Balance drift, diffusion, and electrochemical drive

Take the positive elementary charge as $q>0$ and electric field $\mathbf E$. An electron has charge minus q and low-field drift velocity $-\mu_n\mathbf E$, with positive mobility $\mu_n$. Multiplying electron particle flux by its negative charge makes its drift current point along the field. Holes drift along the field and carry positive charge, so their drift current also points along the field. Opposite particle velocities therefore do not imply opposite drift-current directions.

Diffusion transports particles down their own density gradient. The electron flux is $-D_n\nabla n$; its conventional current consequently points up the electron gradient. The hole diffusion current instead points down the hole gradient. These signs follow directly from charge times particle flux and do not depend on which carrier happens to be the minority. Mobilities have units square metres per volt-second, diffusion coefficients square metres per second, and current densities amperes per square metre when densities use SI units.

For a uniform material at constant temperature, an equilibrium Fermi level is spatially constant. The conduction edge varies as $E_c=E_{c0}-q\phi$, while $\mathbf E=-\nabla\phi$. Differentiating the Boltzmann electron density then gives $\nabla n=-qn\mathbf E/(k_BT)$. Substitution into the electron current makes it vanish only when diffusion divided by mobility equals the thermal voltage. The hole derivation reaches the same ratio despite its opposite diffusion-current sign.

This Einstein relation is a consequence of dilute equilibrium statistics and linear response, rather than a universal conversion independent of regime. A useful generalized equilibrium expression replaces thermal voltage by carrier density divided by its thermodynamic density response. Here the chemical potential is an energy, and must not be confused with mobility despite the shared Greek letter often used for both. Degenerate carriers have a different density response; hot-carrier distributions or strong nonequilibrium can invalidate even a local equilibrium generalization.

A concentration gradient can coexist with zero electrical current because the equilibrium electrostatic potential adjusts to oppose diffusion. Conversely, a uniform density under an applied field carries drift current unless another driving term intervenes. Under injection, quasi-Fermi-level gradients organize the combined drive. They should not be interpreted as a license to set electron and hole densities independently while ignoring continuity equations, recombination, and Poisson electrostatics. A device solution requires transport and charge conservation together.

The local model is most reliable when carrier momentum relaxes over distances short compared with the scale of density and potential variation. Contacts, abrupt interfaces, large fields, and nanometre channels can require boundary or kinetic treatments. Even within the model, a negative net electron current does not mean negative mobility: it can simply mean that diffusion outweighs positive drift current. Compute signed contributions first, then interpret their sum, rather than assigning a direction from the majority-carrier label alone.