Aharonov-Casher and spin Hall effects in mesoscopic ring structures with strong spin-orbit interaction

We study the quantum interference effects induced by the Aharonov-Casher phase in asymmetrically confined two-dimensional electron and heavy-hole ring structure systems, taking into account the electrically tunable spin-orbit (SO) interaction. We have calculated the nonadiabatic transport properties of charges (heavy holes and electrons) in two-probe thin-ring structures and compare how the form of the SO coupling of the carriers affects it. We show that both the SO splitting of the bands and the carrier density can be used to modulate the conductance through the ring. We show that the dependence on carrier density is due to the backscattering from the leads, which shows pronounce resonances when the Fermi energy is close to the eigenenergy of the ring. We also calculate the spin Hall conductivity and longitudinal conductivity in four-probe rings as a function of the carrier density and SO interaction, demonstrating that for heavy-hole carriers both conductivities are larger than for electrons. Finally, we investigate the transport properties of mesoscopic rings with spatially inhomogeneous SO coupling. We show that devices with inhomogeneous SO interaction exhibit an electrically controlled spin-flipping mechanism.

Figure 1

(Color online) Semiconductor quantum well patterned as a ring of radius $r_0$ in the presence of a magnetic field $B_0$ and of spin-orbit coupling. An electron (hole) spin traveling around the ring acquires phase due to the sum of the magnetic fields acting on it. The effective field, $B_{\text{eff}}$, is given by the yellow (○ markings) arrow, the applied out-of-plane magnetic field, $B_0$, is represented by the gray (x markings) arrow, and the momentum-dependent in-plane magnetic field due to spin-orbit interaction, $B_{\text{SO}}$, seen in green (● markings) arrow. The orientation of the spin-orbit field changes at different rates depending on the carriers due to the holes (solid arrow) having a cubic in momentum spin splitting and the electrons (dashed arrow) having a spin splitting that depends linearly on the momentum.

Figure 2

(Color online) The ring attached to two leads can be modeled as two three-way junctions. Each of these three-way junctions connects to one lead and the subsections corresponding to each of the arm in the ring.

Figure 3

(Color online) Calculation of the zero-temperature conductance in a single-mode ring connected to two leads based on the Landauer-Büttiker formula. The conductance in both electron and heavy-hole systems is modulated as a function of the spin-orbit coupling and the carrier concentration. In the left panel we present the result for an electron system and in the right panel for a heavy-hole system; both systems have an effective mass of $m^{\ast }=0.031m_0$.

Figure 4

(Color online) Calculation based on the tight-binding model of the zero-temperature conductance in a single-mode ring connected to two leads using a recursive $S$ matrix method. Each panel shows the conductance modulation as a function of both the spin-orbit interaction and the carrier density of the electrons when backscattering from the exit lead $N$ times. $N\to \infty$ corresponds to the full calculation shown in Fig. 3.

Figure 5

(Color online) Calculation of the zero-temperature spin Hall and longitudinal conductances in four terminal ring structure based on the Landauer-Büttiker formula. The spin Hall conductance (top panels) and longitudinal conductance (bottom panels) in electron and heavy-hole systems are modulated as a function of the spin-orbit coupling and carrier concentration. We present on the left the electron system where the spin Hall conductivity has a maximum value of $A=0.72e/\left(8\pi \right)$ and the longitudinal conductance has a maximum value of $B=1.2e^2/h$. In the heavy-hole system, shown on the right, the maximum value of the spin Hall conductivity is $A=1.0e/\left(8\pi \right)$ and the longitudinal conductance reaches a maximum value of $B=1.7e^2/h$.

Figure 6

(Color online) Calculation of the zero-temperature spin Hall conductance (four terminal device) based on the Landauer-Büttiker formula. The spin Hall conductance in electron systems is modulated as a function of the spin-orbit coupling and electron density. When the spin-orbit coupling is homogeneous (left picture), the spin Hall conductivity has a maximum value of $A=0.72e/\left(8\pi \right)$. When the spin-orbit coupling is present in only one half of the ring structure (right picture), the maximum value of the spin Hall conductivity is $A=1.07e/\left(8\pi \right)$.

Figure 7

(Color online) Calculation of the zero-temperature spin-resolved conductance based on the Landauer-Büttiker formula.