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Spin Hall effect in a simple classical picture of spin forces

Spin Hall effect (SHE) in a 2D-Rashba system has been treated in the spin-dependent precession [J. Sinova et al., Phys. Rev. Letts. 92, 126603 (2004).] and the time-space gauge [T. Fujita et al., New J. Phys. 12, 013016 (2010).] approaches, both yielding SHE conductivity of {\sigma}_y^z=e/8{\pi} . Separate studies based on the concept of spin transverse force [S-Q Shen, Phys. Rev. Lett. 95, 187203 (2005).] provide a heuristic but not a quantifiable indication of SHE. In this paper, we provide a more complete description of the SHE using the Heisenberg approach, unifying the Yang-Mills force, the Heisenberg spin force, and the SHE {\sigma}_y^z under the classical notion of forces and accelerations. Central to this paper is the spin force equations that are satisfied by both {\sigma}_y^z and {\sigma}_x^z, the Yang-Mills, and the Heisenberg spin forces. By linking {\sigma}_y^z to the spin forces, one sees that the physics of SHE in a 2D-Rashba system is also a simple classical Lorentz force picture.


Introduction
Following the first description of the spin Hall effect (SHE) by M.I. Dyakonov and V.I. Perel [1], and an early experimental indication of such effect [2], scientific interest in SHE took a long silence before reemerging only in the 21 st century with numerous experimental verifications [3][4][5] and theoretical discussions [6][7][8]. SHE was discussed in various systems e.g. the Luttinger, Rashba, cubic and linear Dresselhaus and others with underlying spin orbit coupling. In the Rashba two-dimensional-electron-gas (2DEG) system in particular, SHE was described by Sinova et al. [9] with a theory of electron precession and its transverse separation, leading to a universal SHE conductivity of ⁄ . A few conclusions in this theory seem to also find support in separate studies. For example, the need for a momentum change ( ) means that impurity scattering might destroy SHE. This is consistent with Inoue's [10] prediction that SHE is destroyed by excessive impurities scattering (vertex correction) in a diffusive system. It had also been reasoned that impurities scattering is equivalent to a retardation force which results in average vanishing momentum change ( ) which might in turn destroy SHE.

A. Spin Force Equations for a Rashba system
In this section, we will ignore the effect of momentum change ( ), i.e. assuming there is enough scattering to suppress ( ). The spin orbit gauge used in this section is derived from a simple rewriting of the Hamiltonian. Thus, in this section is known as the simple spin orbit gauge.

Classical Yang-Mills
We will give a brief review of SHE physics described by the classical Yang-Mills fields and the spin transverse forces [14-18]. We will begin with the Hamiltonian (vector in bold notation) of a Rashba SOC system in the presence of an external electric field: ( in which is a vertical unit vector. Due to the coefficient, this effective Yang-Mills magnetic field can be treated like it is spin-dependent, i.e. it points up (down) when "seen" by electron of spin down (up).
electrons. For example, if a spin-down electron initially moving along -direction, it will feel a transverse force and move to the left, likewise, spin-up will experience a force in the opposite direction and move to the right, leading to the separation of spins and hence (3) The above is a "force" expression in the classical sense of Lorentz force and Hall effects. One can thus derive the Yang-Mills force as follows The physics is pictorially illustrated in Fig.1 which shows a Lorentz-force effect on electron of one spin orientation. The spin transverse force is denoted by . The above indeed gives a heuristic, and intuitive picture of spin separation due to , but the actual dynamics of electron remains unclear. For example, spin undergoes precession about the spin-orbit effective magnetic field (not the Yang-Mills B of Eq.2). As a result of this imagination, the Yang-Mills B field acting on each electron also flips up or down at the rate of precession, leading to the transverse oscillation of . Thus spin transverse force leads to spin transverse oscillation rather than the permanent spin separation of SHE.

Heisenberg approach
We will now show using the Heisenberg approach that the spin orbit gauge could lead to a set of spin force equations which relate electron acceleration due to to spin conductivity of and . We start with the classical equation of motion for force i.e. . Since in classical mechanics, , one can rewrite the force equation but now taking as an operator Applying the quantum dynamic of Of physical importance is the fact that each force component is perpendicular to a momentum coupled to it. For example, the term { } contains . Combining and , the spin force equations are:

SHE conductivity
We have seen that the Yang-Mills forces have been related to the Heisenberg spin forces. We will now set out to study their relations to the SHE conductivity. The spin current operator, that has been used in Refs [9,11,12] to derive SHE current or conductivity is: where spin current has the dimension of [ ] . One notes that simple vanishes from (12) With this, the SHE conductivity/current has been connected to the Yang-Mills and the Heisenberg spin forces. As an aside, one notes by inspection that the SHE conductivity (current) is related to the spin longitudinal force, but not the spin transverse force. One can then modify Fig.1. and construct a new picture below ( As mentioned earlier that once the link is established through the above, one can proceed to independently derive the explicit expression of or , noting that . The explicit expression of the SHE conductivity (current) will depend on the quantum state of the carrier i.e. | ⟩ , which can be obtained by solving the Hamiltonian. Figure 3 gives a pictorial summary of the theoretical processes in section A.

B. Spin Force Equations for a Rashba System with Time-gauge
This section revisits the theoretical processes in Section A except the spin orbit gauge used here is the time-space version, which describes a Rashba system with finite momentum change . As a matter of fact, with the simple spin orbit gauge of Section A, SHE conductivity (current) as shown in

SHE conductivity
Simple spin orbit gauge with the generator . But this will not produce the effect of momentum change or electric curvature.
On the other hand, the eigenstate of which is | ⟩ will only evolve in dynamic phase. Therefore, to obtain non-vanishing SHE conductivity (current) , one needs the time-dependent treatment [11,12], where an electric curvature (effective magnetic field) appears vertical to the 2D plane via a local gauge transformation in the momentum space-time. A sum field which is the vector sum of the vertical time-field and the in-plane spin orbit field could thus be defined. Electron with spin aligned parallel/antiparallel to the total field is identified with the bands. In summary, one ventures from the original to a time-dependent , and performs the local gauge transformation.
An inverse rotation would yield the Hamiltonian with the electric curvature. In the accelerating frame of electron, one obtains back a time-independent Hamiltonian of (15) where is perpendicular to , and the explicit expression is ( ) . We note in passing that two transformations have been performed to arrive at Eq.(15) see Ref [11,12] for details. In fact, noting that , one could reason with classical physics that as spin aligns to the sum field , a vertical component of spin polarization 〈 〉 ⁄ is generated, is the spin quantum number. Now, it can be derived, noting that ⁄ , and using the time-independent that the time-space spin orbit gauge is: (16) Note that the momentum is in-plane. With Eq.7(b), the Heisenberg spin force equations for the timespace spin orbit gauge is The SHE conductivity (current) of 〈{ }〉 ⟨ | {( ) } | ⟩ can, once again, be fitted with approximations into the spin force equations. Different from the simple Rashba system, in the time-space gauge, the expression ⟨ |{ }| ⟩ is non-vanishing. In fact it can be derived from the polarization expression of 〈 〉 ⁄ , which is non-vanishing, and finite. Therefore, the SHE conductivity (current) will be non-vanishing consistent with previous accounts. Figure 4 summarizes the theoretical processes similar to those in Section A except the above is carried out using the time-space spin orbit gauge, instead of the simple spin orbit gauge.

Classical Yang-Mills approach Heisenberg approach
Ignoring h.o. terms