232 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 21, NO. 3, JUNE 2011 Thermal Properties of Silicon Nitride Beams Below One Kelvin G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C.L. Chang, L. Bleem, A.T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom Abstract—We have investigated thermal properties of 1 thick silicon nitride beams of different lateral dimensions. We measured the thermal conductance by simultaneously employing a TES both as a heater and as a sensor. Based upon these mea- surements, we calculate the thermal conductivity of the beams. We utilize a boundary limited phonon transport model and as- sume a temperature independent phonon mean free path. We find that the thermal conductivity is determined by the frac- tion of diffusive reflection at surface. The following results are obtained from 0.30 K to 0.55 K: the volume heat capacity is . The width dependent phonon mean free path is 6.58 , 9.80 and 11.55 for 10 , 20 and 30 beams respectively at a 29% surface diffusive reflection. Index Terms—Heat transport, phonons, superconducting tran- sition edge sensor, thermal conductivity. I. INTRODUCTION L OW stress silicon nitride film has been widely used as mechanical supports and weak thermal links in bolometric detectors and micro-calorimetric detectors using su- perconducting Transition Edge Sensor (TES). thermal conductivity and the layout geometry define the thermal con- ductance of a cryogenic detector. Heat transport in an insulator such as is realized through phonon propagation. At a low temperature, phonon propagation in film reveals two limits: a phonon radiative Manuscript received August 02, 2010; accepted October 12, 2010. Date of publication November 29, 2010; date of current version May 27, 2011. The work at Argonne National Laboratory, including the use of facility at the Center for Nanoscale Materials (CNM), was supported by the Office of Science and Of- fice of Basic Energy Sciences of the US Department of Energy, under Contract DE-AC02-06CH11357. Technical support from Nanofabrication Group at the CNM, under User Proposal #164, #467 and #750, is gratefully acknowledged. The work at the University of Chicago was supported by the NSF through Grant ANT-0638937 and the NSF Physics Frontier Center Grant PHY-0114422 to the Kavli Institute of Cosmological Physics at the University of Chicago. It also re- ceives generous support from the Kavli Foundation and the Gordon and Betty Moore Foundation. G. Wang, V. Yefremenko, V. Novosad, A. Datesman, and J. Pearson are with Materials Science Division, Argonne National Laboratory, Argonne, IL 60439 USA (e-mail: gwang@anl.gov). R. Divan is with the Center for Nanoscale Materials, Argonne National Lab- oratory, Argonne, IL 60439 USA (e-mail: divan@aps.anl.gov). C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, S. S. Meyer, and J. E. Carlstrom are with Kavli Institute for Cosmological Physics, Chicago, IL 60637 USA (e-mail: clchang@kicp.uchicago.edu). J. McMahon is with Physics Department, University of Michigan, Michigan 48109, USA (e-mail: jeffmcm@umich.edu). J. Ruhl and J. Sayer are with Physics Department, Case Western Reserve Uni- versity, Cleveland, Ohio 44106, USA (e-mail: ruhl@case.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASC.2010.2089407 ballistic limit (also called surface specular reflection limit) and a Casimir [1] limit (also called surface diffusive reflection limit). The radiative ballistic phonon transport [2]–[4] appears at extremely low temperatures, for example, below100 mK. The diffusive limit appears at much higher temperatures [3]. In the phenomenological phonon gas kinetic theory, thermal conductivity is written as (1) where is volume heat capacity, is phonon mean free path, and is the average sound speed cal- culated with its longitudinal and transverse sound speeds [5]. The phonon mean free path could be much less than the beam length for a long narrow beam. Therefore, a phonon dif- fusive formulation can be used approximately for a description of heat transport. The heat flux is defined by Fourier’s law, (2) where is the temperature gradient along the beam. is constant at a steady state. Therefore, the heat conduction power from the hot end to the bath is (3) where is the beam length, is its cross section, is the tem- perature at the hot end, is the bath temperature. For a defined thermal structure at a steady state and by using Lebniz’s rule, its thermal conductance is (4) Similarly, . The minus sign means that decreases when increasing at a fixed TES transition temperature. Empirically, the heat power flow along the beam as a function of bath temperature for a defined thermal structure is fitted as a power law [6]–[9] of (5) where coefficient and temperature power index charac- terize thermal conductance amplitude and temperature depen- dence of heat capacity and phonon mean free path. In our in- vestigation temperature range, we assume a temperature inde- pendent phonon mean free path and a power law dependence of heat capacity on temperature. The thermal conductance is the first derivative of the power over temperature, (6) 1051-8223/$26.00 © 2010 IEEE