Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

tions or the total dopant concentration calculated using formulas and parameters in Figure 4.3 (b).

These effective impurity concentrations can be used in the above calculations. One might guess that the actual background levels of each type of impurity are unimportant. Actually, this is not true, since the ionized impurities tend to decrease carrier mobility by increasing the number of scattering centers as shown in Figure 4.2. Eq. (4.18) uses the conduction band and valence band as the reference point for calculating EF. A more convenient approach is to define an intrinsic Fermi energy, Ei, such that Ei = EF in undoped, i.e. intrinsic, material. Then the Fermi level in doped materials can be referenced to Ei. If we assume that EF in the bandgap is more than several kT away from either the conduction or valence band, Eqs. (4.5-4.6) can be simplified to the form:

which is exactly the Boltzmann statistics. These equations amount to approximating the Fermi-Dirac statistics, Eq. (4.4), with simple exponential distributions. For the intrinsic condition that p = n = ni,

when pn > nr and depletion when pn < nr. There is a general tendency for a semiconductor system to remain in the state of charge-neutrality even under nonequilibrium conditions. In order to facilitate the further discussion, we partition the carrier concentration into two parts - that at the equilibrium and that deviated from the equilibrium value. That is,

where subscript nor P represents the material type and subscript '0' the equilibrium. The charge neutrality condition thus requires b.p = b.n.

It is apparent that in nonequilibrium if one still wishes to use

Eqs. (4.23-4.24) to express Pn and nn in terms of a single Fermi level EF, it simply does not exist. That is, the use of a single Fermi level is valid only for equilibrium. For injection, both Pn and nn increase from their equilibrium values, causing the effective Fermi level for electrons to move closer to the conduction band and the effective Fermi level for holes to move closer to the valence band. Under such conditions we define quasi-Fermi energies for electrons and holes as EFn and EFp, respectively, using equations

where P and n are the nonequilibrium values (with appropriate subscripts for P or n materials).

When the nonequilibrium condition is imposed, a process occurs in the semiconductor which tries to restore equilibrium. For example, if pn > nr (perhaps due to generation of extra hole-electron pairs by radiation) the carrier recombination process must be more vigorous to counter-balance the generation process or an infinite number of holeelectron pairs will be built-up in the material. Steady-state exists when the recombination rate equals the generation rate. This, however, does not mean that p and n are at their equilibrium values. As with most chemical processes, the recombination rate, T, is proportional to the products of species concentrations. Hence we will write

(4.31)

In equilibrium, the thermal excitation is the sole source for generation of electron-hole pairs. The thermal generation rate, 9th, must equal r to maintain equilibrium:

(4.32)

This generation rate should remain constant independent of the state of the semiconductor for it is determined only by thermal conditions in the solid. Hence, under nonequilibrium conditions we can define the net rate of recombination with respect to the thermal generation, Uth, as

(4.33)

The value of Uth can be either positive or negative, and has units of number· cm-3 ·sec-1 .

Although the generation of electron-hole pairs by direct transition between the valence and conduction bands exists in all semiconductors, the indirect band structure, where the conduction band minimum and valence band maximum correspond to different k vectors (momentum), in such semiconductors as Si and Ge makes such transitions very unlikely. A common generation and recombination process is through the intermediate (and localized) levels in the bandgap. These levels are called recombination centers or traps, depending upon their respective capabilities in capturing electrons and holes [4.7]. The equation describing the above process was first proposed by Shockley, Read, and Hall, and the process is frequently referred to as the SRH recombination. The net recombination rate is written as

(4.34)

where Tp and Tn are lifetimes for holes and electrons, respectively, and Et is the energy level for recombination centers. In later discussions, we will consider the dependence of these lifetimes on doping densities. It can be shown that a recombination level is most effective for generationrecombination when Et = Ei, in which a maximum lui is reached for

In this case,

(4.35)

Using the above equation, we can define two useful special cases for either p- or n-type material. Define the low level conditions in nonequilibrium to be when the excess minority carrier concentrations, ~p and ~n, are much less than the concentration of the dominant equilibrium species, (either Ppo or nnO), called majority carriers. Under these conditions, for an n-type material

Eqs. (4.38-4.39) state that for low-level conditions the net recombination rate is proportional to the excess of minority species above the equilibrium value.

4.4Carrier Transport and Conservation

The flow of current is one manifestation of nonequilibrium conditions. For the moment we will consider only de current flow. Two mechanisms

concentration gradient gives rise to diffusion current.

Drift current can be described as follows. For an applied electric

thermal velocity Vth then the drift velocity is proportional to the applied field strength. The constant of proportionality is called mobility, JLn, which has units of velocity per unit field (or em2 IV .sec):

(4.40)

The resulting current due to this directional drift movement (compared with random thermal motion) is given by the number of charged

particles multiplied by their charge and drift velocity. Since the electrons have negative charge and move in the minus-x direction, for an £(x) in the positive-x direction, the current density in the +x direction is:

(4.41)

For holes the drift velocity is given by

(4.42)

Once again, current density is positive, since positively charged particles move in the positive-x direction:

(4.43)

The above conditions apply only for low electric fields. For large electric fields, the drift velocity saturates at a value Vth and becomes independent of electric field.

Diffusion current results when non-uniformities of concentrations exist. Consider the distribution of holes, increasing with x. For this distribution the gradient dpjdx is positive. On the average the carriers move in the negative-x direction, or toward the regions of lower concentration. Since the holes are positively charged, this means that the current density will be in the negative x direction. The constant of proportionality is Dp , the diffusion constant. This current flow can be expressed as:

(4.44)

The above discussion assumes a one-dimensional gradient.

For electrons, the increasing distribution with x again gives rise to a flow of carriers in the negative x direction, but the negative charge of the particles gives a resultant current density of

The drift and diffusion components can be added to find expressions for the total flow of holes and electrons:

(4.46)

E, ----

----.. x

Figure 4.4: Energy band diagrams for a pn junction, (a) prior to contact, (b) after contact illustrating 8EF/8x = 0, band bending with 8Ec/8x = 8Ed8x > 0 across the transition region.

This is the continuity equation for electrons. The appropriate expressions for Jp(x) and In(x) can be obtained from Eqs. (4.47) and (4.46).

4.5The pn Junction - Equilibrium Conditions

When p- and n-type materials are brought together, a rectifying contact is formed. This pn junction is the basis of operation for many semiconductor devices. Suppose that we wish to join the p and n materials to form a pn junction (see Figure 4.4 (a)). The Fermi energy for both materials should match up. If this were not the case, a difference in potential energy would exist between the two metal contacts and a "battery" would result. That is, the potential would appear at the terminals and be usable to "do work." It is certainly not reasonable for an external potential to exist since all processes must typically balance to give no net current within the solid. However, the condition of a constant (or flat) Fermi level across the pn junction introduces the possible problem of discontinuous conduction and valence energy bands at the