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.

Carrier Actions of Diffusion Current

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 flow of current. This flow of 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 of diffusion current

To derive the diffusion current in semi-conductor diode, the depletion layer must be large enough compared to the mean free path. We begin with the net current equation in a semi-conductor diode ,

J_n = q(μ_n E + D_n * dn/dx) .....Equation (1)

Substituting E = -dΦ/dx in the above equation (1) and multiplying both sides with e^(-Φ/V_t), Hence equation (1) becomes as follows :

J_n e^{(-Φ / V_t)} = q D_n[- n / V_t(dΦ/dx + dn/dx)]e^{(-Φ / V_t)} = q D_n d/dx[e^{(-Φ / V_t)}] .....Equation (2)

Integrating equation (2) over the depletion region, Hence giving us :

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

Which can be written as,

J_n = { q D_n N_c e^{(-Φ_B / V_t)}[e^{(V_a / V_t)} - 1]} / (₀ʃ^x_d e^{(- Φ* / V_t)} dx) .....Equation(3)

where Φ^* = Φ_B + Φ_i - V_a

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

Φ = - q N_d / 2E_s (x - x_d)²

Therefore Φ^* can be written as:

Φ^* = [(q N_d * x) / E_s](x_d - x/2) = (Φ_i - V_a)(x / x_d) .....Equation(4)

Since the x << x_d the term "x_d - x/2" is approximately equal to x_d , Using this approximation equation (4) is solved as follows :

(₀ʃ^x_d e^{(-Φ* / V_t)}dx = x_d (Φ_i - V_a) / V_t

Since, (Φ_i – V_a) > V_t. We obtain the equation of current caused due to diffusion :

J_n = [(q² D_n N_c) / V_t] [( 2q( Φ_i - V_a) N_d) / E_s]^½ e^{(- Φ_B / V_t)}[e^{(V_a / V_t)} - 1] .....Equation(5)

From equation (5) we 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 ,

E_max = [(2q (Φ_i - V_a) N_d / E_s]^½ .....Equation(6)

Substituting equation (6) in equation (5) we get ,

J_n = q μ_n E_max N_c e^{(- Φ_B / V_t)} [e^{(V_a / V_t)} - 1] .....Equation(7)

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

References

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

"Differences between diffusion current". Diffusion. http://www.ece.utep.edu/courses/ee3329/ee3329/Studyguide/ToC/Fundamentals/CAction/diffusion.html. Retrieved 10 September 2011. 

"Carrier Actions of Diffusion Current". Diffusion. http://http://ece.utep.edu/courses/ee3329/ee3329/Studyguide/ToC/Fundamentals/CAction/diffusion.html. Retrieved 11 October 2011. 

"derivation of difussion current". http://www.ee.ui.ac.id/~astha/courses/ts/teksem/msdiffus.htm. Retrieved 15 October 2011.