ANHARMONIC EFFECTS
2. General aspects
It is convenient to introduce a density of states AF^CD) which de-scribes the changes in the atomic vibrations when a defect is created in a lattice. Consider a solid with N lattice sites, of which A^f are associated with the creation of a defect. For instance, if the defect is a vacancy, N^f/(N + Ndef) ^ N^/N is the vacancy concentration. In the case of a surface, Ndef may be loosely identified with the number of surface atoms, but see the discussion below. We now write
^totM = Fbulk(co) + (Ndef/N)AFdQf(co), (9.1)
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where Ftot(co) is the total density of states of the actual specimen and FbUik(&>) is its density of states in the absence of the defect under con-sideration, in both cases as an average per atom. The normalisation relations are f Fiot(co) dco = 3 = / Fbu\k(co) dco. Then
/ AFdef(a>)d^ = 0, (9.2)
Jo
i.e. AFdef(&>) varies in sign.
From AFdef(&>) we can calculate, for example, the change in the vibrational heat capacity;
AChar,def(r) = / Chai(hco/kBT)AFdef(co) dco, (9.3) Jo
where comax is the maximum frequency in the presence of lattice de-fects. It may be larger than &>max of the defect-free solid. Char is the heat capacity of a single harmonic oscillator. The high-temperature limit ( £Br > > / ^m a x) o f ( 9 . 3 ) i s
/^max
AChar,def(r) = *B / A Fd e fM dco = 0. (9.4) Jo
This relation simply reflects the fact that the vibrational heat capacity at high temperatures has the classical value of 3kB per atom, for a perfect crystalline atomic arrangement as well as for a structure with defects. Note that there may also be an additional heat capacity associ-ated with the formation of the defect, such as the two-level description of vacancies, eq. (11.7).
Except for point defects, the number Afdef in eq. (9.1) is usually not exactly defined, but there is a strict operational definition of, e.g. the surface contribution to the thermodynamic functions as (half) the differ-ence between the properties of two well separated blocks of a material, and the same blocks joined with perfect atomic matching. Even though TVdef may be poorly defined, it gives a measure of the size, or amount, of defect regions, e.g. the area of an interface, and can be used to estimate how vibrations at defects affect thermodynamic quantities.
Example: surface states in TiN. Rieder and Drexel (1975) found, from neutron scattering experiments, that AFdef(&>) due to surface effects in
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Fig. 9.1. The defect-related vibrational density of states F^Qf(co) for surface states in TiN, expressed as the change in the bulk density of states caused by the surface states (shaded regions), compared with F{co) in the bulk. Arbitrary units and arbitrary scale
factors for F{co) and F^ef(co). After Rieder and Drexel (1975).
TiN has three positive and two negative regions when plotted versus co. Their result is schematically shown in fig. 9.1. A similar result for AFdef(&>) has been obtained by Loram et al. (1993) in an analysis of heat capacity data for the high temperature superconductor YE^CUBC^+X
for varying x.
3. Surfaces
3.1. Elastic waves in a semi-infinite elastic continuum
In an ordinary Debye model, the lattice vibrations are described by elas-tic waves propagating in an infinite medium. In a semi-infinite medium, bounded by a free surface, the classical wave equation d2up/dt2 — c2pS72up — 0 has solutions up which are the usual bulk waves, but also solutions with an amplitude localised to the surface region. The latter modes, known as Rayleigh waves (Rayleigh 1900), propagate with their wave vector q in the surface plane. Their frequency is
Q) = CT\q\%, (9.5)
where Cj is the transverse sound velocity in the bulk. The dimensionless parameter § depends on the ratio C J / C L and lies between 0.874 and 0.955. For a given q, the frequency of the Rayleigh wave is thus less than that of the transverse elastic bulk wave. (In a real solid the sound velocity is anisotropic (Gazis et al. I960).) A brief general discussion of elastic surface waves is found in Landau and Lifshitz (1959). Maradudin (1981) has reviewed the entire field of surface waves.
3.2. Thermal properties of an elastic-continuum surface
We shall calculate the heat capacity in the low temperature limit, i.e.
when only elastic waves are excited. The allowed (qx,qy) for wave propagation parallel to the surface give a density of states which varies linearly with co. The number of (Rayleigh) states is proportional to the area A of the surface. Their density of states is
FR(co) = Akxco. (9.6)
We note that the frequencies of both the bulk and the surface states are linear in the wave number q (for small q). The fact that FR(co) is linear in co, while the bulk F(co) varies as co2, reflects the difference in possible q-vectors sampled in a two- and three-dimensional system. The low frequency part of the density of states, for the states not localised to the surface, is of the bulk form plus a correction Ak^oy1. Hence, approximately,
Ftot(co) = (1 - Ak3)Fhulk(a)) + AkAco. (9.7) This is the "Debye model" in the presence of a surface. For
mathe-matical details, see Wallis (1975), Stratton (1953, 1962), Dupuis et al.
(1960), Maradudin et al. (1963) and Maradudin and Wallis (1966). From eq. (9.7) we obtain the low temperature heat capacity
CUT) = (1 - Ak3)Chulk(T) + Ak5T\ (9.8) where Cbuik ~ T3. The parameters k\,..., ks depend on the elastic
parameters and the mass density of the material. The r2-term in eq.
(9.8) can be observed only at low temperatures and if the surface-to-volume ratio is large enough, i.e. for very small particles. But then the finite size of the particle is important (§3.4), which gives corrections
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to our model of vibrations in an elastic continuum instead of the true discrete atomic lattice. Therefore, eq. (9.8) can only be used for very qualitative estimations.
3.3. Thin slabs
Consider a thin slab formed by N layers of atoms. This is the three-dimensional generalisation of a finite monatomic linear chain. Such a model system has been extensively studied by Allen et al. (1971). The solutions to the equation of motion are of the form
u = u(Rj{z)) exp[/(q • Rj(xy) - cot)]. (9.9)
Rj(z) denotes the z-component (i.e. perpendicular to the surfaces) of the position vector Ry of the 7th atom. R/o^) is a position vector along the slab and q = (qx, qy) is a two-dimensional wave vector. Almost all of the modes of the form eq. (9.9) have amplitudes u(/?/(z)) which are appreciable throughout the width of the slab. However, a mode corresponding to a Rayleigh wave has a displacement u(/?y-(Z)) which decreases rapidly as R7 moves inward from either surface. Allen et al.
(1971) also discovered new surface modes which do not exist in the limit of small q and thus have no elastic-wave counterpart. Similar results have been obtained in a study of the TiN(OOl) surface (Benedek et al.
1984).
3.4. Small particles
The bulk material has a phonon density of states which is quasicon-tinuous and varies as co2 for small co. In a very small sample, on the other hand, the eigenfrequencies form a discrete spectrum. In particular, there is a lowest eigenfrequency com[n which can be estimated crudely as follows. In the bulk, co = Cq, where C is a sound velocity. In a small particle, of diameter d, it is unphysical to consider wavelengths larger than d, i.e. q > 2n/d. With &>D = Cq® ~ 2nC/do, where do is the diameter of an atom, we get co^m ~ (do/d)co^. The discrete nature of the low frequency part of the vibrational spectrum means that we must write out explicitly the first terms in the partition function when T < (do/d)6v, instead of applying the usual integral approxi-mation. This has been recognised by Burton (1970), Chen et al. (1971) and Baltes and Hilf (1973). Nishiguchi and Sakuma (1981) made an
accurate study of the vibrations of a small elastic sphere. The excess heat capacity (above the bulk value) has been measured for ionic solids (NaCl; Barkman et al. (1965)) and metals (Pb, In; Novotny and Meincke (1973)). The theory referred to above is in reasonable agreement with the experiments. Dobrzynski and Leman (1969) developed a frequency-moment representation of the surface phonons, and calculated their contribution to the heat capacity.
4. Point imperfections