Increasing the valley splitting in Si-based heterostructures is critical for improving the performance of semiconductor qubits. This paper establishes 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 nonsymmorphic symmetry of the diamond lattice. This semi-Dirac-node degeneracy gives rise to the ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_{2^{\prime }}$ 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, $\vec{\mathcal{B}}$. These fields are defined by a Zeeman-like coupling $\vec{\mathcal{B}}·\vec{\tau }$ 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 $\vec{\mathcal{B}}$. 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.

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• Qubits
• Semiconductors
• Symmetries
• Group theory