and its linear function of the optical excitation intensity. In this chapter the nonlinear dependence of PCR signals from p-Si wafers on optical excitation intensity and its effect on charge carrier transport properties is presented. Part of the following results were published by J. Tolev, A. Mandelis and M. Pawlak in J. Electrochem. Soc. J. Electrochem. (2007) and Eur. Phys. J. Special Topics (2008) 7.1 Introduction In chapter 6 the PCR signal nonlinearities associated with the existence of a space charge layer in the silicon wafer and the effects of optical biasing and conditioning of this layer in the flatband configuration under low-intensity conditions were presented. If the space charge layer disappears then the PCR signal was considered as linear function on optical excitation intensity [Mandelis et al., 2003] in sense that the amplitude of the signal depends linearly on laser power at low power ranges. This was attributed to the linear relation between the PCR signal and the free carrier absorption coefficient. With this assumption the expression of the PCR signal is given by (4.5). Furthermore, it was also assumed that the free 84 carrier absorption coefficient is a linear function of the carrier concentration ((4.3)). This coefficient can be written in the frame of the Drude model [Smith, 1978] α f (N ) = Nq 2 λ 2 . 8π 2 nc3 m * τ S ( N ) (7.1) where N is carrier concentration τ is the recombination lifetime and other quantities are defined in Abbreviations. From above formula it is seen that the free carrier IR absorption coefficient is a function of the carrier recombination lifetime. Since the carrier recombination lifetime is independent on the carrier concentration in a semiconductor the ambipolar diffusion equation can be described by Eq.3.5. As a consequence the PCR signal is a linear function of the optical excitation intensity (carrier concentration). The CDW field is expressed by the solution of the ambipolar diffusion equation which is given by (3.12). This is true at low carrier concentrations where the recombination lifetime is dominated by the SRH recombination as indicated in Figure 3.5. At high carrier concentrations the recombination lifetime can be affected by other recombination processes such those discussed in Section 3.5. This implies an ambipolar diffusion equation that is described by equation (3.5) instead of Equ. (3.7). The former equation is a nonlinear differential equation that is difficult to solve. The concept of the nonlinear coefficient β (defined later) is introduced to take into account possible effects of the carrier concentration on the recombination lifetimes (and subsequently on the PCR signal). Then the PCR is sensitive only on the emitting photons. Photons (in the spectral detection range of the PCR measurements) are produced by the radiative recombination of an electron to the defects state (monopolar limit) or by an electron-hole pair (bipolar limit). If the recombination lifetime of the carriers is dominated by the radiative recombination, then the recombination lifetime for p-type silicon wafer in high injection level ( ∆n >> p 0 ) can be described by the SRH recombination (monopolar limit) [Schroder ,1998] τ SRH = τ 0 n + τ 0 p , The bipolar recombination can be expressed by τ rad ,band −band = 1 , B∆n 85 The used quantities are defined in Section 3.5. In general, the relation between the effective radiative recombination lifetime and photo-injected carrier concentration can be written as: τ rad = C∆n −γ (7.2) where 0<γ<1 and C are constants. At near-degenerate densities of the order of 1018 cm-3 and higher, it has been found that γ = 1/2 for Ge [Seidel, 1961]. In this range of densities electronic transport is limited by impurity or carrier-carrier scattering. In summary, the nonlinear coefficient is related with the PCR amplitude. Phenomenological, the amplitude of the PCR signal │SPCR(ω)│is a nominally linear function of the incident laser power P0, or intensity I0, but, in view of the N-dependence of the Drude equation (7.1), it can be generalized as: S PCR (ω ) = aP0β = bI 0β (7.3) where the experimental value of β is given by the slope β= ∆ log S PCR (ω ) ∆ log(P0 ) (7.4) Here a and b are constants, and β is the non-linearity coefficient/exponent. Therefore, β = 1 + γ and αf(N) ∝ I01+γ. The concept of the non-linear coefficient was introduced by Guidotti et al. [Guidotti et al, 1988; Guidotti et 1989]. The authors investigated theoretically and experimentally the laser power dependence of the modulated photoluminescence in 10 – 15 Ω cm p-Si corresponding to equilibrium hole density p0 = 1x1015 cm-3 under room temperature conditions, Figure 7.1. 86 Figure 7.1 Power dependence of Y(ω) and Y(2ω) [Guidotti et al., 1988] They found a sharp linear-to-quadratic transition using a 647-nm Krypton-Argon laser with the quadratic threshold at incident power ~ 5 mW corresponding to an excess electronhole density ~ 3x1016 cm-3. The linear range they attributed to recombination of photoexcited carriers via donor (or acceptor) density of states present primarily due to semiconductor doping. Recombination in the presence of impurities and dopants in Si has been known to be a source of room-temperature photoluminescence (PL) since the 1950s [Haynes et al.,1956]. Consequently the quadratic behavior was explained as due to bipolar recombination via photogenerated electron and hole densities of states. A superposition of linear and quadratic dependence on laser intensity of spectrally integrated dc PL emission in layered semiconductors (InP/InGaAs/InP) has been reported by Nuban et al. [Krawczyk and Nuban, 1994; Nuban and Krawczyk, 1997]. 87 7.2 Experimental methodology and materials During the study of nonlinear dependence of PCR signal on the optical excitation intensity two PCR systems were used. The super-bandgap laser intensity and frequency dependence of PCR signals were studied using a set-up with 532 nm wavelength and 18 µm spotsize and another set-up with 830 nm wavelength and 24, 387 and 830 µm spotsizes. 7.2.1 Low resolution PCR system. The PCR system with the diode laser with λ=830 nm operating wavelength is shown in Figure 7.2. FILTER GROUP LENS GROUP AND SPOTSIZE Pos. 1 – NO FILTER Pos. 1 – NO FILTER 24 µm Pos. 2 – DEFOCUSING LENS f’ = 1 m 387 µm Pos. 2 – GROUP NEUTRAL DENSITY FILTERS 830 µm Pos. 3 – VARIABLE NEUTRAL DENSITY FILTER Pos. 3 – DEFOCUSING LENS f’ = 0.5 m SUPER-BAND GAP LASER 830 nm GRADIUM LENS MIRROR OFF-AXIS PARABOLOIDAL MIRRORS LOCK-IN AMPLIFIER FUNCTION GENERATOR COMPUTER LONG PASS FILTER InGaAs DETECTOR SAMPLE X Y Z MOTORIZED STAGES Figure 7.2 Experimental system for PCR measurements with super-bandgap laser at the operating wavelength 830 nm. A 150 mW diode laser (Melles Griot model 561CS115/HS) with a operating wavelength of 450 MHz through voltage input from a function generator (Stanford Research Systems model DS335). Output power modulation is achieved through current stealing so that when a positive voltage is applied some or the entire laser current is shunted through the modulation 88 circuitry. The laser beam is directed to the surface and focused onto the sample surface using a gradium lens. The position of the laser beam is coincident with the focal point of an off-axis paraboloidal mirror that collects a portion of any diffuse back-scattered photons. The collected IR is then focused onto detector. The system is designed such that the specular reflection of the excitation beam is not collected by the paraboloidal mirrors and thus is not focused onto the detector (Chapter 4). The angle of incidence of the excitation beam is ~28°. The signal from the detector is demodulated by a lock in amplifier (EG&G model 5210). All instruments, data acquisition, and data storage are controlled by a computer running MATLAB program with a graphical user interface and real-time display of experimental data. 7.2.2 High resolution PCR system. The PCR system with the λ=532 nm operating wavelength is shown in Figure 7.3. MOTORIZED STAGES Z FUNCTION GENERATOR COMPUTER MIRROR SAMPLE X Y REFLECTING OBJECTIVES LONG PASS FILTER 45° MICRO-MIRROR GRADIUM LENS InGaAs DETECTOR FILTER GROUP LOCK-IN AMPLIFIER Pos. 1 – FREE SPACE AOM DRIVER Pos. 2 – GROUP NEUTRAL DENSITY FILTERS Pos. 3 – VARIABLE NEUTRAL DENSITY FILTER ACOUSTO_OPTIC MODULATOR DIAPHRAGM SUPER-BAND GAP LASER 532 nm MIRROR Figure 7.3 Experimental system for PCR measurements with super-bandgap laser wavelength 532 nm. 89 Radiation from a Coherent Model Verdi V10 diode-pumped laser was harmonically modulated by an acousto-optic modulator (AOM) (ISOMET Model 1205C-2). The modulated beam was directly focused by a Gradium lens (focal length 60 mm) to the polished surface of the wafers following a 90-degree reflection from a 45° micro-mirror attached to the collecting reflecting objective (Ealing/Coherent Model 25-0522). The beam spotsize was 18 µm. Under the high-intensity conditions for this experimental configuration, a signal stabilization period was necessary, during which the PCR amplitude reached a constant level while the phase remained constant, before any measurements were recorded. In order to avoid unnecessary exposure of the sample to laser irradiation leading to PL fatigue, the beam was interrupted between successive measurements. The PCR NIR emissions from the samples were collected and collimated by the silver coated reflecting objective and focused onto an InGaAs Detector ( Thorlabs Model DET410,Chapter 3) with a second reflecting objective (Ealing/Coherent Model 25-0506). The output signal was fed to a lock-in amplifier (Model SRS 850 DSP) and processed by a personal computer. 7.2.3 Materials Experiments were performed with two (100) p-type (boron-doped) Si wafers. One wafer with relatively long lifetime, 6// diameter, thickness 675 ± 20 µm, and resistivity 14 - 24 Ωcm, was labeled W1. The other wafer, with relatively short lifetime, 4// diameter, thickness 525 ± 20 µm, resistivity 20 - 25 Ωcm, was labeled W2. Both wafers had a 500 Å thermal oxide layer and were placed on a mirror sample holder to amplify the PCR signals [Mandelis et al.,2003]. In the previous chapter the influence of the SCL on the PCR signal was found to be important during estimation of the transport parameters at the Si-Si02. It was found that the influence of the SCL is very important for the long lifetime silicon wafer. Since the W1 and the W2 wafers have relatively small recombination lifetimes and the thermal oxide layer is considerably thinner to that in Chapter 6, the effect of the SCL is neglected. 90 7.3. Numerical simulations of the PCR signal as a function of the non-linear coefficient β and photo-injected carriers. If there is no the carrier concentrations depend on the recombination lifetime τ, the one dimensional PCR signal can be expressed by the formula (3.5): L S1D (ω ) = F (λ1 , λ2 )∫ ∆N ( x, ω )dx = F (λ1 , λ2 )E1D (ω )M 1D (ω ) . (7.5) 0 Otherwise the non-linear coefficient β (defined in (7.4)) is introduced to the one-dimensional PCR signal expression and then it is given by L S1D (ω , β ) = F (λ1 , λ2 )∫ ∆N β ( x, ω )dx = F (λ1 , λ2 )E1D (ω , β )M 1D (ω , β ), (7.6) 0 where E1D (ω , β ) = I 0ηα (1 − R ) ( 2hνD * α 2 − σ e2 ( ) ) β ( , (7.6a) ) β γ 1 Γ2 + e −σ e L − γ 2 Γ1e −(α +σ e ) L + e −αL σ e −αL − 1 − e and M 1D (ω , β ) = , α Γ2 − Γ1e − 2σ e L ( ) (7.6b) where existing quantities were defined in Section 3.4. The numerical simulations of formula (7.4) using MATLAB program were performed. The results of these simulations as a non-linear coefficient β set a parameter are shown in Figure 7.4. 91 β=1.0 β=1.2 β=1.4 β=1.6 β=2 -1 10 -0.4 -0.8 -6 10 -1.0 -11 10 PCR phase [deg.] PCR amplitude [a.u.] -0.6 -1.2 -1.4 -16 10 -3 10 -2 10 -1 10 0 1 10 10 -2 Intensity I0 [W/cm ] -3 10 -2 10 -1 10 0 1 10 10 -2 Intensity I0 [W/cm ] Figure 7.4 The PCR amplitude (a) and phase (b) of p-type silicon wafer as a function of the intensity with the different values of nonlinear coefficient. Other parameters were set: τ=50µs ,S1=100cm/s, S2=105 cm/s, Dn* = 30 cm2/s, L = 500 µm,α(λ=514nm) = 7.76×103 cm-1,f=5 kHz. It is clearly seen from Figure 7.4 that with increasing non-linear coefficient the PCR amplitude also increases. Additionally, slopes of the PCR amplitudes increase with increasing non-linear coefficient. This can be explained by the fact that with increasing non-linear coefficient the probability of the radiative recombination processes also increase. As a result also the PCR amplitude increases. Moreover small changes in PCR phase are also observed. It is seen that with increasing non-linear coefficient increase the PCR phase lag. Regardless of the fact that CDW field expressed by Equ. 7.6 doesn’t describe correctly dependence of the carrier concentration on the PCR, some effects of the nonlinearity can be seen in numerical simulations of the linear PCR signal expression with the recombination lifetime, described by formula (3.17). The numerical simulations results are shown in Figure 7.5 and it was performed by means of the MATLAB program. 92 PCR amlitude[a.u.] 10 24 10 19 10 14 β=1.0 β=1.2 β=1.6 β=1.8 β=2.0 9 10 -3 10 -2 -1 10 10 10 0 1 10 2 Intensity [W/cm ] β=1.0 β=1.2 β=1.6 β=1.8 β=2.0 -0,5 -0,6 PCR phase [deg.] -0,7 -0,8 -0,9 -1,0 -1,1 -1,2 -1,3 -1,4 -1,5 1E-3 0,01 0,1 1 10 2 Intensity [W/cm ] Figure 7.5 The PCR amplitude (a) and phase (b) of p-type silicon wafer as a function of the intensity with the different values of nonlinear coefficient. Other parameters were set: τ=50µs ,S1=100cm/s, S2=105 cm/s, Dn* = 30 cm2/s, L = 500 µm,α(λ=514nm) = 7.76×103 cm-1,f=5 kHz and in Chapter 3. The three dimensional expression of PCR signal can be similarly generalized to the one dimensional ones in terms of the non-linear coefficient and can be written as C3D ∞ ~ S PCR ,3 D (ω , β ) = N 3 D (λ , ω , β )J 1 (λa 2 )dλ , πa 2 ∫0 (7.7) 93 where ~ N 3 D (λ ,ω , β ) = E3 D (λ ,ω , β )M 3 D (λ , ω , β ) E3 D (λ , ω , β ) = ( αηI 0 (1 − R )e ( − λ2W 2 4 2hνD * α 2 − ξ e2 )[ (7.8a) (7.8b) ) β ] ( ) 1 − e −ξe L 1 − e −αL and M 3 D (λ , ω , β ) = C1 (λ , ω ) + C 2 (λ , ω )e −ξe L − α ξe β (7.8c) 7.4 Experimental results and discussion 7.4.1 Laser power dependencies In this section the laser power dependence, P0, of the amplitude and phase of the PCR signal is reported. In the frame of this chapter experimental results are presented as a function of power or laser spotsize, instead of optical excitation intensity. This is due to dimensionality of signals which affects the theoretical interpretation and determination of the transport parameters (Chapter 3). Figure 7.6 shows results from wafer W1 made on the Low Resolution PCR system at 10 kHz. 94 a Log(PCR Amplitude [mV]) 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 0.2 0.4 0.6 0.8 1.0 b -30 PCR Phase [degrees] 1.2 -40 -50 -60 0.2 0.4 0.6 0.8 1.0 1.2 Log(Laser Power [mW]) Figure 7.6 Log – Log plot of PCR amplitude (a) and Lin – Log plot of phase (b) vs. laser power at 830 nm and 10 kHz from wafer W1 and 24-µm (■), 387-µm (●) and 830µm (▲) spotsizes of the focused laser beam. Slopes of the corresponding best fits (—) for amplitudes: β = 1.90 (24 µm); 1.82 (387 µm); 1.72 (830 µm). 95 Log(PCR Amplitude [mV]) a 0.0 -0.5 -1.0 -1.5 -2.0 1.1 1.2 1.3 1.4 1.5 1.6 b PCR Phase [degrees] 3 2 1 0 -1 -2 1.1 1.2 1.3 1.4 1.5 1.6 Log(Laser power [mW]) Figure 7.7 Log – Log plot of PCR amplitude (a) and Lin – Log plot of phase (b) vs. laser power at 830 nm and 10 kHz from wafer W2 and 24-µm (■), 387-µm (●) and 830-µm (▲) spotsizes of the focused laser beam. Slopes of the corresponding best fits (—) for amplitudes: β = 1.76 (24 µm); 1.76 (387 µm); 1.60 (830µm) Laser power dependencies of the PCR signal were obtained and slopes in log-log plots of amplitudes (Fig. 7.6a) vs. power were calculated. The laser power was varied in the range 1.8 mW – 15.6 mW. Slopes β were calculated from Eq.(7.6) and were found to be 1.90, 1.82, and 1.72 for spotsizes 24 µm, 387 µm and 830 µm, respectively. It is clear that the values of β decrease with increasing spotsize, that is, with decreasing laser intensity. Except for a rigid phase shift associated with the change in the diffusing carrier-wave dimensionality with changing spotsize shown in Fig. 7.6b, there is also a dependence of the PCR phase on laser power, indicative of the non-linear origin of the PCR signal (simluation). For fixed spotsize, the decrease in phase lag is consistent with enhanced carrier-carrier scattering with increasing laser power leading to accelerated recombination and a decrease in the distance below the 96 wafer surface of the location of the carrier density wave centroid, physically a diffusionlimited free carrier mean free path [Mandelis, 2001]. Decreasing the spotsize affects the dimensionality of the diffuse carrier density wave, enhancing the radial diffusion degrees of freedom and shifting the carrier-wave centroid phase lag farther away from surface. The experimental results of the optical power dependence for the W2 are shown in Fig. 7.7.The laser power was varied in the range 13.6 mW – 37.5 mW and, again, the slopes β were found to decrease with decreasing intensities. Furthermore, they were consistently lower than those measured with the long-lifetime wafer W1. There was no measurable phase shift with laser spotsize or power changes in Fig. 7.7(b). This behavior at the considered frequency (f=10kHz) may be the result of a weaker non-linearity mechanism sample W2 than in W1, as expected from the shorter lifetime and smaller photogenerated root mean square rms freecarrier-density in the former Si wafer. Another set of measurements was made at the High Resolution PCR System using the highly focused set-up shown in Fig. 7.3 at 10 kHz. Figure 7.8(a) shows the power dependence of PCR amplitude for wafer W1 in the range 6 mW – 25 mW. The slope was found to be 1.89. The phase lag exhibited a slight decrease with increasing laser power, Fig. 7.8(b), as expected from enhanced carrier scattering processes. The same measurement was repeated with wafer W2 and the results are shown in Fig. 7.9 97 Log(PCR Amplitude [mV]) -2.6 -2.8 -3.0 -3.2 -3.4 -3.6 -3.8 -4.0 -21.0 PCR Phase [degrees] a 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 b -21.5 -22.0 -22.5 -23.0 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Log(Laser power [mW]) Figure 7.8 Log – Log plot of PCR amplitude (a) and Lin – Log plot of phase (b) vs. laser power (■) at 532 nm and 10 kHz from wafer M 28 and 18-µm spotsize of the focused laser beam. Slope of the corresponding best fit (—) for amplitude: β = 1.89 (18 µm). 98 Log(PCR Amplitude [mV]) a -3.25 -3.50 -3.75 -4.00 0.09 0.10 0.11 0.12 0.13 PCR Phase [degrees] -4.4 0.14 b -4.5 -4.6 -4.7 -4.8 -4.9 0.09 0.10 0.11 0.12 0.13 0.14 Log(Laser power [mW]) Figure 7.9 Log – Log plot of PCR amplitude (a) and Lin – Log plot of phase (b) vs. laser power (■) at 532 nm and 10 kHz from wafer W2 and 18-µm spotsize of the focused laser beam. Slope of the corresponding best fit (—) for amplitude: β = 1.94 (18 µm). Laser power was varied between 8.7 mW and 22.4 mW and the amplitude slope was 1.94. No discernible PCR phase dependence on laser power was found. This result at 532 nm is consistent with the 830-nm results obtained with this wafer. They can both be understood in terms of the short recombination lifetime, which renders the subsurface position of the diffusive carrier-wave density centroid [Mandelis, 2001] essentially independent of lifetime at the frequency of these experiments. The higher slopes than those obtained at 830 nm are due to the increased laser-induced carrier-wave densities as discussed below. 99 7.4.2. Modulation frequency dependence at the low resolution system. Figure 7.10 shows frequency scans obtained from wafer W1 and three laser powers at a spotsize of 830 nm. The corresponding best fits of the 3-D theory, Eq. (7.10) with β = 1 to the data are also shown. The phases, Fig. 7.10(b) nearly overlap and thus they appear not to depend strongly on laser power. Figure 7.6(b) shows a change of ~ 10 degrees between 1.8 – 15.6 mW at 10 kHz. Small differences on the order of 2° exist among phases across the narrower power range in Fig. 7.10(b), however, they are nearly imperceptible over the full 75° phase range. The frequency dependence of the PCR signals using the same laser powers and Log(PCR Amplitude [mV]) 24 µm and 387 µm spotsizes exhibited similar behavior. a 1 0.1 b PCR Phase [degrees] 0 -10 -20 -30 -40 -50 -60 -70 -2 10 -1 10 0 10 1 10 2 10 Log (Frequency [kHz]) 100 Figure 7.10 Experimental frequency scans (10 Hz – 100 kHz) of PCR amplitudes (a) and phases (b) from wafer W1 for various 830-nm laser powers focused to an 830-µm spotsize, and the corresponding multiparameter theoretical 3-D best fits (—) using Eq. 7.10 with β = 1. Laser power: 20 mW (■), 15.2 mW (●) and 9.6 mW (▲). Figure 7.11 shows the frequency dependent PCR signals from wafer W1 using 20 mW and various spotsizes, as well as the corresponding best fits to the 3-D theory, Eq. (7.10), with β = 1. Figure 7.11(b) shows significant phase changes with spotsize at high frequencies due to changes in the dimensionality of the PCR signal [Rodriguez et al., 2000]. Log(PCR Amplitude [mV]) a 1 1E-1 b PCR Phase [degrees] 0 -10 -20 -30 -40 -50 -60 -70 -80 1E-2 1E-1 1 10 100 Log (Frequency [kHz]) 101 Figure 7.11 Experimental frequency scans (10 Hz – 100 kHz) of PCR amplitudes (a) and phases (b) from wafer W1 for 830-nm and 20 mW laser beam focused to various spotsizes: 24 µm (■), 387 µm (●) and 830 µm (▲). Best theoretical 3-D fits (—) using Eq. 7.4 with β = 1 are also shown. The results of best-fitting the 3-D theory to the data and extracting transport parameters are shown in Table 7.1 The frequency dependencies of the PCR signals generated with laser powers 15.2 mW and 9.6 mW and focused on the same spotsizes exhibited similar behaviors. Table 7.1. Best-fitted computational results for sample W1. Experimental data at 830 nm 3-D theory and integer values of the non-linearity coefficient β Power P0 [mW] 9.6 15.2 20 Transport and Beam Parameters 3-D Model, Eq. (7.10) I0 [W/m2] 24 µm 21.2×106 β=1 387 µm 8.2×104 830 µm 1.8×104 β=2 24 µm 21.2×106 ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] I0 [W/m2] 8. ×1018 39.3 0.31 0.07 2.3×104 33.6×106 9.51016 49.3 4.2 980 2.4×104 1.3105 3.9×1016 84.9 9.9 960 3.7×103 2.8×104 6.6×1017 35.9 11.1 4.3×103 1.2×104 33.6×106 ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] I0 [W/m2] 1.5×1019 43.5 0.26 62 4.4×103 44.2×106 1.31017 42.6 4.9 890 4×103 1.7×105 4.3×1016 55.1 5.5 787 6.8×103 3.7×104 9.6×1017 28.8 9.9 4×103 1.2×104 44.2×106 ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] 1.7×1019 37.2 0.26 60 1.7×104 1.7×1017 43.8 5.3 890 6.5×103 5.4×1016 51.6 4.7 740 3.6×104 1.2×1018 26.7 10 4×103 1.2×104 The theoretical best fits in Figs. 7.10 and 7.11 were made using a computational program which minimizes the variance of the combined amplitude and phase fits [Li et al., 2005]. Furthermore, the PCR signals were fitted using a non-linear extension of the foregoing 3-D computation algorithm with β = 2. It is important to recall that fractional powers 1 < β < 2 102 could not be efficiently accommodated in the 3-D PCR theory, Eq. (7.10). A 1-D non-linear best-fit algorithm was generated based on Eq. (7.9) and capable of accommodating arbitrary values 1 ≤ β ≤ 2. The computational algorithm was implemented with the corresponding experimental values for β (Fig. 7.6 for sample W1) and numerical quantities are presented in Table 7.2. Table 7.2. Best-fitted computational results for sample W1. Experimental data at 830 nm 1-D theory and experimental values of non-linearity coefficient β. Power P0 [mW] 9.6 15.2 20 Transport and Beam Parameters 1-D Model, Eq. (7.9) I0 [W/m2] β = 1.82 387 µm 8.2×104 β = 1.72 830 µm 1.8×104 ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] I0 [W/m2] 4×1016 31.1 25 1×104 420 1.3105 3.2×1016 81.4 28.7 7×103 9.1×103 2.8×104 ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] I0 [W/m2] 7.4×1016 27.9 9.2 4.8×104 0.3 1.7×105 4.1×1016 65.8 27.5 6.9×103 9.7×103 3.7×104 ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] 8.7×1016 28 15.3 1.1×104 0.2 5×1016 59.8 24.7 6.5×103 9.9×103 In all tables the photogenerated excess carrier-wave densities (m-3) shown are mean values calculated using the expression: 103 ∆Ν = P0 1 τ −19 2 1.6 × 10 (hω0 ) π [W + LD ( f ) ]2 (1/ α P ) , (7.11a) where P0 is the incident power [W]; hω0 is the incident photon energy expressed in eV ,ω0 = hc/λP is the photon angular frequency at the wavelength of the laser; and αP is the optical absorption coefficient at the excitation wavelength. LD(f) is the magnitude of the complex carrier diffusion length LD ( f ) = Dτ 1 + i 2π f τ . (7.11b) Values of LD and τ used in Eq. (7.11) were calculated from the multiparameter best-fit values obtained from Tables 7.1-7.4. From Table 7.1 the values of lifetimes obtained with the linear 3-D theory, β = 1, and 24-µm spotsize, are reasonable for p-Si [Rodriguez et al., 2000], however, the corresponding diffusion coefficients are outside the typical range for p-type wafers [Rodriguez et al., 2000]. Table 7.1 also shows the results of the same experimental data, Figs. 7.8 and 7.9, fitted with the 3-D theory and β = 2, Eq. (7.10). The experimental slope for 24 µm laser spotsize (β = 1.9) is very close to 2 and validates this fitting procedure. Fitting error was very low, 0.6% – 0.9%. The measured transport parameters for all laser intensities and 24 µm spotsize can be compared for β = 1 and β = 2 from Table 7.1. It is seen that recombination lifetimes consistently decrease in the non-linear fit. This decrease is larger for higher intensities reaching up to 43%. The absolute value of the lifetime monotonically decreases with increasing intensity for β = 2, as expected from the increased carrier-wave densities and the proportionately increased scattering probabilities which limit the diffusion length (or mean free path). In the non-linear fit, carrier diffusivity values increase dramatically from unrealistically low levels at β = 1: They rise to values approx. 10 cm2/s, which is normal medium to high injection conditions (∆N ~ 1018 – 1019 cm-3). Front surface recombination velocities also increase considerably in the non-linear fit, and remain essentially unchanged with increasing intensity, as expected, since the defect / recombination center density on the surface is not affected by the incident photon flux. Back surface recombination velocities do not change as dramatically under the non-linear fit. They are higher than S1, as expected from the matte nature of the back surfaces of our wafers compared to the polished front surfaces. No attempt for non-linear fits to the data corresponding to 387 µm and 830 µm spotsizes was made, as the slopes (β = 1.82 and 1.72) were far from β = 2 to guarantee the validity of the 3-D computation. As shown in Fig. 3.8, the best fits of PCR signals generated with the relatively large spotsizes, 2W ~ 830 µm , yield very similar sets of 104 transport parameters using either 1-D or 3-D linear theories (β = 1) only when τ < 20 µs. This is not the case with sample W1 as seen in Table 7.2. However, for comparison with Table 7.2, the β = 1 parameters were calculated using the 3-D theory and are shown in Table 7.1. Table 7.2 shows results from PCR signals with 2W = 387 µm and 830 µm fitted to the non-linear 1D theory, Eq. (12), with β = 1.82 and 1.72, from Fig. 7.6(a). In view of Fig. 3.7 and the bestfitted values of recombination lifetime under all excitation intensities, the calculated transport parameters with the 1-D model can only be considered to be approximate. The excess photoinjected carrier densities are somewhat lower than the 2W = 24-µm case. Recombination lifetimes under all incident intensities do not change much from the β = 1 values but they decrease monotonically with increased intensity, as expected for enhanced carrier-carrier scattering [Li et al.,1997]. It is noted that the lifetime obtained under 9.6 mW excitation and 2W = 830 µm is 81.4 µs, close to the value 84.9 µs obtained from fitting the same data with the 3-D theory and β = 1. These differences increase with increasing incident intensity. As observed with 2W = 24 µm spotsize, for non-linear fits with β = 1.82 (2W = 387 µm) and β = 1.72 (2W = 830 µm) the values of D increased manifold over the β = 1 values, attaining the normal range of p-Si [Rodriguez et al., 2000]. However, the values of D are larger than those obtained with 2W = 24 µm as expected from the much lower intensities accompanying the larger spotsizes. Only slight decreases are observed with increasing intensity [Li et al.,1997]. Front and back surface recombination velocities do not undergo significant changes for all non-linear fits, β = 2 (2W = 24 µm), β = 1.82 (2W = 387 µm) and β = 1.72 (2W = 830 µm). This is as expected from the intrinsic defect structure of the wafer surfaces. S2 is consistently higher than S1 for all incident intensities. The comparison between Tables 7.1 and 7.2 show that the difference between the β = 1 values and the non-linear results is large and due to the non-linearity exponents β = 1.72 and 1.82. This comparison demonstrates the need for precise knowledge of the degree of non-linearity of the PCR signal in order to obtain physically reasonable and reliable (i.e. self-consistent) measurements of semiconductor transport properties. Figure 7.12 shows two sets of modulation frequency scans obtained from wafer W2 including the corresponding best fits to 3-D theory with β = 1. The phase plots at fixed spotsize nearly overlap, Fig. 7.12b, and are essentially independent of laser power, consistent with Fig. 7.7b. Figure 7.13 shows frequency scans with 38-mW laser power and various spotsizes. The corresponding 3-D theoretical best fits to Eq. (7.10) and β = 1 are also shown. 105 a Log(PCR Amplitude [mV]) 5E-2 4E-2 3E-2 2E-2 1E-2 9E-3 5 b PCR Phase [degrees] 0 -5 -10 -15 -20 -25 1 10 100 Log (Frequency [kHz]) Figure 7.12 Experimental frequency scans (500 Hz – 100 kHz) of PCR amplitudes (a) and phases (b) from wafer W2 for various 830-nm laser powers focused to an 830-µm spotsize and the corresponding theoretical 3-D best fits (—) using Eq. 7.4 with β = 1. Laser power: 38 mW (■) and 12 mW (●). 106 a Log(PCR Amplitude [mV]) 1 1E-1 b PCR Phase [degrees] 0 -5 -10 -15 -20 -25 1 10 100 Log (Frequency [kHz]) Figure 7.13 Experimental frequency scans (500 Hz – 100 kHz) of PCR amplitudes (a) and phases (b) from wafer W2 for 830-nm, 38 mW laser beam focused to various spotsizes: 24 µm (■), 387 µm (●) and 830 µm (▲). Best theoretical 3-D fits (—) using Eq. 7.4 with β = 1 are also shown. The phase shifts at the high-frequency end of Fig. 7.13b are due to changing signal dimensionality from 3-D (2W = 24 µm) to 1-D (2W = 830 µm). The frequency dependence of the PCR signals with excitation power 12 mW and the same spotsizes exhibited similar behavior. The PCR signals of Fig. 7.12 were fitted to 1-D theory using the actual experimental non-linear values for β, Fig. 7.7a. Calculated transport parameters are presented in Table 7.3. 107 Table 7.3. Best -fitted computational results for sample W2. Experimental data at 830 nm with 1-D and 3-D theories, linear and various non-linearity coefficients Power P0 [mW] Transport and Beam Parameters 38 β=1 1-D Fitting Model I0 [W/m ] 387 µm 1×105 830 µm 2.2×104 β = 1.76 387 µm 1×105 ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] I0 [W/m2] 3.6×1015 1.3 24.6 9.3×103 3.9×104 3.2×105 6.2×1014 0.9 34.5 370 2.2×104 7×104 1.6×1015 0.54 23.2 390 1.6×104 3.2×105 3.9×1014 0.55 23.4 390 1.6×104 7×104 ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] 1×1016 1.3 32 8×103 1.5×104 2.2×1015 1.1 32.2 430 2.4×104 4.8×1015 0.51 23.4 407 1.5×104 1.5×1015 0.65 19.5 920 1.6×104 2 12 3-D Fitting Model β = 1.60 830 µm 2.2×104 Unlike sample W1, the short-lifetime (τ ∼ 1 µs) wafer W2 did yield PCR frequency scans which could be considered fully one-dimensional under the expanded laser beam spotsizes of 387 and 830 µm, as the numerical simulations of PCR signal presented on Fig. 4.9. At 2W = 24 µm the supra-linear slope β = 1.76, Fig. 7.7a, did not allow using the 3-D Eq. (7.10) to fit the data and yield representative values of the transport parameters since we could only use integer values of β. Therefore, only fits for the largest two spotsizes are shown in Table 7.3. Here the lifetimes are very short for both linear and non-linear fits and they do not change perceptively with increased laser intensity. This fact may be traced to the relatively low photoexcited carrier-wave densities in the 1014 – 1015 cm-3 range, using Eq. (7.11). These are up to four orders of magnitude lower than those estimated for the sample W1. The best-fitted diffusivities are in the range of acceptable values for p-Si [Rodriguez et al., 2000] and vary little with increased intensity; again, owing to the relatively low ∆Ν. Front-surface recombination velocities are lower than back-surface velocities, as expected. There are large differences between the linear and non-linear fits for the same W which lead to the same conclusion as in the case of wafer W1 in terms of the importance of using the actual nonlinear exponent obtained experimentally in order to calculate the transport parameters. In summary, there were neither large differences in values of the transport parameters between 108 excitation with 12 and 38 mW, nor were clear trends in those parameters established. This would be expected under low-injection conditions, however, in these experiments the ∆Ν range corresponded to intermediate conditions [Bullis and Huff, 1996]. 7.4.3. Modulation frequency dependence at 532 nm Figures 7.14 and 7.15 show the frequency dependent amplitude and phase of the PCR signal from wafers W1 and W2, respectively, using several laser powers and 18 µm spotsize. The corresponding best fits to the 3-D theory, Eq. (7.10), are also shown. Log(PCR Amplitude [mV]) 1E-3 a 1E-4 b PCR Phase [degrees] 0 -10 -20 -30 -40 1E-2 1E-1 1 10 Log (Frequency [kHz]) 100 Figure 7.14 Experimental frequency scans (10 Hz – 100 kHz) of PCR amplitudes (a) and phases (b) from wafer W1 for various 532-nm laser powers focused to an 18-µm spotsize and the corresponding theoretical 3-D best fits (—) using Eq. 7.4 with β = 2. Laser power: 16 mW (■), 13.2 mW (●) and 9.6 mW (▲). 109 a Log(PCR Amplitude [mV]) 2E-4 1E-4 9E-5 8E-5 7E-5 6E-5 b PCR Phase [degrees] 0 -2 -4 -6 -8 -10 -12 1E-2 1E-1 1 10 Log (Frequency [kHz]) 100 Figure 7.15 Experimental frequency scans (10 Hz – 100 kHz) of PCR amplitudes (a) and phases (b) from wafer W2 for various 532-nm laser powers focused to an 18-µm spotsize and the corresponding theoretical 3-D best fits (—) using Eq. 7.4 with β = 2. Laser power: 16 mW (■), 13.2 mW (●) and 9.6 mW (▲). 110 While the amplitudes scale non-linearly with laser power with slopes β = 1.89 (W1, Fig. 7.8) and β = 1.94 (W2, Fig.7.7), the phases at the various laser powers nearly overlap, consistently with Figs. 7.8b and 7.7b. The best fits to Eq. (7.10) presented in Figs. 7.14a and 7.15a were made using β = 2. This value of β is very close to the experimentally measured slopes, therefore it is expected that transport parameters obtained from these fits are close to those that would have been obtained using the exact experimental supra-linear slopes. The multiparameter best-fit results to the data are presented in Table 7.4 for three laser powers. Table 7.4. Best-fitted computational results for samples W1 and W2. Experimental data at 532 nm with 3-D theory. Power P0 [mW] Transport and Beam Parameters 2 9.6 13.2 16 I0 [W/m ] ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] I0 [W/m2] ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] I0 [W/m2] ∆N[1/cm3] τ [µs] D [cm2/s] S1 [cm/s] S2 [cm/s] 3-D Fitting Model for Wafer W1 Β=1 3.8×107 3.6×1020 238 0.46 123 2×104 5.2×107 6.1×1020 259 0.37 184 4.3×104 6.3×107 5.9×1020 200 0.35 241 3.9×104 β=2 3.8×107 3.1×1019 70.5 2.7 5.6×104 9.4×104 5.2×107 3.4×1019 56.7 2.9 4.3×104 8.6×104 6.3×107 3×1019 44.5 3.2 4×104 6.8×104 3-D Fitting Model for Wafer W2 β=1 3.8×107 3×1018 4.2 5.7 6.7×103 4.2×104 5.2×107 4.4×1018 3.5 4.7 6.3×103 3.7×104 6.3×107 5.5×1018 4.1 4.8 7.4×103 6.9×104 β=2 3.8×107 8.3×1017 0.34 8.9 3.7×103 9.8×103 5.2×107 1.2×1018 0.34 8.9 3.7×103 9.9×103 6.3×107 1.6×1018 0.33 6.2 3.6×103 9.6×103 For sample W1 the values of τ decrease with increasing laser intensity, in a manner similar to that observed with 830-nm irradiation, Table 7.1. The relative values of τ for excitation with beam spotsizes 18 and 24 µm at the two distinct wavelengths, Tables 7.1 and 7.4, are quite close, within a factor of two. On the other hand, in Table 7.4 the values of τ obtained with β = 2 for the three incident intensities I0, are more than three times smaller than those obtained 111 with β = 1. Once again, this shows the dramatic difference the non-linearity coefficient value can make in PCR measurements. The diffusion coefficients obtained using β = 1 are far from the range of acceptable values for this parameter for p-type wafers [Rodriguez et al., 2000] and those obtained with β = 2 are marginally within the range of typical values. This is approx. one order of magnitude smaller than the D values obtained in Table 7.1 and is probably due to the much stronger carrier-carrier scattering at high-injection carrier-wave densities in the 1019 – 1020 cm-3 range [Li et al., 1997]. The best-fitted values of S1 obtained with β = 1 are more than 100 times lower than those obtained with β = 2. The latter values are essentially independent of laser intensity (also the case in Table 7.1 under 830-nm excitation) but are clearly higher than their β = 2 counterparts in Table 7.1. This, along with differences in τ (532 nm) and τ (830 nm), is expected, because the optical absorption depth at 523 nm is much shorter (0.96 µm; αP = 10,340 cm-1) than that at 830 nm (15.2 µm; αP = 658.9 cm-1) with the consequence that optical injection samples the defect densities in the near-surface region must more strongly at 523 nm than at 830 nm, thus exhibiting enhanced sensitivity to surface recombination and yielding a larger effective surface recombination velocity S1. The back-surface recombination velocities are all in the range of 104 cm/s for both wavelengths, Tables 7.1 and 7.4. Physically, they should be the same, however our simulations show that the shallower optical penetration depth at 523 nm renders the PCR signal less sensitive to S2 than that at 830 nm. The best fits to Eq. (7.10) resulting from the W2-sample experimental data with β = 1 and 2 are also shown in Table 7.4. The values of τ obtained with β = 2 which is very close to the experimental value β = 1.94 for all laser powers are in the sub-µs range, consistently with those obtained at 830 nm in Table 7.2. In fact, they are approx. two times smaller under 523-nm excitation as expected from a significantly higher intensity range (about three orders of magnitude) and the corresponding high-injection ramifications. The τ values obtained with the linear fit are very unrealistic in comparison with other data in Table 7.4, more than ten times larger than those obtained with β = 2. The carrier diffusivity values in Table 7.4 are considerably smaller than those in Table 7.2 and decrease with increased power 16 mW. These trends are consistent with the shallower penetration depth, higher photoinjected carrier-wave densities and higher laser intensities under 523 nm excitation [Li et al., 1997]. In a manner similar to the behavior of sample W1, the S1 values under 523-nm excitation are higher than under 830-nm, whereas values of S2 are very close. Overall, values of the transport parameters in Table 7.4 fitted with β = 1 show some significant deviations from the non-linear fits, especially for τ and S1. 112 7.5 Summary It is interesting to note for both samples that the ca has three-orders-of-magnitude higher intensities and shorter optical absorption depth at 523 nm which are associated with increased non-linearity exponents which tend to values closer to 2, a limiting value for electron-hole band-to-band bipolar recombination [Guidotti et al, 1988-1989; Nuban and Krawczyk, 1997; Nuban and Krawczyk, 1997]. In that limit, band-to-defect recombination may be dominated by the N×P product of electron-hole density recombination. A different mechanism for quadratic PL dependence on photo-carrier density may have to be considered such as proposed by Guidotti et al. [Guidotti et al, 1988-1989] and Nuban et al. [Nuban and Krawczyk, 1997; Nuban and Krawczyk, 1997] and which have been discussed in the context of Eq. (3.1). 113 Chapter 8: Conclusion and outlook This final chapter summarizes the main conclusions of this dissertation and outlines the directions for the future research. Following the organization of the thesis the last chapter is also devided into two parts. The first section is concerned with the thermal properties of Cd1-xMgxSe single crystals, the second one is devoted to the new developed technique of photocarrier radiometry (PCR). Thermal properties of Cd1-xMgxSe single crystals Pyroelectric experiments have proven to be a valuable tool to characterize the thermal parameters of Cd1-xMgxSe single crystals in a limited temperature range around room temperature. With a home-made apparatus the thermal diffusivity of Cd1-xMgxSe single crystals could be determined experimentally. It was shown on the glassy carbon (reference sample) that the thermal diffusivity can be determined directly from measured PPE amplitudes and phases when sample and detector are thermally thick and optically opaque. From these data the thermal conductivity of these compounds was estimated. It was found that thermal conductivity of Cd1-xMgxSe single crystals decreases by a factor of three as the Mg concentration is increased from zero to about 50 %. This strong variation could be explained by structural effects of the mixed crystal. Future work should concentrate on development of the photopyroelectric cell able to measure in broader temperature range. From the theoretical point of view it is very interesting to develop theoretical model for mixed crystals to elucidate the relative role of the electronic and phonon contributions to thermal conduction and the mass-scattering effects as well. 114 Photocarrier radiometry signal It was shown that in some cases the existence of a space charge layer or bipolar recombination mechanism can be important when the electrical transport properties of a silicon wafers are measured by photocarrier radiometry (PCR). In chapter 6 the interface modulated-charge-density theory developed by Mandelis [Mandelis, 2005a] was experimentally proved. The interface modulated-charge-density theory developed by Mandelis [Mandelis, 2005a] was used for physical simulations involving PCR signals in order to study the effects of the various SCL optoelectronic transport parameters on the PCR amplitude and phase signal channels. Furthermore, an experimental configuration was used involving n- and p-doped Si – SiO2 interfaces and a low-intensity (non-perturbing) modulated laser source as well as a co-incident dc laser of variable intensity acting as interface-state neutralizing optical bias. The application of the theory to the experiments yielded the various transport parameters of the samples as well as the depth profile of the SCL. It was shown that PCR can monitor the complete flattening of the energy bands at the interface of p-Si – SiO2 with dc optical powers up to a 300 mW. The uncompensated charge density at the interface was also calculated from the theory. It was found that this methodology is the relatively low efficient for driving n-Si into the flatband state. The independence of PCR phases from IDC is an additional indicator that the optical bias does not affect the transport properties of the SiO2 - n-Si interface. This behavior was explained by the much lower (ca. 100 times [Aberle, 1992]) minority carrier capture cross-section of the n-type interface. This fact leads to much shorter interface lifetimes, τs [Mandelis, 2005a], making it harder for an optical source to build up the neutralized interface charge coverage observed with p-Si – SiO2 interfaces, Fig. 6.15. The same effect makes the interface recombination velocity essentially independent of excess carrier density at the n-Si – SiO2 interface [Aberle, 1992] which results in the independence of the PCR phase from PDC, Fig. 6.16. In chapter 7 it was shown that the introduced nonlinear coefficient, which could be determined from intensity-scans, can considerably improve estimated values of the carrier transport parameters. It is interesting to note for both samples that the ca. has three-ordersof-magnitude higher intensities and shorter optical absorption depth at 523 nm which are associated with increased non-linearity exponents which tend to values closer to 2, a limiting value for electron-hole band-to-band bipolar recombination [Guidotti et al, 1988-1989; Nuban and Krawczyk, 1997; Nuban and Krawczyk, 1997]. In that limit, band-to-defect 115 recombination may be dominated by the N×P product of electron-hole density recombination. A different mechanism for quadratic PL dependence on photo-carrier density may have to be considered such as proposed by Guidotti et al. [Guidotti et al, 1988-1989] and Nuban et al. [Nuban and Krawczyk, 1997; Nuban and Krawczyk, 1997] and which have been discussed in the context of Eq. (3.1). Altogether it is demonstrated that the photocarrier radiometry technique is a powerful tool for the investigation of electronic properties of silicon wafers. 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Special Topics 153, 317-320 (2008) 122 Curriculum Vitae PERSONAL DETAILS Name Michał Janusz Pawlak Nationality Polish Date and place of birth 30.01.1978, Bydgoszcz, Poland Address Moniuszki 5/3 87-100 Toruń Poland [email protected] Contact tel.: +48 608 349 125 EMPLOYMENT HISTORY Date Since April 2009 Position IT specialist at Kujawsko-Pomorskie Voivodeship, Torun, Poland Nov 2008 – March 2009 Assistant at Ruhr University Bochum, Germany Sept 2007 – April 2008 Postdoctoral Fellow at University of Toronto, Canada May 2007 – August 2007 Assistant at Ruhr University Bochum, Germany April 2004-March 2007 Scholarship holder at Ruhr University Bochum, Germany Oct 2002 – March 2004 PhD student at Nicolaus Copernicus University, Torun, Poland Oct 2001 – June 2002 Assistant at Nicolaus Copernicus University, Torun, Poland 123 EDUCATION Date April 2004-March 2007 Title Scholarship holder Institution Ruhr University Bochum, Germany October 2002 – March 2004 PhD student Nicolaus Copernicus University, Torun, Poland Since April 2004 Electrical Eng. Warsaw University of Technology, Poland October 2000 – June 2002 October 1997 – June 2000 Master of science Nicolaus Copernicus (Physics) University, Torun, Poland Bachelor (Physics) Nicolaus Copernicus University, Torun, Poland PERSONAL QUALITIES Programming in Java, Matlab, Labview; digital image processing, biomedical eng. Languages: Polish - native, English - good, German – basic. 124 List of publications and conference contributions Publications 1. Photoacoustic Study of Zn1-x-yBexMnySe Mixed Crystals J. Zakrzewski, F. Firszt, S. Łęgowski, H. Męczyńska, M. Pawlak, A. Marasek, abstract book 31st Winter School on Molecular and Quantum Acoustics and 7th workshop on Photoacoustics and Photothermics, page. 81, 2002. 2. Thermal Diffusivity Measurements of Zn1-x-yBexMnySe Mixed Crystals by Photoacoustic Method J. Zakrzewski, F. Firszt, S. Łęgowski, H. Męczyńska, M. Pawlak, abstract book 28. Deutsche Jahrestagung fur Akustik, page 114, Bochum, 2002 3. Piezoelectric and pyroelectric study of Zn1-x-yBexMnySe mixed crystal, J. Zakrzewski, F. Firszt, S. Legowski, H. Meczynska, M. Pawlak, and A. Marasek, Rev. of Scienfic Instruments, vol. 74 nr 1, p. 566-568 (January 2003) 4. Photoacoustic investigation of Cd1-XMnXTe mixed crystals J. Zakrzewski, F. Firszt, S. Łęgowski, H. Męczyńska, A. Marasek, and M. Pawlak, Rev. of Scienfic Instruments, vol. 74 nr 1, 572-574 (January 2003) 5. Photoacoustic Study of Cd1-x-yBexMnyTe Mixed Crystals, J. Zakrzewski, F. Firszt, S. Łęgowski, H. Męczyńska, A. Marasek, M. Pawlak, Journal de Physic IV 109 (2003), 123126 6. Growth and characterization of selected wide – gap II – VI ternary solid solutions, F. Firszt, S. Legowski, A.Marasek, H. Meczynska, M.Pawlak and J.Zakrzewski, abstract book, International Symposium on 50th Anniversary of the Death of Prof. Dr. Jan Chochralski, Torun, Poland, p. 4 125 7. Photoelectric and Photothermal Properties of Selected II – VI Compounds Mixed Crystals, F. Firszt, S. Łęgowski, A. Marasek, H. Męczyńska, M. Pawlak and J. Zakrzewski, Optoelectronics Rev. 12 (1), p. 161-164 (2004) 8. Study of optical properties of Zn1-xBexTe mixed crystals by means of combined modulated IR radiometry and photoacoustics, M. Pawlak, J. Gibkes, J. L. Fotsing, J. Zakrzewski, M. Maliński, B. K. Bein, J. Pelzl, F. Firszt and A. Marasek, J. Phys. IV France 117, p. 47-56 (2004) 9. Carrier - density - wave transport and local internal electric field measurements in biased metal-oxide-semiconductor n-Si devices using contactless laser photo-carrier radiometry,A.Mandelis, M.Pawlak, D.Shauhnessy, Semicond.Sci.Technol. 19, p. 12401249 (2004) 10. Growth and Optical Characterization of Cd1-xBe xSe and Cd1-xMgxSe crystals, F. Firszt, A. Wronkowska, A. Wronkowski, S. Łęgowski, A. Marasek, H. Męczyńska, M. Pawlak, W. Paszkowicz, K. Strzałkowski, and J. Zakrzewski, Cryst. Res. Technol. 40, No. 4/5, p. 386-394 (2005) 11. Piezoelectric photothermal study of Cd1-x-yBexZnySe crystals,J. Zakrzewski, F. Firszt, S. Łęgowski, H. Męczyńska, A. Marasek, M. Pawlak, K. Strzałkowski, M. Maliński, and L. Bychto, Journal de Physique IV 125, p. 473-476 (2005) 12. Photoacoustic Study of Cd1-XBeXSe Mixed Crystals, J.Zakrzewski, M.Pawlak, F. Firszt, S. Łęgowski, A. Marasek, H. Męczyńska,M.Malinski, Int. Journal of Thermophysics 26 (1), p. 285 (2005) 13. Non – contacting Laser Photocarrier Radiometric Depth Profilometry Of Harmonically Modulated Band – Bending In The Space Charge Layer At Doped SiO2 – Si Interfaces, A.Mandelis, J. Batista, J.Gibkes ,M.Pawlak, J.Pelzl, Journal Of Applied Physics 97 083507 (2005). 126 14. Space charge layer dynamics at oxide-semiconductor interfaces under optical modulation: Theory and experimentalstudies by non-contacr photocarrier radiometry, A. Mandelis, J.Batista, J.Gibkes, J.Pelzl, J.Phys.IV France 125, 565-567 (2005) 15. Investigations of AII-BVI mixed crystals with the piezoelectric photothermal method, M. Maliński, L. Bychto, J. Zakrzewski, F. Firszt, M. Pawlak, P. Binnebesel, Journal de Physique IV 125, p. 379-382 (2005) 16. Time-domain and lock-in rate-window photocarrier radiometric measurements of recombination processes in silicon,A. Mandelis, M. Pawlak, Ch. Wang, I. DelgadilloHoltfort, J. Pelzl, Journal of Applied Physics 98, p.123518 (2005) 17. Photoacoustic and photothermal radiometry spectra of implanted Si wafers, D.M.Todorovic, M. Pawlak, I. Delgadillo-Holtfort, J. Pelzl, European Physical Journal Special Topics 153,259-262 (2006). 18. Non-linear Dependence of Photocarrier Radiometry Signals from p-Si Wafers on Optical Excitation Intensity, J. Tolev, A. Mandelis and M. Pawlak, J. Electrochem. Soc. J. Electrochem. Soc., Volume 154, Issue 11, pp. H983-H994 (2007) 19. On the non-linear dependence of photocarrier radiometry signals from Si wafers on the intensity of the laser beam, J. Tolev, A. Mandelis and M. Pawlak, Eur. Phys. J. Special Topics 153, 317-320 (2008) 20. Laser Photothermal Radiometric Instrument For Industrial Steel Hardness Inspection, X. Guo, K. Sivagurunathan, M. Pawlak, J. Garcia, A. Mandelis, S. Giunta, S. Milletari and S. Bawa, conference material for 15th International Conference Of Photoacoustic And Photothermal Phenomena in Leuven, Belgium, 2009 (July 2009) 21. Thermal Transport Properties of Cd1-xMgxSe Mixed Crystals Measured by Means of the Photopyroelectric Method, M. Pawlak, F. Firszt, S. Łęgowski, H. Męczyńska, J. Gibkes and J. Pelzl , Intern. Journal of Thermophysics 31, 1, 187-198 (2010) 127 Conference contributions 32nd Winter School on Molecular and Quantum Acoustics and 8th workshop on Photoacoustics and Photothermics) w Szczyrku, Poland, (oral), 2003 Photoacoustic Study of Cd1-x-yBexMnyTe Mixed Crystals International Symposium on 50th Anniversary of the Death of Prof. Dr. Jan Chochralski, 2003, Torun, Poland (2 posters), 2003 Growth and characterization of selected wide – gap II – VI ternary solid solutions XXXII International School on the Physics of Semiconducting Compounds Jaszowiec, Poland, 2003. XVII School of Optoelectronics, Kazimierz Dolny, Poland, 2003 (poster) Photoelectric and Photothermal Properties of Selected II – VI Compounds Mixed Crystals, 33rd Winter School on Molecular and Quantum Acoustics and 9th workshop on Photoacoustics and Photothermics) w Szczyrku, Poland, (oral), 2004, Study of optical properties of Zn1-xBexTe mixed crystals by means of combined modulated IR radiometry and photoacoustics 13th International Conference Of Photoacoustic And Photothermal Phenomena in Rio de Janeiro, Brazil, 2004 Investigation Of AII BVI Mixed Crystals With The Piezoelectric Photothermal Method, oral Piezoelectric and Microphone Photoacoustic Study Of BeZnCdSe Mixed Crystals, poster Space Charge Layer Dynamics At Oxide – Semiconductor Interface Under Optical Modulation: Theory and Experimental Studies By Non – Concact Photocarrier Radiometry, poster 128 34th Winter School on Molecular and Quantum Acoustics and 10th workshop on Photoacoustics and Photothermics) in Szczyrku, Poland, 2005: Determination of the Thermophysical Parameters of Zn1XBeXSe Mixed Crystals by means of Standard Photopyroelectric and Piezoelectric Photothermal Techniques, Gordon Research Conference on "Photoacoustic and Photothermal Phenomena" Trieste, Italy, 2005, poster: Photocarrier Radiometric Imaging of H+ Ions Implanted in Si wafers, 14th International Conference Of Photoacoustic And Photothermal Phenomena in Cairo, Egypt, 2007, poster: The Non-linear Dependence of Photocarrier Radiometry Signals from Si Wafers on the Intensity of the Laser Beam, Michal Pawlak, Jordan Tolev and Andreas Mandelis. 14th Winter Workshop on Photoacoustics and Thermal Waves Methods w Korbielów, Poland, (oral), 2009. 129 Appendix A Current controller In order to drive a Peltier element from PPE experimental set up presented in Fig. (4.13) the dedicated current controller had been designed. The electrical schema of this controller is presented in Figure A.1. 130 Figure A.1. The electrical schema of the current controller. 131