Signatures of Rashba spin-orbit interaction in charge and spin properties of quantum Hall systems

We study the local equilibrium properties of two-dimensional electron gases at high magnetic fields in the presence of random smooth electrostatic disorder, Rashba spin-orbit coupling, and the Zeeman interaction. Using a systematic magnetic length ($l_B$) expansion within a Green's function framework we derive quantum functionals for the local spin-resolved particle and current densities which can be useful for future studies combining disorder and mean-field electron-electron interaction in the quantum Hall regime. We point out that the spin polarization presents a peculiar spatial dependence which can be used to determine the strength of the Rashba coupling by local probes. The spatial structure of the current density, consisting of both compressible and incompressible contributions, also essentially reflects the effects of Rashba spin-orbit interaction on the energy spectrum. We show that in the semiclassical limit $l_B\to 0$ the local Hall conductivity remains, however, still quantized in units of $e^2/h$ for any finite strength of the spin-orbit interaction. In contrast, it becomes half-integer quantized when the latter is infinite, a situation which corresponds to a disordered topological insulator surface consisting of a single Dirac cone. Finally, we argue how to define at high magnetic fields a spin Hall conductivity related to a dissipationless angular momentum flow, which is characterized by a sequence of plateaus as a function of the inverse magnetic field (thus free of resonances).

Figure 1

Quantum (solid blue curve) and semiclassical (dashed purple and green dotted curves) local electronic density $n\left(x\right)$ [in units of $1/\left(2\pi l_B^2\right)$] as a function of the normalized electron position $x/l_B$ for a quadratic 1D potential (shown in units of $\hslash {\omega }_c$ as a shifted dashed-dotted black half parabola) with ${\omega }_0={\omega }_c/8$ and equilibrium chemical potential $\mu =3\hslash {\omega }_c$. In the three cases, relatively high temperature is chosen as $k_{B}T/\left(\hslash {\omega }_c\right)=0.06$. Pertinent numerical values for the dressed SO and Zeeman parameters are taken from Ref. [27], $S=0.88$ and $Z=-0.37$, and correspond to typical values experimentally found in InSb semiconductors, characterized by strong SO interaction. In the semiclassical approximation, comparison is made between Eq. (38) in the presence (dashed purple line) and the absence (green dotted line) of Rashba SO interaction. Note the significant difference in the position and width of the density plateaus between the $S\ne 0$ and the $S=0$ cases.

Figure 2

Quantum (solid blue curve) and semiclassical (dashed purple and green dotted curves) local spin polarization ${\Pi }^z\left(x\right)$ [in units of $\hslash /\left(4\pi l_B^2\right)$] as a function of the electron position $x/l_B$ for a quadratic 1D potential (not shown here) with ${\omega }_0={\omega }_c/8$, equilibrium chemical potential $\mu =3\hslash {\omega }_c$, and relatively high temperature $k_{B}T/\left(\hslash {\omega }_c\right)=0.06$. We use the same numerical values for the magnetic field dressed SO and Zeeman parameters as in Fig. 1. For the semiclassical expression (39) both the physical situations $S\ne 0$ (dashed purple line) and $S=0$ (green dotted line) are presented. The spin polarization oscillates within the leading quantum expression (33), considerably smoothed compared to the semiclassical formula (39). For strong SO interaction, the semiclassical spin polarization can even change its sign at particular spatial regions.

Figure 3

Semiclassical density-gradient (55) and drift (59) current density contributions [in units of $e{\omega }_c/\left(2\pi l_B\right)$] as a function of the normalized electron position $x/l_B$ for a quadratic 1D potential (represented in units of $\hslash {\omega }_c$ by the shifted dashed-dotted parabola) with ${\omega }_0={\omega }_c/8$, $\mu =3\hslash {\omega }_c$, and $k_{B}T/\left(\hslash {\omega }_c\right)=0.06$. Here, the SO and Zeeman parameters $S$ and $Z$ are chosen to be the same as in Fig. 1. The broadened peaks representing the edge states in the density-gradient component of the local current density show the spatial structure of the energy spectrum. Note the alternating sign of each of the contributions that combine to give nonnegligible spatial oscillations of the total current density.

Figure 4

Semiclassical Hall and spin Hall conductivity from Eqs. (61) and (69) (represented by the blue and green solid lines, respectively) expressed in units of $e^2/h$ and $e/4\pi$ as a function of the quantity $\mu /\hslash {\omega }_c$. Here, SO and Zeeman parameters $S$ and $Z$ are the same as in Fig. 1 and the temperature is chosen as $k_{B}T/\left(\hslash {\omega }_c\right)=0.03$. The dashed-dotted black line represents the classical result where the Hall conductivity grows linearly with the inverse of the magnetic field. The perpendicular dotted line marks the jump in the Hall conductivity that can be related to an SO imbalance below the Fermi level (as represented by the schema on top).