Gauge Physics of Spin Hall Effect

Spin Hall effect (SHE) has been discussed in the context of Kubo formulation, geometric physics, spin orbit force, and numerous semi-classical treatments. It can be confusing if the different pictures have partial or overlapping claims of contribution to the SHE. In this article, we present a gauge-theoretic, time-momentum elucidation, which provides a general SHE equation of motion, that unifies under one theoretical framework, all contributions of SHE conductivity due to the kinetic, the spin orbit force (Yang-Mills), and the geometric (Murakami-Fujita) effects. Our work puts right an ambiguity surrounding previously partial treatments involving the Kubo, semiclassical, Berry curvatures, or the spin orbit force. Our full treatment shows the Rashba 2DEG SHE conductivity to be instead of −, and Rashba heavy hole instead of −. This renewed treatment suggests a need to re-derive and re-calculate previously studied SHE conductivity.

. This renewed treatment suggests a need to re-derive and re-calculate previously studied SHE conductivity.
Spin Hall Effect (SHE) [1][2][3][4][5] refers generally to the transverse separation of the electron carriers of opposite spin, quantized along the axis-z, which results in a net accumulation of spin but not charge on the left and right lateral edges of a nanoscale device. There have been many studies of the numerous possible mechanisms that could have given rise to SHE, but the gauge theory approach by Murakami et al. 6 showed for the first time that in the Luttinger spin orbit coupling (SOC) system, SHE physics is related to the adiabatic alignment of electron spin with the spin orbit effective magnetic field in the momentum space. An emergent form of magnetic field, with spin quantization axis along the lab-z axis, can then be defined and linked physically to a transverse velocity component of geometric origin. Following this emergent gauge approach, SHE physics of k-geometric origin could be conveniently extended to many other systems, e.g. the linear and the cubic spin orbit in semiconductor and metal, pseudospin in massless and massive graphene, topological insulator and so forth 7,8 .
On the other hand, Sinova et al. 9 derived the SHE conductivity for a two-dimensional-electron-gas (2DEG) system with linear Rashba SOC. Careful analysis 8,[10][11][12][13] would reveal that the SHE conductivity is in fact related to the velocity of kinetic origin. In 2010, Fujita et al. derived a gauge field in time (t) space that also led specifically to the kinetic velocity contributing to SHE in the 2DEG. The time-space gauge field can, in turn be linked to a t-geometric velocity which has the same form as 13,14 the k-geometric velocity of Murakami. It is thus clear that one now should be particularly mindful of the multiple sources of velocity that contribute to the physics of SHE: kinetic, Murakami k-geometric, and Fujita t-geometric.
On the other hand, a separate body of work [15][16][17][18][19] which study the spin transverse force in terms of the non-Abelian spin orbit gauge, has led to the concepts of spin orbit force and spin orbit velocity. At first glance, one might be tempted to ascribe the transverse spin orbit force to SHE. But it was soon realized that while spin orbit force might contribute to the jittering motion (Zitterbewegung) of the spin carrier, it did not quite contribute to SHE yet. In fact, it is the spin orbit velocity that provides an additional source to the SHE. This results immediately in a SHE velocity originating from an emergent gauge reminiscent of the non-Abelian Yang-Mills gauge.
We are therefore motivated to provide, in this paper, a gauge-theoretic energy framework that unifies SHE velocity of kinetic, Yang-Mills, k-geometric, and t-geometric origins for any SOC system under one equation of motion (EOM). One unified energy system that merges the two spaces of t and k is derived, debunking any previous suspicion of overlapping energy terms. The energy equation with a merged t-k identity is then used to derive the velocity equation-of-motion (EOM) for all SHE systems. Previous efforts 13,14 unified Luttinger and Rashba SHE with respect to the adiabatic physics and the gauge fields, but still it remained that the Luttinger was described in k-space, and the Rashba in t-space.

Results
The main accomplishment in our renewed treatment is that we show that a form-invariant t-k Hamiltonian is a complete energy equation, and can be used to derive a complete EOM that describes spin Hall. The complete EOM reveals in clear-cut, and non-overlapping manner, the velocity components of kinetic, Yang-Mills, and geometric origins. The significance of this result is the prediction of a reversal of sign in the SHE conductivity, suggesting the need to revisit and recalculate previously derived SHE conductivity. The Methods section will describe in details the theoretical techniques used in the process of unification. Unification provides the theoretical basis for a form-invariant, t-k manifestation of the gauge potential, which is summarized in Table 1 above.
We will now use the t-k form-invariant energy to derive a complete spin Hall EOM for the spin carrier. In the energy physics of the SOC systems, we have shown in Methods and summarized in Table 1 the unified, t-k manifestation of the local gauge. It is reasonable to ascribe before merger, the k-gauge to the k-geometric velocity due to Murakami 6 , and the t-gauge to the t-geometric velocity due to Fujita 10 . What we have done, after merger is uniting the two velocities and precluding their simultaneous manifestation. We show that a physically intuitive spin Hall EOM that encompasses the kinetic, Yang-Mills, and Murakami-Fujita velocity can be derived from the locally transformed t-k energy equation. Local transformation in this context is an abstract but useful technique to absorb the physics of spin dynamics into the gauge potential. One is free to view the effective Hamiltonian in either time or momentum space. In time space, one can define an effective magnetic field of We would like to note that the velocity expression 〈 〉 v y z above follows from the more formal expression of y z y z , which in the case of a 2D system in lab frame leads to as used in Eqs. (1)&(2) above. The first term on the RHS is the kinetic velocity. The second term comprises the Yang-Mills and the Murakami-Fujita velocity as shown below The kinetic SHE velocity can also be written as is the simple spin orbit field. The kinetic spin velocity is thus Local gauge transformation and vector potential notation Hamiltonian in the locally rotated Rotation of the z-axis to B(t) clearly showing that the external electric field is required to generate ∂ t n. The expression t relates to the physics of E x producing an effective B field. For illustration in the 2D projected region surrounded by the equator, n Σ points along axis −z in the region of + p y , and axis + z in the region of − p y (see illustration in Fig. 1). In other words, n Σ actually changes sign with p y . Therefore, careful examination of the (+ ) band (Fig. 1a)   . What we would like to establish therefore in this work is that future treatment of SHE should be based on a locally contributing to the physics of SHE.

Discussion
One popular form of SHE exists in device or hetero-structure that exhibits the Rashba spin orbit coupling (RSOC). Examples are the GaAs/AlGaAs/GaAs semiconductor heterostructure, or the oxide/Metal/Pt metal multilayer, both with structural inversion asymmetry. Shown above in Fig. 2 is a schematic of the nanoscale structure where the specific RSOC exists as a result of inversion asymmetry at the interface. The effective magnetic field of the RSOC device is where α is the strength of the Rashba SOC effect. We will first examine a two-dimensional system, where the SHE current density is obtained from the SHE kinetic velocity as follows The coupling constant g represents spin flux for g = /2, or electron charge flux for g = e. It has been shown that, for g = /2, SHE conductivity σ = − π Referring to Eq. (5), we will now proceed to the Yang-Mills velocity. In the case of a linear SOC system, where ef fe c t ive SO C f ield is cont aine d in t he 2D plane, it is e asy to deter mine t hat t he Yang-Mills effect vanishes. We will move on to the last SHE term which is the Murakami-Fujita of , and which can in turn be broken down into two terms.
The first term produces SHE current density . This is in addition to the − π e 8 arising due to 〈 〉 v y z KE giving rise to a total SHE σ = + π y z e 8 . The advantage of physical clarity with the gauge theoretic approach is clearly manifest here. The first contribution to SHE conductivity originates from the kinetic velocity which is effective in the annular region of the 2D concentric circles. The second contribution originates from the Murakami-Fujita velocity that has a geometric origin, and is effective in the degenerate point where the (+ ) and the (− ) bands intersect. The same is carried out for the Rashba heavy hole system where with partial treatments. Both results are summarized in Table 2.  We will make a quick remark on the second part of 〈 〉 v y z MF , which is When integrated via partial fractional reformulation of the integrand, the integral delivers a " π" or a "-π" solution depending on the sequence in which the integration is performed. But a proper treatment referring to the Fubini-Tonelli theorem leads to its vanishing results.
Therefore, the central results in this paper consist of the unified energy and the SHE EOM. On the energy, we have provided a theoretical basis to the existence of the t-k interchangeable, form-invariant Hamiltonian. This Hamiltonian allows the physics of SHE in various SOC systems to be studied under one SHE velocity EOM. The EOM descended from the t-k energy would lead to the kinetic, Yang-Mills, and Murakami-Fujita velocity which give a complete account of all contribution to SHE conductivity in any SOC system. Our work puts right an ambiguity surrounding previously partial treatments of SHE involving the use of the Kubo, semiclassical, Berry curvatures, or the spin orbit gauge. We showed for the Rashba 2DEG and the Rashba heavy hole that full treatment produces SHE conductivity of opposite signs ( Table 2) due to the Murakami-Fujita contribution. More importantly, our renewed treatment can be extended to all other SOC systems to re-derive and re-calculate SHE conductivity.

Methods
Energy in the Unified Time-Momentum (t-k) Space. The Hamiltonian of a system with SOC can be written with the physical clarity of simple magnetism as follows: where B is a momentum dependent effective magnetic field, and γ has the dimension of γ = .

Joule Tesla
In a single-particle system with electric field, the Hamiltonian = + . E r H e p a m 2 2 might seem sufficient, at first glance, to describe a carrier with kinetic and potential energy. But a charge-spin carrier with a constant p in the presence of E field generates an energy term of γσ.B(p). The Dirac relativistic quantum mechanics is needed to account for this SOC energy. In the absence of any retardation effect due to scattering, the carrier would accelerate due to the E field. As a result, the carrier will acquire an E-dependent energy σ . This is a geometric related energy that can only be revealed with the local gauge transformation or time-dependent perturbation treatment.
Momentum Space. The approach that has been used to derive SHE velocity in refs 6,22 is based on a local gauge transformation in the k-space. In the "Schrodinger" picture, transformation applies to the k-space only, but not the t-space, because momentum is time-independent. Local transformation leads to where iU∂ t U † = i∂ t + iU(∂ t U † ), and the second term is contingent upon ∂k/∂t ≠ 0, a condition that would be fulfilled when E field is present in the device. We assign as follows: