Numerical simulation of the heat transfer in amorphous ... - Physics

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4402 Rev. Sci. Instrum., Vol. 74, No. 10, October 2003 Revaz et al. FIG. 14. Variation of the square error in T2 as a function of the parameter c 2D,s for calorimeter with Cr film only. The sum is made for times from 4 to 120 ms, which excludes the first five points. Inset: residue plot of the fit using c 2D,s 0.5239 J/K cm 2 , which gave the lowest square error with c 2D,m 0.2669 J/K cm 2 , k 2D,m 0.194 W/K, k 2D,s 0.243 W/K, c 2D,Pt 0.69 J/K cm 2 ,andk 2D,Pt 0.11 W/K). in the sample area, in particular, two positions within the large area thermometer T1, with the c 2D and k 2D parameters found here, were also made, but only qualitatively matched the data for T1, which contain a complicated averaging due to variations in both its temperature and its temperature sensitivity a result of small compositional gradients in the Nb– Si. Finally, the specific heat of Cr is calculated from c 2D,s 0.257 J/K cm 2 , which gives C3.455 mJ/K g 180 mJ/K mol a density 7.19 g/cm 3 and the molar mass 52.0 g/mol have been used. IV. DISCUSSION Under conditions similar to our standard experimental operating conditions, with an 1800-Å-thick a-Si–N membrane, an 1800 Å Cu thermal conduction layer, and a sample with heat capacity c s , the simulations confirm the usual relaxation method fitting with a single exponential ext , and confirm the differential determination of c s from ext P/T c a , with T measured on either T1 or T2 and c a measured using the same relaxation method ( ext P/T) for the Cu layer alone. We note that c a can be measured on the same device before depositing the sample or on an equivalent device. The absolute accuracy is better than 2%, dependent on the ratio of sample heat capacity to addenda heat capacity. This small systematic error 2% is shown in Figs. 12 and in 11b for an 1800 Å membrane and 1800 Å Cu conduction layer, with the second sample layer being either Cu or any other material, k 2D,s /k 2D,m 100 and c 2D,s /c 2D,m 2. It is important to note that this absolute systematic error of 2% for any sample of any thickness refers only to the systematic errors discussed in this article, not the many other experimental errors inherent in measuring a thin film sample such as uncertainty in sample thickness and noise, both of which will introduce errors that depend on sample thickness. Figure 12 can be used to provide a first-order iterative correction to the experimentally determined heat capacity based on the measured values of c 2D,s and c 2D,m ; this is useful if other uncertainties such as film thickness are better than 2%, or if it is desired to eliminate the Cu layer such as for a sample of moderately high thermal conductivity. In a high precision experiment, c 2D,s for a real sample would be estimated from a traditional difference method: the sample’s estimated 2D heat capacity is determined from c tot (addendasample)c tot (addenda)/A s . Here, addenda sample refers to a device measured with Cu conduction layersample; addenda to a device with conduction layer only. This is then added to the Cu 2D heat capacity c 2D,Cu to give c 2D,s . The resulting estimate for the total c 2D,s /c 2D,m then gives a % correction to c 2D,s based on Fig. 12; this % correction would then be used to correct the sample heat capacity itself, since the Cu conduction layer contribution to c 2D,s did not change between the two measurements the systematic error comes from the changing value of membrane contribution X. When a Cu layer is used, k 2D,s /k 2D,m exceeds 100 for both addenda and addenda plus sample, and the difference in Fig. 12 between k 2D,s /k 2D,m 100 and 5 10 6 is small, hence, the increased value of k 2D,s /k 2D,m on adding the sample is likely to be not important. If no Cu layer is used, the changing value of k 2D,s /k 2D,m would need to be considered, and the errors and their corrections in k, c tot , and changing membrane contribution X calculated directly from Figs. 3, 8, and 10. The availability of two thermometers on opposite sides of the sample area with respect to the heater which is asymmetric gives the possibility of improving the accuracy of the specific heat measurement by making simulations of data as was done here for the Cr film. We note also that the quantitative simulation technique in this thin film limit could be improved by using a simpler design of the thermometry circuits on the microcalorimeter, in particular, matching the geometries of T1 and T2 to the isothermal contour lines. In addition, a quantitative study of the thermal properties of the Pt films would reduce the uncertainty in the thermal parameters used to describe the leads and heater. We briefly comment on the values obtained in this work. The thermal conductivity of low stress a-Si–N is significantly larger than that of vitreous silica and silica-based glasses at 20 K, but is consistent with other values reported for low stress a-Si–N. 15,20,27 We have also found that the plateau commonly observed in the thermal conductivity of insulating glasses is at 30–50 K in a-Si–N, while it is near 10 K in silica glasses, and the specific heat in J/g K of a-Si–N is smaller. 23 The thermal conductivity of the Cr film 23.5 mW/K cm is low, which is a consequence of the disorder of the sputtered film. Measurement of the resistivity of Cr films deposited under similar conditions show a residual resistivity of 15 cm, which gives Lk/T1.7310 8 W K 1 , in rough agreement with the Wiedemann–Franz relation L 2.4510 8 W K 1 . The specific heat of the Cr film measured in this work C Cr 180 mJ/K mol27 mJ/K mol. The error on C Cr is estimated at 15%, taking 5% errors on c m and c s . The same Cr film was measured with the single time Downloaded 13 Jun 2005 to 128.32.228.151. Redistribution subject to AIP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp
Rev. Sci. Instrum., Vol. 74, No. 10, October 2003 Simulation of membrane calorimeter 4403 constant technique, using an 1800-Å-thick Cu film as a thermal conduction layer. 16 In this previous measurement, we found C Cr 234 mJ/K mol70 mJ/K mol at 20 K; the contribution of the Cr film at 20 K was only 10% of the total heat capacity so that the error in c Cr was of order 30%. Using the numerical technique described here, with no Cu layer, c Cr is found with less uncertainty as it represents 50% of c tot .We note that this numerical technique is especially adapted to measurement of the low-temperature specific heat of superconducting films and particles for two reasons: 1 the thermal conductivity of metals is lowered in the superconducting state and 2 using a metallic thermal conduction layer would introduce spurious effects because of the proximity effect. The technique can be straightforwardly adapted to measurement of the specific heat of submicron particles deposited on the membrane as the local thermal parameters can be controlled in the calculation. ACKNOWLEDGMENTS The authors thank B. Woodfield for valuable discussions concerning calorimetry and the NSF, DOE, and Swiss National Fund for Scientific Research for support. 1 G. R. Stewart, Rev. Sci. Instrum. 54, 11983. 2 R. Bachmann, F. J. DiSalvo, Jr., T. H. Geballe, R. L. Greene, R. E. Howard, C. N. King, H. C. Kirsch, K. N. Lee, R. E. Schwall, H.-U. Thomas, and R. B. Zubeck, Rev. Sci. Instrum. 43, 2051972. 3 P. F. Sullivan and G. Seidel, Phys. Rev. 173, 679 1968. 4 D. W. Denlinger, E. N. Abarra, K. Allen, P. W. Rooney, M. T. Messer, S. K. Watson, and F. Hellman, Rev. Sci. Instrum. 65, 946 1994. 5 S. L. Lai, G. Ramanath, L. H. Allen, P. Infante, and Z. Ma, Appl. Phys. Lett. 67, 12291995. 6 F. Fominaya, T. Fournier, P. Gandit, and J. Chaussy, Rev. Sci. Instrum. 68, 4191 1997. 7 O. Riou, P. Gandit, M. Charalambous, and J. Chaussy, Rev. Sci. Instrum. 68, 1501 1997. 8 E. N. Abarra, K. Takano, F. Hellman, and A. E. Berkowitz, Phys. Rev. Lett. 77, 3451 1996. 9 F. Hellman, E. N. Abarra, A. L. Shapiro, and R. B. van Dover, Phys. Rev. B 58, 5672 1998. 10 K. Allen and F. Hellman, J. Chem. Phys. 111, 5291 1999; K. Allen and F. Hellman, Phys. Rev. B 60, 11765 1999. 11 B. L. Zink, E. Janod, K. Allen, and F. Hellman, Phys. Rev. Lett. 83, 2266 1999. 12 B. L. Zink, B. Revaz, R. Sappey, and F. Hellman, Rev. Sci. Instrum. 73, 1841 2002. 13 D. Kim, F. Hellman, and J. M. D. Coey, Phys. Rev. B 65, 214424 2002. 14 M. Zhang, M. Y. Efremov, F. Schiettekatte, E. A. Olson, A. T. Kwan, L. S. Lai, J. E. Greene, and L. H. Allen, Phys. Rev. B 62, 10548 2000; E.A. Olson, M. Yu. Efremov, A. T. Kwan, S. Lai, F. Schiettekatte, J. T. Warren, T. Wisleder, M. Zhang, V. Petrova, and L. H. Allen, Appl. Phys. Lett. 77, 2671 2000; M. Y. Efremov, F. Schiettekatte, M. Zhang, E. A. Olson, A. T. Kwan, R. S. Berry, and L. H. Allen, Phys. Rev. Lett. 85, 3560 2000. 15 B. L. Zink, B. Revaz, and F. Hellman, Rev. Sci. Instrum. submitted. 16 B. Revaz, M.-C. Cyrille, B. L. Zink, Ivan K. Schuller, and F. Hellman, Phys. Rev. B 65, 094417 2002. 17 Partial Differential Equation Toolbox User’s Guide, The MathWorks, Inc. 1995. 18 D. G. Cahill, H. E. Fischer, T. Klitsner, E. T. Swartz, and R. O. Pohl, J. Vac. Sci. Technol. A 7, 1259 1989. 19 T1 and T2 were incorrectly drawn in Ref. 4. 20 W. Holmes, J. M. Gildemeister, P. L. Richards, and V. Kotsubo, Appl. Phys. Lett. 72, 2250 1998. 21 C. Mastrangelo, Y.-C. Tai, and R. S. Muller, Sens. Actuators A 23, 856 1990. 22 L. Kiesewetter, J.-M. Zhang, D. Houdeau, and A. Steckenborn, Sens. Actuators A 35, 153 1992. 23 B. L. Zink and F. Hellman, Solid State Commun. accepted. 24 Specific Heat: Elements and Metallic Alloys, edited by Y. S. Touloukian IFI/Plenum, New York, 1970. 25 I. Y. Guzman, A. F. Demidenko, V. I. Koshchenko, M. S. Fraifel’d, and Y. V. Egner, IEEE Eng. Med. Biol. Mag. 12, 1546 1976. 26 B. Revaz, Ph.D. thesis, University of Geneva 1997 unpublished. 27 R. C. Zeller and R. P. Pohl, Phys. Rev. B 4, 2029 1971. Downloaded 13 Jun 2005 to 128.32.228.151. Redistribution subject to AIP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp
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