Realization of a Valley Superlattice

In a number of widely studied materials, such as Si, AlAs, Bi, graphene, ${\mathrm{MoS}}_2$, and many transition metal dichalcogenide monolayers, electrons acquire an additional, spinlike degree of freedom at the degenerate conduction band minima, also known as “valleys.” External symmetry-breaking fields such as mechanical strain, or electric or magnetic fields, can tune the valley polarization of these materials, making them suitable candidates for “valleytronics.” Here we study a quantum well of AlAs, where the two-dimensional electrons reside in two energetically degenerate valleys. By fabricating a strain-inducing grating on the sample surface, we engineer a spatial modulation of the electron population in different valleys, i.e., a “valley superlattice” in the quantum well plane. Our results establish a novel manipulation technique of the valley degree of freedom, paving the way to realizing a valley-selective layered structure in multivalley materials, with potential application in valleytronics.

Figure 1

(a) For 2D electrons confined to a wide AlAs QW, the degenerate $X$ and $Y$ valleys are equally occupied. (b) Positive symmetry-breaking, uniaxial strain ($ϵ={ϵ}_{[100]}-{ϵ}_{[010]}$) in the QW plane transfers 2D electrons from the $X$ valley to the $Y$ valley, and vice versa for negative $ϵ$. (c) Sample schematic. The surface is partially covered by a grating made of strips of electron-beam resist (blue strips). Each box in (d) represents the valley occupation of the 2DES directly above it, as also indicated by the density plots at the bottom of the panel. Here we portray $n_X$ and $n_Y$ plots as step functions for simplicity; however, a sinusoidal variation may be more realistic. Note that the total density $n$ is constant in all boxes. (e) Magnetoresistance traces for the patterned region (blue) and reference region (black) for $n\simeq 3.7\times {10}^{11}{\mathrm{cm}}^{-2}$. The inset shows the Fourier transform (FT) spectrum of the oscillations (marked by vertical arrows) in the blue trace; the green and red circles mark the expected positions of the maxima in the FT spectrum.

Figure 2

(a) Schematic of the experimental setup for the application of global, uniaxial strain ($ϵ$) with the sample and a strain gauge glued to the opposite faces of a piezoactuator. (b) Piezoresistance of the patterned region as a function of $ϵ$ for $n\simeq 3.7\times {10}^{11}{\mathrm{cm}}^{-2}$. We mark points $\mathbf{A}–\mathbf{E}$ on the trace. (c) Magnetoresistance traces corresponding to points $\mathbf{A}–\mathbf{E}$ in (b). The traces are vertically offset for clarity. Arrows mark the prominent CO minima observed in the black traces. At higher fields, Shubnikov–de Haas oscillations, originating from the formation of well-defined Landau levels, are observed. (d) Schematic of the spatial density profile of the $X$ and $Y$ valleys for points $\mathbf{A}–\mathbf{E}$. Note that the density distributions shown are only qualitative. (e) Examples of FT spectra for different strain values. (f) Plot of the observed FT frequencies squared versus $ϵ$. The dashed green and red lines represent the expected $ϵ$-induced evolution of the CO frequencies from the $X$ and $Y$ valleys, respectively, for $n\simeq 3.7\times {10}^{11}{\mathrm{cm}}^{-2}$.

Figure 3

Schematic of realizing valley helical edge states from a valley superlattice subjected to large perpendicular magnetic fields.