Easier way to do the calculation:
Back to the linear chain model we used for phonons (see chapter 6.1). Now we are talking about
mechanical motion, not about electrons. There are masses connected by springs which we handle with
Newton’s law:
md^2 us
dt^2
=C(us+1− 2 us+us− 1 ).If we write Newton’s law in terms of a matrix equation it looks like shown in eqn. (55). In the diagonal
you always get the 2 Cand one step away theC. The masses further away are not coupled with each
other, so we get zero. For periodic boundary conditions we get theCin the corners.
−w^2 m
A 1
A 2
A 3
A 4
=
− 2 C C 0 C
C − 2 C C 0
0 C − 2 C C
C 0 C − 2 C
A 1
A 2
A 3
A 4
(55)The matrix is exactly the same as before in the tight binding model. The easy way to solve the
tight bindig model is to write down the Hamiltonian and then write down plane waves times the
wave functions. This solves the Schrödinger equation. We know that plane waves times the atomic
wavefunctions is the answer in the tight binding model. Taking that and putting it into the Schrödinger
equation we get the dispersion relationship.
In the tight binding method we want to find the wavefunctions for electrons in a crystal, and we
know the wavefunctions of the electrons on isolated atoms. When the atoms are pretty far apart, the
atomic wavefunctions are good solutions for the electron wavefunctions of the whole system. We try
to write a basis for the crystal wavefunctions in terms of the atomic wavefunctions and we get the big
hamiltonian matrix. Then you have to diagonalize the matrix and the best solution you can make for
electrons moving in a periodic crystal is
Ψk=1
√
N
∑l,m,nei(lka^1 +mka^2 +nka^3 )Ψ(r−la 1 +ma 2 +na 3 ). (56)It is the atomic wavefunction, one on every side (one on every atom) times a phase factor (the same we
used for phonons) that looks like a plane wave. Now we calculate the dispersion curve by calculating
the energy of this particular solution:
Ek=〈Ψk|H|Ψk〉
〈Ψk|Ψk〉. (57)
If you guess a wavefunction and the Hamiltonian you can calculate the corresponding energy. This
does not mean that it is the lowest possible wavefunction. The calculation goes over a lot of terms
for the calculation of the matrix elements. But the wavefunction falls exponentially with the distance,
so the interaction is just important between nearest neighbors and next nearest neighbors. Therefore
three atoms apart the interaction is zero (so the product of these two wavefunctions is zero). Our
calculations now are just for the nearest neighbors. In this part of the calculations (the mean diagonal