Spin transport and dynamic properties of two-dimensional spin-momentum locked states

The electronic states with spin-momentum locking (SML) into mutually perpendicular directions occur at the surface of a topological insulator or at the interface with a strong Rashba interfacial spin-orbit coupling Hasan and Kane (2010); Manchon et al. (2015); Soumyanarayanan et al. (2016). These SML states are two-dimensional itinerant magnetic states without spontaneous magnetic moment in the absence of the magnetic field and there is no magnetic hysteresis response to the magnetic field. Yet, these materials have displayed profound magnetic phenomena. For example, an applied electric current can induce a non-equilibrium spin density known as the Edelstein effect (EE) Edelstein (1990); Kato et al. (2004); Kondou et al. (2016, 2016), and reciprocally, a spin current injection produced by, e.g., spin pumping Tserkovnyak et al. (2002a, b), to such electronic surface states yields an electric charge current termed as the inverse Edelstein effect (IEE) Ganichev et al. (2002); Shen et al. (2014); Sánchez et al. (2013); Shiomi et al. (2014); Rojas-Sánchez et al. (2016); Zhang et al. (2015). These observed EE and IEE phenomena have drawn considerable interest due to their potential applications for spin-based electronic devices Žutić et al. (2004).

In this paper, we theoretically study the time-dependent magnetic and spin transport properties of such SML systems in the presence of the external spin injection and magnetic field. In the conventional magnetic materials, the time-dependent magnetization dynamics is described by the Landau-Lifshitz-Gilbert equation with an additional Slonczewski spin transfer torque Berger (1996); Slonczewski (1996) when a spatially varying spin current is present. For the SML systems, there is no net magnetization without a magnetic field in the equilibrium even though the spin state of an electron with a given momentum is well-defined, i.e., the spin is ordered perpendicularly to the momentum. Such unique ordered spin configuration in the momentum space may be described by three macroscopic variables: a vectorial helical magnetization (VHM) $\boldsymbol{\xi}\equiv<\hat{\bf p}\times{\boldsymbol{\sigma}}>$ and a scalar helical magnetization (SHM) $\eta\equiv<\hat{\bf p}\cdot{\boldsymbol{\sigma}}>$, in addition to the conventional magnetization ${\boldsymbol{m}}\equiv<{\boldsymbol{\sigma}}>$, where $\hat{\bf p}={\bf p}/{\rm p}$ is the direction of momentum and ${\boldsymbol{\sigma}}$ is the Pauli matrix. Clearly, VHM and SHM characterize the relative orientation of the momentum and spin. In equilibrium, these macroscopic variables take simple values for the SML system, $\boldsymbol{\xi}_{\rm eq}\neq 0$, $\eta_{\rm eq}=0$ and ${\boldsymbol{m}}_{\rm eq}=0$.

The main objective of the paper is to determine these variables in the presence of a driving force such as a time-dependent magnetic field or a spin current injected from a nearby contacting metallic layer, and more interestingly, to establish the relations between these variables and transport properties that can be measured experimentally. Due to the special band structure of SML systems, we find that the spin transport and dynamic properties display a number of unique characteristics and the above defined three macroscopic variables provide a convenient way to understand and explain the non-equilibrium processes. This paper is organized as follows. In Sec. II, we present our model and introduce a spinor form of the Boltzmann equation. In Sec. III, we derive the dynamic equations of these three macroscopic variables from the spinor Boltzmann equation, with appropriate simplifications. In Sec. IV, we solve the equations in several cases that are experimentally accessible. In Sec. IV, we consider the effect of the time-dependent motion of the VHM on the spin pumping. Finally, we conclude the paper in Sec. V.

We consider a spin-orbit coupled two-dimensional band structure with a simple dispersion relation given by

$$\hat{\varepsilon}_{\bf p}=\varepsilon_{\bf p}^{0}+\alpha({\hat{\bf z}\times{\bf p})\cdot\boldsymbol{\sigma}}$$

(1)

where $\varepsilon_{\bf p}^{0}$ is the spin-independent part of the electron dispersion which we will take zero for the surface state of a topological insulator and $\varepsilon_{\bf p}^{0}={\bf p}^{2}/2{m^{*}}$ for a Rashba band, where $m^{*}$ is the effective mass, and $\alpha$ is the spin-orbit coupling constant. For a given momentum ${\bf p}$, the eigenvalues take $\varepsilon_{{\bf p}\pm}=\varepsilon_{\bf p}^{0}\pm\alpha{\rm p}$ with the corresponding spin eiegnstates $\chi_{{\bf p}\pm}$ satisfying $(\boldsymbol{\Omega}_{\bf p}\cdot{\boldsymbol{\sigma}})\chi_{{\bf p}\pm}=\pm\chi_{{\bf p}\pm}$, where $\boldsymbol{\Omega}_{\bf p}\equiv(\hat{\bf z}\times\hat{\bf p})$ is the unit vector representing the quantization axis of the spin for a given momentum ${\bf p}$.

The above model describes a simple one-electron SML state at equilibrium. We now turn on an AC magnetic field ${\bf h}_{ex}(t)$ and a DC electric field ${\bf E}_{ex}$, in addition to a possible spin injection from a contacting normal metal (NM), as shown in Fig. 1. The electron Hamiltonian with a given momentum then reads

$$\hat{H}_{\bf p}=\hat{\varepsilon}_{\bf p}+{\bf h}_{ex}(t)\cdot\boldsymbol{\sigma}+e{\bf E}_{ex}\cdot{\bf r}$$

(2)

To calculate the spin transport properties in response to the magnetic and electric fields, we introduce a spinor form of the semiclassical Boltzmann distribution function for the SML band,

$$\displaystyle\hat{f}({\bf p},t)=\hat{f}_{0}({\bf p})+f_{c}({\bf p},t)+{\boldsymbol{f}}_{s}({\bf p},t)\cdot{\boldsymbol{\sigma}}$$

(3)

where $\hat{f}_{0}$ is the equilibrium Fermi distribution function and we take the distribution at $T=0$, i.e., $\hat{f}_{0}({\bf p})=\Theta(\varepsilon_{F}-\hat{\varepsilon}_{\bf p})$ with ${\varepsilon_{F}}$ the Fermi energy. In Eq. (3), the non-equilibrium distribution function is separated into the spin-independent ($f_{c}$) and spin-dependent (${\boldsymbol{f}}_{s}$) parts. The generalized spinor Boltzmann equation is,

$$\frac{\partial\hat{f}}{\partial t}+e{\bf E}_{ex}\cdot\hat{\bf v}\left(-\frac{\partial\hat{f}_{0}}{\partial\hat{\varepsilon}_{\bf p}}\right)+i\Big{[}\hat{H}_{\bf p},\hat{f}\Big{]}=\left(\frac{\partial\hat{f}}{\partial t}\right)_{\rm col}$$

(4)

where $\hat{\bf v}=\partial{\hat{\varepsilon}}_{\bf p}/{\partial{\bf p}}$ is the velocity and the anticommutator is implied for the product between Pauli matrices in this paper.

At this point, we want to emphasize that the presence of the commutator in Eq. (4) between $\hat{H}_{\bf p}$ and the spinor distribution function allows the electron to occupy the states that are not the spin eigenstates at equilibrium; this term represents the procession of the nonequilibrium electron spin around the effective magnetic field (the spin-orbit field $\alpha(\hat{\bf z}\times{\bf p})$ and the external magnetic field ${\bf h}_{ex}$). Thus, we do not assume that $\boldsymbol{f}_{s}$ is parallel to the $\boldsymbol{\Omega}_{\bf p}$. Recall that in a ferromagnetic metal, the spin dependent distribution function ${\boldsymbol{f}}_{s}$ is always confined to the states parallel or antiparallel to the local magnetization ${\bf M}$, i.e., ${\boldsymbol{f}}_{s}\propto{\bf M}$ and thus the Boltzmann equation in the conventional ferromagnet has only two components (spin up and down relative to the local magnetization). The transverse component of spin accumulation or spin current injected to a ferromagnet is absorbed by the ferromagnet within a few atomic distance due to the strong exchange interaction between the transverse spin and local magnetization; the absorption of the transverse spin current is considered as the manifestation of spin transfer torque. In the present case, we argue that the spin orbit coupling responsible for the dephasing of injected spins is much weaker compared to that of the transverse spin in conventional ferromagnets since a typical spin orbit coupling in a Rashba Hamiltonian or TI, which is about tens of meV Zhang et al. (2015); Kondou et al. (2016); Rojas-Sánchez et al. (2016), is much smaller than the exchange interaction in ferromagnets (of the order of eV); we will return to this point later when we model relaxation times. Thus, we do not specify the direction of ${\boldsymbol{f}}_{s}$ and instead we will determine the dynamic equations relevant to it from the above spinor Boltzmann equation. The resulting commutator in Eq. (4) for an arbitrary ${\boldsymbol{f}}_{s}$ is

$$\Big{[}\hat{H}_{\bf p},\hat{f}\Big{]}=2i\Big{[}\Big{(}\alpha\hat{\bf z}\times{\bf p}+{\bf h}_{ex}\Big{)}\times{\boldsymbol{f}_{s}}\Big{]}\cdot\boldsymbol{\sigma}$$

(5)

The collision term on the right-hand side of Eq. (4) has two contributions: one is the internal spin and momentum relaxations of the SML system, and the other is the interfacial scattering with the attached NM layer. We may parameterize these processes below,

	$\displaystyle\left(\frac{\partial\hat{f}}{\partial t}\right)_{\rm col}=$	$\displaystyle-\frac{{f}_{c}+\boldsymbol{f}_{L}\cdot\boldsymbol{\sigma}}{\tau_{p}}-\frac{\boldsymbol{f}_{T}\cdot\boldsymbol{\sigma}}{\tau_{\phi}}$
		$\displaystyle+\sum_{\bf p^{\prime}}\hat{\Gamma}_{{\bf p}{\bf p}^{\prime}}\left[\hat{g}({\bf p}^{\prime},t)-\hat{f}({\bf p},t)\right]$		(6)

where we have introduced two characteristic relaxation times for the SML: the first term is caused by the momentum scattering of electrons, which is responsible for the relaxation of the longitudinal part of spin-dependent distribution function (relative to the spin-orbit field direction), i.e., $\boldsymbol{f}_{L}\equiv({\boldsymbol{f}}_{s}\cdot{\boldsymbol{\Omega}_{\bf p}}){\boldsymbol{\Omega}_{\bf p}}$ with a relaxation time $\tau_{p}$, and the second term is the spin dephasing for the transverse part of the spin-dependent distribution function $\boldsymbol{f}_{T}\equiv\boldsymbol{f}_{s}-({\boldsymbol{f}}_{s}\cdot{\boldsymbol{\Omega}_{\bf p}}){\boldsymbol{\Omega}_{\bf p}}$ with the relaxation time $\tau_{\phi}$. The last term in Eq. (II) represents the contribution from the interfacial scattering between the SML and NM layers with a transition rate $\hat{\Gamma}_{{\bf p}{\bf p}^{\prime}}$ and $\hat{g}({\bf p}^{\prime},t)$ being the spinor distribution function of the NM layer at the interface.

The Boltzmann equation, Eq. (4), along with Eqs. (5) and (II), remains mathematically complicated since the collision terms make the Boltzmann equation an integral equation. To further reduce the mathematical complication, in this Section, we propose to generate the spin-diffusion-like equations from the Boltzmann equation such that resulting simpler equations can be directly used for experimental analysis. The following approximations are made. First, we assume the distribution function of the NM layer $\hat{g}({\bf p}^{\prime},t)$ can be described by a time-independent single parameter, i.e., the spin chemical potential, defined as $\boldsymbol{\mu}_{s}=(1/e\mathcal{N}_{F})\sum_{\bf p}{\rm Tr}_{\sigma}({\boldsymbol{\sigma}}\hat{g})$ with $\mathcal{N}_{F}$ its density of state at the Fermi level, which acts as a bias injecting the spin current into the SML layer. Note that the spin chemical potential in the NM layer may come from the spin injection of the source layer such as the spin pumping of ferromagnetic layer, see Fig. 1. In principle, one should self-consistently determine the distribution function $\hat{g}({\bf p^{\prime}},t)$ or the spin chemical potential $\boldsymbol{\mu}_{s}$; this will involve the Boltzmann equation and boundary conditions for $\hat{g}({\bf p}^{\prime},t)$ at the interface of the NM and spin source layers. For the purpose of deriving the closed form of macroscopic dynamic equations, the presence of the spin chemical potential near the SML layer is sufficient. The second approximation is to assume that the electron transition across the interface conserves the spin and the interface transition rate $\hat{\Gamma}_{{\bf p}{\bf p}^{\prime}}$ is independent of the momentum ${\bf p}^{\prime}$ of electrons in the NM layer; this is the assumption frequently used for modeling electron tunneling or diffusion across a rough interface. It is, however, that $\hat{\Gamma}_{{\bf p}{\bf p}^{\prime}}$ does depend on the momentum ${\bf p}$ of electrons in the SML layer even for the rough interface since the electron spin in the SML layer is coupled to its momentum. The symmetry of the SML states demands $\hat{\Gamma}_{{\bf p}{\bf p}^{\prime}}$ to have the following form,

$$\hat{\Gamma}_{{\bf p}{\bf p}^{\prime}}=\left(\frac{1}{\tau_{c}}\right)+\left(\frac{1}{\tau_{s}}\right)\boldsymbol{\Omega}_{\bf p}\cdot{\boldsymbol{\sigma}}$$

(7)

where $1/\tau_{c}$ and $1/\tau_{s}$ characterize the sum and difference of the transition rate across the interface for two spin subbands $``\pm"$ of the SML, respectively. In the case of a single subband at the Fermi Level, i.e., a topological insulator band, $1/\tau_{c}=1/\tau_{s}$ Zhang and Fert (2016), while in the case of a Rashba band we assume that the transition rates for two subbands are the same in the limit $\varepsilon_{F}\gg\alpha{\rm p}_{F}$ and thus $1/\tau_{s}=0$.

Finally, we assume the momentum relaxation time is much faster than the transverse spin relaxation time ($\tau_{p}/\tau_{\phi}\ll 1$), as we have discussed above. Physically, $\tau_{p}$ is due to the impurity or defect scattering, while $\tau_{\phi}$ involves inelastic or interband scattering. More quantitatively, $\tau_{\phi}$ scales inversely with the strength of spin orbit coupling which is the order of several tens of meV for the TI or Rashba systems, and thus $\tau_{\phi}$ is about picoseconds while the momentum relaxation time is typically a few femtoseconds Shen et al. (2014). The separation between these two time scales are critically important since we are able to treat the dynamics for the longitudinal and transverse spins differently. In fact, we will limit our dynamic description between these two time scales such that the longitudinal spin reaches steady states instantly. Equivalently, we take the limit $\partial f_{c}/\partial t=0$ and $\partial\boldsymbol{f}_{L}/{\partial t}=0$ in the Boltzmann equation, and we focus the time dependence of the distribution function on the $``slow"$ dynamics of the transverse spin part $\boldsymbol{f}_{T}$.

With above simplifications, we can now explicitly establish the dynamic equations for three macroscopic variables by inserting the Boltzmann equation Eq. (4) along with the explicit relations of Eqs. (5), (II) and (7) into the definitions of the VHM, $\boldsymbol{\xi}(t)=\sum_{\bf p}{\rm Tr}_{\sigma}(\hat{\bf p}\times\boldsymbol{\sigma}\hat{f}),$ the SHM, $\eta(t)=\sum_{\bf p}{\rm Tr}_{\sigma}(\hat{\bf p}\cdot\boldsymbol{\sigma}\hat{f}),$ and the conventional magnetization, ${\boldsymbol{m}}(t)=\sum_{\bf p}{\rm Tr}_{\sigma}(\boldsymbol{\sigma}\hat{f})$. After a tedious but straightforward algebra, we obtain,

	$\displaystyle\frac{\partial{\boldsymbol{m}}}{\partial t}=$	$\displaystyle\omega_{0}\Big{[}\hat{\bf z}\times\boldsymbol{\xi}-\eta\hat{\bf z}\Big{]}+2\boldsymbol{h}_{ex}\times{\boldsymbol{m}}-\frac{{\boldsymbol{m}}-{\boldsymbol{m}}_{\rm p}}{\tau}$
		$\displaystyle+g_{\rm int}\Big{[}\big{(}\boldsymbol{\mu}_{s}\cdot\hat{\bf z}\big{)}\hat{\bf z}+\frac{1}{2}\big{(}\hat{\bf z}\times\boldsymbol{\mu}_{s}\big{)}\times\hat{\bf z}\Big{]}$		(8)
	$\displaystyle\frac{\partial\boldsymbol{\xi}}{\partial t}=$	$\displaystyle\omega_{0}\hat{\bf z}\times\Big{[}{\boldsymbol{m}}-{\boldsymbol{m}}_{\rm p}\Big{]}+2\eta\boldsymbol{h}_{ex}-\frac{\boldsymbol{\xi}-\boldsymbol{\xi}_{eq}}{\tau}$		(9)
	$\displaystyle\frac{\partial\eta}{\partial t}=$	$\displaystyle\omega_{0}\hat{\bf z}\cdot{\boldsymbol{m}}-2\boldsymbol{h}_{ex}\cdot\boldsymbol{\xi}-\frac{\eta}{\tau}$		(10)

where $\omega_{0}=2\alpha{\rm p}_{F}$ with ${\rm p}_{F}$ the Fermi momentum, $g_{\rm int}=e\mathcal{N}_{F}/\tau_{c}$ is an interface spin conductance, $1/\tau=1/\tau_{\phi}+1/\tau_{c}$ and $\boldsymbol{m}_{\rm p}=\sum_{\bf p}{\rm Tr}_{\sigma}[(\boldsymbol{f}_{\boldsymbol{L}}\cdot\boldsymbol{\sigma})\boldsymbol{\sigma}]$ reads

$\displaystyle{\boldsymbol{m}}_{\rm p}=$

$\displaystyle\frac{e\mathcal{N}_{F}\tau_{p}}{2(\tau_{p}+\tau_{c})}\hat{\bf z}\times(\boldsymbol{\mu}_{s}\times\hat{\bf z})+\tau_{c}q_{EE}\hat{\bf z}\times{\bf E}_{ex}$

(11)

where $q_{EE}$ is the EE coefficient,

$$q_{EE}=\begin{cases}\frac{\alpha e\tau_{p}m^{*}}{2\pi(\tau_{p}+\tau_{c})}&\mbox{for Rashba,}\\ -{{\rm sgn}(\varepsilon_{F})}\frac{e{\rm p}_{F}\tau_{p}}{4\pi(\tau_{p}+\tau_{c})}&\mbox{for TI}\end{cases}$$

(12)

where ${\rm sgn}(\varepsilon_{F})=+1$ for $\varepsilon_{F}>0$ and ${\rm sgn}(\varepsilon_{F})=-1$ for $\varepsilon_{F}<0$. Eq. (8)-(10) along with Eq. (11) are our main results and they can be broadly used for capturing the dynamics of the SML states in the presence of external fields and spin current injection. We shall point out that the difference between the total magnetization ${\boldsymbol{m}}$ and the ${\boldsymbol{m}}_{\rm p}$: the latter describes the shift of momentum center due to the presence of an electric field and spin current injection. Since we assume a fast relaxation for the electron momentum, ${\boldsymbol{m}}_{\rm p}$ is treated as a steady state solution. Microscopically, ${\boldsymbol{m}}_{\rm p}$ is comprised of the states with the longitudinal spins (relative to the spin-orbit filed direction ${\boldsymbol{\Omega}_{\bf p}}$), while the ${\boldsymbol{m}}$ includes both transverse and longitudinal spin components. It is the transverse component of ${\boldsymbol{m}}$ that gives arise to the time-dependent motion. When a spin current injected from the contact, its longitudinal spin component induces a spin accumulation ($\boldsymbol{m}_{\rm p}$) and thereby converts to an electric current, i.e., the inverse Edelstein effect, while its transverse spin component is equivalent to the spin transfer torque which drives the magnetization dynamics. One might compare this picture with the conventional spin injection to a ferromagnet where the transverse spin current leads to the spin transfer torque on a macros-spin while the longitudinal component generates a magnetoresistance (or spin accumulation).

These equations may be considered as an extension of the Landau-Lifshitz-Gilbert-Slonzcewski (LLGS) to the SML systems. For a conventional itinerant ferromagnet, the LLG equation involves only one macroscopic variable in magnetization; we have three coupled equations for three helix-dependent magnetizations $\boldsymbol{\xi}(t)$, ${\boldsymbol{m}}(t)$ and $\eta(t)$. Therefore, it may be more appropriate to compare Eq. (8)-(10) with the LLGS for antiferromagnets in which the dynamics of the staggered magnetic moment is always coupled to the magnetization because a time-dependent change of the staggered moment is only possible when the magnetic moment of each sublattice does not exactly compensate for each other. In the present case, our VHM and SHM are coupled to the in-plane and out-of-plane direction of the conventional magnetization through the spin-orbit coupling, respectively. In this Section, we solve these equations in two simple cases that can be readily tested experimentally.

We have considered the magnetic and spin transport properties in the presence of the time-dependent magnetic field and spin current injection from a contact. We want to comment on the differences of the SML spin transport compared to other materials.

Up until now, we have not included any interaction among spins, and thus it is more appropriate to classify the SML as a paramagnetic materials with a momentum dependent magnetic field on each particle. The magnetic resonant frequency is given by the strength of spin-orbit magnetic field $\alpha{\rm p}_{F}$, similar to the external magnetic field as the paramagnetic resonance. On the other hand, the spins of electrons of the SML is perfectly ordered in a helical state in the momentum space, similar to the ferromagnetic or antiferromagnetic spins ordered in real space. Although the spin ordering in the SML is not driven by the exchange interaction, there are some shared spin transport properties such as spin dephasing and spin pumping.

In conventional ferromagnetic systems, there are two different degree of freedom for the magnetization dynamics and spin transport. The magnetization density consists all electrons of occupied states while the spin transport is confined to the electron near or at the Fermi level. Thus the dynamics involves the time-dependence of the magnetization and of the conduction electrons. The magnetization dynamics are considered much slower than that of the conduction electrons, even though they are strongly coupled. In the SML system, both magnetization and transport are governed by the states near the Fermi level.

A key observation of the SML is the different spin relaxations for the longitudinal and transverse components. For conventional ferromagnetic metals, the spins have much shorter transverse relaxation time (or dephasing time) than the longitudinal spins, and thus the magnetization dynamics modeled by the LLGS do not address the conduction electrons, but the much slow dynamics of the local magnetization. While for the SML system, there is no spontaneous local magnetization. On the other hand, the transverse spins relax much slower than the longitudinal spins, and thus our dynamic equations address the dynamics of the conduction electrons.

The VHM and SHM provide a useful tool to visualize the spin orientation of the SML. When the VHM deviates from the ground state, the in-plane component of the VHM indicates the degree of the spin tilting away from the perfect perpendicular locking between the momentum of the spin.

In summary, we introduce three macroscopic variables for the SML and establish their equations of motion. Among other things, we have discussed the magnetic resonant frequency, spin injections associated with the EE and IEE, and we propose a spin pumping formalism.

This work was partially supported by US National Science Foundation under Grant No. ECCS-1708180, the National Natural Science Foundation of China(NSFC, Grant No.11434014) and China Scholarship Council (CSC, [2017]3109).