QUASI FERMI LEVELS

So far we have been considering semiconductors that are in thermal equilibrium where pn = n_i² and that the probability of occupancy of an electronic state is govern by Fermi-Dirac distribution function. Under this condition we have a Fermi level which is constant throughout the sample.

When there is current flow or when there is an external stimulation, the sample is no longer in equilibrium, the concept of Fermi level is no longer applicable. However the non-equilibrium carrier concentrations can still be express-ed in familiar way if we define quantities known as quasi-Fermi levels, E_FP for holes and E_FN for electrons:

When there are excess carriers, E_FN is displaced above the equilibrium Fermi level E_F and E_FP is displaced below E_F. For each quasi-Fermi level we can define a quasi-Fermi potential:

.

Often in describing the behaviour of a semiconductor material or device we consider the quasi-Fermi levels instead of the carrier concentrations.

The current equations:

, and

can be written in terms of the quasi-Fermi potentials.

Electric field E at point x is:

= = -E

The variation of E_C is reflected in E_i (the bands are 'parallel'), hence :

= = -E, or

= E,

Substitute this into the hole current equation:

,

Use Einstein relationship, and cancelling terms in E_I, we get:

For electron current we have :

CARRIER GENERATION AND RECOMBINATION

Generation - process whereby an electron-hole pair is created.

Recombination - process whereby an electron and a hole are eliminated when the electron (in the conduction band) loses sufficient energy and occupies the empty energy level in the valence band represented by the hole.

There are many different processes by which generation and recombination can occur. Of most interest to us at present is the processes involving energy levels in the middle of the bandgap - intermediate level centres or generation-recombination centres.

The theory was first proposed by Shockley, Read and Hall and has been very successful in describing carrier generation and recombination in many semiconductor devices. This is known as the Shockley-Read-Hall Recombination Process.

LOW-LEVEL INJECTION

Before going into the details of the theory let us revise the simple model used to describe carrier generation-recombination in semiconductors when we have carrier concentrations not very different from the equilibrium levels. This is usually known as low-level injection. In this case we assume majority carrier concentration to be unchanged from its equilibrium level and only the minority carrier concentration is of interest in this model.

With a p-type material, let n be the (minority) electron concentration and n_o be the equilibrium electron concentration. The recombination rate R (number of electron-hole pairs disappearing per unit time) is assumed to be proportional to the excess electron concentration dn = (n-n_o):

t_n is a constant and is known as the minority carrier lifetime.

R which represents the rate of removal of electron is or Therefore:

The equation indicates that recombination rate increases with excess minority carrier concentration. It tells us that if the carrier concentration is below the equilibrium level there is a spontaneous generation process to try to restore the carrier to its equilibrium level.

It is important to differentiate this spontaneous process from the externally induced generation such as through in introduction of light with photon energy larger than the bandgap energy.

The above equation simple to handle and is used extensively in analytical work.

SHOCKLEY-READ-HALLRECOMBINATION THEORY

In the Shockley Read Hall Theory electrons and holes are

assumed to interact with intermediate level centres, which are electron energy levels in the bandgap.

There are four separate processes:

electron capture by a centre

electron emission "

hole capture "

hole emission "

We have to look at the capture and emission rates of these individual processes.

Rate of electron capture R_a

This is assumed to be proportional to the electron concentration and the concentration of the intermediate level centres that are empty:

R_a = C_n .n.(1-f)N_t

where

n = electron concentration

N_t=concentration of the intermediate level in semiconductor

f = probability of occupancy of a centre by an electron

C_n= a proportionality constant known as capture probability = s_n.v_th, where s_n is the capture cross-section for electrons, and v_th is the thermal velocity of electrons

Rate of electron emission R_b

This is assumed to be proportional to concentration of centres occupied by an electron:

R_b = e_n .f .N_t

where

e_n = the proportionality constant called emission probability.

Rate of hole capture R_c

This is described in a similar fashion as R_a:

R_c = C_p .n.f.N_t

Rate of hole emission R_d :

R_d = e_p .(1-f). N_t

When the semiconductor is in equilibrium the transition to and from the conduction band must be the same: R_a = R_b. This allows us to find the relationship between C_n and e_n . Also in equilibrium:

This gives,

C_n n_o (1- f_o ) Nt = e_n f_o N_t

producing:

e_n = C_nn₁

where

By similar equilibrium consideration for R_c and R_d, we get:

e_p = C_p p₁

Next consider the net rates of electron and hole capture by the intermediate level centres when there is an excess of carriers.

Electron capture rate:

R_n = R_a -R_b = C_nn(1-f) N_t - C_nn₁N_t.

Hole Capture rate:

R_p = R_c –R_d = C_ppfN_t – C_pp₁(1-f) N_t.

Now we consider a steady-state condition where the occupancy of the intermediate level centres does not change with time. This requires

R_n = R_p = R.

This leads to:

,

and this gives

This is the S-H-R recombination rate equation where,