Diffusion current

Introduction

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

Diffusion current versus drift 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.

Derivation of diffusion current

The diffusion current in a metal-semiconductor diode is derived on the basis of assumption that the depletion layer is large enough compared to the mean free path, so that the equations of drift and diffusion currents are valid. We start our derivation from the expression for the total current (for 'n' region of the diode) and then integrate it over the width of the depletion region:

J_n = q(μ_n ɛ + D_n * dn/dx)

which can be rewritten by substituting ɛ = -dΦ/dx and multiplying both sides with exp(-Φ/V_t), Hence equation becomes as follows :

J_n exp(-Φ / V_t) = q D_n[- n / V_t(dΦ/dx + dn/dx)]exp(-Φ / V_t)=q D_n d/dx[exp(-Φ / V_t)]

Next step is to integrate both sides of the equation over the depletion region :

J_n = q D_n n exp(-Φ / V_t)|₀^x_d / [₀ʃ^x_d exp(-Φ / V_t)dx]

J_n = { q D_n N_c exp(-Φ_B / V_t)[exp(V_a / V_t) - 1]} / (₀ʃ^x_d exp(- Φ^* / V_t) dx)

where Φ^* = Φ_B + Φ_i - V_a

The integral in the denominator is solved using the potential obtained from the full depletion approximation, or

Φ = - q N_d / 2ɛ_s (x - x_d)²

Hence, Φ^* can be written as:

Φ^* = [(q N_d * x) / ɛ_s](x_d - x/2) = (Φ_i - V_a)(x / x_d)

The second term of above equation was dropped, since the linear term is dominant because x << x_d. Using this approximation one can solve the integral as :

(₀ʃ^x_d exp(-Φ^* / V_t)dx = x_d (Φ_i - V_a) / V_t

Since, (Φ_i – V_a) > V_t. This gives us the final expression for the current due to diffusion :

J_n = [(q² D_n N_c) / V_t] [( 2q( Φ_i - V_a) N_d) / ɛ_s]^½ exp(- Φ_B / V_t)[exp(V_a / V_t) - 1]

The above expression indicates that the current depends exponentially on the applied voltage, V_a, and the barrier height, Φ_B. The pre-factor can be understood, if one rewrites that term as a function of the electric field intensity at the metal-semiconductor junction, ɛ_max:

ɛ_max =[(2q (Φ_i - V_a) N_d / ɛ_s]^½

therefore,

J_n = q μ_n ɛ_max N_c exp(- Φ_B / V_t) [exp(V_a / V_t) - 1]

Hence the pre-factor equals the drift current at the metal-semiconductor junction, which for zero applied voltage exactly balances the diffusion current.

References

Ben G. Streetman, Santay Kumar Banerjee;Solid State Electronic Devices(6th Edition), Pearson International Edition; pp. 126–135.

"Differences between diffusion current". Diffusion. Retrieved 10 September 2011.

"Diffusion current derivation". Retrieved 20 september 2011. {{cite web}}: Check date values in: |accessdate= (help)