Rashba spin orbit interaction in a quantum wire superlattice

In this work, we study the effects of a longitudinal periodic potential on a parabolic quantum wire defined in a two-dimensional electron gas with Rashba spin-orbit interaction. For an infinite wire superlattice we find, by direct diagonalization, that the energy gaps are shifted away from the usual Bragg planes due to the Rashba spin-orbit interaction. Interestingly, our results show that the location of the band gaps in energy can be controlled via the strength of the Rashba spin-orbit interaction. We have also calculated the charge conductance through a periodic potential of a finite length via the nonequilibrium Green's function method combined with the Landauer formalism. We find dips in the conductance that correspond well to the energy gaps of the infinite wire superlattice. From the infinite wire energy dispersion, we derive an equation relating the location of the conductance dips as a function of the (gate controllable) Fermi energy to the Rashba spin-orbit coupling strength. We propose that the strength of the Rashba spin-orbit interaction can be extracted via a charge conductance measurement.

Figure 1

Schematic of the proposed experimental setup. The top gates produce the parabolic confinement of the quantum wire while the finger gates produce the longitudinal periodic potential. The Rashba spin-orbit interaction is controlled by the backgate.

Figure 2

(a) Energy bands of the zeroth-order Hamiltonian ${\tilde{H}}_0$ for $l=50$ nm, $\lambda =62$ nm $=1.2l$, $\alpha =8$ meV nm ($q_{\text{R}}=0.21l^{-1}$), $k_{\text{c}}^{*}=2.3l^{-1}$, $c_0=1/2$, and $V_{\text{p0}}=0.5$. (b) Crossing of the energy bands corresponding to the $|k,0,-1\rangle$, $|k,1,+1\rangle$, and $|k-2\pi /\lambda ,0,-1\rangle$ states. (c) Crossing of the energy bands corresponding to the $|k,1,-1\rangle$, $|k,2,+1\rangle$, $|k-2\pi /\lambda ,1,-1\rangle$, and $|k-2\pi /\lambda ,0,+1\rangle$ states. (d) Same as (c) but for higher bands, i.e., $m\to m+1$.

Figure 3

(a) Energy bands of the Hamiltonian $\tilde{H}={\tilde{H}}_0+{\tilde{H}}_1$ for the same parameters as used in Fig. 2. The energy bands are calculated via direct diagonalization of $\tilde{H}$. (b) A blow up focusing on the band gaps of the first three crossings at $k_{\text{c}}^{*}$. (c) The band gaps formed at the energy band crossing shown in Fig. 2b. In (d) and (e), the energy band crossings shown in Figs. 2c and 2d, respectively.

Figure 4

Trajectories of the crossing points between the energy bands. The black dashed lines are the linearization of the left moving trajectories around ${\alpha }_0$. We also mark the values ${\alpha }^{*}=8$ meV nm and ${\alpha }_1=12$ meV nm onto the $y$ axis. These values are used for the conductance results in Fig. 6.

Figure 5

Schematic of the system used in our numerical simulations. The system is divided into a central region of length $L_x$, described by the Hamiltonian $H_{\text{C}}$ and two semi-infinite wires, described by the Hamiltonians $H_{\text{L}}$ and $H_{\text{R}}$. The total system is discretized on a grid with mesh size $a$. At the left and right edges of the central region the Rashba spin-orbit coupling is smoothly turned on.

Figure 6

Comparison between energy gaps of the infinite superlattice and conductance dips of the finite periodic potential. In (a) the results from the numerical calculation of the conductance is shown for ${\alpha }_1=12$ meV nm, and ${\alpha }^{*}=8$ meV nm. Note that the conductance curve for ${\alpha }_1$ has been shifted by $2\times 2e^2/h$ for clarity. The results for a wire with a nonperiodic potential $V_{\text{p}}\left(x\right)=1/2V_{\text{p0}}$ is plotted with black solid lines. In (b) and (c) the energy bands of the infinite wire superlattice are plotted for the same $\alpha$ values as in (a), respectively. Note that (c) is the same figure as Fig. 3b, but rotated clockwise by ${90}^{\circ }$. The correspondence between the nine pairs of dips and band gaps are indicated by the labeled arrows.

Figure 7

Plot of the differential conductance $\partial G/\partial E_{\text{F}}$ through the finite superlattice as a function of the Fermi energy $E_{\text{F}}$ and spin-orbit coupling, $\alpha$. The linearized trajectories of the crosspoints between the energy bands corresponding to the $|k,m+1,+1\rangle$ and $|k-2\pi /\lambda ,m-1,+1\rangle$ states are shown with gray solid lines. The trajectories are linearized around ${\alpha }_0=20$ meV nm. The Rashba spin-orbit values, ${\alpha }^{*}=8$ meV nm and ${\alpha }_1=12$ meV nm, corresponding to the curves in Fig. 6 are marked on the right border. Note that $q_{\text{R}}=m^{*}\alpha /{\hslash }^{2}l$ and that the differential conductance $\partial G/\partial E_{\text{F}}$ is in arbitrary units.

Figure 8

Charge conductance $G$ through the finite periodic potential as a function of Fermi energy $E_{\text{F}}$ for (a) different values of $V_{\text{p0}}$ and (b) different numbers of fingergates. All the conductances were calculated with $\alpha =12$ meV nm. Note that the conductances have been separated by $2\times 2e^2/\hslash$.

Figure 9

Plots of the differential conductance $\partial G/\partial E_{\text{F}}$ through the finite periodic potential as a function of Fermi energy $E_{\text{F}}$ and spin-orbit coupling, $\alpha$. Four cases are considered: A system with (a) 320, (b) 80, (c) 40, and (d) 10 fingergates, respectively. In all the plots we introduce a 5$\%$ Gaussian error in the length and height of the potential in each period.