Half-Mirror for Electrons in Quantum Hall Copropagating Edge Channels in a Mach-Zehnder Interferometer

A half-mirror that divides a spin-polarized electron into two parallel copropagating spin-resolved quantum Hall edge channels one half each is presented in this study. The partition process is coherent, as confirmed by observing the Aharonov-Bohm oscillation at high visibility of up to 85% in a Mach-Zehnder interferometer, which comprises two such half-mirrors. The coherence length of the interferometer exceeds $200\text{μ}\mathrm{m}$, which reflects the robust nature of the copropagating channels against the decoherence caused by the coupling to the environment. In addition, the device characteristics are highly stable, making the device promising in the application of quantum information processing. The beam-splitting process is theoretically modeled, and the numerical simulation successfully reproduces the experimental observation. The partition of the electron accompanied by the spin rotation is explained by the angular momentum transfer from the orbital to the spin via spin-orbit interactions.

Figure 1

(a) Schematic of the BS device. Channels 1 and 2 are indicated by the red and blue lines, respectively. (b) Optical micrograph of the sample with the gate and Ohmic contact configuration. Brighter regions indicate the five metallic gates annotated as gate $G_a$ and spin-filter gates ${\mathrm{SF}}_{b,c,d,e}$. Filling factors under the gates are annotated as ${\nu }_{a,b,c,d,e}$. (c) Magnified view of the BS obtained by a scanning electron microscope (gates are yellow). The parameters are set to ${\nu }_a={\nu }_d=0$ and ${\nu }_b={\nu }_c=1$. For the measurement of ${\mathcal{T}}_{\downarrow }$, $V_{\mathrm{ac}}$ is applied to contact C1, and the currents at contact C5 and C3 are measured as $I_{\uparrow }$ and $I_{\downarrow }$, respectively. (d) Magnified view of the MZI. The settings for the interferometry are ${\nu }_c={\nu }_d=1$, ${\nu }_b={\nu }_e=0$; $V_a$ is applied in the range of ${\nu }_a=0$. For the measurement of the interference, $V_{\mathrm{ac}}$ is applied to contact C2, and the current at contacts C7 and C5 are measured as $I_{\uparrow }$ and $I_{\downarrow }$, respectively.

Figure 2

Outputs of the beam splitter. The data are obtained in the configuration shown in Fig. 1 with $V_b=V_c=-0.28\mathrm{V}$ (${\nu }_b={\nu }_c=1$) and $V_d=-0.5\mathrm{V}$ (${\nu }_d=0$). (a) $V_a$ dependence of $G_{\uparrow ,\downarrow }$ at $B=4.1\mathrm{T}$ ($\nu =3.8$). The inset presents the data from $V_a=0.1\mathrm{V}$ to $-0.3\mathrm{V}$. (b) Color plot of $G_{\downarrow }$ as a function of $V_a$ and $B$.

Figure 3

A grayscale plot of the electron density $N$ at a depth of $64\mathrm{nm}$ from the surface, which is calculated with the finite-element method described within the Supplemental Material [38] for various center gate voltages $V_a$. The equipotential lines of $(1/2)(N_0/\nu )$ and $(3/2)(N_0/\nu )$, where $\nu =4$ and $N_0$ are the electron densities in the nongated region, are plotted in red and blue, indicating the position of channels 1 and 2, respectively. The voltage of the spin filters is set to $-0.28\mathrm{V}$ in this simulation, which is the same as that in the experiment. (a) Plot of $V_a=-0.34\mathrm{V}$. (b) Plot of $V_a=-0.5\mathrm{V}$. (c) Plot of $V_a=-0.84\mathrm{V}$.

Figure 4

(a) $G_{\downarrow }$-output of the MZI device on $\nu =4$ plateau as a function of $B$. The gate voltages are set to achieve the channels, as illustrated in Fig. 1. $V_a$ on $G_a$ is fixed to $-0.75\mathrm{V}$. $V_{\mathrm{ac}}=10\text{μ}{\mathrm{V}}_{\mathrm{rms}}$. (b) Red dots indicate the width of the incompressible strip $w$ obtained from the oscillation data as $w=(2/3)(h/e)/L\mathrm{\Delta }B$ at $V_a=-0.75\mathrm{V}$, where $\mathrm{\Delta }B$ is estimated to be twice the distance between the adjacent oscillation peak and dip, and $L=8.28\text{μ}\mathrm{m}$. The blue line indicates $w$ calculated from Eq. (2) with the following parameters: $g^{\ast }=-0.6$ [32]; $ϵ=12.35$ [41].

Figure 5

Color plot of measured $G_{\downarrow }$ as a function of $V_a$ and $B$ obtained for the MZI.

Figure 6

(a) Temperature dependence of the visibility $\mathcal{V}$ at various magnetic fields $B_{av}$ indicated in the figure. The inset shows the original traces of $G_{\downarrow }$ versus $B$ taken at various temperatures $T$ ranging from $14\mathrm{mK}$ to $900\mathrm{mK}$. $V_a$ is fixed to $-0.75\mathrm{V}$. $\mathcal{V}$ is extracted from the adjacent oscillation peak and dip and $B_{av}$ is the center field between them. (b) Coherence length $l_{\phi }$ deduced from the relation $l_{\phi }=2L{T}_0/T$ [54] at $T=20\mathrm{mK}$, where $T_0$ is extracted from the exponential fits according to Eq. (3).

Figure 7

(a) Schematic of the model used for the simulation of the BS. Spin-filter gates ${\mathrm{S}}_{\mathrm{FR}}$ and ${\mathrm{SF}}_L$ correspond to ${\mathrm{SF}}_b$ and ${\mathrm{SF}}_c$ in the experiment, respectively. The potential is set to $V_{\mathrm{SF}}$ in the yellow regions. The blue-gray triangular gate $G_{\mathrm{BS}}$ with potential $V_{\mathrm{BS}}$ corresponds to gate $G_a$ in the experiment. Outgoing normal leads are attached to the left side of the sample. (b) Calculated distribution of the ${\sigma }^z$ component $⟨{\sigma }^z⟩$ of the current carrying state, (c) ${\sigma }^x$ component $⟨{\sigma }^x⟩$, and (d) ${\sigma }^y$ component $⟨{\sigma }^y⟩$.

Figure 8

Calculated transition probability ${\mathcal{T}}_{\downarrow }^{\mathrm{cal}}$ as a function of three different parameters. (a) ${\mathcal{T}}_{\downarrow }^{\mathrm{cal}}$ as a function of the height $H$ of the triangular gate potential, with a fixed width of $W=25a$. The strength of SOI is set to ${\theta }_R=0.02\pi$ and ${\theta }_D=0.01\pi$. $\xi =0.5a$. (b) ${\mathcal{T}}_{\downarrow }^{\mathrm{cal}}$ as a function of the decay length of the potential $\xi$. Other parameters are as follows: $H=70a$, $W=25a$, ${\theta }_R=0.02\pi$, ${\theta }_D=0.01\pi$. (c) ${\mathcal{T}}_{\downarrow }^{\mathrm{cal}}$ as a function of the strength of the Rashba SOI ${\theta }_R$. Other parameters are as follows: $H=70a$, $W=25a$, ${\theta }_D=0.01\pi$, and $\xi =0.5a$.