Lattice waves

The linear model can be extended to include more distant neighbours and to three dimensions. Let us consider an elastic lattice wave with wave vector q. The collective vibrational modes of the lattice are illustrated in Figure 8.6. The formation of small local deformations (strain) in the direction of the incoming wave gives rise to stresses in the same direction (upper part of Figure 8.6) but also perpendicular (lower part of Figure 8.6) to the incoming wave because of the elasticity of the material. The cohesive forces between the atoms then transport the deformation of the lattice to the... 

The atoms of a crystal vibrate around their equilibrium position at finite temperatures. There are lattice waves propagating with certain wavelengths and frequencies through the crystal [7], The characteristic wave vector q can be reduced to the first Brillouin zone of the reciprocal lattice, 0 < q <7t/a, when a is the lattice constant. 

In the case of lattice waves and spin waves, the procedure is different but the principle is the same. The periodic potential is represented with the help of a Fourier series... 

Similarly, during their effort to understand the thermal energy of solids, Einstein and Debye quantized the lattice waves and the resulting quantum was named phonon. Consequently, it is possible to consider the lattice waves as a gas of noninteracting quasiparticles named phonons, which carries energy, E=U co, and momentum, p = Uk. That is, each normal mode of oscillation, which is a one-dimensional harmonic oscillator, can be considered as a one-phonon state. 

Our approach to the study of the Hamiltonian (3) is based on the direct consideration of the processes of electron hopping between neighboring lattice sites with the restriction to the states without doubly occupied lattice sites [23], Let us consider a rectangular lattice strip with 8 sites and 5 electrons and enumerate all the variables of the lattice wave function in succession along the lattice rows beginning from the upper one. 

If nf - p/r is a fraction other than 2, the system is said to be weakly commensurate. There now appears in the LG free-energy functional a weak commensurability term [51] of the form id [ty(x))r + [iji ( )]r = d i >(x) r cos r0, coming from an r-vertex bubble in which r2kF is equal to a reciprocal lattice wave number (2ttpla). Here d 5rCDW( TrvF) 1( o) r+2 where... 

This completes the specification of the pseudopotential as a perturbation in a perfect crystal. We have obtained all of the matrix elements between the plane-wave states, which arc the electronic states of zero order in the pseudopotcntial. We have found that they vanish unless the difference in wave number between the two coupled states is a lattice wave number, and in that case they are given by the pseudopotential form factor for that wave number difference by Eq. (16-7), assuming that there is only one ion per primitive cell, as in the face-centered and body-centered cubic structures. We discuss only cases with more than one ion per primitive cell when we apply pseudopotential theory to semiconductors in Chapter 18. Tlicn the matrix element will be given by a structure factor, Eq. (16-17),... 

FIGURE 16-3 The grid of wave numbers allowed by periodic boundary conditions, as in Fig. 15-2. Also shown are several lattice wave numbers (neavier dots). The relative spacings and geometry correspond to a simple cubic lattice of 6859 ions (see Problem 16-1). A sample wave number k is shown. 

The lattice wave numbers defined in Eq. (16-16) arc the reciprocal lattice vectors familiar in diffraction theory. Only if the change in wave number resulting from the diffraction is equal to a lattice wave number can a wave, whether it be an X-ray or an electron, be diffracted otherwise the wavelets scattered by the different ions interfere with each other and reduce the diffracted intensity to zero. Only for diffraction by q equal to a lattice wave number do the scattered waves add in phase. Thus a wave having wave number k can only be diffracted to final states of wave number k that can be written as k = k -I- q, where q is a lattice wave number. Furthermore, the diffracted wave will have the same frequency as the incident wave if it is an X-ray, or the same energy if it is an electron, from which it follows that k = k. Combining the two conditions, k -I- qp = gives the Bragg condition for diffraction. 

This may be stated as the condition that an electron can only be diffracted if its wave number lies on a plane bisecting some lattice wave number. (We have used the fact that if q is a lattice wave number, then so is —q.) Such planes, called Bragg planes, are illustrated in Fig. 16-4 for the simple cubic lattice. 

The lattice wave numbers and Bragg planes for the system. shown in Fig. 16-3. We imagine, though, a much larger crystal, so the mesh of wave numbers allowed by periodic boundary conditions becomes very fine and is not shown. 

This docs not change the lattice wave numbers. The wave number k shown docs not. satisfy the Bragg condition. 

Since states on opposite faces of the zone are entirely equivalent, one may repeat the bands and the Fermi surface by constructing identical zones around every lattice wave number, as shown in Fig. 16-7. In this representation, called the periodic-zone scheme, the second-band Fermi surface is seen to consist (again, for the simple cubic structure) of three closed lens-shaped segments. The advantage of this representation is that all discontinuous jumps in wavenumber have been eliminated the wave numbers connected by diffractions have been plotted... 

Consider next the evaluation for q differing from a lattice wave number. Then, a sum over all r-, gives zero and the sum over all but r gives... 

The first term is exactly the structure factor for tlie undistorted crystal, giving a value of 1 for the lattice wave numbers and 0 for other wave numbers. Wc may immediately see that the structure factor is corrected at tlie lattice wave number. Notice that the second term vanishes for lattice wave numbers because c " is the same at every site, though (5r, oscillates from one site to another, averaging to zero. The third term, on the other liand, does not vanish c is again the same at every site, but (q f)r,) can be seen to average to 2(q u)(q u ) by squaring Hq. (17-13) and summing over i. Let us then indicate the lattice wave numbers of the perfect crystal by q wc have found that, to second order in u. 

These sums also can be evaluated immediately. They are of the same form as those for the perfect crystal, but with q replaced by q + k. Thus the first will be if q — k is a lattice wave number, and zero otherwise the second will be if q -t- k is a lattice wave number, and zero otherwise. The lattice distortion has given nonzero structure factors at satellites to each of the lattice wave number, as indicated in Fig. 17-3 they lie at wave numbers qo + k and have structure factors -i(qo + k) - u. 

A lattice vibration of wave number k reduces the structure factor at the lattice wave numbers (solid dots), but... 

We immediately have the band-structure energy from the structure factors that were obtained. It is convenient to sum the lattice wave numbers and the satellites together in the form... 

Without tlie periodic array, there arc no longer lattice wave numbers but a distributed structure factor S(q), The phase will vary in a complicated way, but an average measurable S (q)S(q) exists and is spherically symmetric. This is just what is needed for a calculation of the resistivity of the liquid metals. The first such calculation using pseudopotentials (Harrison, 1963b) followed an earlier and conceptually similar calculation by Ziman (1961). It involved the direct substitution of S (q)S(q), obtained by X-ray diffraction experiments on the liquid, into... 

The observed elastic shear constant C44 of aluminum is 2,8 x 10" crg/cm, whereas the electrostatic contribution is 14.8 x 10" erg/cm (Harrison, 1966a, p, 179 the value there was based upon an effective charge 7.9 percent larger tlian the 3,0 appropriate here). The band-structure energy is an estimate of the difference, —12.0 x 10" crg/cm. Even if we sum all terms, the result is approximate because of the neglect ofterms of higher order than two and the use of an approximate pscudopoteiitial. We obtain the effect of the nearest lattice wave numbers here. 

Aluminum has a face-centered cubic structure with cube edge a = 4.04 A, The wave number lattice, therefore, is body-centered cubic, and contains shortest lattice wave numbers of the form [11 l](2rt/a), with all eight combinations of I. A distortion e has been applied to the wave number lattice and modifies these eight lattice wave numbers as indicated in Fig. 17-9. 

The Bragg planes are specified in Fig. 18-3 in terms of the lattice wave number giving rise to them. For example, there are lattice wave numbers [220]27c/fl, with a the cube edge of F ig. 3-1. The plane bisecting this vector is called the (220) plane. The (111) planes, which were shown in Fig. 16-9, are omitted here for clarity. The diagonal lines in part (a) are actually edges made by the intersections of planes of the (202) or (022) type with various combinations of signs of components. 

This view runs into difficulties that have only recently been completely resolved. The principal one is that the pseudopotential form factor happens to be very small for this particular diffraction. In Fig. 18-4 is sketched the pseudopotenlial form factor for silicon obtained from the Solid Stale Table the form factor that gives the [220] diffraction is indicated. Because it lies so close to the crossing, it is small and the diffraction is not expected to be strong. Heine and Jones (1969) noted, however, that a second-order diffraction can take an electron across the Jones Zone this could be a virtual diffraction by a lattice wave number of [1 ll]27t/fl followed by a virtual diffraction by [I lT]27c/a. (Virtual diffraction is an expression used to describe terms in perturbation theory it can be helpful but is not essential to the analysis here.) This second-order diffraction would involve the large matrix elements associated with the [11 l]27t/a lattice wave number indicated in Fig. 18-4, and Heine and Jones correctly indicated that these are the dominant matrix elements. 

If we wish to study a stale at the face of the Jones Zone, we must consider not only the plane wave with wave number at that face, say k,, o = [110]27c/a, and that at the opposite face, /states differs from the others by a lattice wave number, so that if the free-clectron bands were plotted in the reduced-zone scheme, they would all be at the same point, the point [001]27c/fl, which is at the center of one of the square faces of the Brillouin Zone, for example, the point X in... 

In obtaining the final form, we used the fact that q is a lattice wave number for the gallium lattice, to take the sum of c" over the NJ2 gallium atoms as N /2, and... 

The sum over i will be identically zero unless q is a lattice wave number, in which case the sum over each cell is identical ... 

The lattice wave numbers for the perovskite structure, shown in Fig. 19-2,are given by... 

Now, imagine a shear distortion such that an atom at [.v, y, z] is displaced in the j>-direction by i x. (Symmetry appears to rule out internal displacements for this case.) The lattice wave numbers (Eqs. 16-11 and 16-16) may be recomputed for the distorted structure and are... 

To evaluate the sum over the first term in parentheses, wc write e e 9 and sum over Vj, holding tlj fixed. The sum is identically zero unless k + Q — k is zero or is a lattice wave number. In this representation, there is no distinction between normal and umklapp scattering. If k = k 3- Q, the sum is and the remaining sum has become a sum over (L. Similarly, the sum over the second term in parentheses leads to a factor Af., e" and Eq. (19-53) becomes... 

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