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Transport in semiconductors: drift, diffusion and recombination

Mobility and drift current, resistivity, diffusion and Fick's law, the Einstein relation with proof, lifetime, LEDs, the continuity equation and diffusion length.

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Knowing how many carriers there are is not enough: to compute a current you need to know how they move. In a semiconductor they move in two ways: pulled by an electric field (drift) or pushed by a concentration difference (diffusion). How diodes and transistors work comes down to the competition between these two mechanisms and recombination.

Drift current

A free electron in a field E accelerates but keeps colliding with the vibrating lattice and with impurities: on average it reaches a drift velocity proportional to the field,

    \[ v_n = -\mu_n E,\qquad v_p = \mu_p E \]

where \mu is the mobility. In lightly doped silicon \mu_n \approx 1350 cm²/(V·s) and \mu_p \approx 480 cm²/(V·s): holes are almost three times slower, which is one reason circuits favour n-channel transistors. The total current density is

    \[ J_{drift} = q\,(n\mu_n + p\mu_p)\,E = \sigma E \]

which is Ohm’s law in local form, with conductivity \sigma = q(n\mu_n + p\mu_p).

Example. Intrinsic silicon: \rho = 1/\sigma = 1/(1.6\cdot 10^{-19}\cdot 10^{10}\cdot 1830) \approx 3.4\cdot 10^5 Ω·cm. With N_D = 10^{16} cm⁻³: \rho = 1/(1.6\cdot 10^{-19}\cdot 10^{16}\cdot 1350) \approx 0.46 Ω·cm, almost a million times lower. Copper has 1.7\cdot 10^{-6} Ω·cm.

At high fields (above about 10^4 V/cm) the velocity stops growing and saturates at about 10^7 cm/s: this is what limits the speed of the smallest transistors.

Diffusion current

If carriers are more concentrated on one side, their random motion takes them on average towards the less concentrated region, like a drop of ink in water. The flux is proportional to the gradient (Fick’s law):

    \[ J_{n,diff} = q D_n\frac{dn}{dx},\qquad J_{p,diff} = -q D_p\frac{dp}{dx} \]

D is the diffusion coefficient. The signs differ because the charges are opposite: electrons and holes both diffuse towards lower concentration, but the hole current flows in the direction of their motion, the electron current in the opposite direction.

The Einstein relation

Mobility and diffusion are not independent: both depend on collisions with the lattice.

Theorem (Einstein relation).

    \[ \frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q} = V_T \]

Proof. Consider a semiconductor at equilibrium with non-uniform doping. Electrons diffuse and set up a field E = -dV/dx, and at equilibrium the total electron current is zero:

    \[ q n\mu_n E + qD_n\frac{dn}{dx} = 0 \]

At equilibrium the concentration follows Boltzmann statistics, n = n_0\,e^{V/V_T} (electrons gather where the potential is higher), so \frac{dn}{dx} = \frac{n}{V_T}\frac{dV}{dx} = -\frac{n}{V_T}E. Substituting: qn E(\mu_n - D_n/V_T) = 0, that is D_n = \mu_n V_T. The same reasoning holds for holes. ∎

At 300 K V_T = 25.9 mV, the thermal voltage, which will appear in every diode and transistor formula. For electrons in silicon D_n = 1350\cdot 0.0259 \approx 35 cm²/s.

Generation, recombination and lifetime

If excess carriers are injected (by light or through a junction), the law np = n_i^2 no longer holds and the system returns to equilibrium by recombination. For excess minority carriers \Delta p in n-type material, the recombination rate is \Delta p/\tau_p, where \tau_p is the lifetime: from nanoseconds to milliseconds, depending on the purity of the crystal.

In silicon recombination happens almost always through defects and impurities (levels in the middle of the gap) and releases heat. In direct-gap semiconductors such as gallium arsenide and gallium nitride, instead, electron and hole recombine directly by emitting a photon of energy about E_g: this is the principle of LEDs and lasers. The colour depends on the gap: \lambda \approx 1240/E_g nm with E_g in eV.

The continuity equation and diffusion length

The balance of minority carriers in a small volume says that their concentration changes because of current flowing in or out and because of recombination:

    \[ \frac{\partial \Delta p}{\partial t} = D_p\frac{\partial^2 \Delta p}{\partial x^2} - \frac{\Delta p}{\tau_p} \]

(with no field and no generation). In steady state, with an excess \Delta p_0 held at x = 0, the solution that vanishes far away is

    \[ \Delta p(x) = \Delta p_0\,e^{-x/L_p},\qquad L_p = \sqrt{D_p\tau_p} \]

Exponential profile of excess minority carriers diffusing and recombining
The excess holes injected at x = 0 diffuse to the right and recombine along the way: the concentration decays exponentially and after one diffusion length it has dropped by a factor e. The tangent at the origin meets the axis exactly at L_p.

The diffusion length L_p is the average distance a minority carrier travels before recombining: with D_p = 12 cm²/s and \tau_p = 1 µs, L_p \approx 35 µm. This number sets device geometry: in the bipolar transistor the base must be much thinner than L, so that carriers cross it without getting lost.

The full picture

Putting drift and diffusion together:

    \[ J_n = qn\mu_n E + qD_n\frac{dn}{dx},\qquad J_p = qp\mu_p E - qD_p\frac{dp}{dx} \]

These two equations, with the continuity equation and Poisson’s equation for the field, are the model simulators use to compute any semiconductor device. In the next lessons we apply them by hand to the most important cases: the pn junction, the diode, the bipolar transistor and the MOSFET.

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