Semiconductor

The chemical potential difference across p-n junction, induces electrostatic potential difference. The induced potential at equilibrium is called the built-in potential \( V_{\rm bi} \).

\begin{circuitikz} \draw (.5,1) rectangle (2.5,-1); \draw (0,0) node[above] {n} to[short, *-] (.5,0) to[battery2, invert] (1.5,0) to[capacitor] (2.5,0) to[short, -*] (3,0) node[above] {p}; \end{circuitikz}

The capacitor represent the depletion region, and the battery is for the chemical potential of electron. The battery is oriented in the direction that pushes electron from n-type to p-type within the battery.

When at equilibrium the capacitor charges up until it exactly cancels the chemical potential. When you measure the voltage between two electrode, it reads zero, because the voltmeter measures both voltage and electron chemical potential: \[ V_{\rm eff} = V - \mu_{\rm e}/e. \] Note that the pure electrostatic potential difference is \( V_{\rm bi} \).

The inclusion of chemical potential into the circuit diagram is crucial difference between semiconductor and electrochemical cell. In electrochemical cell only the side effect, voltage difference in the other direction, is considered. In semiconductor, the side effect, capacitor, and the chemical potential is included.