A study of symmetry-protected topology of surfaces of maximum entanglement in Si, Ge and GaAs for spin-qubit architectures

Progress on spin qubits depends on a fundamental and precise understanding of electron and hole g factors and their anisotropy.

We evaluate tight-binding analytical expressions to calculate the g-tensor of electron and hole spins for group IV and III-V semiconductors with focus on crystalline Si, Ge and GaAs -- the leading hosts for gate-controlled spin qubits. Recalling that the electron g-factor for bulk Si is +2 but -0.44 for bulk GaAs indicates the need to diagnose the occurrence of large variations in g. Mathematically, these values are the singular values of the asymmetric g tensor. Using the determinant of g,  det(g), as a diagnostic, we see large regions of k-space, for many bands in these semiconductors, where det(g) has an inverted sign. The topology of the surfaces where det(g) = 0 is particularly intricate in the case of the first conduction bands of Si and the second conduction band of Ge. We observe that these surfaces can be as complex as Fermi surfaces and share similar characteristics. We show that, considering just the spin contribution $g_S$ to the g tensor, surfaces where $det(g_S)=0$, which also commonly occur, indicate the occurrence of maximal spin-orbital entanglement in the Bloch states.