Torque and conventional spin Hall currents in two-dimensional spin-orbit coupled systems: Universal relation and hyperselection rule
Abstract
We investigate torque and also conventionally defined spin Hall currents in two-dimensional (2D) spin-orbit coupled systems of spin-1/2 particles within the linear-response Kubo formalism. We obtain some interesting relations between the conventional and torque spin Hall conductivities for the generic effective Hamiltonian H0=γk0+A(k)σx-B(k)σy , where A(k)=ηiAki+ηijAkikj+ηijlAkikjkl+⋯ , B(k)=ηiBki+ηijBkikj+ηijlBkikjkl+⋯ , and η ’s are the specific system-dependent coefficients. Specifically, we find that in the intrinsic case, the magnitude of torque spin Hall conductivity σxyτz(0) is always twice larger than the conventional spin Hall conductivity σxysz(0) , and the two conductivities have the opposite signs, i.e., σxyτz(0)=-2σxysz(0) . This universal relation, therefore, suggests that in the intrinsic case, the total spin Hall conductivity σxyz(0) in the 2D systems is equal to the conventional spin Hall conductivity in magnitude but has the opposite sign, namely, σxyz(0)=σxyτz(0)+σxysz(0)=-σxysz(0) . This universal relation also holds in the presence of a uniform in-plane magnetic field. We also find that if the 2D systems are rotationally invariant, there exists a hyperangular momentum Iz=(k×(∂θ)/(∂k))zsz+Lz which is conserved. Furthermore, the hyperangular-momentum current ⟨(1)/(2){Iz,vx}⟩ vanishes and this leads to a hyperselection rule for the conventional spin Hall current. In particular, in the 2D k -linear Rashba and wurtzite-type systems, Iz=sz+Lz , and the up(down)-spin current would always be accompanied by the down(up)-orbital-angular-momentum (OAM) current. In the 2D k -cubic Rashba, Iz=3sz+Lz , and the hyperselection rule is the same as in the k -linear Rashba system. In the 2D k -linear Dresselhaus system, on the other hand, Iz=-sz+Lz , and the up(down)-spin current would always be followed by the up(down)-OAM current.
- Publication:
-
Physical Review B
- Pub Date:
- March 2009
- DOI:
- 10.1103/PhysRevB.79.125301
- arXiv:
- arXiv:0808.3625
- Bibcode:
- 2009PhRvB..79l5301C
- Keywords:
-
- 71.70.Ej;
- 72.25.Dc;
- 73.63.Hs;
- 85.75.-d;
- Spin-orbit coupling Zeeman and Stark splitting Jahn-Teller effect;
- Spin polarized transport in semiconductors;
- Quantum wells;
- Magnetoelectronics;
- spintronics: devices exploiting spin polarized transport or integrated magnetic fields;
- Condensed Matter - Mesoscopic Systems and Quantum Hall Effect
- E-Print:
- 10 pages, 5 figures