REPORTS Ultra-Low Thermal Conductivity in W/Al2O3 Nanolaminates R. M. Costescu,1 D. G. Cahill,1* F. H. Fabreguette,2 Z. A. Sechrist,2 S. M. George2 Atomic layer deposition and magnetron sputter deposition were used to synthesize thin-film multilayers of W/Al2O3. With individual layers only a few nanometers thick, the high interface density produced a strong impediment to heat transfer, giving rise to a thermal conductivity of ⬃0.6 watts per meter per kelvin. This result suggests that high densities of interfaces between dissimilar materials may provide a route for the production of thermal barriers with ultra-low thermal conductivity. Materials with structure on nanometer-length scales are being studied for their promise of providing novel optical, electrical, magnetic, or mechanical properties. The thermal properties of nanostructured materials, however, have received much less attention (1). In general, internal interfaces impede the flow of heat: The interface disorder scatters phonons at grain boundaries or interfaces between similar materials, and the differences in elastic properties and densities of vibrational states inhibit the transfer of vibrational energy across interfaces between dissimilar materials. Thus, it may be expected that materials engineered with high interface densities should reduce the thermal conductivity and improve the performance of thermal barriers (2) and of materials used in thermoelectric energy conversion (3). Conversely, the thermal resistance of interfaces degrades the performance of materials intended for thermal management, such as polycrystalline diamond (4) and nanoscale composites (5). The interface thermal conductance G is de˙ ⫽ G⌬T, where Q ˙ is the heat flux fined by Q normal to the interface and ⌬T is the temperature drop at the interface. If a nanolaminate made of alternating layers of two materials has an interface spacing ␦ and the series resistance of the interfaces dominates the thermal transport, then the thermal conductivity ⌳ of the nanolaminate is simply ⌳ ⫽ ␦G. Values of G for individual solid-solid interfaces have been measured (6) to be as small as G ⬇ 50 MW m⫺2 K⫺1. If this small conductance could be maintained with ␦ ⫽ 2 nm, the resulting material would have an extremely low thermal conductivity of ⌳ ⫽ 0.1 W m⫺1 K⫺1. For comparison, the thermal conducDepartment of Materials Science and Engineering and Frederick Seitz Materials Research Laboratory, University of Illinois, Urbana, IL 61801, USA. 2Department of Chemistry and Biochemistry, University of Colorado, Boulder, CO 80309, USA.
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*To whom correspondence should be addressed. Email:
[email protected]
tivities of amorphous oxides (7) or strongly disordered crystalline oxides (8, 9) fall in the range 1.3 ⬍ ⌳ ⬍ 3.0 W m⫺1 K⫺1 at temperatures near 300 K. Although several examples of multilayers, superlattices, and nanocrystalline materials (10) have been studied (1), none show conductivities significantly below the values typical of disordered dielectrics. We prepared three sets of nanolaminate samples: two sets were synthesized by atomic layer deposition (ALD) with substrate temperatures of 177°C and 300°C, and a third set was deposited by magnetron sputtering with substrates near room temperature (11). In ALD, a cycle of gas-phase reactants is used to deposit a fixed amount of material per cycle. In the steady state, alternating exposures to Al(CH3)3 and H2O deposit 0.11 nm of amorphous Al2O3 (12), and each alternating exposure to WF6 and Si2H6 deposits ⬃0.5 nm of W (13). The nanolaminate samples are 40 to 70 nm in total thickness. The Si substrates are first coated by ⬃2 nm of Al2O3; the top layer of each nanolaminate is Al2O3. To measure the nanolaminate thermal conductivity, we applied time-domain thermoreflectance (14, 15), but modified the analysis to take
advantage of the extra information in the out-ofphase signal of the lock-in amplifier (11, 16, 17). The thermal model (17) used to analyze the raw data has several parameters, but the only important unknowns are the thermal conductance of the Al/Al2O3 interface at the top of the nanolaminate and the property we seek, the thermal conductivity of the nanolaminate. For example, a 10% change in the assumed value of the heat capacity of the nanolaminate film changes the best-fit thermal conductivity by only 1%. Film thicknesses are measured by picosecond acoustics or x-ray reflectivity; heat capacities are taken from literature values; and the thermal properties of the Si substrate are relatively unimportant, because of the small thermal conductance of the nanolaminate films. We separately measured the thermal conductance of an Al/Al2O3/Si stack using a 2-nm alumina film, and we used this conductance in the model to account for the series thermal conductance of the Al/Al2O3 and Al2O3/Si interfaces. G was 80 MW m⫺2 K⫺1 at room temperature and decreased to 35 MW m⫺2 K⫺1 at 80 K. The heat flow was primarily onedimensional, but our thermal model takes into account the full three-dimensional heat flow in cylindrical coordinates using the algorithm described by Feldman (18). The thermal conductivity of nanolaminate films at room temperature as a function of the interface density 1/␦ (Fig. 1) reveals that all three sets of samples exhibited similar behavior: The thermal conductivity decreased with increasing 1/␦, and the reduction in ⌳ began to saturate at 1/␦ ⬎ 0.4 nm⫺1. Sputtered nanolaminates showed a slightly higher thermal conductivity than ALD samples. At the highest interface densities, the thermal conductivity was a factor of ⬃4 smaller than the series average of the conductivities of the alumina and W layers. We analyzed these data to estimate the thermal conductance G of each individual W/Al2O3 interface. The dashed line in Fig. 1 shows the
Fig. 1. Room temperature thermal conductivity of W/Al2O3 nanolaminates as a function of interface density 1/␦; ␦ is the distance between interfaces. Solid circles, ALD nanolaminates deposited at 300°C; open circles, 177°C ALD deposition; solid triangles, nanolaminates deposited by magnetron sputtering. The dashed line is a fit using a constant W/Al2O3 interface thermal conductance of G ⫽ 260 MW m⫺2 K⫺1; the measured thermal conductivity of the alumina layers, ⌳ ⫽ 1.65 W m⫺1 K⫺1; and the thermal conductivity of W layers calculated from the electrical resistance and the Wiedemann-Franz law, ⌳ ⫽ 6.1 W m⫺1 K⫺1. The error bars reflect the combined uncertainties in film thickness, heat capacities, and reproducibility of the interface conductance at the top and bottom of the nanolaminate film.
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REPORTS Fig. 2. Temperature dependence of the thermal conductivity of the W/Al2O3 nanolaminate deposited at 177°C when ␦ ⫽ 2.9 nm (open circles). Data for a fully dense amorphous Al2O3 film prepared by ion-beam sputtering (solid triangles) (7) are included for comparison. The dashed line is the calculated minimum conductivity ⌳min for alumina; the solid line is ⌳ ⫽ ␦GDMM, where ␦ ⫽ 2.9 nm and GDMM is the calculated conductance of W/Al2O3 interfaces under the diffuse mismatch model (19).
expected thermal conductivity, assuming a constant G ⫽ 260 MW m⫺2 K⫺1 and fixed values for the thermal conductivities of the individual alumina and W layers. For the ALD nanolaminate with 1/␦ ⫽ 0.35 nm⫺1 and a deposition temperature of 177°C, the thermal conductivity was almost wholly dominated by this interface conductance, i.e., ⌳ ⬇ ␦G. This experimental value for G is close to the prediction of the diffuse mismatch model (DMM) (19). In this model of interface thermal transport, lattice vibrations are assumed to be scattered strongly at the interface and to have a transmission coefficient given by the ratio of the densities of vibrational states on either side of the interface. Using a Debye model for the densities of states, we calculated that GDMM ⫽ 320 MW m⫺2 K⫺1 for W/Al2O3. Typically, the DMM overestimates the conductance near room temperature, because this model, when based on a Debye density of states, does not take into account the dispersion of the vibrational modes. We continued the comparison between our data and the GDMM by examining the temperature dependence of the thermal conductivity (Fig. 2). The solid line in Fig. 2 is ⌳ ⫽ ␦GDMM; that is, the solid line shows the thermal conductivity of a hypothetical nanolaminate in which ␦ is 2.9 nm and the thermal conductivity is dominated by the diffuse mismatch value of the thermal conductance of the W/Al2O3 interfaces. The temperature dependence of the data and GDMM are similar, giving further support to our assertion that thermal transport in the nanolaminates is mostly controlled by the conductance of the interfaces. Interfaces between dissimilar materials such as W and Al2O3 are effective in reducing the thermal conductivity of nanostructured materials, but the relatively high interface energy will limit the stability of these materials at the high service temperatures typically required of ther-
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mal barrier coatings. Applications of nanolaminates as thermal barriers at temperatures higher than 1000°C would require the development of material interfaces that satisfy the conflicting demands of low thermal conductance and exceptional thermal stability. References and Notes
1. D. G. Cahill et al., J. Appl. Phys. 93, 793 (2003). 2. N. P. Padture, M. Gell, E. H. Jordan, Science 296, 280 (2002). 3. S. T. Huxtable et al., Appl. Phys. Lett. 80, 1737 (2002).
4. K. E. Goodson, J. Heat Transfer 118, 279 (1996). 5. C.-W. Nan, R. Birringer, D. R. Clarke, H. Gleiter, J. Appl. Phys. 81, 6692 (1997). 6. R. J. Stoner, H. J. Maris, Phys. Rev. B 48, 16373 (1993). 7. S.-M. Lee, D. G. Cahill, T. H. Allen, Phys. Rev. B 52, 253 (1995). 8. D. P. H. Hasselman et al., Am. Ceram. Soc. Bull. 66, 799 (1987). 9. K. W. Schlichting, N. P. Padture, P. G. Klemens, J. Mater. Sci. 36, 3003 (2001). 10. G. Soyez et al., Appl. Phys. Lett. 77, 1155 (2000). 11. Materials and methods are available as supporting material on Science Online. 12. A. C. Dillon, A. W. Ott, J. D. Way, S. M. George, Surf. Sci. 322, 230 (1995). 13. J. W. Klaus, S. J. Ferro, S. M. George, Thin Solid Films 360, 145 (2000). 14. C. A. Paddock, G. L. Eesley, J. Appl. Phys. 60, 285 (1986). 15. D. A. Young, C. Thomsen, H. T. Grahn, H. J. Maris, J. Tauc, in Phonon Scattering in Condensed Matter, A. C. Anderson, J. P. Wolfe, Eds. (Springer, Berlin, 1986), p. 49 –51. 16. D. G. Cahill, K. E. Goodson, A. Majumdar, J. Heat Transfer 124, 223 (2002). 17. R. M. Costescu, M. A. Wall, D. G. Cahill, Phys. Rev. B 67, 54302 (2003). 18. A. Feldman, High Temp. High Pressures 31, 293 (1999). 19. E. T. Swartz, R. O. Pohl, Rev. Mod. Phys. 61, 605 (1989). 20. Supported by NSF grant no. CTS-0319235, U.S. Department of Energy (DOE) grant no. DEFG02-01ER45938, and the Air Force Office of Scientific Research. Sample characterization used the Laser Facility of the Seitz Materials Research Laboratory and the facilities of the Center for Microanalysis of Materials, which is partially supported by DOE under grant no. DEFG02-91-ER45439. Supporting Online Material sciencemag.org/cgi/content/full/303/5660/989/DC1 Materials and Methods References and Notes 17 November 2003; accepted 29 December 2003
Improving the Density of Jammed Disordered Packings Using Ellipsoids Aleksandar Donev,1,4 Ibrahim Cisse,2,5 David Sachs,2 Evan A. Variano,2,6 Frank H. Stillinger,3 Robert Connelly,7 Salvatore Torquato,1,3,4* P. M. Chaikin2,4 Packing problems, such as how densely objects can fill a volume, are among the most ancient and persistent problems in mathematics and science. For equal spheres, it has only recently been proved that the face-centered cubic lattice has the highest possible packing fraction ⫽ /公18 ⬇ 0.74. It is also well known that certain random (amorphous) jammed packings have ⬇ 0.64. Here, we show experimentally and with a new simulation algorithm that ellipsoids can randomly pack more densely— up to ⫽ 0.68 to 0.71 for spheroids with an aspect ratio close to that of M&M’s Candies—and even approach ⬇ 0.74 for ellipsoids with other aspect ratios. We suggest that the higher density is directly related to the higher number of degrees of freedom per particle and thus the larger number of particle contacts required to mechanically stabilize the packing. We measured the number of contacts per particle Z ⬇ 10 for our spheroids, as compared to Z ⬇ 6 for spheres. Our results have implications for a broad range of scientific disciplines, including the properties of granular media and ceramics, glass formation, and discrete geometry. The structure of liquids, crystals, and glasses is intimately related to volume fractions of ordered and disordered (random) hard-sphere
packings, as are the transitions between these phases (1). Packing problems (2) are of current interest in dimensions higher than three
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