Conductance resonances and crossing of the edge channels in the quantum Hall ferromagnetic state of Cd(Mn)Te microstructures

We report on the observation of very high and narrow magnetoconductance peaks which we attribute to the transition to the quantum Hall ferromagnetic (QHFM) state occurring at the edges of the sample. We show that the expected spatial degeneracy of chiral edge channels is spontaneously lifted in agreement with theoretical studies performed within the Hartree-Fock approximation. We indicate also that separated edge currents which flow in parallel may nevertheless cross at certain points, giving rise to the formation of topological defects or one-dimensional magnetic domains. Furthermore, we find that such local crossing of chiral channels can be induced on demand by coupling spin states of a Landau level to different current terminals and applying a dc source-drain voltage.

Figure 1

Energies of spin-split Landau levels $E_n^{\uparrow \downarrow }$ vs magnetic field $B$ (up to $n=6)$ calculated self-consistently using a modified Zeeman term taken from Eq. (1). Spin-up $(n\uparrow )$ and spin-down $(n\downarrow )$ ladders are indicated with blue and red colors, respectively; the black solid line shows the position of the Fermi energy $E_{\mathrm{F}}$. The chemical potential and spin polarization were calculated at temperature $T=0.050\mathrm{K}$ assuming $\mathrm{\Gamma }=0.1\mathrm{meV}$ as the Gaussian broadening of the density of states [15].

Figure 2

Spin-up $E_n^{\uparrow }$ (blue), spin-down $E_n^{\downarrow }$ (red) Landau levels and Fermi energy $E_{\mathrm{F}}$ calculated for magnetic field range 1 T to 2 T. Below, edge current picture of LLs crossings occurring at $B=B_{\mathrm{c}}$ within the shaded bands, as indicated above. (a) Position dependent energies of $(n\uparrow )$ and $(m\downarrow )$ LLs for $B\lesssim B_{\mathrm{c}},y$ is the spatial coordinate, ${\mathrm{\Delta }}^{*}=y_{\downarrow }-y_{\uparrow }$. (b) The same for $B\gtrsim B_{\mathrm{c}}$. (c) ${\mathrm{\Delta }}^{*}$ vs source-drain voltage $V_{\mathrm{sd}}$ (adopted from [7]).

Figure 3

Hall resistance $R_{\mathrm{H}}$ and phase of Hall voltage $\mathrm{\Theta }$ vs magnetic field for $B<0$. An image of the sample with numbered contact areas and separating grooves is shown on the left. Pads 1 and 6 were used as current contacts, pads 2 and 8 as voltage probes. Nonlabeled contacts were not used. Below, the $I\text{−}V$ characteristic of the 1-8 terminal pair is also shown, data were recorded at $B=1.508$ T.

Figure 4

$G_{18,26}$ (in $G_0=2e^2/h$ units) and current $I_{18}$ at a base temperature $T=0.01\mathrm{K}$ vs increasing magnetic field $B$. The relative phase $\mathrm{\Theta }$ of the $V_{\mathrm{ac}}$ and $V_{26}$ signals vs magnetic field $B$ is shown below. Shaded bands represent the magnetic field ranges, where $I_{18}\to 0$; corresponding filling factors $\nu$ are also indicated.

Figure 5

Conductance $G_{18,26}$, voltage $V_{26}$, phase $\mathrm{\Theta }$ (left axis), and current $I_{18}$ (right axis) vs magnetic field $B$. It is an enlarged part of Fig. 4, which corresponds to the transition from filling factor $\nu =3$ to $\nu =2$.

Figure 6

Current carrying outer edge channels (blue) and inner edge channels (red), which are disconnected at contact areas but still coupled to them by the internal capacitors $C_{\mathrm{i}}$. The geometry of our sample is reproduced schematically on the left. The simplified electrical equivalent scheme is shown on the right. For cable capacitance $C_{\mathrm{e}}=400\mathrm{pF}$ and $C_{\mathrm{i}}=5\mathrm{pF}$ [24] we obtained almost exactly $\left|\mathrm{\Theta }\right|=\pi /2$.

Figure 7

Conductance, current and phase vs increasing magnetic field, obtained for (26, 18) and (12, 68) contact configurations at $T=0.01$ K. $B$ values in the left panel and $\mathrm{\Theta }$ values in the right panel were multiplied by $-1$, for clarity.

Figure 8

Detailed view of the magnetoconductance resonances observed around $B=1.5\mathrm{T}$. Current $I_{18}$ data are also shown (as indicated in the upper panel). Phase $\mathrm{\Theta }$ vs $B$ is presented below (right axis) together with the bulk energies of the $E_5^{\downarrow },E_2^{\uparrow }$ and $E_6^{\downarrow }$ Landau levels, along with the position of the Fermi energy $E_{\mathrm{F}}$ (left axis).

Figure 9

Left: $G_{18,26}$ resonances observed around $B=1.5\mathrm{T}$ for two temperatures and for opposite directions of magnetic field change (as indicated). The phase $\mathrm{\Theta }$ vs decreasing magnetic field is shown below. Right: A schematic view of the outermost edge currents at the resonance $\left(B=B_{\mathrm{c}}\right)$ and off-resonance $\left(B\ne B_{\mathrm{c}}\right)$. Interchannel scattering is marked with dotted lines. Edge states, which flow closer to the boundaries, are not shown.

Figure 10

$G_{18,26}$ and $I_{18}$ vs source-drain voltage $V_{\mathrm{sd}}^{*}$ recorded at constant magnetic field tuned slightly off-resonance. The relative change of the phase of the ac voltage signal is shown below. Arrows indicate the opposite directions of the $V_{\mathrm{sd}}^{*}$ sweep. The inset shows a schematic of edge currents at local crossings, which result in $\mathrm{\Delta }\mathrm{\Theta }=\pm \pi$. Channel ordering at the entrance to the voltage probe corresponds to Fig. 2.

Figure 11

Conductance $G_{18,26}$ (upper panel) and relative change of voltage phase (lower panel) vs source-drain voltage $V_{\mathrm{sd}}^{*}$ recorded at magnetic field $B=1.508\mathrm{T}$ for different temperatures. In the upper panel, data are shown for both directions of the $V_{\mathrm{sd}}^{*}$ sweep, and were shifted vertically for clarity. The $\mathrm{\Delta }\mathrm{\Theta }$ data are shown for decreasing source-drain voltage only.