Mesoscopic Transport in Electrostatically Defined Spin-Full Channels in Quantum Hall Ferromagnets

In this work, we use electrostatic control of quantum Hall ferromagnetic transitions in CdMnTe quantum wells to study electron transport through individual domain walls (DWs) induced at a specific location. These DWs are formed due to the hybridization of two counterpropagating edge states with opposite spin polarization. Conduction through DWs is found to be symmetric under magnetic field direction reversal, consistent with the helical nature of these DWs. We observe that long domain walls are in the insulating regime with a localization length of $4–6\mu \mathrm{m}$. In shorter DWs, the resistance saturates to a nonzero value at low temperatures. Mesoscopic resistance fluctuations in a magnetic field are investigated. The theoretical model of transport through impurity states within the gap induced by spin-orbit interactions agrees well with the experimental data. Helical DWs have the required symmetry for the formation of synthetic $p$-wave superconductors. The achieved electrostatic control of a single helical domain wall is a milestone on the path to their reconfigurable network and ultimately to a demonstration of the braiding of non-Abelian excitations.

Figure 1

(a) Calculated energy spectrum of Landau levels in CdMnTe with 1.7% Mn doping and $s-d$ overlap $0.9{\chi }_0$ (solid lines) and $1.1{\chi }_0$ (dashed lines), where ${\chi }_0$ is the overlap for zero gate voltages. $B^{*}$ marks QHFM transitions where the ground state changes from $|0\uparrow ⟩$ to $|1\downarrow ⟩$. (b) Experimentally measured shift of QHFM at $\nu =2$ as a function of front ($V_{\mathrm{fg}}$) and back ($V_{\mathrm{bg}}$) gate voltages. (c) Longitudinal and Hall resistance measured at an elevated temperature of 300 mK. The sharp peak at 7.3 T within the $\nu =2$ shaded region is a QHFM transition between fully polarized and unpolarized states, where the top filled Landau level changes polarization.

Figure 2

(a) Optical image of a sample; dark areas are etched, and yellow areas are covered by a top gate. The inset is an AFM image of a constriction, where the vertical gate boundary is clearly seen. (b) An artistic rendering of an AFM image at $\nu =2$ with a schematic flow of $|0\uparrow ⟩$, $|0\downarrow ⟩$, and $|1\downarrow ⟩$ edge channels assuming that the QHFM transition is gate tuned across the constriction. $|0\uparrow ⟩$ and $|1\downarrow ⟩$ states hybridize forming a helical domain wall. (c) Schematic of the measurement setup.

Figure 3

(a) The upper panel shows QHFM transitions for large ungated and gated areas. $R_{\mathrm{DW}}$ for $L=4$ and $6\mu \mathrm{m}$ constrictions is plotted in the lower panels. Dashed lines mark $B^{*}$ in gated and ungated areas. At low temperatures, the resistance of the $L=6\mu \mathrm{m}$ constriction almost vanishes, while for $L=4\mu \mathrm{m}$ it saturates to a nonzero value. (b),(c) $R_{\mathrm{DW}}(T)$ dependence for constrictions with different $L$ for $\mathrm{\Delta }{B}^{*}=0.11\mathrm{T}$ in device $A$ and $\mathrm{\Delta }{B}^{*}=0.25\mathrm{T}$ in device $B$ are plotted in an Arrhenius plot. Solid lines are fits to $R=R_0+A{e}^{-E_a/kT}$, and dashed lines are fits to thermally activated conduction with a gap $\approx 1\mathrm{K}$.

Figure 4

(a) The resistance of the HDW is symmetric under magnetic field reversal. (b) In the presence of chiral channels formed at a boundary between two different QHE states, the resistance is highly asymmetric under magnetic field reversal (highlighted regions are for the boundaries between $\nu =1$ and 2 and 2 and 3 QHE states).

Figure 5

(a),(b) Mesoscopic fluctuations measured in a device with $L=2\mu \mathrm{m}$ constriction at $T=27\mathrm{mK}$. In (a), the magnetic field was swept within the $\nu =2$ state 6.8–7.5 T. Fluctuations have a similar pattern with a quasiperiod of $\mathrm{\Delta }B\sim 40\mathrm{mT}$. In (b), $B$ was changed in a wide range of 5–10 T, and the fluctuation pattern changes drastically. (c) Energy diagram of a HDW formed at the gate boundary. Wiggling lines indicate schematically the role of disorder, and shaded areas are localized states in the tails of Landau levels. At low temperatures, conduction occurs via localized states in the gap. (d) Schematic of a conducting channel formed by coupled $\nu =2$ edge states. Electron tunneling via magenta in-gap states provide several interfering trajectories resulting in mesoscopic fluctuations of resistance.