Envelope-function theory of inhomogeneous strain for hole-spin qubits in silicon and germanium

Strain is a common feature of semiconductor nanostructures, arising from the lattice mismatch between heterogeneous layers and the cooling down of materials with different thermal-expansion coefficients. It significantly alters the system properties, representing either a problem (if uncontrolled) or a resource, e.g. to enhance the carrier mobility. Inhomogeneous-strain engineering can be used to efficiently improve Rabi frequencies [1] and g-tensors [2] in Si spin qubits. However, the rigorous inclusion of strain within the k-dot-p formalism – the reference framework for the simulation of charge carriers in semiconductors – has been so far limited to the homogeneous case, which is accounted for by the theory of Bir and Pikus [3, 4]. In our work, we have developed a complete theory to include inhomogeneous strain in the k-dot-p framework, by generalizing the Bir-Pikus approach via concepts from differential geometry [5]. We have obtained an envelope-function Hamiltonian that is applicable to a variety of semiconductor systems. It includes several new terms, which depend on the spatial derivatives of the strain tensor, and could not be deduced from previous approaches. I will discuss the implications of our findings for the simulation and strain engineering of hole-spin qubits in Si and Ge quantum dots.

[1] J. C. Abadillo-Uriel, E. A. Rodríguez-Mena, B. Martinez, and Y.-M. Niquet, Hole-spin driving by strain-induced spin-orbit interactions, Phys. Rev. Lett. 131, 097002 (2023).

[2] S. D. Liles, F. Martins, D. S. Miserev, A. A. Kiselev, I. D. Thorvaldson, M. J. Rendell, I. K. Jin, F. E. Hudson, M. Veldhorst, K. M. Itoh, O. P. Sushkov, T. D. Ladd, A. S. Dzurak, and A. R. Hamilton, Electrical control of the g-tensor of the first hole in a silicon MOS quantum dot, Phys. Rev. B 104, 235303 (2021).

[3] G. Bir and G. Pikus, Symmetry and Strain-induced Effects in Semiconductors (Wiley, New York, 1974).

[4] C. Y.-P. Chao and S. L. Chuang, Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells, Phys. Rev. B 46, 4110 (1992).

[5] A. Secchi and F. Troiani, Envelope-function theory of inhomogeneous strain in semiconductor nanostructures, Phys. Rev. B 110, 045420 (2024).