Electronics at the PhD level begins with the connection between microscopic transport and macroscopic circuit behavior. We will treat carriers as quantum particles in solids, derive current-voltage relations from statistical mechanics, and connect band structure to the performance limits of devices such as diodes, MOSFETs, and heterostructure transistors.

In thermal equilibrium, electron and hole densities follow Fermi-Dirac statistics. For non-degenerate semiconductors, the approximations $n \approx N_C e^{-(E_C-E_F)/k_B T}$ and $p \approx N_V e^{-(E_F-E_V)/k_B T}$ are often sufficient. Transport combines drift and diffusion, giving $J_n = q n \mu_n E + q D_n \nabla n$ and $J_p = q p \mu_p E - q D_p \nabla p$ with the Einstein relation $D/\mu = k_B T/q$.

The built-in potential and depletion width arise from the balance of charge density and electrostatic potential through Poisson's equation, $\nabla^2 \phi = -\rho/\epsilon$. Under the depletion approximation, mobile carriers are neglected in the space-charge region, enabling closed-form expressions for barrier height, capacitance, and breakdown scaling.