Diffusion Currents in Semiconductors

The carriers present in a semiconductor move by one of two mechanisms, drift or diffusion.

Similarly to the conduction current, the diffu­sion current, symbolized as I _dif, may be consti­tuted by electrons or by holes. The respective densities are given by:

J_{n dif} = eD_n∆n/∆x and J_{p dif} = –eD_p∆p/∆x, (1.6)

where the terms ∆ n /∆x and ∆ p /∆x are referred to as the concentration gradients, and the quanti­ties D_n and D_p as the diffusion coefficients.

Taking germanium at room temperature, it is D_n = 98 cm² s^-1 and D_p = 47 cm² s^-1. For silicon, the respective figures are D_n = 34 cm² s^-1 and D_p = 12 cm² s^-1.

The excess concentration will decrease exponen­tially, as shown in the plot of Fig. 1.7 for the electron concentration. The time during which the excess concentration is reduced by a factor of 2.7, that is, falls to 0.37 of its original value, n ₀, is called the lifetime of nonequilibrium carriers, τ_n.

Fig.1.7. Time variations in excess charge Fig.1.8. Spatial variations in excess charge concentration concentration

When nonequilibrium carriers (say, electrons) propagate through a specimen by diffusion, their concentration likewise decreases with dis­tance exponentially due to recombination. The distance L_n over which the excess concentration of nonequilibrium (usu­ally, minority) carriers is reduced by a factor of 2.7, that is, falls to 0.37 of its original value n ₀, is called the diffusion length.

Thus, the excess concentration decreases both with time and distance, and so the lifetime τ_n and the diffusion distance L_n are interconnected by a relation of the form:

L_n = (D_nτ_n)^1/2.

2. P-N Junctions

2. 1. A P-N Junction with No External Voltage Applied

The term junction in our case refers to the boundary, or the region of transition, between p -type and n- typesemiconductor materials; hence the name a p-n junction. A p-n junction has an unsymmetrical conductivity.

We assume that no external voltage is applied across the p-n junction (Fig. 2.1). Electrons diffuse from the n -type semiconductor where their concentration is high into the p -type semiconductor where their concentration is low. Conversely, holes diffuse from the p -type semiconductor where their concentration is high into the n -type semi­conductor where their concentration is low. The circles labelled with the " + " and “–” signs represent the donor and acceptor impurity atoms charged positively and nega­tively, respectively.

Fig. 2.1. P-N junction with no external bias applied

As a result are produced a positive immovable charge in the n -region and a negative immovable charge in the p -region.

What is known as the contact potential dif­ference:

φ₀ = φ_n - φ_p

is produced between the two charges which give rise to an electric field.

Assuming an average impurity con­centration in the case of germanium, we obtain φ₀= 0.3-0.4 V and d= 10^-4-10 ^-5 cm. At the not high impurity concentrations produced in silicon devices φ₀ ≈ 0.7V and d = 10^-6cm.

The diffusion of majority carriers across the junction is accompanied by the reverse migration of carriers under the influence of the, electric field set up by the contact potential difference Econt.

The motion of carriers due to diffusion as the diffusion current I_dif and the motion of carriers due to the action of the field as, the drift current I_dr. In a steady state, that is, in a state of dynamic equilibrium, the two currents are equal in magnitude and opposite in sign.

2.2. The Forward-Biased P-N Junction

Let an external voltage source be connected with its positive terminal to the p -type se­miconductor (the anod), and with its negative terminal to the n -type semiconductor (the cathode), that make up a p-n junction. This is called forward bias­ing.

With forward biasing, the applied potentials establish an electric field which drives the maj­ority carriers of each region towards the jun­ction, thereby giving rise to the forward current, I_F, across the junction (Fig. 2.2).

Fig. 2.2. Forward-biased p-n junction

The potential difference at the junction is brought down which is another way of saying that the height of the potential barrier is reduced while the diffusion current builds up because a greater number of carriers can overcome the reduced barrier.

The voltage across the junction may be taken equal to (φ₀-U_F).

With forward biasing, I_dif > I_dr and so the net current across the junction, that is, the forward current:

I_F = I_dif – I_dr > 0.

The introduction of excess charge carriers across the potential barrier reduced in height by forward biasing into the region where they are minority carriers is called the carrier injection.

The resistan­ce in the forward direction, R_F, falls to a very small value (from units to a few tens of ohms).

The total forward current, I_F is:

I_F = I_n – I_p = , (2.1)

where I_S – revers saturation current of diode, φ_T =0.0258V (for T=300K) – termodinamical potential.

2.3. The Reverse-Biased P-N Junction

Now we will connect the " + " side of an external voltage source to the n -region and the "–" side to the p -region of a p-n junction, or diode (Fig. 2.3a ). Thus connected, the junction is said to be reverse-biased. The applied reverse bias voltage U_R causes a very small reverse current, I_R ≈I_S, to cross the junction. The resultant field gains in strength so thatjhe height of the, potential barrier now is (φ₀ + U_R). The removal of minority carriers across a p-n junction by the accelerating electric field set up by the reverse bias voltage is called the carrier extraction.

Fig. 2.3. Reverse-biased p-n junction

The reverse current is very small be­cause the resistance of the barrier layer in a reverse-biased p-n junction (or diode) is very high so that R_R» R_F.

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