Macroscopic electric current in a semiconductor is governed by two complementary microscopic mechanisms: drift, driven by electrostatic potential gradients, and diffusion, driven by statistical concentration gradients. The unification of these processes under thermodynamic equilibrium establishes the Einstein relation and the complete drift-diffusion transport equations.
1. Carrier Drift and Semiclassical Mobility
When an external electric field (mathbf{E}) is applied to a semiconductor, free electrons and holes experience a directional Lorentz acceleration interrupted by frequent stochastic scattering events (with acoustic phonons, optical phonons, and ionized impurities). Under the relaxation time approximation (Drude-Lorentz kinetics), the net drift velocity (mathbf{v}_d) is proportional to the electric field:
[
mathbf{v}_{dn} = -mu_n mathbf{E}, quad mathbf{v}_{dp} = mu_p mathbf{E}
]
where (mu_n) and (mu_p) are the electron and hole mobilities, defined in terms of the momentum relaxation time ( au_m):
[
mu_n = rac{e au_{mn}}{m_{ce}^*}, quad mu_p = rac{e au_{mp}}{m_{ch}^*}
]
The resulting drift current density (mathbf{J}_{ ext{drift}}) is given by Ohm’s differential law:
[
mathbf{J}_{ ext{drift}} = mathbf{J}_n^{ ext{drift}} + mathbf{J}_p^{ ext{drift}} = (-e n mathbf{v}_{dn}) + (e p mathbf{v}_{dp}) = (q n mu_n + q p mu_p) mathbf{E} = sigma mathbf{E}
]
where (sigma = e(n mu_n + p mu_p)) is the macroscopic electrical conductivity.
2. Scattering Mechanisms and Matthiessen’s Rule
Total carrier mobility is limited by multiple independent scattering mechanisms. According to Matthiessen’s rule, the inverse of total relaxation time equals the sum of individual scattering rates:
[
rac{1}{ au_{ ext{total}}} = sum_i rac{1}{ au_i} implies rac{1}{mu} = rac{1}{mu_{ ext{lattice}}} + rac{1}{mu_{ ext{impurity}}}
]
- Acoustic Phonon (Lattice) Scattering: Lattice vibrations perturb the periodic potential via acoustic deformation potentials. The scattering rate increases with temperature due to higher phonon population ((mu_L propto T^{-3/2})).
- Ionized Impurity (Coulomb) Scattering: Screened Coulomb deflections of moving carriers by charged donor ((N_D^+)) and acceptor ((N_A^-)) cores (Brooks-Herring and Conwell-Weisskopf models). Slower carriers at lower temperatures spend more time interacting with the Coulomb field, making scattering more frequent ((mu_I propto T^{+3/2} / N_I)).
3. Carrier Diffusion and Fick’s Laws
Whenever spatial gradients in carrier concentration exist ((
abla n
eq 0,
abla p
eq 0)), random thermal motion drives a net statistical flux of particles from regions of high concentration to regions of low concentration. In accordance with Fick’s first law, the resulting diffusion current densities are:
[
mathbf{J}_n^{ ext{diff}} = -e (-D_n
abla n) = +e D_n
abla n
]
[
mathbf{J}_p^{ ext{diff}} = +e (-D_p
abla p) = -e D_p
abla p
]
where (D_n) and (D_p) are the electron and hole diffusion coefficients (units of ( ext{cm}^2/ ext{s})).
4. The Einstein Relation
To establish the exact physical relationship between mobility (mu) and diffusion coefficient (D), consider a non-uniformly doped semiconductor in complete thermodynamic equilibrium. No external voltage is applied, so the net electric current of each carrier type must vanish identically everywhere:
[
mathbf{J}_n = mathbf{J}_n^{ ext{drift}} + mathbf{J}_n^{ ext{diff}} = e n mu_n mathbf{E} + e D_n rac{dn}{dx} = 0
]
Under non-degenerate statistics, the electron concentration is (n(x) = N_c e^{-(E_c(x) – E_F)/k_B T}). Differentiating spatially while recognizing that the Fermi level (E_F) is spatially constant in equilibrium ((rac{dE_F}{dx} = 0)):
[
rac{dn}{dx} = -rac{n}{k_B T} rac{dE_c}{dx}
]
The electric field is related to the electrostatic potential (psi) and band edge by (E = -rac{dpsi}{dx} = rac{1}{e} rac{dE_c}{dx} implies rac{dE_c}{dx} = e E). Substituting back into the zero-current equilibrium condition:
[
e n mu_n E + e D_n left( -rac{n e E}{k_B T}
ight) = 0 implies e n E left( mu_n – rac{e D_n}{k_B T}
ight) = 0
]
This yields the celebrated Einstein Relation:
[
rac{D_n}{mu_n} = rac{k_B T}{e} = V_t, quad rac{D_p}{mu_p} = rac{k_B T}{e} = V_t
]
where (V_t = k_B T / e approx 25.86 ext{ mV}) at (300 ext{ K}) is the thermal voltage. Combining drift and diffusion yields the foundational drift-diffusion equations that govern all modern semiconductor devices:
[
mathbf{J}_n = q n mu_n mathbf{E} + q D_n
abla n, quad mathbf{J}_p = q p mu_p mathbf{E} – q D_p
abla p
]
