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Phonon scattering in diamond films
Physica B 263—264 (1999) 745—748
Phonon scattering in diamond films V.B. Efimov*, L.P. Mezhov-Deglin Institute of Solid State Physics RAS, Chernogolovka, Moscow distr. 142432, Russia
Abstract We have studied the thermal conductivity i of free standing diamond films with different quality and sizes of the crystallines in bulk. The measurements were performed in the temperature range 10—350 K by steady-state heat flux technique. We have estimated the value and temperature dependence of the phonon mean free path l using the simple gas kinetic equation for thermal conductivity: i+1/3 l C v. The values of heat capacity C and mean sound velocity N v are known from literature for massive crystals. At high (room) temperatures the mean free path of phonons is limited by the phonon—phonon Umklapp interaction and phonon scattering on point defect in bulk. Both these processes lead to increasing of l (¹) with cooling the sample but they have different temperature dependencies. So from l (¹) dependence one can judge the quality (purity) of the sample. At low temperatures the phonon-boundary scattering dominates and l is limited by the crystalline sizes. The mean free path of phonons in the best of samples has reached the value of a few microns at temperatures 20—70 K, which agrees with the middle sizes of crystallites estimated from X-ray measurements and from microphotos. At lower temperatures the value of l is increasing again. It can be attributed to the phonon penetration through the boundaries between crystallines. The temperature dependence of j in this process follows l(¹)&¹\\. 1999 Elsevier Science B.V. All rights reserved. Keywords: Phonon scattering; Diamond film; Thermal conductivity
1. Introduction Investigations of the thermal conductivity of diamond films can give us very important information about the mechanisms of heat transport and scattering processes in bulk. The knowledge of absolute values of the thermal conductivity is necessary for practical applications of diamond films as a high heat transmission substrate.
* Corresponding author.
Diamond is a very convenient model system with very high Debye temperature (h+2200 K). The room temperatures are low for diamond in comparison with other materials. The phonon wave length is of the order of ten lattice constants at room temperatures and of the order of 1000 A at helium temperatures, so at low temperatures the scattering of phonons by point defects is negligibly small. We have studied two diamond films of different quality. The diamond films were obtained from Dr. V. Ralchenko of General Physics Institute RAS
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 8 0 - 0
746
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 745—748
in a microwave plasma enhanced chemical vapor deposition (MW-PECVD) reactor (8 kW power, 2.45 GHz frequency) using CH —O —H reaction gas mixtures as described elsewhere [1]. The polycrystalline diamond was grown on mirror-polished Si substrates of 57 mm diameter, which were then chemically removed to produce a free-standing film. The samples were further cut with a Nd : YAG laser to pieces for characterisation. Oxygen concentration and the substrate temperature were varied to establish optimum parameters to grow diamond films of improved quality. Both the films have showed well-faceted crystallites, however the sample A looked black, while sample B was translucent.
2. Experiments and results The thermal conductivity was measured in the temperature range 8—350 K by the steady-state heat flux technique. One end of the sample was mounted on a copper cooler. A wire resistor was attached to the other end of the free standing diamond film. The two junctions of a 25 lm chromel— constantan differential thermocouple were glued to the sample. The separation between the two junctions was &5—8 mm. The heat loss due to the heater leads and the thermocouple is negligible. The radiation losses at temperatures (250 K are insignificant; however, they can cause small increases in the estimated values of thermal conductivity at higher temperatures. The calculated values of radiation losses allowed us to estimate the absolute value of the thermal conductivity and were taken into account. The temperature dependence of the thermal conductivity of the free standing diamond films and the behaviour of k(¹) for a diamond single crystal reproduced from the literature [2] are presented in Fig. 1. Thermal conductivity of diamond single crystals rises upon cooling down to 80 K, attains a maximum and then decreases. This behaviour can be described by Debye—Peiers phonon model. The thermal conductivity of the film CA monotonically increases upon lowering the temperature. The thermal conductivity of the second sample CB is nearly two times higher than k(¹) for film CA at room
Fig. 1. The temperature dependence of the thermal conductivity of diamond fimls and single crystals.
temperature and had a good visible hump in the temperature region 200—300 K. The reasons for the strong difference in absolute values of the thermal conductivity and the temperature dependence k(¹) of both the films CA, CB and single crystals can indicate to the strong difference in phonon processes in the bulk of the samples. For understanding the main scattering mechanism for phonons we have estimated the phonon mean free path l by using the simple gas kinetic equation for thermal conductivity: k(¹)"1/3 C (¹)vl (¹), where C is the specific heat and l the sound velocity. The absolute values for C were obtained from the experimental work [3]. For we used the average value v "13 430 m/s [4]. The results of these calculations are shown in Fig. 2. For single crystals the phonon mean free path calculated from results of Ref. [2] increases with decreasing ¹ down to temperatures 5—20 K and becomes temperature independent below 5 K. This behaviour can be explained as a reduction of phonon—phonon and phonon—defects interactions. At
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 745—748
747
of the films. In film CA one can see small crystallites with a size d&100 nm inside the bigger crystals (d&1—10 lm). In film CB we cannot find similar small size crystallites. Characteristic dimensions of crystals in this film were of the order 1—10 lm. The micro Raman spectra of the diamond films revealed a significant presence of amorphous carbon in film CA, while the only line in a spectrum of high quality sample CB was narrow, about 3.0 cm\, diamond peak was placed at 1332 cm\ frequency (data of E. Obraztsova, [5]). Our investigations of X-ray diffraction peaks have shown the existence crystallines with dimensions of order 70 nm in film CA and 0.65 lm in film CB.
3. Discussion
Fig. 2. The temperature dependence of phonon mean free path for diamond films and single crystals.
the lowest temperatures the phonon mean free path is limited by the size of the crystal (&mm). The temperature dependence l (¹) for films shows another behaviour. At room temperatures l in both films is close to the value l in single crystals. Sample C A. In the temperature region between 120 and 250 K l is nearly constant, l &40 nm. At lower temperatures the phonon mean free path increases proportional to &¹\, and reaches the value of &2 lm at ¹&10 K. Sample CB. At room temperatures the phonon mean free path is close to l (¹) in single crystals and has the same temperature dependence. At ¹&70300 K l (¹) increase is weaker than in single crys tals and follows &¹\\. In the temperature region 20—70 K the phonon mean free path is nearly temperature independent (l &1.5 lm). At lower ¹ l rises like &¹\. We tried to find the reasons for these differences in the behaviour of phonon scattering in our diamond films. The electron microscope photos of both films have showed the difference in crystalline structure
At high temperatures (room temperature) the thermal conductivity of the best diamond films (like in a perfect single) is limited by phonon—phonon scattering (Umklapp processes). The thermal conductivity of the worst films is reduced by phonon— defects scattering. The phonon mean free path l for both these mechanisms increases with cooling of the sample. The increase of the phonon mean free path is limited by phonon scattering on crystalline boundaries. The characteristic sizes of the crystallites in a polycrystalline film CA was found to be of the order 100 nm (microphoto) and 70 nm (X-ray diffraction). The estimations of the phonon mean free path give the size of crystallites near 40 nm. For film CB the same estimations give the sizes 1—10 lm (photo), 0.65 lm (X-ray) and l &1.5 lm. For single crystals the phonon mean free path is limited by the size of a crystal and its quality and can reach the value of a few mm. It predetermines the very high thermal conductivity of diamond crystals at low (liquid nitrogen and helium) temperatures. The thermal phonon length j is sufficiently big at low temperatures. For example, at ¹"10 K, j&70 nm. It means that phonons can penetrate through crystalline boundaries in polycrystals like sound waves. It can result in the increase of the
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V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 745—748
effective mean free path of phonons. Some estimations of temperature dependence of thermal conductivity were made in Ref. [6]. The accurate calculations of temperature dependence of the phonon mean free path are difficult and demand definite theoretical models for a boundary structure. Our experiments show that the phonon mean free path in the case of phonon-boundary scattering for long wave thermal phonons depends upon the sample quality and at low temperatures increases like l (¹)&¹\!¹\.
References [1] V.G. Ralchenko, A.A. Smolin, V.I. Konov et al., Diamond and Related Materials 6 (1997) 417. [2] G.E.Childs, L.J.Ericks, R.L.Powell, Thermal conductivity of solids at room temperature and below of diamond single crystals, National Bureau of Standards, Boulder, CO, 1973. [3] Y.S. Touloukian, E.H.Buyco (Eds), Specific Heat of Nonmetallic Solids, Thermophysics Data Series, IFI/Plenum, New York, 1970. [4] McScimin, Bond. Phys. Rev. 105 (1957) 116. [5] V.B. Efimov et al., Mol. Matt., in press. [6] S.I. Ivanov et al., ZhETPh (Rus) 102 (1992) 600.
Phonon scattering in diamond films V.B. Efimov*, L.P. Mezhov-Deglin Institute of Solid State Physics RAS, Chernogolovka, Moscow distr. 142432, Russia
Abstract We have studied the thermal conductivity i of free standing diamond films with different quality and sizes of the crystallines in bulk. The measurements were performed in the temperature range 10—350 K by steady-state heat flux technique. We have estimated the value and temperature dependence of the phonon mean free path l using the simple gas kinetic equation for thermal conductivity: i+1/3 l C v. The values of heat capacity C and mean sound velocity N v are known from literature for massive crystals. At high (room) temperatures the mean free path of phonons is limited by the phonon—phonon Umklapp interaction and phonon scattering on point defect in bulk. Both these processes lead to increasing of l (¹) with cooling the sample but they have different temperature dependencies. So from l (¹) dependence one can judge the quality (purity) of the sample. At low temperatures the phonon-boundary scattering dominates and l is limited by the crystalline sizes. The mean free path of phonons in the best of samples has reached the value of a few microns at temperatures 20—70 K, which agrees with the middle sizes of crystallites estimated from X-ray measurements and from microphotos. At lower temperatures the value of l is increasing again. It can be attributed to the phonon penetration through the boundaries between crystallines. The temperature dependence of j in this process follows l(¹)&¹\\. 1999 Elsevier Science B.V. All rights reserved. Keywords: Phonon scattering; Diamond film; Thermal conductivity
1. Introduction Investigations of the thermal conductivity of diamond films can give us very important information about the mechanisms of heat transport and scattering processes in bulk. The knowledge of absolute values of the thermal conductivity is necessary for practical applications of diamond films as a high heat transmission substrate.
* Corresponding author.
Diamond is a very convenient model system with very high Debye temperature (h+2200 K). The room temperatures are low for diamond in comparison with other materials. The phonon wave length is of the order of ten lattice constants at room temperatures and of the order of 1000 A at helium temperatures, so at low temperatures the scattering of phonons by point defects is negligibly small. We have studied two diamond films of different quality. The diamond films were obtained from Dr. V. Ralchenko of General Physics Institute RAS
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 8 0 - 0
746
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 745—748
in a microwave plasma enhanced chemical vapor deposition (MW-PECVD) reactor (8 kW power, 2.45 GHz frequency) using CH —O —H reaction gas mixtures as described elsewhere [1]. The polycrystalline diamond was grown on mirror-polished Si substrates of 57 mm diameter, which were then chemically removed to produce a free-standing film. The samples were further cut with a Nd : YAG laser to pieces for characterisation. Oxygen concentration and the substrate temperature were varied to establish optimum parameters to grow diamond films of improved quality. Both the films have showed well-faceted crystallites, however the sample A looked black, while sample B was translucent.
2. Experiments and results The thermal conductivity was measured in the temperature range 8—350 K by the steady-state heat flux technique. One end of the sample was mounted on a copper cooler. A wire resistor was attached to the other end of the free standing diamond film. The two junctions of a 25 lm chromel— constantan differential thermocouple were glued to the sample. The separation between the two junctions was &5—8 mm. The heat loss due to the heater leads and the thermocouple is negligible. The radiation losses at temperatures (250 K are insignificant; however, they can cause small increases in the estimated values of thermal conductivity at higher temperatures. The calculated values of radiation losses allowed us to estimate the absolute value of the thermal conductivity and were taken into account. The temperature dependence of the thermal conductivity of the free standing diamond films and the behaviour of k(¹) for a diamond single crystal reproduced from the literature [2] are presented in Fig. 1. Thermal conductivity of diamond single crystals rises upon cooling down to 80 K, attains a maximum and then decreases. This behaviour can be described by Debye—Peiers phonon model. The thermal conductivity of the film CA monotonically increases upon lowering the temperature. The thermal conductivity of the second sample CB is nearly two times higher than k(¹) for film CA at room
Fig. 1. The temperature dependence of the thermal conductivity of diamond fimls and single crystals.
temperature and had a good visible hump in the temperature region 200—300 K. The reasons for the strong difference in absolute values of the thermal conductivity and the temperature dependence k(¹) of both the films CA, CB and single crystals can indicate to the strong difference in phonon processes in the bulk of the samples. For understanding the main scattering mechanism for phonons we have estimated the phonon mean free path l by using the simple gas kinetic equation for thermal conductivity: k(¹)"1/3 C (¹)vl (¹), where C is the specific heat and l the sound velocity. The absolute values for C were obtained from the experimental work [3]. For we used the average value v "13 430 m/s [4]. The results of these calculations are shown in Fig. 2. For single crystals the phonon mean free path calculated from results of Ref. [2] increases with decreasing ¹ down to temperatures 5—20 K and becomes temperature independent below 5 K. This behaviour can be explained as a reduction of phonon—phonon and phonon—defects interactions. At
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 745—748
747
of the films. In film CA one can see small crystallites with a size d&100 nm inside the bigger crystals (d&1—10 lm). In film CB we cannot find similar small size crystallites. Characteristic dimensions of crystals in this film were of the order 1—10 lm. The micro Raman spectra of the diamond films revealed a significant presence of amorphous carbon in film CA, while the only line in a spectrum of high quality sample CB was narrow, about 3.0 cm\, diamond peak was placed at 1332 cm\ frequency (data of E. Obraztsova, [5]). Our investigations of X-ray diffraction peaks have shown the existence crystallines with dimensions of order 70 nm in film CA and 0.65 lm in film CB.
3. Discussion
Fig. 2. The temperature dependence of phonon mean free path for diamond films and single crystals.
the lowest temperatures the phonon mean free path is limited by the size of the crystal (&mm). The temperature dependence l (¹) for films shows another behaviour. At room temperatures l in both films is close to the value l in single crystals. Sample C A. In the temperature region between 120 and 250 K l is nearly constant, l &40 nm. At lower temperatures the phonon mean free path increases proportional to &¹\, and reaches the value of &2 lm at ¹&10 K. Sample CB. At room temperatures the phonon mean free path is close to l (¹) in single crystals and has the same temperature dependence. At ¹&70300 K l (¹) increase is weaker than in single crys tals and follows &¹\\. In the temperature region 20—70 K the phonon mean free path is nearly temperature independent (l &1.5 lm). At lower ¹ l rises like &¹\. We tried to find the reasons for these differences in the behaviour of phonon scattering in our diamond films. The electron microscope photos of both films have showed the difference in crystalline structure
At high temperatures (room temperature) the thermal conductivity of the best diamond films (like in a perfect single) is limited by phonon—phonon scattering (Umklapp processes). The thermal conductivity of the worst films is reduced by phonon— defects scattering. The phonon mean free path l for both these mechanisms increases with cooling of the sample. The increase of the phonon mean free path is limited by phonon scattering on crystalline boundaries. The characteristic sizes of the crystallites in a polycrystalline film CA was found to be of the order 100 nm (microphoto) and 70 nm (X-ray diffraction). The estimations of the phonon mean free path give the size of crystallites near 40 nm. For film CB the same estimations give the sizes 1—10 lm (photo), 0.65 lm (X-ray) and l &1.5 lm. For single crystals the phonon mean free path is limited by the size of a crystal and its quality and can reach the value of a few mm. It predetermines the very high thermal conductivity of diamond crystals at low (liquid nitrogen and helium) temperatures. The thermal phonon length j is sufficiently big at low temperatures. For example, at ¹"10 K, j&70 nm. It means that phonons can penetrate through crystalline boundaries in polycrystals like sound waves. It can result in the increase of the
748
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 745—748
effective mean free path of phonons. Some estimations of temperature dependence of thermal conductivity were made in Ref. [6]. The accurate calculations of temperature dependence of the phonon mean free path are difficult and demand definite theoretical models for a boundary structure. Our experiments show that the phonon mean free path in the case of phonon-boundary scattering for long wave thermal phonons depends upon the sample quality and at low temperatures increases like l (¹)&¹\!¹\.
References [1] V.G. Ralchenko, A.A. Smolin, V.I. Konov et al., Diamond and Related Materials 6 (1997) 417. [2] G.E.Childs, L.J.Ericks, R.L.Powell, Thermal conductivity of solids at room temperature and below of diamond single crystals, National Bureau of Standards, Boulder, CO, 1973. [3] Y.S. Touloukian, E.H.Buyco (Eds), Specific Heat of Nonmetallic Solids, Thermophysics Data Series, IFI/Plenum, New York, 1970. [4] McScimin, Bond. Phys. Rev. 105 (1957) 116. [5] V.B. Efimov et al., Mol. Matt., in press. [6] S.I. Ivanov et al., ZhETPh (Rus) 102 (1992) 600.