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Degenerate semiconductors

What if we introduce a lot of donors? Well, the Fermi level will rise more and more towards the conduction band. At some stage the approximations will no longer hold because more than only the tail of the Fermi Dirac function overlaps with the band edge. The approximations break down when the Fermi level gets closer than about 3 kT to one of the band edges. At room temperature this is approximately 75 meV.

Degeneracy is often observed in the so called transparent conductors. These are usually oxides with bandgaps of more than 3 eV which makes them transparent for the visible light. Prominent examples are ZnO doped with any of the group III elements (B, Al, Ga, In), SnO₂ doped with fluorine (FTO) or a mixture of In₂O₃ and SnO₂ (ITO), CdO, and CdSnO₄ (cadmium stannate). A reasonable electric conductivity can be obtained by doping them to degeneracy. For example, for ITO specific resistivities of less than 5 μΩm have been routinely obtained. Metals are only one or two orders of magnitude below, lead and titanium have specific resistivities of 0.2 and 0.45 μΩm, respectively, and the lowest specific resistivity is 0.015 μΩm in silver.

In order to get an idea about the location of the Fermi level we make a crude approximation which is often employed for the description of metals. We simply take the zero temperature case where the Fermi Dirac function is a step function (illustrated in red in the figure below).

With the step function the calculation of the integrals is trivial. In fact we can count the levels like we did earlier in the case of free electrons. The Fermi level is located inside the conduction band, the effective mass accounts for the parabolicity of the band

Burstein noted an important consequence for the light absorption of degenerate semiconductors. We cannot absorb photons with energies exactly of the bandgap energy because all the states around the conduction band minimum are filled. Rather, we need a little extra energy to excite electrons into free states higher up in the conduction band. Thus, in optical absorption measurements the band gap appears to be increased. The phenomenon is called Burstein Moss shift [1].

There is a second effect of the high density of donors. We noted that the energetic position of donor states is inside the band gap, somewhere below the conduction band. From the hydrogen model of the dopant we can estimate the Bohr radius of the dopant electron. If the "orbitals" of the dopants overlap we will observe a splitting of the energy levels, or, for yet higher doping, a whole donor band. At sufficiently high levels of doping the donor band becomes so broad that it actually merges with the conduction band minimum and effectively results in a decreased band gap [2].

Non-parabolicity

For the non-degenerate case the approximation of parabolic bands is quite sufficient, but as soon as we fill considerable parts of the band, the parabolic model begins to fail. The differences arise from two different aspects; first, the energy surface might be more complicated than the assumed spherical or ellipsoidal relation, and second, electrons with higher wavevectors may feel the effects of the periodic crystal more distinctly than those at the conduction band minimum. In this case their wavefunctions are no longer composed of travelling waves, but they will also have contributions of standing waves.

The obvious thing for the first case is to expand the energy surface E(k) to higher orders of the wavevector k, leading to third and fourth rank tensors analogous to the second rank tensor of the effective mass. However, this is not often done because the second effect is usually more pronounced. Thus, energy surfaces of second order are assumed and the non-parabolicity is taken care by a non-linear dependence on the energy.

In order to illstrate the effect more clearly, the figure below shows a part of the band diagram of ZnO [3]. We observe that the parabolic approximation of the conduction band overestimates the energy of electrons with higher wavevectors.

The considerations above translate into the following mathematical procedure: we keep a second order energy surface (a rotational ellipsis or a shpere) and as we increase its volume, the energy no longer increases linearly with E but according to a function γ(E).

Usually the corrections are small and γ itself is expanded in a series and often the first non-linear term is sufficient. Theoretical considerations of the interactions between valence and conduction band suggest that the term γ₂ is equal to the inverse of the bandgap energy [4]. In the case of spherical energy surfaces a scalar term for the effective mass is sufficient, and the non-parabolicity may be absorbed into a new effective mass which is then dependent on the energy:

This so called density of states effective mass is defined as:

There are also other definitions for the effective mass, and the results may be different. For example, there is the "conduction effective mass" and the "band curvature effective mass". It is important to know which effective mass is relevant for the experiment in question (see for example the discussion in [5,6]).

[1] E. Burstein, Phys. Rev.93(3) (1954), 632
[2] A. P. Roth, J. B. Webb, and D. F. Williams, Phys. Rev. B, 25(12) (1982), 7836
[3] A. F. Kohan, G. Ceder, D. Morgan, and C. B. Van de Walle, Phys. Rev. B, 61(22) (2000) 15 019
[4] E. A. Kane, J. Phys. Chem. Solids, 1 (1957), 249
[5] D. M. Szmyd, P. Porro, and A. Mejerfeld, J. Appl. Phys. 68(5) (1990), 2367
[6] D. M. Szmyd, M. C. Hanna, and A. Mejerfeld, J. Appl. Phys 68(5) (1990), 2376

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