ANHARMONIC EFFECTS
3. The quasi-harmonic approximation and phonon Gruneisen parameters
3.1. General aspects and cubic lattice symmetry
The quantities <&^(/, j) in eq. (8.1) have the dimension of force con-stants, measuring the interatomic forces when the potential O is given.
The derivatives with respect to the coordinates (a, /?) are evaluated for atoms labelled (/, j) that have the position vectors R; and R7. If the lat-tice is strained, the atoms take new positions where the derivatives of O should be evaluated. Therefore, the force constants depend on the state of strain. For instance, they will vary with an external pressure. In the quasiharmonic approximation we allow for such a strain dependence of the phonon frequencies co(q,s), but still consider them to be harmonic.
This may be a very good approximation to describe physical phenom-ena, although from a strict mathematical point of view, it is inconsistent for the following reason. If the vibrations are truly harmonic, the third-and higher-order derivatives of O are zero. But variations in the second derivative of O with small variations in the positions R are proportional to these high-order derivatives.
The Gruneisen parameter y(q, s) is now introduced as a measure of how the phonon frequency &>(q, s) is altered under a small change in the geometry of the crystallographic unit cell. The most commonly encountered Gruneisen parameter is that referring to an isotropic change in the volume V;
^ > = - ^ m = - ( ™ ) - < s - 4 >
This expression will be the cornerstone in the simplest theory of thermal expansion, (Chapter 14, §4). It is a special case of a more generally de-fined Gruneisen parameter y (q, s; £;) that expresses how &>(q, s) varies
The quasi-harmonic approximation andphonon Griineisen parameters 139
under a small change in £;, where 8t is a strain parameter specifying the geometry of the unit cell. Here, / = 1, 2, 3 refer to the lengths at of the cell axes and i = 4, 5, 6 refer to the angles between a,- (Chapter 3, §2);
y(q,s;si) = . (8.5) co(q,s) V 3fi| Je>
The derivative is taken with all s[ ^ £; kept constant. Usually y is calculated at the reference state of zero external tension or shear. For clarity we will sometimes add a label V to the most common Griineisen parameter, as defined in eq. (8.4), i.e. y (q, s; V). We remark that other authors often write the Griineisen parameter as yc, with a subscript G. In this book the notation yc is used for the "thermodynamic" Griineisen parameter (eq. (13.23)), which is obtained directly from macroscopic thermodynamic quantities. Griineisen parameters can be defined also for electronic and magnetic contributions to the total free energy of a solid. For brevity the label "phonon" on the Griineisen parameters in this chapter is suppressed.
If, for / = 1, 2, 3, one of the at is changed by a relative amount Adi/di, the corresponding strain is st = Aat/at. Then, from eq. (8.5),
/ Aco(q, s)\ ( Aat\
— ; = - y (q, s; 8i)A£i = -y (q, s; e() , (8.6)
V co(q,s) J V ai J
and we can write
(
31no)(q, s)\a 1 4
• (8.7) 3 In at )
In a lattice with cubic symmetry, ^ 1 = ^ 2 = ^ 3 = ^ and V ~ a3, and therefore 3 In at/3 In V = 1/3 under uniform expansion or contraction of the unit cell. Hence,
t TM / 3 1 n ^ ( q , s ) \
f^\ d\nat )\dlnVj Y VH
where we have introduced yet another Griineisen parameter, y(q, s\ a).
Usually, the Gruneisen parameters are positive and lie in the range 1.5 ± 1. Negative Gruneisen parameters sometimes occur for low-lying frequencies (long-wavelength transverse modes) in open structures like those of Ge, Si and some alkali halides.
3.2. Gruneisen parameters in hexagonal lattice symmetry
The conventional notation for the lattice parameters in hexagonal lat-tices is a\ = #2 = a and a^ = c. Then, for a uniform expansion or contraction of the unit cell (i.e. while keeping c/a fixed)
'dhuo(q,s\ai)\ fdlna{ y(q,s; V) = - 2 '
31nai / \ 3 1 n V /31n&>(q, s; c)\ / 31nc \
= ( l / 3 ) [ 2 y ( q ^ ; a1) + y(q,5;c)]. (8.9) Here, we have to be careful in the notation. In eq. (8.9) we have written
a\ to denote that we only change the strain in one direction, perpendic-ular to the c-axis. It is now natural to introduce Gruneisen parameters Yw and y±, such that y\\ = Y± in the special case of cubic symmetry. We define
Y\\
/ 3 1 n o ; \
(q,s) = Y(q,s;c) = - l ^ — \ , (8.10)
In the last derivative, it is the cell dimension a that is varied (i.e. a\ and
#2 both vary by the same amount), and this gives rise to the prefactor 1/2. With the definitions above, and those to follow in this chapter, the various Gruneisen parameters will be equal in the special case that all y(q, s) of the individual phonon modes are equal and depend only on volume changes, irrespective of shape deformations of the lattice unit cell.
Example: Gruneisen parameter for varying c/a in hep lattices. Some materials with hep structures, e.g. Cd and Zn, have a c/a ratio which deviates strongly from the "ideal" value 1.63, but the atomic volume
The quasi-harmonic approximation and phonon Gruneisen parameters 141
3
2
1
- 3 - 2 - 1 0 1 2 3 4 n
Fig. 8.1. The Gruneisen parameter y(n; V) as a function of n, for C3F2 and SrF2 (Bailey and Yates 1967) and for Zn (Barron and Munn 1967). Because the Zn lattice is
hexagonal, there are two Gruneisen parameters, y±_ and y\\.
is not abnormal (cf. the example in Chapter 19, §2). It is therefore of interest to consider how the phonon frequencies vary with c/a at fixed volume V. One has
/ 9 1 n c y ( q , j ) \ V 91n(c/a) /v
_ /aino>(q,5)\ / ainfl \ /31ncw(q,j)\ / 31nc \
" V a i n c J . V a i n ( c / a ) Jv
- (2/3)[Kn(q,5)-yi.(q^)]. (8.12)
Here, we have used the fact that ca2 ~ V = constant, which yields c/a = (constant)/a3 and, hence, [31n(c/a)/31na]y = —3. Similarly, [d\n(c/a)/dlnc]v = 3/2. In zinc, y± = 2.50 and y\\ = 1-28 when all modes are given equal weights (n = 1, fig. 8.1). Then y(c/a) — —0.81.
In the special case of isotropy, y\\ = y i , w e would get y(q,s; c/a) = 0 as expected, since it was assumed that c/a varied under constant volume.
3.3. Gruneisen parameters for moment frequencies and Debye temperatures
Gruneisen parameters can be defined also for the moment frequencies co(n) and the corresponding Debye temperatures #D(^)- For instance,
/d\n9D(n)\ /d\nco(n)\
T 1 \ i i i i r
Zn Yi
0 Zn YH
J I I I I I I L
In the strict Debye model, coD = CSOund,D^D = &B#D/^, where CSOunci,D is the sound velocity and q& = (6JT2N/V)1/3. Then
y(flb;v)
=- h^v)=- [-3^-) + 3- (814)
Only if all y(q, s; V) are equal, is y(n\ V) independent of n. Often y(q,s; V) of different modes (q, s) differ by as much as a factor of two. The corresponding variation of y(n\ V) is shown in fig. 8.1. The value of y(2; V) has a simple relation to the trace of the dynamical matrix (see Appendix C).
It is not unusual to approximate y(n) by the thermodynamic Gruneisen parameter yo (see eq. (13.23)). This may be too crude an ap-proximation. For instance, in the very anisotropic graphite, y (n) varies very strongly with n (Bailey and Yates 1970).
If y (n; V) depends only weakly on the volume, we can integrate eq.
(8.13) and obtain
eD(n; vb) V v
In hexagonal structures, the corresponding relation is 0p(n;a,c) _ /a^v^n) /CQ\n{n)
9v(n; ao,co) V a / V c /
Example: Slater's form of the Gruneisen parameter y(—3; V). Slater (1940) derived an expression for the GrUneisen parameter, essentially as follows. Expand the volume change V — Vo, due to an external pressure /?, in powers of p and keep only the first two terms; V — Vb = Vo(a\p + a2P2). The average sound velocity CSOUnd,D to be used in #D(—3) is given by 3/(CSOUnd,D)3 = 1/CL + 2 / C | , eq. (6.15). If we neglect the volume dependence of the Poisson ratio and use eqs. (5.9) and (5.10) for CL and Cj expressed in the elastic constants, we get CS0Und,D ~ (KV)l/1. K is the bulk modulus; K~l = -(l/V)(dV/dp)T = -(V/V0)[ai + 2a2(V
— Vo)/(a\ Vo)]. Then, as in eq. (8.14), one obtains Slater's expression y(-3;V) = - ( l / 2 ) ( d l n ^ / d l n V ) - l / 6
= a2/a\ - 2/3 = (l/2)(d£/d/?) - 1/6. (8.17)
The quasi-harmonic approximation andphonon Griineisen parameters 143
The coefficients a\ and ai, which yield din K/din V, can be measured (Gschneidner 1964). At the time of Slater's original derivation, it was unknown to what extent y (q, s) varies with the phonon mode (q, s) and no distinction was made between y(n\ V) for different n. Here we have stated explicitly that Slater's expression is an estimation of y(—3; V).
Thus, it is not equal to the Griineisen parameter y(0; V) ( ^ / G ) that is approximately obtained from the thermal expansion coefficient at mod-erate and high temperatures (Chapter 14, §4.2). The quantity (dK/dp) in the last part of eq. (8.17) is discussed in Chapter 13, §2.
Example: internal pressure from zero-point vibrations. Suppose that we have calculated the atomic volume Q& from a model that considers only the static lattice. We now estimate how much fia is changed due to the zero-point (T = 0) lattice vibrations. The pressure p is related to the energy by p = — (dU/dV)s. If we add to U the zero-point energy (3/2)hco(l) per atom, (eq. (7.17)), the pressure is changed by
pz = -(V/Qa)(3/k/2)[3a>(l)/3V]
= [ 3 / M l M l ) ] / [ 2 na] . (8.18)
V/ Qa is the number of atoms in the solid. An added internal pressure pz gives rise to a relative change in the atomic volume;
A B . = ^ = 3 » o K l )y ( 1 ) t ( 8 1 9 )
&2a Kj z.Kj^l^
where a mass dependence (isotope effect) enters through co(l). At high temperatures, with p — —(dF/dV)r and F as given in eq. (D.7), we find that hco(l)y(l) in eq. (8.19) should be replaced by 2kBTy (0). Then p does not depend on the atomic mass. Figure 8.2 shows how the lattice parameter depends on the isotopic composition and the temperature, for lithium hydrides with different lithium and hydrogen isotopes; after Grimvall (1996). See also Johansson and Rosengren (1975) for a discus-sion of 6Li and 7Li, and Ramdas (1995) and Haller (1995) for reviews of various isotope effects in semiconductors.
Ch. 8. Phonons in real crystals: anharmonic effects
0^^7LiH/7LiT J
— — + ^ 6LiH/6LiD I
* ^
7LiH/7LiD 1
0 200 400 600 T(K)
Fig. 8.2. Effect of isotope mass on the lattice parameter of lithium hydrides.