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Topics on the theory of electron spins in semiconductors DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Nicholas J. Harmon, B. A. Graduate Program in Physics The Ohio State University 2010 Dissertation Committee: William O. Putikka, Advisor John Wilkins Ezekiel Johnston-Halperin Richard Furnstahl T AF c Copyright by Nicholas J. Harmon DR 2010 Abstract As electron spin continues to be sought for exploitation in technological devices, understanding the spin’s coupling to its environment is essential. This dissertation theoretically explores spin relaxation in two systems: the cubic zinc-blende and hexagonal wurtzite crystals. First, bulk systems with the zinc-blende crystal structure are studied. A model that includes both localized and itinerant spins and their interaction successfully explains the observed phenomena. When this model is applied to certain quasi-two-dimensional structures of the same crystal type, it succeeds again after the exciton and exciton-bound-donor spin species are introduced. For the first time a quantitatively accurate explanation of the strange temperature dependence in intrinsic (110)-GaAs quantum wells is given. Second, the properties of the wurtzite crystal are studied in regards to their usefulness in spintronic devices. Research along these line in wurtzite is undeveloped in comparison to zinc-blende. The theory of the D’yakonov-Perel’ and Elliott-Yafet spin relaxation mechanisms is developed. n-ZnO is concentrated on in bulk systems. The impurities must be carefully considered when explaining the experimentally observed spin relaxation times; the presence of a deep donor and a shallow donor give rise to the observed phenomena. Wurtzite quantum wells have properties that could be especially beneficial spintronic devices. The D’yakonov-Perel’ mechanism, which the dominant spin relaxation mechanism in the materials considered here, can be suppressed at low temperatures much more effectively than can be done in zinc-blende due to the difference in spin-orbit fields. Suppression of spin relaxation is also greater in wurtzite than in zinc-blende at room temperature due to the smaller spin-orbit coupling in the examined wurtzite semiconductors. Finally, the role an external magnetic field plays in the interacting localized-itinerant spin picture is examined and used to explain a ‘spin beating’ phenomenon. ii “We usually only want to know something so that we can talk about it; in other words, we would never travel by sea if it meant never talking about it, and for the sheer pleasure of seeing things we could never hope to describe to others.” -Pascal- iii Acknowledgments First I would like to thank my advisor Prof. William O. Putikka, who provided guidance and support through the entire period of my graduate research. Next I would like to thank my advisory committee, Prof. Richard Furnstahl, Prof. Ezekiel Johnston-Halperin, and Prof. John Wilkins who served as my committee members in my final oral exam and in my candidacy exam. I would also like to thank Prof. Robert Joynt and Prof. Jay Kikkawa who were invaluable sources and answered many of my questions over the past several years. The various projects in this dissertation were supported by National Science Foundation through Grant No. NSF-ECS-0523918 and by the Center for Emergent Materials at the Ohio State University, an NSF MRSEC (Award No. DMR-0820414). The list would be incomplete without thanking my friends: Sheldon Bailey, James C. Davis, Kevin P. Driver, Ben Dundee, Michael Fellinger, Dave Gohlke, Robert Guidry, Adam Hauser, William Parker, Patrick D. Smith, Jeffery Stevens, Rakesh Tiwari, Gregory Vieira, and the Dr. K’s softball team for making my graduate student life more enjoyable than it should be. I also thank my friends from back home and from Wooster: Christopher Doherty, Joe Hall, Evan Rae, Josh Ross, Bradley Thomas, and the whole 5:15 crew. The support from 1011H Beverage Company and it’s customers is also appreciated. Many thanks to Dad and Mom who never taught me much physics but taught me the more important things of life. Last but not least, I would like to thank my beloved wife Barbara for putting up with me working late nights and for letting me tell her more about that physics stuff. iv Vita March 7, 1982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Bismarck, North Dakota 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. A., The College of Wooster, Wooster, OH 2004–2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research/Teaching Assistant, Dept. of Physics, The Ohio State University, Columbus, Ohio Publications [1] N.J. Harmon, W.O. Putikka, and R. Joynt, “Prediction of extremely long electron spin relaxation times in wurtzite quantum wells”, in submission. [2] N.J. Harmon, W.O. Putikka, and R. Joynt, “Theory of electron spin relaxation in ndoped quantum wells”, Phys. Rev. B 81, 085320 (2010). [3] N.J. Harmon, W.O. Putikka, and R. Joynt, “Theory of electron spin relaxation in ZnO”, Phys. Rev. B 79, 115204 (2009). [4] J.F. Lindner, M.I. Rosenberry, D.E. Shai, N.J. Harmon, and K.D. Olaksen, “Precession and chaos in the classical two-body problem in a spherical universe”, Internation J. of Bifurcation and Chaos 18, 455-464 (2008). [5] N.J. Harmon, C. Leidel, and J.F. Lindner, “Optimal exit: Solar escape as a restricted three-body problem”, Am. J. Phys. 71, 871-877 (2003). Fields of Study Major Field: Physics v Table of Contents Abstract . . . . . . Dedication . . . . Acknowledgments Vita . . . . . . . . List of Tables . . List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page . ii . iii . iv . v . ix . x Chapters 1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . 1.1.1 Spin field effect transistors . . . . . . 1.2 Organization of dissertation . . . . . . . . . 1.3 What is spin? . . . . . . . . . . . . . . . . . 1.3.1 A brief history of spin . . . . . . . . 1.4 Magnetic dipoles in a constant external field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4 6 7 7 9 2 Spin relaxation 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spin density matrices and spins in a static magnetic field . 2.3 Spin relaxation due to random magnetic fields . . . . . . . . 2.3.1 The Redfield theory of relaxation . . . . . . . . . . . 2.3.2 The Redfield equation in the eigenstate formulation 2.3.3 The Bloch equations . . . . . . . . . . . . . . . . . . 2.3.4 Phenomenology . . . . . . . . . . . . . . . . . . . . . 2.4 The modified Bloch equations . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 13 15 15 17 20 24 24 26 . . . . . . 28 28 28 31 33 40 41 . . . . . . . . . . . . . . . . . . 3 Spin relaxation mechanisms in semiconductors 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.2 Conduction spin relaxation mechanisms . . . . . 3.2.1 The Elliott-Yafet mechanism . . . . . . . 3.2.2 The D’yakonov-Perel’ mechanism . . . . . 3.2.3 Finite temperatures . . . . . . . . . . . . 3.2.4 Hyperfine interaction . . . . . . . . . . . . vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 . . . . . 43 43 45 48 48 4 Phenomenological approach to spin relaxation in semiconductors I; case studies in bulk and quasi-2D zinc-blende crystals 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bulk crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Spin relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Comparison with experiments . . . . . . . . . . . . . . . . . . . . . . 4.3 Quasi-2D nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Spin polarization in quantum wells . . . . . . . . . . . . . . . . . . . 4.3.2 Modified Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Occupation concentrations . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Spin relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Results for GaAs/AlGaAs quantum well . . . . . . . . . . . . . . . . 4.3.6 Results for CdTe/CdMgTe quantum well . . . . . . . . . . . . . . . 4.3.7 Comparison of GaAs and CdTe quantum wells . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 51 52 54 57 59 60 64 65 67 69 73 73 76 3.4 3.5 Localized spin relaxation mechanisms . 3.3.1 Hyperfine interaction . . . . . . . 3.3.2 Anisotropic exchange interaction Cross-relaxation . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Phenomenological approach to spin relaxation in semiconductors II; case studies in bulk and quasi-2D wurtzite crystals 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Bulk crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Elliott-Yafet mechanism in bulk wurtzite crystals . . . . . . . . 5.2.2 The D’yakonov Perel’ mechanism in bulk wurtzite crystals . . . . . . 5.2.3 ZnO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Quasi-2D nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Temperature dependence of DP mechanism in wurtzite and zincblende QWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Comparison between wurtzite and zinc-blende . . . . . . . . . . . . . 5.3.4 Tuning of spin-orbit parameters . . . . . . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 102 106 108 6 Magnetic field effects 109 7 Conclusions 112 77 77 77 79 84 85 95 98 Appendices A Material parameters 125 B An integral involving spherical harmonics 127 vii C Important integrals 128 D The polylogarithm function 130 viii List of Tables Table 3.1 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Page The Dresselhaus spin-orbit Hamiltonians for bulk zinc-blende and wurtzite semiconductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of several semiconductor nanostructures with different growth orientations and their respective crystal axes. Also included is the parameter βD which gives the strength of the linear Dresselhaus terms in the Dresselhaus spin-orbit interaction (see Table 5.2). . . . . . . . . . . . . . . . . . . . . . . The spin-orbit Hamiltonians for several semiconductor QWs with different crystallographic orientations. The parameter βD is tabulated in Table 5.1. . A table of the various quantities needed in determining the DP spin relaxation rate in wurtzite and zinc-blende QWs. The quantities δν and ∆ν are located in Table 5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A table of the various quantities needed in determining the DP spin relaxation rate inP wurtzite and zinc-blende QWs. . . . . . . . . . . . . . . . . . . . . . zz Γzz = ∞ n=−∞ Γn as used in Eq. (5.64). Using αR = 0 for zb-(110). . . . . 1/τDP ,the DP spin relaxation rate, for several types of QWs in both the degenerate and non-degenerate limits. ζ(F ) = 2m∗ kB T(F ) /~2 . . . . . . . . . ∗ ,the minimum DP spin relaxation rate, for several types of QWs in 1/τDP both the degenerate and non-degenerate limits. . . . . . . . . . . . . . . . . Parameters for several semiconductors. . . . . . . . . . . . . . . . . . . . . . D.1 Using ν = 0. ζ(F ) = 2m∗ kB T(F ) /~2 . . . . . . . . . . . . . . . . . . . . . . . ix 31 96 97 98 99 100 101 103 107 131 List of Figures Figure 1.1 1.2 1.3 1.4 1.5 1.6 2.1 2.2 Page The amount of transistors per die have approximately doubled every 24 months in the last several decades. A die is the block of material on which the circuit is fabricated; its size is typically in the hundreds of millimeters squared. For comparison, the Pentium IV chips contains about 108 transistors/cm2 [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In conventional electronics, electrons ‘see’ a potential profile (solid line denoted by E). When the potential is large (OFF), electrons cannot pass through leading to reduced current. The opposite occurs when the potential is low (ON). The potential is controlled by the gate voltage. Figure adapted from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Datta-Das spin transistor [3]. The source and drain are ferromagnets (F). The channel is the two dimensional electron system (2DES) which has a Rashba spin-orbit coupling induced by the gate voltage Vg . . . . . . . . . The Stoner-Wohlfarth model of a ferromagnet’s energy dispersion. Polarization, P , is less than unity since at the Fermi energy, carriers of both spin exist. Notice that at the Fermi level, the Fermi wave vector, kF , is different for the two spin types. The two conduction band minima are offset by the exchange splitting energy, ∆. . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic detailing the basic features of the spin relaxation transistor. The source and drain are antiparallel ferromagnets (F). . . . . . . . . . . . . . . Picture of the magnetic moment µ resultant from a current, I, flowing around a rectangular loop of area A in an applied field B. The direction of the dipole moment is determined from the right hand rule as shown. The torque due to the external field is directed tangentially to the moment and causes it to precess around the magnetic field. The energy of interaction between the moment and the field is lowered if the moment aligned with the field. . . . 2 3 4 5 5 10 A π/2 pulse from an alternating magnetic field can rotate the magnetization into the plane orthogonal to the static field. Relaxation processes will work to restore the equilibrium magnetization. . . . . . . . . . . . . . . . . . . . 12 Depiction of T2 relaxation in rotating reference frame (with frequency ω0 ). In this rotating reference frame, moments precessing at ω0 will appear stationary. 13 x 2.3 2.4 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Occupations of conduction (red) and localized (blue) states for n-GaAs doped at 2 × 1016 cm−3 (solid lines) and 1 × 1016 cm−3 (dashed lines). Parameters used: m∗ = 0.067me , εB = −5.8 meV. Higher temperature are required to deplete localized electrons in the higher doped system. . . . . . . . . . . . Spin relaxation rate versus temperature in n-GaAs doped at 1016 cm−3 . Data is from Ref. [4]. Solid lines are least squares fit using Eq. (2.45). Dasheddotted curve: (nl /nimp )(1/τl ) for B = 0 T. Dashed curve: (nl /nimp )(1/τl ) for B = 4 T. Dotted curve: (nc /nimp )(1/τc ) for B = 0 T. Inset: momentum relaxation times for two different dopings: a) 1016 cm−3 b) 1018 cm−3 . Figure is from Ref. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pictorial descriptions of three common conduction spin mechanisms: Elliott Yafet (EY), D’yakonov Perel’ (DP), and Bir Aronov Pikus (BAP). For the first two, each vertex represents a scattering event. For BAP, electrons and holes are depicted by different colors and the arrows between them signify the exchange interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin polarization depends on k. For each k there are two mutually antiparallel pseudospins (only one of each pair is shown for a select few wave vectors). Scattering that alters the momentum (from k1 to k2 ) also changes the spin orientation. Graphic taken from [1] . . . . . . . . . . . . . . . . . . . . . . . Spin relaxation time versus doping density. Three distinct regimes are observed, indicating three different spin relaxation mechanisms: hyperfine, anisotropic exchange, and D’yakonov-Perel’. Figure taken from [6]. SNS refers to non-invasive spin-noise-spectroscopy measurements while conventional probes are optical orientation experiments. Data point references are located in [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conventional cubic cell of the zinc-blende crystal structure. . . . . . . . . . Band structure of GaAs at 300 K near the Γ-point (k = 0). Spin-splittings of the conduction band due to the Dresselhaus interaction are too small to be seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured and theoretical spin relaxation rates below the metal-insulatortransition (nM IT ≈ 2 × 1016 cm−3 ) in n-GaAs at low temperatures (below 10 K). Symbols are various experiments referenced in [7]. The theory curves contain no fitting parameters. The maximum spin relaxation time appears around 0.15nM IT . A maximum spin relaxation time has been also seen in one study on CdTe [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adapted from [9] showing how the temperature dependence of the spin relaxation depends on the doping density. . . . . . . . . . . . . . . . . . . . . Solid blue circles from Kikkawa using n-GaAs with nimp = 1 × 1016 cm−3 at zero applied field [4]. Solid line is fit with Eq. (2.45). . . . . . . . . . . . . . Solid blue circles from Malajovich et al. using n-ZnSe with nimp = 5 × 1016 cm−3 at zero applied field [10] Solid line is fit with Eq. (2.45). Used same mobility as for GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid blue circles from Sprinzl et al. using n-CdTe with nimp = 4.9 × 1016 cm−3 at zero applied field [8]. Solid line is fit with Eq. (2.45). Transport time taken from mobility measurements of [11] . . . . . . . . . . . . . . . . xi 27 27 30 32 50 52 53 56 57 58 59 60 4.8 Illustration of optical spin pumping in bulk semiconductor. CB and VB are conduction and valence bands respectively. σ ± denotes the helicity of the absorbed and emitted photons (wiggly lines). Photon promotes one electron from VB to CB leaving a hole behind. The hole spin (thick arrows) is assumed to relax much quicker than the electron spin (thin arrows). Sz is total electron spin. Graphic taken from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Illustration of optical spin pumping in QWs when photo-excitation is resonant with the trion formation. When excited at the donor-bound-exciton resonance instead, the picture is similar except that the exciton ‘steals’ an electron from a neutral donor (or is actually captured by the neutral donor) instead of a free electron. The key difference is that after hole spin relaxation and recombination, the neutral donors are left with a net polarization instead of the free electrons. Graphic taken from [12]. . . . . . . . . . . . . . . . . . 4.10 Illustration of optical pumping in QWs when photo-excitation is resonant with exciton formation. The conduction band (CB) starts with no spin polarization as the exciton is created (left most panel). The hole spin bound in the exciton rapidly relaxes (second panel). An antiparallel resident electron spin is grabbed from the electron gas (or the exciton binds to a neutral donor). This leaves the resident electron polarized (third panel). Oppositely oriented electron and hole spins recombine, casting the third electron back into the electron sea adding more net spin moment (rightmost panel). Taken from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.1 78 The wurtzite crystal structure. . . . . . . . . . . . . . . . . . . . . . . . . . xii 61 62 Chapter 1 Introduction 1.1 Motivation The electronic transistor is an ubiquitous staple in early 21st century life. In the little more than half century since John Bardeen, Walter Brattain, and William Shockley’s 1947 discovery that under certain circumstances output power could be larger than input power for electrical contacts on germanium, the transistor has revolutionized every facet of modern life. The world’s reliance on electronic devices stems not only from the original invention but also from the continual improvements and extensions to electronic circuitry that has contributed to the prominence of the transistor. The uncanny progress of the electronic industry is best summarized by Moore’s Law: the number of transistors inexpensively placed on an integrated circuit doubles approximately every two years [13]. As Figure 1.1 displays, this trend has continued for the last 50 years. When this exponential growth will stop has been a matter of speculation and has often been predicted to occur in the near future only to be proved wrong. However there are fundamental limits to how small transistors can be manufactured and still operate satisfactorily. To address the limits of conventional transistors that are rapidly being approached, the chief characteristics of the transistor must first be discussed. The most rudimentary definition of a transistor is a three terminal device where one terminal modulates the flow of current between the other two terminals. The usefulness of the transistor comes from the prospect that a small change in the voltage at one terminal leads to a large modulation of current between the other two terminals; in other words there is gain. The type of transistor to be considered here is the field effect transistor (FET). It is comprised of three key terminal components: a source, drain, and gate. The voltage at the gate controls the current between the source and drain by altering the potential barrier faced by electrons; see Figure 1.2. The metal insulator field effect transistor (MISFET) is a popular type of transistor. In the MISFET, the source and drain are metallic contacts with a semiconducting region in between them. On top of the semiconductor is stacked, perpendicular to the path 1 Figure 1.1: The amount of transistors per die have approximately doubled every 24 months in the last several decades. A die is the block of material on which the circuit is fabricated; its size is typically in the hundreds of millimeters squared. For comparison, the Pentium IV chips contains about 108 transistors/cm2 [1]. between the source and drain, an insulating layer and a metallic gate. Depending on the applied voltage at the gate, a two dimensional electron gas may form at the surface of the insulator and semiconductor which allows for n-type conduction [14]. As exemplified in Figure 1.2, the most simplistic explanation of the MISFET is a resistor whose resistance is controlled by a gate voltage. The issues that are run into when the number of transistor per area is increased are now addressed. A key feature of an operable transistor is the clear delineation between ON and OFF; this means that the conductance of the ON-state should be much greater than conductance of the OFF-state. Current ON to OFF ratios are in the range 106 [15]; channel lengths now are as small as a few tens of nanometers whereas in the early 1970s lengths were around ten microns. Miniaturizing transistors leads to larger leakage currents: unwanted currents between source and drain when the transistor is in the OFF-state. This is also known as standby or static power dissipation. The reason for this is that the potential barrier that prohibits large conductances must be thinner and therefore tunneling through the barrier becomes a possibility. The tunneling can be reduced by increasing the barrier height but there is an energy cost for doing so as well as a lengthening of the time to switch from OFF to ON [2]; the switching energy goes as 12 CVG2 where C is the gate capacitance and VG is the gate voltage. The switching energy plays into the dynamic power dissipation 2 Figure 1.2: In conventional electronics, electrons ‘see’ a potential profile (solid line denoted by E). When the potential is large (OFF), electrons cannot pass through leading to reduced current. The opposite occurs when the potential is low (ON). The potential is controlled by the gate voltage. Figure adapted from [2]. and is sought to be minimized. Additionally a high barrier must be maintained to avoid substantial thermionic emission from the source over the barrier; the amount of electrons thermalized into the channel goes as exp(−eVG /kB T ). So there is a dilemma: transistors can be made smaller but to avoid standby power dissipation barrier heights must be made larger which has the negative consequence of creating larger dynamic power dissipation [2]. The issue of power dissipation in MISFETS is so fundamental that the International Technology Roadmap for Semiconductors has labeled it a ‘red brick wall’ [16]. Bandyopadhyay and Cahay [15] have estimated the power dissipation on a chip in the year 2025 if Moore’s Law holds true; their calculation is sobering. A Pentium IV chip contains about 108 transistors/cm−2 . Each transistor dissipates 1500-2500 eV when switched (ON ↔ OFF). Given a switching speed of 2.8 GHz and 5% activity level at any one time, the power dissipation is around 5 W cm−2 . In 2025, if the density increases to 1013 cm−2 and the clock speed increases to 10 GHz, the dissipation will be 2 MW cm−2 which is equivalent to the thermal load in the nozzle of a rocket ship. Encoding information by charge displacement is an inherently inefficient means since it requires an energy |∆V (Q1 − Q2 )| (charge into the channel, charge out of the channel). Moreover, such encoding works best when Q1 and Q2 are very different in magnitude such that the two states can be clearly distinguished. A much more energy efficient mechanism to switch the transistor would be realizable if the charge in the channel could remain constant. The fundamental enterprise of spin-electronics (spintronics) is to utilize carriers’ spin degrees of freedom, which is a vector quantity, instead of its scalar charge. If altering spin states is an energy cheap process, then different spin states, as opposed to charge states, would be responsible for changing the conductance between source and drain and the problem of thermal dissipation could be largely avoided. 3 1.1.1 Spin field effect transistors In 1990, Datta and Das published the seminal article of semiconductor spintronics. The article is benignly entitled “Electronic analog of the electro-optic modulator” [3]. In this paper, they proposed a device very similar in structure to the traditional charge transistor but functionally very different; current between the source and drain of the FET would be modulated by altering the spin polarization of carriers. A simple schematic of the operation is shown in Figure 1.3. The source and drain in this spin field effect transistor Figure 1.3: The Datta-Das spin transistor [3]. The source and drain are ferromagnets (F). The channel is the two dimensional electron system (2DES) which has a Rashba spin-orbit coupling induced by the gate voltage Vg . (SPINFET) are ferromagnets. In the simple Stoner-Wohlfarth band structure of a ferromagnet (see Figure 1.4 where spin up is the majority carrier), the Fermi wave vectors of p √ the two spin carriers are kF,↑ = 2mEF /~ and kF,↓ = 2m(EF − ∆)/~ where ∆ is the exchange splitting energy which is typically several electron volts. Electrons with majority spin have a higher velocity and encounter less scattering than a minority spin. In other words the amount of resistance experienced by a carrier will depend on that carriers spin. If the carriers’ spins can be changed in the channel between the source and drain, it is then possible to affect transisting action. The use of the semiconductor in the channel is to allow the spins to be manipulated by an effective magnetic field created by the gate voltage. The details of this effective magnetic field are discussed at length in a later chapter. This effective field produces transistor action by rotating the spin of a carrier from majority to minority (as shown in Figure 1.3) which in turn increases the resistance. In general, the angle of rotation is θ = 2m∗ αR L/~2 where αR is a constant giving the strength of the effective field and L is the length of the channel [3]. Recently, the construction of this type of transistor using InAs was reported [17] but it was questioned whether those results actually display transistor action [18]. One main reason prohibiting the creation and utility of a Das-Datta SPINFET is the smallness of the spin-orbit coupling and the inability to change 4 Figure 1.4: The Stoner-Wohlfarth model of a ferromagnet’s energy dispersion. Polarization, P , is less than unity since at the Fermi energy, carriers of both spin exist. Notice that at the Fermi level, the Fermi wave vector, kF , is different for the two spin types. The two conduction band minima are offset by the exchange splitting energy, ∆. the coupling with the application of small voltages [15]. To rotate the spin 180◦ with a small spin-orbit field requires a longer channel which is detrimental to the overall aim of the further miniaturization of transistors. Many other ideas have been proposed and are discussed in [1]. One of these ideas the spin relaxation transistor [2] - is highlighted here. The spin relaxation transistor does not rely on spin precession to modulate the current but instead relies on spin relaxation. Similar to other conceptions, the ON-OFF states are determined by changing a spin-orbit field via a gate voltage. Figure 1.5 shows how the ON and OFF states are differentiated. Figure 1.5: Schematic detailing the basic features of the spin relaxation transistor. The source and drain are antiparallel ferromagnets (F). 5 Rapid spin relaxation is not seen as a negative but instead defines the ON state since in an unpolarized current, half of the carriers will be aligned with the ferromagnetic drain. However the spin relaxation rate must be controlled since the OFF state requires very little spin relaxation such that all spins will be anti-parallel with the drain. A successful spin relaxation transistor entails a relaxation rate that can be tailored by a small gate voltage. Though not implemented, one experiment suggests difficulties for the spin relaxation transistor. This study [19] found that the gate voltage needed to change by 3 V to decrease the spin diffusion length by 2.5%. The aim of this dissertation is to investigate materials in the hope that their spintronic properties may be suitable for utilization in a new class semiconductor devices. 1.2 Organization of dissertation In the remainder of this first chapter, the quantum mechanical concept of spin is introduced. Chapters 2 and 3 can appropriately be labeled as background chapters; in chapter 2, the subject of spin relaxation is discussed. The density matrix formalism is briefly outlined. Finally, the Redfield theory of spin relaxation is derived and the most pertinent results from the theory are discussed. Chapter 3 lays out the important spin relaxation mechanisms in semiconductors and especially those mechanisms relevant to this dissertation. Chapter 4 is divided into two sections: bulk and confined zinc-blende semiconductors. The treatment of bulk zinc-blende is the application of Putikka and Joynt’s earlier work [5] to more semiconductor materials (ZnSe and CdTe). Quasi-two-dimensional zinc-blende nanostructures are also investigated. New ideas were developed to explain the experimental phenomena. The analysis offers insight into the usefulness of certain experimental practices to accurately measure electron spin relaxation times. The work in quasi-two-dimensional nanostructures culminated in an article in Physical Review B [20]. Chapter 5 is set up likewise except for wurtzite semiconductors. The bulk wurtzite semiconductor ZnO is examined carefully for the first time. In order to explain the observed spin relaxation times, the analysis that was sufficient for zinc-blende systems is not so in ZnO. This work led to another publication in Physical Review B [21]. As of now, spin lifetime measurements in wurtzite quantum wells have not been conducted. The novel calculations here show that wurtzite quantum wells may offer many benefits over their zinc-blende counterparts in the emerging technologies that seek to utilize electron spin. This work has been submitted to Physical Review Letters [22]. Chapter 6 includes current work exploring the effects of magnetic fields on spin relaxation. Conclusions are drawn in chapter 7. Several appendices have been included to aide in the coherence of the dissertation and also offer the reader convenient reference. Appendix A contains a list of the notation used throughout. Additionally, all relevant material parameters for the semiconductors studied 6 are compiled. Appendices B and C are guides to derivations and calculations that are too long and would impede the flow of the dissertation if included within the main text. 1.3 What is spin? “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is deep down in relativistic quantum mechanics. This probably means that we do not have a complete understanding of the fundamental principle involved”[23] - Richard Feynman 1.3.1 A brief history of spin Much of the information in the following sections can be found in any standard treatment of quantum mechanics [24, 1]. The lectures by Sin-itrio Tomonaga are especially insightful [25]. In the mid-1910’s, Arnold Sommerfield and Peter Debye refined Niels Bohr’s atomic model to account for observed energy splittings in magnetic fields. They did this by introducing the orbital and magnetic quantum numbers l and ml where −l ≤ ml ≤ l. The angular momentum in a magnetic field was quantized with values m~. These ideas explained most of the possible atomic transitions but not all. In large magnetic fields energies split once more - doubling the amount predicted in the contemporary theory of angular momentum. This was termed the anomalous Zeeman effect. In 1925, Ralph Kronig proposed that another angular momentum, in addition to the orbital angular momentum, must be present that couples to the magnetic field. He postulated that this new angular momentum was due to the electron spinning on its own axis. If the magnitude of this angular momentum was fixed at ~/2 he was able to explain the observed atomic transitions. Kronig realized that his idea contained a serious flaw: it required the electron to spin so rapidly that the surface velocity would exceed the speed of light by over 60 times. He soon publicly pointed out this fallacy when Uhlenbeck and Goudsmit published a similar idea. However Uhlenbeck and Goudsmit also realized that their results were a factor of two off from experiment. L.H. Thomas soon corrected this discrepancy by clarifying the electron’s rest frame. In 1927, Ronald Fraser found further confirmation of the spin-model when he reinterpreted an experiment by Otto Stern and Walther Gerlach in 1922. In this experiment they had measured some sort of angular momentum quantization but had ascribed it to orbital angular momentum. Fraser noted that their system of silver atoms should have no orbital angular momentum in the ground state and hence actually the spin angular momentum had been measured. At about the same time the matrix mechanics and wave mechanics of Heisenberg and 7 Schrödinger, respectively, were being formulated. Pauli figured out how spin should be incorporated. He noted that (1) the components of spin, since it is an angular momentum, should obey commutation relations akin to orbital angular momentum and (2) spin measured along any coordinate axis yielded magnitudes ±~/2. The commutation relations are then easy to predict: [Sx , Sy ] = i~Sz plus the cyclic permutations. The two possible values of spin (as measured in Stern-Gerlach experiment for instance) suggest that the spin operator, Sz , is a 2 × 2 matrix with two eigenvalues of ±~/2. Of course there is nothing intrinsically special about the z-direction - the magnetic field could just as well be oriented along x or y - so the operators Sx and Sy should also have the same eigenvalues as Sz . With this constraint, in addition to the angular momentum commutation relations, Pauli deduced that spin must be an operator of the form S = ~σ/2 where ! ! ! 0 1 0 −i 1 0 σx = , σy = , and σz = 1 0 i 0 0 −1 are the Pauli spin matrices. P.A.M. Dirac extended the Schrödinger equation to incorporate relativity. In the process, spin was introduced naturally and not post facto as Pauli has done. Dirac’s seminal equation is ∂Ψ qA = (cα · (p − ) + βmc2 + qV )Ψ, (1.1) ∂t c where Ψ is a four component wave function (χΦ) and Φ, the large component, and χ, the i~ small component, are two component spinors themselves [24]. p is the momentum operator, c is the speed of light in a vacuum, A is the magnetic vector potential, V is a scalar potential, q is the charge of the particle, and m is its mass. α and β are the Dirac matrices which are ! ! 0 σ I2 0 α= and β = . σ 0 0 −I2 I2 is the 2 × 2 identity matrix. From now on only electrons (spin-1/2) are considered. When considering the Dirac equation to order v 2 /c2 , the presence of the magnetic vector potential naturally leads to the Zeeman Hamiltonian, HZ = −γe S · B = −gµB σ · B where µB = e~ 2mc is the electron Bohr magneton and γe = gµB is the gyromagnetic ratio [24]. In the theory of Dirac, g = 2 - this implies that the gyromagnetic ratio for spin is twice as large as for orbital angular momentum. If electrons really did spin in the physical sense, g would expected to be one. When working to order v 4 /c4 , the spin-orbit effect ‘falls out’ of the Dirac equation: Hs.o. = ~ σ · [∇V × p]. 4m2 c2 8 (1.2) Assuming a spherically symmetric potential, Hs.o. = ~ 1 dV (r) σ · L. 4m2 c2 r dr (1.3) A certain case when the potential is not symmetric is of particular importance; taking V = eF z where F is an electric field in the ẑ direction yields Hs.o. = e~F (σy px − σx py ), 4m2 c2 (1.4) which has the same form as the Rashba interaction to be discussed further in this dissertation. 1.4 Magnetic dipoles in a constant external field Since electrons possess an intrinsic magnetic dipole moment, it is beneficial to recall some results from classical physics. A current carrying loop is succinctly described by its magnetic dipole moment. The moment, µ, is defined as IAn̂ where I is the current in the loop, A is the area enclosed by the loop, and n̂ is a unit vector normal to the plane of the loop. It is well known from the Lorentz force law that a current carrying wire feels a magnetic force when the wire is in an applied field. For a loop of wire, a torque is exerted: T = IAB sin θ where θ is the angle between B and n̂ [26]. The torque generalizes to T = µ × B, (1.5) where µ = IAn̂. From picture in Fig. 1.6, it is evident (use the right hand rule) that the torque causes the dipole to precess around the applied field. If θ = 90◦ is defined as the zero of energy, then the potential energy of the dipole at an arbitrary angle can be determined by calculating the work done to rotate it away from 90◦ . The potential energy R R is Vint = T (θ)dθ = µB sin θdθ = −µB cos θ = −µ · B. Now instead of a loop of wire consider a circling charged particle. The current, I, due to the charge traveling past any point in the circle of radius r is qv/2πr. The dipole moment is found by multiplying by the area of the orbit, giving µ = qvr/2 = (q/2m)mvr = ql/2m where l = mvr is the angular momentum’s magnitude. In vector format, µ = shown to be dµ dt q 2m l. The time evolution of the moment is = −T = −µ × B. For completeness, the quantum mechanical description of a spin in an external field is treated now [27]. Given a field in the ẑ direction, the Zeeman Hamiltonian is HZ = −γe Bz Sz = γe Bz ~σz /2. The eigenstates are just that of Sz : χ↑ and χ↓ . The eigenvalues are ±γe Bz ~/2. Solutions to the time dependent Schrödinger equation (HZ χ = i~χ̇) are of the form χ(t) = aχ↑ e−ε↑ t/~ + bχ↓ e−ε↓ t/~ , 9 (1.6) Figure 1.6: Picture of the magnetic moment µ resultant from a current, I, flowing around a rectangular loop of area A in an applied field B. The direction of the dipole moment is determined from the right hand rule as shown. The torque due to the external field is directed tangentially to the moment and causes it to precess around the magnetic field. The energy of interaction between the moment and the field is lowered if the moment aligned with the field. where a and b are determined by initial conditions. In anticipation of the interpretation to follow, these constants are taken to be cos(α/2) and sin(α/2) respectively. Expectation values of the spin operators can readily be calculated by hSi i = χ(t)† Si χ(t) to obtain: ~ sin α cos(γe Bz t) 2 ~ hSy i = − sin α sin(γe Bz t) 2 ~ hSz i = cos α. 2 hSx i = (1.7) It is now clear that α marks the angle between B and hSi. By Ehrenfest’s theorem, the expectation value of an operator should evolve like its classical analog. Therefore the expectation value of the spin operator should evolve like the classical dipole moment µ, dhSi = γe hSi × B. dt This equation also holds for time dependent magnetic fields [28]. 10 (1.8) Chapter 2 Spin relaxation 2.1 Introduction What is meant by ‘spin relaxation’ ? In non-magnetic systems, there is no net spin polarization under equilibrium conditions when an external magnetic field is absent. Hence along any quantization direction, there is roughly an identical number of up and down spins. When certain perturbations are applied to such a system (to be discussed later), a nonequilibrium non-zero net spin polarization will form. Spin relaxation is the process by which the spins of the system return to equilibrium (i.e. zero net spin polarization). When an external magnetic field is present, there will be a non-zero net spin polarization under equilibrium conditions (to be calculated in Section 2.2); this is the context in which nuclear magnetic resonance (NMR - the spins be that of nuclei) and electron spin resonance (ESR - the spin being that of electrons) is utilized. In NMR experiments a second, alternating, magnetic field is applied in the plane orthogonal to the static field. The effect of this time varying field is to rotate the magnetization away from its equilibrium position. If applied indefinitely, the magnetization continually rotates between up and down. This phenomena is known as Rabi oscillations [24]. If applied for a specific time period, the magnetization is rotated from the z-direction to the x-y plane as shown in Figure 2.1. After the alternating field is shut off, the magnetization precesses around the static field as expected from classical dynamics. Thorough discussions of NMR can be found in many texts [28, 24]. Will the magnetic moment precess indefinitely? No, it will relax - meaning equilibrium will be restored. ‘Relaxation’ can occur along several pathways. Relaxation necessarily implies that the transverse moment (in x-y plane) decreases. The obvious way in which this happens is that the moment reorients along the static field. The timescale for which this occurs is known as T1 . The more subtle way is that some parts of the macroscopic moment may precess either quicker or slower than the Larmor frequency, ω0 . When the microscopic moments are added vectorally, the net moment is seen to decrease as the individual moments dephase. This is shown in Figure 11 2.2. The transverse relaxation timescale is T2 though there are other subtleties regarding transverse relaxation that will be discussed later. Figure 2.1: A π/2 pulse from an alternating magnetic field can rotate the magnetization into the plane orthogonal to the static field. Relaxation processes will work to restore the equilibrium magnetization. The following observations can be made. T1 processes must result in a transfer of energy since it involves magnetic dipoles reorienting in a magnetic field. Quantum mechanically, it is a change of populations between spin-down states to spin-up states which are nondegenerate in a magnetic field. Since the energy is typically gained by the lattice, T1 is frequently termed as the lattice or longitudinal spin relaxation time. Rarely, the term ‘spin relaxation’ is used to only describe T1 . Any longitudinal relaxation necessarily causes transverse relaxation. Hence T2 should be constrained by T1 but later it will be shown that it need not necessarily be less than T1 . T2 processes do not involve an energy transfer since the moment is transverse to the static field. T2 is known as the transverse or spin-spin relaxation time. It may also be referred to as a dephasing or decoherence time. To describe spin relaxation, F. Bloch presented a set of equations which included T1 and T2 as phenomenological parameters in 1946 that now bear his name [29]. Bloembergen, Purcell, and Pound devised a method by which these spin relaxation times could be derived for nuclear systems affected my molecular motions [30]. This theory was further refined by Wangsness and Bloch [31] and Redfield [32] in the 1950’s. It is this Wangsness, Bloch, and Redfield theory (or often simply Redfield theory) that is described in the following sections. Excellent reviews can be found in recent literature [33, 34, 35, 28]. It should be stated that 12 Figure 2.2: Depiction of T2 relaxation in rotating reference frame (with frequency ω0 ). In this rotating reference frame, moments precessing at ω0 will appear stationary. while much of this theory and nomenclature was developed for NMR and ESR, it is also useful for the study of spin dynamics in general. 2.2 Spin density matrices and spins in a static magnetic field In this section density matrices are introduced; the density matrix is a vital tool in the theory of spins and their interactions. A wave function ψ describes some system such that the expected value of some observable is hOi = hψ|O|ψi. If the wave function is expanded in terms of basis states φn (that are complete and orthonormal), we obtain hOi = P ∗ n,m cn cm hφn |O|φm i where the c’s are coefficients in the expansion of ψ. If the system is a mixture of particles in different states, the coefficients will vary. So the expected value of our observable will depend on the distribution of states. This can be expressed now as hOi = X Wk hψk |O|ψk i = k XX k Wk c∗n,(k) cm,(k) hφn |O|φm i (2.1) n,m where Wk denotes the probability of a specific ci,(k) occurring. The idea behind the density matrix is that instead of describing a system in terms of a wave function and its expansion coefficients (cn ), the system’s information is carried in the ensemble average of the product of P coefficients ( k Wk c∗n,(k) cm,(k) ). We set this ensemble average equal to the matrix elements of the density matrix hφm |ρ|φn i. The density matrix can be explicitly written as ρ= XX k Wk c∗n,(k) cm,(k) |φm ihφn |. (2.2) nm What information do the diagonal components of the density matrix carry? hm|ρ|mi = P ρmm = k Wk |cm,(k) |2 which is the probability of finding the system in the state |φm i. The expected value of some observable is expressible in terms of the density matrix P as hOi = n,m hφm |ρ|φn ihφn |O|φm i = Tr(ρO). As shown in standard texts [36, 28], the Schrödinger equation can be redrafted in terms of the density matrix which becomes what 13 is known as the von Neumann or Liouville equation: dρ i = [ρ, H ]. dt ~ (2.3) As an instructive example, spin- 12 particles with magnetic moments µi = ~ 2 γe σ i are considered in an external magnetic field. The Hamiltonian is H0 = −µ·H. We are interested in the expected value of the various magnetic moment components hµi i = ~ 2 γe hσi i. γe is the electronic gyromagnetic ratio and is equal to gµB . The time dependence of a Pauli spin matrix is found by using the properties of the density matrix: i~ dhσi i dt d dρ (Tr(ρσi )) = i~Tr( σi ) = Tr([H0 , ρ(t)]σi ) = Tr([σi , H0 ]ρ(t)) dt dt 1 (2.4) = − γe ~Hz Tr([σi , σz ]ρ(t)). 2 = i~ Any 2×2 matrix can be decomposed into a sum of the identity and Pauli spin matrices as P ρ(t) = 21 (I2 + k mk σk ) where I2 is the 2×2 identity matrix. It is simple to show that hσi i = mi . Substituting for the density matrix in Eq. (2.4), we ascertain i~ X dhσi i 1 = − γe ~Hz Tr([σi , σz ]) + mk Tr([σi , σz ]σk ) . dt 4 (2.5) k The first term in the sum is zero since the commutator of two identical operators is zero and the commutator of two different Pauli spin matrices is traceless. The second term can be compactly written as Tr([σi , σz ]σk ) = 4iizk , where izk is the antisymmetry (or Levi-Civita) tensor defined as ijk if i,j,k are cyclical 1 = . −1 if i,j,k are not cyclical 0 it two indices are identical We therefore are left with X dhσi i dmi = = −γe Hz mk izk . dt dt (2.6) k For an arbitrary magnetic field, the above equation generalizes to X dmi = −γe Hj mk ijk . dt (2.7) j,k Looking at a single component of the spin polarization mx , dmx = −γe (Hy mz − Hz my ) = γe (m × H)x . dt 14 (2.8) This generalizes to the familiar result dm = γe m × H. dt (2.9) Another useful example of the density matrix is from equilibrium statistical mechanics: what is the magnetization of an ensemble of spins in a static magnetic field? The diagonal elements of the density matrix give information on the populations of the eigenstates here spin up and down. In equilibrium, it is assumed that these populations are given P −Em /k T −En /k T B by the Boltzmann factor: e Z B where Z = is the partition function. me The off-diagonal terms are zero by the ‘hypothesis of random phases’ [28]. This assumption implies nothing more than that the magnetization transverse to the external field will vanish. Therefore ρ = 1 −H0 /kB T Ze where H0 = −γe ~Hz σz /2; in matrix form 1 ρ = −γ ~H σ /2k T e z z B e + eγe ~Hz σz /2kB T ! e−γe ~Hz σz /2kB T 0 0 eγe ~Hz σz /2kB T Using the simple results from the theory of density matrices, N µz = γe ~ 2 Tr(ρσz ) . ≈ γe2 ~2 Hz 4kB T in the high temperature limit. This is the familiar Curie law for paramagnetic magnetization [37]. 2.3 Spin relaxation due to random magnetic fields In the previous section, the density matrix formalism was introduced and used in two simple situations: spin dynamics and populations in a uniform magnetic field. While the phenomenological times T1 and T2 have been introduced to describe spin relaxation, no physical mechanisms have been touched upon. The goal of this section is to show in detail how one certain environmental interaction, that of a fluctuating magnetic field, can lead to spin relaxation. Explicit formulae will be derived for T1 and T2 in terms of the interaction parameters. The theory that allows this description is the aforementioned Redfield theory of spin relaxation. It is a semi-classical theory in that spin is accounted for quantum mechanically but the environment (or lattice) is dealt with classically. It is helpful to remember that it is essentially a second order time dependent perturbation theory calculation. It is only the density matrix notation that causes the development of the theory to appear foreign. 2.3.1 The Redfield theory of relaxation Consider the following time dependent Hamiltonian: H (t) = H0 + H1 (t) 15 (2.10) where H0 is due to the presence of a time independent external magnetic field and H1 (t) is a perturbative Hamiltonian that results from a much smaller magnetic field that is a random function of time turned on at t = 0. dρ(t) i = [ρ(t), H1 (t)] dt ~ (2.11) An operator, O(t), in the interaction representation is denoted by Õ(t) = eiH0 t/~ O(t)e−iH0 t/~ . (2.12) dρ̃(t) i = [ρ̃(t), H˜1 (t)] (2.13) dt ~ Integration of the Liouville equation of motion in the interaction representation leads to Z i t ρ̃(t) = ρ̃(0) + [ρ̃(t0 ), H˜1 (t0 )]dt0 . (2.14) ~ 0 This can then be substituted back into Eq. (2.13) which yields dρ̃(t) i 1 = [ρ̃(0), H˜1 (t)] − 2 dt ~ ~ Z t [[ρ̃(t0 ), H˜1 (t0 )], H˜1 (t)]dt0 (2.15) 0 This iterative solution is exact. To proceed, we consider an ensemble of ensembles that are identical at t = 0 but then evolve with different perturbative Hamiltonians H1 (t) [38]. We assume that the ensemble of ensemble average of H1 (t) vanishes for all times. The averaging is denoted by a bar such that H˜1 (t) = H1 (t) = 0. If this is not the case, whatever non-vanishing piece can be subsumed into H0 . So Eq. (2.15) averaged becomes: dρ̃(t) 1 =− 2 dt ~ Z t [[ρ̃(t0 ), H˜1 (t0 )], H˜1 (t)]dt0 . (2.16) 0 The first term of Eq. (2.15) vanishes because ρ̃(0) is independent of the perturbing Hamiltonian so this term is linear in H˜1 (t) and therefore vanishes as mentioned above. To proceed, several assumptions must be made concerning the perturbing Hamiltonian. First a correlation time, τc is introduced. Physically this quantity is the timescale on which the random field fluctuates. Values of the field separated by times longer than τc are therefore uncorrelated. It is assumed that this correlation time is much shorter than the timescale on which ρ̃(t) changes. This is justifiable since ρ̃(t) would be time independent if the perturbing Hamiltonian did not exist (Eq. (2.16)); since the fluctuating field is small, the time evolution of ρ̃(t) is slow [33]. This separation of timescales allows the following: • ρ̃(t0 ) may be replaced by ρ̃(t). Eq. (2.16) implies that ρ̃(t) depends on its history from times t0 = 0 to t0 = t. However we know that the system’s “memory” is erased after a short time τc . As previously shown, ρ̃ weakly depends on time so ρ̃(t − τc ) is not an 16 appreciable change and ρ̃(t0 ) ≈ ρ̃(t). In other words the time rate of change of the density matrix depends on the current density matrix and not on its past history. This is known as the Markov approximation [36, 39, 35]. • Secondly, correlations between ρ̃(t) and H˜1 (t) are neglected due to the difference in time scales. • Thirdly, the upper limit of the integral can be taken to ∞ since the short correlation time implies the integrand vanishes at long times. Hence this theory is limited to describing the dynamics at times longer than τc . As we proceed, these points will be revisited as necessary. With these assumptions in mind, a master equation is ascertained: dρ̃(t) 1 =− 2 dt ~ Z ∞ [[ρ̃(t), H˜1 (t0 )], H˜1 (t)]dt0 . (2.17) 0 Alternative derivations can be found in Refs. [33, 34]. In order to obtain useful results (spin relaxation rates) from the master equation, we will examine the master equation element by element. In doing so, we will derive the Redfield equation. This is known as the eigenstate formulation of Redfield theory. 2.3.2 The Redfield equation in the eigenstate formulation In this section, the Redfield equation is derived in a basis set where the spin eigenstates are of the unperturbed static Hamiltonian. We would like to calculate each element of the master equation in this basis: dρ̃(t)αα0 dhα|ρ̃(t)|α0 i 1 ≡ =− 2 dt dt ~ Z ∞ hα|[H˜1 (t), [H˜1 (t0 ), ρ̃(t)]]|α0 idt0 , (2.18) 0 where |αi is an eigenstate of H0 . The double commutator expands to H˜1 (t)H˜1 (t0 )ρ̃(t) + ρ̃(t)H˜1 (t0 )H˜1 (t) − H˜1 (t0 )ρ̃(t)H˜1 (t) − H˜1 (t)ρ̃(t)H˜1 (t0 ). (2.19) Three complete sets of states are inserted (such that the matrix elements of all three operators can be specified) into the matrix element hα|[H˜1 (t), [H˜1 (t0 ), ρ̃(t)]]|α0 i such that XX β β0 hα| X H˜1 (t)|γihγ|H˜1 (t0 )|βihβ|ρ̃(t)|β 0 ihβ 0 | + γ |βihβ|ρ̃(t)|β 0 ihβ 0 |H˜1 (t0 )|γihγ|H˜1 (t) − H˜1 (t0 )|βihβ|ρ̃(t)|β 0 ihβ 0 |H˜1 (t) − ! H˜1 (t)|βihβ|ρ̃(t)|β 0 ihβ 0 |H˜1 (t0 ) |α0 i. 17 (2.20) By using the relation between the Schrödinger and Interaction representations, hm|H˜1 (t)|ni = hm|H1 (t)|niei(Em −En )t/~ (where Em and En are the eigenvalues of H0 ), we ascertain X 0 hα|H1 (t)|γihγ|H1 (t0 )|βihβ|ρ̃(t)|β 0 ihβ 0 |α0 iei(Eγ −Eβ )t /~ ei(Eα −Eγ )t/~ + β,β 0 ,γ X 0 hα|βihβ|ρ̃(t)|β 0 ihβ 0 |H1 (t0 )|γihγ|H1 (t)|α0 iei(Eβ0 −Eγ )t /~ ei(Eγ −Eα0 )t/~ − β,β 0 ,γ X 0 hα|H1 (t0 )|βihβ|ρ̃(t)|β 0 ihβ 0 |H1 (t)|α0 iei(Eα −Eβ )t /~ ei(Eβ0 −Eα0 )t/~ − β,β 0 X 0 hα|H1 (t)|βihβ|ρ̃(t)|β 0 ihβ 0 |H1 (t0 )|α0 iei(Eβ0 −Eα0 )t /~ ei(Eα −Eβ )t/~ . (2.21) β,β 0 The time t0 can be redefined as t0 = t + τ . Now consider ensemble (or time) averaging as in the master equation. As mentioned earlier, correlations between the perturbing Hamiltonian and the density matrix are neglected. So we are interested in quantities like hα|H1 (t)|βihβ 0 |H1 (t + τ )|α0 i = hα|H1 (t − τ )|βihβ 0 |H1 (t)|α0 i which we define to be correlation functions Gαβα0 β 0 (τ ) which is real and an even function of τ [28]. These results allow us to write the master equation as Z X dρ̃(t)αα0 1 X ∞ =− 2 dτ δα0 β 0 Gαγβγ (τ )ei(Eγ −Eβ )τ /~ ei(Eα −Eβ )t/~ (2.22) dt ~ 0 γ β,β 0 X −i(Eγ −Eβ 0 )τ /~ i(Eβ 0 −Eα0 )t/~ Gβ 0 γα0 γ (τ )e e +δαβ γ −Gαβα0 β 0 (τ )ei(Eα −Eβ )τ /~ ei(Eα −Eβ +Eβ0 −Eα0 )t/~ ! −Gαβα0 β 0 (τ )e−i(Eα0 −Eβ0 )τ /~ ei(Eα −Eβ +Eβ0 −Eα0 )t/~ ρ̃(t)ββ 0 . To proceed, we must use fact that the perturbations are stationary which implies that [28] Gαβα0 β 0 (τ ) = hα|H1 (t)|βihβ 0 |H1 (t + τ )|α0 i = hα|H1 (t − τ )|βihβ 0 |H1 (t)|α0 i = hβ 0 |H1 (t + τ )|α0 ihα|H1 (t)|βi = Gβ 0 α0 βα (−τ ). (2.23) This allows us to write Z X dρ̃(t)αα0 1 X ∞ =− 2 dτ δα0 β 0 Gγβγα (−τ )e−i(Eγ −Eβ )(−τ )/~ ei(Eα −Eβ )t/~ dt ~ γ β,β 0 0 X −i(Eγ −Eβ 0 )τ /~ i(Eβ 0 −Eα0 )t/~ +δαβ Gβ 0 γα0 γ (τ )e e − (2.24) γ Gβ 0 α0 βα (−τ )e−i(α−Eβ )(−τ )/~ ei(Eα −Eβ +Eβ0 −Eα0 )t/~ ! −Gαβα0 β 0 (τ )e−i(Eα0 −Eβ0 )τ /~ ei(Eα −Eβ +Eβ0 −Eα0 )t/~ ρ̃(t)ββ 0 . 18 We now define what are known as spectral density functions: Z ∞ 0 0 0 0 dτ Gαβα0 β 0 (τ )e−i(Eα0 −Eβ0 )τ Jαβα β (Eα − Eβ ) = 2 (2.25) 0 which results in the master equation X dρ̃(t)αα0 1 X Jγβγα (Eγ − Eβ )ei(Eα −Eβ +Eβ0 −Eα0 )t/~ =− 2 δ α0 β 0 dt 2~ γ β,β 0 X +δαβ Jγα0 γβ 0 (Eγ − Eβ 0 )ei(Eα −Eβ +Eβ0 −Eα0 )t/~ − (2.26) γ Jαβα0 β 0 (Eα − Eβ )ei(Eα −Eβ +Eβ0 −Eα0 )t/~ ! −Jαβα0 β 0 (Eα0 − Eβ 0 )ei(Eα −Eβ +Eβ0 −Eα0 )t/~ ρ̃(t)ββ 0 after indices of the second and third terms have been rearranged as allowed. Also the first two terms have been multiplied by different exponential terms which has no effect (multiplying by unity) because of the Krönecker delta in each of those terms. The spectral function is a complex function but the imaginary part which produces the dynamic frequency shift which is in effect a small effective field which can be added to the large static field [34]. This phenomenon is ignored here. These results are summed up in the Redfield equation: X dρ̃(t)αα0 = Rαα0 ββ 0 ei(ωα −ωβ +ωβ0 −ωα0 )t ρ̃(t)ββ 0 dt 0 (2.27) β,β where ωi = Ei /~ and Rαα0 ββ 0 = − X X 1 0β0 Jγα0 γβ 0 (ωγ − ωβ 0 ) − J (ω − ω ) + δ δ γ γβγα β αβ α 2~2 γ γ ! Jαβα0 β 0 (ωα − ωβ ) − Jαβα0 β 0 (ωα0 − ωβ 0 ) (2.28) is known as the Redfield relaxation matrix. Three more steps are in order: • it is common practice to leave off the over-bars which denote ensemble averaging. • terms with ωα − ωβ + ωβ 0 − ωα0 6= 0 oscillate rapidly and average to zero such that ωα − ωβ + ωβ 0 − ωα0 = 0 dominates (secular approximation). The exponential terms also will vanish exactly when the representation is switched to that of Heisenberg (see Section 2.3.3) [38]. • so far the calculation has been done at infinite temperature - physically this means that there would be no equilibrium magnetization along the direction of the static field. This is remedied phenomenologically by ρ̃(t)ββ 0 → ρ̃(t)ββ 0 − ρ̃eq ββ 0 . A quantum mechanical treatment of the lattice has been been reviewed by several authors [33, 35]. 19 In conclusion we write the Redfield equation in its final simplified form X dρ̃(t)αα0 = Rαα0 ββ 0 (ρ̃(t)ββ 0 − ρ̃eq ββ 0 ) dt 0 (2.29) β,β where the relaxation matrix is defined as above. Redfield theory can also be derived through the operator formulation [33, 34, 35, 38]. 2.3.3 The Bloch equations After developing the formalism above, it is now instructive to obtain some useful results from it. The Bloch equations are derived here from a simple model perturbative Hamiltonian. The spin relaxation rates T1 and T2 will be obtained for this model. Recalling that H (t) = H0 + H1 (t) we set forth a general perturbation Hamiltonian that models a fluctuating magnetic field: 1 H1 (t) = ~ ωx (t)σx + ωy (t)σy + ωz (t)σz . 2 (2.30) H0 = 21 ~ω0 σz for a large static field in the z-direction. ω0 = γe H0 is the Larmor frequency and γe = gµB ~ . Several more simplifications must be made to make the problem tractable: • different components of the random field are independent (e.g. ωx (t) is not correlated to ωy (t)). • random fields in the same direction are correlated up to a correlation time τc . • after making these assumptions, we substitute Eq. (2.30) into the correlation function 2 P 0 0 Gαβα0 β 0 (τ ) = hα|H1 (t)|βihβ 0 |H1 (t + τ )|α0 i = ~4 i,i0 ωi (t)ωi0 (t + τ )hα|σi |βihβ |σi0 |α i. • a component of the random field will be correlated for a short time and then at longer times be uncorrelated. A simple model of the correlation function is then ωi (t)ωi0 (t + τ ) = δi,i0 ωi2 e−τ /τc [39, 28]. The correlation and spectral density functions then become 2 P 0 0 2 −τ /τc , Gαβα0 β 0 (τ ) = ~4 i hα|σi |βihβ |σi |α iωi e Jαβα0 β 0 (α0 − β 0 ) = 2 Z ∞ dτ 0 = 2 ~2 4 ~2 X hα|σi |βihβ 0 |σi |α0 iωi2 e−τ /τc cos((ωα0 − ωβ 0 )τ ) 4 i X hα|σi |βihβ 0 |σi |α0 iωi2 i τc , 1 + ωα2 0 β 0 τc2 (2.31) where we have neglected any imaginary portions of the spectral functions. At this juncture, we transform the density matrix back into the Heisenberg representation: dρ̃αα0 dt d d i(ωα −ωα0 )t iH0 t/~ −iH0 t/~ 0 0 = hα|e ρe |α i = e hα|ρ|α i dt dt d = ei(ωα −ωα0 )t hα|ρ|α0 i + i(ωα − ωα0 )ei(ωα −ωα0 )t hα|ρ|α0 i. dt 20 (2.32) The Redfield equation in the Heisenberg representation then becomes X dρ(t)αα0 = i(α0 − α)ρ(t)αα0 + Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq ββ 0 ) dt 0 (2.33) β,β where all the exponential terms of Eq. (2.27) have conveniently vanished without the need to invoke the secular approximation. We now write these differential equations in terms of observables - the magnetization vector m. Recalling the definition of the trace operator in P the familiar expression mi = ~/2Tr(ρσi ), we write mi = ~/2 α hα|ρσi |αi. If a complete set is inserted, then the observable is expressible in terms of the elements of the density matrix: P P mi = ~/2 α,α0 hα|ρ|α0 ihα0 |σi |αi = ~/2 α,α0 ραα0 hα0 |σi |αi. After summing Eq. (2.33) over α and α0 and multiplying by ~hα0 |σi |αi/2, the following is ascertained: dmi dt = ~ X dρ(t)αα0 0 hα |σi |αi = 2 dt 0 α,α = ~ ~iX [ρ, H0 ]αα0 hα0 |σi |αi + 2~ 0 2 α,α = = ~ ~i Tr([ρ, H0 ]σi ) + 2~ 2 X 0 Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq ββ 0 )hα |σi |αi β,β 0 ,α,α0 0 Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq ββ 0 )hα |σi |αi X β,β 0 ,α,α0 ~ i ~ω0 ~ Tr(ρ[σz , σi ]) + 2~ 2 2 X 0 Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq ββ 0 )hα |σi |αi. (2.34) β,β 0 ,α,α0 Recalling Section 2.2, the first term on the right hand side is equal to γe (m × H0 )i . The P second terms can be shown to be ~3 /8 α,β,j hβ|σj |αihα|[[σi , σj ], ρ]|βiωj2 1+ωτ2c τ 2 [28]. Now αβ c consider specific components - dmz /dt for instance. In the sum over j, j = z has no contribution since [σz , σz ] = 0. σx and σy have no diagonal elements so β and α will differ P ωx2 +ωy2 eq 0 by ω0 . In the end β,β 0 ,α,α0 Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq ββ 0 )hα |σi |αi = −τc 1+ω 2 τ 2 (mz − mz ). In 0 c totality, ωx2 + ωy2 dmz = γe (m × H0 )z − τc (mz − meq z ). dt 1 + ω02 τc2 (2.35) This is indeed the Bloch equation for the z-component of the magnetization. The first term describes the precession in the external static field. The second term describes the relaxation due to the small fluctuating magnetic field. The timescale on which the nonequilibrium magnetization is restored to its equilibrium value of meq z is T1 - the longitudinal or spin-lattice spin relaxation time: ωx2 + ωy2 1 τc = τc = (ωx2 + ωy2 ) . 2 2 T1 1 + ω0 τc 1 + ω02 τc2 (2.36) The evolution of mx and my can be treated similarly except that now there will be no equilibrium magnetization for those components since the static field is oriented along z. 21 Also for j = z, t α = β since σz is diagonal. The following are obtained: ωy2 dmx mx − τc ωz2 mx = γe (m × H0 )x − τc dt 1 + ω02 τc2 dmy ωx2 my − τc ωz2 my = γe (m × H0 )y − τc dt 1 + ω02 τc2 (2.37) which is more succinctly expressed as 1 dmx mx = γe (m × H0 )x − dt T2x dmy 1 my = γe (m × H0 )y − − dt T2y (2.38) with ωy2 1 = τc + τc ωz2 T2x 1 + ω02 τc2 1 ωx2 + τc ωz2 . = τc T2y 1 + ω02 τc2 (2.39) Note that both magnetizations decay due to fluctuating fields orthogonal to the magnetization direction. This is sensible since fields parallel to a moment cannot act on the moment. The in-plane magnetization, m⊥ relaxes with a rate 1 T2 = 21 ( T12x + 1 T2y ) which is found by averaging over one precessional period [28]. Combining these results, the final form of the Bloch equations is written as dmx 1 = γe (m × H0 )x − mx dt T2 dmy 1 = γe (m × H0 )y − my dt T2 dmz 1 = γe (m × H0 )z − (mz − meq z ), dt T1 (2.40) where 1 τc = (ωx2 + ωy2 ) T1 1 + ω02 τc2 τc ωx2 + ωy2 1 1 1 1 1 1 = + τc ωz2 = + 0. + τc ωz2 = 2 2 T2 2 1 + ω0 τc 2 T1 2 T1 T2 (2.41) We now will discuss T1 and T2 and in the process gain some physical interpretations to the above results. The term with ωz either aides or inhibits the external field and therefore causes variations in the precession frequency. How this causes transverse relaxation is seen via a simple model [28]. Consider ωz jumping from +ωz to −ωz in increments of the correlation time, τc . A spin experiencing this fluctuation will precess an additional angle 22 according to dφ = ±ωz τc . After n intervals of fluctuating (in a time t), the mean square dephasing angle will be ∆φ2 = ndφ2 = nωz2 τc2 . The condition of relaxation is defined as when the mean square dephasing angle is one radian; this occurs at t = T20 . So 1/T20 = ωz2 τc . T20 is known as secular broadening in the literature. The strange relation 1/T20 ∼ τc is a phenomenon known as motional narrowing; the quicker the field changes, the longer the resulting relaxation time since the phase does not have time to accumulate before the field changes (i.e. dφ is small when τc is small). If the fluctuating field is isotropic (ωx2 = ωy2 = ωz2 ) and the correlation time is short such that ω0 τc 1 (extreme motional narrowing), then T1 = T2 . Since the correlation time is often short, T1 ∼ T2 is applicable in small magnetic fields. If the secular broadening is eliminated (ωz2 = 0), the result T2 = 2T1 holds. Yafet has shown that the restraint T2 ≤ 2T1 is quite general [40]. Lastly, the regime in which the transverse fluctuations are small will cause negligible longitudinal relaxation and all relaxation will be dominated by T2 = T20 . It is to be expected that T2 would be limited in some sense by T1 since any moment that leaves the transverse plane for the longitudinal direction will necessarily cause transverse relaxation. However transverse relaxation can occur in the absence of any longitudinal relaxation; this is pure spin dephasing. It occurs when fluctuations perpendicular to the external magnetic field are suppressed but parallel fluctuations are not. It will also occur when the external field is very large. In later chapters, it will be seen that systems can be fabricated that indeed suppress certain components of the fluctuating field. This will lead to relaxation anisotropy. Relations between T1 and T2 are useful since both are rarely either measured or calculated together. T1 is tractable through calculations while T2 is not [41]. As mentioned earlier, it is not quite T2 that is measured in NMR and ESR experiments. Transverse magnetization can also decay due to inhomogeneous broadening. Inhomogeneous broadening or precessional dephasing results from spatially inhomogeneous magnetic fields which cause different precession rates. It is to be distinguished from true decoherence or relaxation since it is potentially reversible via spin-echo experiments. Inhomogeneous magnetic fields will cause true decoherence if the spin’s orientation loses correlation with its position [42]. The totality of transverse magnetization loss is then summed up in the quantity T2∗ . Since the reversible losses are added to the irreversible losses, T2∗ ≤ T2 necessarily. For localized electrons (on quantum dots or donors), T2∗ has been observed to be much smaller than T2 [41]. Due to the itinerant nature of conduction electrons, spatial inhomogeneities are ‘washed out’ due to motional narrowing and T2∗ ≈ T2 . Since the notation for spin relaxation is extensive, commonly a single quantity τs is used for all types of relaxation when external fields are small. This thesis adopts that notation except when it is necessary to delineate between different types of relaxation. 23 2.3.4 Phenomenology The Bloch equations that were derived in the previous section resulted from a specific Hamiltonian (small random magnetic fluctuations). For different interaction Hamiltonians, one would need to check if Bloch equations formed at all [28]. Typically the Bloch equations are treated exactly how they were originally developed: phenomenologically. For many systems of interest (namely that of itinerant electrons), T1 and T2 are assumed to be comparable (and equal to τs ) and inhomogeneous broadening is neglected. When there is no applied field, dm/dt = −m/τs where the rate 1/τs can be calculated by various means. This relaxation rate could be a sum of different mechanisms but typically one will be dominant and show up in experiments. Chapter 3 will be devoted to describing the arsenal from which mechanisms are chosen. In the next section, a phenomenological modification of the Bloch equations is presented which is necessary when spins in multiple spin species (e.g. itinerant and localized) interact through an exchange interaction. 2.4 The modified Bloch equations An important principle to be discussed in this thesis is the spin exchange between electrons in different environments. Most important for the purposes here, is the exchange interaction between localized and itinerant electrons in semiconductors. In semiconductors with sufficiently high purity, the localized electrons are situated on donor sites. Itinerant electrons are in the conduction band. In 1981 D. Paget found striking experimental evidence that the exchange of spins was necessary to explain the observed spin lifetimes [43]. The evidence was that both conduction and donor spins relaxed similarly by the hyperfine interaction (to be discussed in chapter 3) even though it was expected that only donor spins would experience large enough nuclear fluctuations. Paget posited that the conduction electrons depolarized not by their own interaction with nuclei but by their strong interaction with the electrons bound to donors. In recent years this mechanism has been used to describe phenomena in several experiments on spins in semiconductors [44, 45, 5, 46, 7]. Paget wrote a pair of coupled Bloch equations1 to phenomenologically account for the exchange interaction [43].2 1 The equations in Ref. [5] are identical to Paget’s except that Paget writes his equations in terms of mean magnetizations and not total magnetizations as Ref. [5] does. The notation of Ref. [5] is followed here. 2 For the sake of simplicity, the spin relaxation from the two states τc and τl have been neglected. At the current time, only the effects of cross relaxation want to be examined. 24 dmc (t) 1 = − (nl mc (t) − nc ml (t)) dt γcr dml (t) 1 = − (nc ml (t) − nl mc (t)), dt γcr (2.42) where the quantity3 nimp /γcr = nimp Γcr is the time scale for the mean magnetization of the two populations to become equal which can simply be obtained by setting the time derivatives equal to zero: mc /nc = ml /nl . Magnetization here is defined as mi = ni,↑ − ni,↓ for some i species of particles. The density of that species of particles is ni = ni,↑ + ni,↓ . The mean magnetization is mav i = mi /ni . The subscripts c and l stand for conduction and localized states respectively. The determination of this timescale - the cross relaxation time - will be set aside until later. The solution to Eq. (2.42) is mc (0) nc + nl e−nimp Γcr t nc + nl mc (0)nl ml (t) = 1 − e−nimp Γcr t , nc + nl mc (t) = (2.43) where the following initial conditions have been used: ml (0) = 0 and mc (0) 6= 0. As expected since exchange is a spin preserving interaction, ṁ = ṁc + ṁl = 0 and m(t) is a constant dictated by the initial conditions. At long times, Eq. (2.43) reduces to mc (t)/nc = mc (0)/nimp and ml (t)/nl = mc (0)/nimp which confirms the expectation that the mean magnetizations are equal. When spin relaxation is added, dmc (t) 1 1 = − (nl mc (t) − nc ml (t)) − mc dt γcr τc dml (t) 1 1 = − (nc ml (t) − nl mc (t)) − ml . dt γcr τl (2.44) The solution, though more cumbersome than before, can still be found exactly. When the cross relaxation time is assumed short, the decay constant in the solution simplifies to 1 nc 1 nl 1 = + . τs nc + nl τc nc + nl τl (2.45) Two decades after Paget’s initial work, the group from St. Petersburg, Russia found similar evidence supporting the rapid cross relaxation between the two spin systems [44, 45]. At low temperatures in a low doped n-GaAs quantum well, they were able to increase measured spin lifetimes from a few nanoseconds to nearly 300 nanoseconds by modifying the conduction electron concentration. As can be seen from Eq. (2.45), this will cause the rapid relaxation 3 This quantity is equal to Paget’s 1/τe [43]. 25 from the localized spins (due to hyperfine interaction) to be suppressed while the slower conduction spin relaxation becomes important. The theory of exchanging spins between localized and conduction electrons was examined most carefully by Putikka and Joynt [5] in 2004 when they explained the anomalous temperature dependence of spin relaxation in bulk semi-insulating n-GaAs [4]. By knowing the approximate impurity density to be nimp = 1016 cm−3 , they were able to ascertain the occupations of both states for all temperatures using standard statistical mechanics. The concentration of conduction and donor-bound electrons for a given impurity density is known to be [47] Z ∞ dεg(ε)f0 (ε, µ), nc = nl = 0 1+ nimp 1 (εB −µ)/kB T 2e (2.46) where g(ε) = (2m∗ )3/2 ε1/2 /2π 2 ~3 is the electronic density of states in three dimensions, f0 (ε, µ) = 1/(1 + e(ε−µ)/kB T ) is the Fermi-Dirac function, µ is the chemical potential, and εB < 0 is the binding energy of the electron bound to the donor. nc and nl are constrained by nc + nl = nimp . In general, the chemical potential cannot be solved in closed form in three dimensions. It is not difficult to solve numerically. Once done, the concentrations of both electron types are calculated as a function of temperature which is shown in Fig. 2.3. From Eq. (2.45), it is obvious that the occupations of the states will figure into the observed relaxation times. This is indeed what was seen by Kikkawa and Awschalom (data in Fig. 2.4) [4]. Due to the population statistics, the dominant spin lifetimes observed at low temperatures and high temperatures should be that due to localized and itinerant electrons respectively. What these lifetimes are, is the subject of chapter 3. It is evident from this picture that a non-monotonic dependence of relaxation rate on temperature is possible and indeed confirmed for B = 4 T. This behavior is also seen at zero applied field in n-ZnO, covered in chapter 5. 2.5 Summary In this chapter the framework of what is to come has been laid. The key concepts within the theory of spins in semiconductors have been identified. The Bloch equations have been introduced as phenomenological equations but have also been rigorously shown to be valid given some model interactions. Lastly, it was shown that the exchange interaction is an important feature in semiconductors that must be accounted for in the study of semiconducting electron spins. No mention of the actual spin relaxation types that affect spin polarization in real systems have been identified. This is the subject of the next chapter. 26 1.0 0.6 ncnimp nlnimp 0.8 0.4 0.2 0.0 0 50 100 150 200 250 300 T HKL Figure 2.3: Occupations of conduction (red) and localized (blue) states for n-GaAs doped at 2 × 1016 cm−3 (solid lines) and 1 × 1016 cm−3 (dashed lines). Parameters used: m∗ = 0.067me , εB = −5.8 meV. Higher temperature are required to deplete localized electrons in the higher doped system. Figure 2.4: Spin relaxation rate versus temperature in n-GaAs doped at 1016 cm−3 . Data is from Ref. [4]. Solid lines are least squares fit using Eq. (2.45). Dashed-dotted curve: (nl /nimp )(1/τl ) for B = 0 T. Dashed curve: (nl /nimp )(1/τl ) for B = 4 T. Dotted curve: (nc /nimp )(1/τc ) for B = 0 T. Inset: momentum relaxation times for two different dopings: a) 1016 cm−3 b) 1018 cm−3 . Figure is from Ref. [5]. 27 Chapter 3 Spin relaxation mechanisms in semiconductors 3.1 Introduction As discussed in chapter 2, magnetic field fluctuations that couple to spin can cause spin relaxation. In semiconductors there is an arsenal of several mechanisms to choose from that contribute to spin relaxation. Which mechanism dominates depends on system variables such as temperature, field, and doping. The easiest delineation is between the metallic and insulating regimes. In the metallic regime, electrons are itinerant as they populate the conduction band. Several relaxation mechanisms are found to be especially important in this case; they are called conduction spin relaxation mechanisms and are detailed below. One expects these mechanisms to be predominant in highly doped systems or systems at high temperatures. When a semiconductor is insulating, electrons are localized - typically at donor sites in n-type materials. A different set of mechanisms is important and are entitled localized spin relaxation mechanisms. One expects these mechanisms to be most efficient in low doped systems or systems at low temperatures when electrons are not thermalized and their wave functions are weakly overlapping. 3.2 Conduction spin relaxation mechanisms The three prevalent relaxation mechanisms of conduction spin can be attributed to three different manifestations of the spin-orbit (SO) interaction. They are briefly outlined here and discussed further in sections 3.2.1 and 3.2.2. In 1954 Elliott asserted that spins could relax due to spin-independent scattering [48]. However this is only possible due to the SO interaction which prohibits spin from being a ‘good’ quantum number (the operator Sz does not commute with the SO Hamiltonian). This implies that the spin component of the wave function is not pure but is an admixture of spin eigenstates. The amount of ‘mixing’ depends on the size of the SO interaction. Typically 28 the amount of mixing is small and a pseudospin is defined by the dominant component (up or down). For instance, a pseudospin is up (↑) if hΨk |σz |Ψk i > 0; a pseudospin is down (↓) if hΨk |σz |Ψk i < 0 [49]. For example, a pseudospin down state is labeled as Ψk,↓ . A consequence of this spin admixture is that scattering induced transitions from pseudospin up to pseudospin down states are non-zero probability events as they would be if the spinor were pure. A cartoon that exaggerates the amount of spin flips is shown in Fig. 3.1. Yafet’s contribution came in 1961 when he posited that since the SO interaction is caused by lattice ions, when those ions vibrate the SO interaction is modulated [40]. The Yafet process is also known as the short range interaction where the Elliott process is called the long range interaction [39]. The long range interaction is most important in III-V semiconductors and the short range interaction will not be considered further in this dissertation. Aside from the short range interaction, Yafet also systematized the theory of Elliott by applying it to specific band structures and electron-phonon scattering. Hence, the long range process still goes by the name Elliott-Yafet (EY). Determining the EY relaxation rate will be discussed in section 3.2.1. D’yakonov Perel’ (DP) spin relaxation also originates from the SO effect but in a very different way than EY. Besides the necessity of the SO effect, DP also requires the crystal to be non-centrosymmetric (no inversion symmetry). The importance of lacking inversion symmetry can be seen from quantum mechanics. First consider the time reversal operator, K̂.4 If a system possesses time reversal symmetry (e.g. no external magnetic field), both Ψk and K̂Ψk are orthogonal eigenstates with the same eigenvalue [49]. If the time reversal operator is applied to a semiconducting crystal whose wave function are Bloch functions, KΨk,↑ = Ψ−k,↓ and K̂Ψk,↓ = Ψ−k,↑ since the time reversal operator affects k → −k and also reverses angular momentum. So εk,↑ = ε−k,↓ , εk,↓ = ε−k,↑ . (3.1) This implies that bands are not degenerate but opposite pseudospins (just spin if spin mixing of wave functions is ignored) are related by opposite wave vectors. If spatial inversion is a characteristic of the crystal then k is undifferentiated from −k; hence, εk,↑ = εk,↓ , (3.2) which is the familiar result of doubly (pseudo)spin degenerate bands. Since non-centrosymmetric crystals lack inversion symmetry, Eq. (3.2) does not hold but Eq. (3.1) does hold when external fields are absent. Eq. (3.1) alone suggests that bands are not doubly degenerate (they are spin-split)5 except for certain k values. Obviously there 4 The operator K̂ is represented by iσy Ĉ where Ĉ is the conjugation operator [49] A point worth mentioning because it is rarely stated in the literature: in analytical calculations of DP relaxation, the pseudospin nature is neglected in favor of pure spin eigenstates. Since spin admixture is 5 29 Figure 3.1: Pictorial descriptions of three common conduction spin mechanisms: Elliott Yafet (EY), D’yakonov Perel’ (DP), and Bir Aronov Pikus (BAP). For the first two, each vertex represents a scattering event. For BAP, electrons and holes are depicted by different colors and the arrows between them signify the exchange interaction. is degeneracy at k = 0. The quantity and characteristics of spin-splitting are determined from band structure [39]. The Dresselhaus splitting can be derived using the extended Kane model for the band structure [50, 51, 39]. It is found from this calculation that the conduction band splits as if there is a k-dependent magnetic field. Additionally, the form of the spin-splitting can be found by group-theoretic techniques [52]. Consequently, this effective field is described by a Zeeman-like Hamiltonian: Hso (k) = ~2 Ωk · σ. Table 3.1 contains the Dresselhaus spin-orbit Hamiltonian for bulk zinc-blende and wurtzite semiconductors. Figure 3.1 depicts the evolution of a spin in the effective k-dependent magnetic field which is termed now as the spin-orbit field; between each scattering event, the spin precesses around a field with a random precession axis. Over time, the spin ‘forgets’ its initial direction. φ = Ωk τp is the angle of precession between a scattering event. The mean-square assumed small, this approximation is valid when considering DP [42]. 30 Bulk Material zinc-blende wurtzite H1 H3 0 β1 (ky σx − kx σy ) β3 kx (ky2 − kz2 )σx + ky (kz2 − kx2 )σy + kz (kx2 β3 (bkz2 − kx2 − ky2 )(ky σx − kx σy ) − ky2 )σz Table 3.1: The Dresselhaus spin-orbit Hamiltonians for bulk zinc-blende and wurtzite semiconductors. angle increases as the number of scattering events, n, increases: hφ2 i = hΩ2k iτp2 n. The spin is sufficiently randomized to warrant relaxation when hφ2 i = 1. This happens at time t = τs and after n = τs /τp scattering events. The result for the relaxation rate is then τs−1 ∼ hΩ2k iτp . This brief calculation assumes the the precession period is much smaller than the scattering time; this is known as the motionally narrowed regime. Section 3.2.2 contains a more formal calculation of the DP mechanism. In semiconductors where holes are plentiful, electron polarization can decay due to exchange interactions with holes (see Figure 3.1. This is known as the Bir-Aronov-Pikus (BAP) mechanism [53, 54]. No materials examined in this dissertation are p-type and it is therefore safe to ignore the BAP mechanism [55, 56]. In section 4.3 an intrinsic sample is investigated and the BAP may be important in such a case after photo-excitation when a equivalent amount of electrons and holes are present in the form of excitons. For the time being the relaxation of spin in excitons is left as a phenomenological parameter. In semiconductors consisting of host nuclei with non-zero nuclear spin, the hyperfine interaction causes conduction electron spin relaxation. This is discussed further in section 3.2.4. 3.2.1 The Elliott-Yafet mechanism The wave functions in the presence of spin-orbit coupling are of the form Ψk,↑ = |k, ↑ieik·r = ak | ↑i + bk | ↓i eik·r . (3.3) When a magnetic field is absent, we also the know that the time reversed wave function is K̂|k, ↑ieik·r = a∗k | ↓i − b∗k | ↑i e−ik·r , (3.4) which has the same energy as the prior wave function and is orthogonal to it. What is the expected value of σz in the two states? hΨk,↑ |σz |Ψk,↑ i = |ak |2 − |bk |2 and hΨk,↓ |σz |Ψk,↓ i = −|ak |2 + |bk |2 which says that the pseudospins are antiparallel. Typically the spin-orbit interaction is weak and a b such that the pseudospins approximate pure spin which gives 31 ±1 for the above expectation values.6 When an electron scatters, it transitions from k −→ k0 . As was just pointed out, states with the same k and opposite pseudospin are orthogonal and a spin-independent potential should not induce any change in the spin structure of the wave function. However when k0 6= k, the coefficients of Ψk0 change which allows for the pseudospin to flip at scattering events. This process is shown in Fig. 3.2 Here the framework of calculating the Elliott Yafet relaxation rate is described. It is calculated explicitly for the bulk wurtzite crystal in chapter 5. Figure 3.2: Spin polarization depends on k. For each k there are two mutually antiparallel pseudospins (only one of each pair is shown for a select few wave vectors). Scattering that alters the momentum (from k1 to k2 ) also changes the spin orientation. Graphic taken from [1] In general, the scattering rate from a potential Vscatt. is found by Fermi’s Golden Rule [24] X 2π ~ k0 |hf |Vscatt. |ii|2 , (3.5) where the state vector i and f refer to the initial and final states respectively. Using the notation outlined above, consider the matrix element hk0 , ↓ |Vscatt. |k, ↑i which describes an initially ‘up’ state scattering to a ‘down’ state: Z 0 hk0 , ↓ |V |k, ↑i = ei(k−k )·r Vscatt. dr hk0 , ↓ |k, ↑i , V 6 Wave functions are assumed to be normalized such that |a|2 + |b|2 = 1. 32 (3.6) where the factorization of the integral (over the total volume V ) involving the envelope and potential functions is possible since they are slowly varying over the unit cell which is not necessarily true for the Bloch functions [57]. The first term in parentheses is simply the Fourier transform of the potential [58]; the second in parentheses is an integral of Bloch functions over a single unit cell; so Vk,k0 hk0 , ↓ |k, ↑i is left or more generally Vk,k0 hk0 , s0 |k, si, (3.7) where s and s0 are the initial and final pseudospins. To proceed, the actual conduction band Bloch functions must be known. They are known from k · p theory for both cubic and wurtzite direct gap semiconductors [57, 58, 59]. The Elliott-Yafet mechanism has been studied extensively in zinc-blende semiconductors [57, 58, 60, 61]. In chapter 5, it is discussed in wurtzite crystals for the first time. 3.2.2 The D’yakonov-Perel’ mechanism In this section the derivation of the spin relaxation rate in non-centrosymmetric bulk semiconductors proposed in 1971 by D’yakonov and Perel’ is presented. While we follow their derivation [62, 63], other sources have clarified their work [61, 39]. Spin relaxation by this same mechanism is examined in two-dimensional semiconductors which was taken up in 1986 by D’yakonov and Kachorovskii [64]. In both systems, the mechanism is commonly referred to as the D’yakonov-Perel’ (DP) mechanism. Since the DP mechanism is most important over a significant temperature range (especially at room temperature) for the zinc-blende and wurtzite crystals studied here, the spin relaxation due to DP will be derived here fully for bulk systems in general. The framework in which the DP relaxation rate is solved is the semi-classical spin Boltzmann equation. In the following, the semi-classical spin Boltzmann equation is derived and used to calculate the DP relaxation rate in the motional narrowing regime (scattering time shorter than spin-orbit precession period). The derivation follows that of [39]. Consider the semi-classical evolution of a wave packet’s position and momentum in external fields: ∂εk ~∂k ~k̇ = Fk = −eE − evk × B, ṙ = vk = (3.8) where F is the Lorentz force, E is the electric field, and εk is the electronic dispersion of the conduction band. Spin is introduced quantum mechanically. By assigning each electron wave packet a 2×2 density matrix, the electron’s spin state can be described in general. The single electron density matrix is δk,k0 δ(r−r 0 )ρk (r, t) ≡ ρk (r, t) where we have approximated position and momentum to be diagonal. The same restriction is not placed on spin. The properties of the density matrix are such that the trace of the density matrix in state k at 33 r is the number of electrons in that state. The total electron density is found by summing over all space and momenta: 1 n= Ω Z d3 r X Trρk (r, t). (3.9) k The spin density is found similarly, m(t) = 1 Ω Z d3 r X Tr ρk (r, t)σ . (3.10) k While the position and momenta evolution are evaluated semi-classically, the spin evolution is evaluated quantum mechanically. We consider the spin-orbit Hamiltonian which was discussed earlier: ~ Ωk · σ. 2 The time evolution of the density matrix ρ (≡ ρk (r, t)) is Hs.o. = ∂ρ i = − [Hs.o. (k), ρ]. ∂t ~ (3.11) (3.12) How does the density matrix at time ρ(t − dt) change in a small amount of time dt? ρ(t) − ρ(t − dt) i = − [Hs.o. (k), ρ], dt ~ i ρ(t) = ρ(t − dt) − [Hs.o. (k), ρ]dt. ~ (3.13) This is the first order in dt expansion of the exact evolution ρ(t) = e−iHs.o. dt/~ ρ(t − dt)eiHs.o. dt/~ . (3.14) From this expression of the density matrix, the Boltzmann equation can be derived in the standard way [47]. This is done by Taylor expanding Eq. (3.14) to first order in dt: ρ = ρ(k(t), r(t), t) = e−iHs.o. dt/~ ρ(k(t − dt), r(t − dt), t − dt)eiHs.o. dt/~ ∂ρ ∂ρ ∂ρ i ≈ ρ(r, k, t) − · F dt − · vk dt − dt − [Hs.o. , ρ]dt. ~∂k ∂r ∂t ~ (3.15) Then by taking the time derivative, ∂ρ ∂ρ ∂ρ i + ·F + · vk + [Hs.o. , ρ] = 0. ∂t ~∂k ∂r ~ (3.16) The density matrix at phase space point (r, k) can also change due to electrons scattering into and out of that point. Such collisions are not included in Eq. (3.15); Eq. (3.15) assumes electrons travel from (r − vdt, k − F dt/~) to (r, k) in time dt [47]. In reality, collisions will either facilitate or hinder this motion. The effect of collisions can be added to Eq. (3.15) 34 by hand and simplified to obtain ∂ρ ∂ρk ∂ρk ∂ρk i k , + ·F + · vk + [Hs.o. , ρk ] = ∂t ~∂k ∂r ~ ∂t coll. (3.17) where the k’s have been made explicit again. Due to the Pauli Exclusion Principle the probability per unit time, Wk,k0 , must be modified because certain states (those occupied) will not be allowed to be scattered into. So the modification is Wk,k0 −→ Wk,k0 ρk (1 − ρk0 ) for the probability of scattering k −→ k0 . Electrons can also scatter into state k in which case the rate is Wk0 ,k ρk0 (1 − ρk ). By using the principle of detailed balance [47], we obtain for the collision piece: ∂ρ k ∂t coll. =− X Wk,k0 (ρk − ρk0 ). (3.18) k0 In summary, we write the spin Boltzmann equation now as X ∂ρk ∂ρk ∂ρk i + ·F + · vk + [Hs.o. , ρk ] = − Wk,k0 (ρk − ρk0 ). ∂t ~∂k ∂r ~ 0 (3.19) k By using Eq. (3.10), the transport equations for spin density can be derived though they will not be used here in this general form: X ∂mk ∂mk ∂mk + ·F + · vk − Ωk × mk = − Wk,k0 (mk − mk0 ). ∂t ~∂k ∂r 0 (3.20) k The second term on the left hand side describes spin drift due to external fields. The third terms encompasses the diffusion present in inhomogeneous spin distributions. The fourth term contains the spin precession due to the spin-orbit field. Let us now go back to Eq. (3.19) and make several simplifications that allow an analytic solution for the DP spin relaxation rate. The following approximation/simplifications are used: 1. Elastic scattering (k = k 0 ) 2. Isotropic energy (εk = ~2 k 2 /2m) 3. Homogeneous distributions (∂ρk /∂r = 0) 4. No external fields (F = 0). These conditions simplify Eq. (3.19) to X i ∂ρk + [Hs.o. , ρk ] = − Wk,k0 (ρk − ρk0 ). ∂t ~ 0 (3.21) k We now separate the effects of momentum relaxation from spin relaxation; the density matrix can be decomposed into two parts ρk = ρ + ρ0k where ρ is the isotropic portion of the density matrix, meaning that it is ρk averaged over all directions of k and is therefore independent of k [61, 65, 39]. ρ0k is the part of the density matrix that deviates from the 35 average value. It is called the anisotropic piece. Since ρk = ρ + ρ0k , it follows that ρ0k = 0. ρ0k results from the spin-orbit Hamiltonian Hs.o. and is therefore much smaller than the isotropic piece since the spin-orbit Hamiltonian is assumed to be a perturbation [65]. If Eq. (3.21) is k-averaged, we obtain ∂ρ i + [Hs.o. (k), ρ0k ] = 0, ∂t ~ (3.22) since Hs.o. ρ = 0 because Hs.o. is an odd function of k. The right-hand side of Eq. (3.21) vanishes exactly upon averaging over k. We also substitute the decomposed density matrix into Eq. (3.21) to get X ∂ρ ∂ρ0k i + + [Hs.o. , ρ + ρ0k ] = − Wk,k0 (ρ + ρ0k − ρ − ρ0k0 ). ∂t ∂t ~ 0 (3.23) k We use the expression for the time derivative of the isotropic piece of the density matrix, Eq. (3.22), in Eq. (3.19) to obtain X ∂ρ0 i i i − [Hs.o. (k), ρ0k ] + k + [Hs.o. , ρ] + [Hs.o. , ρ0k ] = − Wk,k0 (ρ0k − ρ0k0 ). ~ ∂t ~ ~ 0 (3.24) k The anisotropic part of the density matrix should return to isotropy on the order of the momentum relaxation time. So in the quasi-static limit ∂ρ0k /∂t = 0. Keeping terms to first order in the spin-orbit coupling, X i Wk,k0 (ρk − ρk0 ). − [Hso (k), ρ] = ~ 0 (3.25) k At this point, it is useful to consider the dimensionality of the electron gas. We look at three and two dimensions separately in the following sections. Three dimensions The first step is to decompose the spin-orbit Hamiltonian into spherical harmonics. The order (l) of the spherical harmonic match the order of the spin-splitting in k. Examples are shown in chapters 4 and 5. For now, we write Hso (k) = l X X l Hl,n Yln (ϑk , ϕk ). (3.26) n=−l Now an ansatz is made for the solution to Eq. (3.25)[63, 61, 39], ρk = − l iX X τl Yln (ϑk , ϕk )[Hl,n , ρ]. ~ l n=−l 36 (3.27) Commonly only one l is used in derivations of the DP mechanism since the spin-splitting is assumed to be solely linear or cubic but not both [61, 39, 56]. In such cases, l X i i Yln (ϑk , ϕk )[Hl,n , ρ] = − τl∗ [Hso (k), ρ]. ρk = − τl∗ ~ ~ (3.28) n=−l The scattering is assumed to be elastic (k = k 0 ) so only the spherical angles of ρk will differ from ρk0 . Using the ansatz of Eq. (3.27), the right-hand side of Eq. (3.25) is recast as X Wk,k0 (ρk − ρk0 ) = − k0 X iX Wk,k0 τl [Hl,n , ρ](Yln (ϑk , ϕk ) − Yln (ϑk0 , ϕk0 )) ~ 0 (3.29) l,n k By converting the sum over final k-states to an integral and using an identity of spherical harmonics (Eq. (B.5) proved in Appendix B), the following is found X Wk,k0 (ρk − ρk0 ) k0 Z π Z iX n 0 W (θ) =− τl [Hl,n , ρ]Yl (ϑk , ϕk ) dΩ − sin θdθW (θ)Pl (cos θ) ~ 2π 0 l,n Z π iX sin θdθW (θ) 1 − Pl (cos θ) (3.30) τl [Hl,n , ρ]Yln (ϑk , ϕk ) =− ~ 0 l,n where the trivial ϕ0 integral was computed in going from line one to two. Also the full P P P summation over n is suppressed but it is understood that l,n = l ln=−l . By going from Wk,k0 → W (θ), it is assumed that the scattering probability depends only on the angle θ between k and k0 . The final equality in Eq. (3.30) is equal to − ~i [Hso (k), ρ] by Eq. (3.25). If both sides are multiplied by Yn∗0 ,l0 (ϑ, ϕ) and then integrated over the solid angle of the unprimed coordinates, Z δn,n0 δl,l0 [Hl,n , ρ] = δn,n0 δl,l0 τl [Hl,n , ρ] sin θdθW (θ)(1 − Pl (cos θ)). This allows the rate τl−1 to be expressed as Z π 1 = sin θdθW (θ)(1 − Pl (cos θ)). τl 0 (3.31) (3.32) Now the ansatz of Eq. (3.27) is substituted into Eq. (3.22) to get a tractable differential equation for the isotropic portion of the density matrix, ∂ρ 1 XX =− 2 τl [Hso (k), Yln (ϑk , ϕk )[Hl,n , ρ]], ∂t ~ n l 37 (3.33) with the τl ’s given by Eq. (3.32). Then expanding the spin-orbit Hamiltonian in terms of spherical harmonics yields ∂ρ 1 XX 0 =− 2 τl [Hl0 ,n0 Yln0 (ϑ, ϕ), Yln (ϑ, ϕ)[Hl,n , ρ]]. ∂t ~ 0 0 (3.34) l,l n,n Recall that the overbar is an angular average. The only angular quantities in the above equation are the spherical harmonic functions so ∂ρ ∂t 1 XX 0 τl Yln0 (ϑ, ϕ)Yln (ϑ, ϕ)[Hl0 ,n0 , [Hl,n , ρ]] 2 ~ l,l0 n,n0 X X Z dΩ 0 1 Yln0 (ϑ, ϕ)Yln (ϑ, ϕ)[Hl0 ,n0 , [Hl,n , ρ]] τl = − 2 ~ 4π 0 0 = − l,l n,n 1 X X (−1)n = − 2 τl δ−n,n0 δl,l0 [Hl0 ,n0 , [Hl,n , ρ]] ~ 4π 0 0 l,l n,n = l 1 X X − (−1)n τl [Hl,−n , [Hl,n , ρ]], 4π~2 l (3.35) n=−l where certain properties of the spherical harmonics are used. See Appendix B for information on spherical harmonics. The coefficients are found by multiplying by Yl∗0 ,n0 (ϑ, ϕ) and integrating over the solid angle, Z Z XX 0 n0 ∗ dΩHso (k)Yl0 (ϑ, ϕ) = dΩ Hl,n Yln (ϑ, ϕ)Yln0 ∗ (ϑ, ϕ) = Hl,n δn,n0 δl,l0 , (3.36) n l which in the end gives Z Hl,n = dΩHso (k)Yln∗ (ϑ, ϕ). (3.37) From our equation for the evolution of the density matrix, we determine how the components of the spin magnetization’s expectation value evolve in time by the relations m = T r(ρk σ) = Tr(ρσ) and ρ = 12 (I2 +mx σx +my σy +mz σz ). Due to the commutators, the identity matrix, I2 , gives no contribution. When ρ is substituted into Eq. (3.35), Bloch-like equations result, dmi 1 =− mj dt τs,ij (3.38) for which the relaxation tensor is 1 τs,ij l X X 1 n = Tr (−1) τ [H , [H , σ ]]σ . j i l l,−n l,n 8π~2 l n=−l 38 (3.39) Two dimensions The calculation in two dimensions proceeds similarly to that in three dimensions [65]. The difference in the derivation lies in the fact that the spin-orbit Hamiltonian is now expanded in terms of plane waves instead of spherical harmonics, Hso (k) = ∞ X Hn einϕ , (3.40) n=−∞ where k = kx x̂ + ky ŷ if we consider motion in the x − y plane only. ϕ = ϕk is the angular measure of the k direction. The rest of the analysis proceeds as in three dimensions. One finds that this allows the rate τl−1 to be expressed as 1 = τl 2π Z dθW (θ)(1 − cos nθ). (3.41) 0 where θ = ϕ0 − ϕ, is the angle between k and k0 . Mirroring the previous calculation, we obtain ∂ρ 1 X =− 2 τn [Hn0 ein0 ϕ , einϕ [Hn , ρ]]. ∂t ~ 0 (3.42) n,n The angular averaging is easier in two dimensions. We find that ∂ρ ∂t 1 X τn ein0 ϕ einϕ [Hn0 , [Hn , ρ]] ~2 0 n,n Z X 1 dϕ in0 ϕ inϕ = − 2 τn e e [Hn0 , [Hn , ρ]] ~ 2π 0 = − n,n 1 X = − 2 τn δ−n,n0 [Hn0 , [Hn , ρ]] ~ 0 n,n = − ∞ 1 X τn [H−n , [Hn , ρ]], ~2 n=−∞ where the coefficients are Z Hn = 0 2π dϕ Hso (k)einϕ . 2π (3.43) (3.44) The relaxation tensor for two dimensions is found to be 1 τs,ij ∞ X 1 = 2 Tr τn [H−n , [Hn , σj ]]σi . 2~ n=−∞ (3.45) Examples of this spin relaxation mechanism are demonstrated in chapters 4 and 5. So far the calculation has been done at T = 0 K. Finite temperatures are considered now. 39 3.2.3 Finite temperatures The previous calculations assume zero temperature. In actuality temperatures are finite and if electron spins are to be used in technology, room temperature is the most important. Typically, one determines the density of particles by summing over all possible energies and the number of electrons occupying each of those energies; this looks like n = R∞ 0 g(ε)f0 (ε, µ)dε where g(ε) is the number of states in an interval ε → ε + dε and f0 (ε, µ) is the Fermi-Dirac distribution where µ is the chemical potential. Since the existence of spin polarization implies an imbalance of populations, the up and down spins, along some axes, should have different chemical potentials. This assumes that equilibrium is achieved much quicker than the spins can relax. The spin along some direction ŝ is s(ε) = ŝ(f+ (ε, µ+ ) − f− (ε, µ− )) where f+ and f− are distributions of electrons with spin projection in ŝ [65]. The total spin density component i, mi , is determined from Z ∞ Z ∞ mi = g(ε)(f+ (ε, µ+ ) − f− (ε, µ− ))dε = g(ε)si (ε)dε. 0 (3.46) 0 The time dynamics are determined by Z ∞ Z ∞ Z ∞ dmi dsi (ε) 1 1 = g(ε) dε = − g(ε) sj dε = − g(ε) (f+ (ε, µ+ )−f− (ε, µ− ))dε, dt dt τ τ s,ij s,ij 0 0 0 (3.47) where Eq. (3.38) has been used. Now the last equation on the right-hand side of Eq. (3.47) can be multiplied by 1 = mj /mj to give R∞ 1 dmi 1 0 g(ε) τs,ij (f+ (ε, µ+ ) − f− (ε, µ− ))dε = − R∞ mj = − mj , dt τs,ij 0 g(ε)(f+ (ε, µ+ ) − f− (ε, µ− ))dε where now the temperature dependent spin relaxation rate is defined as R ∞ g(ε) 1 (f (ε, µ ) − f (ε, µ ))dε + − − 1 τs,ij + 0 . = R∞ τs,ij 0 g(ε)(f+ (ε, µ+ ) − f− (ε, µ− ))dε (3.48) (3.49) Simplifications can be made if |µ+ −µ− | |µ+ |, |µ− |. In such a case, f+ −f− ≈ −∆µ∂f0 /∂ε which simplifies Eq.(3.50) to 1 R∞ = τs,ij 1 ∂f0 /∂εdε g(ε) τs,ij R∞ . 0 g(ε)∂f0 /∂εdε 0 If the DP mechanism is considered, R ∞ g (d) (ε) 1 ∂f0 dε 1 Iˆ(d) [εn τl (ε)] τs,ij ∂ε 0 = R∞ = α . s τs,ij Iˆ(d) [1] g (d) (ε) ∂f0 dε 0 (3.50) (3.51) ∂ε where Iˆ(d) [h] defines an integral operator operating on an arbitrary function h in d dimen40 sions: Iˆ(d) [h] = R∞ 0 0 g (d) (ε) ∂f ∂ε h(ε)dε , R∞ ∂f0 (d) 0 g (ε) ∂ε dε (3.52) and the DP relaxation rate has been decomposed into 1/τs,ij = αs εn τl (ε) to highlight its energy dependence. The density of states is now labeled as g (d) (ε) to highlight its dimensional dependence. It is generally that the case the scattering mechanisms have a power law dependence on energy, τl ∼ εν ; this allows us to write 1/τs,ij = αs εn sl εν where sl is a proportionality constant. As we have seen, the relaxation rate may be a sum of different power laws in which case our Iˆ operator would have to act on the relaxation rate term by term. Since Iˆ(d) [1] = 1, we rewrite Eq. (3.51) as 1 τs,ij = αs Iˆ(d) [εn τl (ε)] (3.53) From now on the angular brackets denoting the thermal averaging are dropped since whether or not a quantity is averaged is evident from the context. The integrals in Eq. (3.51) are important and are computed in Appendix C. The results are used frequently in what follows. The transport time is a useful quantity to define since it can be determined from mobility measurements. It is defined as [66, 67] R ∞ (d) 0 g (ε)ετ1 ∂f Iˆ(d) [εs1 εν ] Iˆ(d) [εν+1 ] ∂ε dε τtr = R0 ∞ = = s . 1 ∂f0 (d) Iˆ(d) [ε] Iˆ(d) [ε] 0 g (ε)ε ∂ε dε (3.54) The idea is to solve for the proportionality constant s1 and then express the scattering time τ1 in terms of the experimental quantity τtr . This is now easily accomplished: τ1 = s1 εν = τtr Iˆ(d) [ε] Iˆ(d) [εν+1 ] εν = εν τtr β ν Id+1 (βµ) , Id+ν+1 (βµ) (3.55) where the results of Appendix C are used. Other scattering times are found similarly, τ3 = γ3 τ1 = γ3 εν τtr β ν Id+1 (βµ) , Id+ν+1 (βµ) (3.56) where the order-unity constant γ3 can be determined from Eq. (3.32) when the scattering mechanism is known. Now that τl (ε) is expressible in terms of the measurable quantity, τtr , it is appropriate to substitute Eq. (3.55) or Eq. (3.56) into Eq. (3.53). The temperature dependence of the DP relaxation rate has been shown; examples of the calculation are given in the proceeding chapters. Similar analysis follows for the other relaxation rates. 3.2.4 Hyperfine interaction Both electronic and nuclear spins relax due to their mutual magnetic interaction. These interactions are collectively called hyperfine interactions [28, 68]. The first type of magnetic 41 interaction is the dipolar coupling between the magnetic dipoles of the electron and nuclear angular momenta. For an electron and nucleus suitably far apart, the dipolar coupling Hamiltonian is [69] µ0 γ e γ n 3 I · S − (I · r)(S · r) , (3.57) 4πr3 r2 where γn is the nuclear gyromagnetic ratio and I is the nuclear spin. For the spherically symHdip. = metric s-state electrons, the expectation value of this Hamiltonian disappears [69]. However the s-states must be considered carefully; the wave function is concentrated on the nucleus (at r = 0) which calls the dipole approximation into question due to the proximity of the two spins (divergence at r = 0) [28]. This problem is remedied by considering the finite size of the nucleus. Assume a uniform magnetization over the spherical nucleus M = γn I/V . The flux density is then 2µ0 M /3 after the demagnetizing field has been subtracted [68]. Thus an electron that enters the nuclear region will experience a magnetic field BN = 2µ0 γn I δ(r). 3V (3.58) Unlike the previous interaction, electrons with l > 0 are not affected by this interaction since they have zero amplitude at r = 0. The field of Eq. (3.58) interacts with the electron spin in what is known as the Fermi contact interaction. The energy cost associated with this interaction is Econtact = 2µ0 γe γn |ψ(0)|2 I · S, 3 (3.59) where the factor V |ψ(0)|2 enters because it gives the probability of the electron to be found at the nucleus’ location (r = 0).7 This energy can be determined from the Hamiltonian Hcontact = 2µ0 γe γn I · Sδ(r), 3 (3.60) by using Econtact = hψ(0)|Hcontact |ψ(0)i.8 The dipolar interaction for l > 0 is typically considerably weaker than the contact interaction (for l = 0) and hence is neglected [70, 68]. The Fermi contact Hamiltonian can be written in a more compact way: Hcontact = Ah.f. VN σn · σe δ(r), (3.61) where Ah.f. = 2µ0 γe γn ~2 /12VN is the hyperfine coupling constant and has units of energy; VN is the nuclear volume and σe,n are the Pauli spin matrix operators for electrons and nuclei. The spin relaxation time due to this interaction is found by the rate of mutual spin flips between the two spin systems; that is electron state |k ↑i and nuclear state |I, µi to |k ↓i and |I, µ + 1i respectively. A.W. Overhauser first did this for metals in 1953 [71]; 7 The wave function ψ(r) is assumed to not vary much over the nucleus which justifies using ψ(0). This quantity is commonly written in Gaussian units in which Eq. (3.60) must be multiplied by a factor of 4π/µ0 [28, 70] 8 42 Fishman and Lampel extended the calculation to non-degenerate semiconductors in 1977 [60]. Following the work of Fishman and Lampel, the transition probability for the mutual spin flip is [60] w↑,µ−→↓,µ+1 = 2 2π 2 A V |ψ(0)| [I(I + 1) − µ(µ + 1)]δ(ε − ε0 ), N h.f. ~2 (3.62) where the term in square brackets is determined by the nuclear spin raising operator, σn+ . If the nuclei are considered to be weakly polarized, each spin state, µ, has an identical probability equal to 1/(2I + 1); then the transition probability for an electron spin flip from up to down due to all identical nuclei is w↑−→↓ = nnuc. V X w↑,µ−→↓,µ+1 , 2I + 1 µ (3.63) where nnuc. is the density of nuclei. The total relaxation rate is found by summing over final k states and accounting for spin-flips of the opposite variety; this yields [60] √ ∗3 1 2 2 2 8I(I + 1) 2m ε = nnuc. (Ah.f. V ) (|ψ(0)| V ) , τh.f. 3 2π~4 (3.64) where the quantities Ah.f. V and |ψ(0)|2 V have either been calculated or measured for several semiconductors [72, 43, 60]. These processes have also been recently studied in semiconductor nanostructures [73, 74, 75, 76]. Spin relaxation due to the hyperfine coupling has been determined to be too weak (τs ≈ 103 − 104 ns [60]) to be observed in the metallic regime in bulk GaAs [77, 7, 39]; however the hyperfine interaction is very relevant in the insulating regime as will be looked at in the upcoming section. 3.3 Localized spin relaxation mechanisms In the previous section, it was seen how conduction spins relax. Those types of relaxation processes depend on the electron state being itinerant. In this section spin relaxation for localized electrons is discussed. Localized relaxation is important at low temperatures and at low impurity concentrations when the semiconductor is insulating. The localization centers are considered to be positively charged donor impurities. The two relaxation mechanisms discussed below are not too foreign to the previous discussions on itinerant spin relaxation; they are again hyperfine coupling and another interaction induced by the spin-orbit effect called the anisotropic spin exchange interaction. 3.3.1 Hyperfine interaction When host nuclei possess a non-zero magnetic moment (proportional to the nucleus’ spin, I), the nuclear spins interact with the electron spin and cause spin relaxation as if the nuclei 43 were randomly oriented magnetic fields. There are two ways this can occur for localized electrons. 1. Frozen nuclear fields [78]. The magnetic moment of nuclei is much smaller than the magnetic moment of electrons; this leads to the fact that nuclear spin evolves much slower than electron spin. First, no applied magnetic field is assumed. The randomly oriented nuclei produce a nuclear magnetic field, BN , that is felt by the electrons. The electron spin will precess in this nuclear field whereas the nuclear spin will be nearly stationary in the presence of the electronic magnetic field, Be . So in essence, the electron ‘sees’ a frozen array of ∼ 105 [70] randomly oriented BN ’s. The magnitude and direction of the nuclear field is normally distributed so W (BN ) = 1 2 2 e−(BN ) /∆B , π 3/2 ∆3B (3.65) where ∆B is the width of the nuclear field distribution. The evolution of a spin moment in a fixed magnetic field is [78] m(t) = (m0 · n)n + m0 − (m0 · n)n cos ωN t + m0 − (m0 · n)n × n sin ωN t, (3.66) where n = BN /BN is the unit vector of the nuclear magnetic field, ωN = γe BN , and m0 is the initial electronic spin moment. When the spin ensemble is averaged over the Gaussian spread of nuclear fields, the polarization is found to decay as ! m0 t 2 −t2 /Tnuc. 2 hm(t)i = 1 + 2 1 − 2( ) e 3 Tnuc. (3.67) where the characteristic dephasing time is Tnuc. = 1 . γe ∆B (3.68) The spin evolves non-exponentially and unexpectedly; the polarization decreases (on the timescale of Tnuc. ) to 33% of its initial value and then is maintained. This remarkable time dependence has been observed in InAs/GaAs quantum dots [79]. Relaxation of this sort is reversible (Tnuc. = T2∗ ); each electron spin is coherent in its nuclear environment and it is the spin ensemble that is seen to decay. This loss of polarization can be reversed via spin echo experiments. When this is done, it is a more difficult problem to determine the much longer electron spin relaxation in the slowly varying nuclear fields [80, 39]. 2. Motional narrowing. When strong orbital or spin correlations are present, the effect of the nuclear fields may be motionally narrowed [41]. Examples include electrons hopping from site to site, spin exchange between donors, and spin exchange with free electrons in 44 which the spin precession is interrupted after the correlation time. This regime is met under 2 i1/2 τ 1 where hB 2 i1/2 is the root mean square of the nuclear field. the condition γe hBN c N The relaxation time is then [7] 1 2 ≈ γe2 hBN iτc , τs 2 i1/2 ≈ 54 G in GaAs [7]. where hBN (3.69) As the metallic regime is approached, the hyperfine interaction becomes negligible [41]. This is mainly because of strong motional narrowing; delocalized electrons sample many nuclear fields for a short amount of time leading to no substantial effect. 3.3.2 Anisotropic exchange interaction The spin-orbit interaction is very important for understanding spin relaxation of itinerant electrons. How does this interaction manifest itself in insulating systems? Since the spinorbit interaction ‘ties’ together the electron’s spin and position variables, the wave function is expected to be affected by the spin-orbit interaction. This is indeed the case; each component of the spinor becomes a different function of coordinates [7]. As before, the spinorbit interaction in systems lacking inversion symmetry leads to the Hamiltonian Hs.o. = ~ 2 Ωk · σ where Ωk is odd in wave vector. For localized electrons, wave vectors are obviously small and near the localization center the effect of the spin-orbit Hamiltonian on the wave functions is negligible in comparison to the binding potential. Interestingly though, far from the potential, in the asymptotic region, the wave function can be altered significantly since the spin-orbit interaction is now comparable to the binding potential. The asymptotic form of the wave function is determined in the Wentzel-Kramers-Brillouin (WKB) or semiclassical approximation [24, 81, 7]. An electron moving in a constant one dimensional potential, V , with energy ε has p a wave function Ψ ∼ e±ikx where k = 2m(ε − V )/~2 is the wave vector and the ± sign indicates the possibility of right and left moving waves. The wave vector can also be considered the phase shift per unit length. If the potential is not quite constant but varies slowly, the wave function over a small region in which the potential has not changed drastically is modified such that the wave vector depends on the potential at each x such p that k(x) = 2m(ε − V (x))/~2 . Over a traversed distance then (x = 0 → x = x0 for Rx instance), the wave function acquires a phase shift 0 k(x0 )dx0 . More generally, the wave function becomes Ψ ∼ ei Rr 0 k(r 0 )·dr 0 . (3.70) In a classically forbidden region such that ε < V (r), k becomes imaginary and the wave function assumes a decaying exponential as expected. In the absence of spin-orbit coupling 45 ~2 k02 + V (r) = ε, (3.71) 2m where k0 specifies the unperturbed wave vector (with no spin-orbit effect). So k0 is simply p k0 = 2m(ε − V (r))/~2 . Since we want to consider the asymptotic region where r is large, p k0 ≈ i 2m|εB |/~2 because ε−V (r) < 0. When spin-orbit coupling is included, this changes to ~2 k 2 ~ + Ωk · σ = ε − V (r), (3.72) 2m 2 where now k = k0 +∆k where ∆k is the piece of the wave vector that is due to the spin-orbit term. Expanding k around k0 yields ~2 k02 ~2 k0 ∆k ~2 ∆k 2 ~ + V (r) + + + Ωk0 +∆k · σ = ε; 2m m 2m 2 (3.73) the first two terms are the unperturbed energy which should be close to ε since the spin-orbit energy is small. So the first two terms cancel with the right hand side leaving ~2 k0 ∆k ~2 ∆k 2 ~ + + Ωk0 +∆k · σ = 0 m 2m 2 (3.74) Since ∆k k0 , the second term can be ignored. Also the spin-orbit term can be assumed to depend only on k0 for similar reasons: ~2 k0 ∆k ~ + Ωk0 · σ = 0. m 2 (3.75) Solving for the perturbed wave vector, ∆k = mΩk0 · σ mΩk · σ = √ 0 , 2~k0 2i 2mεB (3.76) where ∆k is real since Ω is an odd function of k0 . The asymptotic form of the wave function is Ψ∼e q mΩk ·σ 2mεB i √ 0 r − 2i 2mεB 2 r e ~ . (3.77) The first exponential is familiar; it is the ground state hydrogenic wave function. The second exponential is due to the spin-orbit interaction and it has the form of a finite rotation on the spin σ, exp(iγ(r) · σ/2) where [24, 7] mΩk r γ(r) = √ 0 , i 2mεB (3.78) which is real. The effect can be described as follows: near the localizing center the spin has a projection 1/2 on some axis. At a distance r away the spin will have the same projection on an axis rotated by the angle γ(r) [82, 7]. If an electron tunnels between two sites with displacement Rij = |Ri − Rj |, then the spin rotates an angle γ(Rij ) ≡ γij . The presence 46 of the spin-orbit coupling changes the nature of the spinor: e−r/aB eiγ·σ/2 χ, (3.79) since the exponential operator acts on the spin part of the wave function. It is helpful p to define a spin-orbit length Ls.o. such that hγ(Ls.o. )2 i = 1 where the angular brackets denote an averaging over the directions of γ. This leads to hγ 2 (r)i1/2 = r/Ls.o. . If hopping is considered, the spin of the hopping electron rotates at each hopping event. If the angle of rotation is small the spin is considered relaxed when the mean square of the accumulated P 2 angle is one radian: hγi,j i = 1. For the cubic Dresselhaus Hamiltonian9 in Table (3.1), √ the spin-orbit length becomes Ls.o. = 35 ~2 1 2 2m |k0 |2 β3 . For a quantum well confined in the z-direction, the linear Dresselhaus term gives a spin-orbit length [83] Ls.o. = ~2 1 2m∗ β3 hkz2 i where hkz2 i is due to the quasi-2D confinement and is of the form β 2 L−2 ; for infinite well confinement β = π. When both linear and cubic terms are present, interference between them may lead to diverging spin-orbit lengths similar to what occurs in DP spin relaxation. It has been shown though that hopping is an inefficient mechanism for this type of spin relaxation [7]. A stronger type of effect exists that has been shown to be more effective in destroying spin polarization; this is the anisotropic exchange effect. In semiconductors with the type of spin-orbit interaction discussed extensively in this chapter, there exists a correction to the standard isotropic exchange interaction between two electrons at different sites. The form of this correction is anisotropic in spin space and has been worked out in the lab frame to be [82, 83] Hex. = Hiso. +Haniso. = 2Jij cos γij Si ·Sj +sin γij γ̂ij ·Si ×Sj +(1−cos γij ) γ̂ij ·Si γ̂ij ·Si , (3.80) where the first term on the right hand side is the isotropic exchange and the second two terms make up anisotropic portions of the exchange interaction: the Dzyaloshinskii-Moriya and the pseudo-dipole interactions. When spins are exchanged between sites, the anisotropic correction acts to rotate the spin through the angle γ and much like in the case of hopping the polarization can relax over a series of exchanges. The spin relaxation due to the anisotropic exchange interaction can be considered in the regime of motional narrowing if the spin-orbit interaction is sufficiently small; in such a 2 i1/2 /~ is the spin precession frequency in the anisotropic case Ωτc 1 where Ω ≈ hJij ihγij field and τc ≈ ~/hJij i is the time between exchange interactions between localized electrons [81, 7, 39, 56]. Using the heuristic equation for motional narrowing spin relaxation, τs−1 ∼ 9 In some literature [82, 83, 7] a dimensionless parameter αs.o is often used instead of β1 or β3 . In zincblende crystals where p there is no intrinsic linear spin splitting, the relationship between these two notations is β3 = αs.o. ~3 /m∗ 2m∗ Eg . 47 hΩ2 iτc , we obtain [7, 39] 2 τs−1 ∼ hΩ2 iτc ∼ hJij ihγij i/~. (3.81) The exchange integral for hydrogen-type atoms has been determined in two and three dimensions to be [84, 85, 86] r 5/2 ij e−2rij /aB aB r 7/4 ij Jij = 15.2εB e−4rij /aB aB Jij = 0.82εB where εB = m∗ Ryd m2r (3D) (3.82) (2D), (3.83) is the effective binding or Rydberg energy. r is the dielectric constant and Ryd = 13.6 eV is the hydrogen atom’s Rydberg energy in three dimensions. Reduced dimensionality reduces the Rydberg energy by a factor of four in two dimensions. These formulae are accurate for rij > aB . The exchange constant must be averaged over the cluster of donors (inter-donor distances vary); this is achieved in the bulk by substituting −1/3 rij −→ αnimp /2 where α is the constant 1.73 [7]. 3.4 Cross-relaxation Chapter 2 introduces the possibility of cross relaxation between localized and itinerant states. This phenomenon is also referred to as spin-exchange scattering; a free electron with some spin scatters from a bound electron (to donor or hole) with opposite spin which can lead to a mutual spin-flip [46]. This process is responsible for the Kondo Effect in metals and is characterized by the interaction JSl · Sc where J is the exchange integral and the subscripts denoted the l ocalized and conduction spins respectively. Mahan and Woodworth recently calculated this rate in detail and the their results will not be repeated here [46]. It is extremely rapid; in bulk n-GaAs with nimp = 1016 cm−3 , the cross relaxation time is ∼ 100 fs. The net effect of this process, is that moments rapidly equilibrate such that the average magnetizations in the two states are equal (ml /nl = mc /nc ). Spin relaxation takes affect on a longer time scale than cross relaxation. Hence the spin relaxes according to Eq. (2.45) which is the weighted average of the two environments’ relaxation times. The presence of an average total spin relaxation rate as oppose to two single rates was first observed by Paget in n-GaAs [43]. This idea reoccurs in the future chapters of this dissertation. 3.5 Summary Figure 3.3 shows various measurements of the spin relaxation time versus doping concentration in bulk n-GaAs. It is characterized by three distinct regimes: below the metal-insulator transition (MIT), near the MIT, and above the MIT which can also be identified as the insulating, donor band, and metallic regimes respectively. Dzhioev et al. and Müller et al. 48 point out that different mechanisms dominate in each of these regimes [77, 6]. The low doping regime consists of hyperfine induced relaxation. As the impurity content is augmented the hyperfine relaxation gets motionally narrowed and longer times are observed. The spin-noise-spectroscopy (SNS) measurements yield shorter times because there production of free electrons that lead to further motional narrowing as seen in the optical orientation experiments (conventional probes). The longest times, near the MIT, are due to anisotropic exchange relaxation. In the metallic regime, the donor wave functions strongly overlap and become conducting which leads to DP relaxation. Some researchers have argued in the past that the DP mechanism is responsible for the non-monotonic impurity and temperature dependences shown in Figures 2.4 and 3.3 [87, 88, 55]. Fully microscopic kinetic Bloch equation calculations do suggest a non- monotonic dependence but the relaxation times are very different than those measured [55]. Also, such calculations did not account for donor-bound spins which are now realized to be of vital importance [7, 56]. Also the recent emergence of spin-noise-spectroscopy (SNS) experimental technique of relaxation time measurements [9, 6] have elucidated the intrinsic spin relaxation times in semiconductors. The method is non-invasive in that electrons and holes are not photo-excited and therefore cannot ‘influence’ the relaxation times. Instead SNS measures statistical fluctuations of spin polarization by Faraday rotation of linearly polarized laser light at frequencies below the band gap. The fact that SNS measures the same trend in relaxation times (as shown in Figure 3.3), points out that the general features are not artifacts of photo-electrons. The next two chapters examine spin lifetime measurements and deduce which mechanisms from the options listed in this chapter are relevant. 49 Figure 3.3: Spin relaxation time versus doping density. Three distinct regimes are observed, indicating three different spin relaxation mechanisms: hyperfine, anisotropic exchange, and D’yakonov-Perel’. Figure taken from [6]. SNS refers to non-invasive spin-noise-spectroscopy measurements while conventional probes are optical orientation experiments. Data point references are located in [6]. 50 Chapter 4 Phenomenological approach to spin relaxation in semiconductors I; case studies in bulk and quasi-2D zinc-blende crystals 4.1 Introduction In this chapter and the next, a phenomenological approach to explaining spin relaxation rates is undertaken in zinc-blende (this chapter) and wurtzite (chapter 5) semiconductors. Each study will be divided into bulk and quasi-2D constituents. The approach is made up of the following steps: 1. First the question is asked: what mechanisms are responsible for a given set of measured spin relaxation rates? This is not an easy question to answer and it is shown in these chapters that sometimes the existing experiments are inadequate in providing a clear answer. In many instances, an answer can be given though several parameters must be accounted for: temperature, magnetic field, and doping density among other experimental factors. 2. The approach is phenomenological in that it begins with the modified Bloch equations of Eq. (2.44) which yield the spin relaxation rate, Eq. (2.45), 1 nc 1 nl 1 = + . τs nc + nl τc nc + nl τl (4.1) From the experimentally measured impurity density nimp , nc and nl are determined as described in chapter 2. 3. The spin relaxation times for the localized and conduction electrons must be determined. By knowing the qualitative characteristics of the mechanisms described in chapter 51 3, most candidates can be eliminated when the experimental parameter space is known. The mechanism can be determined quantitatively by a least squares fit of Eq. (2.45) to the data. 4. As shown in the following sections and chapter, occasionally the modified Bloch equations must be altered to match the experimental set-up in which case Eq. (2.45) will change accordingly. Also when the theory of certain spin relaxation mechanisms is lacking, it must be developed as will be done for the wurtzite structure in the next chapter. The analysis in zinc-blende (zb) bulk is similar to that done in n-GaAs by Putikka and Joynt [5]. In addition to n-GaAs, also n-ZnSe and n-CdTe are examined in the following section. In section 4.3, this phenomenological approach outlined above is applied to quasi2D semiconductors for the first time as presented in [20]. 4.2 Bulk crystals The actual material zinc-blende refers to ZnS (zinc sulfide) but many other semiconductor compounds form similarly under ambient conditions; they include GaP, AlP, InP, InSb, InAs, and GaAs to name a few. It is similar to the diamond structure (tetrahedral bonds and cubic) except that each ion has four opposite ions as nearest neighbors as shown in Figure 4.1. GaAs has been the archetypal zinc-blende semiconductor for spintronic study. Figure 4.1: Conventional cubic cell of the zinc-blende crystal structure. It is a direct band gap material with band structure shown in Figure 4.2. Zinc-blende materials lack inversion symmetry which gives rise to Dresselhaus spin-splitting though this splitting between the up and down spin conduction bands is too small to be seen in the 52 Figure 4.2: Band structure of GaAs at 300 K near the Γ-point (k = 0). Spin-splittings of the conduction band due to the Dresselhaus interaction are too small to be seen. figure. The Dresselhaus Hamiltonian for bulk zinc-blende is expressed as HD (k) = ~ D ω (k) · σ 2 3 (4.2) where ω3D (k) = kx (ky2 − kz2 ) 2β3 ky (kz2 − kx2 ) , ~ kz (kx2 − ky2 ) (4.3) where β3 is the cubic Dresselhaus parameter and gives the strength of Dresselhaus splitting. The functional form of ω3D (k) in Eq. (4.3) is true for only zinc-blende. It is different for semiconductors with different crystal symmetries as discussed in chapter 5. The Dresselhaus spin-orbit splitting is determined in quasi-2D structures by spatially averaging the Hamiltonian in Eq. (4.2) along the direction of confinement [64]. A common quantum well orientation is the (001) plane; the Dresselhaus Hamiltonian then becomes HD (k) = ~ D ~ ω1 (k) · σ + ω3D (k) · σ 2 2 (4.4) where ω1D (k) = −kx 2β3 2 hkz i ky ~ 0 53 (4.5) and ω3D (k) = kx ky2 2β3 −ky kx2 . ~ 0 (4.6) (110)-quantum wells have engendered interest because the Hamiltonian is HD (k) = ~ D ~ ω1 (k) · σ + ω3D (k) · σ 2 2 (4.7) where ω1D (k) = 2β3 2 hk i ~ z 0 0 (4.8) −kx /2 and ω3D (k) = 2β3 ~ 0 0 , kx (kx2 − 2ky2 )/2 (4.9) which is interesting because the effective magnetic field is always in the z-direction and should not cause spin relaxation for spins parallel to it. Both (001) and (110) quantum wells are discussed in section 4.3. 4.2.1 Spin relaxation EY has been studied extensively in bulk zb systems [60, 58, 89, 5, 55]. Calculations of the EY spin relaxation rate have shown it to be much too weak to produce the measured rates in III-V zb-bulk semiconductors such as those considered here [89, 5, 55]. The BAP mechanism is also neglected here since it is irrelevant in n-type systems where the amount of holes is minuscule [89, 55]. This leaves the DP mechanism which is shown below to give the correct qualitative and quantitative description of the higher temperature data. Let us calculate the explicit temperature dependence of the DP mechanism in bulk zinc-blende crystals. First recall section 3.2.2 that in bulk, 1 τs,ij = l X X 1 n Tr (−1) τ [H , [H , σ ]]σ . j i l l,−n l,n 8π~2 l (4.10) n=−l In bulk zinc-blende, Hs.o. ∼ k 3 so only l = 3 will yield a non-zero result. τ3 can easily be related to τ1 by using Eqs. (3.32) for l = 1, 3 once the scattering type is discerned: τ3 = γ3 τ1 54 where γ3 is a constant that depends on the scattering type. When the sum is expanded, 3 2 X n (−1) τ Tr [H , [H , σ ]]σ = 3 3,−n 3,n j i 8π~2 n=1 τ3 − Tr [H , [H , σ ]]σ − Tr [H , [H , σ ]]σ 3,−1 3,1 j i 3,−3 3,3 j i 4π~2 +Tr [H3,−2 , [H3,2 , σj ]]σi . (4.11) The traces can quickly be computed; the first trace is −64πm∗3 β32 ε3 /21~6 , the second trace is −64πm∗3 β32 ε3 /35~6 , and the third trace is 512πm∗3 β32 ε3 /105~6 . We obtain10 256 1 m∗3 ε3 = γ3 τ1 β32 , τs 105 ~8 (4.12) where spin relaxation can be identified as the diagonal part only of the relaxation tensor since the relaxation is isotropic (τxx = τyy = τzz and all other components are zero). We now proceed through the details of the thermal averaging: 256 m∗3 1 = γ3 β32 8 Iˆ(3) [τ1 ε3 ]. τs 105 ~ (4.13) After substituting for the scattering time and using the processes developed in Section 3.2.3 and Appendix C, we ascertain the following: 1 τs = = = I7/2+ν (βµ) s1 ˆ(3) 256m∗3 β32 3+ν 256m∗3 s1 γ3 β32 γ3 I [ ε ] = (kB T )3+ν 2 6 8 ~ 105~ 105~ I1/2 (βµ) 256m∗3 τtr (kB T )−ν I I3/2 (βµ) 3/2+ν (βµ) γ3 β32 105~8 (kB T )3+ν I7/2+ν (βµ) I1/2 (βµ) I3/2 (βµ)I7/2+ν (βµ) 256m∗3 τtr γ3 β32 (kB T )3 . 8 105~ I3/2+ν (βµ)I1/2 (βµ) (4.14) Unfortunately in three dimensions, unlike two, the chemical potential cannot be solved exactly in an ideal Fermi gas. The chemical potential, when Fermi-Dirac statistics are used, R ∞ √ eβµ e−x is intractable in the expression for the electron density, n = n0 0 dx x 1+e βµ e−x where n0 = 3/2 1 2m∗ k T , B 4 ~2 π T 3/2 4 n F √ . = n0 T 3 π (4.15) In the non-degenerate limit, when Boltzmann statistics is valid, n = n0 eβµ and the chemical potential can be determined analytically [90]. However analytic approximations exist in the general case that allow expressions for the chemical potential to be valid even at relatively high levels of degeneracy. The most famous of these is the Joyce-Dixon approx10 This is equivalent to 3 32 γ τ α2 ε 105 3 1 c ~2 Eg which is a formulation often seen in the earlier literature [61]. 55 Figure 4.3: Measured and theoretical spin relaxation rates below the metal-insulatortransition (nM IT ≈ 2 × 1016 cm−3 ) in n-GaAs at low temperatures (below 10 K). Symbols are various experiments referenced in [7]. The theory curves contain no fitting parameters. The maximum spin relaxation time appears around 0.15nM IT . A maximum spin relaxation time has been also seen in one study on CdTe [8]. imation [91]. However even better approximations have been devised by using the L/M Padé approximation [92, 90]: n n n µ = kB T ln , + 4.897 ln(0.045 + 1) + 0.133 n0 n0 n0 (4.16) In the non-degenerate regime, Boltzmann statistics is valid, and the integrals of Eq. (4.14) reduce to 5 I3/2 (βµ)I7/2+ν (βµ) T TF ( 32 )!( 72 + ν)! 37 −→ 1 3 = +ν +ν . I3/2+ν (βµ)I1/2 (βµ) 2 2 2 ( 2 )!( 2 + ν)! (4.17) The final result for the DP relaxation rate at high temperatures is then 8Qm∗3 τtr β32 (kB T )3 , 105~8 where Q = 16 35 γ3 (7/2 + ν)(5/2 + ν). (4.18) Q is of order unity for bulk scattering mechanisms [61]. Localized mechanisms are expected to be dominant at low temperatures when impurity sites are near full occupation. The two pertinent mechanisms are the hyperfine and anisotropic exchange interactions detailed in chapter 3. Which of these two prevails depends 56 Figure 4.4: Adapted from [9] showing how the temperature dependence of the spin relaxation depends on the doping density. on the impurity concentration and effective Bohr radius of the impurities [77, 7]. In GaAs (aB = 10.4 nm), the hyperfine interaction dominates the spin relaxation for impurity concentrations below ∼ 4 × 1015 cm−3 and the anisotropic exchange interaction affects larger rates in the range 4 × 1015 < nimp < 2 × 1016 cm−3 = nM IT = (0.25/aB )3 where nM IT is the impurity concentration at the metal-insulator-transition (MIT). This is shown in Figs. 3.3, 4.3, and 4.4. For even higher concentrations, a metallic regime is entered. ZnSe and CdTe have smaller Bohr radii (4.6 nm and 5.3 nm, respectively) and as a consequence the above regimes are expected be at higher concentrations than for GaAs. Unfortunately these two materials have not been studied experimentally nearly as much as GaAs but one study does suggest this to be true for CdTe [8]. 4.2.2 Comparison with experiments Figures 4.5,4.6, and 4.7 present three experiments (solid circles) in which the spin relaxation rate was measured over a temperature range in the following n-type semiconductors: GaAs, ZnSe, and CdTe respectively. In GaAs, the theory curves in Fig. 4.3 agree with the large amount of studies which suggest that the anisotropic exchange mechanism is operative in the doping range of the data shown in Figure 4.5 [77, 5, 7]. The lack of data for ZnSe 57 1T2* Hns-1L 10 1 0.1 0.01 0.001 5 10 20 50 100 200 T HKL Figure 4.5: Solid blue circles from Kikkawa using n-GaAs with nimp = 1 × 1016 cm−3 at zero applied field [4]. Solid line is fit with Eq. (2.45). and CdTe makes determinations of the precise mechanism difficult at low temperatures. This is because both likely mechanisms - motionally narrowed hyperfine and anisotropic exchange (see section 3.3) - are exponential in nature when the correlation time for the hyperfine interaction is assumed to be due to exchange. So in GaAs we can be certain that anisotropic exchange is the largest effect at nimp = 1 × 1016 cm−3 whereas for the impurity concentrations in ZnSe and CdTe we are not sure which is dominant and it is possible both could be similar in effect. Due to this uncertainty in any fit, we leave the localized relaxation rate as a fitting parameter in Figs. 4.6 and 4.7. The situation is much clearer at high temperatures when the DP mechanism takes charge. Eq. (4.18) can be inserted into Eq. (2.45) for 1/τc . There is some variance in the experimental and theoretical determinations of the Dresselhaus (cubic) spin splitting coefficient β3 . In GaAs, by using a value near the lower end of the recently measured and calculated spectrum (6.5 meV nm3 ) [94, 95], a least squares fit results in Q = 2.8 which agrees well with acoustic phonon scattering (Q = 2.7) [61]. The results of the fits for the three materials is summarized in Table 4.2.2. The fits are strikingly good in determining Q. The fits also suggest that the localized relaxation mechanism is not strongly temperature dependent. The conclusion of this section is that the modified Bloch equation approach, which 58 1T2* Hns-1L 10.00 5.00 1.00 0.50 0.10 0.05 0.01 5 10 20 50 100 200 T HKL Figure 4.6: Solid blue circles from Malajovich et al. using n-ZnSe with nimp = 5 × 1016 cm−3 at zero applied field [10] Solid line is fit with Eq. (2.45). Used same mobility as for GaAs. takes into account exchange interactions between localized and itinerant electrons, provides a good description of the observed spin relaxation times in bulk zinc-blende semiconductors in no field. In the next section this idea will be extended to structures with reduced dimensionality. 4.3 Quasi-2D nanostructures In recent years, uniformly doped quantum wells (QWs) have generated increasing interest due to the long relaxation times measured therein [96, 97, 98]. The long relaxation times are due to spins localized on donor centers. While similar relaxation times have been measured in modulation doped systems, their duration has not been as reliable due to the weaker binding energy of localized states and potential fluctuations from remote impurities [99, 12, 100]. Localization is either not seen at all [99] or localization centers thermally ionize rapidly with increasing temperature due to a small binding energy [100, 12]. QWs uniformly doped within the well have the advantage of being characterized by well defined impurity centers with a larger binding energy. The experimental control in the amount of doping 59 1T2* Hns-1L 100.0 50.0 10.0 5.0 1.0 0.5 0.1 5 10 20 50 100 200 T HKL Figure 4.7: Solid blue circles from Sprinzl et al. using n-CdTe with nimp = 4.9 × 1016 cm−3 at zero applied field [8]. Solid line is fit with Eq. (2.45). Transport time taken from mobility measurements of [11] and well size make doped QWs particularly appealing to the study of quasi-two-dimensional spin dynamics. Much of the theoretical study of spin relaxation in semiconducting systems (QWs in particular) has either focused solely on itinerant electrons [101, 102, 67] or solely on localized electrons [83, 7] without regard for either the presence of the other state or the interaction between the two states. Recently the existence and interaction between itinerant and localized states has been dealt with in bulk systems in [5, 21, 46]. The results of these calculations are in very good quantitative and qualitative agreement with experimental observations [4, 103] in bulk n-GaAs and n-ZnO. In the following section, the theory of two interacting spin subsystems is applied to QWs. 4.3.1 Spin polarization in quantum wells In QWs at low temperatures the creation of non-zero spin polarization, in the conduction band and donor states, proceeds from the formation of trions (charged excitons, X ± ) and exciton-bound-donor complexes (D0 X) respectively, from the absorption of circularly po60 Bulk material GaAs ZnSe CdTe γ3 (meV nm3 ) 6.5 1.3 8.5 1/τl (ns−1 ) (fit) 0.010 0.017 0.43 Q (fit) 2.8 3.7 5.4 Q (th.) 2.7 2.7 2.7 Table 4.1: Results of fits in Figs. 4.5, 4.6, and 4.7. Acoustic phonon scattering is assumed for each (give Q = 2.7). Mobilities for GaAs come from [5] and references therein. Mobilities of ZnSe are not available so that of GaAs were used. Mobilities for CdTe come from [93]. larized light. Figure 4.8 and 4.9 illustrates the absorption and polarization process in bulk semiconductors and QWs via the trion route. Figure 4.8: Illustration of optical spin pumping in bulk semiconductor. CB and VB are conduction and valence bands respectively. σ ± denotes the helicity of the absorbed and emitted photons (wiggly lines). Photon promotes one electron from VB to CB leaving a hole behind. The hole spin (thick arrows) is assumed to relax much quicker than the electron spin (thin arrows). Sz is total electron spin. Graphic taken from [12]. Polarization via the trion avenue is most relevant for modulation doped QWs where donor centers in the well are sparse [99, 98]. Due to the modulation doping outside the well, the number of conduction electrons in the well may be plentiful. In such cases, assuming incident σ + pump pulse, a + 32 hole and − 12 electron are created. These bind with a resident − electron from the electron gas in the QW to form a trion (X3/2 ). The trion’s binding energy is ∼ 2 meV [100, 12]. The ‘stolen’ electron will be + 21 to form a singlet state with the exciton’s electron. Hence, the electron gas will be left negatively polarized since the excitons are preferentially formed with spin up resident electrons. If the hole spin relaxes faster than the trion decays the electron gas will remain polarized [99]. Selection rules 61 dictate + 32 (− 32 ) holes will recombine only with − 21 (+ 12 ) electrons. Therefore if the hole spins relax rapidly, the released electrons will have no net polarization and the polarized electron gas will remain predominantly negatively oriented. Figure 4.9: Illustration of optical spin pumping in QWs when photo-excitation is resonant with the trion formation. When excited at the donor-bound-exciton resonance instead, the picture is similar except that the exciton ‘steals’ an electron from a neutral donor (or is actually captured by the neutral donor) instead of a free electron. The key difference is that after hole spin relaxation and recombination, the neutral donors are left with a net polarization instead of the free electrons. Graphic taken from [12]. A very similar picture is given for the polarization of donor bound electrons in uniformly doped QWs where the donor bound electrons play the role of the resident electrons [96, 97]. At low temperatures the donors are nearly all occupied and the density of the electron gas is negligible. When excitations are tuned at the exciton-bound-donor resonance, instead of photo-excitons binding with the resident electron gas, they bind with neutral donors to form the complexes D0 X3/2 . This notation implies that a + 32 hole - − 21 electron exciton is bound to a + 12 donor bound electron. The D0 X3/2 ’s binding energy is ∼ 4.5 meV [104]. Once again for very short hole relaxation times, the donor bound electrons can be spin polarized. The measured long spin relaxation times in uniformly doped QWs imply that spin polarization remains after short time processes such as X and D0 X recombination have completed. In other words, the translational degrees of freedom thermalize much more quickly than the spin degrees of freedom. The occupational statistics of itinerant and localized electrons are important and can be determined from equilibrium thermodynamics. As the temperature is increased, the electrons bound to donors thermally ionize and become itinerant. As the number of electrons in the conduction states increases, the spin that exists on the donors equilibrates by cross relaxing to conduction states by the isotropic exchange 62 Figure 4.10: Illustration of optical pumping in QWs when photo-excitation is resonant with exciton formation. The conduction band (CB) starts with no spin polarization as the exciton is created (left most panel). The hole spin bound in the exciton rapidly relaxes (second panel). An antiparallel resident electron spin is grabbed from the electron gas (or the exciton binds to a neutral donor). This leaves the resident electron polarized (third panel). Oppositely oriented electron and hole spins recombine, casting the third electron back into the electron sea adding more net spin moment (rightmost panel). Taken from [12]. interaction much like what occurs in bulk. If cross relaxation is rapid enough, the total spin, which is conserved by exchange, exists in the donor and conduction states weighted by their respective equilibrium densities [5, 46, 21]. The polarized electron moments will then proceed to relax via different processes for the localized and itinerant states - the total relaxation given again by Eq. (2.45). The above description is complicated when the photo-excitation energy is at the exciton resonance and not the exciton-bound-donor resonance. In such a case, the excitons may recombine, the electron-in-exciton spin may relax before recombination, or the electron spin may cross-relax to the localized or free electrons. One expects the low temperature spin relaxation to reflect also the exciton spin dynamics instead of the donor electron spin dynamics alone [100]; see Figure 4.10. In essence, the electrons in an exciton represent a third spin environment with a characteristic spin relaxation time scale different from that of the localized donor and itinerant electrons. Because of the electron’s proximity to a hole, relaxation may result from electron-hole spin exchange or recombination. Therefore to understand the spin dynamics in QWs, it is imperative to examine the relaxation processes that affect the polarized spin moments of the various spin systems. 63 4.3.2 Modified Bloch equations First photo-excitation energies near the exciton-bound-donor resonance are considered. After rapid exciton-donor-bound complex formation, recombination, and hole relaxation, we model the zero field spin dynamics of the system in terms of modified Bloch equations as initially discussed in chapter 2: 1 dmc nl nc + cr mc + cr ml =− dt τc γc,l γc,l dml nl 1 nc = cr mc − + cr ml dt γc,l τl γc,l (4.19) where mc (ml ) are the conduction (localized) magnetizations, nc (nl ) are the conduction (localized) equilibrium occupation densities, τc (τl ) are the conduction (localized) spin recr = τ laxation times, and γc,l exch nimp is a parameter describing the cross relaxation time between the two spin subsystems. Mahan and Woodworth [46] have shown the cross relaxation time between impurity and conduction electron spins to be much shorter than any of the other spin relaxation times relevant here. We shall assume below that the same is true for the cross relaxation between electrons bound in an exciton and conduction or impurity electron spins. The motivation of these modified Bloch equations is set forth in [5, 21]. Eqs. (4.19) is valid for photo-excitation energies that do not cause free exciton formation (only two relevant spin systems). It is important to note that Eqs. (4.19) hold only for time scales that are long compared with laser pulse times, energy relaxation times that determine subsystem populations, and donor-bound-exciton formation times. Fortunately, these time scales are the ones probed in the experiments. Standard methods can be used to solve these differential equations with initial conditions mc (0) and ml (0). We assume that the initial spin polarization is perpendicular to the QW’s growth plane and that the excitation density, Nx , is small enough such that the resultant spin relaxation time, τs , will not depend strongly on Nx [97]. The solutions yield a time dependence of the total magnetization m(t) = mc (t)+ml (t) to be a sum of two exponentials - one of which is exp(−t/τs ) and the other of which has a time constant proportional to the cross relaxation time. In the case of rapid cross relaxation (faster than all spin relaxation mechanisms), only one exponential survives and the total relaxation rate is expressed as 1 nl 1 nc 1 = + τs nimp τl nimp τc (4.20) where nimp = nl + nc is the total impurity concentration. This model, or variations of it, has been successfully applied to bulk n-GaAs and bulk n-ZnO [5, 21]. If the photo-excitation energy is set near the exciton energy, the Bloch equations must be modified to take into account exciton spin relaxation and multiple cross relaxations: 64 γi,j for i, j ∈ c, l, x for conduction, localized, and exciton spins respectively. The exciton spin relaxation is modeled as electron-in-exciton spin relaxation [105] and the hole spin relaxation is assumed to be very rapid. Eq. (5.54) generalizes to 1 dmc nl nx nc + Nx − nx nc + Nx − nx =− + cr + cr mc + ml + mx cr cr dt τc γc,l γc,x γc,l γc,x 1 dml nx nl nl nc + N x − nx + ml + cr mx = cr mc − + cr cr dt γc,l τl γc,l γl,x γl,x dmx nc + N x − nx nx nx 1 nl + = cr mc + cr ml − + cr mx , cr dt γc,x γl,x τx γc,x γl,x (4.21) where τx represents spin lifetime of an electron bound to a hole. nx (mx ) is the number (magnetization) of electrons bound in an exciton. Nx is the initial density of photo-excited electrons and the quantity Nx − nx is the number of photo-excited electrons that do not participate in an exciton. We assume quasi-equilibrium such that nx is determined from thermodynamics (see Section 4.3.3). It should be stated that Eq. (4.21) is valid only for times shorter than the recombination time; in other words, on a time scale where Nx can be assumed to not change significantly. Recombination times have been measured [106] in similar systems as to those studied here to be longer than the observed spin relaxation times so this approximation seems justified. In Section 4.3.5, we find that the effects of recombination of free carriers can be added to 1/τc to obtain excellent agreement with the experimental data. If the system of equations in Eq. (4.21) is solved as done for Eq. (4.19), the following relaxation rate is obtained: nl 1 nc + Nx − nx 1 nx 1 1 = + + . τs nimp + Nx τl nimp + Nx τc nimp + Nx τx (4.22) For both Eqs. (4.20) and (4.22), we allow τl , τc , and τx to be phenomenological paP rameters of the form τi−1 = j 1/τj where j refers to a type of spin relaxation mechanism. From the experimental constraints and results, the important relaxation mechanisms are determined. 4.3.3 Occupation concentrations As shown above, the relative occupations of localized and itinerant states play an important role in our theory. Fortunately, in two dimensional systems, the occupation probabilities of the two states (nl /nimp and nc /nimp ) can be determined exactly. However the densities we are interested in are dilute enough such that the non-degenerate limit (Boltzmann statistics) can be utilized. The probability for a donor to be singly occupied (only the ground state needs to be 65 Figure 4.11: Occupation probabilities of localized (solid line) and conduction (dash-dotted line) states with impurity density nimp = 4 × 1010 cm−2 determined from Eqs. (4.25, 4.26). Other parameters for GaAs are a∗B = 10.4 nm and m∗ = 0.067m. considered [107]) is [47] nl 1 = 1 (ε −µ)/k T . B B nimp +1 2e (4.23) The density of itinerant states is given by nc = Nc eµ/kB T (4.24) where Nc = m∗ kB T /~2 π and the conduction band edge is taken to be zero energy. The chemical potential µ can be found using the constraint nl nc + = 1. nimp nimp (4.25) p 1 + Q(T, nimp ) − 1 nl =p , nimp 1 + Q(T, nimp ) + 1 (4.26) 8nimp −εB /kB T e . Nc (4.27) Using the result for µ, one obtains where Q(T, nimp ) = An example of the temperature dependence of these occupation probabilities is shown for a GaAs QW in Figure 4.11 where nimp = 4 × 1010 cm−2 . At the lowest temperatures, 66 the donors are fully occupied. As the temperature increases, nl decreases and nc increases to where at around 50 K, the two occupation probabilities are equal. From Eqs. (4.20, 4.22), it is evident that these occupational statistics have ramifications in the measured spin relaxation times. The results here are also applied to the excitons in quasi-equilibrium. 4.3.4 Spin relaxation We now discuss the relevant spin relaxation mechanisms for both localized and conduction electrons. The electron-in-exciton spin relaxation, τx , is a combination of electron-hole recombination and electron-hole exchange relaxation. Due to its complicated nature, calculation of τx is deferred to future work. Here we treat it as a phenomenological parameter. Localized spin relaxation First we discuss spin relaxation via the anisotropic spin exchange for donor bound electrons. This has been treated extensively elsewhere [108, 109, 82, 83]. Most recently it has been examined by Kavokin in [7]. It is his treatment that we detail below for semiconducting QWs. Kavokin argues [7] that some portion of localized relaxation results from spin diffusion due to the exchange interaction between donors. Anisotropic corrections to the isotropic exchange Hamiltonian cause a spin to rotate through an angle γi,j when it is transferred between two donor centers located at positions ri and rj . The angle-averaged rotation angle 2 i1/2 = hr 2 i1/2 /L is hγi,j s.o. where Ls.o. is the spin orbit length [7]. The spin is relaxed when i,j P 2 the accumulated rotation angle Γ becomes on the order of unity such that Γ2 = hγi,j i= P 2 hri,j i/L2s.o. = 1. It is assumed that the spin rotates in a motionally narrowed fashion such that Eq. (3.81) holds. The results of section 3.3.2 for the exchange integral and spin-orbit length combine to obtain the relaxation rate of Eq. (3.81) in terms of a dimensionless impurity separation scale, x: 1 β 2 hk 2 i2 m∗ 2 7/4 −4x = 2 · 15.21 3 z3 hx ihx e i τex ~ (4.28) where x = ri,j /aB and the exponential dependence is noted. Only linear-in-k spin-splitting is considered.How ri,j is to be determined will be discussed in Section 4.3.5. Localized electron spins may also relax due to nuclear fields as described in section 3.2.4. A localized electron is coupled to many nuclear spins by the hyperfine interaction. To the electrons, these nuclear spins appear as a randomly fluctuating field but these nuclear fields can be assumed quasi-stationary since the nuclear evolution time is much longer than electron evolution time due to the contrast in magnetic moments [7]. Merkulov et al. [78] 67 find a dephasing rate, Eq. (3.68), for quantum dots to be s P 16 j Ij (Ij + 1)A2j 1 = Tnuc 3~2 NL (4.29) where the sum over j is a sum over all nuclei in the unit cell, Ij is the nuclear spin, Aj is the hyperfine constant, and NL is the number of nuclei in the electron’s localized volume. If the hyperfine interaction is motionally narrowed by exchange, the relaxation time would tend to be longer since a spin would freely precess in the nuclear field a shorter amount of time. Conduction spin relaxation Conduction band states undergo ordinary impurity and phonon scattering. Each scattering event gives a change in the wave vector k, which in turn changes the effective magnetic field on the spin that comes from spin-orbit coupling. This fluctuating field relaxes the spin. This is known as the D’yakonov-Perel’ (DP) spin relaxation mechanism [110, 64] and is treated extensively in chapter 3. The effective field strength is proportional to the conduction band splitting. We are interested in conduction spin relaxation in (001) and (110) oriented QWs. For (001) QWs the spin relaxation rate results from a spin-orbit term in the Hamiltonian, Hs.o = ~2 ωD (k|| ) · σ where [67] ωD (k|| ) = kx (ky2 − hkz2 i) 2β3 ky (hkz2 i − kx2 ) . ~ 0 The angular brackets denote spatial averaging across the well width. β3 is a band parameter that governs the magnitude of the spin-orbit splitting. For GaAs, β3 ∼ 17 meV nm3 [111]. The QWs have been grown symmetrically and therefore any Rashba contribution is negligible [112]. The resulting spin relaxation has been worked out in detail by Kainz et al. in [67]. For the experiment [113] we compare to, we find the non-degenerate limit to be applicable and hence use the relaxation rate for spin oriented in the z-direction, " ∗ 1 4 β32 hkz2 i 2m∗ kB T 2 2 2 2 2m kB T = τ (T ) β hk i − j2 + p 3 z τz ~2 ~2 2 ~2 # ∗ 3 2 1 + τ3 /τ1 2m kB T β3 j3 16 ~2 (4.30) where j2 ≈ 2 and j3 ≈ 6 depend on the type of scattering mechanism. We assume Type I scattering as defined in [67]. The ratio τ3 /τ1 is unity for Type I scattering. τp (T ) is the 68 momentum relaxation time which can be extracted from mobility measurements. A more interesting case is that of (110) QWs where the spin-orbit Hamiltonian is [114] Hs.o. = −β3 σz kx 1 2 1 hkz i − (kx2 − 2ky2 ) 2 2 (4.31) which is obtained from the (001) Hamiltonian by transforming the coordinate system such that x||[110], y||[001], and z||[110]. As can be seen from the form of this Hamiltonian, the effective magnetic field is in the direction of the growth plane. Hence, spins oriented along the effective field will experience no spin relaxation. Conduction spins also relax due to the Elliott-Yafet (EY) mechanism [48, 40] which arises from spin mixing in the wavefunctions. Due to spin-orbit interaction, when a conduction electron is scattered by a spin-independent potential from state k to k0 , the initial and final states are not eigenstates of the spin projection operator Sz so the process allows for spinflips. In bulk, the relaxation rate is known to be of the form 1/τEY = αEY T 2 /τp (T ) where αEY is a material-dependent parameter and τp is the momentum relaxation time [58]. However the EY mechanism in quasi-two dimensions will not take the same form since k will be quantized in one direction (the direction of confinement). The treatment in bulk [59] has been extended to QWs to obtain [115] 1 τEY ≈ ∆s.o. 2 m∗ 2 Ec kB T 1 1− , ∆s.o. + Eg m Eg2 τp (T ) (4.32) where ∆s.o. is the spin-orbit splitting energy and Ec is the QW confinement energy. Spins may also relax due to the Bir-Aronov-Pikus (BAP) mechanism [53] which arises from the scattering of electrons and holes. This relaxation mechanism is commonly considered efficient only in p-type materials when the number of holes is large [89]. We fit the experimental data in Section 4.3.5 without consideration of this mechanism. We now examine how these relaxation mechanisms are manifest in two different QWs. 4.3.5 Results for GaAs/AlGaAs quantum well We apply our method to measured spin relaxation times of two GaAs/AlGaAs QWs by Ohno et al. [113, 116]: (100) n-doped QW with doping nimp = 4 × 1010 cm−2 , well width L = 7.5 nm; and a (110) undoped QW with well width L = 7.5 nm. In both (pump-probe) experiments, the pump or photo-excitation energy was tuned to the heavy hole exciton resonance and normally incident on the sample. As mentioned in Section 4.3.1, the exciton spin becomes important at low temperatures for such excitation energies. The experimental spin relaxation times as a function of temperature are displayed (solid circles) in Figures 4.12 and 4.13.11 11 The experimental rates of Ohno et al. depicted here are twice that of what is reported in the actual experiments by that group. This is due to different definitions of spin relaxation time [67]. 69 Figure 4.12: Spin relaxation versus temperature in undoped (110) GaAs QW. Points are experiment of [116]. Dash-dotted line: Using only conduction portion of Eq. (4.33) and 1/τc = 1/τEY + 1/τr . Intersubband spin relaxation is also important when DP is suppressed [117]. Dashed line: using only excitonic portion of Eq. (4.33). Solid line: Eq. (4.33). Spin relaxation rate of excitons decreases with temperature increase due to thermal ionization. Conduction spin relaxation is longer in (110) QW than in other oriented QWs due to vanishing DP mechanism. For the undoped (110) QW, Eq. (4.22) is modified to become 1 Nx − nx 1 nx 1 = + . τs Nx τc Nx τx (4.33) For this sample, at low temperatures, nx = Nx so the τs = τx ≈ 0.15 ns which is seen from viewing Figure 4.12. At higher temperatures, recombination (in time τr ) and EY act to relax conduction spins since DP relaxation is significantly reduced for the (110) QW orientation. To account for the quasi-two dimensional nature of the QW, we use an intermediate value (between 2D and 3D values) for the exciton’s binding energy [118]. Eq. (4.33) (solid line) fits the data (points) with excellent agreement in Figure 4.12 when Nx = 1.5 × 1010 cm−2 and τr = 2 ns which are near the experimentally reported values (Nx ≈ 1010 cm−2 and τr ≈ 1.6 ns) [106]. The contributions from the excitons and conduction electrons are also shown (dashed and dash-dotted lines respectively). The trend in the data is well described by our theory using Eq. (4.33) - at low temperatures excitons predominate and the spin relaxation time is τx . When the temperature increases, the excitons thermally ionize leading to net moment in the conduction band. Since the conduction band spin relaxation time is longer than the exciton spin relaxation time, the measured relaxation time increases with 70 temperature as described in Eq. (4.33). We expect the relaxation times to eventually level out as the excitons disappear. Eventually, the relaxation time will decrease as the temperature dependence of EY takes effect. For the doped (100) QW, Eq. (4.22) should be used to describe the temperature dependence of the relaxation rate. Using the values from above and nimp = 4 × 1010 cm−2 , τs = 0.35 ns, we can extract the approximate value of τl . In doing so we obtain τl ≈ 0.5 ns. We stress that this value has considerable uncertainty due to the uncertainty in the parameters (namely Nx ) that determine τl . The presence of impurities has lengthened the observed low temperature spin relaxation time by more than a factor of two. The relaxation time in the doped sample can be further increased by reducing the excitation density. As the temperature is increased, donors become unoccupied and conduction electrons will play a larger role in relaxation as expressed in Eq. (4.33). The main conduction spin relaxation mechanism is determined by investigating its temperature dependence. Figure 4.13: Spin relaxation versus temperature in n-doped (100) GaAs QW. Points are from [113]. Dashed line: excitonic contribution in Eq. (4.22). Dotted line: localized contribution in Eq. (4.22). Dash-dotted line: conduction contribution in Eq. (4.22). Solid black line: Eq. (4.22). Both exciton and localized spin relaxation contribute to the observed low temperature spin relaxation. Conduction spin relaxation is the most strong contributor to the observed relaxation at higher temperatures. We are now left with the task of determining what the localized and conduction spin relaxation mechanisms are. We plot the relaxation rate for the n-doped GaAs QW as a function of temperature in Figure 4.13. The dashed, dotted, and dash-dotted lines refer to 71 the three terms of Eq. (4.33) - the density weighted average of the respective relaxation rates. The solid line is the sum of all three terms. We begin by calculating spin relaxation due to spin exchange diffusion in Eq. (4.28). This is difficult due to the exponential dependence on ri,j . Parameters for GaAs can be found in Appendix A. To calculate hkz2 i = β 2 /L2 , we need to know the band offsets and assume a finite square well. The potential depth for a AlGaAs QW is about V0 = 0.23 eV. This comes from ∆Ec ∆Eg = 0.62 and ∆Eg = 0.37 eV in GaAs [119] . From this we can determine β which will also depend on the well width L. For L = 7.5 nm, β = 2.19. In the limit of V0 → ∞, β → π. What remains to be determined is ri,j which is proportional to the −1/2 inter-donor separation ri,j = γnimp . For average inter-donor spacing in two dimensions, γav = 0.564. When we allow γ to be fitting parameter, we obtain ri,j = 19.5 nm which corresponds to γ = 0.4. We now determine the relaxation rate due to the hyperfine interaction using Eq. (4.29). Since nearly all nuclei have the same spin [120] (I = 3/2), Eq. (4.29) is expressed as s P 5 j A2j 1 =2 , (4.34) Tnuc ~2 NL P with j A2j = 1.2 × 10−3 meV2 and NL ∼ 2.1 × 105 [78]. This yields Tnuc = 3.9 ns. Due to the donor’s confinement in the QW, its wave function may shrink thereby reducing the localization volume and therefore also reducing NL and Tnuc [118]. Motional narrowing by exchange interactions would lengthen this time, taking it further from the experimental value. In Figure 4.13, we find find excellent agreement with experiment over a large temperature range when τp (T ) in Eq. (4.30) is made a factor of three smaller than what is reported in [67]. We attain approximately the same quantitative accuracy as in [67] but since we also take into account the localized spins, we find excellent qualitative agreement as well. It should be emphasized that the DP’s quadratic and cubic terms of Eq. (4.30) are important in the high temperature regime. The EY rate is qualitatively and quantitatively different from the data. For instance, 1/τEY ≈ 0.1 ns−1 at 300 K so is ruled out of contention. We also now ignore recombination of carriers since an appreciable amount of equilibrium carriers exist (n-doped system) leading to recombination of primarily non-polarized spins. One would not expect these results to agree with spin relaxation measurements in modulation doped QWs. In modulation doped systems, the occupation densities nl and nc cannot be calculated as we have done here. In such systems different spin relaxation dependencies are seen [106, 117]. 72 4.3.6 Results for CdTe/CdMgTe quantum well The experiment by Tribollet et. al. on a n-CdTe QW offers an instructive complement to the previous experiments on GaAs. In their experiment, Tribollet et al. measure spin relaxation times τs ≈ 20 ns for CdTe/CdMgTe QWs with nimp = 1 × 1011 cm−2 . Importantly, they excited with laser energies at the donor bound exciton frequency instead of the heavy hole exciton frequency. Parameters for CdTe can be found in Appendix A. To obtain potential well depth for CdTe QW, Eg (xM g ) = 1.61 + 1.76xM g where xM g gives fraction of Mg in Cd1−x Mgx Te [121]. If xM g = 0.1, V0 = 0.12eV which leads to β = 2.18. We now determine the relaxation rate due to the hyperfine interaction. Since all nuclei with non-zero spin have the same spin [120] (I = 1/2), Eq. (4.29) is sP 2 1 j Aj Pj , =2 Tnuc ~2 NL (4.35) where Pj has been appended to account for isotopic abundances [96]. The natural abundances of spin-1/2 Cd and Te nuclei dictate that PCd = 0.25 and PT e = 0.08. The remaining isotopes are spin-0. NL = 1.8 × 104 , ACd = 31 µeV, and ATe = 45 µeV which yields Tnuc = 4.4 ns [96]. The confined donor wave function in CdTe shrinks less than in GaAs since the effective Bohr radius is half as large. This value is within an order of magnitude of what is calculated for relaxation due to the hyperfine interaction. We can also compare the experimental time to what we obtain for spin exchange diffusion. When we allow γ to be a fitting parameter, we obtain ri,j = 19.3 nm which corresponds to γ = 0.61. Unfortunately no relaxation measurements have been performed at higher temperatures in n-doped CdTe QWs that we are aware of. We are also not aware of mobility measurements in n-doped CdTe QWs. The prevalent mechanism (DP or EY) will depend on the mobility so we forgo determining the more efficient rate. However, in analogy to bulk systems, we expect the CdTe QW mobilities to be less than the GaAs QW mobilities [5, 93]. In the next section we analyze CdTe’s spin relaxation rate for (110) grown crystal so DP can be ignored. 4.3.7 Comparison of GaAs and CdTe quantum wells First we discuss the low temperature spin relaxation. Interestingly, the localized relaxation time in CdTe is about 20 times longer than in GaAs. This can be explained by the spin exchange relaxation despite the larger spin orbit parameter in the CdTe. This is more than offset by the smaller effective Bohr radius in CdTe (5.3 nm vs. 10.4 nm) and the exponential behavior of the anisotropic exchange relaxation. However due to the exponential factor, 73 any discrepancy between the two QWs can be explained by adjusting their respective γs appropriately, though the fitted γ’s do fall near γavg . The discrepancy in times is difficult to explain by the hyperfine interaction since the two calculated relaxation times are very near each other. Additionally, no plateau effect is seen that is indicative of frozen-field hyperfine dephasing [96, 79]. Another possibility is that one QW is governed by relaxation from spin exchange and the other from hyperfine interactions. The metal-insulator-transitions for the two materials is nM IT (GaAs) ≈ 6 × 1010 cm−2 and nM IT (CdTe) ≈ 2 × 1011 cm−2 which are just above the experimental doping densities. Without additional experimental data, answering these questions is difficult. It is our hope that further experiments will be done to sort out these questions. However, we can propose ways in which these answers can be discovered. Relaxation by anisotropic spin exchange is strongly dependent on the impurity density. By altering the impurity doping within the well, one should see large changes in the spin relaxation time if this mechanism is dominant. From Eq. (4.28) we see that this mechanism will also depend on the confinement energy. Hence this mechanism should also be affected by changing the well width. The hyperfine dephasing mechanism should be largely unaffected by impurity concentration differences as long as they are not so extensive as to cause the correlation time to become very short and enter a motional narrowing regime. Varying the well width will have an effect on the donor wavefunctions, but as long as they are not squeezed too thin as to approach the motional narrowing regime the effect should not be dramatic. For spin relaxation at higher temperatures, DP prevails in (100) GaAs QWs as mentioned earlier. Whether DP or EY is more efficient in CdTe depends on the momentum relaxation time. By changes in momentum relaxation times (by changing well width or impurity concentration), we predict the the possibility to induce a clear ‘dip’ in the temperature dependence which we see in Figure 4.14. This same non-monotonicity has been observed bulk GaAs and ZnO [4, 103, 5, 21]. Using our results we propose that n-doped (110) QWs should optimize spin lifetimes (when excited at exciton-bound-donor frequency) since DP is suppressed. Figure 4.15 displays our results for GaAs and CdTe (110) QWs at impurity densities nimp = 4 × 1010 cm−2 and nimp = 1 × 1011 cm−2 respectively. The decrease seen in GaAs is now due to depopulation of donor states instead of exciton thermalization. The depopulation is much slower in CdTe since the doping is higher. The up-turn in the CdTe curve as room temperature is reached is due to EY which is too weak to be seen in GaAs. We plot the data points from the undoped (110) GaAs QW for comparison. By avoiding the creation of excitons and their short lifetimes, long spin relaxation times can be achieved. 74 Figure 4.14: Spin relaxation in GaAs (100) QWs with different well widths (all other parameters, including τl and τx , do not change). Points are from Ref. [113] where L0 = 7.5 nm. Dotted: 2L0 ; dash-dotted: 3L0 /2; solid: L0 ; dashed: L0 /2. Figure 4.15: Spin relaxation in (110) GaAs (nimp = 4 × 1010 cm−2 ): dashed-dotted line. Spin relaxation in (110) CdTe (nimp = 1 × 1011 cm−2 ): solid line. Points from undoped (110) GaAs QW experiment[116] are included for comparison. For both systems, τp (T ) from [67] were used. EY is too weak over the temperature range depicted to be seen in the GaAs system. However EY is the cause of the increase in spin relaxation rate for the CdTe system. 75 4.4 Summary The spin relaxation times in n-doped bulk and QW semiconductors are well described by a theory invoking the exchange interaction between spin species. In undoped (110) QWs, where DP is absent, we find that exciton spin relaxation is important and leads to the observed surprising temperature dependence. We predict that a similar temperature dependence (though with longer relaxation times) should be observed in n-doped (110) QWs when excited at the exciton-bound-donor frequency. The DP mechanism is the dominant spin relaxation mechanism at high temperatures except for (110)-QWs when it is completely suppressed. We have suggested future experimental work to resolve what mechanisms relax spin localized on donors in n-doped GaAs and CdTe QWs. The theory allows us to predict experimental conditions that should optimize the measured spin relaxation times in GaAs and CdTe QWs. 76 Chapter 5 Phenomenological approach to spin relaxation in semiconductors II; case studies in bulk and quasi-2D wurtzite crystals 5.1 Introduction This chapter is similar to the previous one except that wurtzite (w) semiconductors are studied instead of zinc-blende (zb). The enterprise of the previous chapter is first closely followed for bulk n-ZnO in which the cross-over from localized to conduction spin relaxation is observed to be much like in the systems of chapter 4. However the phenomenological approach requires new additions due to the new system as shown in section 5.2.3 and reported in [21]. The experimental and theoretical situation in w-QWs is not nearly as developed as in zb-QWs; for this reason section 5.3 develops the DP mechanism in w-QWs for the first time as recently reported in [22]. 5.2 Bulk crystals The actual material wurtzite refers to ZnO but several other semiconductor compounds form similar crystal structures under ambient conditions; they include GaN, AlN, and InN among others. The wurtzite structure is similar to zinc-blende in that it is non-centrosymmetric and its bonds are tetrahedral; however its stacking sequence is different which leads to hexagonal as opposed to face centered cubic symmetry (see Figures 4.1 and 5.1). The hexagonal nature of the lattice leads to two lattice constants: c is the height of the hexagonal cylinder and a is the basal length of an edge. The four tetrahedral bonds in wurtzite are not identical and this leads to spontaneous electric polarization, in the absence of strain, which has recently 77 Figure 5.1: The wurtzite crystal structure. generated interest in the nitrides [59]. For bulk wurtzite, the spin-orbit Hamiltonian due to bulk inversion asymmetry (BIA) is [122, 123] HD (k) = ~ D ~ ω1 (k) · σ + ω3D (k) · σ, 2 2 (5.1) where ky 2β1 −kx , ~ 0 ky 2β3 (bkz2 − k||2 ) −kx , ω3D (k) = ~ 0 ω1D (k) = (5.2) (5.3) where k|| = kx2 + ky2 . In an asymmetric quantum well (with structural inversion asymmetry - SIA), spin-splitting will also occur due to the Rashba effect which will present another linear-in-k term (ω1R ) above such that β1 → β1 + αR . If the QW is confined along the z-direction [0001], after a spatial average along the direction of confinement H = HR + HD = αR + βD − β3 k||2 ) (ky σx − kx σy ) is obtained where π2 , (5.5) L2 is the Rashba coupling and is proportional to the βD = β1 + bβ3 hkz2 i ≈ β1 + bβ3 where σ are the Pauli matrices; αR (5.4) 78 external electric field in the z-direction, produced either by electrodes or by asymmetry in the structure; β3 is the cubic-in-k coupling, while βD = β3 kz2 is a Dresselhaus-type term that is controlled by confinement. kz2 is the expectation value of the operator kz in the QW wave function. If L denotes the well width, then kz2 ∼ (π/L)2 for the lowest electric subband and small structural asymmetry. αR is proportional to the electric field Ez and can thus be tuned by applying a gate voltage or producing structures with varying asymmetry; βD depends on L; β3 depends only on the material and cannot be turned off. When elastic scattering is assumed, k||2 ∝ ε is a conserved quantity. 5.2.1 The Elliott-Yafet mechanism in bulk wurtzite crystals In addition to creating the Rashba and Dresselhaus interactions, the spin-orbit interaction prevents Sz from being a good quantum number since this operator will not commute with the spin-orbit Hamiltonian. Wave functions are then impure spin mixtures which may lead to EY spin relaxation when scatterers are present as discussed in chapter 3. While the DP mechanism has proved to be dominant in zinc-blende semiconductors [55], the situation is not clear in wurtzite since EY and DP have yet to be calculated. The goal of the next two sections is to calculate the Elliott-Yafet and D’yakonov-Perel’ spin relaxation rates in bulk wurtzite semiconductors. Conduction band wave functions in bulk wurtzite semiconductors The energies and wave functions of conduction and valence band electrons can be found using the k · p approximation. The method is described for zinc-blende semiconductors in many texts [66] and the review by Fabian et al. [39] is especially detailed in its treatment. It has also been extended to the lower symmetry wurtzite crystal and those results [59] are used in the following discussion. The full calculations are not repeated here but the key differences between the zinc-blende and wurtzite solutions are noted. In k · p theory, the Hamiltonian equation is where H0 = ~2 k 2 ~ H0 + + k · p + Hs.o 2m m p2 2m un (r, k ↑) un (r, k ↓) ! = εn un (r, k ↑) un (r, k ↓) ! , (5.6) + V (r) is the kinetic energy and periodic crystal potential; Hs.o. is the spin-orbit Hamiltonian in Eq. (1.2) and is written here as Hs.o. = Hx σx + Hy σy + Hz σz ; un (r, k) are Bloch functions with band index n. Eight bands are included in this simple version of the theory: three degenerate valence bands and one conduction band, all doubled for spin. The eight basis functions are uc1 = |iS ↑i, u1 = | − X + iY X − iY √ ↑i, u2 = | √ ↑i, u3 = |Z ↑i, 2 2 79 (5.7) uc2 = |iS ↓i, u4 = | X − iY X + iY √ ↓i, u5 = | − √ ↓i, u6 = |Z ↓i, 2 2 (5.8) where S, X, Y , and Z are angular quantities expressible in terms of spherical harmonics [66]. The spin quantization axis is along the c-axis, [0001]. The wave function of a conduction electron is a linear combination of the eight basis functions. By solving Eq. (5.6), the conduction band wave function is |k ↑i = auc1 + bu1 + cu2 + du3 + f u5 + gu6 ; (5.9) the wave function with the opposite spin can be found by applying the time reversal operator K̂ (see section 3.2). The following is obtained: |k ↓i = auc2 + f u2 − g ∗ u3 − b∗ u4 − c∗ u5 + du6 , (5.10) where the arrows in the bra-ket refer to the pseudospin of the wave function. Due to the lower symmetry of the wurtzite crystal (compared to zinc-blende), the spin-orbit matrix elements are not all equal: hX|Hz |Y i = −i∆2 6= hY |Hx |Zi = hZ|Hy |Xi = −i∆3 . For the same reasons hZ|H0 |Zi = Ev 6= hX|H0 |Xi = hY |H0 |Y i = Ev + ∆1 where ∆1 is the crystal field splitting energy which is zero in cubic crystals. Ev is the valence band edge when there is no spin-orbit effect. Also the momentum matrix elements are hS|pz |Zi = mP1 /~ 6= hS|pz |Y i = hS|pz |Xi = mP2 /~. The cubic case can be retained when ∆1 = 0, ∆2 = ∆3 , and P1 = P2 . The lower symmetry in wurtzite forces the coefficients of Eqs (5.9, 5.10) to be more complicated than zinc-blende’s; the wurtzite coefficients are a= Ea Eb − 2∆23 −k− P2 (Ea Eb − 2∆23 ) k + P2 E a √ ,b = ,c = √ D 2Eg D 2D √ kz P1 Eb kz P1 2∆3 k+ P2 ∆3 d= , e = 0, f = ,g = , D D D (5.11) (5.12) where D2 (k) = (Ea Eb − 2∆23 )2 (1 + 2 P2 2 2 k⊥ 2 2 2 k⊥ P2 ) + (E + 2∆ ) + (Eb2 + 2∆23 )kz2 P12 , a 3 2Eg2 2 (5.13) with Ea = Eg + ∆1 + ∆2 , Eb = Eg + 2∆2 , P1 ≈ Eg ~2 (1 − m/m∗z )/2m∗z , and P2 ≈ Eg ~2 (1 − m/m∗⊥ )/2m∗⊥ . The effective mass is not strongly anisotropic so it is reasonable to make the simplification P1 ∼ P2 = P which will be done from now on [59]. From the defined wave functions it is determined that hk0 ↓ |k ↑i = cf 0 + gd0 − dg 0 − f c0 where the primed coefficients refer to coefficients of the final wave vector k0 ; this shows that states with antiparallel pseudospins are not necessarily orthogonal. 80 EY relaxation time The calculation starts with Fermi’s Golden Rule: Wk,σ→k0 ,σ0 = 2π |hf |U (r)|ii|2 δ(εk0 − εk ) ~ (5.14) where Wk,σ→k0 σ0 is the scattering rate from an initial state, |ii, with wave vector k and spin σ to the final state, |f i, k0 and spin σ 0 . Using the approximations in section 3.2.1, the spin-flip rate is 2π |hexp −ik0 · r|U (r)| exp ik · ri|2 |hk0 ↓ |k ↑i|2 δ(εk0 − εk ) ~ 4π 2 U 0 |hk0 ↓ |k ↑i|2 δ(εk0 − εk ). ~V 2 |k −k| Wspin−f lip = 2 = (5.15) (5.16) The spin-orbit interaction is assumed to be small such that the wave function is nearly one type of spin. Hence, the rate at which an electron scatters and does not flip spin is approximated by the total scattering rate: Wtotal ≈ Wno−spin−f lip = 4π 2 U 0 |hk0 ↑ |k ↑i|2 δ(εk0 − εk ). ~V 2 |k −k| (5.17) From the wave functions of Eqs. (5.9, 5.10), the inner products can be expressed as hk0 ↑ |k ↑i = B(k, k0 ) (5.18) hk0 ↓ |k ↑i = A(k, k0 ) sin θeiφ , (5.19) and where we have chosen k to be along the polar axis so θ is the angle between k0 and k and we have assumed that the wave vector is conserved throughout such that k 0 = k. φ is the azimuthal angle of k0 . Most generally, A(k, k0 ) = (2Eg + ∆1 + 3∆2 ) P1 P2 k 2 ∆3 D(k)D(k0 ) (5.20) and B(k, k0 ) = 0 k + k k 0 )P 2 ) + 2(2k k 0 P 2 + k 0 k P 2 )∆2 2Eg4 + Eg2 (2kz kz0 P12 + (k− + − + 2 z z 1 − + 2 3 0 2D(k)D(k ) (5.21) The spin relaxation time will be Wspin−f lip summed over all possible final k-states such that X 1 = Wspin−f lip −→ τs 0 k 81 Z d3 k 0 V Wspin−f lip (2π)3 (5.22) Making the necessary substitutions for W , Z sin θdθφk 02 dk 0 V 4π 2 1 = U 0 |A(k, k0 )|2 sin2 θδ(εk0 − εk ) τs (2π)3 ~V 2 |k −k| By using δ(εk0 − εk ) = m∗ /(k 0 ~2 )δ(k 0 − k), Z 1 sin θdθdφk 02 dk 0 V 4π 2 = U 0 |A(k, k0 )|2 sin2 θδ(k 0 − k)m∗ /(k 0 ~2 ). τs (2π)3 ~V 2 |k −k| (5.23) (5.24) Now we do the trivial integrals over k 0 and φ to obtain Z sin θdθk 8π 2 2 1 (5.25) = U 0 |A(k, k0 )|2 sin2 θm∗ /~2 . τs (2π)3 ~V |k −k| √ Realizing that the three dimensional density of states is g(ε) = 2m∗ εm/π 2 ~3 and using √ k = 2m∗ ε/~ we ascertain Z sin θdθ 8π 4 2 1 = U 0 |A(k, k0 )|2 sin2 θkm∗ /π 2 ~2 (5.26) τs (2π)3 ~V |k −k| Z sin θdθ 8π 4 2 U 0 |A(k, k0 )|2 sin2 θg(ε) (5.27) = (2π)3 ~V |k −k| and by simplifying we get 1 π g(ε) = τs ~V Z sin θdθU 2 (k, θ)|A(k, k0 )|2 sin2 θ (5.28) The momentum relaxation time will be similar except a transport factor is added in the standard way so it can be compared to mobility measurements [119], Z 3 0 Z 1 d kV π = Wtotal (1 − cos θ) = g(E) sin θdθU 2 (k, θ)|B(k, k0 )|2 (1 − cos θ) (5.29) τ̃p (2π)3 ~V We divide the spin relaxation rate by the momentum relaxation rate to find R sin θdθU 2 (k, θ)|A(k, θ)|2 (1 − cos2 θ) τ̃p = R τs sin θdθU 2 (k, θ)|B(k, θ)|2 (1 − cos θ) (5.30) We now make a first round of approximations: the spin-orbit and crystal field energy splittings are much less than the bandgap so we neglect those terms. This is obviously true for the wurtzite materials with parameters listed in Appendix A. Also, as suggested by Ridley [59], we can take P1 = P2 = P and assume isotropic effective mass. The matrix element, P , can be expressed as [59] P2 = Eg ~2 m∗ (1 − ). 2m∗ m 82 (5.31) These simplifications yield D2 (k) = Eg4 + Eg2 P 2 k 2 = Eg4 + (1 − m∗ 3 )E ε, m g (5.32) ∗ 2(1 − mm )Eg2 ε∆3 2Eg P 2 k 2 ∆3 A(k, k ) ≈ 4 = , ∗ Eg + Eg2 P 2 k 2 Eg4 + (1 − mm )Eg3 ε 0 and (5.33) ∗ 2Eg4 + 2Eg2 P 2 k 2 cos θ Eg4 + (1 − mm )Eg3 ε cos θ B(k, k ) ≈ = , ∗ 2(Eg4 + Eg2 P 2 k 2 ) Eg4 + (1 − mm )Eg3 ε 0 (5.34) where the fact that kx = ky = 0 and k = k 0 have been used to greatly simplify matters. All this allows Eq. (5.30) to be simplified to τ̃p = τs m∗ 2 2(1 − )E ε∆3 m g !2 sin θdθU 2 (k, θ)(1 − cos2 θ) . (5.35) ∗ sin θdθU 2 (k, θ)(Eg4 + (1 − mm )Eg3 ε cos θ)2 (1 − cos θ) R R Rearrangement of the previous result leads to τ̃p m∗ 2 ε = 4(1 − ) τs m Eg !2 ∆3 Eg !2 sin θdθU 2 (k, θ)(1 − cos2 θ) . ∗ sin θdθU 2 (k, θ)(1 + (1 − mm ) Eεg cos θ)2 (1 − cos θ) (5.36) R R The ratio of integrals is a factor that is of order unity; call it Q. Since we are mainly interested in an order of magnitude estimate, we are not concerned with correctly evaluating it as it depends on the scattering potential. Since ∆3 = ∆0 /3 [124, 125, 59], we obtain 1 m∗ 2 ε 4 ) ≈ (1 − τs 9 m Eg !2 ∆0 Eg !2 Q , τ̃p (5.37) where ∆0 is valence band spin-splitting energy (between heavy/light hole band and split-off band; see Fig. 4.2). To obtain an order of magnitude estimate of the EY relaxation rate at high temperatures, kB T is substituted for ε: 4 m∗ 2 kB T 1 ≈ (1 − ) τs 9 m Eg !2 ∆0 Eg !2 Q . τp (5.38) Eq. (5.38) is nearly identical to the expression found in zinc-blende semiconductors [56]. Using the numbers from Appendix A, τs ∼ 3 ms in GaN at 300 K if the scattering time is taken to be 1 ps. This time is much longer than what is found in GaAs for at room temperature: τs ∼ 60 ns. The massive discrepancy is due to the much smaller spin-orbit splitting and the larger band gap in GaN. Appendix A shows that this is true for other wurtzite semiconductors as well. 83 5.2.2 The D’yakonov Perel’ mechanism in bulk wurtzite crystals The conduction band states undergo ordinary impurity and phonon scattering. Each scattering event gives a change in the wavevector k, which in turn changes the effective magnetic field on the spin that comes from spin-orbit coupling. This fluctuating field relaxes the spin. The effective field strength is proportional to the conduction band spin splitting. Bulk zincblende crystals have conduction band splittings cubic-in-k due to bulk inversion asymmetry (Dresselhaus effect) [52]. In addition to cubic terms, bulk wurtzite conduction bands also possess spin splittings proportional to linear terms in k due to the hexagonal c axis which gives bulk wurtzite a reflection asymmetry similar to the Rashba effect [112, 126, 111, 127]. Let us calculate the explicit temperature dependence of the DP mechanism in bulk wurtzite crystals. First recall section 3.2.2 that in bulk, 1 τs,ij l X X 1 n (−1) τ [H , [H , σ ]]σ . = Tr j i l l,−n l,n 8π~2 l (5.39) n=−l In bulk wurtzite, unlike zinc-blende, Hs.o. ∼ k and k 3 so both l = 1, 3 will yield a non-zero result. τ3 can easily be related to τ1 by using Eqs. (3.32) as was done for zinc-blende in the previous chapter: τ3 = γ3 τ1 . Calculation of Eq. (5.39) yields 1 τs,xx = 1 τs,yy = 2 ∗2 1 8τ1 m∗ 2 4β1 β3 m∗ τ3 2 4β3 m = (β ε+ (b−4)ε + (7(4−b)2 +8(1+b)2 )ε3 ), 1 4 2 4 2τs,zz 3~ 5~ 175~ τ1 (5.40) where the spin relaxation tensor contains a slight anisotropy between the in-plane and outof-plane components. The constant b has been determined [128, 129] to be near four which simplifies the rates to 1 τs,xx = 1 τs,yy = 1 8τ1 m∗ 2 256γ3 τ1 β32 m∗3 3 ≈ β ε + ε , 1 2τs,zz 3~4 21~8 (5.41) which effectively eliminates the interference term (quadratic in ε) and one of the cubic terms as demonstrated implicitly in [21] and confirmed explicitly in [129]. Due to the identical symmetry of the Rashba and linear Dresselhaus terms, the presence of an electric field along ẑ results in the substitution: β1 → β1 + αR . The temperature dependence can be found by using Eq. (C.12), Id+1 (βµ) In+ν+d (βµ) , Iˆ(d) [τ1 εn ] = τtr β −n Id (βµ) Id+ν+1 (βµ) (5.42) to obtain I3/2 (βµ) 256m∗3 τtr γ3 β32 I3/2 (βµ)I7/2+ν (βµ) 1 8m∗ τtr β12 = k T + (kB T )3 , B 4 8 τs 3~ I1/2 (βµ) 21~ I3/2+ν (βµ)I1/2 (βµ) 84 (5.43) where d = 1/2 has been used since the density of states goes as ε1/2 in three dimensions. The temperature dependent momentum relaxation time, τtr , can be determined from electron mobility (µe ) measurements from µe = eτtr /m∗ where e is the charge of an electron. With the EY and DP mechanisms in the arsenal now, they are compared to experiments to determine which is operable in n-doped ZnO in the following section. 5.2.3 ZnO Zinc oxide has been the subject of considerable experimental and theoretical investigation for many years [130]. Its band gap is in the near ultraviolet, making it useful as a transparent conductor and as sunscreen. Its piezoelectricity opens up transduction applications. The activity has intensified more recently because of the possibility that ZnO might be useful for spintronics or spin-based quantum computation. It has been predicted to be a roomtemperature ferromagnet when doped with Mn [131]. Furthermore, its spin-orbit coupling is generally thought to be very weak compared with GaAs. The usual measure of the strength of spin-orbit coupling in semiconductors is the energy splitting at the top of the valence band. It is said that the spin-orbit coupling is negligible in ZnO because the valenceband splitting is −3.5 meV [125], as opposed to 340 meV for GaAs. Smaller spin-orbit coupling should lead to long spin relaxation times. Long relaxation times are required if spin information is to be transported over appreciable distances. The spin relaxation time τs has been measured by Ghosh et al. [103] to be about 20 ns from 0 to 20 K in optical orientation experiments. τs is sometimes called T2∗ even in the absence of an external field. Since the data from [103] used here were taken at zero field, the relaxation time is taken to be τs to avoid confusion with experiments conducted at finite field. The data show two surprising features. First, the relaxation times are actually somewhat shorter than the longest relaxation times in GaAs, which are about 100 ns [4]. One might expect the opposite given the relative strength of spin-orbit coupling in the two materials. Second, τs shows a non-monotonic temperature dependence, first increasing slightly and then rapidly decreasing with increasing temperature - but increasing temperature usually promotes spin relaxation. We show that the theory previously developed for τs in GaAs [5] can account for these observations. The theory must be modified to take account of the different impurity levels and binding energies of ZnO. This is important, because, in spite of intensive investigation, the nature of the impurities that govern the electrical properties of ZnO remains controversial, and our analysis sheds some light on this issue. Even more interestingly, it turns out that the wurtzite crystal structure has very important consequences for the D’yakonovPerel (DP) [63, 110] scattering that dominates the relaxation at higher temperatures. Thus the crystal structure must be taken into account fully. The final message will be that the “weak” spin-orbit coupling of ZnO is not negligible for spin relaxation, and it does not lead 85 to long relaxation times. In ZnO produced by the hydrothermal method, it is generally thought that there are two sets of impurity states, one shallow and quasi-hydrogenic, one deep and very well localized [132, 133]. Their precise physical nature is not known. In the case of the deep impurity, it is believed that a lattice defect accompanies the chemical impurity. The binding energies are in the range of a few 10s of meV for the shallow impurity and a few 100s of meV for the deep impurity. We shall demonstrate below that the optical orientation data can put bounds on these numbers. ZnO crystallizes in the wurtzite structure rather than the zinc-blende structure familiar from the III-V compounds. This has very important implications for the conduction band states. The spin-orbit interaction lifts the spin degeneracy in the conduction band. In zincblende structures crystal symmetry implies that the splitting is cubic in the magnitude of the wave vector k, but in the wurtzite structure the splitting is linear [126]. However, the spin relaxation time of the low-lying conduction band states depends mainly on the spin splitting near the conduction band minimum, and this is larger in ZnO than in GaAs for small enough k. In optical orientation experiments, electrons are excited from the valence band to the conduction band by circularly polarized light tuned close to the bandgap energy (pump step). The population of conduction electrons so created is spin-polarized [70]. Energy relaxation then occurs on a short time scale (≤ 1 ns), but most of this relaxation is from spin-conserving processes, so there is a longer time scale (or time scales) on which the spin of the system relaxes. This longer time scale is measured using Faraday or Kerr rotation (probe step). The important physical point is that the fast energy relaxation leads to a thermal charge distribution for the electrons by the time 1 ns has elapsed, but the spin distribution relaxes on longer time scales. The thermal charge distribution means that the localized donor impurity states are mostly full at the relatively low temperatures of the experiment. The spins of the localized electrons must be included along with the conduction electron spins. The spins of localized and extended states can be interchanged by the exchange coupling, a process we call cross-relaxation. This is often a rather fast process and is particularly important when the relaxation times of the localized and extended states are very different in magnitude. In GaAs this process is important in all the regimes of temperature, applied field, and impurity density that have been studied, and it is important in ZnO as well. In the following section a set of modified Bloch equations is derived to describe the aforementioned spin dynamics. 86 Modified Bloch equations We consider a conduction electron in the semi-classical approximation. It moves as a wave packet with a well-defined momentum and scatters from impurities and phonons at time intervals of average length τp , where τp is the momentum relaxation time. Its spin operator is sc . The spin-dependent part of its Hamiltonian in the absence of an applied external magnetic field is: H c = H1c + H2c = − 1 X µB J (r − Ri ) si · sc − g b (t) · sc . 2} ~ (5.44) i The first term, H1c , is the exchange interaction with impurity spins si located at an positions Ri . It is the same interaction that is responsible for the Kondo effect, but the temperatures here are all much greater than the Kondo temperature. The range of the function J (r − Ri ) is roughly aB , where aB is the effective Bohr radius. The second term, H2c , represents other spin relaxation mechanisms that we model as a small random classical field b(t) with a correlation time much shorter than τs . An analogous Hamiltonian H l can be written for a localized electron. First, we concentrate on the spin dynamics resulting from the spin-spin term and ignore 1/3 the second term. In the dilute limit (aB nimp 1), a conduction electron encounters impurities with randomly aligned spins if no short-range order is present in the impurity system. An effective field from the impurity spin affects the conduction electron when it is within ∼ aB of the impurity. When |r − Ri | > aB , the conduction electron proceeds unhindered by the effective field. This effective field is a result of the exchange potential. An itinerant electron will spend an average time of aB /v within the range of the effective field where v is the velocity of the electron. Thus the time between encounters is 1/nl a2B v [11]. In a semi-classical picture the spin of the itinerant electron undergoes precession of magnitude ∆φ = JaB /2v through a random angle during each encounter with an impurity. The spin of the impurity electron also precesses but with angle −∆φ. Since the sum of spins, sc + sl , commutes with H1c + H1l , the total spin in the system must be conserved. However the spin in each subsystem may shift between one another; this is cross-relaxation. It turns out for the parameters of the system under consideration that ∆φ ∼ 1, and we then find that τccr ∼ 1 nl a2B v (5.45) which implies that the spin is essentially randomized after one impurity encounter. If we consider an ensemble of conduction electrons with a net magnetization mc , this magnetization is exchanged at a rate of 1/τccr . As previously mentioned, any magnetization lost from the conduction electrons must be gained by the localized electrons and vice-versa. 87 For clarity we write 1/τccr = nl /γ cr and 1/τlcr = nc /γ cr where γ cr = 1/a2B v. Mahan and Woodworth find a similar expression for low velocity electrons using a quantum mechanical argument [46]. We now examine the second term of the Hamiltonian 1 H2c (t) = − gµB bx (t)σx + by (t) σy + bz (t) σz . 2 (5.46) This Hamiltonian relaxes the conduction electron spin. To extract a relaxation rate from this Hamiltonian, we use the equation of motion i dρ(t) = [ρ(t), H2c (t)] dt ~ (5.47) where ρ(t) is the 2 × 2 spin density matrix for an electron of a given momentum. We assume that the total density matrix for the conduction electron factorizes; we neglect off-diagonal terms that come from correlations. By iteration, we can write this equation as dρ (t) = dt Z t i 1 c 0 [ρ (0) , H2 t ] − 2 ρ t0 , H2c t0 , H2c (t) dt0 ~ ~ 0 (5.48) where the angular brackets indicate averaging over all orientations of b(t). To simplify notation, from now on angular brackets will be suppressed on the density matrix. Since hbi (t)i = 0, the first term is zero. We assume that different directions of bi are uncorrelated and (since the external field is zero) different direction are equivalent. hbi (t)bj (t0 )i = hb(t)b(t0 )iδi,j . Then we have Therefore, Eq. (5.48) reduces to g 2 µ2B dρ (t) =− dt 2~2 Z tX 0 [ρ(t0 ), σi ]σi hb(t)b(t0 )idt0 . (5.49) i The correlation function is assumed to be stationary in time so hb(t)b(t0 )i = g(t0 − t) = g(τ ).[28] If the correlation time of the b-fluctuations, τe , is short, ρ will not change on that timescale and g(τ ) will be nearly a δ- function. Eq. (5.49) can then be written as 2g 2 µ2B 1 X dρ (t) =− [ρ(t), σi ]σi dt ~2 4 i Z ∞ b(t)b(t0 ) dt0 . (5.50) 0 The integral is approximated by hb2 iτe . Define the relaxation time scale τc by gµ 2 1 B =2 hb2 iτe τc ~ (5.51) dρ (t) 1 X =− [ρ(t), σi ]σi . dt 4τc (5.52) giving i 88 The density matrix can be expanded in Pauli spin matrices 1 1X ρ (t) = I + mi (t) σi . 2 2 (5.53) i where I is the 2 × 2 identity matrix and mi = T r(σi ρ) is the expected value of the magnetization. Inserting Eq. (5.53) in Eq. (5.52) and matching coefficients of Pauli matrices gives a set of equations for the dynamics of m. For instance for conduction electron magnetization mc in the x-direction, dmc /dt = T r(σx dρ/dt) = −mc /τc . As with H1c , similar expressions for the localized magnetization ml can be found: dml /dt = T r(σx dρ/dt) = −ml /τl . By combining the effects of H1 = H1c + H1l and H2 = H2c + H2l , the modified Bloch equations for the magnetizations can be expressed as 1 dmc nl nc =− + cr mc + cr ml dt τc γ γ 1 dml nl nc = cr mc − + ml . dt γ τl γ cr (5.54) for two spin systems - itinerant and localized spins. τc and τl in Eq. (5.54) are now described in terms of well known relaxation mechanisms which will be discussed in the next section. This model was successfully applied to GaAs [5]. For ZnO, these Bloch equations are easily extended to account for the multiple-type impurities present. Method We now seek to write equations like those of Eq. (5.54) with regard given to the two types of impurities in ZnO - shallow and deep. As mentioned above, we find that the cross-relaxation is important to understand the data. These rates come from the Kondolike Jsl · sc interaction between an impurity spin sl and a conduction band spin sc . An expression for J in terms of tight-binding parameters can be derived using the SchriefferWolf transformation [134]. One expects that the cross-relaxation between conduction and shallow donor electrons to be much more rapid than the cross-relaxation between conduction and deep donor electrons because of the greater binding energy of the deep impurity and its larger on-site Coulomb energy. This is confirmed by the fit to the data. In fact we find that terms involving cross-relaxation between the deep donors and either the conduction band electrons or the shallow donor electrons can be neglected. With these simplifications, 89 for ZnO Eq. (5.54) extends to 1 nls dmc nc + cr mc + cr mls =− dt τc γc,s γc,s 1 nls nc dmls = cr mc − + cr mls dt γc,s τls γc,s dmld 1 = − mld . dt τld (5.55) In this equation, mc , mls , and mld stand for the magnetizations of the conduction electrons, the electrons on shallow impurities, and the electrons on deep impurities, respectively. The n’s denote the corresponding volume densities. Each of the populations has a relaxation time τc , τls , and τld . From Eq. (5.55), we find the magnetization as a function of time. Standard methods are used to solve these differential equations. The solutions yield a time dependence of the total magnetization, m(t) = mc (t) + mls (t) + mld (t), to be a sum of three exponentials, exp(−Γ+ t), exp(−Γ− t), and exp(−Γd t) where ! 1 1 1 nc + nls 1 Γ± = + + ± S , Γd = cr 2 τc τls γc,s τld (5.56) with S given by v u u S=t 1 1 nc − nls − + cr τls τc γc,s !2 + 4nc nls . cr 2 γc,s (5.57) No net moment can exist on the deep donor sites since no moment is excited into the deep states on account of them being significantly below the conduction band, and no net moment cross relaxes into these states. Therefore Γd can be ruled out as being the observed cr 1/τ , 1/τ , the rate Γ simplifies to relaxation rate. In the regime that (nls + nc )/γc,s c + ls cr and is very rapid and the rate Γ is slower, (nc + nls )/γc,s − Γ− = nc 1 nls 1 + . nc + nls τc nc + nls τls (5.58) We fit the data with this equation and associate it with τs . We see that the relaxation rate depends on two factors: the thermodynamic occupations of the shallow donors (the deep donors are always nearly full in the temperature range studied here) and the form of the relaxation rates for the conduction and localized shallow states. The densities can be computed using standard formulas from equilibrium statistical mechanics, since we deal only with time scales long compared to the fast energy relaxation scale. As a function of temperature T , the ratio nc /nls naturally increases rapidly as T → |εls |/kB , where εls is the binding energy of the shallow impurity. |εld | is so large that these states are always occupied at the experimental temperatures, which range from 5 K to 80 K. 90 τc is fairly complicated to calculate because there are several mechanisms that can relax the conduction electron spins. The simplest such mechanism is the Elliot-Yafet (EY) process [48] that arises from spin mixing in the wavefunctions. When a conduction electron is scattered by a spin-independent potential from state k to state k0 , the initial and final states are not eigenstates of the spin projection operator Sz so the process relaxes the spin. The rate of relaxation due to the EY process is well known to be of the form: 1/τEY = αEY T 2 /τp (T ) where αEY is a material dependent parameter and τp is the momentum relaxation time [58]. We estimate αEY (th) = 4.6 × 10−15 K−2 . The Bir-Aronov-Pikus (BAP) mechanism [53] arises from the scattering of electron and holes. This relaxation mechanism is commonly considered to be negligible in n-type materials like those under consideration here since the number of holes is small [89]. The D’yakonov-Perel’ (DP) mechanism [110] arises from the ordinary scattering of conduction-band states. Since this has not previously been calculated in a wurtzite structure, we devote the next section to it. This calculation yields an expression for τc as a function of temperature. τls and τld are due to non-spin-conserving anisotropic exchange (Dzyaloshinski-Moriya) interactions [108, 109]. The anisotropic exchange term is important. It arises from spin-orbit coupling and produces a term proportional to d · s1 × s2 where d is related to the interspin separation and the exchange integral between the wave function on sites 1 and 2. However, it is not possible to calculate it in detail when the nature of the impurities is not well known. We estimate the rate as 1/τDM = αDM (nimp,s + nimp,d ) where nimp,s and nimp,d are the total impurity concentrations of the shallow and deep impurity respectively and αDM has a weak temperature dependence that we neglect. The main contribution comes from the the overlap of the shallow impurity wavefunctions, which we take to be hydrogenic, with the deep impurity wavefunctions, which we take to be well-localized on an atomic scale. The details of how to estimate the resulting relaxation may be found in [5, 82, 83]. The numerical value we find from theory is αDM (th) = 1.12 × 10−20 cm3 ns−1 . When nuclei possess nonzero magnetic moments, the hyperfine interaction between electron and nuclear spin is a source of spin relaxation for localized electrons [41]. However, zero nuclear spin isotopes of Zn and O are 96% and 99.5% naturally abundant respectively. Therefore we rule out the hyperfine interaction from being an observed relaxation mechanism in [103]. DP mechanism in ZnO There had been no calculations of the DP mechanism in wurtzite crystals until the publication of [21]. The general DP equation of Eq. (5.43) can be used to determine the DP relaxation rate in bulk ZnO: I3/2 (βµ) 256m∗3 τtr γ3 β32 1 8m∗ τtr β12 3 I3/2 (βµ)I7/2+ν (βµ) = k T + (k T ) , B B τs 3~4 I1/2 (βµ) 21~8 I3/2+ν (βµ)I1/2 (βµ) 91 (5.59) the factors of I reduce to I3/2 (βµ) T TF 3 −→ , I1/2 (βµ) 2 5 I3/2 (βµ)I7/2+ν (βµ) T TF 3 7 −→ +ν +ν I3/2+ν (βµ)I1/2 (βµ) 2 2 2 (5.60) in the non-degenerate limit. In anticipation of curve fitting, we write 1 (1) τDP (1) where αDP (th) = 4m∗ β12 kB ~4 (3) = αDP τtr T + αDP τtr T 3 , (3) and αDP (th) = 3 m∗3 β 2 40QkB 3 7~8 (5.61) 16 35 γ3 (7/2 + ν)(5/2 + ν) 1.1 × 10−4 eV-nm which with Q = being of order unity. β1 has been calculated in ZnO [126] to be (1) gives a theoretical value of αDP (th) = 34.6 K−1 ns−2 . β3 has been calculated in ZnO [126] (3) to be 3.3 × 10−4 eV-nm3 which yields αDP (th) = 9.3 × 10−4 K−3 ns−2 . The sample from which the momentum relaxation times τtr (T ) were extracted [135] was hydrothermally grown by the same company as the Ghosh et al. sample in [103]. Results and discussion In Figure 5.2 we show that temperature dependence of τs as measured in a bulk ZnO sample and our fit (using Eq. (2.45)) to the data. It is seen immediately that the temperature dependence is not monotonic and that this is well-reproduced by the theory. The reason is simple. At low temperatures T |εls |/kB nearly all the electrons are in localized states. These states relax by the temperature-independent DM mechanism: 1/τls = 1/τDM . This mechanism alone determines the T = 0 values. When T approaches |εls |/kB , the deep impurities are all occupied but the rest of the population is shared by shallow localized and conduction band states. Initially, the conduction band electrons have a longer spin lifetime at low temperatures as seen from Eq. (5.61) so the DP mechanism that relaxes them is not very effective. However, the DP mechanism increases rapidly as T increases and the τs curve turns around. At T |εls |/kB , the shallow impurity level is empty and the relaxation (1) is dominated by the DP mechanism in the conduction band: 1/τc = 1/τDP (T ). At this point it is necessary to point out why only the linear-in-T DP mechanism is needed to explain the observed conduction spin relaxation. The other two viable candidates (cubic DP and EY) for relaxation are much too weak to explain the observed relaxation (1) (3) times in ZnO. We use the calculated values for αDP (th) and αDP (th) in the previous section to obtain the relative relaxation efficiencies between the linear and cubic DP mechanism terms: (1) 1/τDP (3) 1/τDP (1) = αDP (th) (3) αDP (th)T 2 = 3.72 × 104 K2 T2 (5.62) which demonstrates that the efficiency of the cubic-in-T term does not become comparable to the linear-in-T term at temperatures below 200 K which is far above the temperature 92 range investigated here. For this reason we can ignore the cubic-in-T DP mechanism term in our fit though it will surely be important at higher temperatures than experimentally probed. The crystal structure of ZnO therefore makes its spin relaxation qualitatively different from spin relaxation in bulk n-GaAs. We also compare the efficiencies of the DP and EY mechanisms: (1) (1) α (th)τp2 (T ) 7.5 × 1015 τp2 (T ) K ns−2 1/τDP = DP = . 1/τEY αEY (th)T T (5.63) Even if the momentum relaxation time taken to be unrealistically low, say 1 fs, the DP mechanism is still nearly two orders of magnitude more efficient at relaxing spins than the EY mechanism in the temperature range studied here. Due to the drastic qualitative and quantitative differences between relaxation mechanisms, we have unequivocally determined the relevant conduction electron spin relaxation mechanism in ZnO. -1 1/τs (ns ) 0.5 0.1 0.01 10 T (K) 50 100 Figure 5.2: Plot of 1/τs vs. temperature. Points are experiment of [103]. Dashed curve: [nls /(nc + nls )](1/τDM ). Dotted curve: [nc /(nc + nls )](1/τDP ). Solid curve: total 1/τs . nimp,s = 6.0 × 1014 cm−3 , nimp,d = 5.0 × 1017 cm−3 , εls = −23 meV, and εld = −360 meV. The fit of theory to the experimental data is clearly very good. We found that no reasonable fit was possible using only a single impurity level, though this worked very well for GaAs [5], so we used two levels. A good fit by this method was possible by adjusting the (1) coefficients αDP (exp) and αDM (exp), and the binding energies εls , εld and concentrations 93 nimp,s , nimp,d of the two donors, subject to the constraint that the room temperature carrier density should equal the measured [103] value of 1.26 × 1015 cm−3 . Qualitatively, one finds that nimp,d nimp,s and |εld | |εls | to get the right order of magnitude of the relaxation at low T. Physically, the deep impurity spins are important because they relax the shallow impurity spins by the DM mechanism, and the strength of the low T relaxation implies that the deep impurities must be quite numerous. Quantitatively, a least squares fit to the data (1) of [103] yields αDP (exp) = 134.5 K−1 ns−2 , αDM (exp)nimp,d = 0.06 ns−1 , |εld | = 360 meV, |εls | = 23 meV, and nimp,s = 6.0 × 1014 cm−3 . (1) (1) αDP (exp) is about four times larger than the theoretical value of αDP (th) given above, possibly due to strain effects. We also note that the values of τp that we used were taken from a different sample. If we take nimp,d to be near the highest values measured for the deep donor (see below) then αDM (exp) = 12 × 10−20 cm3 ns−1 is about one order of magnitude larger than the theoretical estimate αDM (th) given above. In view of the very poor understanding of the impurity wavefunctions, and the exponential dependence of αDM on the overlaps, this is perhaps not too disturbing. The presence of a shallow donor and a very deep donor has been seen in hydrothermally grown ZnO samples of the type investigated here [133, 136]. Donor concentrations up to nearly 5.0 × 1017 cm−3 (nimp,d ) have been measured for donors 330 − 360 meV (|εld |) deep [137, 133, 136]. Donors as shallow as 13 − 51 meV (|εls |) have been measured [132] at lower concentrations ∼ 5.0 × 1014 cm−3 (nimp,s ). Comparison with our values indicates that the parameters extracted from the fit are very reasonable for this material. From this analysis, we predict that in ZnO samples with fewer deep impurities, the relaxation time at low temperatures can be increased. As the impurities of ZnO vary greatly between different growth techniques [138], this prediction could be tested by further optical orientation experiments on different samples. We have found that τs in bulk ZnO can be understood by invoking previously known spin relaxation mechanisms. The dominant mechanisms in the material turn out to be the DP (scattering) relaxation of the conduction electron spins for T > 50 K and the DM (anisotropic exchange) mechanism for the localized spins for T < 50 K. In addition, it is very important to include the cross-relaxation between localized and conduction states previously proposed for GaAs. These physical ingredients explain quantitatively the relatively fast relaxation at low temperatures as being due mainly to the DM mechanism which in turn depends on having both deep and shallow impurity states. At high temperatures, the conduction states are dominant, and the DP mechanism gives an excellent fit to the data. The combination explains the very surprising non-monotonic temperature dependence of τs . Finally, there are two aspects of the data in [103] that we have not addressed here: the 94 applied magnetic field dependences on the spin relaxation and the spin relaxation observed in ZnO epilayers. We plan on addressing the former issue in a future publication. As for the latter issue, the epilayers are doped three to four orders of magnitude higher than in the bulk case. At such high dopings, spin glass effects become important and localized donor states coalesce to produce donor bands; we do not expect our theory to be applicable in such a regime. The theory has now been sufficiently developed that optical orientation experiments can actually serve as a characterization tool for doped semiconductors, giving information about the binding energies and concentrations of the electrically active impurities in n-type materials. 5.3 Quasi-2D nanostructures Much of semiconductor research in recent years has focused on the electron spin degree of freedom [41]. Electron spins in quantum dots can serve as a qubit - the electron spin couples relatively weakly to the environment and provides an ideal two-level system. For many quantum even classical spintronics applications, however, one requires mobile electrons. Even as a carrier of classical information the spin of mobile electrons offers advantages: translation and rotation are in principle dissipationless, offering great potential advantages over charge motion. For all types of applications, long spin coherence times and ease of spin manipulation at room temperature are of crucial importance. Several types of devices based on mobile spins have been proposed: ballistic [3] and non-ballistic [139, 140] spin field effect transistors, and double-barrier structures [2]. A central insight was that tuning of the spin-orbit (SO) parameters is possible by applying a gate voltage to a quantum well (QW) to vary the Rashba coupling. In addition, systematic variation of the well width has made it possible to independently tune other couplings and observe momentum-dependent relaxation times in (001)-GaAs wells [95]. Possible realization of the Datta-Das device [3] has recently been achieved in an InAs heterostructure [17]. Generally speaking, experimental studies relevant to the realization of these devices have been carried out at rather low temperatures T or short spin lifetimes τs . The spin effects in InAs disappear at about T = 40 K and the disappearance was attributed to additional scattering [17]. In the GaAs work, signs of enhanced lifetime (due to the persistent spin helix [141]) decreased rapidly with temperature and at T = 300 K, τs was only slightly above 100 ps; the loss of coherence is due to cubic-in-k terms in the spin-orbit Hamiltonian that relax the spin by the D’yakonov-Perel’ (DP) mechanism [62, 63]. Long spin lifetimes are fundamentally limited by the strength of the SO interaction in direct band gap zinc-blende semiconductors. Wurtzite QWs offer great advantages over the aforementioned zinc-blende structures in that long coherence times can be maintained at room temperature due to their 95 Material w-(0001) zb-(001) zb-(111) zb-(110) x [100] [110] [112] [110] y [010] [110] [110] [010] z [001] [001] [111] [110] βD β1 + bβ3 hkz2 i β3 hkz2 i cβ3 hkz2 i β3 hkz2 i/2 Table 5.1: Table of several semiconductor nanostructures with different growth orientations and their respective crystal axes. Also included is the parameter βD which gives the strength of the linear Dresselhaus terms in the Dresselhaus spin-orbit interaction (see Table 5.2). small spin-orbit couplings. This body of work, and indeed nearly all of the theory and experiment along this research direction, has been carried out on materials with the zinc-blende (zb) structure. However, there has also been work on bulk wurtzite (w) materials because their SO splittings are small [142] and there is hope of room temperature ferromagnetism in magnetically doped w-GaN and w-ZnO [143]. Spin dynamics in bulk w-GaN has been studied by Beschoten et al. [142, 129], and there has been experimental [103, 144] and theoretical [21, 145] work on spin lifetimes in w-ZnO. However, it is often difficult to separate contributions coming from localized and mobile electrons in bulk doped semiconductors. For example, the rather long spin lifetimes measured in bulk zb-GaAs at low temperatures [4] were shown to come from localized spins [5]. This strongly suggests that w-QWs should be studied, and indeed there is some very recent work along these lines. Experimentally, the SO splittings have been measured by Lo et al. [146] in Alx Ga1−x N/GaN QWs by either Shubnikov-de Haas (SdH) or weak antilocalization (WAL) measurements. WAL measurements unambiguously point to SO coupling [147] and such measurements are found to agree with theory [148]. There are calculations for w-ZnO wells that did not consider tuning [145]. There are striking differences between the zb symmetries and the hexagonal symmetry of wurtzite materials such as GaN, ZnO, and AlN. The Dresselhaus Hamiltonians for (001), (110), (111)-zb and (0001)-w are tabulated in Table 5.2. For (0001) QWs with wurtzite symmetry we have [146, 128] (w) w HSO = αR + βD − β3 kk2 (ky σx − kx σy ) , (w) where βD = β1 + bβ3 kz2 and kk = (kx , ky ) is the in-plane wavevector. This form is clearly very different from the (001) and (110)-zb case, and the difference has been confirmed experimentally [147]. The (111)-zb case is similar to (0001)-w. As before, αR can be tuned by applying a gate voltage or otherwise varying the asymmetry; however, β1 6= 0 even in the absence of an applied electric field - the wurtzite structure does not have the mirror 96 H1 H3 2 −β3 k|| (ky σx Material w-(0001) zb-(001) αR (ky σx − kx σy ) + βD (−kx σx + ky σy ) zb-(111) (αR + βD )(ky σx − kx σy )) zb-(110) (αR + βD )(ky σx − kx σy ) − kx σy ) β3 (kx ky2 σx − ky kx2 σy ) − 2β√33 k||2 (ky σx − kx σy ) + β3 (−kx2 /2 αR (ky σx − kx σy ) + βD kx σz + √ β3√ 2 k (3kx2 2 3 y 2 ky )kx σz − ky2 )σz Table 5.2: The spin-orbit Hamiltonians for several semiconductor QWs with different crystallographic orientations. The parameter βD is tabulated in Table 5.1. (w) symmetry z ↔ −z. βD can be tuned by changing L. zb ’s and H w : There are important formal differences between the HSO SO (zb) • (001)-zb : When αR = ±βD the linear-in-k terms produce a k-independent effective magnetic field in the [110] or [110] direction [139, 140]. This can enhance spin coherence for spins oriented along the effective magnetic field. Aside from other possible mechanisms, the spin relaxation time will be limited by the cubic-in-k terms of the Dresselhaus Hamiltonian. w differs in that the linear-in-k term can be eliminated for spins oriented in any direction HSO (w) when αR = −βD . Second, at the cubic-in-k level for a circular Fermi surface and elastic scattering, |kk | is conserved and we can set kk = kF , the Fermi wavevector. This gives the possibility of canceling the effective field all the way to third order by enforcing the condition (w) αR + βD − β3 kF2 = 0, eliminating what appears to be the major source of spin decoherence in the experiments to date [146, 128]. Note that the final term can be independently tuned by changing the electron density. The ability to ‘tune’ away the spin-orbit terms up to cubic-in-k gives the wurtzite structure an advantage over zinc-blende. zb is very similar to H w ; the linear terms have the exact same • (111)-zb : This HSO SO form and one of the cubic terms can be canceled√ out for zinc-blende as happens for wurtzite. However an additional cubic term remains, β3√ 2 k (3kx2 2 3 y − ky2 )σz ; since the effective field of this ‘left-over’ piece is always in the z-direction, it will not relax z-oriented spins. Spin not w . oriented along ẑ will relax and this is different than what happens for HSO • (110)-zb : This structure has engendered much interest [149] since in the absence of structural inversion asymmetry (SIA), the spin-orbit Hamiltonian is Zeeman-like with the field in a single direction, ẑ. Spins aligned in the z-direction do not relax though other components will. In the presence of the Rashba term, no lifetime is infinite and the relaxation rate tensor can only be diagonalized in a basis dependent on the SIA and BIA couplings [140, 150]. As pointed out by Lo et al., these properties are what makes their Alx Ga1−x N/GaN structure an excellent candidate for the non-ballistic spin FET [146]. In this section, we compute the temperature-dependent spin relaxation by the DP mechanism, varying the 97 Exact ˆ 1 E] I[τ ˆ 1E2] I[τ ˆ 1E3] I[τ T TF T TF τtr kB TF ” “ 2 1 + TT 2 δν F ” “ 2 3 τtr kB TF3 1 + TT 2 ∆ν τtr kB T T =0 0) τtr kB T II10 (βµ (βµ0 ) 0 )Iν+2 (βµ0 ) τtr (kB T )2 II10 (βµ (βµ0 )Iν+1 (βµ0 ) 0 )Iν+3 (βµ0 ) τtr (kB T )3 II10 (βµ (βµ0 )Iν+1 (βµ0 ) τtr kB TF τtr (kB TF ) 2 2 τtr kB TF2 τtr (kB TF )3 (ν + 2)τtr (kB T )2 (ν + 3)(ν + 2)τtr (kB T )3 F Table 5.3: A table of the various quantities needed in determining the DP spin relaxation rate in wurtzite and zinc-blende QWs. The quantities δν and ∆ν are located in Table 5.4. w . parameters in HSO The DP mechanism is dominant at room temperature in bulk zb- GaAs [5] and w-ZnO [21] and this is expected to be true in modulation doped QWs as well. We discuss other mechanisms that may take over when DP is suppressed. We also discuss the feasibility of the needed tunings. 5.3.1 Formalism Conduction band states undergo ordinary impurity and phonon scattering. Each scattering event gives a change in the wave vector k, which in turn changes the effective magnetic field on the spin that comes from spin-orbit coupling. This fluctuating field relaxes the spin. This is known as the D’yakonov-Perel’ (DP) spin relaxation mechanism [63, 64] . The effective field strength is proportional to the conduction band splitting. Following the treatment in chapter 3, the dynamics of elastically scattered 2D electrons with spin polarization S is described by Ṡi = − τ1ij Sj where i, j ∈ {x, y, z} and ∞ 1 1 X = 2 τij 2~ n=−∞ R∞ ji 0 R dE(F+ − F− )τn Γn ∞ 0 dE(F+ − F− ) (5.64) where Γji n = −Tr{[H−n , [Hn , σj ]]σi }, and Hn = Z 0 2π dφk H e−inφk , 2π 1 = τn Z (5.65) π dθWk,k0 (1 − cos nθ). (5.66) 0 The angle φk is between the x-axis and k; Wk,k0 is the spin-conserving scattering rate between an initial wave vector k to a final wave vector k0 . θ denotes the angle between k ˆ 1 E n ] is needed in the and k0 . Other symbols are defined in section 3.2.3 The quantity I[τ determination of the DP rate’s temperature dependence and is computed in the limiting cases in Tables 5.3 and 5.4. 98 ν 0 δν 1 2π 2 1 3 1+π 2 T 2 /3TF2 1+7T 2 /15T 2 π 2 1+π2 T 2 /T 2F F 2 ∆ν π2 π2 3 2 2 5π 2 1+7T /25TF 3 1+π 2 T 2 /3TF2 2 2 7π 2 1+T /TF 3 1+π 2 T 2 /TF2 Table 5.4: A table of the various quantities needed in determining the DP spin relaxation rate in wurtzite and zinc-blende QWs. 5.3.2 Temperature dependence of DP mechanism in wurtzite and zincblende QWs Experimental situations are neither at T = 0 nor exactly non-degenerate. Since the behavior at T = 0 and at high temperatures are dramatically different, it is important to examine how the relaxation rate changes as the temperature and electron density vary in general. In general, we find from Eq. (5.64): " ∗ 1 1 2τtr I1 (βµ0 ) (w) 2 2m = = (α + β ) kB T R D (w) 2 τz(w) (T ) ~2 I0 (βµ0 ) ~2 τx,y (T ) 1 (5.67) 4m∗2 Iν+2 (βµ0 ) 8m∗3 Iν+3 (βµ0 ) (w) − 2(αR + βD )β3 4 (kB T )2 + β32 6 (kB T )3 ~ Iν+1 (βµ0 ) ~ Iν+1 (βµ0 ) # where In (βµ) is a function related to the polylogarithm defined in Appendix C and is tabulated in Table D.1. This simplifies at zero temperature, where the substitution E → εF = kB TF can be made, to 1 4τtr m∗ εF 1 = = (w) 2 τz(w) (T ) ~4 τx,y (T ) 1 αR + (w) βD 2m∗ β3 εF − ~2 2 . εF is the Fermi energy. TF is the Fermi temperature and is related to the electron density by TF = ~2 πn/kB m∗ . Clearly the T = 0 relaxation times diverge when the tunable quantity (w) αR + βD − 2m∗ β3 εF /~2 vanishes. This divergence is cut off by finite temperatures. All other components of relaxation tensor vanish at all T. Preliminary expressions (before thermal average) for zb-QWs can be determined similarly and can be found in Table 5.5. For zb-QWs the ratio η = τ3 /τ1 is required: ( (2−ν)(3−ν) , ν ≤ 3/2 τ3 ν 2 −ν+6 η= = τ1 1/9, ν > 3/2 which is only true in two-dimensional systems. It should be stated that additional k 6 terms should exist due to the product of linear-in-k and quintic-in-k terms in the above formalism. 99 Γzz Material 2 4τ1 k|| ~2 w-(0001) zb-(001) 4τ1 k2 ~2 αR + βD − β3 k||2 2 zb-(110) Material zb-(110) 0 Γxx,yy 2 2τ1 k|| w-(0001) zb-(111) β 2 (1+η) 4 k 2 + β 2 − k β3 βD + 3 αR D 2 16 β32 2 2 4τ1 k2 αR + βD − 2√3 k ~2 zb-(111) zb-(001) 2 ~2 αR + βD − β3 k||2 2 2 β 2 (1+η) 2τ1 k2 (±αR − βD )2 + k β3 (±α2 R −βD ) + 3 16 k 4 ~2 2 β 2 (1+2η) 4 +β ) 2τ1 k2 2 − k β3 (α √R D + 3 (α + β ) k R D 2 12 ~ 3 β32 (1+9η) 4 k2 β3 βD 2τ1 k2 2 β − + k 2 D 4 64 ~ Table 5.5: Γzz = P∞ zz n=−∞ Γn as used in Eq. (5.64). Using αR = 0 for zb-(110). We are not aware of expressions for the spin-orbit Hamiltonian to such a high order in k. T TF (degenerate regime) The degenerate regime must he dealt with carefully. The low temperature regime is determined from the content in Appendices C and D. Following the notation of the Appendices, −z = −eβµ0 = 1 − eTF /T ≈ −eTF /T where µ0 = β −1 ln(eεF β − 1) is an exact expression for the chemical potential in a two dimensional electron gas [67]. In the degenerate regime, one can enforce Lin (1 − eTF /T ) ≈ Lin (−eTF /T ). At T = 0, one expands Lin (−eTF /T ) to first order which is −(TF /T )n /n!. However since the cancellation occurs in the terms for wurtzite, we should also look at the second leading terms when the polylogarithm is expanded. The higher order terms of the polylogarithm must be retained especially near (w) αR + βD − 2m∗ β3 εF /~2 = 0 where the first order terms vanish completely. Table 5.6 shows the low temperature spin relaxation rates for wurtzite and several zincblende quantum wells. The factorization that allows the spin relaxation time to vanish at T = 0 does not occur at a small finite temperature; unfortunately, due to the thermal averaging we cannot get infinite spin lifetime for any ν. What must the Rashba coefficient be tuned at to achieve a maximum relaxation time? By ∂ 1 ∂αR τDP ∗ , the = 0, in the degenerate regime, we obtain the elements of Table 5.7 for 1/τDP ∗ . The maximum minimum DP relaxation rate, and the corresponding Rashba coefficient, αR spin lifetimes can have different behaviors depending on the QW and its orientation. Figure 5.3 illustrates this fact. 100 Material w-(0001) 1/τx zb-(001) 1/τz zb-(001) 1/τ− zb-(111) 1/τz zb-(111) 1/τx zb-(110) 1/τz zb-(110) 1/τx Material Rate w-(0001) 1/τx zb-(001) 1/τz zb-(001) 1/τ− zb-(111) 1/τz zb-(111) 1/τx zb-(110) 1/τz zb-(110) T TF Rate 1/τx 2τtr ζF ~2 “ αR + βD − β3 ζF 2 + 2 2τtr ζ 2 β3 ~2” β3 ζF ∆ν − 2(αR + βD )δν “ ” 2 β (1+η) (1+η) 4τtr ζF 2 2 αR + βD − β23 βD ζF + 3 16 ζF2 + 4τtr~ζ2 β3 β3 ζF16 ∆ν − β2D δν ~2 “ ” “ ” 2 β (1+η) 2 2τtr ζF (αR +βD ) β 2τtr ζ 2 β3 β3 ζF (1+η) (αR + βD )2 − 23 (αR + βD )ζF + 3 16 ζF + ∆ν − δν 2 2 16 2 ~ 4τtr ζF ~2 2τtr ζF ~2 αR + β D − β√3 ζ 2 3 F 2 + 4τtr ζ 2 β3 ~2 ~ β3 ζF 12 ∆ν − (αR +βD ) √ δν 3 “ “ ” β 2 (1+2η) 2 ” (αR +βD ) β 2τtr ζ 2 β3 β3 ζF (1+2η) √ (αR + βD )2 − √3 (αR + βD )ζF + 3 12 ζF + ∆ν − δν 2 12 3 2τtr ζF ~2 2 − βD β3 βD 4 ζF 3 ~ + β32 (1+9η) 2 ζF 64 0 + 2τtr ζ 2 β3 ~2 1+9η 64 β3 ζF ∆ν − 41 βD δν T TF (αR + βD − 2(ν + 2)(αR + βD )β3 ζ 2 + (ν + 3)(ν + 2)β32 ζ 3 2 4τtr ζ 2 + β 2 − (ν + 2) β3 βD ζ 2 + (ν + 3)(ν + 2) β3 (1+η) ζ 3 α 2 R D 2 16 ~ β32 (1+η) 3 β3 (αR +βD ) 2 2τtr 2 (α + β ) ζ − (ν + 2) ζ + (ν + 3)(ν + 2) ζ R D 2 2 16 ~ β32 3 (ν+2)β3 4τtr 2 2 √ (α + β ) ζ − (α + β )ζ + (ν + 3)(ν + 2) ζ R D R D 2 12 ~ 3 2 2τtr 3 2 ζ − (ν+2)β 2 + (ν + 3)(ν + 2) β3 (1+2η) ζ 3 √ (α + β ) (α + β )ζ R D R D 12 ~2 3 2τtr ~2 )2 ζ 0 2τtr ~2 2ζ βD − (ν + 2) β34βD ζ 2 + (ν + 3)(ν + 2) β32 (1+9η) 3 ζ 64 Table 5.6: 1/τDP ,the DP spin relaxation rate, for several types of QWs in both the degenerate and non-degenerate limits. ζ(F ) = 2m∗ kB T(F ) /~2 . 101 105 1Τ*DP 1000 10 0.1 0.001 1 2 5 10 20 T HKL 50 100 200 Figure 5.3: Maximum DP relaxation times for several QWs: w-(0001), τx solid black, zb(001) τz solid blue, zb-(001) τ− dashed blue, zb-(111) τz solid red, zb-(111) τx dashed red. ∗ These times are determined from 1/τDP in Table 5.7. Wurtzite parameters are for GaN; zinc-blende parameters are for GaAs (see Appendix A); TF = 20 K. T TF (non-degenerate regime) Since In /Im → n!/m! when T TF , we obtain the expressions in the lower half of Table 5.6 for the relaxation rates at high temperatures. The maximum times achievable for various QWs are given in the lower half of Table 5.7. 5.3.3 Comparison between wurtzite and zinc-blende From the preceding analysis, it is apparent that the spin relaxation times can be modified by altering the Rashba coupling, αR , and thereby causing interference with Dresselhaus couplings. The same could be done by changing βD though this would have to be done by changing the QW geometry (e.g. well width) which would be impractical for devices. The interference between spin-orbit terms is well known in zinc-blende semiconductors [65] and can also be seen in Table 5.6. In this section we contrast the aforementioned tabulated results for zb-QWs and w-QWs with actual material parameters. For zb we use GaAs and for wurtzite we examine both GaN and AlN. First, in Figure 5.4 displays DP relaxation times at zero temperature. There are a few points to be made: both τx,z for w-(0001) and τz for zb-(111) diverge when tuned correctly. Despite this divergence, the zb-(001) crystal has garnered more attention [139, 140] though the cubic term prevents divergence at zero temperature. τ± for zb-(001) suppresses spin when αR = ±βD but the τz component causes decay very rapidly in comparison. 102 Material ∗ :1/τDP w-(0001) :1/τx at α∗R zb-(001) :1/τz 8τtr m∗ kB ~4 T TF 16τtr m∗3 (∆ν −2δν ) 3 2 δν2 T2 2 1− k T T τ β tr F 8 2 3 B ∆ν−2δν TF ~ 2m∗ kB TF T2 −βD + β3 1 + δν T 2 ~2 F h “ ”i 2 2 T3 ∗ 2 ∗ m∗ kB TF β 2 (1+η) m∗2 kB 2 F + m kB T β3 β3 m kB TF (1+η) ∆ − β δ TF βD − β3 βD + 3 4 ν D ν 2 4 2 2 ~ ~ zb-(001) :1/τ− at α∗R zb-(111) :1/τz at α∗R zb-(111) :1/τx at α∗R τtr m∗3 3 3 2 kB TF β3 η ~8 ∗ at α∗R zb-(111) :1/τz at α∗R zb-(111):1/τx at α∗R 4 F δν2 T 4 2η TF4 F at α∗R zb-(001) :1/τ− 2 2 kB T −βD + m√3~ 2 β3 (ν + 2) 2 ∆ν 8τtr m∗3 3 3 2 kB TF β3 η 1 + ( 2η + ∆ν − δν ) TT 2 − 3~8 F ∗ T2 B TF −βD + m√k3~ 2 β3 1 + δν T 2 ∗ :1/τDP w-(0001) :1/τx at α∗R 1 + η1 ((1 + η)∆ν − 2δν ) TT 2 − δην TT 4 F F ∗k T T2 B F −βD + m 2~ β 1 + δ 3 ν T2 2 F 8τtr m∗3 (∆ν −δν ) 3 2 δν2 T2 2 k T T τ β 1 − F tr 3 B ∆ν −2δν T 2 3~8 Material zb-(001) :1/τz 4~ ~ 0 at α∗R 4τtr ζ ~2 T TF 16τtr m∗3 3 3 2 kB T β3 (ν + 2) ~8 ∗ −βD + 2m ~k2B T β3 (ν + 2) 2 − β β (ν + 2)ζ 2 + βD 3 D 6β32 (1+η)(ν+3)(ν+2) 3 ζ 16 0 τtr m∗3 3 3 2 kB T β3 (ν + 2)(1 + η(ν + 3)) ~8 ∗ −βD + m 2~kB2 T β3 (ν + 2) 8τtr m∗3 3 3 2 kB T β3 (ν + 2) 3~8 ∗k T B −βD + m√3~ 2 β3 (ν + 2) 4τtr m∗3 3 3 2 kB T β3 (ν + 2 + 2(ν + 2)(ν + 3)η) 3~8 ∗k T B −βD + m√3~ 2 β3 (ν + 2) ∗ ,the minimum DP spin relaxation rate, for several types of QWs in both Table 5.7: 1/τDP the degenerate and non-degenerate limits. 103 ΤDP HnsL 1000 10 0.1 0.001 -4 -3 -2 -1 ΑR HmeV nmL 0 1 Figure 5.4: DP relaxation times for several QWs at T = 0 K with TF = 25 K: w-(0001), τx solid black, zb-(001) τz solid blue, zb-(001) τ− dashed blue, zb-(111) τz solid red, zb-(111) τx dashed red. These times are determined from upper half of Table 5.7 by setting T = 0 K. Wurtzite parameters are for GaN; zinc-blende parameters are for GaAs (see Appendix A). The low temperature regime (T = 5 K) is graphed in Figure 5.5. The infinite lifetimes seen at T = 0 K no longer exist. As seen in Table 5.7, the maximum time is limited by the cubic parameter β3 . Thus the much longer times in wurtzite are a consequence of the smaller SO coupling (see Table 5.8); AlN achieves the longest spin lifetime (> 1 ms). The non-degenerate regime (T = 300 K) is shown in Figure 5.6. The elements in the lower half of Table 5.6 are used in the figure. As was the case at low temperatures, β3 again limits the maximum spin lifetime for all QWs (see expressions in lower half of Table 5.7). Room temperature behavior is most important for device functions; here w-AlN yields what would be the longest spin lifetime (∼ 0.5µs) for mobile electrons at room temperature if other mechanisms are assumed to be even weaker. Appropriately tuned zb-GaAs reaches a longest time of just over 100 ps - three orders of magnitude shorter than w-AlN. The intermediate β3 (Table 5.8) of w-GaN places its spin relaxation time between w-AlN and zb-GaAs. A desired feature for a semiconductor suitable for spintronic applications is an easy to manipulate spin relaxation time. As discussed in the first chapter of this dissertation, a device’s ON/OFF state is dictated by carrier spin orientations at a ferrogmagnetic drain. It is highly desirable to change the spin lifetime from long to short without needing to change the SO coupling drastically. In light of Figure 5.6, it appears that the two wurtzite plots in the left panel are ‘sharper’ than the zinc-blende plots in the right panel. This is better seen 104 7 10 (a) 6 10 5 10 τx w-AlN w-GaN τz, zb-(001) τ-, zb-(001) τz, zb-(111) τx, zb-(111) 4 10 3 τs (ns) 10 2 10 T=5K T = 5 K (b) zb-GaAs 1 10 0 10 -1 10 -2 10 -3 10 -4 10 -2 -1.5 -1 -0.5 0 0.5 αR (meV nm) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 αR(meV nm) Figure 5.5: Low temperature DP spin relaxation times versus Rashba coupling, αR for wurtzite (left panel) and zinc-blende (right panel) semiconductors. AlN: TF = 15 K, GaN: TF = 25 K, GaAs: TF = 80 K (all Fermi temperatures correspond to an electron density ∼ 2 × 1011 cm−2 . Entries from top half of Table 5.6 are used in the plot. in linear-linear axes which is shown in Figure 5.7 where the relaxation rate is plotted instead of the relaxation time. The ‘sharpness’ of the relaxation rate versus αR is governed by the 2 . For all types of QWs, the curvature goes as ∼ m∗ T . This informs curvature or ∂ 2 τs−1 /∂αR us that in a spin relaxation transistor, device performance due to a large ON/OFF ratio should be optimized at high temperatures and large effective masses. AlN in the wurtzite phase has an effective mass ∼ 6 times larger than zb-GaAs. To reduce the maximum spin relaxation time by 105 , one needs to change the Rashba coefficient by ∆αR = ±0.7 meV nm. To do the same in (001)-GaAs, ∆αR = ±300 meV nm. Symmetric zb-(110) structures are unique from the others because the DP mechanism can be completely suppressed for spin oriented along the growth axis [110]. Spin oriented off that axis do relax. The prospect of going between extremely long and very short relaxation times is very appealing and has fostered interest in this type of QW. However other relaxation mechanisms present themselves when DP is suppressed and inhibit the theorized long times. In 2004 Döhrmann et al. discovered a new spin relaxation mechanism they termed intersubband spin relaxation (ISR) [117]. Despite the suppression of DP in their (110)-GaAs QW, they measured relaxation times on the order of only a few nanoseconds at temperatures greater than 100 K. Much longer times are expected due to the absence of DP 105 3 10 (a) 2 10 τx w-AlN w-GaN (b) zb-GaAs 1 τs (ns) 10 T = 300 K τz, zb-(001) τ-, zb-(001) τx, zb-(111) τz, zb-(111) T = 300 K 0 10 -1 10 -2 10 -3 10 -2 -1 0 1 αR (meV nm) -3 -2 -1 0 αR(meV nm) 1 Figure 5.6: High temperature DP spin relaxation times versus Rashba coupling, αR for wurtzite (left panel) and zinc-blende (right panel) semiconductors. Entries from bottom half of Table 5.6 are used in the plot. and the inefficiency of EY. A limiting mechanism was proposed where due to spin mixing there is a probability for the spin of an itinerant electron to flip when scattered to a different electronic subband [117, 153]. The amount of mixing depends quadratically on the size of the SO coupling and the energy gap between subbands. Their estimate of this relaxation time agreed well with experiment. This mechanism does not erode the results mentioned in this chapter; due to the SO coupling discrepancy, ISR in AlN is 107 times weaker than in GaAs which means ISR times around several milliseconds - longer than what is found from DP at room temperature. Additionally, ISR is weakened by narrowing the QW and creating larger energy gaps between different subbands. Reducing the well width increases hkz2 i which could affect larger DP relaxation but this can be tuned away along with all the other linear terms of the SO Hamiltonian in Tables 5.1 and 5.2. 5.3.4 Tuning of spin-orbit parameters We now address the tunability of possible devices. We note first that in zb-GaAs it has been possible to achieve quite substantial variations in the appropriate parameters (zb) [95, 154]. For example, Koralek et al. were able to change βD by making structures with different well widths and to change αR by adjusting the dopant concentrations on 106 Material w-GaN w-AlN w-ZnO zb-GaAs β3 (meV nm3 ) −0.32 [111] −0.01 [128, 152, 151] 0.33 [111] 6.5 − 30 [95] β1 (meV nm) 0.90 [111] 0.09 [151] −0.11 [111] 0 m∗ /m0 0.2 0.4 0.25 0.067 Eg (eV) 3.2 6.2 3.3 1.5 Table 5.8: Parameters for several semiconductors. 5000 1ΤDP Hns-1L 4000 3000 2000 1000 0 -6 -4 -2 0 ΑR HmeV nmL 2 4 Figure 5.7: Spin relaxation rates for w-AlN τx (black) and zb-GaAs (111) τx (red). Plot is linear-linear to demonstrate the different curvatures which go as m∗ T . the sides of the well, corresponding to a maximum electric field of 5.4 × 10−3 V/nm. In (zb) this way a range of αR /βD of about 0.25 to 1.25 was achieved, without even needing a gate. β3 was inaccessible and remained constant for all structures, setting a upper limit on spin lifetimes. Significant experimental tuning of SO coupling in w-GaN has not yet been achieved. However, calculations have been done [155] for w-GaN, which produce the correct magnitude for the spin splittings overall (∼ 5 meV at Fermi wavevector of typical structures). These authors do not compute αR , β1 , and β3 explicitly, but their computed spin splittings at a typical Fermi wavevector shows that changes in spin splittings by a factor of 4 or so can be achieved by changing the well width from 10 to 2 unit cells; to achieve the same sort of change due to external electric fields required very strong fields of order 1 V/nm. This suggests that changing well width and electron density rather than electric field will be the most favorable route for tuning of wurtzite QWs. 107 5.4 Summary An important quantitative difference between zinc-blende and wurtzite semiconductors is the strength of the SO interaction. The wurtzites AlN, GaN, and ZnO have considerably less nuclear charge than GaAs and therefore also a much smaller SO coupling (see Table 5.8. Of prime importance is the fact that the material w-AlN possesses a much smaller SO coupling than its wurtzite relatives GaN and ZnO. This is understandable given the positive charge of Al and N to be 13 and 7 respectively; Zn and O have 30 and 8; Ga and N have 31 and 7. As has 33 protons which makes GaAs a much heavier compound though it is the lightest of the direct band gap zinc-blende semiconductors. AlN is the lightest direct band gap wurtzite material. By this simple analysis, it is not surprising that the wurtzites have considerably longer spin relaxation times when SO coupling is the main reason for spin relaxation. However wurtzite crystals offer enough flexibility such that short lifetimes can also be achieved through the existence of linear-in-k terms (αR 6= ±βD ). Recently the idea that strain may allow further tuning of the Dresselhaus Hamiltonian suggests the possibility of eliminating even the cubic-in-k term at all temperatures [156]; if possible much longer lifetimes would be achievable at room temperature. In conclusion, the findings of this dissertation assert that semiconductors possessing the wurtzite crystal structure offer enough promising features that could lead to it supplanting the current zinc-blende paradigm of semiconductor spintronics. 108 Chapter 6 Magnetic field effects The physics of spin relaxation in applied magnetic fields in the insulating regime is unclear. The recent review of Wu et al. and the references therein consider many of the quandaries [56]. These mysteries are not discussed here; instead the dynamics of the localized and itinerant spin systems are considered when a field is present. When magnetic fields are present, the different gyroscopic ratios of the localized and itinerant electrons must be included. If the magnetic moments, instead of the magnetizations, are considered, the Bloch equations are dµl nc nl γl = γl µl × B − µl + µc dt γcr γcr γc (6.1) nl nc γl dµc = γc µc × B − µc + µl , dt γcr γcr γc (6.2) where the relaxation mechanisms specific to the two environments are omitted for convenience. The difference between γc and γl is small (∼ 5%). The term γncrl µc γγcl can be written as γncrl µc γc +(γγcl −γc ) = γncrl µc 1 + (γl − γc )/γc ≈ γncrl µc . The same is done for the last term of Eq. (6.1), giving dµl nc nl = γl µl × B − µl + µc dt γcr γcr (6.3) nl nc dµc = γc µc × B − µc + µl , dt γcr γcr (6.4) where γl = gl∗ µ~B and γc = gc∗ µ~B . When no field is present, the total magnetization is constant (in the absence of relaxation mechanisms) since ṁl + ṁc = 0. With the two above equations, d(µl + µc ) = γl µl × B + γc µc × B, dt (6.5) which describes precessing moments as expected in an external field. However, unlike in a 109 single g ∗ case, µl + µc will depend on γcr because dµl /dt − dµc /dt depends explicitly on γcr . In a reference frame rotating at ω, µi × γi B → µi × (γi B − ω). (6.6) So if we take ω = (γ1 + γ2 )B/2 which is the average angular precession frequency, γl µl × B → γl − γc µl × B 2 (6.7) γl − γc µc × B. (6.8) 2 can be substituted into the above equations. In this rotating reference frame, we see that γc µc × B → − d(µl + µc ) γl − γc = (µl − µc ) × B dt 2 (6.9) and it is clear that the cross relaxation rate does not affect the total spin vector only if γl = γc . For γl 6= γc , µl and µc can both be solved for analytically in the lab frame (non-rotating). The solutions are sums of sinusoidal exponential products with exponential dependence as e−Γi t where Γi is a relaxation rate: where S= i 1 Γ1 = (nl + nc )Γcr − (γl + γc )B + S 2 2 (6.10) 1 i Γ2 = (nl + nc )Γcr − (γl + γc )B − S, 2 2 (6.11) 1p ((nc − nl )Γcr − i(γc − γl )B)2 + 4nl nc Γ2cr , 2 (6.12) −1 . and Γcr = γcr Let us try to simplify by going to the regime ni Γcr (γc − γl )B; this forces nl,c to be appreciable and we assume nl ∼ nc . By expanding S to second order in B, 1 (γc − γl )2 B 2 nl nc nc − nl i S ≈ nimp Γcr − − (γc − γl )B . 2 2 Γcr nimp nimp 2 nimp (6.13) We then obtain approximate relaxation rates: Γ2 ≈ i i nc − nl Γ1 ≈ nimp Γcr − (γl + γc )B − (γc − γl )B 2 2 nimp (6.14) i (γc − γl )2 B 2 nl nc i nc − nl − (γl + γc )B + (γc − γl )B , Γcr nimp n2imp 2 2 nimp (6.15) 110 where the imaginary part gives the precession frequency of the moment. The frequency here is ∼ (γl + γl )B/2 = ωavg. , the average frequency of the two moments. The correction frequency is small since γl − γc is small as is nc − nl . This result is sensible because the spin moment rapidly distributes itself evenly among the two similarly populated electron states; thus the total moment reacts to the field like an average of the two moments. Hence, in this regime of quick cross-relaxation and low field, we would not expect to see any beating phenomena. The real part of the above Γ1 is a very rapid rate which disappears quickly. The real part of the Γ2 is much slower. It will be largest when nl = nc ; (γc − γl )B ≈ 2 ns−1 at B = 1 T in GaAs. This rate is then R(Γ2 ) ≈ 1 nimp Γcr ns−2 . As stipulated earlier, the limit nimp Γcr (γc − γl )B is under consideration. If the cross-relaxation time is taken to be 1 ps [5], 1/R(Γ2 ) ≈ 1000 ns which is far longer than what has been measured; hence other mechanisms overshadow this effect. Even at B = 6 T, 1/R(Γ2 ) ≈ 25 ns which again is too large to match experiments. If the cross-relaxation time is longer the rate due to this mechanism will be faster and may be observable. If other mechanisms could be suppressed, this effect offers a way to experimentally determine the cross-relaxation rate nimp Γcr which has never been measured. In high fields (such that precession is quicker than cross-relaxation), the two environments act independently and two Larmor frequencies are predicted: γl B and γc B. If the cross-relaxation time is again taken to be 1 ps, then the field would have to be 100s of tesla to have large enough precession frequencies. However beating has been observed for fields > 3 T [157]; see Figure 6.1. If the cross-relaxation time is slower, ∼ 100 ps as suggested by [43], then beating may be observable in fields of a few tesla. Future research is aimed at an accurate calculation of the cross-relaxation rate. Figure 6.1: Left: Faraday rotation data from [157] showing spin beating. Right: Fourier transform of data depicting two distinct Larmor frequencies. T = 1.8 K, B = 6 T, nimp = 3 × 1016 cm−3 . 111 Chapter 7 Conclusions This dissertation studies several topics regarding electron spin relaxation in semiconductors. In chapter 4, a model that incorporates cross-relaxation between localized and itinerant electrons in a Bloch equation approach explains the temperature dependence of the spin relaxation in several bulk zinc-blende crystals. Due to the differing thermal occupations, localized and conduction spin relaxation mechanisms take affect at low and high temperatures respectively. The same idea applies to zinc-blende quantum wells though some modifications are necessary. One important factor is the excitation process - whether electrons are photo-excited into donor-bound-exciton or just exciton states. The measured spin lifetimes reflect this difference since the two spin environments are different and hence have different relaxation times. Also important is the impurity concentration and quantum well growth direction as shown for i-GaAs confined in the [110]-plane and n-GaAs confined in the [001]plane. The Bloch equation model phenomenologically describes the experiments in both systems though the operative localized and exciton spin relaxation mechanisms is not yet a settled issue. Chapter 5 focuses on applying some of the mechanisms described in chapter 3 to the wurtzite crystal. Previous research has been in zinc-blende crystals only. The work in this dissertation finds that the D’yakonov-Perel’ mechanism is qualitatively and quantitatively different in wurtzite crystals. The phenomenological model utilized in chapter 4 is used to describe the temperature dependent spin relaxation in bulk n-ZnO. Due to the more complex impurity profile in ZnO, a three spin environments are required: itinerant, shallow donor, and deep donors. This work in ZnO demonstrates that this method can be used as a characterization tool for semiconductor samples, yielding information on impurity concentrations and donor binding energies. The chapter concludes by examining the D’yakonov-Perel’ mechanism in wurtzite quantum wells and showing much longer lifetimes are achievable and therefore may be of use in future semiconductor spintronic devices. Chapter 6 explores certain anomalous effects seen in the magnetic field dependence in the spin relaxation rates. One such effect is the ‘spin beat’ phenomenon seen in some 112 optical orientation experiments. The theory of two interacting subsystems (itinerant and localized) suggest that the two different g ∗ ’s of the electron species cause the two interfering frequencies observed in the time-resolved experiments. Also the modified Bloch equations result in a new relaxation mechanism when an external field is present. 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Elsevier North Holland, New York, 1981. 124 Appendix A Material parameters zb-CdTe aB = 5.3 nm (Iodine donor) εB = 12.4 meV (Iodine donor) β1 = 0 meV nm β3 = 8.5 meV nm3 Eg = 1.606 eV m∗ = 0.11me zb-GaAs aB = 10.4 nm (Silicon donor) εB = 5.8 meV (Silicon donor) β1 = 0 meV nm β3 = 6.5 − 30 meV nm3 ∆0 = 340 meV Eg = 1.51 eV m∗ = 0.067me zb-ZnSe aB = 4.6 nm (Chlorine donor) εB = 14 meV (Chlorine donor) β1 = 0 meV nm β3 = 1.3 meV nm3 Eg = 2.82 eV 125 m∗ = 0.13me w-AlN β1 = 0.09 meV nm β3 = −0.01 meV nm3 ∆0 = 19 meV Eg = 6.1 eV m∗ = 0.4me w-GaN β1 = 0.9 meV nm β3 = −0.32 meV nm3 ∆0 = 13 meV Eg = 3.2 eV m∗ = 0.2me w-ZnO β1 = 0.11 meV nm β3 = −0.33 meV nm3 ∆0 = −3.5 meV Eg = 3.3 eV m∗ = 0.2me 126 Appendix B An integral involving spherical harmonics Spherical harmonics have several different definitions depending on how authors choose to deal with the normalization factor and orthogonality relation. We follow the definition of Jackson [69]: s Yln (θ, φ) = (2l1 )(l − n)! n P (cos θ)einφ . 4π(l + n)! l (B.1) Pln (cos θ) are associated Legendre polynomials that can be determined for positive and negative n from Rodrigues’ formula: Pln (x) = l+n (−1)n 2 n/2 d (1 − x ) (x2 − 1)l . 2l l! dxl+n (B.2) From this recursive formula, it is simple to show Pl−n (x) = (−1)n (l − n)! n P (x). (l + n)! l (B.3) It is tempting to think to use the standard orthogonality relation of spherical harmonics, R 0 dΩYln0 (θ, φ)Yln∗ (θ, φ) = 1, to compute the angular average in Eq. (3.35). However the equation actually reads 0 Yln0 (θ, φ)Yln (θ, φ) Z = dΩ n0 Y 0 (θ, φ)Yln (θ, φ). 4π l However by using the above definitions, it can be shown that Z dΩ n0 (−1)n Yl0 (θ, φ)Yln (θ, φ) = δn0 ,−n δl0 ,l . 4π 4π R 2π i(n0 +n)ϕ = 1. For the two dimensional situation, ein0 ϕ einϕ = 0 dϕ 2π e 127 (B.4) (B.5) Appendix C Important integrals Often we would like to compute the following quantity: R∞ n ∂f0 0 g(ε)τ1 ε ∂ε dε = Iˆ(d) [τ1 εn ]. R∞ ∂f0 g(ε) dε ∂ε 0 (C.1) where f0 is the Fermi-Dirac function and g(ε) is the density of states. To do so, other quantities such as Iˆ(d) [εn ] need to be determined. First we look at the top integral in Eq. (C.1). Constant factors in the density of states will be discarded since they will eventually cancel with the denominator. So we are left with ∞ Z εd εn+ν s1 0 e−βε+βµ dε, (1 + e−βε+βµ )2 (C.2) where d is 1/2 for three dimensions and 0 for two dimensions. By defining x = βε and z = exp(βµ), Z s1 β n+ν+d+1 ∞ xd xn+ν 0 ze−x dx, (1 + ze−x )2 (C.3) which is the final integral we want to solve. The integral can also be written as [67] R∞ s1 xd xn+ν dx. It is not a simple integral so we look to expand the inteβ n+ν+d+1 0 4 cosh2 ((x−ln z)/2) grand in a series of z [90]. This yields ∞ Z s1 β n+ν+d+1 d n+ν x x 0 ∞ X (j + 1)z j+1 e−x(j+1) dx. (C.4) j=0 When integrating term by term the following series is obtained: − s1 β (n + ν + d)! n+ν+d+1 ∞ X (−z)j . j n+ν+d (C.5) j=1 The series is defined as the polylogarithm function of order n + ν + d, Lin+ν+d (−z) [158]. 128 We define the dimensionless integral as In+ν+d (βµ) = −(n + ν + d)!Lin+ν+d (−z) so Z 0 ∞ εd εn+ν e−βε+βµ s1 s1 dε = − n+ν+d+1 (n + ν + d)!Lin+ν+d (−z) = n+ν+d+1 In+ν+d (βµ), −βε+βµ 2 (1 + e ) β β (C.6) where z = exp(βµ). The bottom integral of Eq. (C.1) is now trivial to solve: Z 0 ∞ ∂f0 g(ε) dε = ∂ε Z 0 So the quantity in Eq. (C.1) is finally R∞ n ∂f0 0 g(ε)τ1 ε ∂ε dε = Iˆ(d) [τ1 εn ] = R∞ ∂f0 g(ε) dε ∂ε 0 ∞ εd e−βε+βµ Id (βµ) dε = d+1 . −βε+βµ 2 (1 + e ) β s1 I (βµ) β n+ν+d+1 n+ν+d Id (βµ) β d+1 = s1 n+ν β In+ν+d (βµ) Id (βµ) (C.7) (C.8) where the s1 is yet to be determined. As shown in Eq. (3.55) in the main text, s1 = τtr Iˆ(d) [ε] Iˆ(d) [εν+1 ] , (C.9) so we need to detemined the result of the integral operator acting on the energy to an arbitrary power. This has actually been done already in the above analysis so we just show the final result: Id+i (βµ) Iˆ(d) [εi ] = β −i . Id (βµ) (C.10) s1 is then simply Id+1 (βµ) Id (βµ) τtr I (βµ) β −(ν+1) d+ν+1 Id (βµ) β −1 s1 = = τtr β ν Id+1 (βµ) . Id+ν+1 (βµ) (C.11) We are now in a position to express the temperature dependence of our desired quantity, Id+1 (βµ) In+ν+d (βµ) Iˆ(d) [τ1 εn ] = τtr β −n . Id (βµ) Id+ν+1 (βµ) 129 (C.12) Appendix D The polylogarithm function The polylogarithm is a special function defined by the series ∞ X zk Lin (z) = kn (D.1) k=1 with |z| ≤ 1. By analytic continuation, the polylogarithm can be defined over a larger range of z. An important result, known as the inversion equation, is[158] bnc 2 X lnn−2k (z) 1 Li2k (−1), Lin (−z) + (−1)n Lin (−1/z) = − lnn (z) + 2 n! (n − 2k)! (D.2) k=1 whereb n2 c is the greatest integer contained in n/2. For the purposes of this article, we use the inversion equation in the quasi-degenerate regime and obtain n Lin (−eTF /T ) ≈ −(TF /T )n /n! + 2 b2c X (TF /T )n−2k k=1 (n − 2k)! Li2k (−1). (D.3) To obtain this result we have used TF /T 1 such that Lin (1 − eTF /T ) ≈ Lin (−eTF /T ). Also, Lin (−e−TF /T ) is small and therefore neglected. Note that the first term on the right hand side is what we would obtain at T = 0. We ascertain These results are equivalent to the Sommerfield expansion. 130 n 0 1 2 3 4 5 Lin (−1) −π 2 /12 −7π 2 /720 Lin (1 − eTF /T ) −1 −TF /T 2 − 12 ( TTF )2 − π6 2 − 16 ( TTF )3 − π6 TTF 2 2 1 TF 4 ( T ) − π12 ( TTF )2 − 7π − 24 360 2 2 TF 1 TF 5 ( T ) − π36 ( TTF )3 − 7π − 120 360 T In (βµ0 ) 1 TF /T 2 ( TTF )2 + π3 ( TTF )3 + π 2 TTF 2 TF 4 ( T ) + 2π 2 ( TTF )2 + 7π 15 2 2 TF 3 7π TF ( TTF )5 + 10π 3 ( T ) + 3 T Table D.1: Using ν = 0. ζ(F ) = 2m∗ kB T(F ) /~2 131