Equi-spin-splitting distribution near the minimum-spin-splitting surface under biaxial strains in bulk wurtzite materials

The spin-splitting energies in biaxially strained bulk wurtzite materials are calculated using the linear combination of atomic orbital method, and the equi-spin-splitting distributions in $k$-space near the minimum-spin-splitting (MSS) surfaces are illustrated. These data are compared with those derived analytically using the two-band \textbf{k} $\mathbf{\cdot}$ \textbf{Dp} (2KP) model. It is found the results from these two methods are in good agreement for small $k$. However, the ellipsoidal MSS surface under compressively biaxial strain predicted by the 2KP model does not exist, due to the data points are far from the $\Gamma $ point in this case. As a result, from compressively to tensilely biaxial strain, only three types of shapes of the MSS surface exist in the wurtzite Brillouin zone; that is, a hyperboloid of two sheets, a hexagonal cone and a hyperboloid of one sheet.