Strain-Controlled Valley Splitting in Si-SiGe Heterostructures

A splitting between equivalent valleys larger than the spin splitting was recently reported in a gate confined electron system in thin Si films grown on SiGe substrate [1]. This degeneracy lifting reduces scattering and improves the coherence time. The valley splitting larger than the spin splitting opens a way to build spin qubits, which makes silicon-based quantum devices promising for future applications in quantum computing. In this work we propose an alternative mechanism to create and control the valley splitting in Si-SiGe heterostructures by applying shear strain. Uniaxial stress is already used for performance enhancement of modern MOSFETs, and its application to control the valley splitting is technologically possible. We have adopted the two-band k·p model for analytical calculations [2-4]. In contrast to the usually assumed parabolic energy dispersion for the conduction band valleys with the transversal mass mt and the longitudinal mass ml, the two-band k·p model is able to describe the conduction band structure in the presence of shear strain. A good agreement of the analytical two-band k·p model with the results of numerical pseudo-potential band structure calculations (EPM) with parameters from [5] is shown in Fig.1. Within the two-band k·p model the subband structure for an infinite square well potential can be found analytically. The dispersion in the first unprimed subband is compared against the parabolic approximation with strainand thickness-dependent transversal effective masses in Fig.2. The valley degeneracy, however, is not removed even when the dispersion is not parabolic (Fig.3, Fig.4), and the unprimed subbands remain two-fold degenerate within the two-band k·p approach. To go beyond the k·p theory, we introduce an auxiliary tight-binding model defined on a lattice of sites each containing two localized orbitals in such a way that it mimics the dispersion obtained within the twoband k·p model [6]. Considering symmetric and asymmetric wave functions including the Bloch amplitudes for a finite array of 2N sites [6], one obtains two sets of equations determining the subband energies of the unprimed subband ladder (Fig.5). It then follows that for shear strain 0 = xy ε , when the bands are purely parabolic, the valley splitting is exactly zero. Applying uniaxial [110] stress to [001] ultra-thin Si films generates the valley splitting proportional to strain xy ε . For small xy ε the valley splitting is: