Spin Dynamics Of Two-Dimensional Electrons In Semiconductors,Graphene, Topological Insulators,and Multiferroic Oxides

Spintronics refers to a technology exploiting the spin degree of freedom instead of or in addition to the charge degree of freedom of electrons, especially in solid-state systems. Its aim is to offer opportunities for a new generation of devices combining standard mi-croelectronics with spin-dependent effects that arise from the interaction between electron spin and the optical, electrical, or magnetic properties of the materials or external fields. To achieve this object, it is quite essential to understand the spin dynamics, including the spin relaxation and transport, of different electron systems in various hosts. For this rea-son, the two-dimensional electron system, especially in the semiconductor quantum wells or heterostructurs, has been extensively studied in the past decades. Recently, partly due to the two dimensionality, the single-layer or bilayer graphene, the surface of topological in-sulators, and also the interface of insulating oxides such as LaAlO3/SiTiO3have attracted much attention. This dissertation focuses on the theoretical study on spin relaxation and transport of the two-dimensional electron system in semiconductors, graphene, topological insulators and multiferroic oxides. It is organized as follows.In the introduciton (Chapter1), we first briefly review the background of spintronics. The key factors in realizing spintronic devices, such as the spin generataion and detection, are reviewed. We then summarize the main spin relaxation mechanisms in the time domain, including the D’yakonov-Perel’, Elliot-Yafet and Bir-Aronov-Pikus mechanisms as well as the spin-flip scattering due to the randomness of the spin-orbit coupling. We also specify the mechansim for spin relaxation during transport, i.e., in the spatial domain.In Chapter2, we introduce the materials and systems studied in this dissertation. The band structure and effective Hamiltonian are given for the usual zincblende group III-V semiconductors such as GaAs. The single-layer grahene, which is deemed to be strictly two dimensional, is also introduced, with the effective Hamiltonian for the massless Dirac Fermions at high symmetry points given from a viewpoint of tight binding model. We then turn to the topological insulators, e.g., the representative two dimensional HgTe quantum wells and bulk Bi2Se3. The metallic and helical edge or surface states in the topological insulators are presented, with the effective Hamiltonian introduced from the k· p and invariant methods. After that, we briefly introduce the multiferroic materials. In Chapter3, we review the status of research on spin dynamics in graphene.In Chapter4, we present at first the kinetic spin Bloch equations, based on which all the studies on spin dynamics included in this dissertation are performed. Then starting from these equations, we explain the spin relaxation mechanisms in both time and spatial domains in detail, based on a microscopic viewpoint. From Chapter5to10, we present our studies on spin dynamics in two dimensional electron systems formed in the semiconduc-tors, the single-layer graphene, the surface of topological insulators and the multiferroic oxide interface, respectively.We first study the spin relaxation in semiconductor quantum wells in Chapter5.The carrier density dependence of electron spin relaxation in an intrinsic (001) GaAs quantum well at room temperature is investigated both experimentally and theoretically. The experimental data are from the time-resolved circularly polarized pump-probe spec-troscopy, and indicate that the spin relaxation time first increases and then slightly de-creases with the increase of electron (hole) density. Our fully microscopic calculation with both the D’yakonov-Perel’and the Bir-Aronov-Pikus mechanisms included reproduces the observed phenomenon very well. It is revealed that the spin relaxation time first increases with density in the relatively low density regime as the linear Dresselhaus spin-orbit cou-pling terms are dominant, and then tends to decrease when the density is large as the cubic Dresselhaus spin-orbit coupling terms become important.We then study the multi-valley spin relaxation in n-type (001) GaAs quantum wells with an in-plane electric field at high temperature. We demonstrate that L valleys play the role of a "drain" of the total spin polarization due to the large spin-orbit coupling there and the strong T-L inter-valley scattering. With the increase of the electric field, the spin relaxation time first increases due to the hot-electron effect and then decreases due to both the enhanced inhomogeneous broadening in T valley and the increase in occupation of electrons in higher L valleys where the spin relaxation takes place fast.Apart from electrons, we also investigate the hole spin relaxation in (001) strained asymmetric Si/Sio.7Geo.3(Ge/Si0.3Ge0.7) quantum wells under gate voltage. The effective Hamiltonian, including the Rashba spin-orbit coupling, of the lowest hole subband is ob-tained by the subband Lowdin perturbation method starting from the six-band Luttinger k· p model. It is found that the lowest hole subband in Si/SiGe (Ge/SiGe) quantum wells is light (heavy)-hole like. The spin relaxation time is calculated to be of the or-der of1～100ps (0.1～10ps) in Si/SiGe (Ge/SiGe) quantum wells, for the temperatures, carrier/impurity densities and gate voltages of our consideration. Our study reveals that the hole-phonon scattering is very weak, making the hole-hole Coulomb scattering become very important in the impurity-free samplesx. With the change of temperature, the hole system in Si/SiGe quantum wells is generally in the strong scattering limit, while that in Ge/SiGe quantum wells can be in either the strong or weak scattering limit. In the absence of impurities, the Coulomb scattering leads to a peak in the temperature dependences of spin relaxation time in both the Si/SiGe and Ge/SiGe quantum wells, located around the crossover from the degenerate to nondegenerate regimes. Besides, the Coulomb scattering also leads to a valley in the temperature dependence of spin relaxation time in Ge/SiGe quantum wells, around the crossover from the weak to strong scattering limit.In the above studies, phonons are assumed to be equilibrium phonons, following the widely adopted approximation in the literature. However, in fact, when the carriers are far away from the equilibrium, phonons can be driven away from the equilibrium by carriers as well and in turn affect the electron dynamics. To look into this effect, we perform a study on hot-electron spin relaxation in n-type (001) GaAs quantum wells under the electric field, with the longitudinal optical phonons considered to be nonequilibrium. It is found that when the phonons are treated as the nonequilibrium rather than the equilibrium ones, the spin relaxation time is increased since the electron heating is enhanced and hence the electron-phonon scattering is strengthened. However, the frequency of spin precession, which is roughly proportional to the electron drift velocity, can be either increased or decreased, depending on the electric field strength and/or the lattice temperature.We then go on to study spin transport in semiconductor quantum wells in Chap-ter6. The spin transport in quantum wells, in the presence of the Dresselhaus and/or Rashba spin-orbit coupling and hence the D’yakonov-Perel’spin relaxation mechanism, has already been investigated in group III-V semiconductors. Here we carry out the study in symmetric Si/SiGe quantum wells where the D’yakonov-Perel’spin relaxation mecha-nism is absent but with a static magnetic field in the Voigt configuration. Through this study, we emphasize that even without the momentum dependent effective magnetic field from the spin-orbit coupling, the static magnetic field alone can still cause inhomogeneous broadening in spin precessions in the spatial domain. This inhomogeneous broadening together with the scattering leads to an irreversible spin relaxation along with the spin transport. This mechanism exactly applies to the experiment on spin transport in bulk Si with a magnetic field by Appelbaum et al.[Nature447,295(2007)].In Chapter7, we turn to the single-layer graphene where the electrons are mass-less Dirac Fermions with linear dispersion. The dominant spin relaxation mechanism in graphene is under debate and our study aims to understand the main spin relaxation mechanism there. As the Rashba spin-orbit coupling induced by the gate voltage and/or curvature leads to a spin relaxation time about three orders larger than the experimen-tal measurement (-100-1000ps), we take into account the effect of adatoms, which can enhance the Rashba spin-orbit coupling locally and substantially. Besides, the adatoms also serve as Coulomb potential scatterers. Due to the random distribution of adatoms, the Rashba field is actually fluctuating. The randomness of the Rashba field causes spin relaxation by spin-flip scattering, manifesting itself as an Elliott-Yafet-like mechanism. In our study, both the D’yakonov-Perel’and the Elliott-Yafet-like mechanisms are consid-ered. By fitting and comparing the experiments from the Groningen group [Jozsa et al, Phys. Rev. B80,241403(R)(2009)] and Riverside group [Pi et al., Phys. Rev. Lett.104,187201(2010); Han and Kawakami, ibid.107,047207(2011)] which show either D’yakonov-Perel’-(with the spin relaxation rate being inversely proportional to the mo-mentum scattering rate) or Elliott-Yafet-like (with the spin relaxation rate being propor-tional to the momentum scattering rate) properties, we suggest that the D’yakonov-Perel’ mechanism dominates the spin relaxation in graphene. The later experimental finding of a nonmonotonic dependence of spin relaxation time on diffusion coefficient by Jo et al.[Phys. Rev. B84,075453(2011)] is also well reproduced by our model. At the end of this study, we also introduce the newest experimental progress on spin relaxation in graphene and rediscuss the possibly dominant spin relaxation. After that, we study spin relaxation in low-mobility rippled graphene. In this structure, the ripples does not only lead to Rashba spin-orbit coupling, but also induces a Zeeman-like spin-orbit coupling with opposite effective magnetic fields in two valleys. The joint effect of this Zeeman-like spin-orbit coupling and the intervalley electron-optical phonon scattering opens a spin re-laxation channel, which manifests itself in low-mobility samples with the electron mean free path being smaller than the ripple size. This spin relaxation channel contributes to spin relaxation effectively around room temperature, leading to a spin relaxation time around100ps.In Chapter8the spin transport in graphene is studied in the framework of the D’yakonov-Perel’ mechanism with the Rashba spin-orbit coupling enhanced by the fluc-tuating substrate as well as the adatoms. By fitting the Au doping dependence of spin relaxation from Pi et al.[Phys. Rev. Lett.104,187201(2010)], the Rashba spin-orbit coupling coefficient is found to increase approximately linearly from0.15to0.23meV with the increase of Au density. With this strong spin-orbit coupling, the spin diffusion or transport length is comparable with the experimental values (～μm). We find that in the strong scattering limit (dominated by the electron-impurity scattering), the spin diffusion is uniquely determined by the Rashba spin-orbit coupling strength and insensitive to the temperature, electron density as well as scattering. However, with the presence of an elec-tric field along the spin injection direction, the spin transport length can be modulated by either the electric field or the electron density. It is also shown that the spin diffusion and transport show an anisotropy with respect to the polarization direction of injected spins. This anisotropy differs from the one given by the simple two-component drift-diffusion model.After studying the spin dynamics of massless Dirac Fermions in single-layer graphene, we proceed with the hot-carrier transport and spin relaxation on the surface of topological insulators in Chapter9. We investigate the charge and spin transport under high elec-tric field (up to several kV/cm) on the surface of topological insulator Bi2Se3, where the electron-surface optical phonon scattering dominates except at very low temperature. Due to the spin mixing of conduction and valence bands, the electric field not only accelerates electrons in each band, but also leads to inter-band precession. In the presence of the electric field, electrons can transfer from the valence band to the conduction one via the inter-band precession and inter-band electron-phonon scattering. Besides, we find that due to the spin-momentum locking, a transverse spin polarization, with the magnitude proportional to the momentum scattering time, is induced by the electric field. Our inves-tigation also reveals that due to the large relative static dielectric constant, the Coulomb scattering is too weak to establish a drifted Fermi distribution with a unified hot-electron temperature in the steady state under the electric field. After turning off the electric field in the steady state, the hot carriers cool down in a time scale of energy relaxation which is very long (of the order of100-1000ps) while the spin polarization relaxes in a time scale of momentum scattering which is quite short (of the order of0.01-0.1ps).The dissertation is closed with the study on spin diffusion at the interface of multi-ferroic oxides, i.e., in the two-dimensional electron gas at the interface of LaAlO3/SrTiO3grown on multiferroic TbMnO3, at15K (Chapter10). The spiral magnetic moments of Mn3+in TbMnO3couple with the diffusing spins at the LaAlO3/SrTiO3interface via the Heisenberg exchange interaction. Our study demonstrates that due to this Heisenberg exchange interaction, the spin diffusion length is always finite, despite the polarization direction of the injected spins. This result corrects the claim by Jia and Berakdar [Phys. Rev. B80,014432(2009)] that there is a persistent spin current at the interface when the injected spin’s are polarized perpendicular to the spiral plane of the magnetic moments of Mn3+in TbMnO3.At last we summarize in Chapter11.