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Anharmonic phonon decay in TeO2: confirmation of Herring's theory
Physica B 263—264 (1999) 713—715
Anharmonic phonon decay in TeO : confirmation of Herring’s theory E.P.N. Damen, A.F.M. Arts, H.W. de Wijn* Faculty of Physics and Astronomy, Debye Research Institute, Utrecht University, P.O. Box 80000, 3508 TA Utrecht, The Netherlands
Abstract The decay time q of longitudinal phonons propagating along the [0 0 1] axis in a single crystal of paratellurite (TeO ) is found to vary with the frequency and the temperature as 1/q"u?¹@ with a"1.84$0.20 and b"2.81$0.20 below about 53 K, and as u?¹ with a"1.87$0.20 above this temperature. These results confirm Herring’s theory for anharmonic phonon decay in anisotropic crystals. 1999 Elsevier Science B.V. All rights reserved. Keywords: Phonon scattering; Anharmonic decay
As was pointed out by Herring [1], the decay of phonons in elastically anisotropic media is governed by three-phonon processes that are different from the processes prevailing in the isotropic approximation. In particular, longitudinal phonons are shown to scatter predominantly from transverse phonons in the three-phonon process L# STPFT. Using group-theoretical considerations, Herring established that the decay rate of L phonons against this process depends on the angular frequency u and the temperature ¹ as q\"Au?¹@,
(1)
provided u is in the acoustic range (small wavevector approximation) and ¹ is well below the Debye temperature. The exponent a is determined
* Corresponding author. Fax: #31-30-2532403; e-mail: [email protected].
by the crystal class, the direction off the phonon wave vector, and the presence of dispersion, while a and b satisfy the Herring sum rule a#b"5. The sum rule is quite well accepted [2—4], but direct experimental verification of the individual a and b has remained elusive because of the scarcity of phonon sources delivering a phonon beam of welldefined frequency and wave vector. With regard to thermal conduction, which self-evidently depends on a temperature-averaging summation over all frequencies, theoretical treatments [5,6] usually assume the a#b"5 rule to be valid, and in keeping with this rule adopt u and ¹ dependences for the various phonon relaxation rates, such q\J u¹ for relaxation of L phonons by normal processes. This generally leads to consistency with the experiment [7,8]. In the present paper, the exponents a and b are determined for the case of L phonons propagating along the [0 0 1] axis in paratellurite (TeO ) by
0921-4526/99/$ — 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 4 8 1 - 1
714
E.P.N. Damen et al. / Physica B 263—264 (1999) 713—715
measuring the phonon decay in the ballistic regime versus the temperature at well-defined frequencies. This is achieved by making use of a recently developed technique for the generation of monochromatic Fresnel-diffracted phonon beams in the gigahertz range [9]. The method relies on cw laserinduced thermomodulation of a metallic transducer evaporated onto the crystal. In addition to monochromaticity and tunability, the technique features substantial narrowness of the phonon beam with diffraction-limited divergence. In combination with detection by Brillouin scattering, which is wave-vector selective, diffusive processes and thermal background are therefore largely eliminated. TeO was chosen for its strongly anisotropic transverse phonon branches, which ensures satisfying the small wave-vector approximation, and its high Brillouin yield. The specimen was a single crystal of synthetic TeO , 10.5;9.5;10 mm in size. For the generation of phonons, a gold transducer consisting of steps of four different thicknesses (220—660 nm) was deposited onto an outer surface oriented parallel to (0 0 1). A highly directional monochromatic phonon beam of longitudinal polarization, only 40 lm in diameter and propagating along [0 0 1] with a divergence of about 2°, was generated by modulated heating of the transducer surface [9]. This thermomodulation is brought about by two interfering cw single-frequency ring dye lasers, aligned such that their beams are collinear. The thermomodulation causes a strain at the laser difference frequency, which ultimately is injected into the crystal. The transducer thickness was chosen such as to benefit from acoustic resonances. A helium-gas-flow cryostat was used. The acoustic attenuation was measured as a function of the temperature with reliance on antiStokes Brillouin scattering for various phonon frequencies in the range 3.0—5.0 GHz and a selection of distances from the transducer. The Brillouin spectrometer is of conventional design, employing a quintuple-pass Fabry—Perot interferometer, a monochromator to eliminate laser plasma lines, and photon counting techniques. The detection volume has the shape of a pencil as narrow as 10 lm in diameter, and the Brillouin condition selects wave vectors within a cone of about 1°.
Fig. 1. Logarithm of the Brillouin intensity, i.e., z/vq, arising from acoustic phonons in TeO versus the temperature. The intensity was collected at a distance of 2.0 mm from the transducer for u/2p"3.0, 4.2, and 4.9 GHz. The data at 4.2 and 4.9 GHz are shifted downward for clarity. The curves and straight lines represent the low-temperature dependence Eq. (1) and the high-temperature dependence Eq. (2).
A total of 17 fixed-frequency variable-temperature scan were performed. Fig. 1 shows a few examples of the results. At very low temperatures, the generated L phonons travel virtually ballistically, but their mean free path shortens when the temperature is raised. Before analyzing the data, we consider the values of a and b to be expected from Herring’s theory on the basis of the crystal class. The crystal structure of TeO has been examined in detail by Leciejewicz [10] by the use of neutron diffraction, to find that the space group is D. According to Table 2 of Herring’s paper the crystal class D implies a"4, but a"2 in case phonon dispersion is negligible. A second circumstance favoring a quadratic u dependence is the only minute departure of the TeO structure from the more symmetric rutile structure (space group D). For the crystal class D we F F have a"2 for any phonon frequency. In the present case of L acoustic phonons propagating along
E.P.N. Damen et al. / Physica B 263—264 (1999) 713—715
[0 0 1] in TeO , therefore, we expect a"2 and b"3. In the detailed analysis of the data, the signal intensity is assumed to depend exponentially on the inverse relaxation time according to I"I e\X TO, in which z is the distance the acoustic wave has covered from the transducer, v is the sound velocity, and the zero-temperature intensities I refer to the respective temperature scans. In Fig. 1, therefore, log (I/I )"!1 corresponds to a mean free path C equal to z"2.0 mm. For L phonons propagating along [0 0 1], we have v"4.202$0.010 km/s [13]. We have carried out a recursive multiparameter least-squares fit to all 17 temperature scans with a, b, A, and the I as adjustable parameters. Note that the I pertaining to a particular z, but various u, can be interrelated in the zero-temperature limit. A good fit was obtained, provided the data are limited to temperatures below about 53 K (cf. Fig. 1). The output values for the coefficients are a"1.84$ 0.20 and b"2.81$0.20. We furthermore find A"(4.0$0.5);10\ s?\K\@. The experimental a and b are indeed in conformity with Herring’s predictions, while their sum is close to five. We finally consider the high-temperature regime. According to Herring [1], Eq. (1) is valid only well below the Debye temperature, the attenuation becoming linear according to q\"Bu?¹,
(2)
when most phonon modes are thermally excited. In Fig. 1, it is indeed observed that the temperature dependence changes over from cubic to linear above about 53 K. A fit of Eq. (2), adjusting a, B, and the I to all data from 53 to 100 K, yielded a"1.87$0.20, to be compared with Herring’s
In elastically isotropic media, the relevant anharmonic processes are L#TPL and L#LP L. The associated decay rates vary as q\Ju¹ [4] and q\Ju¹ [11,12]. These processes are at variance with the present experiment.
715
a"2. We furthermore find B"(2.6$0.5);10\ s?\K\. The crossover from the cubic to the linear dependence thus occurs at ¹ +53 K, very much independent of the frequency (Fig. 1). This crossover temperature seems reasonable in relation to the Debye temperature of TeO , which is estimated at roughly 200 K. In summary, we have measured the attenuation of longitudinal phonons propagating along the [0 0 1] axis in a single crystal of TeO as a function of the frequency and the temperature, to find confirmation of the theory for phonon attenuation in anisotropic crystals by Herring. The work was supported by the Netherlands foundations FOM and NWO. References [1] C. Herring, Phys. Rev. 95 (1954) 954. [2] J.W. Tucker, V.W. Rampton, Microwave Ultrasonics in Solid State Physics, North-Holland, Amsterdam, 1972. [3] J.W. Tucker, V.W. Rampton, in: W. Eisenmenger, A.A. Kaplyanskii (Eds.), Nonequilibrium Phonons in Nonmetallic Crystals, North-Holland, Amsterdam, 1986. [4] G.P. Srivastava, The Physics of Phonons, Adam Hilger, Bristol, 1990. [5] J. Callaway, Phys. Rev. 113 (1959) 1046. [6] M.G. Holland, Phys. Rev. 132 (1963) 2461. [7] Y.-J. Han, P.G. Klemens, Phys. Rev. B 48 (1993) 6033. [8] M. Asen-Palmer, K. Bartkowski, E. Gmelin, M. Cardona, A.P. Zhernov, A.V. Inyushkin, A. Taldenkov, V.I. Ozhogin, K.M. Itoh, E.E. Haller, Phys. Rev. B 56 (1997) 9431. [9] E.P.N. Damen, A.F.M. Arts, H.W. de Wijn, Phys. Rev. Lett. 74 (1995) 4249. [10] J. Leciejewicz, Z. Kristallogr 116 (1961) 345. [11] I.S. Ciccarello, K. Dransfeld, Phys. Rev. 134 (1964) A1517. [12] H.J. Maris, Philos. Mag. 9 (1964) 901. [13] Y. Ohmachi, N. Uchida, J. Appl. Phys. 41 (1970) 2307.
Anharmonic phonon decay in TeO : confirmation of Herring’s theory E.P.N. Damen, A.F.M. Arts, H.W. de Wijn* Faculty of Physics and Astronomy, Debye Research Institute, Utrecht University, P.O. Box 80000, 3508 TA Utrecht, The Netherlands
Abstract The decay time q of longitudinal phonons propagating along the [0 0 1] axis in a single crystal of paratellurite (TeO ) is found to vary with the frequency and the temperature as 1/q"u?¹@ with a"1.84$0.20 and b"2.81$0.20 below about 53 K, and as u?¹ with a"1.87$0.20 above this temperature. These results confirm Herring’s theory for anharmonic phonon decay in anisotropic crystals. 1999 Elsevier Science B.V. All rights reserved. Keywords: Phonon scattering; Anharmonic decay
As was pointed out by Herring [1], the decay of phonons in elastically anisotropic media is governed by three-phonon processes that are different from the processes prevailing in the isotropic approximation. In particular, longitudinal phonons are shown to scatter predominantly from transverse phonons in the three-phonon process L# STPFT. Using group-theoretical considerations, Herring established that the decay rate of L phonons against this process depends on the angular frequency u and the temperature ¹ as q\"Au?¹@,
(1)
provided u is in the acoustic range (small wavevector approximation) and ¹ is well below the Debye temperature. The exponent a is determined
* Corresponding author. Fax: #31-30-2532403; e-mail: [email protected].
by the crystal class, the direction off the phonon wave vector, and the presence of dispersion, while a and b satisfy the Herring sum rule a#b"5. The sum rule is quite well accepted [2—4], but direct experimental verification of the individual a and b has remained elusive because of the scarcity of phonon sources delivering a phonon beam of welldefined frequency and wave vector. With regard to thermal conduction, which self-evidently depends on a temperature-averaging summation over all frequencies, theoretical treatments [5,6] usually assume the a#b"5 rule to be valid, and in keeping with this rule adopt u and ¹ dependences for the various phonon relaxation rates, such q\J u¹ for relaxation of L phonons by normal processes. This generally leads to consistency with the experiment [7,8]. In the present paper, the exponents a and b are determined for the case of L phonons propagating along the [0 0 1] axis in paratellurite (TeO ) by
0921-4526/99/$ — 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 4 8 1 - 1
714
E.P.N. Damen et al. / Physica B 263—264 (1999) 713—715
measuring the phonon decay in the ballistic regime versus the temperature at well-defined frequencies. This is achieved by making use of a recently developed technique for the generation of monochromatic Fresnel-diffracted phonon beams in the gigahertz range [9]. The method relies on cw laserinduced thermomodulation of a metallic transducer evaporated onto the crystal. In addition to monochromaticity and tunability, the technique features substantial narrowness of the phonon beam with diffraction-limited divergence. In combination with detection by Brillouin scattering, which is wave-vector selective, diffusive processes and thermal background are therefore largely eliminated. TeO was chosen for its strongly anisotropic transverse phonon branches, which ensures satisfying the small wave-vector approximation, and its high Brillouin yield. The specimen was a single crystal of synthetic TeO , 10.5;9.5;10 mm in size. For the generation of phonons, a gold transducer consisting of steps of four different thicknesses (220—660 nm) was deposited onto an outer surface oriented parallel to (0 0 1). A highly directional monochromatic phonon beam of longitudinal polarization, only 40 lm in diameter and propagating along [0 0 1] with a divergence of about 2°, was generated by modulated heating of the transducer surface [9]. This thermomodulation is brought about by two interfering cw single-frequency ring dye lasers, aligned such that their beams are collinear. The thermomodulation causes a strain at the laser difference frequency, which ultimately is injected into the crystal. The transducer thickness was chosen such as to benefit from acoustic resonances. A helium-gas-flow cryostat was used. The acoustic attenuation was measured as a function of the temperature with reliance on antiStokes Brillouin scattering for various phonon frequencies in the range 3.0—5.0 GHz and a selection of distances from the transducer. The Brillouin spectrometer is of conventional design, employing a quintuple-pass Fabry—Perot interferometer, a monochromator to eliminate laser plasma lines, and photon counting techniques. The detection volume has the shape of a pencil as narrow as 10 lm in diameter, and the Brillouin condition selects wave vectors within a cone of about 1°.
Fig. 1. Logarithm of the Brillouin intensity, i.e., z/vq, arising from acoustic phonons in TeO versus the temperature. The intensity was collected at a distance of 2.0 mm from the transducer for u/2p"3.0, 4.2, and 4.9 GHz. The data at 4.2 and 4.9 GHz are shifted downward for clarity. The curves and straight lines represent the low-temperature dependence Eq. (1) and the high-temperature dependence Eq. (2).
A total of 17 fixed-frequency variable-temperature scan were performed. Fig. 1 shows a few examples of the results. At very low temperatures, the generated L phonons travel virtually ballistically, but their mean free path shortens when the temperature is raised. Before analyzing the data, we consider the values of a and b to be expected from Herring’s theory on the basis of the crystal class. The crystal structure of TeO has been examined in detail by Leciejewicz [10] by the use of neutron diffraction, to find that the space group is D. According to Table 2 of Herring’s paper the crystal class D implies a"4, but a"2 in case phonon dispersion is negligible. A second circumstance favoring a quadratic u dependence is the only minute departure of the TeO structure from the more symmetric rutile structure (space group D). For the crystal class D we F F have a"2 for any phonon frequency. In the present case of L acoustic phonons propagating along
E.P.N. Damen et al. / Physica B 263—264 (1999) 713—715
[0 0 1] in TeO , therefore, we expect a"2 and b"3. In the detailed analysis of the data, the signal intensity is assumed to depend exponentially on the inverse relaxation time according to I"I e\X TO, in which z is the distance the acoustic wave has covered from the transducer, v is the sound velocity, and the zero-temperature intensities I refer to the respective temperature scans. In Fig. 1, therefore, log (I/I )"!1 corresponds to a mean free path C equal to z"2.0 mm. For L phonons propagating along [0 0 1], we have v"4.202$0.010 km/s [13]. We have carried out a recursive multiparameter least-squares fit to all 17 temperature scans with a, b, A, and the I as adjustable parameters. Note that the I pertaining to a particular z, but various u, can be interrelated in the zero-temperature limit. A good fit was obtained, provided the data are limited to temperatures below about 53 K (cf. Fig. 1). The output values for the coefficients are a"1.84$ 0.20 and b"2.81$0.20. We furthermore find A"(4.0$0.5);10\ s?\K\@. The experimental a and b are indeed in conformity with Herring’s predictions, while their sum is close to five. We finally consider the high-temperature regime. According to Herring [1], Eq. (1) is valid only well below the Debye temperature, the attenuation becoming linear according to q\"Bu?¹,
(2)
when most phonon modes are thermally excited. In Fig. 1, it is indeed observed that the temperature dependence changes over from cubic to linear above about 53 K. A fit of Eq. (2), adjusting a, B, and the I to all data from 53 to 100 K, yielded a"1.87$0.20, to be compared with Herring’s
In elastically isotropic media, the relevant anharmonic processes are L#TPL and L#LP L. The associated decay rates vary as q\Ju¹ [4] and q\Ju¹ [11,12]. These processes are at variance with the present experiment.
715
a"2. We furthermore find B"(2.6$0.5);10\ s?\K\. The crossover from the cubic to the linear dependence thus occurs at ¹ +53 K, very much independent of the frequency (Fig. 1). This crossover temperature seems reasonable in relation to the Debye temperature of TeO , which is estimated at roughly 200 K. In summary, we have measured the attenuation of longitudinal phonons propagating along the [0 0 1] axis in a single crystal of TeO as a function of the frequency and the temperature, to find confirmation of the theory for phonon attenuation in anisotropic crystals by Herring. The work was supported by the Netherlands foundations FOM and NWO. References [1] C. Herring, Phys. Rev. 95 (1954) 954. [2] J.W. Tucker, V.W. Rampton, Microwave Ultrasonics in Solid State Physics, North-Holland, Amsterdam, 1972. [3] J.W. Tucker, V.W. Rampton, in: W. Eisenmenger, A.A. Kaplyanskii (Eds.), Nonequilibrium Phonons in Nonmetallic Crystals, North-Holland, Amsterdam, 1986. [4] G.P. Srivastava, The Physics of Phonons, Adam Hilger, Bristol, 1990. [5] J. Callaway, Phys. Rev. 113 (1959) 1046. [6] M.G. Holland, Phys. Rev. 132 (1963) 2461. [7] Y.-J. Han, P.G. Klemens, Phys. Rev. B 48 (1993) 6033. [8] M. Asen-Palmer, K. Bartkowski, E. Gmelin, M. Cardona, A.P. Zhernov, A.V. Inyushkin, A. Taldenkov, V.I. Ozhogin, K.M. Itoh, E.E. Haller, Phys. Rev. B 56 (1997) 9431. [9] E.P.N. Damen, A.F.M. Arts, H.W. de Wijn, Phys. Rev. Lett. 74 (1995) 4249. [10] J. Leciejewicz, Z. Kristallogr 116 (1961) 345. [11] I.S. Ciccarello, K. Dransfeld, Phys. Rev. 134 (1964) A1517. [12] H.J. Maris, Philos. Mag. 9 (1964) 901. [13] Y. Ohmachi, N. Uchida, J. Appl. Phys. 41 (1970) 2307.