- Email: [email protected]
Model for low- and intermediate-temperature properties of glasses
Physica B 169 (1991) North-Holland
316-321
Model for low- and intermediate-temperature glasses
properties
of
James P. Sethna”, Eric R. Grannanb and Mohit Randeria’ “Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501, hAT & T Bell Laboratories, Murray Hill, NJ, USA ‘Physics Department. State University of New York, Stony Brook, NY, USA The invited
USA
talk was given by J.P. Sethna.
Glasses look largely alike below SO K. Since their properties are universal, most theories have been grand in scope. applying to all materials equally. Below 1 K, this has been successful, but between 1 K and SOK many striking features have escaped convincing explanation. Inspired by the glassy crystal KBr : KCN, we have developed a model glass of interacting elastic dipoles. Numerical simulations and perturbation theory are used to calculate the specific heat, the thermal conductivity. and the dielectric loss. The universal features in KBr : KCN are explained using experimentally known constants. The thermal properties of vitreous silica can be fit to neutron scattering data. Thus. we are able to explain the universal properties in a particular system. We will attempt to connect our work to recent scaling approaches.
Glasses, below around 50 K, all have pretty much the same thermal properties. Crystals in this temperature range all share a T” law in their thermal conductivities and specific heats. In a similar way, glasses (from window glass to hard candy to polymer glass) all share a linear term in the specific heat c below 1 K, and an anomalous bump in c/T3 centered around 10K. The thermal conductivities in glasses are T* below 1 K and have a plateau around 10 K. In a rather unsubstantiated analogy with critical exponents at second-order phase transitions, these thermal properties are called “universal”. There have been two approaches to studying these universal properties. The first is the synthetic approach. Since the properties are independent of material, one concentrates on the big picture. looking for a global explanation. The universal properties of glasses below 1 K have succumbed to the synthetic approach. All glasses are thought to have ‘two-level tunneling systems’: atoms or groups of atoms which have two metastable configurations and which can 0921.4526191116fl7.SO
fi
1991 - E.lsevier
Science
Publishers
tunnel between them. Distributions of these tunneling systems have had substantial success in predicting new phenomena, and fair success in providing quantitative fits to experimental data. The synthetic approach has failed to explain the intermediate temperature universal properties so far. We [l, 2,3] have taken the reductionist approach - we have pounded one special case to dust. The difficulty with this approach is that one can become overwhelmed with details. The advantage is that one can find definitive answers for a particular system. The role of this paper is to look back, now that the dust has settled, to find clues about the big picture. The big picture is simple. Just as the properties below 1 K are due to tunneling between the lowest two energy levels of atoms in doublewell potentials, we find that the intermediatetemperature properties of glasses are determined by the small oscillations within the double wells. Our special case is the poisoned salt crystal, This crystal has the simple KBr,XKCN,_X.
B.V. (North-Holland)
J. P. Sethna et al. I Model for low- and intermediate-T properties of glasses
sodium-chloride structure, so the bromine and cyanide ions are randomly placed on an fee lattice. For concentrations not too close to x = 0 or 1, the cyanides freeze into a glassy configuration as the crystals are cooled, and at low temperatures this doped crystal shows all of the universal thermal properties of glasses. In this conference proceeding, we will concentrate on X = 0.5. Unlike glasses, we understood the microscopic structure of (KBr),,,(KCN),,.,. Even so, isolating the key ingredients for a model was a substantial experimental and theoretical project. The first stage of the project was a mean-field analysis [4] of experimental data [5] on the dielectric loss peak in pure KCN. There are three types of interactions affecting the motion of the cyanide ions: electric, elastic, and local field interactions. The local (crystal) field is known to be small (barriers of 5 K) for dilute CN impurities in KBr. The relative strengths of the electric and elastic interactions are known from experiments on pure KCN: the barriers due to elastic [6] interactions, for example, are 1855 K where the asymmetry energy due to electric interactions [7] is about 340 K. The glass transition is a continuation of the ferroelastic transition at higher concentrations; no electrical order is seen in the glassy phases or the neighboring ordered elastic phases. Thus keeping only the elastic forces is a good first approximation: our meanfield model, including only elastic forces, found that the dielectric loss could be explained as the cyanide ions flipping between two nearly degenerate energy minima, separated by roughly 180”. The second stage in isolating the physics was to study the tunneling centers [8,9, lo]. We found that cyanide ions hipping over via quantum tunneling provided, within a factor of two, the number of two-level systems needed to explain the low-temperature thermal conductivity and the time-dependent specific heat. Our ideas here were tested cleanly by Nicholls et al. [ll] using a different system. This group knew that argon doped with CO also had glassy lowtemperature properties, and wondered what would happen if they substituted NZ for CO. We predicted that, while many properties would be
317
unchanged, the quantum tunneling asymmetries would disappear and the properties below 1 K should be drastically different with NZ. They found no difference! Clearly, at least in this system, the tunneling centers must be more complicated, collective excitations. The third stage was the study of small oscillations of the cyanides [12], treating each as sitting in a fixed external elastic strain field. We found there that there was a peaked density of states for these oscillations, and that when reasonantly coupled to phonons they gave an excellent plateau in the thermal conductivity. Unfortunately, their contribution to the specific heat was much too large. Again, the collective nature of the small oscillation normal modes was a key missing ingredient. We were led by this preparatory work to develop a proper numerical model of cyanide ions interacting with phonons. Skepticism by our colleagues [13] that reasonant scattering off local modes could explain real glasses led us later to apply the model as well to vitreous silica. We have no microscopic interpretation of our cyanide rotors in silica; a more plausible model would be nice. On the other hand, it is illuminating that our model can be used to fit the behavior: perhaps there is a ‘defect’ in silica interacting via elastic forces too. Unfortunately for a simple description, the phonons would not fit into the simulation. The wavelength of a thermal phonon at 1 K, cubed, contains so many cyanide ions that our supercomputer could not handle them. Fortunately for the physics, we can separate the phonons from the defects, and study their effects into two pieces. First, the phonons mediate the elastic interactions between the cyanides. This interaction we can incorporate into our Hamiltonian directly. Secondly, the phonons scatter off the harmonic modes of the dipoles. Since these modes (we will see) are at rather low fequency, the coupling to the phonons is fairly weak (separation of length scales), so we treat it within perturbation theory [ 141. The strain field around a cyanide ion can be written in a multipole expansion. The lowest interesting term is an ‘elastic dipole’ strain field,
318
J.P.
which is described example,
by a 3 x 3 matrix
Q=
ir
-+
Sethna et al.
i Model
for low-
Q.
For
$1
is the elastic dipole tensor for a cyanide pointing in the i direction. Of course, the description looks just like that for an electric quadrupole, leading to some confusion in the literature (quadrupolar glass, . .). Ignore the words and keep the physical description firmly in mind: spherical holes cut out of a block of rubber, and American footballs (or rugby balls) of the same volume inserted into the openings. The interaction energy between elastic dipoles, for an infinite, isotropic medium, is well known. There was much water under the dam [2] before we developed a sensible method for finding the interaction between dipoles in a finite, possibly anisotropic, system with periodic boundary conditions. The k = 0 mode also was subtle, and both were important to get the coupling to long-wavelength phonons right. All this work went into finding the form of Jilki in the Hamiltonian. H = - 4
2
Q,,(x)J,,,,(x
and intermediatr-T
properties
of glasws
factor of two lower than the barriers found with the neighbors frozen. Using experimentally known parameters from the dilute and pure systems, we can compare the results of our simulawith tion the dielectric loss peaks in KBr,KCN,_,. As shown in fig. 1, the numerical peaks have the same shapes. means, widths, and concentration dependences as the experimental system. This dielectric loss in peak (KBr),, i(KCN),, i is analogous to the so-called p-relaxation peaks in other glasses. It has long been speculated that P-relaxation is associated with the double wells which also form the tunneling centers: our model of (KBr),, ,(KCN),, T confirms this association. We find the small oscillation harmonic modes by explicitly diagonalizing the quadratic form for the expansion of the energy about the ground state. Since our dipoles have l/r3 interactions. these ‘defect phonons’ are known [lS] to be at least weakly extended. The modes in our system appear highly disordered, and (nonetheless) reasonably localized in the low-frequency tail which dominates the intermediate-temperature properties. When we calculate the thermal conductivity. we do not include the direct transfer of energy through the cyanide sublattice. (Unlike a
- x’)Q&‘)
For our purposes, the only useful information about this interaction is that for two cyanides, the lowest energy configuration is a ‘Tee’, with one cyanide pointing at the second, and the second pointing at right angles to the first. A moment’s thought shows that three cyanides in a triangle cannot pairwise all form Tees: elastic dipoles are frustrated. We use a heat-bath Monte Carlo algorithm to cool our system into a good local minimum of the potential (a glassy ‘ground state’). We find the barriers to flipping a dipole 180” by minimizing the energy of the flipping dipole and its neighbors subject to the constraint that the flipping dipole is pointing somewhere in the circle halfway between its two lowest energy states. These ‘relaxed’ barrier heights are roughly a
0.002
9 E 0.001
0.000 0
1000
500
1500
V (K) Fig.
I.
Barrier
Gausians)
heights
and from
from
dielectric
the simultioa
loss
[ 191
(jagged curves).
(smooth
J. P. Sethna et al. I Model
for low-and
crystal, where harmonic extended phonons lead to an infinite thermal conductivity, energy in a disordered harmonic system diffuses. Our assumption that it diffuses slowly is at least sensible.) Figure 2 introduces our second physical system: vitreous silica. The solid curve shows the total density of states for our model, which is dominated in this frequency range by the harmonic modes of the dipoles. In this figure, two parameters have been varied to match the properties of vitreous silica: the concentration of dipoles and one coupling constant. No further parameters need to be set in order to calculate the specific heat and thermal conductivity. The next step is to use linear algebra and perturbation theory to rejoin the low-frequency phonons to the defects. This immediately produces some interesting, simple experimental consequences. Since the cyanides can absorb some the low-frequency of the external strain, phonons soften. This leads to a large feature in the frequency-dependent velocity of sound, and a large ultrasonic attenuation peak which may develop a mobility edge (leading to localized phonons). Unfortuantely, all of these effects ap-
Fig. 2. Density of low-frequency modes measured from neutron scattering of vitreous silica [20] (rough curve) and the density of states for our model (smooth curve) with the defect density and coupling chosen to describe silica glass.
intermediate-T properties of glasses
319
pear in the terahertz frequency domain, in which these experiments are not feasible at this time. To compare with existing experiments, we must suffer through some other real-world effects. We must introduce the ugly tunneling center distributions, which we do in the traditional way. We get both resonant and relaxational scattering from the tunneling centers. Cyanide and bromine have different masses, and so fluctuations in the cyanide concentration lead to Rayleigh scattering. Fluctuations in the bond strengths could introduce more scattering (which would help us) but not much more. Finally, we must pick a dispersion relation for the phonons. We could do a good job of this, but to do so would force us to make unnecessary explanations to a skeptical audience. We use a Debye spectrum, tuned so that the (softened) elastic constants agree with the experimental ones [16]. As shown in figs. 3-6, our model describes the intermediate-temperature properties both of and of vitreous silica. (The (KBrMKCN),,., agreement below 2 K is due to the tunneling centers. The plateau and the bumps in c/T3 around 10 K would dispappear without the harmonic modes.) All of the parameters for
0
5
10 T (K)
Fig. 3. Specific heat [21] for (KBr),, ,(KCN),, our theoretical prediction (solid curve).
15
20
i, compared
to
J.P. Sethnu et al. I Model
320
0
5
15
T1& Fig. 3. Specific heat [20] for vitreous theoretical prediction (solid curve).
10-5 --LLUtLvILU0.01
for low- and intermediate-T properties of glasses
20
1
10
100
T (K) silica. compared
to our
LlLLLLLLuL 10
0.1
0.1
100
T ;K) Fig. 5. Thermal conductivity [22] for (KBr),, ,(KCN),, compared to our theoretical prediction (solid curve).
i’
(KBr),,,, (KCN),,,, are known experimentally (within ranges); two parameters for silica glass were fit to the neutron scattering data. The silica glass works better for the specific heat largely because the bonds are stiffer: our Debye model for the phonons is less inadequate because the phonons matter less. The plateau in the thermal conductivity actually always happens in our
Fig. 6. Thermal conductivity [23] for vitreous silica, pared to our theoretical prediction (solid curve).
com-
model. A simple scaling analysis [l] shows that, independent of the coupling constant or the concentration, elastic dipoles interacting with phonons will always cause a plateau at some temperature. In a real system, local field effects ignored in our model can mess up the scaling argument. Finally, In our we return to the big picture. model for glasses, (i) The ‘p-relaxation relaxalow-frequency barrier tion peak is due to thermally activated crossing in the same double wells which form the tunneling centers. (ii) The plateau and the anomalous specific heat peak are due to disordered harmonic modes peaked in a rather low-frequency range. The harmonic modes are associated with the same degrees of freedom as are the double wells. (iii) The harmonic modes and the barrier crossings are collective, with many cyanides participating. (iv) Key ingredients of other models for the properties (fractal intermediate-temperature structure in the glass [17], retardation effects in the phonon interactions [IS]) are not necessary for a quantitative, microscopic description of ( KBr),,_i( KCN),, j or a quantitative. model description of silica glass.
J. P. Sethna et al. I Model for low
Acknowledgements We gratefully acknowledge NSF support through the Materials Science Center at Cornell University. We also acknowledge use of the Cornell National Supercomputer Facility, a resource of the Center for Theory and Simulation in Science and Engineering at Cornell University, which is funded in part by the NSF, the State of New York, and the IBM Corporation.
References
1’1 E.R.
Grannan. M. Randeria and J.P. Scthna. Phys. Rev. Lett. 60 (1088) 1402. PI E.R. Grannan, M. Randeria and J.P. Sethna. Phys. Rev. B 41 (1090) 7784. M. Randeria and J.P. Sethna, Phys. ]31 E.R. Grannan. Rev. B 41 (1990) 7799. Phys. I41 J.P. Sethna. S.R. Nagel and T.V. Ramakrishnan, Rev. Lett. 53 (1984) 2489. Phys. Rev. Lett. 50 (1983) 151 F. Liity and J. Ortiz-Lopez. 1289. Crystals. V.M. Tur]61 F. Luty, in: Defects in Insulating kevich and K.K. Swartz, eds. (Springer, Berlin. 1982) p. 60. ]‘I S. Nagel. private communication. W. Knaak. J.P. Sethna, KS. Chow, J.J. ]81 M. Meissner, DrYoreo and R.O. Pohl, Phys. Rev. B 32 (1985) 6091. 5 (1985) 191 J.P. Sethna and K.S. Chow, Phase Transitions 317. 1101J.P. Sethna, in: Dynamical Aspects of Structural Change in Liquids and Glasses. Ann. N.Y. Acad. of Sci. 484 (1985) 130. 1111C.I. Nicholls, L.N. Yadon. D.G. Haase and M.S. Conradi, Phys. Rev. Lett. 59 (1987) 1317.
and intermediate-T propertie
of glasses
321
[121M.
Randeria and J.P. Sethna. Phys. Rev. B 41 (1988) 12607. ]131 A.J. Leggett. private communication. theory breaks down ]141 Near the plateau, this perturbation (the Ioffe-Regel criterion is close to being satisfied. so multiple scattering leads to localization effects). A selfconsistent treatment [12] of the multiple scattering. however. leads to no change in the thermal conductivity because the phonon scattering in this frequency range is so high that no heat is transported in either approximation. Phys. Rev. Lett. 64 (1090) 547; ibid.. ll51 L.S. Levitov. preprint. [I61 Actually, we do spherical averages at various points of the calculation, so we make the softened elastic constants agree with a spherical average of the experimental elastic constants. This is just more dust to obscure the big picture. C. Laermans. R. Orbach and H.M. ]171 S. Alexander, Rosenberg. Phys. Rev. B 28 (1983) 4615; S. Alexander. 0. Entin-Wohlman, and R. Orbach, Phys. Rev. B 34 (1986) 2726. Comm. Condens. Mater. ]181 C.C. Yu and A.J. Leggett, Phys. 14 (108X) 231; ibid., in: Proc. of the 10th Int. Conf. on Low Temperature Physics, Part III, D.S. Bctts, ed., Physica B 169 (North-Holland. Amsterdam, 1991). 1191N.O. Birge. Y.H. Jeong. S.R. Nagel, S. Bhattacharya and S. Susman. Phys. Rev. B 30 (1984) 2306; L. Wu. R.M. Ernst, Y.H. Jeong, S.R. Nagel and S. Susman, Phys. Rev. B 37 (1988) 10444. PO1U. Buchenau. M. Prager, N. Nucker. A.J. Dianoux, N. Ahmad and W.A. Phillips, Phys. Rev. B 34 (1986) 5665; U. Buchenau, H.M. Zhou, N. Nucker. K.S. Gilroy and W.A. Phillips, Phys. Rev. Lett. 60 (1988) 1318. See also U. Buchenau, N. Nucker, and A.J. Dianoux. Phys. Rev. Lett. 53 (1984) 2316. WI S.K. Watson. D.G. Cahill and R.O. Pohl, preprint. WI J.J. De Yoreo. W. Knaak. M. Meissner and R.O. Pohl, Phys. Rev. B 34 (1986) 8828, and references therein. v31 From the Pohl group at Cornell.
316-321
Model for low- and intermediate-temperature glasses
properties
of
James P. Sethna”, Eric R. Grannanb and Mohit Randeria’ “Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501, hAT & T Bell Laboratories, Murray Hill, NJ, USA ‘Physics Department. State University of New York, Stony Brook, NY, USA The invited
USA
talk was given by J.P. Sethna.
Glasses look largely alike below SO K. Since their properties are universal, most theories have been grand in scope. applying to all materials equally. Below 1 K, this has been successful, but between 1 K and SOK many striking features have escaped convincing explanation. Inspired by the glassy crystal KBr : KCN, we have developed a model glass of interacting elastic dipoles. Numerical simulations and perturbation theory are used to calculate the specific heat, the thermal conductivity. and the dielectric loss. The universal features in KBr : KCN are explained using experimentally known constants. The thermal properties of vitreous silica can be fit to neutron scattering data. Thus. we are able to explain the universal properties in a particular system. We will attempt to connect our work to recent scaling approaches.
Glasses, below around 50 K, all have pretty much the same thermal properties. Crystals in this temperature range all share a T” law in their thermal conductivities and specific heats. In a similar way, glasses (from window glass to hard candy to polymer glass) all share a linear term in the specific heat c below 1 K, and an anomalous bump in c/T3 centered around 10K. The thermal conductivities in glasses are T* below 1 K and have a plateau around 10 K. In a rather unsubstantiated analogy with critical exponents at second-order phase transitions, these thermal properties are called “universal”. There have been two approaches to studying these universal properties. The first is the synthetic approach. Since the properties are independent of material, one concentrates on the big picture. looking for a global explanation. The universal properties of glasses below 1 K have succumbed to the synthetic approach. All glasses are thought to have ‘two-level tunneling systems’: atoms or groups of atoms which have two metastable configurations and which can 0921.4526191116fl7.SO
fi
1991 - E.lsevier
Science
Publishers
tunnel between them. Distributions of these tunneling systems have had substantial success in predicting new phenomena, and fair success in providing quantitative fits to experimental data. The synthetic approach has failed to explain the intermediate temperature universal properties so far. We [l, 2,3] have taken the reductionist approach - we have pounded one special case to dust. The difficulty with this approach is that one can become overwhelmed with details. The advantage is that one can find definitive answers for a particular system. The role of this paper is to look back, now that the dust has settled, to find clues about the big picture. The big picture is simple. Just as the properties below 1 K are due to tunneling between the lowest two energy levels of atoms in doublewell potentials, we find that the intermediatetemperature properties of glasses are determined by the small oscillations within the double wells. Our special case is the poisoned salt crystal, This crystal has the simple KBr,XKCN,_X.
B.V. (North-Holland)
J. P. Sethna et al. I Model for low- and intermediate-T properties of glasses
sodium-chloride structure, so the bromine and cyanide ions are randomly placed on an fee lattice. For concentrations not too close to x = 0 or 1, the cyanides freeze into a glassy configuration as the crystals are cooled, and at low temperatures this doped crystal shows all of the universal thermal properties of glasses. In this conference proceeding, we will concentrate on X = 0.5. Unlike glasses, we understood the microscopic structure of (KBr),,,(KCN),,.,. Even so, isolating the key ingredients for a model was a substantial experimental and theoretical project. The first stage of the project was a mean-field analysis [4] of experimental data [5] on the dielectric loss peak in pure KCN. There are three types of interactions affecting the motion of the cyanide ions: electric, elastic, and local field interactions. The local (crystal) field is known to be small (barriers of 5 K) for dilute CN impurities in KBr. The relative strengths of the electric and elastic interactions are known from experiments on pure KCN: the barriers due to elastic [6] interactions, for example, are 1855 K where the asymmetry energy due to electric interactions [7] is about 340 K. The glass transition is a continuation of the ferroelastic transition at higher concentrations; no electrical order is seen in the glassy phases or the neighboring ordered elastic phases. Thus keeping only the elastic forces is a good first approximation: our meanfield model, including only elastic forces, found that the dielectric loss could be explained as the cyanide ions flipping between two nearly degenerate energy minima, separated by roughly 180”. The second stage in isolating the physics was to study the tunneling centers [8,9, lo]. We found that cyanide ions hipping over via quantum tunneling provided, within a factor of two, the number of two-level systems needed to explain the low-temperature thermal conductivity and the time-dependent specific heat. Our ideas here were tested cleanly by Nicholls et al. [ll] using a different system. This group knew that argon doped with CO also had glassy lowtemperature properties, and wondered what would happen if they substituted NZ for CO. We predicted that, while many properties would be
317
unchanged, the quantum tunneling asymmetries would disappear and the properties below 1 K should be drastically different with NZ. They found no difference! Clearly, at least in this system, the tunneling centers must be more complicated, collective excitations. The third stage was the study of small oscillations of the cyanides [12], treating each as sitting in a fixed external elastic strain field. We found there that there was a peaked density of states for these oscillations, and that when reasonantly coupled to phonons they gave an excellent plateau in the thermal conductivity. Unfortunately, their contribution to the specific heat was much too large. Again, the collective nature of the small oscillation normal modes was a key missing ingredient. We were led by this preparatory work to develop a proper numerical model of cyanide ions interacting with phonons. Skepticism by our colleagues [13] that reasonant scattering off local modes could explain real glasses led us later to apply the model as well to vitreous silica. We have no microscopic interpretation of our cyanide rotors in silica; a more plausible model would be nice. On the other hand, it is illuminating that our model can be used to fit the behavior: perhaps there is a ‘defect’ in silica interacting via elastic forces too. Unfortunately for a simple description, the phonons would not fit into the simulation. The wavelength of a thermal phonon at 1 K, cubed, contains so many cyanide ions that our supercomputer could not handle them. Fortunately for the physics, we can separate the phonons from the defects, and study their effects into two pieces. First, the phonons mediate the elastic interactions between the cyanides. This interaction we can incorporate into our Hamiltonian directly. Secondly, the phonons scatter off the harmonic modes of the dipoles. Since these modes (we will see) are at rather low fequency, the coupling to the phonons is fairly weak (separation of length scales), so we treat it within perturbation theory [ 141. The strain field around a cyanide ion can be written in a multipole expansion. The lowest interesting term is an ‘elastic dipole’ strain field,
318
J.P.
which is described example,
by a 3 x 3 matrix
Q=
ir
-+
Sethna et al.
i Model
for low-
Q.
For
$1
is the elastic dipole tensor for a cyanide pointing in the i direction. Of course, the description looks just like that for an electric quadrupole, leading to some confusion in the literature (quadrupolar glass, . .). Ignore the words and keep the physical description firmly in mind: spherical holes cut out of a block of rubber, and American footballs (or rugby balls) of the same volume inserted into the openings. The interaction energy between elastic dipoles, for an infinite, isotropic medium, is well known. There was much water under the dam [2] before we developed a sensible method for finding the interaction between dipoles in a finite, possibly anisotropic, system with periodic boundary conditions. The k = 0 mode also was subtle, and both were important to get the coupling to long-wavelength phonons right. All this work went into finding the form of Jilki in the Hamiltonian. H = - 4
2
Q,,(x)J,,,,(x
and intermediatr-T
properties
of glasws
factor of two lower than the barriers found with the neighbors frozen. Using experimentally known parameters from the dilute and pure systems, we can compare the results of our simulawith tion the dielectric loss peaks in KBr,KCN,_,. As shown in fig. 1, the numerical peaks have the same shapes. means, widths, and concentration dependences as the experimental system. This dielectric loss in peak (KBr),, i(KCN),, i is analogous to the so-called p-relaxation peaks in other glasses. It has long been speculated that P-relaxation is associated with the double wells which also form the tunneling centers: our model of (KBr),, ,(KCN),, T confirms this association. We find the small oscillation harmonic modes by explicitly diagonalizing the quadratic form for the expansion of the energy about the ground state. Since our dipoles have l/r3 interactions. these ‘defect phonons’ are known [lS] to be at least weakly extended. The modes in our system appear highly disordered, and (nonetheless) reasonably localized in the low-frequency tail which dominates the intermediate-temperature properties. When we calculate the thermal conductivity. we do not include the direct transfer of energy through the cyanide sublattice. (Unlike a
- x’)Q&‘)
For our purposes, the only useful information about this interaction is that for two cyanides, the lowest energy configuration is a ‘Tee’, with one cyanide pointing at the second, and the second pointing at right angles to the first. A moment’s thought shows that three cyanides in a triangle cannot pairwise all form Tees: elastic dipoles are frustrated. We use a heat-bath Monte Carlo algorithm to cool our system into a good local minimum of the potential (a glassy ‘ground state’). We find the barriers to flipping a dipole 180” by minimizing the energy of the flipping dipole and its neighbors subject to the constraint that the flipping dipole is pointing somewhere in the circle halfway between its two lowest energy states. These ‘relaxed’ barrier heights are roughly a
0.002
9 E 0.001
0.000 0
1000
500
1500
V (K) Fig.
I.
Barrier
Gausians)
heights
and from
from
dielectric
the simultioa
loss
[ 191
(jagged curves).
(smooth
J. P. Sethna et al. I Model
for low-and
crystal, where harmonic extended phonons lead to an infinite thermal conductivity, energy in a disordered harmonic system diffuses. Our assumption that it diffuses slowly is at least sensible.) Figure 2 introduces our second physical system: vitreous silica. The solid curve shows the total density of states for our model, which is dominated in this frequency range by the harmonic modes of the dipoles. In this figure, two parameters have been varied to match the properties of vitreous silica: the concentration of dipoles and one coupling constant. No further parameters need to be set in order to calculate the specific heat and thermal conductivity. The next step is to use linear algebra and perturbation theory to rejoin the low-frequency phonons to the defects. This immediately produces some interesting, simple experimental consequences. Since the cyanides can absorb some the low-frequency of the external strain, phonons soften. This leads to a large feature in the frequency-dependent velocity of sound, and a large ultrasonic attenuation peak which may develop a mobility edge (leading to localized phonons). Unfortuantely, all of these effects ap-
Fig. 2. Density of low-frequency modes measured from neutron scattering of vitreous silica [20] (rough curve) and the density of states for our model (smooth curve) with the defect density and coupling chosen to describe silica glass.
intermediate-T properties of glasses
319
pear in the terahertz frequency domain, in which these experiments are not feasible at this time. To compare with existing experiments, we must suffer through some other real-world effects. We must introduce the ugly tunneling center distributions, which we do in the traditional way. We get both resonant and relaxational scattering from the tunneling centers. Cyanide and bromine have different masses, and so fluctuations in the cyanide concentration lead to Rayleigh scattering. Fluctuations in the bond strengths could introduce more scattering (which would help us) but not much more. Finally, we must pick a dispersion relation for the phonons. We could do a good job of this, but to do so would force us to make unnecessary explanations to a skeptical audience. We use a Debye spectrum, tuned so that the (softened) elastic constants agree with the experimental ones [16]. As shown in figs. 3-6, our model describes the intermediate-temperature properties both of and of vitreous silica. (The (KBrMKCN),,., agreement below 2 K is due to the tunneling centers. The plateau and the bumps in c/T3 around 10 K would dispappear without the harmonic modes.) All of the parameters for
0
5
10 T (K)
Fig. 3. Specific heat [21] for (KBr),, ,(KCN),, our theoretical prediction (solid curve).
15
20
i, compared
to
J.P. Sethnu et al. I Model
320
0
5
15
T1& Fig. 3. Specific heat [20] for vitreous theoretical prediction (solid curve).
10-5 --LLUtLvILU0.01
for low- and intermediate-T properties of glasses
20
1
10
100
T (K) silica. compared
to our
LlLLLLLLuL 10
0.1
0.1
100
T ;K) Fig. 5. Thermal conductivity [22] for (KBr),, ,(KCN),, compared to our theoretical prediction (solid curve).
i’
(KBr),,,, (KCN),,,, are known experimentally (within ranges); two parameters for silica glass were fit to the neutron scattering data. The silica glass works better for the specific heat largely because the bonds are stiffer: our Debye model for the phonons is less inadequate because the phonons matter less. The plateau in the thermal conductivity actually always happens in our
Fig. 6. Thermal conductivity [23] for vitreous silica, pared to our theoretical prediction (solid curve).
com-
model. A simple scaling analysis [l] shows that, independent of the coupling constant or the concentration, elastic dipoles interacting with phonons will always cause a plateau at some temperature. In a real system, local field effects ignored in our model can mess up the scaling argument. Finally, In our we return to the big picture. model for glasses, (i) The ‘p-relaxation relaxalow-frequency barrier tion peak is due to thermally activated crossing in the same double wells which form the tunneling centers. (ii) The plateau and the anomalous specific heat peak are due to disordered harmonic modes peaked in a rather low-frequency range. The harmonic modes are associated with the same degrees of freedom as are the double wells. (iii) The harmonic modes and the barrier crossings are collective, with many cyanides participating. (iv) Key ingredients of other models for the properties (fractal intermediate-temperature structure in the glass [17], retardation effects in the phonon interactions [IS]) are not necessary for a quantitative, microscopic description of ( KBr),,_i( KCN),, j or a quantitative. model description of silica glass.
J. P. Sethna et al. I Model for low
Acknowledgements We gratefully acknowledge NSF support through the Materials Science Center at Cornell University. We also acknowledge use of the Cornell National Supercomputer Facility, a resource of the Center for Theory and Simulation in Science and Engineering at Cornell University, which is funded in part by the NSF, the State of New York, and the IBM Corporation.
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