How does doping move the Fermi level?

Adding donors supplies electrons and pushes the Fermi level toward the conduction band; adding acceptors creates holes and pushes it toward the valence band, with charge neutrality fixing the exact position for a given dopant concentration.

Carrier Statistics and Doping

The equilibrium concentrations of electrons and holes follow from the density of states and Fermi-Dirac statistics, so the position of the Fermi level, fixed by doping, determines how many carriers a semiconductor has.

Definition

Carrier statistics is the determination of equilibrium electron and hole concentrations from the band density of states and Fermi-Dirac occupation; doping shifts the Fermi level so that, subject to charge neutrality, the product of electron and hole concentrations stays fixed by the law of mass action.

Scope

This topic develops the quantitative statistics of carriers in semiconductors: the effective density of states in the conduction and valence bands, the Fermi-Dirac and Boltzmann (nondegenerate) approximations, the electron and hole concentrations as functions of the Fermi level, the law of mass action, and the charge-neutrality condition that fixes the Fermi level given the dopant concentrations. It quantifies the qualitative doping picture and provides the carrier densities used in device physics.

Core questions

How do the density of states and Fermi-Dirac statistics give the equilibrium carrier concentrations?
When is the nondegenerate Boltzmann approximation valid, and when must full Fermi-Dirac statistics be used?
What is the law of mass action, and why does the carrier product stay constant?
How does charge neutrality fix the Fermi-level position for a given doping?

Key concepts

Effective density of states
Fermi-Dirac and Boltzmann statistics
Law of mass action
Charge-neutrality condition
Fermi-level position and degeneracy

Key theories

Law of mass action for carriers: In thermal equilibrium the product of electron and hole concentrations equals the square of the intrinsic concentration, independent of doping, so increasing one carrier type by doping necessarily suppresses the other.

Clinical relevance

Quantitative carrier statistics let engineers compute the conductivity, built-in potentials, and operating characteristics of devices from the doping profile; the Fermi-level bookkeeping developed here is essential to designing junctions, transistors, and the doping schedules of integrated-circuit fabrication.

History

Fermi-Dirac statistics, formulated in 1926, became the basis for the equilibrium theory of carriers in semiconductors developed by Wilson, Shockley, and others through the 1930s and 1940s, providing the quantitative foundation codified in Shockley's 1950 treatise on semiconductors.

Key figures

Enrico Fermi
Paul Dirac
William Shockley

Seminal works

sze2007
ashcroft1976

Frequently asked questions

Why does the product of electron and hole concentrations stay fixed?: In equilibrium generation and recombination balance, which ties the two concentrations together; the result, the law of mass action, holds the product equal to the intrinsic concentration squared regardless of doping, at a given temperature.
How does doping move the Fermi level?: Adding donors supplies electrons and pushes the Fermi level toward the conduction band; adding acceptors creates holes and pushes it toward the valence band, with charge neutrality fixing the exact position for a given dopant concentration.