Pure Semiconductors

 $$E=q\varphi$$

$$q=\pm e$$

$$Z_1=e^{\beta e\varphi}+e^{-\beta e\varphi}$$

$$Q=\left(\frac{\partial F}{\partial\varphi}\right)_T=Ne\tanh(\frac{e\varphi}{k_B T})$$

$$C\approx \frac{Ne^2}{k_B T}$$

Ensemble of capacitors?