Fig. 1: Strain-assisted Wiggle Well (STRAWW) proposal.
a Unstrained silicon cubic unit cell of length a0, composed of two sublattices (red and blue). In the absence of strain, the sublattices are interchanged by a screw-symmetry operation. b Assuming periodicity in the crystallographic [100] and [010] directions, the three dimensional (3D) lattice can be reduced to an effective 1D lattice27, whose primitive unit cell length is given by a0/4, due to the screw symmetry. c Low-energy Si band structure for the 1D lattice, expressed in a Brillouin zone (BZ) of length 8π/a0 along the [001] reciprocal axis kz. The 2k0 valley coupling is achieved in two steps: (i) a 4π/a0 coupling induced by shear strain; (ii) a 2k1 coupling provided by the Wiggle Well. d In the absence of valley coupling (left side), the states are two-fold degenerate due to the two-fold valley degeneracy in (c). Valley coupling breaks this degeneracy (right side), leading to a valley splitting EVS. e Projection of the unstrained cubic unit cell onto the x–y plane. f Schematic deformation of the unit cell under shear strain along [110]. g Silicon tetrahedral bonds in the absence of strain. h Shear strain affects bonds differently, depending on the crystal plane, causing a lattice deformation along [001]. i, Shear strain reduces the translation symmetry of the 1D lattice, resulting in a primitive unit cell of length a0/2. j The corresponding BZ is then folded in half, and the shear-strain coupling kz = 4π/a0 is manifested as avoided crossings at the BZ boundary. Here, the green and blue arrows correspond to the same arrows in c, and the two black dots denote equivalent kz separated by a reciprocal lattice vector 4π/a0. Notice that the shear strain coupling (green arrow) conserves momentum in the reduced BZ. (c and j are both calculated using the sp3d5s* tight-binding model described in Methods.) k Typical Ge concentration profile of a Wiggle Well heterostructure25,26. Here, the Ge oscillation wavelength, λ = π/k1 ≈ 1.68 nm, is chosen to couple the valleys, as in (c. l, m), Cartoon depiction of shear strain (εxy) in a quantum well, before (l) and after (m) mechanical bending. Cartoon depiction of shear strain induced by lithographic etching, before (n) and after (o) lattice deformation caused by cooling to cryogenic temperatures.
