2.1.1 Extrinsic Dopant Diffusion

At usual diffusion temperatures the dopants are often ionized and the so released electrons cause an electric field which operates as an additional drift term to the normal diffusion flux.

$$\overrightarrow{J} \overrightarrow{v}$$ (2.4)

The velocity of the charged particles can be described with

$$\overrightarrow{v} \mu \overrightarrow{E} \mu \left(\vphantom{ \psi }\right. \psi \left.\vphantom{ \psi }\right)$$ (2.5)

where $\overrightarrow{E}$ is the electric field and $\psi$ the electrostatic potential. The mobility $\mu$ is related to the diffusion coefficient by Einstein's relation

$$\mu {\frac{q}{kT}}$$ (2.6)

where q denotes the elementary charge and k is known as the Boltzmann constant. Introducing (2.6) and (2.5) into (2.4) and taking more than one dopant into account Fick's law can be extended to

$$\overrightarrow{J_A} \left(\vphantom{ grad(C_A)+z_A\cdot \frac{q}{kT}\cdot C_A \cdot grad(\psi)}\right. {\frac{q}{kT}} \psi \left.\vphantom{ grad(C_A)+z_A\cdot \frac{q}{kT}\cdot C_A \cdot grad(\psi)}\right)$$ (2.7)

where z_A denotes the charge state of the belonging dopant (+1 for singly charged acceptors and -1 for singly charged donors).

The electrostatic potential $\psi$ is determined by the Poisson equation

$$\left(\vphantom{\varepsilon\cdot grad(\psi)}\right. \varepsilon \psi \left.\vphantom{\varepsilon\cdot grad(\psi)}\right)$$ (2.8)

with

$$\sum_{A}^{}$$ (2.9)

where the quantity C_net represents the net concentration of all ionized dopants and n, p the concentration of electrons and holes, respectively.

Under the assumption of thermodynamic equilibrium the carrier concentrations n and p can be obtained with

n ^. p = n_i² (2.10)

$$\psi {\frac{q}{kT}}$$ (2.11)

$$\psi {\frac{q}{kT}}$$ (2.12)

In case of global charge neutrality

n - p - C_net = 0 (2.13)

and by means of the equilibrium carrier concentrations (2.11)(2.12) the electrostatic potential can explicitly be calculated by

$$\psi {\frac{kT}{q}} \left(\vphantom{\frac{C_{net}}{2n_i}}\right. {\frac{C_{net}}{2n_i}} \left.\vphantom{\frac{C_{net}}{2n_i}}\right)$$ (2.14)

With the explicit form of the gradient of $\psi$,

$$\psi {\frac{kT}{q}} {\frac{1}{\sqrt{C_{net}^2+4n_i^2}}} \sum_{i}^{}$$ (2.15)

substituting into (2.7) the flux for dopant C_A now depends also on the gradients of the concentrations C_i of all other dopants.

$$\overrightarrow{J_A} \left(\vphantom{ 1 + \frac{C_A}{\sqrt{C_{net}^2+4n_{i}^2}} }\right. {\frac{C_A}{\sqrt{C_{net}^2+4n_{i}^2}}} \left.\vphantom{ 1 + \frac{C_A}{\sqrt{C_{net}^2+4n_{i}^2}} }\right)$$

$${\frac{z_A\cdot C_A}{\sqrt{C_{net}^2+4n_{i}^2}}} \sum_{i\neq A}^{}$$ (2.16)

If only one dopant is present (2.16) simplifies to

$$\overrightarrow{J_A} \left(\vphantom{ 1 + \frac{C_A}{\sqrt{C_A^2+4n_{i}^2}} }\right. {\frac{C_A}{\sqrt{C_A^2+4n_{i}^2}}} \left.\vphantom{ 1 + \frac{C_A}{\sqrt{C_A^2+4n_{i}^2}} }\right)$$ (2.17)

Comparing Fick's first law (2.1) and (2.17) an effective diffusion coefficient with the field enhancement factor $\alpha$ can be extracted

$$\left(\vphantom{ 1 + \frac{C_A}{\sqrt{C_A^2+4n_{i}^2}} }\right. {\frac{C_A}{\sqrt{C_A^2+4n_{i}^2}}} \left.\vphantom{ 1 + \frac{C_A}{\sqrt{C_A^2+4n_{i}^2}} }\right) \alpha$$ (2.18)

For intrinsic conditions ( C_A/n_i $\ll$ 1) this factor has a value close to one and for high concentrations ( C_A/n_i $\gg$ 1) a value close to two.