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Quasidiffusive propagation of phonons in vitreous silica
ELSEVIER
Physica B 219&220 (1996) 751 753
Quasidiffusive propagation of phonons in vitreous silica Hiroshi Ikari Faculty of Education, Shizuoka University, 836 Ohya, Shizuoka 422, Japan
Abstract
The exponential decay of phonon intensity observed in both experiments and simulations for the quasidiffusive propagation regime reflects the elastic and inelastic scattering process of high frequency phonons. In search of the predicted quasidiffusive behaviour, we have removed the contact with liquid helium in the heat pulse measurement of vitreous silica, and greatly reduced the excitation density. The observed exponential decay qualitatively agrees with the Monte Carlo simulation with the simple model proposed by Maris (1990). The resulting strength of the Rayleigh scattering is approximately equal to that needed to fit the thermal conductivity data in amorphous SiO2.
The thermal conductivity plateau of glasses in the intermediate temperature range has been a subject of considerable interest, but as yet no completely satisfactory explanation of the important physical processes has been found [1 4]. There have been many attempts to explain the plateau in terms of propagating phonons including elastic scattering of phonons from Rayleigh scattering due to density fluctuations [5,6], or nonpropagating vibrational modes [-3,4]. The estimated strength B of the Rayleigh scattering (BE 4) ranges from 104 to 106 which depends on the model [2-6], where E is the phonon energy. In this report, we are intended to estimate experimentally the strength of the Rayleigh scattering in the vitreous silica, which will be useful for resolving the plateau problem. Elastic scattering and anharmonicity characterize the phonon transport at a time much later than the ballistic time of flight. The phonon propagation with both elastic and inelastic scatterings is quasidiffusive [7, 8]. Recently, the exponential decay for late arrival times observed experimentally is well reproduced by Monte Carlo simulations, and both elastic and inelastic scattering rates in Si have been quantitatively established [-9, 10]. Therefore, by comparing the observed exponential decay for late arrival time with the Monte Carlo simulation, we will be able to estimate experimentally the strength
of the Rayleigh scattering responsible for the thermal conductivity plateau. This is the main purpose of the present work. Time-of-flight measurements of vitreous silica at 2.3 K have been performed in the low exitation energy where the contact with liquid helium on the excitation surface has been removed [9, 11]. Fig. 1 shows the experimental configuration used to isolate the excitation surface from the liquid He bath. The excitation density is greatly reduced in order to avoid formation of a phonon hot spot mediated by phonon-phonon interactions [,12,13]. Disk-shaped samples of Suprasil W of 300 ~tm thickness were used. Heat pulses were generated with a 50 ~ constantan film (1.8 x 1.8 mm 2) on one face, and detected at the other face with a similarly evaporated thin-film indium-tin alloy bolometer (1.2× 1.2mm2). The input pulse width is 60 ns throughout this work. Removing the contact to the liquid helium is critical. Otherwise, high frequency phonons scattering diffusively near the excitation surface are lost into the bath when they encounter the surface [-I 1]. Under these conditions a broad temporal distribution of phonons has been observed. The peak arrival times in the diffusive profiles are constant below the power of 0.1 W mm 2, and are gradually retarded when the excitation energy is increasing over the power of 0.2 W mm - 2. This retardation effect of
0921-4526/96/$15.00 (~ 1996 Elsevier Science B.V. All rights reserved SSD! 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 8 7 4 - 8
752
I-~ Ikari/Physica B 219&220 (1996) 751-753 I
I
I
I
I
"-'=~c10° i0-~
Sample
I
I
Vacuum \ Heater
I
I
c" ~
I
.-- 100 C i;
~'
Tou=2OOps
210
o~
m (-
10-2 0 I
I
I
I
4- Simulotion
gl C
Fig. 1, The experimental configuration used to isolate the excitation surface from the liquid He bath at 2.3 K. The thickness of the sample of vitreous silica is 300 ~tm.
r-
Tau = 160 ps
I,..
Detector\
.6
:);'
.6
'He
I
40 80 120 160 Micro Seconds
Fig. 2. Intensity of the phonon signal versus time in vitreous silica of 300 ~tm thickness for log plots. The tail of the pulse is approximately exponential with the time constant shown. The power density is greatly reduced at 0.07 W mm - 2
the peak arrival time is considered to be due to formation of the hot spot in the excitation surface at high excitation powers [11, 12]. In this work, we have measured the heat pulse profiles of vitreous silica at 0.07 W m m - 2. Fig. 2 shows the distribution of phonons traversing a 300 Mm thick vitreous silica at 0.07 W m m - ~ . The experimental heat pulse in Fig. 2 displays a long tail which decays approximately exponentially with a decay constant of ~ = 200 Ms. The applicability of a quasidiffusive model is seen by comparing the experimental heat pulses with the Monte Carlo simulation in Fig. 3. This calculation of the heat pulse
I
102 0
I
I
I
I
40 80 120 160 Micro Seconds
Fig. 3. Monte Carlo simulation of the phonon signal versus time together with the experimental data. The simulated late time decay constant is about 1601.ts and is qualitatively consistent with the experimental decay constant in Fig. 2.
resulting from injecting a nonequilibrium p h o n o n distribution assumes the spontaneous p h o n o n decay for vitreous silica calculated analytically from nonlinear elasticity theory [14]. A simplified one-branch model where p h o n o n polarizations are approximated to have the same isotropic spectrum with Debye velocity vo is employed. In the low frequency range we can write the decay rate for the elastic scattering [7] 1/TR = B E 4 with B = 6.2 x 106 s - 1 K - 4 [2] where E is the energy in degrees Kelvin. For anharmonic decay let the decay rate averaged over p h o n o n polarizations and k directions be I/TA = A E 5 with A = 1.0 x 10 3 s - 1 K - 5 [14]. The initial energy Ei,it assigned for each p h o n o n is the same and the relative population of the initial phonons of each mode is specified to be proportional to the density of states. As the strength A of the inelastic scattering ( A E 5) at helium temperature, we tentatively adopted the value three times larger than that at room temperature, because it has been predicted that the calculated inelastic decay rates at liquid helium temperature are larger than those at room temperature by a factors of 2-3 in case of Ge and Si [14]. On the other hand, several different values of the strength B of Rayleigh scattering have been reported and these determined so as to fit the thermal conductivity data [2-6]. The first model [4] gives B = 1.4 x 104. The second [2, 6] gives B = 6-7 x 105, where the elastic scattering is assumed to be due to the Rayleigh scattering, and we used this value of B in the simulation. The third [3] gives B = 2 x 10 6. The initial energy Einit assigned for each p h o n o n is the same and 50 K in our simulation. The detected p h o n o n energy in the simulation is about 15 K on average, which
H. lkari/Physica B 219&220 (1996) 751 753
is consistent with the result predicted in the quasidifl'usion theory. Fig. 3 shows a preliminary result. Our simulation requires very long calculation time. The simulated decay time is about 160 its, which is qualitatively consistent with our experimental result. Our result suggests that the exponential decay for late arrival time observed experimentally in the vitreous silica is well reproduced by the simplified one-branch model simulation. The resulting strength of the Rayleigh scattering is approximately equal to that needed to fit the thermal conductivity data in amorphous SiO2, References [l] C.C. Yu and A.J. Leggett, Comments Condensed Matter Phys. 14 (1988) 231. [2] J.E. Graebner, B. Golding and LC. Allen, Phys. Rev. B 34 (1986) 5696.
753
[3] C.C, Yu and J.J. Freeman, Phys. Rev. B 36 (1987) 7620. [4] E.R. Grannan, M. Randeria and J.P. Sethna, Phys. Rev. B 41 (1990) 7799. [5] D.P. Jones, N. Thomas and W.A. Phillips, Philos. Mag. B 32 (1978) 271. [6] J. Jackle, in: Proc. 4th Int. Conf. on Physics of Noncrystalline Solids, Clanthal-Zellerfeld, ed. G.H. Frichat (Trans Tech Publications, 1976). [7] H.J. Maris, Phys. Rev. B 41 (1990) 9736. [8] W.E. Bron, Y.B. Levinson and J.M. O'Conner, Phys. Rev. Lett. 49 (1982) 209. [9] M.E. Msall, S. Tamura, S.E. Esipov and J.P. Wolfe, Phys. Rev. Lett. 70 (1993) 3463. [10] S. Tamura, Phys. Rev. B 48 (1993) 13502. [11] J.A. Shields, M.E. Msall, M.S. Carrol and J.P. Wolfe, Phys. Rev. B 47 (1993) 12510. [12] J.A. Shields and J.P. Wolfe, Z. Phys. B 75 (1989) 11. [13] Y.B. Levinson, in: Phonons 89, eds. S. Hunklinger, W. Ludwig and G. Weiss (World Scientific, Singapore, 1990). [14] S. Tamura, Phys, Rev. B 31 (1985) 2574.
Physica B 219&220 (1996) 751 753
Quasidiffusive propagation of phonons in vitreous silica Hiroshi Ikari Faculty of Education, Shizuoka University, 836 Ohya, Shizuoka 422, Japan
Abstract
The exponential decay of phonon intensity observed in both experiments and simulations for the quasidiffusive propagation regime reflects the elastic and inelastic scattering process of high frequency phonons. In search of the predicted quasidiffusive behaviour, we have removed the contact with liquid helium in the heat pulse measurement of vitreous silica, and greatly reduced the excitation density. The observed exponential decay qualitatively agrees with the Monte Carlo simulation with the simple model proposed by Maris (1990). The resulting strength of the Rayleigh scattering is approximately equal to that needed to fit the thermal conductivity data in amorphous SiO2.
The thermal conductivity plateau of glasses in the intermediate temperature range has been a subject of considerable interest, but as yet no completely satisfactory explanation of the important physical processes has been found [1 4]. There have been many attempts to explain the plateau in terms of propagating phonons including elastic scattering of phonons from Rayleigh scattering due to density fluctuations [5,6], or nonpropagating vibrational modes [-3,4]. The estimated strength B of the Rayleigh scattering (BE 4) ranges from 104 to 106 which depends on the model [2-6], where E is the phonon energy. In this report, we are intended to estimate experimentally the strength of the Rayleigh scattering in the vitreous silica, which will be useful for resolving the plateau problem. Elastic scattering and anharmonicity characterize the phonon transport at a time much later than the ballistic time of flight. The phonon propagation with both elastic and inelastic scatterings is quasidiffusive [7, 8]. Recently, the exponential decay for late arrival times observed experimentally is well reproduced by Monte Carlo simulations, and both elastic and inelastic scattering rates in Si have been quantitatively established [-9, 10]. Therefore, by comparing the observed exponential decay for late arrival time with the Monte Carlo simulation, we will be able to estimate experimentally the strength
of the Rayleigh scattering responsible for the thermal conductivity plateau. This is the main purpose of the present work. Time-of-flight measurements of vitreous silica at 2.3 K have been performed in the low exitation energy where the contact with liquid helium on the excitation surface has been removed [9, 11]. Fig. 1 shows the experimental configuration used to isolate the excitation surface from the liquid He bath. The excitation density is greatly reduced in order to avoid formation of a phonon hot spot mediated by phonon-phonon interactions [,12,13]. Disk-shaped samples of Suprasil W of 300 ~tm thickness were used. Heat pulses were generated with a 50 ~ constantan film (1.8 x 1.8 mm 2) on one face, and detected at the other face with a similarly evaporated thin-film indium-tin alloy bolometer (1.2× 1.2mm2). The input pulse width is 60 ns throughout this work. Removing the contact to the liquid helium is critical. Otherwise, high frequency phonons scattering diffusively near the excitation surface are lost into the bath when they encounter the surface [-I 1]. Under these conditions a broad temporal distribution of phonons has been observed. The peak arrival times in the diffusive profiles are constant below the power of 0.1 W mm 2, and are gradually retarded when the excitation energy is increasing over the power of 0.2 W mm - 2. This retardation effect of
0921-4526/96/$15.00 (~ 1996 Elsevier Science B.V. All rights reserved SSD! 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 8 7 4 - 8
752
I-~ Ikari/Physica B 219&220 (1996) 751-753 I
I
I
I
I
"-'=~c10° i0-~
Sample
I
I
Vacuum \ Heater
I
I
c" ~
I
.-- 100 C i;
~'
Tou=2OOps
210
o~
m (-
10-2 0 I
I
I
I
4- Simulotion
gl C
Fig. 1, The experimental configuration used to isolate the excitation surface from the liquid He bath at 2.3 K. The thickness of the sample of vitreous silica is 300 ~tm.
r-
Tau = 160 ps
I,..
Detector\
.6
:);'
.6
'He
I
40 80 120 160 Micro Seconds
Fig. 2. Intensity of the phonon signal versus time in vitreous silica of 300 ~tm thickness for log plots. The tail of the pulse is approximately exponential with the time constant shown. The power density is greatly reduced at 0.07 W mm - 2
the peak arrival time is considered to be due to formation of the hot spot in the excitation surface at high excitation powers [11, 12]. In this work, we have measured the heat pulse profiles of vitreous silica at 0.07 W m m - 2. Fig. 2 shows the distribution of phonons traversing a 300 Mm thick vitreous silica at 0.07 W m m - ~ . The experimental heat pulse in Fig. 2 displays a long tail which decays approximately exponentially with a decay constant of ~ = 200 Ms. The applicability of a quasidiffusive model is seen by comparing the experimental heat pulses with the Monte Carlo simulation in Fig. 3. This calculation of the heat pulse
I
102 0
I
I
I
I
40 80 120 160 Micro Seconds
Fig. 3. Monte Carlo simulation of the phonon signal versus time together with the experimental data. The simulated late time decay constant is about 1601.ts and is qualitatively consistent with the experimental decay constant in Fig. 2.
resulting from injecting a nonequilibrium p h o n o n distribution assumes the spontaneous p h o n o n decay for vitreous silica calculated analytically from nonlinear elasticity theory [14]. A simplified one-branch model where p h o n o n polarizations are approximated to have the same isotropic spectrum with Debye velocity vo is employed. In the low frequency range we can write the decay rate for the elastic scattering [7] 1/TR = B E 4 with B = 6.2 x 106 s - 1 K - 4 [2] where E is the energy in degrees Kelvin. For anharmonic decay let the decay rate averaged over p h o n o n polarizations and k directions be I/TA = A E 5 with A = 1.0 x 10 3 s - 1 K - 5 [14]. The initial energy Ei,it assigned for each p h o n o n is the same and the relative population of the initial phonons of each mode is specified to be proportional to the density of states. As the strength A of the inelastic scattering ( A E 5) at helium temperature, we tentatively adopted the value three times larger than that at room temperature, because it has been predicted that the calculated inelastic decay rates at liquid helium temperature are larger than those at room temperature by a factors of 2-3 in case of Ge and Si [14]. On the other hand, several different values of the strength B of Rayleigh scattering have been reported and these determined so as to fit the thermal conductivity data [2-6]. The first model [4] gives B = 1.4 x 104. The second [2, 6] gives B = 6-7 x 105, where the elastic scattering is assumed to be due to the Rayleigh scattering, and we used this value of B in the simulation. The third [3] gives B = 2 x 10 6. The initial energy Einit assigned for each p h o n o n is the same and 50 K in our simulation. The detected p h o n o n energy in the simulation is about 15 K on average, which
H. lkari/Physica B 219&220 (1996) 751 753
is consistent with the result predicted in the quasidifl'usion theory. Fig. 3 shows a preliminary result. Our simulation requires very long calculation time. The simulated decay time is about 160 its, which is qualitatively consistent with our experimental result. Our result suggests that the exponential decay for late arrival time observed experimentally in the vitreous silica is well reproduced by the simplified one-branch model simulation. The resulting strength of the Rayleigh scattering is approximately equal to that needed to fit the thermal conductivity data in amorphous SiO2, References [l] C.C. Yu and A.J. Leggett, Comments Condensed Matter Phys. 14 (1988) 231. [2] J.E. Graebner, B. Golding and LC. Allen, Phys. Rev. B 34 (1986) 5696.
753
[3] C.C, Yu and J.J. Freeman, Phys. Rev. B 36 (1987) 7620. [4] E.R. Grannan, M. Randeria and J.P. Sethna, Phys. Rev. B 41 (1990) 7799. [5] D.P. Jones, N. Thomas and W.A. Phillips, Philos. Mag. B 32 (1978) 271. [6] J. Jackle, in: Proc. 4th Int. Conf. on Physics of Noncrystalline Solids, Clanthal-Zellerfeld, ed. G.H. Frichat (Trans Tech Publications, 1976). [7] H.J. Maris, Phys. Rev. B 41 (1990) 9736. [8] W.E. Bron, Y.B. Levinson and J.M. O'Conner, Phys. Rev. Lett. 49 (1982) 209. [9] M.E. Msall, S. Tamura, S.E. Esipov and J.P. Wolfe, Phys. Rev. Lett. 70 (1993) 3463. [10] S. Tamura, Phys. Rev. B 48 (1993) 13502. [11] J.A. Shields, M.E. Msall, M.S. Carrol and J.P. Wolfe, Phys. Rev. B 47 (1993) 12510. [12] J.A. Shields and J.P. Wolfe, Z. Phys. B 75 (1989) 11. [13] Y.B. Levinson, in: Phonons 89, eds. S. Hunklinger, W. Ludwig and G. Weiss (World Scientific, Singapore, 1990). [14] S. Tamura, Phys, Rev. B 31 (1985) 2574.