Original research April 2013

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. The full treatment shows the Rashba 2DEG SHE conductivity to be instead of , and Rashba heavy hole instead of . PACS: 03.65.Vf, 73.63.-b, 73.43.-f Corresponding author: Seng Ghee Tan Email: [email protected] Original research April 2013 S.G. Tan Page 2 Spin Hall Effect (SHE) Spin Hall Effect (SHE) [1-5] 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-dimensionalelectron-gas (2DEG) system with linear Rashba SOC. Careful analysis [8, 10-13] 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 tgeometric 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 [15-19] 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. Original research April 2013 S.G. Tan Page 3 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-ofmotion (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. Our work is thus divided into three sections. Section 2 is dedicated to unifying the locally gauge transformed energy landscape of a SOC system, providing the theoretical basis for a form-invariant, t-k manifestation of the gauge potential. Local transformation in this context is an abstract but useful technique to absorb the physics of spin dynamics into the gauge potential. Section 3 is dedicated to using the t-k from-invariant energy to derive the EOM for the SHE of the spin carrier. We show here that only a properly transformed, form-invariant t-k energy should be used to derive the SHE EOM that produces in clear-cut, and non-overlapping manner, the velocity components of kinetic, Yang-Mills, and geometric origins. Section 4 is to provide, using specifically the two-dimensional hetero-structure with Rashba SOC, a physical illustration of SHE resulting from all contributing velocity. Original research April 2013 S.G. Tan Page 4 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: