The approach taken in Ref. 9 (and originally derived by Wannier) is inspired by
Equation 3-24. The unperturbed wave function can be written in the form:
where W(r-R) is the Wannier function. Solutions of the form:
are sought. Here Ψm(
R) is a function that is employed to weight the Wannier functions in an expansion of the perturbed wave function. In the limit
H1→0,
Ψm(
R)
→exp(i
k⋅R)/
√N.
This equation is premultiplied by W*(
r-
R′) and integrated over the crystal:
where the fact that the Wannier function is localized around R or
R′ and the assumption that
H1 is a constant on this length scale is employed.
Finally the H0 term is considered. Using the definition of the Wannier function given in
Equation 3-23, this term is written as:
where the orthogonality of the unperturbed wave functions is used and the dummy variable, R′′=
R′-
R is defined. Next we note that ground state energy,
E0(
k), is periodic in k-space (see for example
Figure 3-2), and can be written as a Fourier series in the form:
where R is the set of real space lattice vectors (real space forms a reciprocal space for k-space). The coefficients of the series
ER are given by:
where the integral over the Brillouin zone (with volume Ω) is replaced with a summation over the individual k-states in the zone for consistency with the notation employed in this section. Note that there are
N states in the Brillouin zone, as shown in
The Density of States in a Periodic Potential. Recognizing that the final term in
Equation 3-29 takes the same form as the definition of
EK* given in
Equation 3-31, it is possible to write
Equation 3-29 in the form:
Finally, expanding Ψm(
R′ − R′′) in a Taylor series about the point
R′′ gives:
Next an operator E0(-
i∇) is introduced that results from replacing every instance of
k in the function
E0(
k) with -
i∇. Comparing
Equation 3-32 with equation
Equation 3-30 shows that:
Assembling the terms derived above into Equation 3-28 (with
R′→r) gives the following equation for
Ψm(
r):
Equation 3-33 is an equation for
Ψm(
r), similar in form to the Schrödinger equation, with the perturbing potential
H1 appearing as the potential energy and the operator
E0(-
i∇) replacing the kinetic energy operator. This equation can be used to derive the semiclassical model, which in turn determines the transport properties of electrons.