Kinetic Spin Bloch Equation Research Articles

Valley depolarization dynamics and valley Hall effect of excitons in monolayer and bilayerMoS2

We investigate the valley depolarization dynamics and valley Hall effect of exciton due to the electron-hole exchange interaction in mono- and bilayer MoS$_2$ by solving the kinetic spin Bloch equations. The effect of the exciton energy spectra by the electron-hole exchange interaction is explicitly considered. For the valley depolarization dynamics, in the monolayer MoS$_2$, it is found that in the strong scattering regime, the conventional motional narrowing picture is no longer valid, and a novel valley depolarization channel is opened. For the valley Hall effect of exciton, in both the mono- and bilayer MoS$_2$, with the exciton equally pumped in the K and K' valleys, the system can evolve into the equilibrium state where the valley polarization is parallel to the effective magnetic field due to the exchange interaction. With the drift of this equilibrium state by applied uniaxial strain, the exchange interaction can induce the {\it momentum-dependent} valley/photoluminesence polarization, which leads to the valley/photoluminesence Hall current. Specifically, the disorder strength dependence of the valley Hall conductivity is revealed. In the strong scattering regime, the valley Hall conductivity decreases with the increase of the disorder strength; whereas in the weak scattering regime, it saturates to a constant, which can be much larger than the one in Fermi system due to the absence of the Pauli blocking.

Spin diffusion in ultracold spin-orbit-coupledK40gas

We investigate the steady-state spin diffusion for ultracold spin-orbit coupled $^{40}$K gas by the kinetic spin Bloch equation approach both analytically and numerically. Four configurations, i.e., the spin diffusions along two specific directions with the spin polarization perpendicular (transverse configuration) and parallel (longitudinal configuration) to the effective Zeeman field are studied. It is found that the behaviors of the steady-state spin diffusion for the four configurations are very different, which are determined by three characteristic lengths: the mean free path $l_{\tau}$, the Zeeman oscillation length $l_{\Omega}$ and the spin-orbit coupling oscillation length $l_{\alpha}$. It is analytically revealed and numerically confirmed that by tuning the scattering strength, the system can be divided into {\it five} regimes: I, weak scattering regime ($l_{\tau}\gtrsim l_{\Omega}, l_{\alpha}$); II, Zeeman field-dominated moderate scattering regime ($l_{\Omega}\ll l_{\tau}\ll l_{\alpha}$); III, spin-orbit coupling-dominated moderate scattering regime ($l_{\alpha}\ll l_{\tau}\ll l_{\Omega}$); IV, relatively strong scattering regime ($l_{\tau}^c\ll l_{\tau}\ll l_{\Omega}, l_{\alpha}$); V, strong scattering regime ($l_{\tau}\ll l_{\Omega}, l_{\alpha},l_{\tau}^c$), with $l_{\tau}^c$ representing the crossover length between the relatively strong and strong scattering regimes. In different regimes, the behaviors of the spacial evolution of the steady-state spin polarization are very rich, showing different dependencies on the scattering strength, Zeeman field and spin-orbit coupling strength. The rich behaviors of the spin diffusions in different regimes are hard to be understood in the framework of the simple drift-diffusion model or the direct inhomogeneous broadening picture in the literature. ...

Hot-electron effect in spin relaxation of electrically injected electrons in intrinsic Germanium

The hot-electron effect in the spin relaxation of electrically injected electrons in intrinsic germanium is investigated by the kinetic spin Bloch equations both analytically and numerically. It is shown that in the weak-electric-field regime with E ≲ 0.5 kV cm−1, our calculations have reasonable agreement with the recent transport experiment in the hot-electron spin-injection configuration (2013 Phys. Rev. Lett. 111 257204). We reveal that the spin relaxation is significantly enhanced at low temperature in the presence of weak electric field E ≲ 50 V cm−1, which originates from the obvious center-of-mass drift effect due to the weak electron–phonon interaction, whereas the hot-electron effect is demonstrated to be less important. This can explain the discrepancy between the experimental observation and the previous theoretical calculation (2012 Phys. Rev. B 86 085202), which deviates from the experimental results by about two orders of magnitude at low temperature. It is further shown that in the strong-electric-field regime with 0.5 ≲ E ≲ 2 kV cm−1, the spin relaxation is enhanced due to the hot-electron effect, whereas the drift effect is demonstrated to be marginal. Finally, we find that when 1.4 ≲ E ≲ 2 kV cm−1 which lies in the strong-electric-field regime, a small fraction of electrons (≲5%) can be driven from the L to Γ valley, and the spin relaxation rates are the same for the Γ and L valleys in the intrinsic sample without impurity. With the negligible influence of the spin dynamics in the Γ valley to the whole system, the spin dynamics in the L valley can be measured from the Γ valley by the standard direct optical transition method.

Theory of optical excitation spectra and depolarization dynamics in bilayerWS2from the viewpoint of excimers

We investigate the optical excitation spectra and the photoluminescence depolarization dynamics in bilayer WS$_2$. A different understanding of the optical excitation spectra in the recent photoluminescence experimentby Zhu {\em et al.} [arXiv:1403.6224] in bilayer WS$_2$ is proposed. In the experiment, four excitations (1.68, 1.93, 1.99 and 2.37 eV) are observed and identified to be indirect exciton for the $\Gamma$ valley, trion, A exciton and B exciton excitations, respectively, with the redshift for the A exciton energy measured to be 30$\sim$50 meV when the sample synthesized from monolayer to bilayer. According to our study, by considering there exist both the intra-layer and charge-transfer excitons in the bilayer WS$_2$, with inter-layer hopping of the hole, there exists excimer state composed by the superposition of the intra-layer and charge-transfer exciton states. Accordingly, we show that the four optical excitations in the bilayer WS$_2$ are the A charge-transfer exciton, ${\rm A}'$ excimer, ${\rm B}'$ excimer and B intra-layer exciton states, respectively, with the calculated resonance energies showing good agreement with the experiment. In our picture, the speculated indirect exciton, which involves a high-order phonon absorption/emission process, is not necessary. Furthermore, the binding energy for the excimer state is calculated to be 40 meV, providing reasonable explanation for the experimentally observed energy redshift of the A exciton. Based on the excimer states, we further derive the exchange interaction Hamiltonian. Then the photoluminescence depolarization dynamics due to the electron-hole exchange interaction is studied in the pump-probe setup by the kinetic spin Bloch equations. We find that ......

Electron spin diffusion in monolayerMoS2

Electron spin diffusion is investigated in monolayer ${\mathrm{MoS}}_{2}$ in the absence of external electric and magnetic fields. The electron-impurity scattering, which is shown to play a negligible role in spin relaxation in time domain in this material, has a marked effect on the in-plane spin diffusion due to the anisotropic spin precession frequency in the spatial domain. With the electron-impurity and intervalley electron-phonon scatterings separately included in the scattering term, we study the intra- and intervalley diffusion processes of the in-plane spins by analytically solving the kinetic spin Bloch equations. The intravalley process is found to be dominant in the in-plane spin diffusion, in contrast to the case of spin relaxation in time domain, where the intervalley process can be comparable to or even more important than the intravalley one. For the intravalley process, we find that the in-plane spin diffusion is suppressed with the increase of impurity density but effectively enhanced by increasing electron density in both the degenerate and nondegenerate limits. We also take into account the electron-electron Coulomb scattering in the intravalley process. Interestingly, we find that in the nondegenerate limit, the intravalley spin-diffusion length presents an opposite trend in the electron-density dependence compared to the one with only electron-impurity scattering.

Ultrafast electron spin dynamics in bulk CdTe investigated by femtosecond pump-probe reflection spectroscopy

The transient spin polarization dynamics in bulk cadmium telluride (CdTe) at 70 K is investigated by the circularly polarized pump-probe reflection technique. A general expression is derived from the rate equations of a two-level system with small signal approximation to describe the light-helicity-dependent reflection spectrum. The initial degree of electron spin polarization in the excited state and the electron spin relaxation time in bulk CdTe at low temperature with different carrier density are analyzed according to this model. Our experimental results reveal that the Dʼyakonov–Perelʼ mechanism based on a fully microscopic kinetic spin Bloch equation (microscopic KSBE) approach dominates in the electron spin relaxation process in bulk CdTe crystal. • The transient spin polarization dynamics in bulk CdTe at 70 K is investigated. • A general expression is derived from the rate equations of a two-level system. • P s 0 and τ s in bulk CdTe are analyzed according to this model. • The DP mechanism based on KSBE dominates the electron spin relaxation process.

Electron spin dynamics study of bulk p-GaAs: The screening effect

In this work, the electron spin dynamics of bulk p-GaAs doped with Be grown by molecular beam epitaxy is investigated by using the time-resolved magneto-optical Kerr rotation technique. The spin relaxation/dephasing times T1 and T2* are systematically investigated as a function of hole doping density and photo-excitation density as well as temperature. The complex hole doping density dependence of spin relaxation times T1 and T2* is observed experimentally, which agrees well with predictions of the kinetic spin Bloch equation theory published previously [J. Jiang and M. Wu, Phys. Rev. B 79, 125206 (2009)]. D'yakonov-Perel's mechanism [M. Dyakonov and V. Perel, Sov. Phys. Solid State 13, 3023 (1972)] is discussed to dominate the electron spin relaxation process in p-GaAs, with the effect of hole screening proven to play an important role.

Microscopic theory for Doppler velocimetry of spin propagation in semiconductor quantum wells

We provide a microscopic theory for the Doppler velocimetry of spin propagation in the presence of spatial inhomogeneity, driving electric field, and the spin orbit coupling in semiconductor quantum wells in a wide range of temperature regime based on the kinetic spin Bloch equation. It is analytically shown that under an applied electric field, the spin density wave gains a time-dependent phase shift $\ensuremath{\phi}(t)$. Without the spin-orbit coupling, the phase shift increases linearly with time and is equivalent to a normal Doppler shift in optical measurements. Due to the joint effect of spin-orbit coupling and the applied electric field, the phase shift behaves differently at the early and the later stages. At the early stage, the phase shifts are the same with or without the spin-orbit coupling. While at the later stage, the phase shift deviates from the normal Doppler one when the spin-orbit coupling is present. The crossover time from the early normal Doppler behavior to the anomalous one at the later stage is inversely proportional to the spin diffusion coefficient, wave vector of the spin density wave, and the spin-orbit coupling strength. In the high temperature regime, the crossover time becomes large as a result of the decreased spin diffusion coefficient. The analytical results capture all the quantitative features of the experimental results, while the full numerical calculations agree quantitatively well with the experimental data obtained from the Doppler velocimetry of spin propagation [Nat. Phys. 8, 153 (2012).]. We further predict that the coherent spin precession, originally thought to be broken down at high temperature, is robust up to the room temperature for narrow quantum wells. We point out that one has to carry out the experiments longer to see the effect of the coherent spin precession at higher temperature due to the larger crossover time.

Electron spin relaxation in GaAs1−xBix: Effects of spin-orbit tuning by Bi incorporation

The electron spin relaxation in n-type and intrinsic GaAs1−xBix with Bi composition 0≤x≤0.1 is investigated from the microscopic kinetic spin Bloch equation approach. The incorporation of Bi is shown to markedly decrease the spin relaxation time as a consequence of the modification of the spin-orbit interaction, which shows the feasibility of GaAs1−xBix in the applications of spintronic devices. We demonstrate that the density and temperature dependences of spin relaxation time in GaAs1−xBix resemble the ones in GaAs. Meanwhile, the Bir-Aronov-Pikus mechanism is found to be negligible compared to the D'yakonov-Perel' mechanism in intrinsic sample. Due to the absence of direct measurement of the electron effective mass in the whole compositional range under investigation, we further explore the effect of a possible variation of electron effective mass on the electron spin relaxation.

Hole spin relaxation inp-type (111) GaAs quantum wells

Hole spin relaxation in $p$-type (111) GaAs quantum wells is investigated in the case with only the lowest hole subband, which is heavy-hole like in (111) GaAs/AlAs and light-hole like in (111) GaAs/InP quantum wells, being relevant. The subband L\"{o}wdin perturbation method is applied to obtain the effective Hamiltonian including the Dresselhaus and Rashba spin-orbit couplings. Under a proper gate voltage, the total in-plane effective magnetic field in (111) GaAs/AlAs quantum wells can be strongly suppressed in the whole momentum space, while the one in (111) GaAs/InP quantum wells can be suppressed only on a special momentum circle. The hole spin relaxation due to the D'yakonov-Perel' and Elliott-Yafet mechanisms is calculated by means of the fully microscopic kinetic spin Bloch equation approach with all the relevant scatterings explicitly included. For (111) GaAs/AlAs quantum wells, extremely long heavy-hole spin relaxation time (upto hundreds of nanoseconds) is predicted. In addition, we predict a pronounced peak in the gate-voltage dependence of the heavy-hole spin relaxation time due to the D'yakonov-Perel' mechanism. This peak origins from the suppression of the unique inhomogeneous broadening in (111) GaAs/AlAs quantum wells. Moreover, the Elliott-Yafet mechanism influences the spin relaxation only around the peak area due to the small spin mixing between the heavy and light holes in quantum wells with small well width. We also show the anisotropy of the spin relaxation. In (111) GaAs/InP quantum wells, a mild peak, similar to the case for electrons in (111) GaAs quantum wells, is also predicted in the gate-voltage dependence of the light-hole spin relaxation time. The contribution of the Elliott-Yafet mechanism is always negligible in this case.

Electron spin rephasing in n-type (0 0 1) GaAs quantum wells

We investigate the electron spin relaxation in the presence of spin phase recovered by a serried π-pulse sequence in n-type (0 0 1) GaAs quantum wells in a wide range of temperature and density regimes. Our numerical calculation is based on the kinetic spin Bloch equation approach with all the relevant scatterings explicitly included. We find that the rephasing pulse sequence with a long inter-pulse spacing only has a marginal influence on the spin lifetime in both the strong and weak scattering limits. We show that the spin lifetime can be significantly increased by reducing the inter-pulse spacing. More interestingly, we show that the temperature and density dependences of the spin lifetime in the case of short inter-pulse spacing coincide with those of the momentum scattering time in the low temperature regime, where nonmonotonic behaviors can appear. The origin of this feature is that the scattering under the quick rephasing manipulation mainly performs as the source of the relaxation channel instead of the key to suppress the inhomogeneous broadening. The contributions of the relevant scattering mechanisms are also discussed.

Electron spin relaxation in graphene with random Rashba field: comparison of the D'yakonov–Perel' and Elliott–Yafet-like mechanisms

To understand the main spin relaxation mechanism in graphene, we investigate the spin relaxation with random Rashba field (RRF) induced by both adatoms and a substrate using the kinetic spin Bloch equation approach. The charged adatoms, on the one hand, enhance the Rashba spin–orbit coupling locally and, on the other hand, serve as Coulomb potential scatterers. Both effects contribute to spin relaxation limited by the D'yakonov–Perel' (DP) mechanism. In addition, the RRF also causes spin relaxation by spin-flip scattering, manifesting itself as an Elliott–Yafet (EY)-like mechanism. Both mechanisms are sensitive to the correlation length of the RRF, which may be affected by environmental parameters such as electron density and temperature. Fitting and comparing the experiments of the Groningen group (Józsa et al 2009 Phys. Rev. B 80 241403(R)) and the Riverside group (Pi et al 2010 Phys. Rev. Lett.104 187201; Han and Kawakami 2011 Phys. Rev. Lett.107 047207), which show either DP (with the spin relaxation rate being inversely proportional to the momentum scattering rate) or EY-like (with the spin relaxation rate being proportional to the momentum scattering rate) properties, we suggest that the DP mechanism dominates the spin relaxation in graphene. The latest experimental finding of a nonmonotonic dependence of spin relaxation time on diffusion coefficient by Jo et al 2011 (Phys. Rev. B 84 075453) is also well reproduced by our model.

Multivalley spin relaxation inn-type bulk GaAs in the presence of high electric fields

Multivalley spin relaxation in $n$-type bulk GaAs in the presence of high electric field is investigated from the microscopic kinetic spin Bloch equation approach with the $\ensuremath{\Gamma}$ and $L$ valleys included. We show that the spin relaxation time decreases monotonically with the electric field, which differs from the two-dimensional case and is recognized due to to the cubic form of the Dresselhaus spin-orbit coupling of the $\ensuremath{\Gamma}$ valley in bulk. In addition to the direct modulation of the spin relaxation time, the electric field also strongly influences the density and temperature dependences of the spin relaxation. In contrast to the monotonic decrease with increasing lattice temperature in the field-free condition, the spin relaxation time is shown to decrease more slowly under the influence of the electric field and even to increase monotonically in the case with small electron density and high electric field. We even predict a peak in weakly doped samples under moderate electric field due to the anomalous lattice-temperature dependence of the hot-electron temperature. As for the $L$ valleys, we show that instead of playing the role of a ``drain''of the total spin polarization as in quantum-well systems, in bulk they serve as a ``momentum damping area'' that prevents electrons from drifting to higher momentum states in the $\ensuremath{\Gamma}$ valley. This tends to suppress the inhomogeneous broadening and hence leads to an increase of the spin relaxation time.

Electron spin diffusion at the interface of multiferroic oxides

We study the spin diffusion in a two-dimensional electron gas at the interface of oxide heterostructure LaAlO$_3$/SrTiO$_3$ grown on multiferroic TbMnO$_3$ at 15 K by means of the kinetic spin Bloch equation approach. The spiral magnetic moments of Mn$^{3+}$ in TbMnO$_3$ interact with the diffusing spins at the LaAlO$_3$/SrTiO$_3$ interface via the Heisenberg exchange interaction. It is demonstrated that the spin diffusion length is always finite, despite the polarization direction of the injected spins. Our study also reveals the important role played by the Coulomb scattering, which can effectively suppress the spin diffusion.

Electron spin diffusion and transport in graphene

We investigate the spin diffusion and transport in a graphene monolayer on SiO${}_{2}$ substrate by means of the microscopic kinetic spin Bloch equation approach. The substrate causes a strong Rashba spin-orbit coupling field $~0.15$ meV, which might be accounted for by the impurities initially present in the substrate or even the substrate-induced structure distortion. By surface chemical doping with Au atoms, this Rashba spin-orbit coupling is further strengthened as the adatoms can distort the graphene lattice from ${\mathit{sp}}^{2}$ to ${\mathit{sp}}^{3}$ bonding structure. 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 from 0.15 to 0.23 meV with the increase of Au density. With this strong spin-orbit coupling, the spin diffusion or transport length is comparable with the experimental values. In the strong scattering limit (dominated by the electron-impurity scattering in our study), the spin diffusion is uniquely determined by the Rashba spin-orbit coupling strength and insensitive to the temperature, electron density, as well as scattering. With the presence of an electric field along the spin injection direction, the spin transport length can be modulated by either the electric field or the electron density. It is shown that the spin diffusion and transport show an anisotropy with respect to the polarization direction of injected spins. The spin diffusion or transport lengths with the injected spins polarized in the plane defined by the spin-injection direction and the direction perpendicular to the graphene are identical, but longer than that with the injected spins polarized vertical to this plane. This anisotropy differs from the one given by the two-component drift-diffusion model, which indicates equal spin diffusion or transport lengths when the injected spins are polarized in the graphene plane and relatively shorter lengths when the injected spins are polarized perpendicular to the graphene plane.

Spin diffusion in [formula omitted]-type (111) GaAs quantum wells

We utilize the kinetic spin Bloch equation approach to investigate the steady-state spin diffusion in n -type (111) GaAs quantum wells, where the in-plane components of the Dresselhaus spin–orbit coupling term and the Rashba term can be partially cancelled by each other. A peak of the spin diffusion length due to the cancellation is predicted in the perpendicular electric field dependence. It is shown that the spin diffusion length around the peak can be markedly controlled via temperature and doping. When the electron gas enters into the degenerate regime, the electron density also leads to observable influence on the spin diffusion in the strong cancellation regime. Furthermore, we find that the spin diffusion always presents strong anisotropy with respect to the direction of the injected spin polarization. The anisotropic spin diffusion depends on whether the electric field is far away from or in the strong cancellation regime.

Erratum: Electron-spin relaxation in bulk III-V semiconductors from a fully microscopic kinetic spin Bloch equation approach [Phys. Rev. B79, 125206 (2009)

Erratum: Electron-spin relaxation in bulk III-V semiconductors from a fully microscopic kinetic spin Bloch equation approach [Phys. Rev. B<b>79</b>, 125206 (2009)

Existence of vertical spin stiffness in Landau-Lifshitz-Gilbert equation in ferromagnetic semiconductors

We calculate the magnetization torque due to the spin polarization of the itinerant electrons by deriving the kinetic spin Bloch equations based on the $s$-$d$ model. We find that the first-order gradient of the magnetization inhomogeneity gives rise to the current-induced torques, which are consistent to the previous works. At the second-order gradient, we find an effective magnetic field perpendicular to the spin stiffness filed. This field is proportional to the nonadiabatic parameter $\beta$. We show that this vertical spin stiffness term can significantly modify the domain-wall structure in ferromagnetic semiconductors and hence should be included in the Landau-Lifshitz-Gilbert equation in studying the magnetization dynamics.

Temperature and electron density dependence of spin relaxation in GaAs/AlGaAs quantum well

Temperature and carrier density-dependent spin dynamics for GaAs/AlGaAs quantum wells (QWs) with different structural symmetries have been studied by using time-resolved Kerr rotation technique. The spin relaxation time is measured to be much longer for the symmetrically designed GaAs QW comparing with the asymmetrical one, indicating the strong influence of Rashba spin-orbit coupling on spin relaxation. D'yakonov-Perel' mechanism has been revealed to be the dominant contribution for spin relaxation in GaAs/AlGaAs QWs. The spin relaxation time exhibits non-monotonic-dependent behavior on both temperature and photo-excited carrier density, revealing the important role of non-monotonic temperature and density dependence of electron-electron Coulomb scattering. Our experimental observations demonstrate good agreement with recently developed spin relaxation theory based on microscopic kinetic spin Bloch equation approach.

Spin relaxation in n-type (111) GaAs quantum wells

We investigate the spin relaxation limited by the D’yakonov–Perel’ mechanism in n-type (111) GaAs quantum wells, by means of the kinetic spin Bloch equation approach. In (111) GaAs quantum wells, the in-plane effective magnetic field from the D’yakonov–Perel’ term can be suppressed to zero on a special momentum circle under the proper gate voltage, by the cancellation between the Dresselhaus and Rashba spin-orbit coupling terms. When the spin-polarized electrons mainly distribute around this special circle, the in-plane inhomogeneous broadening is small and the spin relaxation can be suppressed, especially for that along the growth direction of quantum well. This cancellation effect may cause a peak (the cancellation peak) in the density or temperature dependence of the spin relaxation time. In the density (temperature) dependence, the interplay between the cancellation peak and the ordinary density (Coulomb) peak leads to rich features of the density (temperature) dependence of the spin relaxation time. The effect of impurities, with its different weights on the cancellation peak and the Coulomb peak in the temperature dependence of the spin relaxation, is revealed. We also show the anisotropy of the spin relaxation with respect to the spin-polarization direction.