Zeeman-like coupling to valley degree of freedom in Si-based spin qubits

TL;DR

This paper reveals the fundamental nature of valley degrees of freedom in Si-based qubits, showing they are connected to non-symmorphic symmetries and can be manipulated via magnetic and strain fields, enhancing qubit control.

Contribution

It introduces the concept of valleyors, classifies their transformation properties, and identifies new mechanisms for controlling valley states in silicon qubits.

Findings

Valley degrees of freedom originate from semi-Dirac-node degeneracy at the X-point.

Valleyors transform differently from spinors, leading to new classification.

Magnetic and strain fields can induce valley-magnetic coupling, enabling control of valley states.

Abstract

Increasing the valley splitting in Si-based heterostructures is critical for improving the performance of semiconductor qubits. This paper demonstrates that the two low-energy conduction band valleys are not independent parabolic bands. Instead, they originate from the X-point of the Brillouin zone, where they are interconnected by a degeneracy protected by the non-symmorphic symmetry of the diamond lattice. This semi-Dirac-node degeneracy gives rise to the $\Delta_1$ and $\Delta_{2'}$ bands, which constitute the valley degrees of freedom. By explicitly computing the two-component Bloch functions $X_1^\pm$, using the wave vector group at the X-point, we determine the transformation properties of the object $(X_1^+,X_1^-)$. We demonstrate that these properties are fundamentally different from those of a spinor. Consequently, we introduce the term "valleyor" to emphasize this fundamental distinction. The transformation properties of valleyors induce corresponding transformations of the Pauli matrices $\tau_1,\tau_2$ and $\tau_3$ in the valley space. Determining these transformations allows us to classify possible external perturbations that couple to each valley Pauli matrix, thereby identifying candidates for valley-magnetic fields, ${\mathsf B}$. These fields are defined by a Zeeman-like coupling ${\mathsf B}\cdot\vec\tau$ to the valley degree of freedom. In this way, we identify scenarios where an applied magnetic field $\vec B$ can leverage other background fields, such as strain, to generate a valley-magnetic field ${\mathsf B}$. This analysis suggests that beyond the well-known mechanism of potential scattering from Ge impurities, there exist additional channels (mediated by combinations of magnetic and strain-induced vector potentials) to control the valley degree of freedom