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Mathematics for Physics

28 Charge Carrier

A mobile charged particle that is used to carry electric current through a material, where a charged particle is a particle that possesses electric charge.

The charge of a carrier. A charge carrier has a definite electric charge, an integer multiple of the elementary charge. The elementary charge is the magnitude of the charge of the electron. This principle is used to count current as a flow of discrete carriers.

Charge transport. The current density is the product of the carrier density, the charge of one carrier, and the drift velocity. Carrier density is the number of free carriers per unit volume. This principle is used to relate microscopic motion to the macroscopic current.

The current density of drifting carriers is

\[ \mathbf{J} = nq\mathbf{v}_{d} \]

where

  • \(\mathbf{J}\) is the current density.
  • \(n\) is the number of carriers per unit volume.
  • \(q\) is the charge of one carrier.
  • \(\mathbf{v}_{d}\) is the drift velocity.

Drift velocity. Drift velocity is the average net velocity of the carriers along the driving field after many scattering collisions. This principle is used to separate the slow net flow that makes current from the much faster random thermal motion.

Mobility. Mobility is the ratio of drift speed to the applied electric field. This principle is used to measure how easily carriers move through a lattice of vibrating atoms and impurities.

The mobility is

\[ \mu = \dfrac{v_{d}}{E} \]

where

  • \(\mu\) is the mobility.
  • \(v_{d}\) is the drift speed.
  • \(E\) is the electric field magnitude.

Electrons and holes. An electron in the conduction band is a negatively charged carrier. A hole is a vacant positively charged state in the valence band that moves as a positive carrier. This principle is used to describe conduction in metals and in semiconductors. See Semiconductor.

The Hall sign of the carriers. A magnetic field deflects drifting carriers to one side of a conductor. The sign of the resulting Hall voltage reveals the sign of the dominant carriers. This principle is used to tell electron conduction from hole conduction. See Hall Effect.

Note: Also called a carrier.

28.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — drift velocity, \(J=nqv_{d}\), and mobility.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. §5.1.3 — Hall effect and the sign of the carriers.
  3. Kittel, C. Introduction to Solid State Physics. Wiley — mobility, electrons, and holes.
  4. Simon, S. H. The Oxford Solid State Basics. Oxford University Press, 2013. — carrier density and semiconductor transport.
  5. Jaffe, R. L., & Taylor, W. The Physics of Energy. Cambridge University Press, 2018. — charge transport in conductors and semiconductors.