Question #43895

A Zener diode is fabricated from a semiconductor crystal rod of length $L$. The rod is rigidly fixed between two immovable, perfectly insulating walls at an initial temperature $T_0 = 300 \text{ K}$. The diode is connected in series with a resistor $R = 100 \ \Omega$ and a DC voltage source $V_S = 12.0 \text{ V}$. At $T_0$, the diode operates in the breakdown region with a Zener voltage $V_{Z0} = 6.0 \text{ V}$.

The intrinsic temperature coefficient of the Zener voltage (at zero stress) is $C_T = \frac{\partial V_Z}{\partial T} = -4.0 \times 10^{-3} \text{ V/K}$. The crystal has a Young's modulus $Y = 1.5 \times 10^{11} \text{ N/m}^2$ and a coefficient of linear thermal expansion $\alpha = 2.0 \times 10^{-6} \text{ K}^{-1}$. The Zener voltage also depends on the longitudinal compressive stress $\sigma_c$ developed in the rod due to bandgap deformation, with a coefficient $K = \frac{\partial V_Z}{\partial \sigma_c} = +2.0 \times 10^{-8} \text{ V/Pa}$.

If the temperature of the system is uniformly raised to $T = 350 \text{ K}$, calculate the new steady-state Zener current $I_Z$. Assume the walls remain perfectly rigid and the diode remains in the breakdown region.