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Dielectric dispersion of SiO2 glass at low temperatures
Physica B 263—264 (1999) 333—335
Dielectric dispersion of SiO glass at low temperatures T. Ozaki*, T. Ogasawara, T. Kosugi, T. Kamada Department of Physical Science, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Abstract Complex dielectric constants e* of SiO glass were measured at frequencies of 10, 100 Hz, 1 and 10 kHz at temperatures from 3 to 100 K. They were best described by the relaxation time q which obeys the Vogel—Fulcher law q"q exp[E/(¹!¹ )] with ¹ "!0.5$0.5 K, q "(1.9$0.7);10\ s and E distributed below 3000 K. This shows that q substantially obeys the Arrhenius law q"q exp(E/¹), and hence the cooperative interaction between relaxators is too weak to cause a dipole glass transition. An additional contribution of phonon-assisted tunneling to e* is estimated by normalizing the distribution function of E. 1999 Elsevier Science B.V. All rights reserved. Keywords: SiO glass; Dielectric dispersion; Relaxation time
1. Introduction Low-frequency elastic and dielectric anomalies appear in SiO glass below 100 K [1,2]. This is attributed to the thermally activated motion of relaxators caged in double-well potentials, which is microscopically related to the libration of SiO tetrahedra coupled in the network [3]. It is interesting to know whether elastic and/or electrostatic interaction between relaxators is cooperative enough to cause a phase transition. It is known that an average of the relaxation time obeys the Arrhenius law [1,2]. This is negative for the onset of the phase transition. In order to experimentally clarify it, however, we should investigate a slight deviation of the relaxation time q from the
* Corresponding author. Fax: [email protected].
81-824-24-7397;
e-mail:
Arrhenius law in a low-frequency and low-temperature region. This is because Yu predicts that the interaction between elastic dipoles via a strain field causes a dipole glass transition followed by their antiferroelastic ordering at about 5 K [4]. The aim of the present study is to clarify the temperature dependence and the distribution of q in SiO glass by measuring its dielectric dispersion. An additional contribution of phonon-assisted tunneling [5,6] to the dispersion is also estimated.
2. Experimental and analysis We used disks cut out from an SiO glass rod produced by a soot method and containing OH less than 1 ppm. Each disk has a diameter of 8 mm, a thickness of 200 lm and evaporated-gold electrodes. The admittance of four disk capacitors connected in parallel was measured by using a three-terminal
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 2 3 3 - 2
334
T. Ozaki et al. / Physica B 263—264 (1999) 333—335
method with a capacitance bridge (GR1621 Precision Capacitance Measurement System). A signal field was 7.5 V/mm. Complex dielectric constants e*"e !ie were obtained at frequencies of 10, 100 Hz, 1 and 10 kHz at temperatures from 3 to 100 K. We analyzed e* on the following assumptions. (1) q obeys the Vogel—Fulcher law q"q exp[E/ (¹!¹ )], which coincides with the Arrhenius law q"q exp(E/¹) at ¹ "0 K. It is shown that in some glasses the Vogel—Fulcher temperature ¹ takes the same value as the critical temperature ¹ of the dynamical scaling law q&(¹!¹ )\XJ in a narrow frequency region [7]. (2) e* is expressed as superposition of the Debye formula having the weight of a normalized distribution function f (E) of an energy parameter E,
f (E) dE, (1) 1#iuq where e is the static dielectric constant. (3) At 3 K, Q e* is independent of frequency u/2p between 10 Hz and 10 kHz. We obtained f (E) widely distributed in Eq. (1) by using the relations expressed by measurable values under uq+1 [8],
e*(u, ¹)!e "(e !e ) Q
f (E)"2e (l,¹)/[p(¹!¹ )(e !e )] and Q E"(¹!¹ )ln(l /l), (2) where a relaxation frequency l"(2pq)\ and an attempt frequency l "(2pq )\. The strength of the relaxator, e !e , is insensitive to the deterQ mination of ¹ , but sensitive to the normalization of f (E). We determined ¹ and l which minimized the standard deviation p of f (E) obtained at 10, 100 Hz and 10 kHz from f (E) obtained at 1 kHz, in the temperature region above 20 K where an average of l is higher than 10 Hz.
Fig. 1. Temperature dependence of e and e at 10, 100 Hz, 1 and 10 kHz. The inset shows e below 300 K.
Fig. 2. The square of the standard deviation, p, of f (E) and the attempt frequency l vs. the Vogel—Fulcher temperature ¹ .
3. Results The dielectric dispersion shown in Fig. 1 is weak but widely extends below 100 K. The frequency dependence of the peak temperature of e confirms that an average of q is approximately expressed by the Arrhenius law with q "2.6;10\ s and E" 570 K. The value of e !e was estimated to be Q
2.4;10\ after taking account of e at high temperatures in the inset of Fig. 1. In Fig. 2, as ¹ lowers, p decreases and takes a minimum at ¹ "!0.5$0.5 K and l "(9.6$3.4);10 Hz. This demonstrates that q substantially obeys the Arrhenius law, and hence
T. Ozaki et al. / Physica B 263—264 (1999) 333—335
335
Fig. 3. Distribution function f (E) of the activation energy E. The line (1) represents f (E)"7.61;10\, (2) E"256 K, and (3) f (E)"a+1#tanh[b(E#c)], with a"3.54;10\, b"!1.11;10\ K\ and c"163 K.
Fig. 4. Comparison between the measured and the reproduced e . Two sets of lines, (1) and (2), are reproduced by f (E) normalized, respectively, by two straight lines (1) and (2) in Fig. 3.
the dipole glass transition does not occur in SiO glass. The activation energy E shows a wide distribution but no peak of f (E) in Fig. 3. The integration from 0 to 3000 K of the line (3) best fitted to f (E) exceeds unity, indicating another contribution to the dielectric dispersion at low temperatures.
40 K. On the other hand, the additional contribution in case (2) linearly increases from 5 K to each cutoff temperature ¹ between 10 and 15 K. This agrees with the internal friction [6] derived from the relaxation due to the incoherent tunneling strongly assisted by phonons. Contrary to the prediction of Ref. [6], however, ¹ is not proportional to u in Fig. 4. In conclusion, the present study clarifies that the dipole glass transition does not occur in SiO glass. This indicates that the interaction between elastic dipoles is short-range in glasses as compared to crystals, and the magnitude of the electric dipole moment is small.
4. Discussion In order to normalize f (E), we suppose two limiting cases (1) and (2) represented by two straight lines in Fig. 3. Case (1) requires a constant distribution below E"776 K, while case (2) the absence of a distribution below E"256 K. e was reproduced by substituting each normalized f (E) into Eq. (2). The reproduced e are shown by two sets of lines, (1) and (2), in Fig. 4. In case (1), the difference between the measured e and the reproduced e (1) remains constant below, but decreases to zero above, each peak temperature. This additional contribution inherits the plateau region characteristic below 5 K [1,2] where the relaxation due to the coherent tunneling assisted by one phonon is active [5]. However, it is an open question whether the coherent tunneling greatly contributes to e over
References [1] S. Hunklinger, M.V. Schickfus, in: W.A. Phillips (Ed.), Amorphous Solids: Low Temperature Properties, Springer, Berlin, 1981, pp. 81—105. [2] S.H. Mahle, R.D. McCammon, Phys. Chem. Glasses 10 (1969) 222. [3] U. Buchenau, H.M. Zhou, N. Nucker, K.S. Gilroy, W.A. Phillips, Phys. Rev. Lett. 60 (1988) 1318. [4] C.C. Yu, Phys. Rev. Lett. 69 (1992) 2787. [5] J. Ja¨ckle, Z. Phys. 257 (1972) 212. [6] P. Neu, A. Wu¨rger, Europhys. Lett. 27 (1994) 457. [7] J. Souletie, J.L. Tholence, Phys. Rev. B 32 (1985) 516. [8] E. Courtens, Phys. Rev. B 33 (1986) 2975.
Dielectric dispersion of SiO glass at low temperatures T. Ozaki*, T. Ogasawara, T. Kosugi, T. Kamada Department of Physical Science, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Abstract Complex dielectric constants e* of SiO glass were measured at frequencies of 10, 100 Hz, 1 and 10 kHz at temperatures from 3 to 100 K. They were best described by the relaxation time q which obeys the Vogel—Fulcher law q"q exp[E/(¹!¹ )] with ¹ "!0.5$0.5 K, q "(1.9$0.7);10\ s and E distributed below 3000 K. This shows that q substantially obeys the Arrhenius law q"q exp(E/¹), and hence the cooperative interaction between relaxators is too weak to cause a dipole glass transition. An additional contribution of phonon-assisted tunneling to e* is estimated by normalizing the distribution function of E. 1999 Elsevier Science B.V. All rights reserved. Keywords: SiO glass; Dielectric dispersion; Relaxation time
1. Introduction Low-frequency elastic and dielectric anomalies appear in SiO glass below 100 K [1,2]. This is attributed to the thermally activated motion of relaxators caged in double-well potentials, which is microscopically related to the libration of SiO tetrahedra coupled in the network [3]. It is interesting to know whether elastic and/or electrostatic interaction between relaxators is cooperative enough to cause a phase transition. It is known that an average of the relaxation time obeys the Arrhenius law [1,2]. This is negative for the onset of the phase transition. In order to experimentally clarify it, however, we should investigate a slight deviation of the relaxation time q from the
* Corresponding author. Fax: [email protected].
81-824-24-7397;
e-mail:
Arrhenius law in a low-frequency and low-temperature region. This is because Yu predicts that the interaction between elastic dipoles via a strain field causes a dipole glass transition followed by their antiferroelastic ordering at about 5 K [4]. The aim of the present study is to clarify the temperature dependence and the distribution of q in SiO glass by measuring its dielectric dispersion. An additional contribution of phonon-assisted tunneling [5,6] to the dispersion is also estimated.
2. Experimental and analysis We used disks cut out from an SiO glass rod produced by a soot method and containing OH less than 1 ppm. Each disk has a diameter of 8 mm, a thickness of 200 lm and evaporated-gold electrodes. The admittance of four disk capacitors connected in parallel was measured by using a three-terminal
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 2 3 3 - 2
334
T. Ozaki et al. / Physica B 263—264 (1999) 333—335
method with a capacitance bridge (GR1621 Precision Capacitance Measurement System). A signal field was 7.5 V/mm. Complex dielectric constants e*"e !ie were obtained at frequencies of 10, 100 Hz, 1 and 10 kHz at temperatures from 3 to 100 K. We analyzed e* on the following assumptions. (1) q obeys the Vogel—Fulcher law q"q exp[E/ (¹!¹ )], which coincides with the Arrhenius law q"q exp(E/¹) at ¹ "0 K. It is shown that in some glasses the Vogel—Fulcher temperature ¹ takes the same value as the critical temperature ¹ of the dynamical scaling law q&(¹!¹ )\XJ in a narrow frequency region [7]. (2) e* is expressed as superposition of the Debye formula having the weight of a normalized distribution function f (E) of an energy parameter E,
f (E) dE, (1) 1#iuq where e is the static dielectric constant. (3) At 3 K, Q e* is independent of frequency u/2p between 10 Hz and 10 kHz. We obtained f (E) widely distributed in Eq. (1) by using the relations expressed by measurable values under uq+1 [8],
e*(u, ¹)!e "(e !e ) Q
f (E)"2e (l,¹)/[p(¹!¹ )(e !e )] and Q E"(¹!¹ )ln(l /l), (2) where a relaxation frequency l"(2pq)\ and an attempt frequency l "(2pq )\. The strength of the relaxator, e !e , is insensitive to the deterQ mination of ¹ , but sensitive to the normalization of f (E). We determined ¹ and l which minimized the standard deviation p of f (E) obtained at 10, 100 Hz and 10 kHz from f (E) obtained at 1 kHz, in the temperature region above 20 K where an average of l is higher than 10 Hz.
Fig. 1. Temperature dependence of e and e at 10, 100 Hz, 1 and 10 kHz. The inset shows e below 300 K.
Fig. 2. The square of the standard deviation, p, of f (E) and the attempt frequency l vs. the Vogel—Fulcher temperature ¹ .
3. Results The dielectric dispersion shown in Fig. 1 is weak but widely extends below 100 K. The frequency dependence of the peak temperature of e confirms that an average of q is approximately expressed by the Arrhenius law with q "2.6;10\ s and E" 570 K. The value of e !e was estimated to be Q
2.4;10\ after taking account of e at high temperatures in the inset of Fig. 1. In Fig. 2, as ¹ lowers, p decreases and takes a minimum at ¹ "!0.5$0.5 K and l "(9.6$3.4);10 Hz. This demonstrates that q substantially obeys the Arrhenius law, and hence
T. Ozaki et al. / Physica B 263—264 (1999) 333—335
335
Fig. 3. Distribution function f (E) of the activation energy E. The line (1) represents f (E)"7.61;10\, (2) E"256 K, and (3) f (E)"a+1#tanh[b(E#c)], with a"3.54;10\, b"!1.11;10\ K\ and c"163 K.
Fig. 4. Comparison between the measured and the reproduced e . Two sets of lines, (1) and (2), are reproduced by f (E) normalized, respectively, by two straight lines (1) and (2) in Fig. 3.
the dipole glass transition does not occur in SiO glass. The activation energy E shows a wide distribution but no peak of f (E) in Fig. 3. The integration from 0 to 3000 K of the line (3) best fitted to f (E) exceeds unity, indicating another contribution to the dielectric dispersion at low temperatures.
40 K. On the other hand, the additional contribution in case (2) linearly increases from 5 K to each cutoff temperature ¹ between 10 and 15 K. This agrees with the internal friction [6] derived from the relaxation due to the incoherent tunneling strongly assisted by phonons. Contrary to the prediction of Ref. [6], however, ¹ is not proportional to u in Fig. 4. In conclusion, the present study clarifies that the dipole glass transition does not occur in SiO glass. This indicates that the interaction between elastic dipoles is short-range in glasses as compared to crystals, and the magnitude of the electric dipole moment is small.
4. Discussion In order to normalize f (E), we suppose two limiting cases (1) and (2) represented by two straight lines in Fig. 3. Case (1) requires a constant distribution below E"776 K, while case (2) the absence of a distribution below E"256 K. e was reproduced by substituting each normalized f (E) into Eq. (2). The reproduced e are shown by two sets of lines, (1) and (2), in Fig. 4. In case (1), the difference between the measured e and the reproduced e (1) remains constant below, but decreases to zero above, each peak temperature. This additional contribution inherits the plateau region characteristic below 5 K [1,2] where the relaxation due to the coherent tunneling assisted by one phonon is active [5]. However, it is an open question whether the coherent tunneling greatly contributes to e over
References [1] S. Hunklinger, M.V. Schickfus, in: W.A. Phillips (Ed.), Amorphous Solids: Low Temperature Properties, Springer, Berlin, 1981, pp. 81—105. [2] S.H. Mahle, R.D. McCammon, Phys. Chem. Glasses 10 (1969) 222. [3] U. Buchenau, H.M. Zhou, N. Nucker, K.S. Gilroy, W.A. Phillips, Phys. Rev. Lett. 60 (1988) 1318. [4] C.C. Yu, Phys. Rev. Lett. 69 (1992) 2787. [5] J. Ja¨ckle, Z. Phys. 257 (1972) 212. [6] P. Neu, A. Wu¨rger, Europhys. Lett. 27 (1994) 457. [7] J. Souletie, J.L. Tholence, Phys. Rev. B 32 (1985) 516. [8] E. Courtens, Phys. Rev. B 33 (1986) 2975.