Patterson, Bailey - Solid State Physics Introduction to theory

54 2 Lattice Vibrations and Thermal Properties

Then make the change n′ → n′ + 1 in the dummy variable of summation. Because of periodic boundary conditions, no change is necessary in the limits of summation. We obtain

Comparing (2.36) and (2.38) we see that if pun = 0, then pun+1 = 0. If pf = 0 had only one solution, then it follows that

where eiqa is some arbitrary constant K, that is, q = ln(K/ia). Equation (2.39) is an expression of a very important theorem by Bloch that we will have occasion to discuss in great detail later. The fact that we get all solutions by this assumption follows from the fact that if pf = 0 has N solutions, then N linearly independent linear combinations of solutions can always be constructed so that each satisfies an equation of the form (2.39) [75].

By applying (2.39) n times starting with n = 0 it is readily seen that

If we wish to keep un finite as n → ± ∞, then it is evident that q must be real. Further, if there are N atoms, it is clear by periodic boundary conditions that un = u0, so that

where m is an integer.

Over a restricted range, each different value of m labels a different normal mode solution. We will show later that the modes corresponding to m and m + N are in fact the same mode. Therefore, all physically interesting modes are obtained by restricting m to be any N consecutive integers. A common such range is (supposing N to be even)

−(N / 2) +1 ≤ m ≤ N / 2 .

This range of q is called the first Brillouin zone.

Substituting (2.40) into (2.36) shows that (2.40) is indeed a solution, provided that ωq satisfies

56 2 Lattice Vibrations and Thermal Properties

Restricting q = 2π/λ to the first Brillouin zone is equivalent to selecting the range of q to have as small a |q| or as large a wavelength as possible. Letting q become negative just means that the direction of propagation of the wave is reversed. In Fig. 2.2, (a) is a first Brillouin zone description of the wave, whereas

(b) is not.

It is worthwhile to get an explicit solution to this problem in the case where only nearest-neighbor forces are involved. This means that

This is the dispersion relation for our problem. The largest value that ω can have is

By (2.48) it is obvious that increasing q by 2π/a leaves the value of ω unchanged. By (2.35), (2.40), (2.41), and (2.48), the displacement of the nth atom in the mth normal mode is given by

This is also invariant to increasing q = 2πm/Na by 2π/a.

A plot of the dispersion relation (ω versus q) as given by (2.48) looks something like Fig. 2.3. In Fig. 2.3, we imagine N → ∞ so that the curve is defined by an almost continuous set of points.

For the two-atom case, the theory of small oscillations tells us that the normal coordinates (X1, X2) are found from the transformation

Fig. 2.2. Different wavelengths describe the same vibration in a discrete lattice. (The dots represent atoms. Their displacement is indicated by the distance of the dots from the horizontal axis.) (a) q = π/2a, (b) q = 5π/2a

ω

ωc

Fig. 2.3. Frequency versus wave vector for a large one-dimensional crystal

If we label the various components of the eigenvectors (Ei) by adding a subscript, we find that

58 2 Lattice Vibrations and Thermal Properties

The equations of motion of each Xi are harmonic oscillator equations of motion. The normal coordinate transformation reduced the two-atom problem to the problem of two decoupled harmonic oscillators.

We also want to investigate if the normal coordinate transformation reduces the N-atom problem to a set of N decoupled harmonic oscillators. The normal coordinates each vibrate with a time factor eiωt and so they must describe some sort of harmonic oscillators. However, it is useful for later purposes to demonstrate this explicitly.

By analogy with the two-atom case, we expect that the normal coordinates in the N-atom case are given by

where 1/N 1/2 is a normalizing factor. This transformation can be inverted as follows:

In (2.54), the sum over m′ runs over any continuous range in m′ equivalent to one Brillouin zone. For convenience, this range can be chosen from 0 to N − 1. Then

60 2 Lattice Vibrations and Thermal Properties

Equation (2.65) is practically the correct form. What is needed is an equation similar to (2.65) but with the X real. It is possible to find such an expression by making the following transformation: Define u and v so that

Since Xm* = X−m, it is seen that um = u−m and vm = −v−m. The second condition implies that v0 = 0, and also because Xm = Xm+N that vN/2 = 0 (we are assuming that N is even). Therefore the number of independent u and v is 1+2(N/2−1)+1=N, as it should be.

If the definitions z0 = u0

z−1 = 2v1,…, z−(N / 2)+1 = 2v(N / 2)−1

are made, then the z are real, there are N of them, and the Hamiltonian may be written, by (2.65), (2.66), and (2.67),

Equation (2.68) is explicitly the Hamiltonian for N uncoupled harmonic oscillators. This is what was to be proved. The allowed quantum-mechanical energies are then

By relabeling, the sum in (2.69) could just as well go from 0 to N − 1. The Nm are integers.

2.2.3 Specific Heat of Linear Lattice (B)

We will use the canonical ensemble to derive the specific heat of the onedimensional crystal.9 A good reference for the use of the canonical ensemble is Huang [11]. In a canonical ensemble calculation, we first calculate the partition function. The partition function and the Helmholtz free energy are related, and by use of this relation we can calculate all thermodynamic properties once the partition function is known.

If the allowed quantum-mechanical states of the system are labeled by EM, then the partition function Z is given by

Z = ∑M exp(− EM kT ) .

If there are N atoms in the linear lattice, and if we are interested only in the harmonic approximation, then

9The discussion of 1D (and 2D) lattices is perhaps mainly of interest because it sets up a formalism that is useful in 3D. One can show that the mean square displacement of atoms in 1D (and 2D) diverges in the phonon approximation. Such lattices are apparently inherently unstable. Fortunately, the mean energy does not diverge, and so the calculation of it in 1D (and 2D) perhaps makes some sense. However, in view of the divergence, things are not as simple as implied in the text. Also see a related comment on the Mermin–Wagner theorem in Chap. 7 (Sect. 7.2.5 under Two Dimensional Structures).

62 2 Lattice Vibrations and Thermal Properties

The result (2.71) is a consequence of a general property. Whenever we have a set of independent systems, the partition function can be represented as a product of partition functions (one for each independent system). In our case, the independent systems are the independent harmonic oscillators that describe the normal modes of the lattice vibrations.

Since 1/(1 − α) = ∑∞0 αn if |α| < 1, we can write (2.71) as

The relation between the Helmholtz free energy F and the partition function Z is given by

Equation (2.77) without the zero-point energy can be arrived at by much more intuitive reasoning. In this formulation, the zero-point energy (ћ/2 ∑Nn=1 ωn) does not contribute anything to the specific heat anyway, so let us neglect it. Call each energy excitation of frequency ωn and energy ћωn a phonon. Assume that the phonons are bosons, which can be created and destroyed. We shall suppose that the chemical potential is zero so that the number of phonons is not conserved. In this situation, the mean number of phonons of energy ћωn (when the system has a temperature T) is given by 1/[exp(ћωn/kT − 1)]. Except for the zero-point energy, (2.77) now follows directly. Since (2.77) follows so easily, we might wonder if the use of the canonical ensemble is really worthwhile in this problem. In the first place, we need an argument for why phonons act like bosons of zero chemical potential. In the second place, if we had included higher-order terms (than the second-order terms) in the potential, then the phonons would interact and hence have an interaction energy. The canonical ensemble provides a straightforward method of including this interaction energy (for practical cases, approximations would be necessary). The simpler method does not.

The zero-point energy has zero temperature derivative, and so need not be considered for the specific heat. The indicated sum in (2.77) is easily done if N → ∞. Then the modes become infinitesimally close together, and the sum can be replaced by an integral. We can then write

where n(ω)dω is the number of modes (with q > 0) between ω and ω + dω. The factor 2 arises from the fact that for every (q) mode there is a (−q) mode of the same frequency.

n(ω) is called the density of states and it can be evaluated from the appropriate dispersion relation, which is ωn = ωc |sin(πn/N)| for the nearest-neighbor approximation. To obtain the density of states, we differentiate the dispersion relation

dωn =πωc cos(πn / N ) d(n / N ), = ωc2 −ωn2π d(n / N ) .

Therefore

Nd(n / N ) = (N /π)(ωc2 −ωn2 )−1/ 2 dωn ≡ n(ωn )dωn ,

or

Combining (2.78), (2.79), and the definition of specific heat at constant volume, we have

Equation (2.81) is just a one-dimensional expression of the law of Dulong and Petit, which is also the classical limit.