Controlled Coupling of Spin-Resolved Quantum Hall Edge States

We introduce and experimentally demonstrate a new method that allows us to controllably couple copropagating spin-resolved edge states of a two-dimensional electron gas (2DEG) in the integer quantum Hall regime. The scheme exploits a spatially periodic in-plane magnetic field that is created by an array of Cobalt nanomagnets placed at the boundary of the 2DEG. A maximum charge or spin transfer of $28\pm 1\%$ is achieved at 250 mK.

Figure 1

(a) Schematics of the device. The Cobalt fingers (blue bars) produce a fringing field (yellow lines) resulting in an in-plane, oscillatory, magnetic field ${\overrightarrow{B}}_{\parallel }$ at the level of the 2DEG (textured gray) residing below the top surface. The field induces charge transfer between the spin up ${\Psi }_{\uparrow }$ SRES (red line) and spin down ${\Psi }_{\downarrow }$ SRES (blue line). (b) Density plot of the modulus ${\overrightarrow{B}}_{\parallel }$ in the proximity of the magnetic fingers on 2DEG plane. The dashed line indicates the end of the finger array at $0.2\mu \mathrm{m}$ from the physical edge of the mesa (white stripe). (c) Measurement setup: The ${\Psi }_{\uparrow }$ channel is excited by a bias voltage $V$, while ${\Psi }_{\downarrow }$ is grounded at the contact denoted by $G$. The SRESs can be reversibly decoupled by negatively biasing the array with a voltage $V_a$ ($G1$ and $G2$ are contacts for the top gates). (d) Optical image of the device showing four sets of magnetic fingers with different periodicity $\lambda$ placed serially at the mesa boundary (the yellow elements are gold eletrical contacts). Zoomed region is the scanning electron microscopic image of the array of periodicity $\lambda =400\mathrm{nm}$: it is nearly $6\mu \mathrm{m}$ long and has an overlap on the mesa of $0.2\mu \mathrm{m}$.

Figure 2

(a) Plot of the two-terminal magneto-current (2TMC) measured at 250 mK. The value of magnetic field $B=4.75\mathrm{T}$, indicated by an arrow, is used to place the 2DEG approximately at the center of the $\nu =2$ plateau. (b) Plot of the currents $I_{\uparrow }$ (red) and $I_{\downarrow }$ (blue) measured at the current terminals red and blue, respectively, [Fig. 1c] with the voltage $V_G$ applied to the gates $G1$ and $G2$, while the nanomagnets are deactivated by applying a voltage bias of $V_a=-3\mathrm{V}$ to all the arrays. The value of $V_G$ is set to $V_G^{*}$, indicated by an arrow, for separately contacting the spin-resolved edge states [see Fig. 1c]. (c) Temperature dependence of $I_{\uparrow }$ (red) and $I_{\downarrow }$ (blue) currents shows enhancement of relaxation between SRESs with increasing temperature. Thermally mediated mixing of currents becomes negligible at $T=250\mathrm{mK}$.

Figure 3

Upper panel: Plot of the transmitted currents $I_{\uparrow }$ (red) and transferred current $I_{\downarrow }$ (blue) as a function of the inverse periodicity of the activated (by applying $V_a=0$) set of nanofingers at the working point $B=4.75\mathrm{T}$ and $T=250\mathrm{mK}$. The measured current $I_{\uparrow }$ and $I_{\downarrow }$ are guided by the dashed line which demonstrates selectivity of nanomagnet at periodicity between $\lambda =400\mathrm{nm}$ and 500 nm. The inset shows a numerical simulation of transferred current which in the absence of the static disorder and/or inelastic mechanisms predicts a width of the peak that scales inversely on $\Delta X$ as in Eq. (1). Lower Panel: measured transferred current $I_{\downarrow }$ as a function of the perpendicular magnetic field $B$ for the nanomagnet array of periodicity $\lambda =400\mathrm{nm}$.

Figure 4

Upper panel: Dependence of the transmitted current $I_{\uparrow }$ (red) and transferred current $I_{\downarrow }$ (blue) upon the voltage $V_a$ applied to the activated nanofinger set of periodicity $\lambda =400\mathrm{nm}$ at $B=4.75\mathrm{T}$. For $V_a<-2.0\mathrm{V}$ the fingers are effectively decoupled from the SRESs with negligible transfer current; for $V_a\simeq 0$ instead the edges feel the presence of the fingers and a nonzero transfer of current is evident. For intermediate values of $V_a$ a series of pronounced peaks in $I_{\downarrow }$ are evident. Lower panel: contour plot of $I_{\downarrow }$ upon $V_a$ and $B$ for the nanofinger set of periodicity $\lambda =400\mathrm{nm}$. The horizontal dashed line indicates the center of the $\nu =2$ plateau ($B=4.75\mathrm{T}$).