Valley-Polarized Quantum Hall Phase in a Strain-Controlled Dirac System

In multivalley systems, the valley pseudospin offers rich physics going from encoding of information by its polarization (valleytronics), to exploring novel phases of matter when its degeneracy is changed. Here, by strain engineering, we reveal fully valley-polarized quantum Hall phases in the ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}\mathrm{Se}$ Dirac system. Remarkably, when the valley energy splitting exceeds the fundamental band gap, we observe a “bipolar quantum Hall phase,” heralded by the coexistence of hole and electron chiral edge states at distinct valleys in the same quantum well. This suggests that spatially overlaid counterpropagating chiral edge states emerging at different valleys do not interfere with each other.

Figure 1

(a) Brillouin zone of ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}\mathrm{Se}$ projected onto the (111) surface. The $\overline{\mathrm{\Gamma }}$ and $\overline{M}$ valleys are shown in black and red. (b) Structure of the investigated QW samples. (c) Valleytronic phase diagram as a function of Fermi level and in-plane biaxial strain. Valley-polarized (red and black) and bipolar (purple) QH phases manifest themselves by different plateau filling factors. $N$ denotes the Landau level index. The investigated samples $A$, $B$, and $C$ are placed according to their properties.

Figure 2

QH effect measured in sample $A$ (holes) (a) and $B$ (electrons) (b) at $T=1.6\mathrm{K}$. The dashed red lines indicate the calculated plateau values. The insets illustrate the strained multivalley band structures with the corresponding Fermi level.

Figure 3

(a),(b) ARPES band maps of the confined subbands in strained 20 nm ${\mathrm{Pb}}_{1-\mathrm{x}}{\mathrm{Sn}}_{\mathrm{x}}\mathrm{Se}$ QWs at the $\overline{\mathrm{\Gamma }}$ and $\overline{M}$ points for different strain values. The valley splitting ${\mathrm{\Delta }}_{l-o}$ is indicated by the horizontal black and red lines. (c) Valley splitting as a function of in-plane biaxial strain deduced from ARPES and transport. The linear fit with Eq. (1) and error bars on $D_u$ are represented by the solid line and the shaded area. (d) QW gaps versus Sn content determined by magneto-optics. Blue dashed line is a guide for the eye. The black line corresponds to the calculated valley splitting. The crossing between blue and black lines delimits the bipolar regime (purple shaded area).

Figure 4

(a) QH measurements of sample $C$ at $T=1.6\mathrm{K}$. The inset illustrates the multivalley band alignment and the Fermi level. (b) Corresponding calculated Landau levels emerging at $\overline{M}$ (red at the front) and $\overline{\mathrm{\Gamma }}$ (black at the back). The Fermi level as well as the observed filling factor series are indicated. (c) Schematic illustration of the 2D Brillouin zone with the counterpropagating edge channels in different valleys giving $\nu =-2$. (d) Scheme of the $N=0$ Landau levels in real space at $\overline{M}$ (red at the front) and $\overline{\mathrm{\Gamma }}$ (black at the back) and the Fermi level leading to $\nu =-2$. The right-handed (left-handed) chiral edge states are indicated by filled (empty) circles.