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Mar 1, 2002 - thermal conductivity, measured by the 3-ω method or the photothermal deflection ... mation on the thermal conductivity of porous silica xerogel.
JOURNAL OF APPLIED PHYSICS

VOLUME 91, NUMBER 5

1 MARCH 2002

Processing dependent thermal conductivity of nanoporous silica xerogel films Anurag Jain, Svetlana Rogojevic, Shom Ponoth, William N. Gill, and Joel L. Plawskya) Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180

Eva Simonyi and Shyng-Tsong Chen IBM, T. J. Watson Research Center, Yorktown Heights, New York 10598

P. S. Ho University of Texas at Austin, Austin, Texas 78712

共Received 5 November 2001; accepted for publication 10 December 2001兲 Sintered xerogel films 共porous SiO2 兲 show a much higher thermal conductivity than other low dielectric constant 共low-K兲 materials available for the same value of K. The thermal conductivity of xerogels which we have processed using different methods is compared with that of other low-K materials such as silica hybrid 共silsesquioxanes兲 and polymeric low-K materials. The methods used were: 共1兲 single solvent 共ethanol兲 method, 共2兲 binary solvent 共mixture of ethanol and ethylene glycol兲 method, 共3兲 sintering. For the xerogel films, we show that process history is as important as the chemistry of the solid matrix or the porosity in determining the thermal conductivity. The thermal conductivity, measured by the 3-␻ method or the photothermal deflection method, is affected by phonon scattering, which in turn is effected by the size and distribution of pores and particles and the presence of imperfections such as interfaces, substituted chemical species, impurities, microcracks, and microporosity. The thermal conductivity extrapolated to zero porosity for porous sintered xerogel films approaches that of thermally grown SiO2 indicating the least phonon scattering of all processing methods. For these films, the elastic modulus is proportional to thermal conductivity squared, in agreement with theories developed for materials with few defects and a connected matrix. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1448407兴

also a problem for chemical mechanical planarization 共CMP兲 processes as frictional heating in the CMP process may give rise to hot spots resulting in localized stress and cap delamination.3 Small decreases in the thermal conductivity of dielectrics can cause sudden increases in the interconnect temperature.4 Thus, an efficient thermal design of integrated circuits is required and it is essential to understand the heat transport phenomena in thin dielectric films. Goodson and Ju,5 Cahill,6 and Goodson et al.7 have given excellent reviews on this subject. In this article, we present new information on the thermal conductivity of porous silica xerogel films and the relationship between thermal conductivity and elastic modulus. We also compare the thermal conductivity of porous silica xerogel films processed in different way with other low-K 共polymeric and silsesquioxanes兲 dielectrics. The differences are explained qualitatively on the basis of disorder in the films which affects phonon scattering.

I. INTRODUCTION

An important problem faced by current and future integrated circuit 共IC兲 generations is that signals through metal interconnects are delayed by the resistance R of the metal lines and the capacitance C between adjacent lines. This RC delay becomes increasingly important when metal interconnects are designed in a higher packing density. To optimize the performance of ICs at a higher packing density, low dielectric constant materials 共low-K兲 such as polymeric, hybrid, and porous materials are being introduced to replace the conventional dielectric, which is dense SiO2 deposited by plasma enhanced chemical vapor deposition 共PECVD兲. Implementation of low-K dielectrics reduces the capacitance between adjacent lines for a given IC layout and, therefore, reduces the interconnect signal delay. Before low-K materials can be implemented in one of the next generations of ICs, many problems must be solved. One important issue is the lower thermal conductivity of low-K materials compared to conventional SiO2 . The increased density of transistors per chip causes more Joule heating.1 Though metals are efficient conductors of heat, the major resistance to heat transfer resides in the interlayer dielectric. Metal lines and vias, as currently used, are not sufficient, nor designed, for heat removal from chips.2 The low conductivity of dielectrics is

II. THEORY OF THERMAL TRANSPORT PROCESSES IN DIELECTRIC FILMS

This section gives an overview of the heat transport mechanisms in dielectric films at temperatures near or above room temperature. However, only qualitative conclusions can be drawn from this theory. The conduction of heat in disordered dielectric solids may be considered as the propagation of anharmonic elastic waves through a continuum. This propagation occurs via interaction between the quanta of thermal energy called phonons. The frequency 共␻兲 of lat-

a兲

Author to whom all correspondence should be addressed; Electronic mail: [email protected]

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© 2002 American Institute of Physics

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tice waves with velocity v covers a wide range and the thermal conductivity 共k兲, in general form, can be written in terms of a superposition of these waves as k⫽

1 3

冕␳

C p共 ␻ 兲 ␯ l 共 ␻ 兲 d ␻ ,

共1兲

where C p ( ␻ ) is the contribution to the specific heat per frequency interval for lattice waves of that frequency and l ( ␻ ) is the attenuation length or the mean free path for the lattice waves. At temperatures above 50 K, heat transfer throughout disordered dielectric films can be considered to be a diffusion process. The mean free paths in such materials are very short 共⬃ few Å兲 and the temperature 共T兲 dependent thermal diffusivity 共␣兲 can be written as

␣ 共 T 兲 ⫽k 共 T 兲 / 共 ␳ C p 兲 ⫽ 共 1/3兲 ␯ l 共 T 兲 ,

共2兲

where the macroscopic density ␳, the specific heat C p of the material, the transport velocity, ␯ , of the lattice waves 共or phonons兲 and the phonon mean free path, l , are the factors that determine the thermal conductivity. For dielectric films, the temperature dependence is very weak for ␳, C p , and ␯ , and so the mean free path is the main factor that affects the conductivity. If l is equal to the separation distance between constituent atoms, the resulting conductivity is referred to as the minimum thermal conductivity.6 The minimum conductivity is the lower limit to the conductivity of a dense material at room temperature. The minimum thermal conductivity model assumes that long-wavelength 共␭兲 phonons with long mean free paths do not contribute significantly to heat transfer in strongly disordered materials. For amorphous disordered solids like glasses, the mean free path is almost constant at room temperature8 and is limited to several interatomic spacings. The mean free path of fused silica at room temperature is 5.6 Å,9 which is about the size of an elementary silicate ring in the disordered structure of glass.10 Conductivities in excess of the minimum result from additional mechanisms for transport. There are two regimes of heat transport by lattice vibrations. At low vibrational energies, the lattice vibrations are wave-like, and phonons exist with well defined wavelengths 共␭兲, wave vectors ( ␬ ) ⫽2 ␲ /␭ and velocities ( ␯ ). In this regime, the kinetic theory equation, similar to Eq. 共1兲 is applicable: k 共 ␻ 兲 ⫽ ␳ C p 共 ␻ 兲 ␯ l 共 ␻ 兲 /3.

共3兲

This provides the minimum thermal conductivity. As the vibrational energy of the lattice waves increases, the Rayleigh scattering regime is approached. If ␶ i is the average time between the collisions of the phonons with the imperfections in the material, then:

␶ i⫽

1 , ␤ ␴␳ d ␯

共4兲

where ␴ is the scattering cross section, ␤ is a constant of the order of unity, ␳ d is the defect density per unit volume, and ␯ is the velocity of lattice waves 共phonons兲. A shorter ␶ i implies a shorter l . As the vibrational energy of the lattice waves increases, the scattering cross section 共␴兲, which is inversely proportional to l and depends on the size of imperfections, approaches zero,11 and Rayleigh scattering oc-

curs. Rayleigh scattering increases as ␻ 4 until l ⫺1 ⬇ ␬ / ␲ (2/␭). The maximum wavelength of the phonon that could exist in a thin film dielectric is equal to its thickness. The phonon mean free path is affected by a number of factors. The dominant factor in most solids is inelastic phonon–phonon scattering 共Umklapp processes兲. Such scattering is absent in amorphous materials with no long-range order such as silica glass. However, various other factors cause anharmonicities and scattering. These factors include interactions between phonons and any defects or imperfections in the films such as interfaces, microvoids, microcracks, particles, and pores, etc.5 This factor is also evident from Eq. 共4兲 where increasing the defect density, ␳ d , decreases ␶ i and hence l . The presence of atoms of different size or impurities leads to increased phonon scattering. The increased scattering is due to differences in the mass of the elements, differences in the binding force of the substituted atom, and the elastic strain field around the substituted atom. Thus organic–inorganic silica hybrid dielectric materials 共silsesquioxanes兲 are expected to have lower conductivities than pure silica films. Polymeric low-K materials with their inherently low density and specific heat have an inherently lower thermal conductivity than inorganic materials. The thermal conductivity of organic films also depends on additional factors such as the degree of orientation of crystalline regions and of molecular strands in the amorphous regions.5 The effective mean free path is found by adding the contribution of each of the aforementioned effects as a sum of resistances in parallel 共i.e., adding 1/l for each mechanism兲. Thus, the more factors that cause anharmonicities, or the more disorder in the films, the shorter the effective mean free path and the lower the conductivity of the material. At sufficiently high temperatures, generally above room temperature, scattering due to imperfections is independent of the temperature and frequency. If imperfections are reduced or eliminated, the thermal conductivity will increase. Although it is difficult to determine accurately the mean free path for an amorphous dielectric 共with or without porosity兲 material, the concept is very useful in explaining the results we have obtained with our porous silica xerogel films. If we alter the process conditions by which a low-K material 共such as porous silica兲 is made, we can radically change the disorder in the film and alter the thermal conductivity of the film. In porous materials with a broad pore size distribution like xerogels, the dominant scattering mechanism would be from the pores of smallest sizes and the maximum phonon wavelength would be limited to the largest pore size. A material with uniformly sized 10 nm spherical pores such as porous Vycor glass 共porosity ⬃30%兲 is found to have a thermal conductivity12 predicted by the effective medium theory given by Landauer.13 This suggests that the maximum wavelength of phonons is limited to ⬃10 nm. If the spherical pores are assumed to have zero conductivity, and the matrix is a fully dense silica with conductivity k d , then the effective medium theory predicts the thermal conductivity 共for ␾ ⬍2/3兲 to be k⫽k d (1 – 1.5␾ ) where ␾ is the volume fraction of voids. This implies that for ␾ ⬎2/3 there is no connected solid path for thermal conduction.

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J. Appl. Phys., Vol. 91, No. 5, 1 March 2002

III. THERMAL CONDUCTIVITY OF XEROGELS

The thermal conductivity of xerogels varies widely depending upon the porosity and the method of processing. In fact, high porosity xerogels 共⬎90%兲 are very efficient insulators. For low-K applications, lower porosity xerogels 共25%–75%兲 are potential candidates and their thermal conductivities are correspondingly higher. For xerogels of densities, ␳, greater than 100 kg/m3, the relation given by Gross and Fricke for thermal conductivity, k is;14 k⫽C ⬘ ⫻k d ⫻ 关 ␳ / ␳ s 兴 1.88,

共5兲

where k d is the conductivity of the solid matrix 共silica兲. This relation was obtained by measurement of sound velocities in bulk xerogels made using a single solvent process and correlating the thermal conductivity to the sound velocity. The relation indicates that the thermal conductivity is a highly nonlinear function of the density 共or porosity兲. The value of the constant C ⬘ and the exponent 共1.88 here兲 can change depending upon the density range and the method of processing. These scaling relationships are commonly used to describe the variation of properties 共thermal conductivity or the elastic modulus兲 of silica xerogels and those properties can vary by a 1–3 order of magnitude depending upon the density. IV. EXPERIMENTAL DETAILS A. Films processing

Porous silica xerogel films were processed according to the procedure described in previous publications15,16 and their refractive indices and thicknesses were characterized by variable angle spectroscopic ellipsometry. The porosity 共and density兲 was correlated with the refractive index measurements15,16 using effective medium theory and hence small errors may result if the organic content in the film is high. Two methods of porosity control were used in this work. In the first case, the porosity was controlled by controlling the partial pressure of ethanol 共solvent in the reacting sol–gel solution兲 in a closed bowl spin coater. This controls the amount of solids content at the point of gelation and hence the final film porosity. In the second method, the binary solvent method, a mixture of ethylene glycol 共nonvolatile兲 and ethanol 共volatile兲 was used as the solvents and the spinning was done in an open-bowl spin coater. During spinning ethanol evaporates completely and the amount of nonvolatile ethylene glycol in the sol controls the porosity. The higher the amount of ethylene glycol, the higher the porosity 共lower the density兲. The way the spin coating is done and the way ethylene glycol is added were the only differences in the films made with this method. Film porosity could be controlled in the range 25%– 80% and the film thickness in the range 0.5–2.5 ␮m. The structure of the films processed by either method consists of random interconnected chains of small silica spheres 共1–20 nm兲 with both microporosity 共⬍2 nm兲 and macroporosity 共2–25 nm兲. Our processing methods ensured that strong bonds, in the form of small necks, exist between these spheres. The as-prepared films had residual organic content in the form of unreacted ethoxy groups and methyl

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groups from the surface passivation used to make the films hydrophobic. The organic content was at most 10–15 at. %.17 Some of the films made by the binary solvent method were sintered slowly 共at 2 °C/min兲 to 900 °C for 2 h and then cooled down slowly to room temperature. This high temperature annealing process burned off the residual organics in the film, removed microporosity produced a pure, more regularly connected silica backbone,18 and thus reduced the disorder in the structure of the xerogel films. Refractive index measurements after the sintering process gave the density and porosity of these films.

B. Thermal conductivity measurements

There have been various techniques reported to measure the thermal conductivity of thin dielectric films.5,6,19–21 In this study the 3-␻ technique and the photothermal deflection method were used. The details of 3-␻ technique can be found in the literature.22–28 The details of the 3-␻ setup used in this work are given in our previous publication.22 The photothermal deflection technique is based on the periodic heating of a sample by a modulated laser pump beam. There are various configurations of this method. A surface reflection technique29,30 was used here where the absorption of the pump laser beam caused a local-temperature rise, which in turn lead to a local surface deformation, the magnitude of which could be less than 0.1 nm. The surface deflection signal was detected by the probe beam and related to the thermal conductivity of the sample. This detection method of thermal waves is called the photothermal deformation30 technique. The pump laser used was a 325 nm He–Cd single mode Gaussian beam and generated thermal waves on the surface of copper dots 共thickness 3000 Å and 6 mm diameter兲 deposited on the top of porous xerogel films. The modulation of the pump beam was accomplished by a mechanical chopper with a frequency of 20 000 Hz. The detection of the thermal waves was accomplished by a 632.8 nm He–Ne laser probe beam, which was reflected from the sample surface. The probe beam monitored the slope of the surface deformation caused by the thermal expansion effect. Since the pump beam was modulated, the slope of the deformation, and hence the direction of the reflected probe beam, was also modulated. The beam deflection is detected by a quad-cell detector, which records deflection components in both directions normal and parallel to the sample surface. The resulting signals were sent to a pair of lock-in amplifiers, which selectively amplified the ac part of the signal. The measurements of both the 3-␻ and the photothermal method are frequency dependent. It is important that the frequency of measurement is such that the measured thermal conductivity is independent of the film thickness. This was ensured by choosing the thickness of the films and the frequency in such a way that the thermal diffusion length 冑2 ␣ / ␻ was much greater than the film thickness, where ␣ is the thermal diffusivity. This ensured that the dielectric did not store any thermal energy and a one-dimensional quasisteady state thermal heat conduction situation occurred.

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FIG. 2. Thermal conductivity of the xerogel films as a function of dielectric constant. For all films, the extrapolated thermal conductivity at K⫽1 would approach that of air 共0.026 W/mK兲. FIG. 1. Thermal conductivity of xerogel films as a function of density; 共a兲 sintered films; 共b兲, and films made with mixture of ethanol and ethylene glycol 共EG兲 solvents 共binary solvent process兲; 共c兲. Films made with ethanol 共EtOH兲 and no EG. The density was calculated from the porosity 共P兲 using ( P⫽1⫺ ␳ / ␳ s ), with ␳ s ⫽2.1 gm/cm3 for cases 共a兲 and 共b兲 and 2.2 gm/cm3 for 共c兲. Unsintered xerogel films have lower density because of the presence of up to 10 at. % carbon. Solid lines are power law fits.

V. RESULTS A. Experimental results for thermal conductivity of xerogel films

Figure 1 shows the thermal conductivity measurements of the three different types of porous silica films processed at Rensselaer plotted as a function of increasing density 共or reducing porosity兲. The as-prepared films made with ethanol as the solvent have the lowest conductivity of the three types of films. This is because these films have a very broad pore size distribution 共including microporosity兲, microcracks, impurities 共unreacted ethoxy groups兲, defects, and hanging and loose ended silica chains, all of which contribute to significant phonon scattering and limit the mean free path. The thermal conductivity of these films varies with density as a power law relationship with an exponent of 1.65. This is close to what has been reported for bulk aerogels 共1.88兲 prepared by a similar process.14 The mean free path of fused dense silica at room temperature is 5.6 Å9 and is comparable to the size of the micropores present. Hence, phonon scattering would be predominantly from the microporosity. For films made with the binary solvent technique, the pore size distribution is narrower than for films made with the single solvent 共ethanol兲 process.31 This explains their higher conductivity. The anharmonicities due to scattering from micropores are reduced in the binary solvent method. However, more data is needed to support this conclusion since we have uncertainties in the calculation of the density 共the predicted densities are lower, hence only positive error bars兲 of these films due to their higher organic content.31 The power law fit with an exponent of 1.0 is only valid for the density range for which the thermal conductivity data of these films is available. The sintering process heals the microstructure, removes all microcracks, microporosity,32 residual organics, and causes the structure to become more uniform and

beam-like.33 Thus, phonon scattering due to the aforementioned imperfections is reduced. The thermal conductivity of sintered porous silica films is highest and is almost a linear function of the film density. Zeng et al.34 formulated a geometrical models for xerogels and found that for models where the microstructure consists of cylindrical or square cross section rods, the thermal conductivity varies linearly with density. The linear dependence of the thermal conductivity for sintered films is consistent with Zeng et al.’s34 model. Sintered films could contain traces of moisture if they are exposed to the ambient environment for a long time and moisture absorption would cause a small increase in the thermal conductivity. However, the dielectric constant measured by the metal–insulator–metal capacitor method15 before and after sintering did not show any difference, so we believe moisture absorption in our films was negligible. Another way to compare process effects on the thermal conductivity of xerogel films is to plot the thermal conductivity as a function of dielectric constant. This is shown in Fig. 2. The dielectric constant was measured for all three different kinds of xerogel films. The thermal conductivity of the sintered porous xerogel films when extrapolated to the dielectric constant of pure silica 共⬃4.0兲 has the value of ⬃1.35 W/mK, which is close to that of thermally grown oxide films. On the other hand, the thermal conductivity of xerogel films made with single and binary solvent processes extrapolated to K⫽4.0 is lower than that of sintered films but higher than 1.1 W/mK, thermal conductivity of PECVD oxide5 films. B. Comparison with other low-K materials

Figure 3 displays a comparison of the thermal conductivity of the sintered xerogel films and xerogels made using ethanol as the solvent with that of polymeric, hydrogen silsesquioxane 共HSQ兲, and methylsilsesquioxane 共MSQ兲 low-K materials.35–37 The data for our films made using the binary solvent process is omitted for the purpose of clarity. The thermal conductivity of porous and dense MSQ, (K dense ⫽2.75) and dense HSQ (K⫽3.1) was measured by Ju et al.38 The thermal conductivity of the sintered porous xerogel films is much higher than any other low-K material.

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J. Appl. Phys., Vol. 91, No. 5, 1 March 2002

FIG. 3. Comparison of thermal conductivity of three classes of low-K materials. Sintered xerogel films have the highest thermal conductivity for a given dielectric constant of all materials. For polymeric low-K; 1: Benzocyclobutene 共BCB兲, 2: SiLK™ 共Dow Chemical’s silicon-application low-K兲, and 3: polyimide.

Figure 3 also shows that for a given dielectric constant, xerogels 共both prepared using ethanol or sintered兲 have better thermal conductivity than dense SiLK™, BCB, or polyamide materials. Hence, making these polymeric materials porous would only deteriorate their thermal conductivity. The hybrid organosilicates 共or silsesquioxanes兲 have conductivities comparable to xerogel films made using the single solvent 共ethanol兲 method. As discussed in Sec. II, the presence of organic–inorganic chemical specie interfaces contributes to additional phonon scattering and reduces the conductivity of the silsesquioxane materials. For dense SiO2 films, the measured thermal conductivity is also process dependent5 and increases in the following order: evaporated, low pressure chemical vapor deposition 共LPCVD兲, phosphosilicate glass 共7%兲, sputtered, PECVD, and thermal oxide. This sequence correlates with the increased temperature of deposition for the different kinds of silicate films. For films deposited by LPCVD or PECVD, a high temperature anneal increases the thermal conductivity. There is also a progressive increase in the thermal conductivity with annealing temperature. This behavior is similar to the increased thermal conductivity of sintered xerogel films as compared to as-prepared xerogel films. Thus, the degree of disorder is reduced after high temperature annealing and the thermal conductivity approaches those of thermally grown silicon dioxide films. C. Relationship between thermal conductivity, density and Young’s modulus of porous silica xerogel films

In a recent article,33 we showed that the elastic modulus 共E兲 of porous silica xerogel films measured by nanoindentation is also affected by processing history. The modulus varies as a power law with density and the scaling exponents were 2.41 for films made with ethanol, 1.88 for films made with a binary solvent method, and 2.02 for the sintered films. These exponents are lower than the those reported for bulk aerogels 共2.7–3.8兲. However, for a given density, the thinfilm moduli measured are higher than those reported for bulk

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FIG. 4. Scaling relations for the variation of the elastic modulus with thermal conductivity for three types of xerogel films. 共a兲 Sintered films, 共b兲 films made with EG, and 共c兲 films made with only ethanol as the solvent.

aerogels. This indicates that stronger bonding between the particles is obtained by the improved aging procedures15 used in making our xerogel films. The same processes should improve the heat transfer characteristics of the films, too. For a given density, the modulus is highest for the sintered films and films made with the binary solvent method show higher values of the modulus than films made with a single solvent method. Thus, the elastic modulus and thermal conductivity depend on the process history in the same way. Using the relationships between the elastic modulus and density from the previous article33 and the results from Fig. 1, scaling relationships between elastic modulus and thermal conductivity of xerogel films are plotted in Fig. 4. The sintered films show a scaling exponent of 1.95 for the variation of modulus 共E兲 versus thermal conductivity. This value is close to that obtained by Lu et al.39 (1.8⫾0.2) for carbon aerogels. Emmerling and Fricke40 have reported an exponent of 1.87 and 2.03 for silica and carbon aerogels, respectively. The higher value of the scaling exponent of the modulus variation with the density as compared to that for the thermal conductivity variation with density is explained on the basis of the higher tensorial order of the force transmission process as compared to the thermal conduction process.41 This means that any factor that influences the modulus would affect all the elastic constants isotropically in all three directions and would affect both the elastic and shear modulus.42 An analogy exists between the force transmission problem 共to calculate the effective modulus兲 and heat transfer. Heat transmission in dielectrics is affected by phonon scattering processes and similarly, force transmission is effected by the scattering of elastic waves. However, elastic waves are long-wavelength acoustic waves and are not scattered by small imperfections. This effect is used to measure the elastic modulus of polycrystalline43 and porous44 solids by acoustic methods where a relationship of the form E⫽ ␳␯ 2 is used. Similarly, long wavelength acoustic waveguiding was used to measure the intrinsic acoustic velocity in thin film xerogels.45 In these techniques, the mean free path of elastic waves are much longer than any defects or imperfections in the material to make sure that the measurements are not af-

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fected by scattering. If we substitute E⫽ ␳ v 2 in Eq. 共2兲 then the following relation is obtained E⫽



9

␳ l 2 C 2p



k 2.

共6兲

Equation 共6兲 shows that for the same density 共and same specific heat兲, E⬀k 2 if the terms in parenthesis are constant. The scattering of short wavelength excited phonons causes a reduction in the thermal conductivity and this explains why the thermal conductivity of as-prepared films 共made by either the single or binary solvent method兲 rises more slowly with density as compared to the elastic modulus. The thermal conductivity is affected more than the elastic modulus by imperfections in the microstructure that reduce the effective mean free path of phonons. Although microstructural effects are always important in the stress distribution, the elastic modulus is not affected since the longwavelength elastic waves are not scattered. A lower matrix density, irregular and unconnected structure of the asprepared films 共unsintered兲 is responsible for the reduction in the elastic moduli. There is evidence in the literature also that points to this fact. Emmerling and Fricke40 used a simulation procedure to produce three-dimensional gel structures, from which two important parameters can be extracted: 共i兲 the fraction ␣ of interconnected mass of the gel network and 共ii兲 the ratio ␥ of Pythagorean distance to the minimum path length on the gel backbone. The product ␣␥, which enters important macroscopic parameters such as the modulus or solid thermal 共and electrical兲 conductivity, was found to scale with an exponent that is only a function of the mass fractal dimension, a factor largely influenced by processing. Young’s modulus is predicted to vary as the square of the thermal conductivity according to this concept, which is the relation we have observed with our sintered films. Dielectric constant and refractive index measurements of xerogels are also found to be linearly related to density 共or porosity兲.15 The dielectric constant of our films was measured at 1 MHz using metal–xerogel–metal capacitors and the refractive index was measured using ellipsometry with a He–Ne laser of wavelength 632.8 nm. The wavelength of both these electromagnetic waves are much larger than the scale of the microstructural features and hence the linear dependence with density is observed. Percolation theory41 is also used pervasively to explain the variation of the transport properties with density of porous materials. Thermal conductivity and the elastic modulus scale with ( ␳ - ␳ c ), where ␳ c is the density at the percolation threshold. For a xerogel-type microstructure, a connective network would still exist for very low density and hence ␳ c would be close to zero. For a continuous fractal structure like xerogel, continuum percolation 共Swiss-cheese兲 models are applicable and it was shown that modulus varies with conductivity with an exponent of 1.8.42 Thus, density 共porosity兲 is the main factor in determining the thermal and mechanical properties of the foam-type materials like xerogels. If the mean free path of phonons is constant and not affected by microstructural features, then the thermal conductivity of porous films is proportional to density as given by Eq. 共2兲. The elastic modulus is propor-

tional to the square of density for connected beam-like structures.33 These criteria are satisfied for the sintered xerogel films and hence the elastic modulus varies as square of thermal conductivity, the relation we have observed experimentally. The aforementioned observations show that there are some relationships between the factors that govern the mechanical 共modulus兲 and the thermal properties 共conductivity兲 of low-K dielectrics. Reducing defects in the matrix and having ordered microstructures improve both properties tremendously as shown by the results of the sintered films in Fig. 4. Thus, more interconnected, regular structures are desired to obtain the best thermal and mechanical properties for openpore materials. VI. CONCLUSIONS

The thermal conductivity of low-dielectric constant thin films is affected by the mechanisms that influence the phonon scattering and the phonon mean free path. The thermal conductivity of the porous silica xerogel films is affected by the method of processing. Sintered silica films with a pure silica backbone 共absence of any organics, impurities, microcracks, and microporosity兲, for a given dielectric constant, have the highest thermal conductivity of all materials studied. Thus for porous materials, a single component inorganic matrix gives the highest value of thermal conductivity. A narrow pore size distribution obtained by using ethylene glycol as the additional solvent in the processing also leads to higher thermal conductivity than the films with very broad pore size distribution made using only ethanol as the solvent. These factors tend to reduce anharmonicities and increase the thermal conductivity. Polymers have inherently lower thermal conductivity than inorganic dielectrics because of their low density and the specific heat. In hybrid low-K materials, the presence of different size atoms and chemical interfaces in their backbone contribute to the enhanced phonon scattering and the reduced thermal conductivity. The thermal conductivity of silsesquioxane films is comparable to the porous xerogel films made with ethanol as the solvent indicating the presence of a number of anharmonicities and enhanced phonon scattering. Efficient thermal and structural design of future generation IC’s with porous low-K materials should take these factors into consideration. Y.-L. Shen, J. Vac. Sci. Technol. B 17, 2115 共1999兲. W.-Y. Shih, J. Levine, and M. Chang, Proceedings of Adv. Metall. and Interconnect Systems for ULSI Applications, Boston, MA, October 1996, p. 479. 3 M. Fury, Solid State Technol. 42, 33 共1999兲. 4 Y. Morand, Microelectron. Eng. 50, 391 共2000兲. 5 K. E. Goodson and Y. S. Ju, Annu. Rev. Mater. Sci. 29, 261 共1999兲. 6 D. G. Cahill, in Microscale Energy Transport, edited by C.-L. Tien, A. Majumdar, and F. M. Gerner 共Taylor & Francis, Washington DC, 1998兲, p. 95. 7 K. E. Goodson, Y. S. Ju, and M. Asheghi, in Microscale Energy Transport, edited by C.-L. Tien, A. Majumdar, and F. M. Gerner 共Taylor & Francis, Washington, DC, 1998兲, p. 229. 8 J. M. Ziman, Electrons and Phonons 共Oxford University Press, New York, 1960兲. 9 W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed. 共Wiley, New York, 1976兲, p. 617. 1 2

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