Einstein relation; mobility, conductivity, resistivity

Einstein relation

The Einstein relation links the diffusion coefficient $D$ and the drift mobility $\mu$ of a carrier: $\dfrac{D}{\mu} = \dfrac{kT}{e}$.

Derivation. Take a non-uniformly doped n-type bar in thermal equilibrium (no external voltage). The donor and electron concentration $n(x)$ falls with $x$.

 n(x) high ---> low
 ---------------------------
 |  diffusion: e- move ->   |
 |  field E builds up        |
 |  drift:   e- move <-     |
 ---------------------------

1. Electrons diffuse from high to low concentration. Electron diffusion current density:

$$J_{diff} = eD_e\frac{dn}{dx}$$

2. The donors left behind are positive, so a built-in field $\mathcal{E}$ appears; it causes a drift current

$$J_{drift} = en\mu_e\mathcal{E}$$

3. In equilibrium there is no net current:

$$en\mu_e\mathcal{E} + eD_e\frac{dn}{dx} = 0$$

4. In equilibrium the Fermi level is flat, so $n(x) = N_c\exp[-(E_c(x)-E_F)/kT]$ and

$$\frac{dn}{dx} = -\frac{n}{kT}\frac{dE_c}{dx}$$

5. The field bends the band: $dE_c/dx = e\mathcal{E}$. Hence

$$\frac{dn}{dx} = -\frac{ne\mathcal{E}}{kT}$$

6. Substitute in step 3:

$$\begin{aligned} en\mu_e\mathcal{E} - eD_e\frac{ne\mathcal{E}}{kT} &= 0\\ \frac{D_e}{\mu_e} &= \frac{kT}{e} \end{aligned}$$

Similarly for holes $D_h/\mu_h = kT/e$. At 300 K, $kT/e = 0.0259$ V; e.g. for Si electrons $D_e = 1350\times0.0259 \approx 35\ \text{cm}^2\text{s}^{-1}$.

Electron mobility

Drift mobility is the drift velocity per unit applied field:

$$\mu_e = \frac{v_d}{\mathcal{E}} = \frac{e\tau}{m_e^*}$$

Unit: m² V⁻¹ s⁻¹ (or cm² V⁻¹ s⁻¹). It shows how easily electrons move; it falls with lattice scattering (higher T) and impurity scattering (heavier doping).

Conductivity

Conductivity $\sigma$ is the ratio of current density to electric field, $J = \sigma\mathcal{E}$:

$$\sigma = e(n\mu_e + p\mu_h)$$

Unit: S m⁻¹ (Ω⁻¹ m⁻¹).

Resistivity

Resistivity is the reciprocal of conductivity, i.e. the resistance of a unit cube of material:

$$\rho = \frac{1}{\sigma} = \frac{RA}{L}$$

Unit: Ω m.