Question #83311

A cubic semiconductor crystal has independent elastic compliance constants $S_{11}$, $S_{12}$, and $S_{44}$ (in $\text{Pa}^{-1}$). A uniform uniaxial compressive force $F$ is applied to the crystal along the crystallographic direction $[111]$. The cross-sectional area of the crystal perpendicular to the force is $A$.

According to Deformation Potential Theory, the shift in the energy band gap $\Delta E_g$ due to lattice strain is given by $\Delta E_g = a \cdot \text{Tr}(\epsilon)$, where $a$ is the hydrostatic deformation potential (in eV) and $\epsilon$ is the strain tensor.

Assuming the crystal remains in the linear elastic regime and the effective masses of electrons and holes are unaffected by the deformation, what is the isothermal shift in the intrinsic Fermi level relative to the valence band maximum, $\Delta E_F$?

Note: The intrinsic Fermi level is given by $E_F = E_v + \frac{E_g}{2} + \frac{3}{4} k_B T \ln\left(\frac{m_h^*}{m_e^*}\right)$.