matrices de transición entre estados relativos al desajuste

673 672 671 678 9 973    393  !"#

Figure 9.3: How a Diatomic Dispersion Becomes a Monatomic Dispersion When the Two Different Atoms Become the Same. (black) Dispersion relation of vibrations of the one dimensional diatomic chain in the extended zone scheme with κ2 not too different from κ1. (blue) Dispersion relation

when κ2 = κ1. In this case, the two atoms become exactly the same, and we have a monatomic

chain with lattice spacing a/2. This single band dispersion precisely matches that calculated in chapter 8 above, only with the lattice constant redefined to a/2.

other, to think about the dispersion as being a small perturbation to a situation where all atoms are identical. When the atoms are made slightly different, a small gap opens up at the zone boundary, but the rest of the dispersion continues to look mostly as if it is the dispersion of the monatomic chain. This is illustrated in Fig. 9.3.

9.3

Summary of Vibrations of the One Dimensional Di-

atomic Chain

A number of key concepts are introduced in this chapter as well • A unit cell is the repeated motif that comprises a crystal.

• The basis is the description of the unit cell with respect to a reference lattice. • The lattice constant is the size of the unit cell (in 1d).

86 CHAPTER 9. VIBRATIONS OF A ONE DIMENSIONAL DIATOMIC CHAIN • One of these modes is an acoustic mode, meaning that it has linear dispersion at small k,

whereas the remaining M − 1 are optical meaning they have finite frequency at k = 0. • For the acoustic mode, all atoms in the unit cell move in-phase with each other, whereas for

optical modes, they move out of phase with each other.

• Except for the acoustic mode, all other excitation branches have zero group velocity for k = nπ/a for any n.

• If all of the dispersion curves are plotted within the first Brillouin zone |k| 6 π/a we call this the reduced zone scheme. If we “unfold” the curves such that there is only one excitation plotted per k, but we use more than one Brillouin zone, we call this the extended zone scheme. • If the two atoms in the unit cell become identical, the new unit cell is half the size of the old

unit cell. It is convenient to describe this limit in the extended zone scheme.

References

• Ashcroft and Mermin, chapter 22 (but not the 3d part) • Ibach and Luth, section 4.3

• Kittel, chapter 4

• Hook and Hall, sections 2.3.2, 2.4, 2.5 • Burns, section 12.3

Chapter 10

Tight Binding Chain (Interlude

and Preview)

In the previous two chapters we have considered the properties of vibrational waves (phonons) in a one dimensional system. At this point, we are going to make a bit of an excursion to consider electrons in solids again. The point of this excursion, besides being a preview of much of the physics that will re-occur later on, is to make the point that all waves in periodic environments (in crystals) are similar. In the previous two chapters we considered vibrational waves. In this chapter we will consider electron waves (Remember that in quantum mechanics particles are just as well considered to be waves!)

10.1

Tight Binding Model in One Dimension

We described the molecular orbital, or tight binding, picture for molecules previously in section 5.3.2. We also met the equivalent picture, or LCAO (linear combination of atomic orbitals) model of bonding for homework. What we will do here is consider a chain of such molecular orbitals to represent orbitals in a macroscopic (one dimensional) solid.

a

|1i |2i |3i |4i |5i |6i

Fig. 10.1.1

In this picture, there is a single orbital on atom n which we call |ni. For convenience we will assume that the system has periodic boundary conditions (i.e, there are N sites, and site N

88 CHAPTER 10. TIGHT BINDING CHAIN (INTERLUDE AND PREVIEW) is the same as site 0). Further we will assume that all of the orbitals are orthogonal to each other.

hn|mi = δn,m (10.1)

Let us now take a general trial wavefunction of the form |Ψi =X

n

φn|ni

As we showed for homework, the effective Schr¨odinger equation for this type of tight-binding

model can be written as _X

m

Hnmφm= Eφn (10.2)

where Hnm is the matrix element of the Hamiltonian

Hnm= hn|H|mi

As mentioned previously when we studied the molecular orbital model, this Schr¨odinger equation is actually a variational approximation. For example, instead of finding the exact ground state, it finds the best possible ground state made up of the orbitals that we have put in the model. One can make the variational approach increasingly better by expanding the Hilbert space and putting more orbitals into the model. For example, instead of having only one orbital |ni at a given site, one could consider many |n, αi where α runs from 1 to some number p. As p is increased the approach becomes increasingly more accurate and eventually is essentially exact. This method of using tight-binding like orbitals to increasingly well approximate the exact Schr¨odinger equation is known as LCAO (linear combination of atomic orbitals). However, one complication (which we treat only in one of the additional homework assignments) is that when we add many more orbitals we typically have to give up our nice orthogonality assumption, i.e., hn, α|m, βi = δnmδαβ

no longer holds. This makes the effective Schr¨odinger equation a bit more complicated, but not fundamentally different. (See comments in section 5.3.2 above).

At any rate, in the current chapter we will work with only one orbital per site and we assume the orthogonality Eq. 10.1.

We write the Hamiltonian as

H = K +X

j

Vj

where K = p2_{/(2m) is the kinetic energy and V}

j is the Coulomb interaction of the electron with

the nucleus at site j,

Vj = V (r − rj)

where rj is the position of the jth nucleus.

With these definitions we have

H|mi = (K + Vm)|mi +

X

j6=m

Vj|mi

Now, we should recognize that K + Vm is the Hamiltonian which we would have if there were

only a single nucleus (the mth _{nucleus) and no other nuclei in the system. Thus, if we take the}

tight-binding orbitals |mi to be the atomic orbitals, then we have (K + Vm)|mi = atomic|mi

10.2. SOLUTION OF THE TIGHT BINDING CHAIN 89