Valley physics in the two-band k · p model for SiGe heterostructures and spin qubits

We discuss the choice and implementation of inter-valley potentials in the so-called two bands $\mathbf{k}\cdot\mathbf{p}$ model for the opposite $X$, $Y$ or $Z$ valleys of silicon. We focus on the description of valley splittings in Si/SiGe heterostructures for spin qubits, with a particular attention to alloy disorder. We demonstrate that the two bands $\mathbf{k}\cdot\mathbf{p}$ model reproduces the valley splittings of atomistic tight-binding calculations in relevant heterostructures (SiGe spikes, wiggle wells...), yet at a much lower cost. We show that the model also captures the effects of valley-orbit mixing and yields the correct inter-valley dipole matrix elements that characterize manipulation, dephasing and relaxation in spin/valley qubits. We simulate a realistic Si/SiGe spin qubit device as an illustration, and discuss electron-phonon interactions in the two bands $\mathbf{k}\cdot\mathbf{p}$ model. Beyond spin qubits, this model enables efficient simulations of SiGe heterostructure devices where spin and valley physics are relevant.