Carrier Transport: Drift, Diffusion, Mobility & Conductivity

How carriers move and make current. Learn drift in an electric field, mobility and its dependence on doping (ionised-impurity scattering) and temperature (lattice scattering), Matthiessen's rule, velocity saturation at high fields, conductivity and resistivity, sheet resistance, diffusion driven by concentration gradients, the diffusion coefficient and the Einstein relation D = µkT/q, drift + diffusion current densities for electrons and holes, and built-in fields in non-uniform doping — with Python computing Caughey-Thomas mobility, resistivity, drift and diffusion currents.

Current in a semiconductor comes from two mechanisms: drift (carriers pushed by an electric field) and diffusion (carriers spreading from high to low concentration). Transistors and diodes are built by controlling which dominates where.

Mobility links field to velocity and sets how fast devices can be; it depends on the material, doping, temperature and field.

An analogy

Drift is like people walking down a sloping street — the slope (field) pushes them. Diffusion is like a crowd spreading out of a packed stadium into empty streets — nobody pushes, they just move from crowded to empty areas.

Mobility and scattering

Carriers accelerate between collisions (mean free time τ_m): µ = qτ_m/m*. Scattering sources:

Diffusion and the Einstein relation

Random thermal motion causes net flow from high to low concentration (Fick's law). Mobility and diffusion both arise from the same scattering, linked by D/µ = kT/q = 25.85 mV at 300 K. In diodes and bipolar transistors minority carriers move mainly by diffusion; in MOSFET channels, majority carriers move mainly by drift (except in subthreshold, where diffusion dominates).

High-field effects

At fields above ~10⁴ V/cm carriers gain energy faster than they lose it; velocity saturates (Si electrons ~1 × 10⁷ cm/s, holes ~8 × 10⁶). In GaAs, electrons transfer to a heavier valley, causing negative differential mobility (Gunn effect). Very short devices show velocity overshoot and near-ballistic transport. Hot carriers can be injected into oxides, causing reliability degradation.

Built-in fields and equilibrium

In non-uniformly doped material, diffusion would move carriers down the gradient; the resulting charge separation builds a field whose drift current exactly cancels diffusion in equilibrium (J = 0). This built-in field is used in graded-base bipolar transistors and back-surface fields in solar cells.

Formulas

Worked example: Resistivity

Given: P-type Si, N_A 10¹⁶, µ_p 429

Find: ρ.

Answer: 1.46 Ω·cm

Worked example: Diffusion coefficient

Given: µ_n 1350 cm²/V·s

Find: D_n.

Answer: 34.9 cm²/s

Worked example: Sheet resistance

Given: ρ 0.01 Ω·cm, t 1 µm

Find: R_s and a 100 □ resistor.

Answer: 100 Ω/□ → 10 kΩ

Drift vs diffusion

  Drift (field E →)                    Diffusion (gradient)
   + ────────────────── −              high n ░░░▒▒▒▓▓▓ low n
   holes  →  (with E)                  electrons move → (down the gradient)
   electrons ← (against E)             current J_n = +qD_n dn/dx (direction opposite to e⁻ flow sign conventions!)
   J = σE (both add)                   holes diffuse the same way; J_p = −qD_p dp/dx

Electrons and holes drift in opposite directions, but because their charges are opposite both contribute current in the direction of E. Diffusion depends only on concentration gradients — the sign conventions for electron and hole diffusion currents differ, which is a common source of errors.

Mobility, resistivity, drift and diffusion currents

# Drift and diffusion: doping-dependent mobility (Caughey-Thomas), resistivity, currents, Einstein relation
q, VT = 1.602e-19, 0.02585
def mu_n(N): return 68.5 + (1414 - 68.5) / (1 + (N / 9.2e16) ** 0.711)
def mu_p(N): return 44.9 + (470.5 - 44.9) / (1 + (N / 2.23e17) ** 0.719)
for N in (1e14, 1e16, 1e18):
    print(f"N = {N:.0e}: mu_n = {mu_n(N):6.0f}, mu_p = {mu_p(N):5.0f} cm^2/Vs, rho(n-type) = {1/(q*N*mu_n(N)):.3e} ohm*cm")
# Drift current in a bar: Nd 1e16, 1 V across 100 um, area 100 um x 10 um
Nd, V, L, A = 1e16, 1.0, 100e-4, 100e-4 * 10e-4
E = V / L
J = q * Nd * mu_n(Nd) * E
print(f"E = {E:.0f} V/cm, drift velocity {mu_n(Nd)*E:.2e} cm/s, J = {J:.1f} A/cm^2, I = {J*A*1e3:.3f} mA")
# Diffusion: hole concentration falls from 1e14 to 0 over 10 um
Dp = mu_p(1e16) * VT
Jdiff = q * Dp * 1e14 / 10e-4
print(f"Dp = mu*VT = {Dp:.2f} cm^2/s; diffusion current density = {Jdiff:.2f} A/cm^2")
print(f"velocity saturation: at 1e5 V/cm drift model gives {mu_n(Nd)*1e5:.1e} cm/s but electrons saturate near 1e7 cm/s")
# N = 1e+14: mu_n =   1404, mu_p =   469 cm^2/Vs, rho(n-type) = 4.447e+01 ohm*cm
# N = 1e+16: mu_n =   1184, mu_p =   429 cm^2/Vs, rho(n-type) = 5.273e-01 ohm*cm
# N = 1e+18: mu_n =    277, mu_p =   153 cm^2/Vs, rho(n-type) = 2.254e-02 ohm*cm
# E = 100 V/cm, drift velocity 1.18e+05 cm/s, J = 189.6 A/cm^2, I = 1.896 mA
# Dp = mu*VT = 11.10 cm^2/s; diffusion current density = 0.18 A/cm^2
# velocity saturation: at 1e5 V/cm drift model gives 1.2e+08 cm/s but electrons saturate near 1e7 cm/s

Mobility falls from ~1400 to ~280 cm²/V·s as doping rises from 10¹⁴ to 10¹⁸ because ionised impurities scatter carriers. A modest 100 V/cm field drives almost 190 A/cm² in 10¹⁶ n-type silicon. The linear drift model fails at high fields: electrons saturate near 10⁷ cm/s — crucial in short-channel MOSFETs.

Common mistakes

Applications

Key takeaways

Going further