Interference in spin-orbit coupled transverse magnetic focusing: Emergent phase due to in-plane magnetic fields

Spin-orbit (SO) interactions in two dimensional systems split the Fermi surface and allow for the spatial separation of spin states via transverse magnetic focusing (TMF). In this paper, we consider the case of combined Rashba and Zeeman interactions, which leads to a Fermi surface without cylindrical symmetry. While the classical trajectories are effectively unchanged, we predict an additional contribution to the phase, linear in the applied in-plane magnetic field. We show that this term is unique to TMF and vanishes for magnetic (Shubnikov de Haas) oscillations. Finally we propose some experimental signatures of this phase.

Figure 1

The focusing setup with focusing length $l$. We choose axis $x$ and $y$ such that $x$ is aligned along the axis of injection. We locate the source at $(x,y)=(0,0)$. The in plane magnetic field angle $\phi$ is measured from the $y$ axis. The dashed line indicates the magnetic field orientation.

Figure 2

A cartoon of the trajectories in momentum space with a Rashba spin orbit interaction, with an applied in plane magnetic field, ${\mathbf{B}}_{\left|\right|}$. Red (blue) surface has $s=-1\left(s=1\right)$. The azimuthal angle $\varphi$ of the momentum, $\pi$, is presented, with ${\pi }_x$ and ${\pi }_y$ given by Eq. (9).

Figure 3

Trajectories of spin-orbit coupled holes, with $s=1$ in red, and $s=-1$ in blue. We use ${\tilde{\gamma }}_3=0.25$ and ${\tilde{g}}_1=1$ with an in plane magnetic field, $B_{\left|\right|}=4\mathrm{T}$. The Fermi energy is ${\varepsilon }_F=1.9\mathrm{meV}$. The upper panel has a in plane magnetic field orientation $\phi =0$, while the lower panel has a field orientation $\phi =\pi /2$. We present the trajectories normalized with the cyclotron radii, with $r_c\approx l/2$. The dashed lines in the upper and lower panel have $B_{\left|\right|}=0\mathrm{T}$. Note that the change in the trajectory is very small. We note that the injection with velocity directed fully along $x$ does not correspond to ${\varphi }_i=0$.

Figure 4

The two trajectories of injection angle $\theta$ and $-\theta$, measured from the $x$ axis at the point of injection.

Figure 5

Interference pattern calculated for pointlike source and detector from Eq. (28). We use $l=1500\mathrm{nm},k_F=0.107\times {10}^{-1}{\mathrm{nm}}^{-1}$ with a Rashba splitting ${\tilde{\gamma }}_3=0.3{\varepsilon }_F$, and ${\tilde{g}}_1{\mu }_B{B}_{\left|\right|}=0.15{\varepsilon }_F$, corresponding to $B_{\left|\right|}\sim 4–5\mathrm{T}$ for ${\tilde{g}}_1=1$ and ${\varepsilon }_F\approx 2\mathrm{meV}$. For clarity we present only a single spin state. The interference spectrum is calculated from Eq. (28), with a black vertical line indicating the location of the classical cutoff. Red plots in-plane field with $\phi =\pi$; black has $\phi =0$.

Figure 6

Interference patterns versus ${\tilde{g}}_1{\mu }_B{B}_{\left|\right|}$, with range $-0.1<{\tilde{g}}_1{\mu }_B{B}_{\left|\right|}<0.1$, with $\phi =0$. Here $l=2000\mathrm{nm},w=150\mathrm{nm}$. Positive ${\tilde{g}}_1{\mu }_B{B}_x$ is presented in red, while negative ${\tilde{g}}_1{\mu }_B{B}_x$ are in blue. The $B_x=0$ TMF spectrum is a solid black curve. Vertical black lines indicate the location of the classical maximum, $B_{\mathrm{focusing}}=2\hslash k_s/el$.