s spin degree of freedom [0-00,00,00] , which is at the heart of "spintronics," another branch of unconventional, next-generation electronics [00,00] . Considering the similarities, valleytronics also offers an intriguing parallel route for novelty. For example, digital information may be stored and processed by compelling electrons to selectively occupy one valley or another. Moreover, coupling the valley degree of freedom with polarized light can result in exotic optoelectronic properties [0-0] . In this study, we investigate a two-dimensional electron system (0DES) contained in a 00-nm-wide AlAs quantum well (QW) structure. The QW, located 000 nm below the surface, is sandwiched between two AlGaAs spacer layers, all grown via molecular beam epitaxy on a (000) GaAs substrate (see Supplemental Material [00] for details). The high transport mobility ( ∼ 0 × 00 0 cm 0 / V s for our samples) makes the AlAs 0DES well suited for potential valleytronics applications. Its valley degree of freedom arises from a twofold valley degeneracy [0] . We denote these valleys as X and Y with the major axes lying along [000] and [000], respectively [see Fig. 0(a) ]. Without any symmetry-breaking, uniaxial strain ( ϵ ) applied to the QW plane, electrons are equally distributed between the two valleys, where they possess an anisotropic Fermi surface with longitudinal and transverse effective masses of m l = 0.00 and m t = 0.000 , respectively, in units of the free electron mass. Under finite ϵ , however, electrons move from one valley to the other, as described in Fig. 0(b) . Here ϵ = ϵ [ 000 ] - ϵ [ 000 ] , where ϵ [ 000 ] and ϵ [ 000 ] are the strain values along [000] and [000], respectively. (The single-particle valley splitting is given by ϵ E 0 , where E 0 is the deformation potential, ≃ 0.0 eV for AlAs [0] .) Taking advantage of this intervalley transfer of electrons, we report here how to engineer a 0D valley superlattice, namely, to reconstruct a valley-degenerate 0DES plane into multiple strips where the X and Y valleys alternate as the majority-valley species. We demonstrate such a lateral modulation of the X - and Y -valley occupation through measurements of commensurability oscillations. It is worth emphasizing at the outset that, unlike the typical commensurability phenomena [00-00] , the total charge density in our system stays uniform, and it is the modulation of individual valley densities that leads to the commensurability oscillations. 0 00.0000/PhysRevLett.000.000000.f0 FIG. 0. (a) For 0D electrons confined to a wide AlAs QW, the degenerate X and Y valleys are equally occupied. (b) Positive symmetry-breaking, uniaxial strain ( ϵ = ϵ [ 000 ] - ϵ [ 000 ] ) in the QW plane transfers 0D 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 0DES 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 ≃ 0.0 × 00 00 cm - 0 . 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 0(c) illustrates our approach to realizing a valley superlattice. We partially pattern the surface of a standard Hall bar sample along [000] with a grating (shown as blue strips) of 000-nm periodicity, made of negative electron-beam resist. When cooled to low temperatures, strain develops at the interface of each resist strip and the sample surface, thanks to their different thermal contraction coefficients [00-00] . The strain field can then couple to the AlAs QW through the deformation potential. Now, assuming the 0DES below each resist strip is under negative ϵ , it must be under positive ϵ between two strips. The periodic grating thus should subject the 0DES to varying ϵ of the same periodicity, locally breaking the valley degeneracy. As a result, the 0DES (with total density n ) should separate into multiple 000-nm-wide regions, each of which is partially valley polarized with the majority electron population periodically alternating between the X and Y valleys (of density n X and n Y ). Such a density profile is depicted in the blue boxes in Fig. 0(d) , each corresponding to the 0DES section directly above it, as well as the n X and n Y plots. While we expect n X > n Y below each resist strip and n Y > n X in between, the total density n = n X + n Y is independent of strain [0] and should stay uniform for the entire patterned region. Note that AlAs and GaAs are generally piezoelectric. However, they belong to the symmetry group 00m and are not piezoelectric along [000] [00,00] , the direction of the surface grating in our sample. This rules out any electric field modulation due to the periodic surface strain along [000], implying a uniform total 0DES density in that direction [00,00] , which is in agreement with the constant total density profile depicted in Fig. 0(d) . For the unpatterned (reference) region, the 0DES remains valley degenerate ( n X = n Y = n / 0 ). To establish the existence of a valley superlattice in the patterned region, one needs to probe the spatial valley densities, for which we employ the commensurability (also known as Weiss) oscillations (COs) technique [00-00] . By passing current along the grating direction ([000]) under a perpendicular magnetic field ( B ⊥ ), we look for low-field magnetoresistance minima, expected whenever the cyclotron orbit diameter ( 0 R C ) of electrons becomes commensurate with the period ( a ) of the 0DES density modulation. The exact condition for commensurability is 0 R C / a = i - 0 / 0 , where i is an integer [00-00] . The B ⊥ positions of the CO minima depend on the 0DES density according to 0 R C = 0 k F / e B ⊥ , where k F = 0 π n for a single-valley, isotropic 0DES. Before discussing the COs for the bivalley, anisotropic AlAs 0DES, we emphasize an important point. The prerequisite for COs is a periodic density modulation, usually rendered in the total density by various means such as an optical interference pattern [00] , electrostatic gating [00] , or strain-inducing surface superlattice in piezoelectric materials (along the appropriate crystal direction) [00] . In stark contrast to the typical CO scenario, the total density in our samples should be uniform for strain modulation along [000], thanks to the underlying crystal symmetry mentioned previously. This compelling difference guarantees that COs we report here, as discussed below, must originate from the modulation in the individual valley densities [see Fig. 0(d) ], i.e., a valley superlattice . Our transport measurements are carried out in a He 0 cryostat at a temperature of 0.0 K. By varying the illumination time at low temperatures, we tuned n between 0.0 and 0.0 × 00 00 cm - 0 . Figure 0(e) shows the magnetoresistance from the patterned (blue trace, R pat ) and reference (black trace, R ref ) regions for n ≃ 0.0 × 00 00 cm - 0 , as current is passed along [000]. Near B ⊥ = 0 , R pat exhibits a pronounced V -shaped resistance minimum followed by multiple minima (vertical arrows) at slightly higher B ⊥ . The absence of such minima in R ref points to their COs