Longitudinal and spin-Hall conductance of a two-dimensional Rashba system with arbitrary disorder

We calculate the longitudinal and spin-Hall conductances in four-lead bridges with Rashba-Dresselhaus spin-orbit interactions. Numerical results are obtained both within the Landauer-Büttiker formalism and by the direct evaluation of the Kubo formula. The microscopic Hamiltonian is obtained in the tight-binding approximation in terms of the neareast-neighbor hopping integral $t$, the Rashba spin-orbit coupling $V_R$, the Dresselhaus spin-orbit coupling $V_D$, and an Anderson-type, on-site disorder energy strength $W$. We reconfirm that below a critical disorder threshold, the spin-Hall effect is present. Further, we study the effect on the two conductivities of the Fermi energy, Rashba and/or Dresselhaus coefficient ratio and system size.

Figure 1

Graphical depiction of the lattice model used for computing the spin-Hall conductance. Four metallic leads (represented as the dashed regions) acting as injector (1), detector (2), and voltage probes (3 and 4) are attached to the 2DEG. Position and spin dependence are not explicitly decided for the hopping integral.

Figure 2

(a) The Fermi energy dependence of the spin-Hall conductance (SHC) of a two-dimensional four-probe bridge in the clean limit, for a fixed Rashba coupling $V_R=0.5$ and for different Dresselhauss energies. (b) SHC dependence on $V_D$ for different Rashba couplings, for Fermi energy $E_F=-2t$ in the clean limit. For $V_R=V_D$, the SHC vanishes. The system size is $20\times 20$. (c) The SHC represented as function of $E_F$ for $V_R=0.06$ and $V_D=0.0$.

Figure 3

The longitudinal conductance as function of the Fermi energy for different Dresselhaus SO couplings, as indicated. The system size is $20\times 20$.

Figure 4

(Color online) Longitudinal conductance plotted as a function of $V_R$ and $V_D$ for a system size $20\times 20$ and for a Fermi energy $E_F=-2t$. The spectrum is antisymmetric along the $V_D=V_R$ line. The spin-Hall conductance is positive for $V_R>V_D$, negative for $V_R<V_D$, and vanishes for $V_D=V_R$.

Figure 5

(Color online) The spin-Hall conductance plotted as a function of $V_R$ and $V_D$ for a system size $20\times 20$ and for a Fermi energy $E_F=-2t$. The spectrum is antisymmetric along the $V_D=V_R$ line. The spin-Hall conductance is positive for $V_R>V_D$, negative for $V_R<V_D$, and vanishes for $V_D=V_R$.

Figure 6

(Color online) Linear system size dependence of the spin-Hall conductance for a Fermi energy $E_F=2.0t$, with $V_R=0.5$ and for $V_D=\{0.0\left(엯\right),0.3\left(◇\right),0.6\left(▿\right),\text{and}0.9\left(▵\right)\}$.

Figure 7

(Color online) (a) Disorder strength dependence of the SHC with Rashba spin-orbit strength $V_R=0.4$. (b) SHC as function of Rashba coupling for different disorder strengths $W$. (c) SHC as function of Fermi energy and for different disorder strengths $W$. Electron-hole symmetry is preserved in the presence of disorder. In the upper panels Fermi energy is $E_F=-2.0t$. System size is $16\times 16$ in all figures.

Figure 8

(Color online) Longitudinal conductance as function of disorder strength (a), and as function of Rashba spin-orbit coupling strength (b) for different disorder amplitude. (c) Effect of disorder on the energy dependence of the longitudinal conductance. The system size is $16\times 16$. Disorder average is over 1000 samples.

Figure 9

(Color online) Longitudinal (a) and spin-Hall conductance (b) for a system size of $16\times 16$. Solid lines represent results obtained using the Kubo formalism while the dashed lines are obtained using the Landauer-Büttiker method. Results are average over 500 samples. Energy is measured in units of $t$.