The generation of electrical current in semiconductors is fundamentally driven by two mechanisms: drift and diffusion. These processes are essential for the functionality and performance of semiconductor-based devices.

Drift Current:

The drift of charge carriers is started by an external electric field (E). Charged particles, such as electrons and holes, experience an acceleration between collisions with lattice atoms. For electrons, this results in a drift velocity (v_d) given by:

Where μ_e is the electron mobility and E is the electric field intensity.

The current density (J) due to drift for electrons (J_n) and holes (J_p) can be expressed as:

Where q is the elementary charge, n and p, are the concentrations of electrons and holes, respectively, and μ_n and μ_p are the mobilities of electrons and holes. The total drift current density (J_total) is the sum of the electron and hole current densities:

The conductance (σ) is then the sum of the products of charge density, mobility for each type of carrier:

Diffusion Current:

Diffusion occurs due to the thermal motion of carriers, moving from regions of higher concentration to regions of lower concentration. The current density (J_diffusion) is:

D_n and D_p are the diffusion coefficients for electrons and holes, respectively, and dn/dx and dp/dx are the concentration gradients for electrons and holes.

The Einstein relations link mobility and diffusion coefficient for both electrons and holes:

Where k is the Boltzmann constant, and T is the absolute temperature.

When both an electric field and a concentration gradient are present, the total current density is the sum of the drift and diffusion components. In real-world applications, these phenomena are analyzed using the semiconductor equations, a set of differential equations that describe the behavior of charge carriers in a semiconductor.