Electrostatic control of quantum Hall ferromagnetic transition: A step toward reconfigurable network of helical channels

Ferromagnetic transitions between quantum Hall states with different polarization at a fixed filling factor can be studied by varying the ratio of cyclotron and Zeeman energies in tilted magnetic field experiments. However, an ability to locally control such transitions at a fixed magnetic field would open a range of attractive applications, e.g., formation of a reconfigurable network of one-dimensional helical domain walls in a two-dimensional plane. Coupled to a superconductor, such domain walls can support non-Abelian excitations. In this paper we report development of heterostructures where quantum Hall ferromagnetic (QHFm) transition can be controlled locally by electrostatic gating. A high mobility two-dimensional electron gas is formed in CdTe quantum wells with engineered placement of paramagnetic Mn impurities. A gate-induced electrostatic field shifts the electron wave function in the growth direction and changes an overlap between electrons in the quantum well and $d\text{−}\mathrm{shell}$ electrons on Mn, thus controlling the $s\text{−}d$ exchange interaction and the field of the QHFm transition. The demonstrated shift of the QHFm transition at a filling factor $\nu =2$ is large enough to allow full control of spin polarization at a fixed magnetic field.

Figure 1

(a) In a QHE regime a potential barrier creates counterpropagating edge channels with the same polarizations, while (b) a filling factor gradient ${\nu }_2>\nu$ results in a formation of a chiral domain wall, $c$-DW. (c) A local change of the topmost Landau-level polarization results in the formation of a helical domain wall, $h$-DW, where counterpropagating edge channels have opposite polarization. Coupled to a superconducting contact (green), these $h$-DWs should support non-Abelian excitations (magenta dots). (d) Schematic of a reconfigurable $h$-DW in a multigate device.

Figure 2

(a) Energy spectrum of Landau levels in a CdTe QW with 1.5% of Mn calculated for $T=25$ mK. For the filling factor $\nu =2$ (gray shadow) electron gas undergoes ferromagnetic phase transition at $B^{*}\left(V_g\right)$. Field dependence of spin sub-bands [Eq. (1)] is plotted in the inset. (b) Spectrum for composite fermion $\mathrm{\Lambda }$ levels for $x_{\text{eff}}=0.15\%$. QHFm transitions at $\nu =5/3$ and 4/3 have been experimentally observed [22].

Figure 3

(a) The band diagram of a 30-nm CdTe QW heterostructure device is modeled using the ${\mathrm{nextnano}}^3$ package [29]. Electron probability density (square of the modulus of the electron wave function) is calculated for different voltages on the top (b) and back (c) gates. In (d) an integral overlap $\chi \left(V_g\right)$ between Mn-doped regions $\left[{\text{Mn}}_1\right]$ and $\left[{\text{Mn}}_2\right]$, normalized to the value at zero gate voltage $\chi \left(0\right)$, is plotted as a function of the 2D gas density change for front (FG) and back (BG) gates. (e) Mn doping distribution (red regions) in different wafers.

Figure 4

(a) Longitudinal ($R_{xx}$) and Hall ($R_{xy}$) magnetoresistances in wafer A measured at $T=400$ mK for $V_{fg}=V_{bg}=0$. A peak at $B=7$ T is a QHFm transition between $|1\uparrow \rangle$ and $|0\downarrow \rangle$ states. (b) Magnetoresistance in wafer C measured at $T\approx 30$ mK for various $V_{bg}$ from $-200$ V (bottom trace) to $+200$ V (top trace); the traces are offset proportional to $V_{bg}$. The blue arrow marks evolution of the $m=2$ node; the red arrow marks evolution of SdH peaks. In (c) and (d) $R_{xx}$ in wafer A is plotted as a function of $V_{bg}$ or $V_{fg}$ at a fixed $V_{fg}=0$ or $V_{bg}=100$ V, respectively. Position of the QHFm transition is highlighted by a white line. For $B=7$ T polarization of the top LL can be switched between $\uparrow$ and $\downarrow$ by the gate. Both plots have the same color scale. Measurements are performed at $T=300$ mK.

Figure 5

(a) Gate dependence of the measured effective Mn concentration, ${\chi }_{\text{eff}}\left(V_G\right)$ for wafers A-D for front (open symbols) and back (solid symbols) gates. Efficiency of $s\text{−}d$ exchange control depends on the $|d{x}_{\text{eff}}/dn|$ slope: for QHFm transition in the integer QHE regime $d{B}^{*}/dn=d{x}_{\mathrm{eff}}/dn$. (b) Arrhenius plot of the $R_{xx}$ vs $T$ dependence at the QHFm transition; the activation energy is 0.096 meV. Top inset: Temperature dependence of $R_{xx}$ near $\nu =2$. Spin texture in the vicinity of an anticrossing of $|0,\uparrow \rangle$ and $|1,\downarrow \rangle$ levels calculated using the spin-orbit Hamiltonian (see text).