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

TL;DR

The paper reframes the silicon valley problem by introducing the valleyor, a two-component valley degree of freedom rooted in the X-point non-symmorphic symmetry, and shows that valleys do not behave like a spinor under crystal operations. It derives an explicit $X$-point basis and a comprehensive symmetry analysis that yields a new, symmetry-allowed topological term $\kappa k_x k_y \tau_2$ in the low-energy Hamiltonian, producing a nodal line along the $XW$ directions and a Berry flux of $\pi$. The work further identifies a Zeeman-like valley-magnetic coupling through composite fields $\mathcal B\cdot\vec\tau$, enabling valley splitting via combinations of strain, magnetic fields, and dot geometry beyond conventional scalar-potential scattering. These insights offer new routes to stabilize and control valley splitting in Si-based qubits, with direct implications for robust quantum dot and donor architectures across device platforms.

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 $Δ_1$ and $Δ_{2'}$ 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 $τ_1,τ_2$ and $τ_3$ 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 ${\mathsf B}\cdot\vecτ$ 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

Figures (1)

• Figure 1: (Left) Brillouin zone of the face-centered cubic lattice. The $X$ point (cyan) lies at the center of a square face, with four equivalent $W$ points indicated by black dots around each $X$ point. The two Si valleys (green ellipsoids) are located at $\pm k_1$ from the $X$ point along the longitudinal ($z$) direction. (Right) Energy dispersion of the $\Delta_1$ (red) and $\Delta_{2'}$ (blue) bands near the $X$ point in Si, shown as a function of longitudinal momentum $p_\ell = p_z$. These bands, which form the Si valleys, emerge from the two-dimensional representation of the space group at the $X$ point when moving away from this high-symmetry location. The energy of the $X$ point is about $80$ meV above the degenerate conduction band minimum.