- Email: [email protected]
Interacting defects in glasses
ELSEVIER
Physica B 219&220 (1996) 311 313
Interacting defects in glasses H.M. Carruzzo a, E.R. Grannan b, C.C. Y u a, • a Department of Physics and Astronomy, University of California, Irvine, CA 92717-4575, USA b Lehman Brothers, One Broadgate, London, UK
Abstract
We find that recent low-temperature nonequilibrium dielectric experiments indicate that glasses have strongly interacting defects. While many of the features found in the experiments can be explained by the standard model of noninteracting twolevel systems, we find that the frequency dependence cannot. Using a Monte Carlo simulation of a nearest-neighbor Ising spin glass, we show that interactions between defects can qualitatively explain the experiments because they lead to the formation of clusters and a hole in the distribution of local fields.
The low-temperature properties of glasses, such as the specific heat and the thermal conductivity, have been traditionally explained by the model of two-level systems (TLS) which are usually thought of as tunneling centers [1]. The standard model of TLS assumes for the most part that the TLS do not interact with one another. Recently, however, several groups have suggested that interactions between defects may be crucial to our understanding of the universal properties seen in amorphous materials [1]. Until recently, both interacting and noninteractingmodels have been able to explain the low-temperature experiments, so there has been no way of clearly determining whether or not interactions between defects are truly important. However, recent nonequilibrium dielectric experiments [2] have shown that interactions do indeed play an important role in the low-temperature physics of glasses. Salvino et al. put thin films (1-3 I.tm) of glass such as SiO2, SiOx (x ~- 2.1 ), and polymers in a capacitor which they cooled down to low temperatures (20-1000 mK). Then they applied a large DC electric field ( ~ 106-10 7 V/m). Using an AC capacitance bridge, they watched the capacitance jump up and then decay logarithmically with time t after the DC field had been applied. They found that the magnitude of the slope of the logarithmic decay (]dC/d In t], where C is the capacitance) decreased with increasing temperature and increasing frequency. * Corresponding author. 0921-4526/96/$15.00 (~ 1996 Elsevier Science B.V. All rights reserved SSDI0921-4526(95)OO730-X
To understand this behavior, we can start by asking the following question. To what extent can these experiments be explained by the standard model of noninteracting TLS? To answer this, we use the analogy between an electric dipole in an electric field and a spin in a magnetic field to write the Bloch equations. Solving these equations gives us the time dependence of the polarization, the dielectric susceptibility, and the capacitance [ 1]. Averaging over the TLS parameters gives a logarithmic time decay of the capacitance. In fact, independent TLS theory and experiment agree qualitatively on the time, voltage, and temperature dependence of the decay. However, the standard model predicts that the slope of the logarithmic decay will be independent of frequency because the frequency is much smaller than the effective TLS energy splitting. This lack of frequency dependence contradicts experiment. This discrepancy leads us to consider interactions between defects mediated by the strain field. There are two important consequences of interactions. First interactions lead to clusters of strongly interacting defects which make the capacitance frequency dependent. At low frequencies large slow clusters as well as small fast clusters contribute to the response, while at high frequencies only small fast clusters have time to respond. This is consistent with the experimental observation of the jump in the capacitance being bigger and the slope ]dC/dlnt] being larger at low frequencies than at high frequencies.
312
H.M. Carruzzo et al./ Physica B 219&220 (1996) 311-313
Interactions also prevent the distribution of local fields at defect sites from being flat. In particular, stability arguments have been used to show that there must be a hole in the distribution of local fields [1]. To be more specific, think of a spin glass with long-range interactions [1]. Suppose you find the ground state spin configuration and suppose that the distribution of local magnetic fields, P(h), is finite at h - 0. This means that those spins with no local field can flip without changing their energy. But if they do so, other spins have their field altered and so some of them will flip. This in turn causes others to flip and so on. This avalanche means that the supposed ground state is not stable. In order to have a stable ground state, the distribution of local fields must go to zero as the local field h-~0. The Stanford experiments [2] are the first to test these long-standing ideas about a hole in the local field distribution of glasses because the strong DC field enables them to probe P(h) away from h = 0. Consider the situation before the application of the DC field. Defects whose local fields are small can flip easily in response to the AC field; these are the main contributors to the capacitance. When the DC field is applied, a new set of defects find themselves in zero local field. The applied DC field effectively shifts the local field distribution along the local field axis. Thus the susceptibility, and hence capacitance, suddenly jumps up. Once the DC field is switched on, a new hole develops and the capacitance decreases with time. To test these ideas about interactions, we have performed Monte Carlo simulations on a three-dimensional nearestneighbor Ising spin glass. The spins, representing the interacting electric dipoles, are on the sites of a cubic lattice of size 103 . In order to simulate the conditions of the experiment, we run for N/2 Monte Carlo steps per spin with no applied field, then switch on a DC field (poEDc ~ 0.4 and po - 1) and run for another N/2 Monte Carlo steps, where N is the total number of Monte Carlo steps. We used N = 12 000 and 25 000 Monte Carlo steps. Up to 8000 samples were used to average over the disorder. This system is known to have a transition temperature of about 0.9 K [1]. We have run at temperatures ranging from 0.1 to 0.9 K. We need not achieve complete thermal equilibrium at low temperatures since we are interested in nonequilibrium phenomena. The results of our simulations give qualitative agreement with experiment. The application of a DC field causes the capacitance to jump up and then decay logarithmically with time. In addition, the simulations give qualitatively the correct frequency and temperature dependence for the logarithmic slope of the decay as shown in Figs. 1 and 2. To summarize, the frequency dependence found in recent nonequilibrium dielectric experiments cannot be explained with the independent TLS model. However, we obtain
2400
=01..00sa.,es
I I T=0.2, 4000 samples ~T=0.1, 4000 samples
2200
2000
_= .:~ 1800 "0 X
%
i
1600
1400
1200
....
Independent Two Level
SsTm:
~ ' ~ .
0.01
.
.
.
.
.
.
.
.
E*-------~ .
.
.
.
.
.
.
.
.
.
.
010
frequency
Fig. 1. Magnitude of the slope ]dz~/d In t] versus frequency, where Z~ is the real part of the dielectric susceptibility and is proportional to the capacitance. The frequency is the inverse of the period in Monte Carlo steps (MCS) per spin. The slope has been multiplied by a factor of 10 6. The top curve (+) was done at T = 0.2 K with 4000 samples and N = 12 000 MCS per spin. The middle curve ( × ) was done at T = 0.1 with 4000 samples and N = 12000MCS per spin. The bottom curve ( O ) was done at T = 0.1 with 8000 samples and N = 25000 MCS per spin. The same bond configuration was used for the curves averaged over 4000 samples, but a different bond configuration was used for the 8000 sample curve. Comparing the bottom two curves gives some idea of the fluctuations resulting from taking the second derivative d2M/dEAc d In t, where M is the polarization.
2500
2000
e-
1500 X
1000
500
0.0
0.2
0.4
Temperature
0,6
Fig. 2. Magnitude of the slope [dzZ/dln t[ versus temperature T at different frequencies f . The slope has been multiplied by a factor of 106. f = ~4 MCS per spin (O), f = ~ MCS per spin (E3), and f = ~I MCS per spin (+). The second curve from the bottom ( × ) represents P(0) which is independent of frequency. Each point represents an average over 4000 samples with 12 000 MCS per spin. The same bond configuration was used for all the data.
H.M. Carruzzo et al./Physica B 219&220 (1996) 311 313
qualitative agreement with experiment when we include interactions between defects. We would like to thank Doug Osheroff, Sven Rogge, Dora Salvino, Ben Tigner, and Tony Leggett for helpful discussions. This work was supported in part by ONR grant N000014-91-J-1502. CCY is an Alfred P. Sloan Research Fellow.
313
References [I] H.M. Carruzzo, E.R. Grannan and C.C. Yu, Phys. Rev. B 50 (1994) 6685 and references therein. [2] D.J. Salvino, S. Rogge, B. Tigner and D.D. Osheroff~ Phys. Rev. Lett. 73 (1994) 268.
Physica B 219&220 (1996) 311 313
Interacting defects in glasses H.M. Carruzzo a, E.R. Grannan b, C.C. Y u a, • a Department of Physics and Astronomy, University of California, Irvine, CA 92717-4575, USA b Lehman Brothers, One Broadgate, London, UK
Abstract
We find that recent low-temperature nonequilibrium dielectric experiments indicate that glasses have strongly interacting defects. While many of the features found in the experiments can be explained by the standard model of noninteracting twolevel systems, we find that the frequency dependence cannot. Using a Monte Carlo simulation of a nearest-neighbor Ising spin glass, we show that interactions between defects can qualitatively explain the experiments because they lead to the formation of clusters and a hole in the distribution of local fields.
The low-temperature properties of glasses, such as the specific heat and the thermal conductivity, have been traditionally explained by the model of two-level systems (TLS) which are usually thought of as tunneling centers [1]. The standard model of TLS assumes for the most part that the TLS do not interact with one another. Recently, however, several groups have suggested that interactions between defects may be crucial to our understanding of the universal properties seen in amorphous materials [1]. Until recently, both interacting and noninteractingmodels have been able to explain the low-temperature experiments, so there has been no way of clearly determining whether or not interactions between defects are truly important. However, recent nonequilibrium dielectric experiments [2] have shown that interactions do indeed play an important role in the low-temperature physics of glasses. Salvino et al. put thin films (1-3 I.tm) of glass such as SiO2, SiOx (x ~- 2.1 ), and polymers in a capacitor which they cooled down to low temperatures (20-1000 mK). Then they applied a large DC electric field ( ~ 106-10 7 V/m). Using an AC capacitance bridge, they watched the capacitance jump up and then decay logarithmically with time t after the DC field had been applied. They found that the magnitude of the slope of the logarithmic decay (]dC/d In t], where C is the capacitance) decreased with increasing temperature and increasing frequency. * Corresponding author. 0921-4526/96/$15.00 (~ 1996 Elsevier Science B.V. All rights reserved SSDI0921-4526(95)OO730-X
To understand this behavior, we can start by asking the following question. To what extent can these experiments be explained by the standard model of noninteracting TLS? To answer this, we use the analogy between an electric dipole in an electric field and a spin in a magnetic field to write the Bloch equations. Solving these equations gives us the time dependence of the polarization, the dielectric susceptibility, and the capacitance [ 1]. Averaging over the TLS parameters gives a logarithmic time decay of the capacitance. In fact, independent TLS theory and experiment agree qualitatively on the time, voltage, and temperature dependence of the decay. However, the standard model predicts that the slope of the logarithmic decay will be independent of frequency because the frequency is much smaller than the effective TLS energy splitting. This lack of frequency dependence contradicts experiment. This discrepancy leads us to consider interactions between defects mediated by the strain field. There are two important consequences of interactions. First interactions lead to clusters of strongly interacting defects which make the capacitance frequency dependent. At low frequencies large slow clusters as well as small fast clusters contribute to the response, while at high frequencies only small fast clusters have time to respond. This is consistent with the experimental observation of the jump in the capacitance being bigger and the slope ]dC/dlnt] being larger at low frequencies than at high frequencies.
312
H.M. Carruzzo et al./ Physica B 219&220 (1996) 311-313
Interactions also prevent the distribution of local fields at defect sites from being flat. In particular, stability arguments have been used to show that there must be a hole in the distribution of local fields [1]. To be more specific, think of a spin glass with long-range interactions [1]. Suppose you find the ground state spin configuration and suppose that the distribution of local magnetic fields, P(h), is finite at h - 0. This means that those spins with no local field can flip without changing their energy. But if they do so, other spins have their field altered and so some of them will flip. This in turn causes others to flip and so on. This avalanche means that the supposed ground state is not stable. In order to have a stable ground state, the distribution of local fields must go to zero as the local field h-~0. The Stanford experiments [2] are the first to test these long-standing ideas about a hole in the local field distribution of glasses because the strong DC field enables them to probe P(h) away from h = 0. Consider the situation before the application of the DC field. Defects whose local fields are small can flip easily in response to the AC field; these are the main contributors to the capacitance. When the DC field is applied, a new set of defects find themselves in zero local field. The applied DC field effectively shifts the local field distribution along the local field axis. Thus the susceptibility, and hence capacitance, suddenly jumps up. Once the DC field is switched on, a new hole develops and the capacitance decreases with time. To test these ideas about interactions, we have performed Monte Carlo simulations on a three-dimensional nearestneighbor Ising spin glass. The spins, representing the interacting electric dipoles, are on the sites of a cubic lattice of size 103 . In order to simulate the conditions of the experiment, we run for N/2 Monte Carlo steps per spin with no applied field, then switch on a DC field (poEDc ~ 0.4 and po - 1) and run for another N/2 Monte Carlo steps, where N is the total number of Monte Carlo steps. We used N = 12 000 and 25 000 Monte Carlo steps. Up to 8000 samples were used to average over the disorder. This system is known to have a transition temperature of about 0.9 K [1]. We have run at temperatures ranging from 0.1 to 0.9 K. We need not achieve complete thermal equilibrium at low temperatures since we are interested in nonequilibrium phenomena. The results of our simulations give qualitative agreement with experiment. The application of a DC field causes the capacitance to jump up and then decay logarithmically with time. In addition, the simulations give qualitatively the correct frequency and temperature dependence for the logarithmic slope of the decay as shown in Figs. 1 and 2. To summarize, the frequency dependence found in recent nonequilibrium dielectric experiments cannot be explained with the independent TLS model. However, we obtain
2400
=01..00sa.,es
I I T=0.2, 4000 samples ~T=0.1, 4000 samples
2200
2000
_= .:~ 1800 "0 X
%
i
1600
1400
1200
....
Independent Two Level
SsTm:
~ ' ~ .
0.01
.
.
.
.
.
.
.
.
E*-------~ .
.
.
.
.
.
.
.
.
.
.
010
frequency
Fig. 1. Magnitude of the slope ]dz~/d In t] versus frequency, where Z~ is the real part of the dielectric susceptibility and is proportional to the capacitance. The frequency is the inverse of the period in Monte Carlo steps (MCS) per spin. The slope has been multiplied by a factor of 10 6. The top curve (+) was done at T = 0.2 K with 4000 samples and N = 12 000 MCS per spin. The middle curve ( × ) was done at T = 0.1 with 4000 samples and N = 12000MCS per spin. The bottom curve ( O ) was done at T = 0.1 with 8000 samples and N = 25000 MCS per spin. The same bond configuration was used for the curves averaged over 4000 samples, but a different bond configuration was used for the 8000 sample curve. Comparing the bottom two curves gives some idea of the fluctuations resulting from taking the second derivative d2M/dEAc d In t, where M is the polarization.
2500
2000
e-
1500 X
1000
500
0.0
0.2
0.4
Temperature
0,6
Fig. 2. Magnitude of the slope [dzZ/dln t[ versus temperature T at different frequencies f . The slope has been multiplied by a factor of 106. f = ~4 MCS per spin (O), f = ~ MCS per spin (E3), and f = ~I MCS per spin (+). The second curve from the bottom ( × ) represents P(0) which is independent of frequency. Each point represents an average over 4000 samples with 12 000 MCS per spin. The same bond configuration was used for all the data.
H.M. Carruzzo et al./Physica B 219&220 (1996) 311 313
qualitative agreement with experiment when we include interactions between defects. We would like to thank Doug Osheroff, Sven Rogge, Dora Salvino, Ben Tigner, and Tony Leggett for helpful discussions. This work was supported in part by ONR grant N000014-91-J-1502. CCY is an Alfred P. Sloan Research Fellow.
313
References [I] H.M. Carruzzo, E.R. Grannan and C.C. Yu, Phys. Rev. B 50 (1994) 6685 and references therein. [2] D.J. Salvino, S. Rogge, B. Tigner and D.D. Osheroff~ Phys. Rev. Lett. 73 (1994) 268.