intermediate Semiconductor Physics

Carrier Transport & Mobility

Animate electron and hole motion under electric fields. See drift velocity, diffusion, the Einstein relation, and how doping and temperature affect mobility.

E-field0 V/cm

Temperature300 K

Doping Nd10^16.0 cm⁻³

Electrons

μn = 1107 cm²/V·s

vd = 0.00e+0 cm/s

D = 28.6 cm²/s

Holes

μp = 477 cm²/V·s

vd = 0.00e+0 cm/s

D = 12.3 cm²/s

Blue ● = electrons drift opposite to E; Red ○ = holes drift with E. Random walk represents thermal diffusion. Einstein relation: D = μkT/q.

At E = 0 V/cm, both electrons (blue) and holes (red circles) move randomly — this is thermal diffusion. The random walk has no net direction.

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Thermal energy kT gives each carrier an average kinetic energy of (3/2)kT. The random motion averages out to zero net current.
Set E = +200 V/cm. Electrons drift left (opposite to E), holes drift right (with E). The arrows on the canvas show the field direction.

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Drift current density: Jn = q·n·μn·E (electrons), Jp = q·p·μp·E (holes). Both contribute to current in the SAME direction despite carriers moving in opposite directions.
Observe the mobility values in the metrics panel. At Nd = 10^16 cm⁻³, μn ≈ 1200 cm²/V·s and μp ≈ 450 cm²/V·s. Increase doping to 10^18 cm⁻³ — both mobilities drop due to ionised impurity scattering.

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Mobility decreases at high doping because more charged impurities scatter the carriers. This is described by the Caughey-Thomas model.
Increase temperature from 300 K to 450 K. Mobility decreases (lattice scattering dominates at high T). At very low temperatures, mobility would increase (fewer phonons).

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Phonon scattering: μ ∝ T^(-3/2). Impurity scattering: μ ∝ T^(+3/2). Total mobility is determined by Matthiessen's rule: 1/μ = 1/μphonon + 1/μimpurity.
The diffusion coefficient D is shown below the mobility. Notice D/μ = kT/q — this is the Einstein relation. At 300 K, D/μ = 0.02585 V (the thermal voltage).

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The Einstein relation links diffusion to drift — both stem from the same carrier scattering physics. It's fundamental to the design of bipolar transistors and diodes.