Diffusion current

Derivation

In a region where n and p vary with distance, a diffusion current is superimposed on that due to conductivity. This diffusion current is governed by Fick's law:

${\displaystyle F=-D_{\text{e}}\mathrm{\nabla }n}$

where:

F is flux.

D_e is the diffusion coefficient or diffusivity

${\displaystyle \mathrm{\nabla }n}$ is the concentration gradient of electrons

there is a minus sign because the direction of diffusion is opposite to that of the concentration gradient

The diffusion coefficient for a charge carrier is related to its mobility by the Einstein relation:

${\displaystyle D_{\text{e}}=\frac{{\mu }_{\text{e}}{k}_{\text{B}}T}{e}}$

where:

k_B is the Boltzmann constant

T is the absolute temperature

e is the electrical charge of an electron

Now let's focus on the diffusive current in one-dimension along the x-axis:

${\displaystyle F_x=-D_{\text{e}}\frac{dn}{dx}}$

The electron current density J_e is related to flux, F, by:

${\displaystyle J_{\text{e}}=-eF}$

Thus

${\displaystyle J_{\text{e}}=+e{D}_{\text{e}}\frac{dn}{dx}}$

Similarly for holes:

${\displaystyle J_{\text{h}}=-e{D}_{\text{h}}\frac{dp}{dx}}$

Notice that for electrons the diffusive current is in the same direction as the electron density gradient because the minus sign from the negative charge and Fick's law cancel each other out. However, holes have positive charges and therefore the minus sign from Fick's law is carried over.

Superimpose the diffusive current on the drift current to get

${\displaystyle J_{\text{e}}=e{\mu }_{\text{e}}nE+e{D}_{\text{e}}\frac{dn}{dx}}$ for electrons

and

${\displaystyle J_{\text{h}}=e{\mu }_{\text{h}}pE-e{D}_{\text{h}}\frac{dp}{dx}}$ for holes

Consider electrons in a constant electric field E. Electrons will flow (i.e. there is a drift current) until the density gradient builds up enough for the diffusion current to exactly balance the drift current. So at equilibrium there is no net current flow:

${\displaystyle e{\mu }_{\text{e}}nE+e{D}_{\text{e}}\frac{dn}{dx}=0}$

Example

To derive the diffusion current in a semiconductor diode, the depletion layer must be large compared to the mean free path. One begins with the equation for the net current density J in a semiconductor diode,

where D is the diffusion coefficient for the electron in the considered medium, n is the number of electrons per unit volume (i.e. number density), q is the magnitude of charge of an electron, μ is electron mobility in the medium, and E = −dΦ/dx (Φ potential difference) is the electric field as the potential gradient of the electric potential. According to the Einstein relation on electrical mobility ${\displaystyle D=\mu V_t}$ and ${\displaystyle V_t=kT/q}$. Thus, substituting E for the potential gradient in the above equation (1) and multiplying both sides with exp(−Φ/V_t), (1) becomes:

Integrating equation (2) over the depletion region gives

${\displaystyle J=\frac{qDn{e}^{-\mathrm{\Phi }/V_t}{|}_0^{x_d}}{{\int }_0^{x_d}{e}^{-\mathrm{\Phi }/V_t}dx}}$

which can be written as

where

${\displaystyle {\mathrm{\Phi }}^{\ast }={\mathrm{\Phi }}_B+{\mathrm{\Phi }}_i-V_a}$

The denominator in equation (3) can be solved by using the following equation:

${\displaystyle \mathrm{\Phi }=-\frac{q{N}_d}{2E_s(x-x_d{)}^2}}$

Therefore, Φ* can be written as:

Since the x ≪ x_d, the term (x_d − x/2) ≈ x_d, using this approximation equation (3) is solved as follows:

${\displaystyle {\int }_0^{x_d}{e}^{-{\mathrm{\Phi }}^{\ast }/V_t}dx=x_d\frac{{\mathrm{\Phi }}_i-V_a}{V_t}}$,

since (Φ_i − V_a) > V_t. One obtains the equation of current caused due to diffusion:

From equation (5), one can observe that the current depends exponentially on the input voltage V_a, also the barrier height Φ_B. From equation (5), V_a can be written as the function of electric field intensity, which is as follows:

Substituting equation (6) in equation (5) gives:

From equation (7), one can observe that when a zero voltage is applied to the semi-conductor diode, the drift current totally balances the diffusion current. Hence, the net current in a semiconductor diode at zero potential is always zero.

The equation above can be applied to model semiconductor devices. When the density of electrons is not in equilibrium, diffusion of electrons will occur. For example, when a bias is applied to two ends of a chunk of semiconductor, or a light is shining in one place (see right figure), electrons will diffuse from high density regions (center) to low density regions (two ends), forming a gradient of electron density. This process generates diffusion current.