Effects of scattering area shape on spin conductance in a four-terminal spin-Hall setup

We study spin conductance in a ballistic and quasiballistic two-dimensional electron system with Rashba spin-orbit coupling. The system has a four-terminal geometry with round corners at the connection to the leads. It is found that by going from sharp corners to more round corners in the ballistic system the energy-depended spin conductance goes from being relatively flat to a curve showing a series of minima and maxima. It is also found that when changing the size of the terminal area by modifying the roundness of the terminal corners the maxima and minima in the transverse spin conductance are shifted in energy. This shift is due increased (decreased) energy for smaller (larger) terminal area. These results were also found to be reasonably stable in quasiballistic systems.

Figure 1

Schematic picture of the four-terminal system. Sharp corners are replaced by round corners with radii $R_{\text{C}1}$ and $R_{\text{C}2}$, see Fig. 1. The whole system has a length of $L_x=102a=600\text{nm}$ and width of $L_y=L_x$. Leads 1 and 2 are the main leads into the system with a width of $W_y=41a=241\text{nm}$ while leads 3 and 4 are the transverse leads into the system with a width of $W_x=18a=106\text{nm}$. The Rashba SO coupling is turned on smoothly on a $10a=58.8\text{nm}$ thick strip framing the system (hatched area in the figure). Inside the scattering region is shown a circle of width $D$ which we use to estimate the size of the terminal area.

Figure 2

(Color online) Spin conductance through lead 3 plotted as a function of Fermi energy. (a) correspond to the four-terminal shape where $R_{\text{C}}=29a$ and (b) correspond to $R_{\text{C}}=26a$. In both figures the spin conductance through a $R_{\text{C}}=0a$ system (sharp corners) is plotted for comparison. The bottom of the energy bands of transverse leads (with energy scaled in $E_{\text{t}}$) have been marked onto the figure. Note that the $y$ scale is unique for each of the figures.

Figure 3

(Color online) Schematic picture showing the effective shift in the transverse Rasbha energy bands, see (a), when the system gets wider, see (b). In (a) the left energy bands corresponds to the system where the width is $W_{\text{I}}$ and the right energy bands correspond to the system where the width is $W_{\text{II}}$. Here $k_x$ is the longitudinal $k$ vector of the electrons. The red lines (shifted to the right) correspond to spin-up states and the blue lines (shifted to the left) to spin-down states. At Fermi energies $E_1$ and $E_2$ we get additional states with low longitudinal velocities $\left(v_x\approx 0\right)$, marked with white dots, in addition to the states with positive velocities, marked with red and blue dots. In the terminal area the electrons from the original positive longitudinal velocity states can scatter into these newly open states.

Figure 4

(Color online) Spin conductance through lead 3 plotted as a function of radii of the corners and Fermi energy. In (a) all corners are equally varied while in (b) two corners are varied equally, the $R_{\text{C}1}$ corners, and two corners are kept fixed, $R_{\text{C}2}=30a$. The black dashed lines show the shift of $G_{\text{sH}}$ maxima in energy according to Eq. (11). The horizontal black lines mark where the spin conductance through lead 3 in Figs. 2a, 2b lie and the black circles show the ($R_{\text{C}}$, $E_{\text{F}}$) coordinates of the spin densities shown in Fig. 5. The vertical black lines mark the bottom of the transverse energy bands scaled in $E_{\text{t}}=5.46E_0$.

Figure 5

(Color online) Density of the spin $z$ component in the ballistic system at $E_{\text{F}}=56E_0$ for three different geometries. The shape in (a) has $R_{\text{C}}=29a$, the shape in (b) has $R_{\text{C}}=26a$, and the shape in (c) has $R_{\text{C}}=0a$ (sharp corners). These $\left(E_{\text{F}},R_{\text{C}}\right)$ values correspond to the spin-Hall values marked with black circles in Fig. 4a.

Figure 6

(Color online) Scattering shape using $R_{\text{C}}=29a$ corner radii and with $N_{\text{I}}=20$ static impurities added on random locations in the whole box of the scattering area. All impurities are set to have range ${\Gamma }_{\text{I}}=2a$, and the impurity strength is set at $V_{{\text{I}}_0}=1E_0$ for impurities in (a), $V_{{\text{I}}_0}=5E_0$ for impurities in (b), $V_{{\text{I}}_0}=10E_0$ for impurities in (c), and $V_{{\text{I}}_0}=50E_0$ for impurities in (d).

Figure 7

(Color online) The spin conductance through lead 3 and 4 as a function of energy for the cases shown in Fig. 6 compared to the spin conductance of the clean system, see Fig. 2a. Note that the $G_{\text{sH}}$ scale is shifted incrementally for each $V_{{\text{I}}_0}$ curve from the $V_{{\text{I}}_0}=1E_0$ curve.

Figure 8

(Color online) The spin conductance through lead 3 as a function of Fermi energy for a range of disorder strengths $U$. To compare we also plot he spin conductance of the ballistic system which we mark with $U=0t$. These results have been averaged over 1000 runs and the disorder strength is scaled in the tight-binding hopping term $t=171E_0$.