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Thermophysical properties of thin films on substrate
Materials Science and Engineering A292 (2000) 189 – 193 www.elsevier.com/locate/msea
Thermophysical properties of thin films on substrate Ichiro Hatta a,*, Kazuhiko Fujii a, Sok Won Kim b a
Department of Applied Physics, Graduate School of Engineering, Nagoya Uni6ersity, Nagoya 464 -8603, Japan b Department of Physics, Uni6ersity of Ulsan, Ulsan 680 -749, South Korea
Abstract In the analysis of thermophysical properties in a thin film, it is important to take into account those in the interfacial region between a thin film and a substrate on which the film was grown. In order to analyze them, an ac calorimetric thermal diffusivity measurement is very useful. In this measurement, the thermal diffusivity of the component parallel to the surface of a thin film is obtained as a function of film thickness for a single film and also as a function of layer thickness for a superlattice. In the present paper, an analytical method is proposed and it is applied to the analysis of thermophysical properties in diamond films and in AlAs/GaAs superlattices. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Thermal conductivity; Thermal diffusivity; Nano scale region; Interface; Ac calorimetry
1. Introduction Although a lot of measurements of thermophysical properties have been performed in thin films, there still remain unresolved problems. In the measurement of thermophysical properties in a thin film grown on a substrate, we have to consider those of at least two distinct regions. One is the thermophysical property of an interfacial region between a thin film and a substrate. Usually the interfacial region has much smaller thermal conductivity than that of the other remaining part of a thin film. The other is the thermophysical property of the remaining part that might be close to that of the single crystal. However, it is more or less inferior to that of a perfect single crystal. In fact, the above phenomena have been pointed out in some experiments. Graebner et al. [1] have measured the thermal conductivity of CVD diamond films in the direction parallel to the surface as a function of the thickness using a two-heater heated-bar technique. As the thickness increases, the observed thermal conductivity rises. They have shown a method to analyze the above results and estimated the local thermal conductivity as a function of distance from the bottom that was the interface between the grown diamond film and the substrate. Thus estimated local thermal conductivity * Corresponding author. Fax: +81-52-7893706. E-mail address: [email protected] (I. Hatta).
rises with increasing the distance from the bottom and saturates almost at the thermal conductivity reported for single-crystal Type IIa diamond. On the other hand, using an ac calorimetric technique Yao [2] has measured the thermal diffusivity of AlAs/GaAs superlattices in the direction parallel to superlattice surface at various layer thicknesses of GaAs and AlAs, where the both thicknesses are the same. He has found that, as the layer thickness increases, the observed thermal conductivity rises. He also has pointed out that at the limit of the short-period superlattice the observed thermal conductivity becomes close to that of Al0.5Ga0.5As alloy. We will propose a method to analyze the thermophysical properties of such materials. Based upon the analytical method, it will be able to estimate the thermophysical properties of diamond films and GaAs/ AlAs superlattices, in both of which the thermal diffusivity measurement have performed by an ac calorimetric technique. Finally, we will point out the importance of the thermal diffusivity measurement along the surface of thin films and superlattices.
2. Analytical method In the analysis of thermophysical properties in thin films and superlattices, we will point out that the measurement of thermal diffusivity parallel to the surface
0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S0921-5093(00)00995-3
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gives rise to fruitful evidence. As typical systems, we will consider diamond films with a variety of thicknesses which were originally synthesized on a substrate in the same manner and superlattices with a variety of layer thicknesses (or superlattice constants) also in the same manner. First, let us consider a free-standing film, such as a diamond film, removed from a substrate on which the film has been grown primarily. In such a free-standing film, it is likely that the microscopic structure changes gradually from the bottom surface of the substrate side to the upper surface of the opposite side. In fact, in a diamond film made in such a way, the crystalline size is small near the substrate and becomes larger with growth [1]. One can assume that such a functionally graded material is composed of the superposition of a number of layers in which thermal conductivity successively increases layer by layer from the layer that lies near the substrate to the opposite free end. As shown in Fig. 1, the thermal conductivity, the heat capacity per unit volume and the thickness of the layers are denoted by k0, c0 and d0 for the 0-th layer, k1, c1 and d1 for the first layer, and those up to the n-th layer. The free end of the n-th layer lies at the bottom surface that is the counter part of the substrate. In an ac calorimetric technique, the modulated heat is applied to a part of the surface of the multilayer and the ac temperature waves propagate along the direction parallel to the surface outside the periodically heated region. If n
% kjdj 51,
(1)
j=0
and/or when the ac temperature is detected at a position far from the heated region [3,4], the observed total thermal diffusivity is expressed by a simple relation of Eq. (2). In Eq. (1), kj = (v/2Dj), where v is angular frequency and Dj( = kj/cj) is the thermal diffusivity of the j-th layer.
Fig. 1. Model for a thin film with graded thermophysical properties. The thin film is composed of a number of layers. The 0-th layer lies in a free surface and the n-th layer lies in the other surface that is removed from a substrate. The layer thickness is d0 for 0-th layer, d1 for first layer and so on. The total thickness d is given by the sum of these layers.
n
% kjdj D=
j=0 n
(2)
% cjdj
j=0
It is worth to point out that at room temperature usually the heat capacity of each layer in the above model is almost the same independent from the crystalline quality [5]. This might be the case in diamond films. Heat capacity of per unit volume in diamond is 1.80 J cm − 3 K − 1 for Type I, Type IIa and Type IIb [5]. Therefore, we can assume that c0 : c1 : c2 : …: cn
(3)
We define the total thickness d as d= d0 + d1 + d2 + …+ dn
(4)
Then, one can rewrite Eq. (2) approximately as D= D0 −
Es , d
(5)
where D0 =
k0 , c0
(6)
n
% (k0 − kj)dj Es =
j=1
c0
(7)
In the above analysis, we assumed that the thicknesses of d1 to dn are kept unchanged, when one reduces the thickness of d0. Then, only the portion of the 0-th layer becomes larger and therefore D reaches the saturated value D0. On the other hand, when the thickness d decreases in Eq. (5), the observed thermal diffusivity becomes smaller, but it should terminate at a certain thickness where the thermal diffusivity reaches the minimum value in the material with the same composition. Second, let us consider a superlattice. The situation seems to be quite similar to that of the above single film. In a superlattice, for instance, since atoms diffuse through the interface during the crystal growth, the composition graded region takes place and it results in a disorderd interfacial region. As a result, the thermal diffusivity in the interfacial region becomes very small and finally reaches that of the amorphous state. On the other hand, since the diffusion of atoms does not reach at the far distance from the interface, the thermal conductivity becomes larger. In the ac calorimetric measurement, the similar condition of Eq. (1) is required. For simplicity, we consider a multilayer system composed of the periodical arrangement of a set of layer A, one interface layer i, layer B and another interface layer i as shown in Fig. 2, where the total number of the sets is n. Here, we consider the case that the thicknesses of layer A and layer B are the same thickness d0 and the thickness of the interface is di. The
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191
As seen in Eq. (11), the thickness dependence of the observed thermal diffusivity is similar to Eq. (5). Therefore, when the layer thickness d increases, from Eq. (11) one can obtain the saturated thermal diffusivity. On the other hand, when the layer thickness d decreases, the observed thermal diffusivity becomes small, but terminates at a certain value. This fact results from the concept of the minimum thermal conductivity proposed by Einstein [6,7]. From a point of today’s view, an amorphous state of a material has the smallest thermal conductivity. It is very likely that the interfacial region is in an amorphous state and then, if ki is the thermal conductivity of the amorphous state, we can estimate the thickness of the interfacial region di from Eq. (11). 3. Analysis of data
Fig. 2. Model for a superlattice, which is composed of the periodic arrangement of a material A, an interfacial region i between A and B, a material B, and an interfacial region i between B and A. The layer thicknesses of a material A and B are the same, d0. The thickness of the interfacial region i is di. Then, the layer thickness d is given by the sum of d0 and di.
heat capacities per unit volume of layer A, layer B and interface layer i are cA, cB and ci, respectively, and the thermal conductivities are kA, kB and ki, respectively. Then, the observed thermal diffusivity is given by D= =
kA(nd0)+kB(nd0) + ki(2ndi) cA(nd0)+cB(nd0) +ci(2ndi)
kAd0 + kBd0 + 2kidi cAd0 +cBd0 + 2cidi
(8)
We also assume the following relation [5]: c =cA cB ci.
(9)
We define the layer thickness d as d= d0 +di.
(10)
Then, one can rewrite Eq. (8) approximately as D= DAB −
Em , d
(11)
where DAB = Em =
kA + kB , 2c
(kA − ki)di +(kB −ki)di . 2c
(12) (13)
CVD diamond films made by Sumitomo Electric Co. (Itami, Japan) were used for the ac calorimetric measurement. There are three samples with thickness of 0.16, 0.23 and 0.53 mm. The thicker samples were grown successively under the same condition and then, we assume that in Fig. 1, the thickness of the 0-th layer increases as the sample becomes thicker. In such a situation, the observed thermal diffusivities are fitted to Eq. (5) as D0 and Es are fitting parameters. Then, the following results are obtained: Da[cm2 s − 1]= 12.59 0.4−
0.4890.09 , d[mm]
(14)
Dp[cm2 s − 1]= 12.09 0.4−
0.3790.08 , d[mm]
(15)
where Da and Dp are analyzed from the data on the amplitude decay and the phase shift in the ac temperature, respectively. The fitting curves are shown in Fig. 3. Both the results are consistent as those expected from the principle of the ac calorimetric technique. We reanalyzed the data for AlAs/GaAs superlattices reported by Yao [2] based upon the present analytical method. We assume that as shown in Fig. 2 the superlattices are composed of the periodic arrangement of a set of AlAs layer, one interfacial region, GaAs layer and another interfacial region. The observed thermal diffusivities are fitted to Eq. (11) where DAB and Em are fitting parameters. Then, the following result is obtained: D[cm2 s − 1]= 0.179 0.02−
0.590.1 . d[nm]
(16)
The fitting curve is shown in Fig. 4. Although the fitting curve is drawn down to D=0, it should terminate at a certain value as mentioned in the previous section. The saturated value of 0.17 cm2 s − 1 in Eq. (16) is the arithmetical average of the thermal conductivities of AlAs and GaAs as given in Eq. (12).
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4. Discussion From the analysis for diamond films measured in the present experiment, it was found that the saturated magnitude of the thermal diffusivity agrees with the thermal diffusivity of a Type IIa diamond, which is the highest magnitude in a natural diamond. Therefore, at the free surface of a CVD diamond film the thermal
Fig. 3. Dependence of thermal diffusivities Da and Dp of CVD diamond films on thickness. The data was obtained by the ac calorimetric technique. The thermal diffusivity is the component parallel to the surface of the films. Da is the data obtained from the amplitude decay of ac temperature waves and Dp is the data obtained from the phase shift of ac temperature waves. The solid and dotted curves show the analytical result (see text).
diffusivity reaches the highest magnitude when the thickness becomes about 1 mm. On the other hand, below the thickness of 0.05 mm the thermal diffusivity becomes extremely small. This fact might be due to the originally small grain size at the other surface which is the counter part of the substrate. In comparison with the image of the fracture cross section taken by a scanning electron microscope, in the case of diamond films we can clearly exhibit the correlation between the structure and the thermophysical properties. In the analysis for AlAs/GaAs superlattices, the saturated magnitude of the arithmetic average thermal diffusivity is 0.17 9 0.02 cm2 s − 1 as shown in Eq. (16). From the thermal conductivity of AlAs and GaAs for bulk materials [8], the average thermal diffusivity is estimated to be 0.37 cm2 s − 1. It is considerably larger than that from the present experiment. This fact indicates that the crystalline quality of the superlattice is inferior to that of the bulk materials. Assuming cA = 1.8 J cm − 3 K − 1 and cB = 1.8 J cm − 3 K − 1 for GaAs and AlAs, respectively, from Eq. (12) we can roughly estimate kA = 0.22 J s − 1 cm − 1 K − 1 for the GaAs region and kB = 0.39 J s − 1 cm − 1 K − 1 for the AlAs region. On the other hand, as seen in Fig. 4, the thermal diffusivity decreases markedly as the layer thickness d becomes smaller. But, as pointed out in the previous section, the thermal diffusivity should terminate at a certain value. From the structural characteristic of AlAs/GaAs superlattice, the interfacial region consists of compositionally disordered structure and therefore, we can assume that the interfacial region has the thermophysical properties of the anomalous state. Then, putting D=ki/ci where ki = 0.09 J s − 1 cm − 1 K − 1 and ci = 1.8 J cm − 3 K − 1 for the anomalous state, Ga0.5Al0.5As, into Eq. (16), we obtain di = 4 nm. This is the first result for the thickness of the interfacial region of AlAs/GaAs superlattice obtained from the measurement of the thermophysical properties. Based upon the above results for AlAs/GaAs superlattices, we can estimate phonon mean free path l in each region. In a simple manner, thermal conductivity k is related to phonon mean free path l as follows: 1 k= c6l, 3
Fig. 4. Dependence of thermal diffusivity of AlAs/GaAs superlattices on layer thickness (superlattice constant in this case). The experimental data points were measured using the ac calorimetric technique by Yao [2]. These thermal diffusivities are the component parallel to the surface of the films. The solid curve shows the analytical result (see text).
(17)
where 6 is sound velocity. The sound velocity is 3.8× 105 cm s − 1 for GaAs and 4.5× 105 cm s − 1 for AlAs. Although the sound velocity of the anomalous state, Ga0.5Al0.5As, has not been reported yet, we assume that the sound velocity is about 5× 105 cm s − 1 for the amorphous state which is a little larger than that for the crystal state. Using the other quantities indicated above, we can estimate: lA = 10 nm for the GaAs region, lB = 14 nm for the AlAs region,
I. Hatta et al. / Materials Science and Engineering A292 (2000) 189–193
li =3 nm for the interfacial region. The results for AlAs/GaAs superlattice offer a thermophysical picture of an interfacial region. From the measurement normal to the surface, it is difficult to distinguish two cases; one is an interface with thermal resistance and the other is an interfacial region in which disordered structure takes place. On the other hand, from the measurement along the surface it has clearly been shown the existence of the interfacial region which consists of a disordered structure with a certain thickness.
5. Conclusion To analyze thermophysical properties in a thin film, it has been shown that the measurement of thermal diffusivity along the surface of the film gives rise to clear evidence, that is, we can know the feature of the interfacial region between the film and a substrate. For this purpose, the ac calorimetric method is quite useful because it can be applied to a variety of thin films and furthermore, the quantitative evaluation of the interfacial region is possible. Based upon the results on CVD diamond films and AlAs/GaAs superlattices obtained by the ac calorimetric method, it has clearly been shown the existence of
.
193
the interfacial region, especially in the latter that with nano-scale thickness. In addition, the phonon mean free path in the interfacial region is extremely small in comparison with that in the crystal. Therefore, the multi-phonon-scattering process takes place in the interfacial region. Then, the picture that phonon scatters at the interface is no longer realistic. Acknowledgements We (IH and SWK) acknowledge support through Joint Research Project under the Japan-Korea Scientific Promotion Program organized by both Japan Society for the Promotion of Science and Korea Science and Engineering Foundation. References [1] J.E. Graebner, S. Jin, G.W. Kammlott, J.A. Herb, C.F. Gardinier, Appl. Phys. Lett. 60 (1992) 1576. [2] T. Yao, Appl. Phys. Lett. 51 (1987) 1798. [3] I. Hatta, Intern. J. Thermophys. 11 (1990) 293. [4] T. Yamane, S. Katayama, M. Todoki, J. Appl. Phys. 80 (1996) 2019. [5] I. Hatta, Netsu Bussei 4 (1990) 112 (in Japanese). [6] A. Einstein, Ann. Phys. (Leipzig) 35 (1911) 679. [7] D.G. Cahill, S.K. Watson, R.O. Pohl, Phys. Rev. B 46 (1992) 6131. [8] M.A. Afromowitz, J. Appl. Phys. 44 (1973) 1292.
Thermophysical properties of thin films on substrate Ichiro Hatta a,*, Kazuhiko Fujii a, Sok Won Kim b a
Department of Applied Physics, Graduate School of Engineering, Nagoya Uni6ersity, Nagoya 464 -8603, Japan b Department of Physics, Uni6ersity of Ulsan, Ulsan 680 -749, South Korea
Abstract In the analysis of thermophysical properties in a thin film, it is important to take into account those in the interfacial region between a thin film and a substrate on which the film was grown. In order to analyze them, an ac calorimetric thermal diffusivity measurement is very useful. In this measurement, the thermal diffusivity of the component parallel to the surface of a thin film is obtained as a function of film thickness for a single film and also as a function of layer thickness for a superlattice. In the present paper, an analytical method is proposed and it is applied to the analysis of thermophysical properties in diamond films and in AlAs/GaAs superlattices. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Thermal conductivity; Thermal diffusivity; Nano scale region; Interface; Ac calorimetry
1. Introduction Although a lot of measurements of thermophysical properties have been performed in thin films, there still remain unresolved problems. In the measurement of thermophysical properties in a thin film grown on a substrate, we have to consider those of at least two distinct regions. One is the thermophysical property of an interfacial region between a thin film and a substrate. Usually the interfacial region has much smaller thermal conductivity than that of the other remaining part of a thin film. The other is the thermophysical property of the remaining part that might be close to that of the single crystal. However, it is more or less inferior to that of a perfect single crystal. In fact, the above phenomena have been pointed out in some experiments. Graebner et al. [1] have measured the thermal conductivity of CVD diamond films in the direction parallel to the surface as a function of the thickness using a two-heater heated-bar technique. As the thickness increases, the observed thermal conductivity rises. They have shown a method to analyze the above results and estimated the local thermal conductivity as a function of distance from the bottom that was the interface between the grown diamond film and the substrate. Thus estimated local thermal conductivity * Corresponding author. Fax: +81-52-7893706. E-mail address: [email protected] (I. Hatta).
rises with increasing the distance from the bottom and saturates almost at the thermal conductivity reported for single-crystal Type IIa diamond. On the other hand, using an ac calorimetric technique Yao [2] has measured the thermal diffusivity of AlAs/GaAs superlattices in the direction parallel to superlattice surface at various layer thicknesses of GaAs and AlAs, where the both thicknesses are the same. He has found that, as the layer thickness increases, the observed thermal conductivity rises. He also has pointed out that at the limit of the short-period superlattice the observed thermal conductivity becomes close to that of Al0.5Ga0.5As alloy. We will propose a method to analyze the thermophysical properties of such materials. Based upon the analytical method, it will be able to estimate the thermophysical properties of diamond films and GaAs/ AlAs superlattices, in both of which the thermal diffusivity measurement have performed by an ac calorimetric technique. Finally, we will point out the importance of the thermal diffusivity measurement along the surface of thin films and superlattices.
2. Analytical method In the analysis of thermophysical properties in thin films and superlattices, we will point out that the measurement of thermal diffusivity parallel to the surface
0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S0921-5093(00)00995-3
190
I. Hatta et al. / Materials Science and Engineering A292 (2000) 189–193
gives rise to fruitful evidence. As typical systems, we will consider diamond films with a variety of thicknesses which were originally synthesized on a substrate in the same manner and superlattices with a variety of layer thicknesses (or superlattice constants) also in the same manner. First, let us consider a free-standing film, such as a diamond film, removed from a substrate on which the film has been grown primarily. In such a free-standing film, it is likely that the microscopic structure changes gradually from the bottom surface of the substrate side to the upper surface of the opposite side. In fact, in a diamond film made in such a way, the crystalline size is small near the substrate and becomes larger with growth [1]. One can assume that such a functionally graded material is composed of the superposition of a number of layers in which thermal conductivity successively increases layer by layer from the layer that lies near the substrate to the opposite free end. As shown in Fig. 1, the thermal conductivity, the heat capacity per unit volume and the thickness of the layers are denoted by k0, c0 and d0 for the 0-th layer, k1, c1 and d1 for the first layer, and those up to the n-th layer. The free end of the n-th layer lies at the bottom surface that is the counter part of the substrate. In an ac calorimetric technique, the modulated heat is applied to a part of the surface of the multilayer and the ac temperature waves propagate along the direction parallel to the surface outside the periodically heated region. If n
% kjdj 51,
(1)
j=0
and/or when the ac temperature is detected at a position far from the heated region [3,4], the observed total thermal diffusivity is expressed by a simple relation of Eq. (2). In Eq. (1), kj = (v/2Dj), where v is angular frequency and Dj( = kj/cj) is the thermal diffusivity of the j-th layer.
Fig. 1. Model for a thin film with graded thermophysical properties. The thin film is composed of a number of layers. The 0-th layer lies in a free surface and the n-th layer lies in the other surface that is removed from a substrate. The layer thickness is d0 for 0-th layer, d1 for first layer and so on. The total thickness d is given by the sum of these layers.
n
% kjdj D=
j=0 n
(2)
% cjdj
j=0
It is worth to point out that at room temperature usually the heat capacity of each layer in the above model is almost the same independent from the crystalline quality [5]. This might be the case in diamond films. Heat capacity of per unit volume in diamond is 1.80 J cm − 3 K − 1 for Type I, Type IIa and Type IIb [5]. Therefore, we can assume that c0 : c1 : c2 : …: cn
(3)
We define the total thickness d as d= d0 + d1 + d2 + …+ dn
(4)
Then, one can rewrite Eq. (2) approximately as D= D0 −
Es , d
(5)
where D0 =
k0 , c0
(6)
n
% (k0 − kj)dj Es =
j=1
c0
(7)
In the above analysis, we assumed that the thicknesses of d1 to dn are kept unchanged, when one reduces the thickness of d0. Then, only the portion of the 0-th layer becomes larger and therefore D reaches the saturated value D0. On the other hand, when the thickness d decreases in Eq. (5), the observed thermal diffusivity becomes smaller, but it should terminate at a certain thickness where the thermal diffusivity reaches the minimum value in the material with the same composition. Second, let us consider a superlattice. The situation seems to be quite similar to that of the above single film. In a superlattice, for instance, since atoms diffuse through the interface during the crystal growth, the composition graded region takes place and it results in a disorderd interfacial region. As a result, the thermal diffusivity in the interfacial region becomes very small and finally reaches that of the amorphous state. On the other hand, since the diffusion of atoms does not reach at the far distance from the interface, the thermal conductivity becomes larger. In the ac calorimetric measurement, the similar condition of Eq. (1) is required. For simplicity, we consider a multilayer system composed of the periodical arrangement of a set of layer A, one interface layer i, layer B and another interface layer i as shown in Fig. 2, where the total number of the sets is n. Here, we consider the case that the thicknesses of layer A and layer B are the same thickness d0 and the thickness of the interface is di. The
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191
As seen in Eq. (11), the thickness dependence of the observed thermal diffusivity is similar to Eq. (5). Therefore, when the layer thickness d increases, from Eq. (11) one can obtain the saturated thermal diffusivity. On the other hand, when the layer thickness d decreases, the observed thermal diffusivity becomes small, but terminates at a certain value. This fact results from the concept of the minimum thermal conductivity proposed by Einstein [6,7]. From a point of today’s view, an amorphous state of a material has the smallest thermal conductivity. It is very likely that the interfacial region is in an amorphous state and then, if ki is the thermal conductivity of the amorphous state, we can estimate the thickness of the interfacial region di from Eq. (11). 3. Analysis of data
Fig. 2. Model for a superlattice, which is composed of the periodic arrangement of a material A, an interfacial region i between A and B, a material B, and an interfacial region i between B and A. The layer thicknesses of a material A and B are the same, d0. The thickness of the interfacial region i is di. Then, the layer thickness d is given by the sum of d0 and di.
heat capacities per unit volume of layer A, layer B and interface layer i are cA, cB and ci, respectively, and the thermal conductivities are kA, kB and ki, respectively. Then, the observed thermal diffusivity is given by D= =
kA(nd0)+kB(nd0) + ki(2ndi) cA(nd0)+cB(nd0) +ci(2ndi)
kAd0 + kBd0 + 2kidi cAd0 +cBd0 + 2cidi
(8)
We also assume the following relation [5]: c =cA cB ci.
(9)
We define the layer thickness d as d= d0 +di.
(10)
Then, one can rewrite Eq. (8) approximately as D= DAB −
Em , d
(11)
where DAB = Em =
kA + kB , 2c
(kA − ki)di +(kB −ki)di . 2c
(12) (13)
CVD diamond films made by Sumitomo Electric Co. (Itami, Japan) were used for the ac calorimetric measurement. There are three samples with thickness of 0.16, 0.23 and 0.53 mm. The thicker samples were grown successively under the same condition and then, we assume that in Fig. 1, the thickness of the 0-th layer increases as the sample becomes thicker. In such a situation, the observed thermal diffusivities are fitted to Eq. (5) as D0 and Es are fitting parameters. Then, the following results are obtained: Da[cm2 s − 1]= 12.59 0.4−
0.4890.09 , d[mm]
(14)
Dp[cm2 s − 1]= 12.09 0.4−
0.3790.08 , d[mm]
(15)
where Da and Dp are analyzed from the data on the amplitude decay and the phase shift in the ac temperature, respectively. The fitting curves are shown in Fig. 3. Both the results are consistent as those expected from the principle of the ac calorimetric technique. We reanalyzed the data for AlAs/GaAs superlattices reported by Yao [2] based upon the present analytical method. We assume that as shown in Fig. 2 the superlattices are composed of the periodic arrangement of a set of AlAs layer, one interfacial region, GaAs layer and another interfacial region. The observed thermal diffusivities are fitted to Eq. (11) where DAB and Em are fitting parameters. Then, the following result is obtained: D[cm2 s − 1]= 0.179 0.02−
0.590.1 . d[nm]
(16)
The fitting curve is shown in Fig. 4. Although the fitting curve is drawn down to D=0, it should terminate at a certain value as mentioned in the previous section. The saturated value of 0.17 cm2 s − 1 in Eq. (16) is the arithmetical average of the thermal conductivities of AlAs and GaAs as given in Eq. (12).
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I. Hatta et al. / Materials Science and Engineering A292 (2000) 189–193
4. Discussion From the analysis for diamond films measured in the present experiment, it was found that the saturated magnitude of the thermal diffusivity agrees with the thermal diffusivity of a Type IIa diamond, which is the highest magnitude in a natural diamond. Therefore, at the free surface of a CVD diamond film the thermal
Fig. 3. Dependence of thermal diffusivities Da and Dp of CVD diamond films on thickness. The data was obtained by the ac calorimetric technique. The thermal diffusivity is the component parallel to the surface of the films. Da is the data obtained from the amplitude decay of ac temperature waves and Dp is the data obtained from the phase shift of ac temperature waves. The solid and dotted curves show the analytical result (see text).
diffusivity reaches the highest magnitude when the thickness becomes about 1 mm. On the other hand, below the thickness of 0.05 mm the thermal diffusivity becomes extremely small. This fact might be due to the originally small grain size at the other surface which is the counter part of the substrate. In comparison with the image of the fracture cross section taken by a scanning electron microscope, in the case of diamond films we can clearly exhibit the correlation between the structure and the thermophysical properties. In the analysis for AlAs/GaAs superlattices, the saturated magnitude of the arithmetic average thermal diffusivity is 0.17 9 0.02 cm2 s − 1 as shown in Eq. (16). From the thermal conductivity of AlAs and GaAs for bulk materials [8], the average thermal diffusivity is estimated to be 0.37 cm2 s − 1. It is considerably larger than that from the present experiment. This fact indicates that the crystalline quality of the superlattice is inferior to that of the bulk materials. Assuming cA = 1.8 J cm − 3 K − 1 and cB = 1.8 J cm − 3 K − 1 for GaAs and AlAs, respectively, from Eq. (12) we can roughly estimate kA = 0.22 J s − 1 cm − 1 K − 1 for the GaAs region and kB = 0.39 J s − 1 cm − 1 K − 1 for the AlAs region. On the other hand, as seen in Fig. 4, the thermal diffusivity decreases markedly as the layer thickness d becomes smaller. But, as pointed out in the previous section, the thermal diffusivity should terminate at a certain value. From the structural characteristic of AlAs/GaAs superlattice, the interfacial region consists of compositionally disordered structure and therefore, we can assume that the interfacial region has the thermophysical properties of the anomalous state. Then, putting D=ki/ci where ki = 0.09 J s − 1 cm − 1 K − 1 and ci = 1.8 J cm − 3 K − 1 for the anomalous state, Ga0.5Al0.5As, into Eq. (16), we obtain di = 4 nm. This is the first result for the thickness of the interfacial region of AlAs/GaAs superlattice obtained from the measurement of the thermophysical properties. Based upon the above results for AlAs/GaAs superlattices, we can estimate phonon mean free path l in each region. In a simple manner, thermal conductivity k is related to phonon mean free path l as follows: 1 k= c6l, 3
Fig. 4. Dependence of thermal diffusivity of AlAs/GaAs superlattices on layer thickness (superlattice constant in this case). The experimental data points were measured using the ac calorimetric technique by Yao [2]. These thermal diffusivities are the component parallel to the surface of the films. The solid curve shows the analytical result (see text).
(17)
where 6 is sound velocity. The sound velocity is 3.8× 105 cm s − 1 for GaAs and 4.5× 105 cm s − 1 for AlAs. Although the sound velocity of the anomalous state, Ga0.5Al0.5As, has not been reported yet, we assume that the sound velocity is about 5× 105 cm s − 1 for the amorphous state which is a little larger than that for the crystal state. Using the other quantities indicated above, we can estimate: lA = 10 nm for the GaAs region, lB = 14 nm for the AlAs region,
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li =3 nm for the interfacial region. The results for AlAs/GaAs superlattice offer a thermophysical picture of an interfacial region. From the measurement normal to the surface, it is difficult to distinguish two cases; one is an interface with thermal resistance and the other is an interfacial region in which disordered structure takes place. On the other hand, from the measurement along the surface it has clearly been shown the existence of the interfacial region which consists of a disordered structure with a certain thickness.
5. Conclusion To analyze thermophysical properties in a thin film, it has been shown that the measurement of thermal diffusivity along the surface of the film gives rise to clear evidence, that is, we can know the feature of the interfacial region between the film and a substrate. For this purpose, the ac calorimetric method is quite useful because it can be applied to a variety of thin films and furthermore, the quantitative evaluation of the interfacial region is possible. Based upon the results on CVD diamond films and AlAs/GaAs superlattices obtained by the ac calorimetric method, it has clearly been shown the existence of
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the interfacial region, especially in the latter that with nano-scale thickness. In addition, the phonon mean free path in the interfacial region is extremely small in comparison with that in the crystal. Therefore, the multi-phonon-scattering process takes place in the interfacial region. Then, the picture that phonon scatters at the interface is no longer realistic. Acknowledgements We (IH and SWK) acknowledge support through Joint Research Project under the Japan-Korea Scientific Promotion Program organized by both Japan Society for the Promotion of Science and Korea Science and Engineering Foundation. References [1] J.E. Graebner, S. Jin, G.W. Kammlott, J.A. Herb, C.F. Gardinier, Appl. Phys. Lett. 60 (1992) 1576. [2] T. Yao, Appl. Phys. Lett. 51 (1987) 1798. [3] I. Hatta, Intern. J. Thermophys. 11 (1990) 293. [4] T. Yamane, S. Katayama, M. Todoki, J. Appl. Phys. 80 (1996) 2019. [5] I. Hatta, Netsu Bussei 4 (1990) 112 (in Japanese). [6] A. Einstein, Ann. Phys. (Leipzig) 35 (1911) 679. [7] D.G. Cahill, S.K. Watson, R.O. Pohl, Phys. Rev. B 46 (1992) 6131. [8] M.A. Afromowitz, J. Appl. Phys. 44 (1973) 1292.