Question #43901

A semiconductor rod of length $L_0$, cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear thermal expansion $\alpha$ is placed between a rigid wall and a block attached to a linear spring of stiffness $K$. At temperature $T_0$, the spring is relaxed and the rod exactly fits the space. The bandgap energy $E_g$ of the semiconductor depends on its actual macroscopic strain $\epsilon = \Delta L / L_0$ as $E_g = E_{g0} - \Xi \epsilon$, where $E_{g0}$ is the bandgap at $T_0$, and $\Xi > 0$ is the deformation potential constant. The system is uniformly heated to a temperature $T = T_0 + \Delta T$. Assuming the intrinsic carrier concentration is given by $n_i(T) = C T^{3/2} \exp\left(-\frac{E_g}{2k_B T}\right)$ (where $C$ is a constant and $k_B$ is the Boltzmann constant), what is the ratio of the intrinsic carrier concentration in this partially constrained rod to that in an identical, completely unconstrained rod at the same temperature $T$?