n-band model for the band structure
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- The Anderson Hamiltonian
- The Green's function of the system
- The n-band Hamiltonian
- The group velocity
The Anderson Hamiltonian
In a more rigorous approach to the hybridization interpretation, the BAC model discussed earlier was derived from the Anderson many-impurity model [1] by Wu et al [2] using the coherent potential approximation (CPA) [3]. The Anderson Hamiltonian can be written
where
is the sum of the unperturbed Hamiltonians for the extended (host) states (labelled by k) and localised states (labelled by j). The En are the energy eigenvalues, whilst b+n and bn are creation and annihilation operators respectively. The term V describes the interaction of the extended and localised states:
where NC is the number of primitive cells in the crystal and the Vkj specify the hybridization strength.
The Green's function of the system
The Green's function of the system for a single localised state has been derived by Wu et al [2]. We have generalised this result for the many-impurity case [4], finding
where E is the energy of the perturbed system and Dj is interpreted as an energy broadening on the jth nitrogen site.
The n-band Hamiltonian
From the poles of the Green's function, we can construct an n-band model Hamiltonian after the fashion of the BAC model
Due to the broadenings Dj, the energy eigenvalues of the system now become complex. One effect of the broadenings is to soften the non-parabolicity of resulting dispersion relations. For transport calculations we have found that broadenings less than around 100 meV do not alter the results significantly, so for the sake of simplicity we shall take these to tend to zero.
The group velocity
We can obtain the dispersion relations from the characteristic equation of the Hamiltonian above. It should be borne in mind, however, that whilst we can obtain a mapping between wavevector k and energy E via the EM(k), the nitrogen states, being localised in real space are de-localised in k-space. Hence the resulting energy eigenvalues cannot be assigned a unique momentum. Indeed, if we try to derive the density of states from such dispersion relations, we rapidly run into problems as we shall see. On the other hand, since a given eigenstate will still have some degree of k localisation, we argue that the group velocity can still be defined.
It is useful to define a function of energy equivalent to the matrix semiconductor energy
The group velocity can then be written
where Ñk is the gradient in k-space and v0 is the group velocity in the matrix semiconductor.
Figure: Group velocity calculated using the
2-band BAC model.
Click on image to view larger graph
References
[1] P.W. Anderson, Phys. Rev., 124, 41 (1961) abstract
[2] J. Wu, W. Walukiewicz and E.E. Haller, Phys. Rev. B, 65, 233210 (2002) abstract
[3] Paul Soven, Phys. Rev., 156, 809 (1966) abstract
[4] M.P. Vaughan and B.K. Ridley, Phys. Rev. B, 75, 195205 (2007) abstract; M.P. Vaughan and B.K. Ridley, Phys. Stat. Sol. (c), 4, 686 abstract; M.P. Vaughan, PhD Thesis, University of Essex (2007)
M.P. Vaughan (2007)