Rashba spin-orbit interaction and birefringent electron optics in graphene

Electron optics exploits the analogies between rays in geometrical optics and electron trajectories, leading to interesting insights and potential applications. Graphene, with its two dimensionality and photon-like behavior of its charge carriers, is the perfect candidate for the exploitation of electron optics. We show that a circular gate- or doping-controlled region in the presence of Rashba spin-orbit interaction in graphene may indeed behave as a Veselago electronic lens but with two different indices of refraction. We demonstrate that this birefringence results in complex caustics patterns for a circular gate, selective focusing of different spins, and the possible direct measurement of the Rashba coupling strength in scanning-probe experiments.

Figure 1

(a) Electron flux in graphene coming along the $x$ direction onto a circular gate potential covering an area of radius $R$; the normal electric field reverses the carrier character from electron to hole and also generates a Rashba spin-orbit field. (b) Three-dimensional map of ${\left|\psi \right|}^2$ near and inside gated region centered at $(x,y)=(0,0)$ (normalized to incident flux). The electronic patterns produced inside the gated region are analogous to those produced by the refraction of light in a circular medium with refractive index $n=-1$ (Veselago lens). The electronic index of refraction is determined by the ratio of wave numbers in and out of the gated obstacle: both regions are characterized by a linear electronic dispersion (upper-left inset) and, in this case, $kR=300$ (electron) and $k^{\prime }R=300$ (hole). High-intensity maxima correspond to caustics with $p=1$ $(x<0)$ and $p=2$ $(x>0)$, where $p-1$ is the number of internal reflections. $x$ and $y$ coordinates are in units of the radius $R$. (c) In the presence of Rashba spin-orbit interaction the electronic dispersion in the gated region is modified, so that scattering particles have access to two different wave numbers (upper-right inset). Scattering produces electronic patterns as shown in the three-dimensional map ${\left|\psi \right|}^2$, with $kR=300$, $VR/\left(\hslash v_F\right)=600$, ${\lambda }_{R}R/\left(\hslash v_F\right)=100$, for $\uparrow$-spin incoming flux. The $p=1$ and $p=2$ caustics and cusps are doubled, associated with states of different chirality in graphene. The pattern can be described by the refraction from a medium with two (negative) indices, $n_{\pm }$. Here, the degree of birefringence is $\Delta n=n_{-}-n_{+}=0.71$. The dual pattern persists even for spin-unpolarized incidence.

Figure 2

Probability density patterns (scale bars show log of amplitudes, normalized to incident flux) resulting from the scattering of an incoming $\uparrow$-spin electron wave along the $x$ direction with $kR=300$, in the presence of a gate potential $\mathit{\text{VR}}/\left(\hslash v_F\right)=600$, and Rashba coupling ${\lambda }_{R}R/\left(\hslash v_F\right)=3$. This arrangement leads to a system with a small degree of birefringence $\Delta n=n_{-}-n_{+}=0.02.$ (a) The spin-preserving component $|{\psi }_{\uparrow \uparrow }{|}^2$ and (b) spin-flip component $|{\psi }_{\uparrow \downarrow }{|}^2$ display Rashba oscillations with a wavelength covering a large region of the scattering target, due to the small ${\lambda }_R$. (c) Total wave function ${\left|\psi \right|}^2=|{\psi }_{\uparrow \uparrow }{|}^2+{\left|{\psi }_{\uparrow \downarrow }\right|}^2$ displays caustics and cusps that remain almost unchanged from the case in which the Rashba interaction is absent. (d) Net spin $\eta \sim |{\psi }_{\uparrow \uparrow }{|}^2-{\left|{\psi }_{\uparrow \downarrow }\right|}^2$ quantifies the predominance of a given spin in different regions of the system. The large wavelength of Rashba oscillations leads to a cusp at $x\approx -0.5$ with a net $\downarrow$ spin (blue) while the annular region near $x\simeq 0$ has a net $\uparrow$ spin (red).

Figure 3

Wave-function maps (scale bar shows log of amplitudes; normalization is to incident flux) similar to those in Fig. 2 but for larger ${\lambda }_R$. Here, $kR=250$, $\mathit{\text{VR}}/\left(\hslash v_F\right)=600$, and ${\lambda }_{R}R/\left(\hslash v_F\right)=30$. This arrangement leads to a system with a degree of birefringence $\Delta n=n_{-}-n_{+}=0.24$. The different energy of the incoming wave leads to modified positions of caustics and cusps. (a) $|{\psi }_{\uparrow \uparrow }{|}^2$ and (b) $|{\psi }_{\uparrow \downarrow }{|}^2$ show clearer and sharper Rashba oscillations with shorter wavelength. Notice that the number of oscillations inside the gated region is approximately ${\lambda }_{R}R/\left(2\hslash v_F\right)=15$ for both spins, as shown by the rings around the main cusp. (c) The total wave function ${\left|\psi \right|}^2$ displays clear duplicate sets of caustics and cusps manifesting the birefringent character of the scattering, due to the large Rashba interaction and associated different group velocities. (d) The net spin $\eta$ shows variation of the two spin components along the the different cusps and caustics.

Figure 4

Probability density patterns (scale bar shows log of ${\left|\psi \right|}^2$ amplitudes) near and inside gated region centered at $(x,y)=(0,0)$. Parameter values are identical to those in Fig. 1 in main text. This figure shows the contour/color map representation of the 3d version shown in text. (a) Corresponds to Fig. 1b in text, with ${\lambda }_R=0$; (b) corresponds to Fig. 1c in text, ${\lambda }_{R}R/\hslash v_F=100$.