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On the excess specific heat of vitreous silica at low temperatures
Volume 86A, number 1
PHYSICS LETTERS
26 October 1981
ON THE EXCESS SPECIFIC HEAT OF VITREOUS SILICA AT LOW TEMPERATURES J. ZIMMERMANN and G. WEBER Institut für Festkörperphysik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Fed. Rep. Germany Received 16 June 1981
Glasses at low temperatures exhibit an excess specific heat roughly but not exactly linear in T. It is shown for vitreous silica, that the observed deviations from linearity are well described by the standard tunneling model, if recent results of time-dependent specific heat measurements are taken into account.
The specific heat measured in dielectric glasses below 1 K may approximately be described by [11 =
1i-~,
3
3
~
where v = 0.2—0.5. The last term is the Debye contribution. The first two terms are peculiar to the amorphous state and have been investigated intensively during the last years. The second of these terms, the “excess T3-dependent specific heat”, is comparable in magnitude to the Debye contribution. Its origin is not known. The first term is the “linear specific heat”, often taken to be the trademark of the amorphous state. This roughly linear term is best explained in terms of the widely accepted tunneling model of Anderson et al. [21 and Phillips [3]. Also in this case, the microscopic nature of the underlying excitations is not fully understood. As a unique feature of the tunneling model, the specific heat should depend on the measurement time [2], because the wide distribution of tunneling probabilities inherent in the model causes a wide distribution of thermal relaxation times for energy transfer between tunneling systems and Debye phonons. There is a growing body of experimental evidence from measurements on vitreous silica [4—61that the specific heat in the temperature regime of the linear term does indeed depend on the measurement time up to at least 10~s. To our knowledge this effect has not yet been taken into account in the discussion of the long time specific heat experiments carned out on a 10 s time scale. Below we show that upon inclusion of this effect the tunneling model does
not, as is usually assumed, imply a strictly linear specific heat, and, moreover, that the deviations from linearity expected from the model are in accordance with existing experimental data. In the standard tunneling model, two-level systems are supposed to originate from the tunneling of atoms or molecular complexes through an energy barrier between two almost equivalent sites. The number of twolevel systems P per gram, per erg and per tunneling parameter interval is taken to be a constant independent of the parameters of the double-well potential. According to Rammal and Maynard [7] and to Black [8} the model then gives a time-dependent specific heat for measurement time t ~ c(T, t) = (1r2/l2)k~TPln(4t/r*) (2) .
r’~is the lower limit of relaxation times for the most strongly coupled two-level systems. For vitreous silica r” has been obtained by Black in a doiiiinant phonon approximation [81: ~ = 0.46 X lo—~T—3 K3 ~ (3) .
Lacking information on the time dependence of the specific heat, it has generally been assumed that the measurement time exceeds the range Tm~of available thermal relaxation times, and an integral density of states = ~Pin (4rm~/r*)has been defined and set constant. One then obtains [8] the usual specific heat linear in T: c(T)
=
~7r2k~
n 0T.
32
(4)
0031-9163/81/0000—0000/s 02.75 © 1981 North-Holland
/,/~ 0
Volume 86A, number 1
100
PHYSICS LETTERS
I
I
I
I
Suprasil W (—‘ 1,5 ppm OH) Suprasit 1(1200 ppm OH) 2 2 PTIn 4lOs +a T 3 +a —c-—k
—
-
o
-
‘_
-
it
12 10
1(T)
B
0T
exc
-
—
,-~
/
.
II
I 1
-
/
/‘
—
—
-
-
c
-
~y ~-
°
I
—
0.02
K’
/
/
$
-
~
• 0.1
-
0.05
I
I
0.1
/ 1 0.2
I
~
0.5
1 T(K)
Specific heat of vitreous Si0 points are from Lasjaunias et al. [1].2 The below dashed 1 K. The line shows data the Debye contribution. The drawn curves are fits based on the standard tunneling model for 10 s measurement times and two additional T -dependent contributions. For fit parameters see text. Fig. 1.
The experimental results for two types of vitreous silica are shown in fig. 1. The data are from Lasjaunias et al. [1] on Suprasil W with an 0H content < 5 ppm and Suprasil I with an 0H content of about 1200 ppm. The results of other workers [9—11]on vitreous silica agree well with these data. In the temperature region of interest here, i.e. below 0.3 K, the specific heat has been described [1] as in eq. (1) by power laws c T1~’,with v = 0.30 for Suprasil Wand ~ = 0.22 for Suprasil I. Two approaches have been proposed to account for the deviation from the expected linearity according to eq. (4): (i) introduction of an energy-dependent integral density [1] of states n (E); ‘-~
26 October 1981
(ii) introduction of a gap [12] inn0 at low energy E < kB X 16 mK, justified by assuming an upper limit of the barrier height between the two potential wells. However, the recent measurements of the timedependent specific heat [6] have shown that for both types of Suprasil, eq. (2) is followed up to at least l0~s in the temperature regime of the linear term. In consequence, we substitute eq. (2) for the linear term in eq. (1) and obtain: c(T, t) = (i~2/12)k~.~Tln [4t/r*(T)] 3+aDT3 . (5) +aexcT For comparison with the experiment we need the value of the measurement time t, which is not easily estimated from the literature. We take t = 10 s as a reasonable value. The particular value used will not greatly influence the results as t enters only logarithmically. r” may be taken from eq. (3), and we set aD = 8 erg/g K4 as derived from sound velocity measurements [1]. This leaves us with only ~ and aexc in eq. (5) as adjustable parameters. The drawn curves in fig. 1 are fits of eq. (5) to the data with J~= 2.4 X 1031 erg~g~and aexc = 36 erg g1 K—4 for Suprasil W and .~= 34 X 1031 erg—1 g~and aexc = 8 erg g~K—4 for Suprasil I. In both cases we obtain good agreement with the data. The value of~from the fit for Suprasil 1 g—1 as W derived agrees[8] rather by application well with ]~ of= the 1.0 tunneling X 1031 erg— model to the phonon scattering experiments of Golding and Graebner [13], and to F = 1.5 X 1031 erg—1 g1, as derived from our long-time heat relaxation experiments [6]. The excess T3-dependent specific heats are comparable to those found in both types of vitreous silica by Loponen et al. [15] and Meissner et al. [4],respectively. These time-dependent experiments show, moreover, that the excess T3-depen~entterms relax on is time scales and can be considered instantaneous on the time scale of 10 s experiments. In the case of Suprasil W the fit according to a T3 law is only valid below 1 K and includes the rising edge of a bump in the specific heat, which can be removed by thermal treatment [11]. This bump has been discussed [10] in terms of a Schottky anomaly with maximum around 1.5 K, but this discussion does not bear on our problem. In conclusion, we find that the original tunneling model assumption of a constant density of states for the tunneling systems responsible for the linear specific 33
Volume 86A, number 1
PHYSICS LETTERS
heat is adequate to describe the “long-time” specific heat experiments. In our opinion this can only be taken as a relative statement. There is evidence [14] that different subsets of excitations exist, each having its mdividual distribution and contributing differently to different physical properties. These individual distributions might well be smeared out considerably by strain interactions [15], so that the assumption of a constant density of states, as originally proposed, is still as good as any to describe the integral specific heat data. References J.C. Lasjaunias, A. Ravex, M. Vandorpe and S. Hunklinger, Solid State Commun. 17 (1975)1045. [2] P.W. Anderson, B.I. Halperin and C.M. Varma, Phios. [1]
Mag. 25 (1972) 1.
[3] W.A. Phillips, J. Low Temp. Phys.
34
7 (1972) 351.
26 October 1981
[4] M. Meissner and K. Spitzmann, Phys. Rev. Lett. 46 (1981) 265.
[5] M.T. Loponen, R.C. Dynes, V. Narayanamurti and J.P.
Garno, Phys. Rev. Lett. 45 (1980) 457. Zimmermann and G. Weber, Phys. Rev. Lett. 46(1981) 661. [7] R. Rammal and R. Maynard, J. de Phys. (Paris) 39 (1978) C6970. [8] J.L. Black, Phys. Rev. B17 (1978) 2740. [9] R.C. Zeller and R.O. Pohl, Phys. Rev. B4 (1971) 2029. [10] H. v. Löhneysen and M. Platte, Z. Phys. B36 (1979) 113. [11] J.C. Lasjaunias, G. Penn, A. Ravex and M. Vandorpe, J. de Phys. (Paris) Lett. 41(1980) L-131.
[6) J.
[12] J.C. Lasjaunias, R. Maynard and M. Vandorpe, J. de Phys. (Paris) 39 (1978) C6-973. [13] B. Golding and i.E. Graebner, Phys. Rev. Lett. 37(1976) 852. [141 S. Hunklinger, L. Piché, J.C. Lasjaunias and K. Dransfeld, J. Phys. C8 (1975) L423. [15] M.W. Klein, B. Fischer, A.C. Anderson and P.J. Anthony, Phys. Rev. B18 (1978) 5887.
PHYSICS LETTERS
26 October 1981
ON THE EXCESS SPECIFIC HEAT OF VITREOUS SILICA AT LOW TEMPERATURES J. ZIMMERMANN and G. WEBER Institut für Festkörperphysik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Fed. Rep. Germany Received 16 June 1981
Glasses at low temperatures exhibit an excess specific heat roughly but not exactly linear in T. It is shown for vitreous silica, that the observed deviations from linearity are well described by the standard tunneling model, if recent results of time-dependent specific heat measurements are taken into account.
The specific heat measured in dielectric glasses below 1 K may approximately be described by [11 =
1i-~,
3
3
~
where v = 0.2—0.5. The last term is the Debye contribution. The first two terms are peculiar to the amorphous state and have been investigated intensively during the last years. The second of these terms, the “excess T3-dependent specific heat”, is comparable in magnitude to the Debye contribution. Its origin is not known. The first term is the “linear specific heat”, often taken to be the trademark of the amorphous state. This roughly linear term is best explained in terms of the widely accepted tunneling model of Anderson et al. [21 and Phillips [3]. Also in this case, the microscopic nature of the underlying excitations is not fully understood. As a unique feature of the tunneling model, the specific heat should depend on the measurement time [2], because the wide distribution of tunneling probabilities inherent in the model causes a wide distribution of thermal relaxation times for energy transfer between tunneling systems and Debye phonons. There is a growing body of experimental evidence from measurements on vitreous silica [4—61that the specific heat in the temperature regime of the linear term does indeed depend on the measurement time up to at least 10~s. To our knowledge this effect has not yet been taken into account in the discussion of the long time specific heat experiments carned out on a 10 s time scale. Below we show that upon inclusion of this effect the tunneling model does
not, as is usually assumed, imply a strictly linear specific heat, and, moreover, that the deviations from linearity expected from the model are in accordance with existing experimental data. In the standard tunneling model, two-level systems are supposed to originate from the tunneling of atoms or molecular complexes through an energy barrier between two almost equivalent sites. The number of twolevel systems P per gram, per erg and per tunneling parameter interval is taken to be a constant independent of the parameters of the double-well potential. According to Rammal and Maynard [7] and to Black [8} the model then gives a time-dependent specific heat for measurement time t ~ c(T, t) = (1r2/l2)k~TPln(4t/r*) (2) .
r’~is the lower limit of relaxation times for the most strongly coupled two-level systems. For vitreous silica r” has been obtained by Black in a doiiiinant phonon approximation [81: ~ = 0.46 X lo—~T—3 K3 ~ (3) .
Lacking information on the time dependence of the specific heat, it has generally been assumed that the measurement time exceeds the range Tm~of available thermal relaxation times, and an integral density of states = ~Pin (4rm~/r*)has been defined and set constant. One then obtains [8] the usual specific heat linear in T: c(T)
=
~7r2k~
n 0T.
32
(4)
0031-9163/81/0000—0000/s 02.75 © 1981 North-Holland
/,/~ 0
Volume 86A, number 1
100
PHYSICS LETTERS
I
I
I
I
Suprasil W (—‘ 1,5 ppm OH) Suprasit 1(1200 ppm OH) 2 2 PTIn 4lOs +a T 3 +a —c-—k
—
-
o
-
‘_
-
it
12 10
1(T)
B
0T
exc
-
—
,-~
/
.
II
I 1
-
/
/‘
—
—
-
-
c
-
~y ~-
°
I
—
0.02
K’
/
/
$
-
~
• 0.1
-
0.05
I
I
0.1
/ 1 0.2
I
~
0.5
1 T(K)
Specific heat of vitreous Si0 points are from Lasjaunias et al. [1].2 The below dashed 1 K. The line shows data the Debye contribution. The drawn curves are fits based on the standard tunneling model for 10 s measurement times and two additional T -dependent contributions. For fit parameters see text. Fig. 1.
The experimental results for two types of vitreous silica are shown in fig. 1. The data are from Lasjaunias et al. [1] on Suprasil W with an 0H content < 5 ppm and Suprasil I with an 0H content of about 1200 ppm. The results of other workers [9—11]on vitreous silica agree well with these data. In the temperature region of interest here, i.e. below 0.3 K, the specific heat has been described [1] as in eq. (1) by power laws c T1~’,with v = 0.30 for Suprasil Wand ~ = 0.22 for Suprasil I. Two approaches have been proposed to account for the deviation from the expected linearity according to eq. (4): (i) introduction of an energy-dependent integral density [1] of states n (E); ‘-~
26 October 1981
(ii) introduction of a gap [12] inn0 at low energy E < kB X 16 mK, justified by assuming an upper limit of the barrier height between the two potential wells. However, the recent measurements of the timedependent specific heat [6] have shown that for both types of Suprasil, eq. (2) is followed up to at least l0~s in the temperature regime of the linear term. In consequence, we substitute eq. (2) for the linear term in eq. (1) and obtain: c(T, t) = (i~2/12)k~.~Tln [4t/r*(T)] 3+aDT3 . (5) +aexcT For comparison with the experiment we need the value of the measurement time t, which is not easily estimated from the literature. We take t = 10 s as a reasonable value. The particular value used will not greatly influence the results as t enters only logarithmically. r” may be taken from eq. (3), and we set aD = 8 erg/g K4 as derived from sound velocity measurements [1]. This leaves us with only ~ and aexc in eq. (5) as adjustable parameters. The drawn curves in fig. 1 are fits of eq. (5) to the data with J~= 2.4 X 1031 erg~g~and aexc = 36 erg g1 K—4 for Suprasil W and .~= 34 X 1031 erg—1 g~and aexc = 8 erg g~K—4 for Suprasil I. In both cases we obtain good agreement with the data. The value of~from the fit for Suprasil 1 g—1 as W derived agrees[8] rather by application well with ]~ of= the 1.0 tunneling X 1031 erg— model to the phonon scattering experiments of Golding and Graebner [13], and to F = 1.5 X 1031 erg—1 g1, as derived from our long-time heat relaxation experiments [6]. The excess T3-dependent specific heats are comparable to those found in both types of vitreous silica by Loponen et al. [15] and Meissner et al. [4],respectively. These time-dependent experiments show, moreover, that the excess T3-depen~entterms relax on is time scales and can be considered instantaneous on the time scale of 10 s experiments. In the case of Suprasil W the fit according to a T3 law is only valid below 1 K and includes the rising edge of a bump in the specific heat, which can be removed by thermal treatment [11]. This bump has been discussed [10] in terms of a Schottky anomaly with maximum around 1.5 K, but this discussion does not bear on our problem. In conclusion, we find that the original tunneling model assumption of a constant density of states for the tunneling systems responsible for the linear specific 33
Volume 86A, number 1
PHYSICS LETTERS
heat is adequate to describe the “long-time” specific heat experiments. In our opinion this can only be taken as a relative statement. There is evidence [14] that different subsets of excitations exist, each having its mdividual distribution and contributing differently to different physical properties. These individual distributions might well be smeared out considerably by strain interactions [15], so that the assumption of a constant density of states, as originally proposed, is still as good as any to describe the integral specific heat data. References J.C. Lasjaunias, A. Ravex, M. Vandorpe and S. Hunklinger, Solid State Commun. 17 (1975)1045. [2] P.W. Anderson, B.I. Halperin and C.M. Varma, Phios. [1]
Mag. 25 (1972) 1.
[3] W.A. Phillips, J. Low Temp. Phys.
34
7 (1972) 351.
26 October 1981
[4] M. Meissner and K. Spitzmann, Phys. Rev. Lett. 46 (1981) 265.
[5] M.T. Loponen, R.C. Dynes, V. Narayanamurti and J.P.
Garno, Phys. Rev. Lett. 45 (1980) 457. Zimmermann and G. Weber, Phys. Rev. Lett. 46(1981) 661. [7] R. Rammal and R. Maynard, J. de Phys. (Paris) 39 (1978) C6970. [8] J.L. Black, Phys. Rev. B17 (1978) 2740. [9] R.C. Zeller and R.O. Pohl, Phys. Rev. B4 (1971) 2029. [10] H. v. Löhneysen and M. Platte, Z. Phys. B36 (1979) 113. [11] J.C. Lasjaunias, G. Penn, A. Ravex and M. Vandorpe, J. de Phys. (Paris) Lett. 41(1980) L-131.
[6) J.
[12] J.C. Lasjaunias, R. Maynard and M. Vandorpe, J. de Phys. (Paris) 39 (1978) C6-973. [13] B. Golding and i.E. Graebner, Phys. Rev. Lett. 37(1976) 852. [141 S. Hunklinger, L. Piché, J.C. Lasjaunias and K. Dransfeld, J. Phys. C8 (1975) L423. [15] M.W. Klein, B. Fischer, A.C. Anderson and P.J. Anthony, Phys. Rev. B18 (1978) 5887.