Density of States and Carrier Concentrations

The additional electrons and holes occupy energy states in the conduction and valence bands, respectively. Before discussing the rules of occupation of energy levels, the energy distribution of the available energy states must first be derived, as follows. 

In momentum space, the density of allowed points is uniform. Assuming that the surfaces of constant energy are spherical, then the volume of k space between spheres of energy E and E -i- AE is 4n / dk (see ref. [4]). Since a single level occupies a volume of Sjt /V (E = crystal volume) in momentum space and there are two states per level, the density of states is given by 

It has been assumed here that the volume is unity (e.g. 1 cm Inserting Eq. (1.13), one obtains 

The total density of energy states up to a certain energy level is obtained by integration of Eq. (1.22). The result is 

Semiconductor single crystals grown from extremely pure material exhibit a low conductivity because of low carrier density. The latter can be increased by orders of magnitude by doping the material. The principal effect of doping is illustrated 

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Density of States and Carrier Concentrations 115 where the density of states, N, around the top of the valence band is given by... 

Originally developed for amorphous inorganic semiconductors characterized by an exponential density of states, the author discusses the extension of these concepts to Gaussian densities of states under special consideration of state- and carrier-concentrations, electric fields, and temperafures. 

In an intrinsic semiconductor, charge conservation gives n = p = where is the intrinsic carrier concentration as shown in Table 1. Ai, and are the effective densities of states per unit volume for the conduction and valence bands. In terms of these densities of states, n andp are given in equations 4 and... 

Here Nc is the density of states in the conduction band, g the level degeneracy factor, n the carrier concentration in the band, A the activation energy of the level, Boltzmann s constant, and T the temperature. Now, in general, except at fairly low temperatures, the occupancy for shallow levels (with/ = /s) will be small, i.e., fs 1, and consequently... 

In an exact calculation of the distribution of the electrostatic potential, the carrier densities and their currents, (4.81)-(4.87) are solved simultaneously, bearing in mind that only the sum of the diffusion and drift currents has physical significance. Due to the complexity of the above relations and in particular due to the coupling of electron and hole concentrations by Poisson s equation, analytical solutions exist only for a few, very specific conditions. Generally, the determination of local carrier concentrations, current densities, recombination rates, etc., requires extensive numerical procedures. This is especially true if they vary with time, but even in the steady state context. 

The expressions for the components of the total faradaic current at the semiconductor surface as given in eqns. (173) (176) show that this current is given as the product of factors intrinsic to the electron transfer process taking place, to the concentration and thermal energy distribution of the redox couple, and to the concentration of carriers or the density of states. If we restrict attention to an n-type semiconductor and assume that only electron transfer to and from the conduction band is significant, then the nett current can be written... 

We have assumed up to now that besides lattice absorption, intrinsic semiconductors were essentially transparent to photon energies less than the band gap at RT and below. Now, like electrons in metals, the free carriers in semiconductors can absorb electromagnetic radiation to increase their energies. In the calculation of the intrinsic free-carrier concentrations in the VB and CB of a semiconductor, one has to consider the effective densities of states... 

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