17.3. Carrier Transport Phenomena#

In this section:

17.3.1. Currents in semiconductors#

Two main processes:

  1. diffusion due to non-uniform density

  2. drift due to an electric field \(\mathbf{e}(\mathbf{r})\)

The overall current has the contribution of the motion of both electrons (\(n\), for negative charges) and holes (\(p\), for positive “charges”),

\[\mathbf{j}(\mathbf{r},t) = \mathbf{j}_n(\mathbf{r},t) + \mathbf{j}_p(\mathbf{r},t) \ ,\]

and both these contributions have their diffusion and drift part,

\[\begin{split}\begin{aligned} \mathbf{j}_n & = \mathbf{j}_{n,diff} + \mathbf{j}_{n,drift} \\ \mathbf{j}_p & = \mathbf{j}_{p,diff} + \mathbf{j}_{p,drift} \\ \end{aligned}\end{split}\]

For the contribution of the electrons, the diffusion current follows a Fick’s law for the number density \(n\) (multiplied by the constant charge \(-q\) of the elementary charge), while the drift coefficient is written as the product of the negative charge density and their drift velocity,

(17.2)#\[\begin{split}\begin{aligned} \mathbf{j}_n & = \mathbf{j}_{n,diff} + \mathbf{j}_{n,drift} = \\ & = - D_n (-q) \nabla n + \underbrace{(-q) n}_{\rho_n} \mathbf{v}_n = \\ & = D_n q \nabla n + q \mu_n n \mathbf{e} \ , \end{aligned}\end{split}\]

being the average drift velocity, \(\mathbf{v}_n = - \mu_n \mathbf{e}\), a function on the local electric field through the electron mobility \(\mu_n\). Here the minus sign follows the definition of the electron mobility with positive value, and from the opposite direction of the local electric field \(\mathbf{e}\) and the drift velocity of negative charges.

For the contribution of the holes,

(17.3)#\[\begin{aligned} \mathbf{j}_p & = - D_p q \nabla p + q \mu_p p \mathbf{e} \ . \end{aligned}\]

\(\mu_n \, [\dots]\)

\(\mu_p \, [\dots]\)

Silicon

1350

480

Gallium Arsenide

8500

400

Germanium

3900

1900

Remark. While charges in free-space have acceleration proportional to the electric field, in solids they have drift velocity proportional to the electric field, due to collisions with the lattice. This mechanism can be described in terms of conductivity \(\sigma\) (or its inverse, resistivity \(\rho_R\)) of the medium, similarly to Ohm’s law

\[\mathbf{j} = \sigma \mathbf{e} \quad , \quad \mathbf{e} = \rho_R \mathbf{j} \ .\]

Using the expression of the drift current, as the sum of the \(n\) and \(p\) contributions in (17.2) and (17.3) respectively,

\[\mathbf{j}_{drift} = q ( \mu_n n + \mu_p p ) \mathbf{e} \ ,\]

it immediatley follows the formula for the resistivity (and the conductivity)

(17.4)#\[\rho_R = \frac{1}{\sigma} = \frac{1}{q (\mu_n n + \mu_p p)} \ .\]

todo See discussion about this relation in 5.1.4. Velocity Saturation

17.3.2. Einstein relation#

…

(17.5)#\[\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{k T}{q} \ .\]

17.3.3. Balance equations#

The density of free electrons and holes are governed by the following PDEs

(17.6)#\[\begin{split}\begin{aligned} & \partial_t n - \nabla \cdot \left( \frac{\mathbf{j}_n}{q} \right) = ( G_n - R_n ) \\ & \partial_t p + \nabla \cdot \left( \frac{\mathbf{j}_p}{q} \right) = ( G_p - R_p ) \\ \end{aligned}\end{split}\]

where the expression of current densities \(\mathbf{j}_n\), \(\mathbf{j}_p\) are given in (17.2), (17.3) respectively, and the terms \(G_{n,p}\), \(R_{n,p}\) represent source and sink terms representing generation or recombination of free charges and holes.

As for a new free electron there’s a new hole, then \(G_n = G_p\). As recombination occurs between the same number of free electrons and holes, \(R_n = R_g\).

Multiplying the first and the second equation in (17.6) by \(-q\) and \(q\) respectively, the balance equation for the free charge density \(\rho_f = - q n + q p\),

\[\partial_t \rho_f + \nabla \cdot \mathbf{j} = 0 \ .\]