# Diffusion current

Diffusion current is a current in a semiconductor caused by the diffusion of charge carriers (holes and/or electrons). Diffusion current can be in the same or opposite direction of a drift current, that is formed due to the electric field in the semiconductor. At equilibrium in a p–n junction, the forward diffusion current in the depletion region is balanced with a reverse drift current, so that the net current is zero. The diffusion current and drift current together are described by the drift–diffusion equation.[1]

The diffusion constant for a doped material can be determined with the Haynes–Shockley experiment.

## Introduction

### Diffusion current versus drift current

Following are comparisons of the two forms of current:

Diffusion current Drift current
Diffusion current occurs even though there isn't an electric field applied to the semiconductor. Drift current depends on the electric field applied on the p-n junction diode.
It depends on constants Dp and Dn, and +q and −q, for holes and electrons respectively but it is independent of permittivity. It depends upon permittivity.
Direction of the diffusion current depends on the change in the carrier concentrations, not the concentrations themselves. Direction of the drift current depends on the polarity of the applied field.

### Carrier actions

No external electric field across the semiconductor is required for the diffusion of current to take place. This is because diffusion takes place due to the change in concentration of the carrier particles and not the concentrations themselves. The carrier particles, namely the holes and electrons of a semiconductor, move from a place of higher concentration to a place of lower concentration. Hence, due to the flow of holes and electrons there is a current. This current is called the diffusion current. The drift current and the diffusion current make up the total current in the conductor. The change in the concentration of the carrier particles develops a gradient. Due to this gradient, an electric field is produced in the semiconductor.

## Derivation

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

$J = q n \mu E + q D\frac{dn}{dx}$

(1)

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 charge of one 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. We know that D=µ*Vt with Vt=k*T/q (Einstein relation on electrical mobility). Thus, substituting E for the potential gradient in the above equation (1) and multiplying both sides with exp(−Φ/Vt), (1) becomes:

$J e^{-\Phi / V_t} = q D\left(\frac{-n}{V_t}*\frac{d\Phi}{dx} + \frac{dn}{dx}\right)e^{-\Phi / V_t} = q D \frac{d}{dx}(n e^{-\Phi / V_t})$

(2)

Integrating equation (2) over the depletion region gives

$J = \frac{q D n e^{-\Phi / V_t}\big|_0^{x_d}}{\int_0^{x_d} e^{-\Phi / V_t}dx}$

which can be written as

$J = \frac{ q D N_c e^{-\Phi_B/ V_t}\left[e^{V_a/ V_t} - 1\right]}{\int_0^{x_d} e^{-\Phi^*/ V_t} dx}$

(3)

where

$\Phi^* = \Phi_B + \Phi_i - V_a$

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

$\Phi = - \frac{q N_d}{2E_s (x - x_d)^2}$

Therefore, Φ* can be written as:

$\Phi^* = \frac{q N_d x}{E_s} \left(x_d - \frac{x}{2}\right) = (\Phi_i - V_a)\frac{x}{x_d}$

(4)

Since the x << xd the term (xdx/2) ≈ xd, using this approximation equation (3) is solved as follows:

$\int_0^{x_d} e^{-\Phi^* / V_t}dx = x_d \frac{\Phi_i - V_a}{V_t}$,

since (ΦiVa) > Vt. One obtains the equation of current caused due to diffusion:

$J = \frac{q_2 D N_c}{V_t} \left[\frac{2q}{E_s}( \Phi_i - V_a) N_d\right]^{1/2} e^{-\Phi_B / V_t}(e^{V_a / V_t} - 1)$

(5)

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

$E_\mathrm{max} = \left[\frac{2q}{E_s} (\Phi_i - V_a) N_d\right]^{1/2}$

(6)

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

$J = q \mu E_\mathrm{max} N_c e^{-\Phi_B / V_t}(e^{V_a/V_t} - 1)$

(7)

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.