Electrons in a Perturbed Periodic Potential
Semiconductor devices are almost never homogeneous, and consequently it is necessary to consider the effect of perturbations to the periodic potential. First consider solutions of the Schrödinger equation for the unperturbed problem, with a Hamiltonian, H0:
For simplicity, the band number n is dropped from the wave function in the above equation.
The perturbed problem has a set of solutions in the form:
(3-25)
where H=H0+H1.
The perturbed wave functions do not necessarily have a constant wave vector, so the subscript k is dropped and a different quantum number, m replaces it. In many practical applications H1 varies slowly on the length scale of the lattice and this assumption is subsequently made.
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:
(3-26)
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(ik⋅R)/√N.
Substituting Equation 3-26 into Equation 3-25 gives:
(3-27)
This equation is premultiplied by W*(r-R′) and integrated over the crystal:
(3-28)
Consider first the right-hand side of Equation 3-28:
where both the orthogonality of the Wannier functions is employed.
The term in H1 on the left-hand side of Equation 3-28 gives:
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:
(3-29)
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:
(3-30)
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:
(3-31)
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:
where the final step introduces an operator based on the Taylor series expansion of the exponential function. Equation 3-29 can be written as:
(3-32)
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):
(3-33)
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.