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Submitted on 1 Jan 1981
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ENTROPY OF DISORDERED SOLIDS AT LOW
TEMPERATURES
R. Zeyher, R. Dandoloff
To cite this version:
JOUREAL DE PHYSIQUE
CoZZoque C6, suppZ6ment au n012, Tome 42, cle'cembre 1981 page c6-40
ENTROPY O F DISORDERED S O L I D S A T LOW TEMPERATURES
R. Zeyher and R. Dandoloff
Max-PZanck-Insti tut far FestkBrperforschng, 7000 Stuttgart 80, F. R. G.
Abstract.- It is shown that the complete set of low-lying collec- tive excita599ns in a disordered solid yields in general a lead-
ing term =T in the entropy and the specific heat at low tem-
peratures.
Omitting electronic degrees of freedom the Hamiltonian of a solid has the form
The index A counts the N atoms and M(X) is the mass of the atom A
.
;(A) denotes the displacement of the atom X from its rest position
;t, (of (A), p(A) are the corresporiding momenta and V is a general poten- 3 tial. Eq. ( 1 ) assumes implicitly that the t's are finite and there- fore that no mass transport is possible. As a result Eq. ( 1 ) cannot describe for instance an ionic conductor at high temperatures. At low
temperatures all the atoms of a solid however can be assumed to be
localized ( in an "ideal" ionic conductor for instance the atoms are localized because of Anderson localization caused by unavoidable small fluctuations in the periodic potential and the large mass). The Hamiltonian ( 1 ) is therefore quite general for the discussion of low temperature properties. It includes in particular cases where no har-
monic approximation exists or where the density of lattice sites and
the density of particles must be considered as independent variables. The Hamiltonian of Eq. ( 1 ) has several branches of collective, low-lying excitations caused by conservation laws and broken symme-
-+ +
tries: a) The Fourier transform of the momentum density p(k) is con- served in the limit g+o; b) The introduction of rest positions 2'0) (A) breaks the translational symmetry which is restored again by the displacement variables
2($)
in the limit $+o; c) The energy densi-+'
ty e(k) is conserved for z+o; d) Assuming an one-component solid the
+ +
total mass density n(k) is conserved for k+o.
As a result of a) and b) there are three branches of propagating sound waves with linear dispersion and imaginary part ak2. At low tem-
peratures they yield a contribution a~~ to the entropy which is the
usual Debye law. As a result of c) solids exhibit a collective diffu- sion mode describing the dissipation of entropy. This mode has a weight -a2 (T)
,
where a is the thermal expansion coefficient. If the third law of thermodynamics is valid, +$$a= o and the corresponding contri- bution to S is unimportant at low temperatures. Ref. 1 assumes,$&g
a(T) C o. It follows then from general thermodynamic arguments thatS a T. However we can show that under rather general assumptions this
case will not occur in real solids ~ 8 ~ w i l l discard it in the following.
d) yields ( for an one-component solid) an additional diffusion peak
which describes structural rearrangments of the atoms and has been discussed in Refs. 2 and 3. Assuming that the corresponding suscepti- bilities and the transport coefficient is nonzero at low temperatures we calculate in the following the contribution of this diffusion mode to S at low temperatures.
The leading term in the entropy at low temperatures can be calcula- 4
ted from the relation (kg=?l=l):
Gag(X,X',w+i~) is the retarded Green's function associated with the
displacements ua (A) and ug
( 2 )
.
The inverse G" is to be taken with respect to the labels (ha, X ' 6 ),
In is the logarithm and Im the imagi- nary part. It is also understood that the spectral functions of G are to be taken in the limit T+o so that the T-dependence is due to the first argument of A in Eq.(2).JOURNAL DE PHYSIQUE
a1 anda2 are static susceptibiliites:
f12
and describe thebare and the renormalized sound velocity, respectively. D is the trans-
port coefficient associated with the extra mode.
Transforming Eq. (3) into k-space and inserting Eq. (4) the k-inte- gration can be carried out in the complex plane. The leading contribu- tion AS to S is for a,<<"
-
2'
where 5 is Riemann's zeta function and V is the volume of the solid. Because of CV = T b S / b T we find that the leading term in CV should also be e ~ ~ / ~ . The next term in S is and contains in particular the usual Debye contribution.
Experimentally extra contributions to S of the form a ~ " with
1 ,$ a 5 3/2 have been found in glasses, amorphous semiconductors and
metals, polymers, superionic conductors, and f erroelectrica? A charac-
teristic feature of these solids is that some of the atoms can make "large" (i.e. of the order of the bond length) displacements. This also is just a condition that the above extra mode should be impor- tant213. Nevertheless the application of our theory to the above solids is difficult because a1 and D are presently unknown and have so far also not been calculated within theoretical models.
References:
1. P.V. Giaquinta, N.H. March, M. Parinello, and M.P. Tosi, Phys.
Rev. Letters =,41 (1977)
2. P.C. Martin, 0. Parodi, and P.S. Pershan, Phys. Rev. ~,2401(1972)
3. C. Cohen, P. Fleming, and J.H. Gibbs, Phys. Rev. BG,866(1976) 4. P. Fulde and H. Wagner, Phys. Rev. Lett. ~,1280(1971)
