Density of states

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• Problem with the density of states
• Green's function approach
• 2D density of states

Problem with the density of states

A significant problem with the BAC model or its generalisations (such as the n-band model) is that attempts to derive the density of states from the dispersion relations quickly run into problems. To see this, consider the general form for the density of states

where d is the dimensionality and the integration is over a constant energy surface S. Using the chain rule with g(E) = E_M as defined earlier, we have

where N₀(E_M) is the density of states in the matrix semiconductor. If we now integrate this from the bottom of a band to a nitrogen energy E_i, we have

where we have expanded N₀(E_M) as a power series in g(E). But from the earlier definition of g(E) we see that at a nitrogen energy level, this expression becomes infinite. This implies that there are an infinite number of states with energy less than E_i, which is clearly physically invalid.

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Green's function approach

This problem can be addressed by deriving the density of states from the imaginary part of the Green's function. For finite energy broadening, we find the 3D density of states to be [1]

where m₀* is the effective mass of the matrix semiconductor,

Examples of this density of states are shown in the figure below.

Figure: 3D density of states (3-band model).
Click on image to view larger graph

If we take the limit as the broadenings tend to zero, then the imaginary part of the Green's function becomes the Lorentzian representation of the Dirac delta function [1]

This gives us a general prescription for finding the density of states if that of the matrix semiconductor is known – we just have N(E) = N₀(g (E)).

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2D density of states

In 2D, for finite broadening, we have for an infinite square well

where the sum is over sub-bands and E_n is the unperturbed sub-band edge (i.e. in the matrix semiconductor). Taking the limit as the broadenings tend to zero, we have

where q is the Heaviside step function. Note that for a single sub-band, the magnitude of the density of states is not changed, only the band-edges (see figure below).

Figure: 2D density of states in a single sub-band (3-band model).
Click on image to view larger graph

Performing the sum over sub-bands, we find that the singularities at the nitrogen energies in the 3D density of states re-emerge (see below). This is due to higher band states being redistributed beneath the nitrogen energy.

Figure: 2D density of states summed over all sub-bands (2-band model, no broadening, x = 0.01).
Click on image to view larger graph

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