The semiclassical theory for spin dynamics in a disordered system

When applying an external electric field in a spin-orbit coupled system, lateral spin motion is generated and spin accumulates at the edges of the sample, which is known as the spin-Hall effect Hirsch1999 . The spin-Hall effect has been extensively studied theoretically and through experiments over the past decades Sinova2015 . The mechanism of the intrinsic spin–Hall effect stems from the Berry curvature Mura2003 ; Chang2010 or the adiabatic evolution of spin around the effective magnetic field Sinova2004 . A challenging issue is how to define the spin current because each component of spin is not a conserved quantity. In Ref. Shi2006 , a generic definition of spin current (SZXN spin current) is proposed, and the spin current is composed of the conventional definition of spin current and the spin-torque dipole current. The conventional and spin-torque dipole currents were found to be related to the Berry phase in k-linear systems Shen2004 ; Chen2014 , while no such relationship exists for spin-orbit coupled systems with higher order momentum. Finally, the spin-Hall conductivity obtained from the SZXN spin current in response to the electric field is a universal constant in the k-linear system, regardless of the strength of the spin-orbit coupling, such as in pure Rashba  Rashba1984 ; Sinova2004 and Rashba–Dresselhaus systems  Dress1955 ; Chen2006 . This makes it difficult to compare theoretical results with experimental observations.

In semiclassical theory, the lateral motion of spin would be attributed to the force exerted on the spin. From this point of view, the intrinsic force due to the k-linear spin-orbit coupling was found to be related to the conventional definition of spin current Shen2005 . In the presence of disorder, the spin force in the Rashba–Dresselhaus system vanishes by accounting for the vertex correction Chen2009 . However, in the Drude model regime, an effective Lorentz force can be generated by the crystal field Eugene2007 , and it was shown to be related to the spin-Hall effect. Importantly, the spin-Hall conductivity generated by the effective Lorentz force is proportional to the charge conductance, and the resulting spin current fits experimental values observed for metals Eugene2007 , which sheds light on the evaluation of the observable spin motion in the Drude model regime. It also worths noting that the semiclassical analysis of spin dynamics in the intrinsic Luttinger system was investigated by using adiabatic approximation Jiang2005 .

In this study, we attempt to develop spin dynamics for two-dimensional spin-orbit coupled systems in the Drude model regime. The spin dynamics is treated as semiclassical motion governed by the Heisenberg equation of motion. The spin precession is ascribed to the rotation around the successive change in the effective magnetic field. To preserve the magnitude of the spin, the spin precession up to the first order of the electric field is comprised of Larmor and non-Larmor precessions. The former is a periodic motion and the latter is an exponential decay function. The spin-Hall conductivity (SHC) was also calculated by using the two components.

We find that in the regime $t/\tau\ll 1$, the non-Larmor SHC is exactly equal to the result obtained from the Kubo formula. The formalism of the Kubo formula does not consider the Larmor precession. At finite $\tau$, we find that both the Larmor SHC and non-Larmor SHC vanish when the elapsed time is much larger than the relaxation time $\tau$. In the strong disorder limit (finite $\tau$ and vanishing Rashba coupling), both the Larmor and non-Larmor SHCs vanish independently over a finite time. In the intrinsic limit (infinite $\tau$ compared to the Larmor frequency), the Larmor SHC exactly cancels the non-Larmor SHC in the Kubo formula regime ($t/\tau\ll 1$), which solves the problem of determining the universal constant of the SHC in previous studies. We also calculated the time-averaged spin-Hall conductivity for the k-cubic Rashba system Wund2005 . The result is close to the experimental value. We note that the time-averaged quantum Hall conductivity is also investigated in Liu2019 .

The remainder of this paper is organized as follows. In Sec. II, the influence of the momentum relaxation time on the spin-orbit coupled system is investigated. The spin motion up to the first order of the electric field is composed of Larmor and non-Larmor precessions. The boundary condition is also determined by using preservation of the spin magnitude. The analytical solutions for the Larmor and non-Larmor precessions are presented in Sec. III. The relationship between the non-Larmor precession and the Kubo formula is discussed. In Sec. IV, the spin-Hall conductivities for both Larmor and non-Larmor precessions are calculated. In the intrinsic Rashba system, the resulting spin-Hall conductivity drops to zero when the Rashba coupling drops to zero. We also calculate the Larmor and non-Larmor spin-Hall conductivities for the k-cubic Rashba system. The resulting spin-Hall conductivity is shown to be close to the experimental value. Finally, the conclusion is presented in Sec. V.

The two-dimensional spin-orbit coupled Hamiltonian under consideration is given by

where $\bm{\sigma}=(\sigma_{x},\sigma_{y},0)$ with $\bm{\sigma}$ been Pauli matrices and $\bm{\Omega}_{0}=(2d_{x}/\hbar,2d_{y}/\hbar,0)$ is the Larmor frequency. The terms $d_{x}$ and $d_{y}$ depend on the spin-orbit interaction and two-dimensional momentum $\mathbf{k}=(k_{x},k_{y})$. There is no term with $d_{z}$, which implies that the spin is forced to lie along the plane. For example, in the Rashba–Dresselhaus system, we have $d_{x}=\alpha k_{y}+\beta k_{x}$ and $d_{y}=-\alpha k_{x}-\beta k_{y}$. For the k-cubic Rashba-Dresselhaus system Bulaev2005 , the Hamiltonian is given by $H_{0}=\epsilon_{k}+i\alpha(\sigma_{+}k_{-}^{3}-\sigma_{-}k_{+}^{3})-\beta(\sigma_{+}k_{-}k_{+}k_{-}+\sigma_{-}k_{+}k_{-}k_{+})$. In this case, $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$ and $k_{\pm}=k_{x}\pm ik_{y}$. The corresponding $d_{x}$ and $d_{y}$ are defined as $d_{x}=[\alpha\sin(3\phi)+\beta\cos(\phi)]k^{3}$, and $d_{y}=[-\alpha\cos(3\phi)+\beta\sin(\phi)]k^{3}$, where $\phi=\tan^{-1}(k_{y}/k_{x})$. The eigenenergies and eigenvectors are obtained by solving $H_{0}|n\mathbf{k}\rangle=E_{n\mathbf{k}}|n\mathbf{k}\rangle$, which is given by

where $d=\sqrt{d_{x}^{2}+d_{y}^{2}}$ and $n=\pm$ is the energy band index. In the two-dimensional spin-orbit coupled systems, it is easy to show that the spin z component in the unperturbed Hamiltonian Eq. (1) is zero, i.e., $\langle{n\mathbf{k}}|\sigma_{z}|{n\mathbf{k}}\rangle$=0.

We consider a simple model in which the momentum and the Fermi-Dirac distribution eventually approach a finite displacement through the collision between electrons and impurities. In this paper, an electric charge is denoted as $-q$ and $q>0$. The equation of motion of momentum in the presence of a constant electric field is given by

where $\mathbf{k}$ is the electron’s momentum before the electric field is switched on, and $\tau$ is the momentum relaxation time. Eq. (3) can be exactly solved, and the result is given by Jone1985

where

Note that $t^{\ast}(0)=0$ and $t^{\ast}(\infty)=\tau$. When $\tau$ is very large (the clean limit), we have $e^{-t/\tau}=1-t/\tau+t^{2}/2\tau^{2}+\cdots$ and $t^{\ast}(t/\tau\ll 1)\rightarrow t$. This implies that $\mathbf{k}_{t}=\mathbf{k}-q\mathbf{E}t/\hbar$ if $t\ll\tau$, which is the same result as when impurities are not present. We refer to the limit $t/\tau\ll 1$ as the clean limit. In this limit $t/\tau\ll 1$, equilibrium cannot be achieved.

As time $t$ is much larger than $\tau$, the momentum has a finite displacement of $-\tau q\mathbf{E}/\hbar$. The Fermi-Dirac distribution $f_{n\mathbf{k}}$ in $\mathbf{k}$-space is also displaced by the electric field such that ($t\gg\tau$)

The first order correction to the distribution function is proportional to the relaxation time which gives the Drude’s result of charge conductivity Jone1985 . Furthermore, we note that the quantum kinetic equations including spin-orbit coupling has been investigated in Ref. Bryksin2006 , in which the first order correction of distribution function is related to the spin density matrix. However, in this paper, the first order correction of distribution function is irrelevant in the linear response regime. This is because the spin precession around the effective magnetic field is also caused by the applied electric field and moreover the unperturbed spin z component is zero [see Eqs. (37) and (38)].

The Heisenberg equation of motion of spin $\mathbf{s}^{0}\equiv\langle{n\mathbf{k}}|e^{iH_{0}t/\hbar}\bm{\sigma}e^{-iH_{0}t/\hbar}|{n\mathbf{k}}\rangle$ is given by

where $\bm{\Omega}_{0}$ depends on momentum $\mathbf{k}$. In the presence of an applied electric field, the change in momentum would lead to the change in the effective magnetic field and the spin orientation varies due to the precession. Semiclassically, we assume that the effect of disorder and the applied electric field are included in the effective magnetic field and the spin dynamics is still governed by the Heisenberg equation of motion. Thus, the resulting Hamiltonian can be written as

where $\mathbf{k}_{t}$ is given by Eq. (4), $\bm{\Omega}(t)=2\mathbf{d}(t)/\hbar$ and $\mathbf{d}(t)=(d_{x}(\mathbf{k}_{t}),d_{y}(\mathbf{k}_{t}),0)$. We require that the Hamiltonian Eq. (8) satisfies the Schrodinger equation

The time-dependent spin $\mathbf{s}(t)$ is defined as $\mathbf{s}(t)\equiv\langle\psi(t)|\bm{\sigma}|\psi(t)\rangle$ which is the averaged spin in the presence of disorder and an electric field. Apply the first derivative of time to $\mathbf{s}(t)$, we have

where Eq. (8) was used. By using the commutator for Pauli matrices $[\sigma_{i},\sigma_{j}]=2i\epsilon_{ijk}\sigma_{k}$, we obtain the Heisenberg equation of motion for the averaged spin,

where $[\bm{\sigma},\mathbf{k}_{t}^{2}]=0$ was used. We emphasize that the equation of motion for momentum is derived by using semiclassical Drude model (Eq.(4)), not the expectation value of $\mathbf{k}$ with respect to the state $|\psi(t)\rangle$, i.e., $\mathbf{k}_{t}\neq\langle\psi(t)|\mathbf{k}|\psi(t)\rangle$.

We now return to the momentum dependence of the effective magnetic field. The influence of the disorder in the presence of an electric field will lead to a change in the effective magnetic field, that is, $d_{i}(t)=d_{i}(\mathbf{k}_{t})$. Therefore, up to the linear order of the electric field,

Similar to the Fermi-Dirac distribution in the presence of disorder, when $t\rightarrow\infty$, the $\mathbf{d}$ vector changes its direction from $d_{i}$ to another fixed direction $d_{i}-(\tau eE_{a}/\hbar)(\partial d_{i}/\partial k_{a})$. We define $\bm{\Omega}(t)=2\mathbf{d}(t)/\hbar=\bm{\Omega}_{0}+\bm{\Omega}^{\prime}(t)$, where $\bm{\Omega}_{0}=(2d_{x}/\hbar,2d_{y}/\hbar,0)$ and

which also lies on the 2D plane. The change in the effective magnetic field will cause the spin to precess around the new direction of the effective magnetic field. The spin’s z component will be non-zero at this time. The trajectory of $\mathbf{d}(t)$ in $d$-space is a straight line up to the first order of the electric field. The resulting angular velocity of $\mathbf{d}$ measured from the origin must vary with time. Therefore, we cannot use a rotating frame with fixed spin because the resulting Hamiltonian does not commute at different times. Instead, we use the perturbative method defined in Eq. (11). The unitarity is broken and the magnitude of the spin is preserved only up to the first order of an electric field. Namely, we have

where $\mathbf{s}^{0}$ and $\mathbf{s}^{\prime}$ correspond to the solutions without an electric field and with an electric field, respectively. Inserting this result into Eq. (11),

where the following unperturbed Eq. (7) was used. The solution of Eq. (7) (using $|\mathbf{s}^{0}|=1$) is given by

where the angle $\theta_{0}$ is the angle between $\bm{\Omega}_{0}$ and $\mathbf{s}^{0}$, and $\Theta$ is defined as $\tan\Theta=d_{y}/d_{x}$. Eq. (7) implies that $\mathbf{s}^{0}$ behaves as the spin in the intrinsic system $H_{0}$, where the Larmor frequency is $\bm{\Omega}_{0}$. The spin should be aligned with $\bm{\Omega}_{0}$ without applying an electric field. In this sense, we can assume that $\theta_{0}=0$, and we have

To obtain results for different energy bands, we can simply use the replacement $\mathbf{s}_{0}\rightarrow(-n)\mathbf{s}_{0}$ in Eq. (17), where $n=\pm$ is the band index. Substituting Eq. (17) with the appropriate band index into Eq. (15), we have ($\lambda_{a}\equiv qE_{a}$)

for the spin-z component, and

for the spin x and y components, where $\Gamma^{z}_{a}$ is defined as

We observe that the term $\bm{\Omega}^{\prime}(t)\times\mathbf{s}^{0}$ in Eq. (15) behaves as a torque such that $\mathbf{s}^{\prime}$ not only has a Larmor rotation (periodic rotation) but also a non-Larmor rotation (non-periodic motion). Therefore, similar to the derivations in a previous study, the spin will be composed of non-Larmor (N) and Larmor (L) precession terms.

The term $\bm{\Sigma}^{L}$ is called the Larmor component and is governed by

Eq. (22) is similar to Eq. (7); however, $\bm{\Sigma}^{N}$ contains the response of the electric field, and we expect that the direction of $\bm{\Sigma}^{L}$ is not parallel to the static Larmor frequency $\bm{\Omega}_{0}$. On the other hand, the term $\bm{\Sigma}^{N}$ is called the non-Larmor component, which satisfies

It must be emphasized that Eqs. (22) and (23) hold only if the magnitude of spin $\mathbf{s}(t)$ is valid up to the second order of electric field. The magnitude of spin is given by $|\mathbf{s}|^{2}=|\mathbf{s}^{0}+\mathbf{s}^{\prime}+o(E_{a}^{2})|^{2}=|\mathbf{s}^{0}|^{2}+2\mathbf{s}^{0}\cdot\mathbf{s}^{\prime}+o(E_{a}^{2})$. To preserve the magnitude of spin up to the second order of $E_{a}$, we require that

at any time $t$. Furthermore because the time dependence of the Larmor component is different from that of non-Larmor components, Eq. (24) implies that

On the other hand, at $t=0$, the precession of spin begins from the in-plane and ends as an out-of-plane orientation. In this sense, the z component $s_{z}^{\prime}(0)$ must be zero, although its rate of change is non-zero. This can be described by

where Eq. (15) at $t=0$ was used. We note that a similar derivation was proposed in Ref. Paul2018 , in which the effective magnetic field is replaced by a pseudo-magnetic field derived from the spin force. In the following section, we solve $\mathbf{s}^{\prime}$ by acquiring the boundary conditions for Eqs. (25) and (26).

The solutions of Eq. (22) would be the same as Eq. (16). That is, we have

However, the difference is that $\bm{\Sigma}^{L}$ is in linear order with the applied electric field in the disordered system. The angle $\theta$ between $\bm{\Sigma}^{L}$ and $\bm{\Omega}_{0}$ depends on the electric field and relaxation time $\tau$. First, by using the requirement in Eq. (25), we have $\Sigma^{L}_{x}\cos\Theta+\Sigma^{L}_{y}\sin\Theta=0$, which implies that $\Sigma^{L}\cos\theta=0$. Because $\Sigma^{L}\neq 0$, we obtain $\theta=\pi/2$, which means that $\bm{\Sigma}^{L}$ is always perpendicular to $\bm{\Omega}_{0}$. Eq. (27) then becomes

Applying the time derivative to the z component of Eq. (23), and noting that $dt^{\ast}/dt=e^{-t/\tau}$, we obtain

where $\Omega_{0}=2d/\hbar$ was used. The solution of Eq. (29) is given by

where $A_{n}$ is the dimensionless quantity

Eq. (30) plays an important role in the spin-Hall effect. We will return to this point in the next section. We now use Eq. (26), which is $\Sigma^{N}_{z}(0)+\Sigma^{L}_{z}(0)=0$. Thus, $A_{n}-\Sigma^{L}=0$, and the resulting z component of $\bm{\Sigma}^{L}$ is given by

and the x and y components of $\bm{\Sigma}^{L}$ are given by

Substituting Eqs. (30) and (31) into Eq. (23) for the x and y components, we obtain

Arbitrary time-independent constants $C_{x}$ and $C_{y}$ can be added to $\Sigma^{N}_{x}$ and $\Sigma^{N}_{y}$, respectively. The two constants must obey the condition $d_{x}C_{y}-d_{y}C_{x}=0$. Furthermore, the preserved magnitude of spin $|\mathbf{s}(t)|=1+o(E_{a}^{2})$ (see Eq. (25)) implies that $d_{x}C_{x}+d_{y}C_{y}=0$. The only solution is $C_{x}=C_{y}=0$ if $d\neq 0$. On the other hand, it can be shown that Eqs. (30), (32), (33), and (34) satisfy the second boundary condition of Eq. (26).

When $t\rightarrow\infty$, the momentum has a constant shift and the effective magnetic field does not change its direction over time. The system achieves an equilibrium state. We find that the in-plane components $\Sigma^{N}_{x}$ and $\Sigma^{N}_{y}$ do not change with time, and the non-Larmor spin z component $\Sigma^{N}_{z}$ vanishes. Only the $\bm{\Sigma}^{L}$ Larmor components survive in the system. In the next section, we calculate the Larmor SHC and non-Larmor SHC in the Rashba system. We close this section by discussing the relationship between our results and the Kubo formula in the intrinsic system.

If the energy gap is much larger than the broadening of the energy band due to the disorder, the system is said to be within the intrinsic limit, that is, $2d\gg\hbar/\tau$. This implies that the intrinsic limit is given by $\Omega_{0}\tau\gg 1$ when $\mathbf{k}\neq 0$. When $\tau$ is finite, the intrinsic limit implies that the system is in the strong spin-orbit coupling regime and cannot be zero unless $\tau\rightarrow\infty$. Therefore, the intrinsic limit leads to the result that the system displays a physical evolution only for a very short time compared to a finite $\tau$. In the following, we will demonstrate that at the intrinsic limit (and clean limit), the non-Larmor spin z component $\Sigma^{N}_{z}$ is exactly equal to the result obtained from the Kubo formula.

If we take the intrinsic limit $\Omega_{0}\tau\gg 1$, then $\Sigma^{N}_{i}$ is proportional to $(e^{-t/\tau}-1)\Omega_{0}\tau$. Then, taking the clean limit $t/\tau\ll 1$, we have $(e^{-t/\tau}-1)\Omega_{0}\tau\approx-\Omega_{0}t$. Therefore, considering the limit $\Omega_{0}\tau\gg 1$ and then $t/\tau\ll 1$, we have

Note that in the limit $\Omega_{0}\tau\gg 1$ with $t/\tau\ll 1$, the non-Larmor spin x and y components exhibit no physical growth in time. Furthermore, Eq. (35) can be exactly derived from the Heisenberg equation of motion Chen2019 . If we substitute Eq. (35) back into Eq. (23), we obtain

Interestingly, because of the linear time dependence of the spin x and y components, the response of the non-Larmor spin z component is a constant only for a very short time. It has been shown that Eq. (36) is exactly the same as the result obtained from the Kubo formula in the intrinsic case Chen2014 ; Chen2019 . The spin-Hall current from the Kubo formula using the result in Eq. (36) is a universal constant in the Rashba system. The universal constant also leads to the problem that the spin x and y components have no physical growth with time, as has been demonstrated in Refs  Chen2006 ; Chen2019 ; Dim2005 ; Chalaev2005 . Nevertheless, as discussed in the above results, the regime of the Kubo formula in the intrinsic spin-orbit coupled system is valid only for a very short time limit. In particular, the short time limit corresponds to the time when the spin adiabatically aligns its orientation with the effective magnetic field. Consequently, the change in the magnitude of the effective magnetic field (and thus the spin in that direction) yields a non-zero spin z component in order to preserve the spin magnitude Sinova2004 .

The spin-Hall current deduced from the spin-dynamics would be semiclassically obtained by using the conventional definition of the spin current, which is the simple multiplication of spin $(J\hbar/2)s_{z}$ and the lateral velocity $\hbar k_{x}/m$ in which the electric field is applied in y-direction, where $J=1$ for the Rashba–Dresselhaus system and $J=3$ for the k-cubic Rashba system. The time-dependent spin-Hall current is given by

We note that $s^{0}_{z}=0$ (see Eq. (17)) in the present systems under consideration. Up to the first order of the applied electric field, the time-dependent spin-Hall current Eq. (37) becomes

The linear response of the spin z component $s^{\prime}_{z}$ is given by $s_{z}^{\prime}=\Sigma^{L}_{z}(t)+\Sigma^{N}_{z}(t)$ [see Eqs. (30) and (32)]. The spin-Hall conductivity $\sigma^{z}_{xy}$ can now be written as the Larmor component $\sigma^{L}_{xy}$ and non-Larmor component $\sigma^{N}_{xy}$, that is,

We solved the spin dynamics of two-dimensional spin-orbit coupled systems in the Drude model regime, in which the momentum relaxation time is taken into account. By considering the change in the effective magnetic field induced by the applied electric field, the spin will precess around the successive change of the effective magnetic field. The spin dynamics were investigated by using the perturbation method. The spin response to the linear order of the electric field is found to be composed of Larmor and non-Larmor precessions. The Larmor motion is the precession occurring around the unperturbed effective magnetic field, and the non-Larmor motion of spin is due to the extra torque induced by the electric field. The time-dependent spin-Hall conductivity (SHC) of the Rashba system was then calculated. In the presence of disorder, when time grows larger than the relaxation time, we found that both the Larmor and non-Larmor SHCs drop to zero. However, the non-Larmor SHC decays faster than the Larmor SHC. This occurs because the presence of the disorder forces the system to achieve equilibrium, and the effective magnetic field eventually does not change over time. On the other hand, we found that the non-Larmor motion in the short time limit is exactly equal to the result obtained from the Kubo formula in the intrinsic case. In the intrinsic Rashba system, we found that the spin-Hall conductivity is not a universal constant. Furthermore, the spin-Hall conductivity vanishes when the Rashba coupling vanishes. We also calculated the Larmor and non-Larmor SHCs for the k-cubic Rahsba system. To meaningfully compare our results with the experimental value, we used the time-averaged spin-Hall conductivity. By comparing the calculated results to the experimental results, we found that the $6\times 2\pi$ rotation for the spin wave function and the preservation of the magnitude of spin lead to the experimental result.