numerical simulation of the heat transfer in...

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 74, NUMBER 10 OCTOBER 2003

Numerical simulation of the heat transfer in amorphous silicon nitridemembrane-based microcalorimeters

B. Revaz,a) B. L. Zink, D. O’Neil, L. Hull, and F. Hellmanb)

Department of Physics, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093

~Received 9 December 2002; accepted 7 July 2003!

Numerical simulations of the two-dimensional~2D! heat flow in a membrane-basedmicrocalorimeter have been performed. The steady-state isotherms and time-dependent heat flowhave been calculated for a wide range of sample and membrane thermal conductivities and heatcapacities. In the limit of high internal thermal conductivity and low membrane heat capacity, thesample heat capacity determined using the relaxation method with a single time constant is shownto be exact. The fractional contribution of the square 2D membrane border to the total heat capacityis calculated~;24%!. Analysis of the steady-state isotherms provide the 2D geometric factor~10.33! linking membrane thermal conductance to thermal conductivity, allowing extraction of thethermal conductivity of either the membrane itself or a sample deposited everywhere on themembrane. For smaller internal thermal conductivity and/or larger membrane heat capacity,systematic errors are introduced into the determination of heat capacity and thermal conductivity ofa sample analyzed in the standard~single time constant! relaxation method, as has been previouslyshown for one dimension. These errors are due to both the changing contribution of the membraneborder and to deviations from the ideal semiadiabatic approximation of the relaxation method. Theerrors are here calculated as a function of the ratios of thermal conductivity and heat capacity ofsample and membrane. The differential method of measurement in which the sample heat capacityis taken as the difference between a relaxation method measurement with and without the sample isshown to give significantly smaller errors than the absolute errors of a single measurement. Understandard usage, high internal thermal conductivity is guaranteed by use of a thermal conductionlayer such as Cu. The systematic error in this case is an underestimate of true sample heat capacityby less than 2%. The simulation was extended to thermal conditions where a single time constantrelaxation approximation cannot be used, specifically, for a sample with low thermal conductivity.Because of the highly precise geometry of these micromachined devices, a comparison betweenmeasured and simulated steady-state and time-dependent temperatures is demonstrated to allowextraction of the heat capacity and thermal conductivity of this sample with less uncertainty due toelimination of the Cu heat capacity. ©2003 American Institute of Physics.@DOI: 10.1063/1.1605498#

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I. INTRODUCTION

Understanding the thermal behavior of mesoscopic stems and thin films is a critical issue of solid-state scienboth for fundamental and technological reasons. Conventally, specific heat is measured by thermally isolatingsample from the environment, introducing a known amoof heat, and measuring the resulting temperature rise ofsample. Thermal conductivity is measured by inducing ameasuring a temperature gradient along a sample witwell-defined geometry. Both of these measurements beccomplicated for thin films and small samples.1 Issues thatarise include an inability to completely thermally isolate tsample from the environment due to significant heat flthrough the electrical leads used for measuring the tempture and the large background contributions from substrathermometers, and heaters which overwhelm the sm

a!Present address: Department of Condensed Matter Physics, UniversGeneva, 1211 Geneva 4, Switzerland.

b!Electronic mail: [email protected]

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sample signal. The first problem has been addressed inous ways, including the ac method and the relaxatmethod.2,3 These both require a semiadiabatic arrangemenwhich the sample is thermally well connected to a heaterthermometer, and thermally well-enough isolated fromenvironment to permit a separation of internal and exterthermal time constants,t int!text. The second problem, duto the large background contribution~called the addenda!, isdealt with by increasing the precision of the measuremand/or reducing the background by utilizing small thermoeters, substrates, etc.

The development of nanofabrication and Smicromachining techniques in the last 10–15 years has lethe use of thin membranes for substrates and thin film hers, thermometers, and electrical leads, enormously reduthe addenda.4–7 Membrane-based devices are now beiused in commercial calorimeters, and a program to devetheir use in high magnetic fields, including pulsed fields, hbegun at the National High Magnetic Field Laboratory.continuous membrane-based devices, a thermal conduclayer is generally used to give an isothermal sample area

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9 © 2003 American Institute of Physics

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4390 Rev. Sci. Instrum., Vol. 74, No. 10, October 2003 Revaz et al.

heat flow, however, less clearly meets the desired thercondition oft int!text and the contributions of the membranborder and electrical leads to the addenda heat capacitynot obviously defined. In particular, the low thermal diffusiity of the membrane (5k/Cr, for thermal conductivityk,specific heatC, and densityr! connecting sample to environment is significantly different than the more usual meleads which have much higher diffusivity. Despite thecreasingly widespread use of membrane-based calorimethere has not been a quantitative model of the twdimensional~2D! heat flow.

We have for many years used an amorphous~a-! Si–Nmembrane-based calorimeter with thin film heater~Pt!, ther-mometers~Pt for high temperatures;a-Nb–Si alloys for lowtemperature!, and electrical leads~Pt! to measure the specifiheat of a wide variety of thin film and small single crystsamples over a wide temperature range in magnetic fieldto 8 T.8–13 Somewhat similar membrane-based devicesused by Allen’s group for high-temperature scanning calometric studies, e.g., of the melting of nanoparticles.14 Thegeometry of our membrane-based calorimeter is shownFig. 1. The sample is deposited onto the central 0.2530.25 cm area of one side of the 1800-Å-thick 0.5 cm30.5cm continuous membrane; the heater and thermometerson the opposite side, electrically isolated but thermally wconnected to the sample~this point will be further discussedbelow!. Thermal isolation of the sample from the enviroment~long text) is achieved by the low thermal conductanof the thin amorphous membrane surrounding the censample area. Good internal thermal conductivity~short t int)is created by depositing a film of high thermal conductiv~e.g., Ag, Au, Al, and Cu! under or over the sample, in thsame central 0.25 cm30.25 cm area.

We use the relaxation method for measurements, alling us to test that we have met the conditiont int!text. Inthis method, a known amount of powerP is dissipated in theheater, causing the sample temperature to riseDT above theenvironment temperatureT0 . At time t50, P is set to zeroand the time dependence of the relaxation of the sampleperatureDT(t) is measured. In the limit,t int!text, where asingle time constantDTe2t/text is seen in the temperaturdecay, the heat capacity of the sample plus addendacs1ca isgiven bytextk, wherek5P/DT ~in W/K! is the thermal linkconnecting sample area to environment.~In this article,c willrefer to an extensive heat capacity andC to an intensivespecific heat per mole or per g!. The sample specific heat idetermined by making a separate measurement of a dewith thermal conduction layer only, givingca5text8 k8. Ex-perimentally, it has been shown thatk85k to the accuracy ofthe measurement~;1%!, for devices taken from the samprocessing wafer, when corrections are made for variatiin the thickness of the Pt leads~based on the measured rsistance of the Pt heater and precise knowledge of its awithout this correction, variations are several %!.15,16

This differential technique of measurement, in whichca

is measured separately fromcs1ca is a crucial part of therelaxation method, not simply a mathematical constructiWe will show from the simulations that the systematic er~for high but not infinite internal thermal conductivity! made


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FIG. 1. ~a! Photolithographic layout of calorimeter, including outer 1 cm31cm Si frame~shaded! and 0.5 cm30.5 cm membrane area~clear!. Shadedsquare at center is 0.25 cm30.25 cm sample area. Dark lines are Pt; serpetine paths are Pt heater and thermometer, straight lines are leads to thesto the two Nb–Si thermometers T1 and T2~cross-hatched regions!; ~b!layout used for numerical simulation; 0.5 cm30.5 cm membrane area onlyThe heater is drawn in the central sample area as well as the Pt lconnecting the Nb–Si T2 thermometer; Pt thermometer and leads to T1not included in most simulations as they did not significantly changeresults but added computation time. The position of the thermomete~0;10.1! ~T1! and~0;20.1! ~T2!; and~c! mesh used to solve the heat eqution consisting of 6000 nodes. All lengths in cm.


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4391Rev. Sci. Instrum., Vol. 74, No. 10, October 2003 Simulation of membrane calorimeter

in measuringcs differentially is less than the systematic errmade in each measurement separately.

Adding the thermal conduction layer is essential to guanteeingt int!text, but has the disadvantage of increasithe background heat capacityca . The specific heat and themal conductivity of the sample also determine the relaxatDT(t) even without the thermal conduction layer althouthe relaxation cannot then be fit by a single time constaSince the geometry of the microcalorimeters is known wgreat accuracy, it is possible to perform a numerical simution of the heat transfer and to adjust the thermal parameuntil the numerical results fits the measured steady staterelaxation response.

In this article, we present numerical simulations of tsteady-state and time-dependent heat flow in the Smembrane-based calorimeter both with and without a thmal conductivity layer. These simulations yield three imptant results. First, with the thermal conduction layer, in tlimit of high ratios of sample/membrane thermal conductity and specific heat, the relaxation method using a sintime constanttext is shown to be exact, and the exact contbutions of the 2D membrane border and the@not quite one-dimensional~1D!# electrical leads to the addenda heat capity ca are derived. In this same limit, the 2D geometry facconnecting the intrinsic thermal conductivity of the membrane to the thermal linkk is derived~including the effectsof the Pt leads!. This 2D geometric factor allows us to extrathe thermal conductivity of a sample deposited onto theen-tire 0.5 cm30.5 cm membrane~in the limit that the thermalconductance of sample and membrane add in parallelthat radiation can be neglected!. Second, as the ratios osample to membrane k andC are reduced, we derive thsystematic error in the standard relaxation method of demining cs , and show a method for making a first order corection to compensate for this systematic error. We also donstrate that using a differential technique~relaxationmethod measurements of microcalorimeter before and aadding the sample! significantly reduces this systematic errcompared to calculations based on a single measuremMeasurements with microcalorimetry devices under standoperating conditions (a-Si–N membrane and Cu thermconduction layer of comparable thickness! are shown to havean absolute accuracy of better than 2% at all temperatubased on the ratio ofa-Si–N to Cu thermal conductivity andspecific heat. Third, the steady state and time dependenDTfor a sample of low thermal conductivity~with no parallelthermal conduction layer, wheret int;text) is analyzed anddemonstrated to allow extraction ofcs with somewhatgreater accuracy than the usual single time constant mepermits due to reducedca . In this limit, an improved ther-mometry design is proposed which would increase the acracy still further.

II. SIMULATION AND EXPERIMENTAL DETAILS

The numerical simulation of heat flow used the Matlabpackage~Mathworks! pdetool.17 As the thickness of themembrane is 1.831025 cm, orders of magnitude smallethan the characteristic lengths in the plane, the tempera


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can be safely approximated with the solution of the 2D htransfer equation. In this limit, the time constant associawith thermal diffusion in the perpendicular direction is mufaster than the relaxation in the in-plane direction. The teperature is, therefore, assumed not to depend on the vercoordinate and the three-dimensional~3D! heat transferequation can be integrated along the vertical direction. Trelevant local coefficients for the 2D differential equatiobecome k2D5k•t, c2D5r•t•C and P2D5Ptot /S wherek (W/K cm) is the thermal conductivity,C (J/Kg) is the spe-cific heat per unit mass,c2D ~J/K cm2) the heat capacity peunit surface area,r ~g/cm3! is the density,t (cm) is the thick-ness of the film,Ptot (W) is the total power dissipated in thheater, andS(cm2) the surface area of the heater. When twfilms are superposed, it is assumed that the superpositiolinear, meaning that the local conductivityk2D ~W/K) ~orheat capacityc2D) is the sum of the two separate contribtions. For example, for positions (x,y) where a sample~suchas Cr or Cu! is on the a-Si–N membrane,k2D5k2D,s

1k2D,m , and c2D5c2D,s1c2D,m . ~When both a samplesuch as Cr,andCu are on the membrane,c2D,s andk2D,s willrefer to thesumof their separate contributions!. At very lowtemperatures, these two assumptions—2D approximatiothe heat transfer and linear superposition ofk2D andc2D fortwo or more layers—would need to be reconsidered dueKapitza thermal boundary resistance and possible changsurface scattering once the phonon mean free path excthe membrane thickness~experimentally estimated to occusomewhat below 10 K for thickness 1.831025 cm).15,18

An expanded view of the geometry used for the calcution is shown in Fig. 1~b!. The 0.5 cm30.5 cm square iscovered witha-Si–N and the sample~the term ‘‘sample’’will include a Cu conduction layer in the sample space! islocated on the 0.25 cm30.25 cm square at the center~depos-ited directly on thea-Si–N). Power is assumed to be disspated uniformly in the areas occupied by the Pt heater@theserpentine path in Fig. 1~b!#. The a-Si–N membrane is deposited on a single crystal Si~100! chip ~with good thermalconductivity!, which is attached to a sample holder whotemperature is regulated at a block temperatureT0 . We,therefore, fixed the external boundary of the 0.5 cm30.5 cmsquare to be at a constant temperatureT0 . Experiments areperformed under high vacuum, so convective heat tranand conduction through a gas can be neglected~experimen-tally, high vacuum is essential as heat transfer throughgas is highly variable and irreproducible!. Radiative heattransfer is here neglected, but in fact contributes significanat temperatures above approximately 100 K; an analysithe radiative contribution is given in Refs. 4 and 15. Tthermal contributions of the Pt heater and wide heater le@shown in Fig. 1~b!# were included in all calculations.

The temperature is considered at position~0;10.1! and~0;20.1!, corresponding to the centers ofa-NbxSi12x ther-mometers T1 and T2, respectively. The lower-temperatulower-resistance thermometer T1 is a wide, short resispath ~;230.03 mm2!; the higher-temperature, higheresistance thermometer T2 is a narrower, longer resistor~;0.830.2 mm2!.19 Any nonuniformity of temperaturewithin the sample area will have a bigger impact on T1 th


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T2 due to this difference in aspect ratios. The thermal ctribution from the narrow Pt leads of thermometer T2 weincluded but changed the results very little. The contributfrom the narrow leads of thermometer T1 and from a secnarrow Pt thermometer@the interleaved serpentine path sein Fig. 1~a!# affected the calculations even less~less than 1%change at position T1!, and were, therefore, neglectedtheir inclusion more than doubled the calculation time.

The 2D heat transfer equation

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was solved using the finite element method as describeRef. 17, using a 6000 node mesh drawn on the geometrFig. 1~b! using an adaptive mesh generator, as shown in1~c!. The relative error tolerance, which controls the numof correct digits, was set to 1028 for all calculations. Thisvalue corresponds to the accuracy of the computed soluThe absolute error tolerance, which determines the accuwhen a solution approaches zero, was set to 1029.

To provide a comparison with the simulations, at teperatureT0520.3 K, measurements of both steady-statetime-dependent temperatures were made for:~1! an empty~bare! membrane device with heaters and thermometersno sample or thermal conduction layer;~2! a similar deviceafter depositing a 1035-Å-thick Cr sample; and~3! the samedevice after depositing an 1800-Å-thick Cu thermal condtion layer onto the Cr sample. Measurements~2! and ~3!were performed on the same device; measurement~1! on adifferent device from the same wafer which limits the vartion to a few percent.4,16 The Cr film was prepared by asputtering. Its thickness was measured on a Si substratcated close to the calorimeter during deposition by Dekprofilometry. The specific heat of the Cr sample was preously reported from 4 to 300 K in Ref. 16, derived using tstandard single time constant analysis of device~3! with theCu conduction layer. The thickness of thea-Si–N membrane~1800 Å! was determined by ellipsometry. The Cu was dposited by thermal evaporation and its thickness determby Dektak profilometry on a neighboring substrate. For edevice, powerP50.576mW was dissipated in the 1500V Ptheater, causing an increase of temperature of approxima0.3 K. This temperature increase is kept small so thattemperature variation of the thermal parameters is negligiThe temperature of the sample area is obtained fromresistance of the highly sensitivea-NbxSi12x thermometersT1 and T2, which are calibrated during each measuremagainst a commercial Cernox thermometer~see Ref. 4 fordetails!. Measurements were made both in steady state~con-stantP! and as a function of time after turning off the heapower.

The parameters needed for the simulation,k2D,Pt, k2D,m ,k2D,s , c2D,Pt, c2D,m , andc2D,s , all depend on temperatureThis parameter space is made even larger by the neesimulate the effects of adding a second sample layer on


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first layer ~the differential method!, with these two layershaving typically quite different properties~e.g., Cu conduc-tion layer and sample!. Only the ratios of these numbers aimportant to the primary goals of this work~establishinglimits of accuracy of the relaxation method and contributioto the addenda!, but this still leaves a large parameter spaSimulations were therefore performed using values ofk2D,Pt,k2D,m , c2D,Pt, and c2D,m determined at 20.3 K, withk2D,s

andc2D,s for both the first and second layers allowed to vaThe results obtained will be plotted as a functionk2D,s /k2D,m and c2D,s /c2D,m , since these are the importanparameters that control the absolute accuracy of the reation method. The parameter space is too large to alsoclude all possible values ofk2D,Pt andc2D,Pt. We will showthat the effects of the Pt are relatively small and are lineaadditive. The majority of simulations were therefore peformed with fixed Pt values. The results obtained cangeneralized to other temperatures and other values ofk2D,m

andc2D,m , as long as the Pt~and Nb–Si or other thermometers! remains a relatively small~,20%! contribution to thetotal heat capacity and thermal conductance.

The thermal conductivity and specific heat of the mebrane at 20.3 K will be determined experimentally by coparison with simulations on an empty membrane devicewell as one with a Cu conduction layer, as discussed inResults section below. The thermal conductivity of Pt at 2K was determined from the measured resistivity of theheater at 20.3 K, where it has empirically reached a limitresidual resistivity. Assuming the validity of thWiedemann–Franz relation in the residual resistivity limwe obtain kPt522 mW/K cm andk2D,Pt50.110mW/K ~Ptthickness is 500 Å!. The Pt heat capacitycPt was calculatedfrom a Debye temperature of 240 K and a coefficient oflinear electronic term of 6.80 mJ/K2 mol, giving cPt

51.23 J/K mol andc2D,Pt50.69mJ/K cm2 at 20.3 K. The ef-fect of the Pt contributions was explicitly studied by allowing this to vary in some simulations.

III. RESULTS

A. Steady state: TÕtÄ0

In this section, we first calculate the isothermal contolines in steady state for constant powerP for various ratios ofinternal to external thermal conductivity. We then compathese simulations with the experimentally determined valof DT using both thermometers T1 and T2, for an empmembrane, a membrane with a low conductivity sam~Cr!, and a membrane with the standard high thermal cductivity Cu sample. These experiments and the comparwith the simulations allow us to extract the thermal condutivity of the a-Si–N comprising the membrane and of theand give us the 2D geometry factor connecting the therconductivityk with the thermal conductancek. The system-atic error introduced by a less than infinite internal thermconductivity is then derived and discussed, and the ratiointernal to external thermal conductivity under typical eperimental conditions~a Cu thermal conduction layer! isplotted as a function of temperature, giving an upper limitthe systematic error in k of less than approximately 1.5under typical conditions.


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FIG. 2. Isothermal contour plots for (k2D,s

1k2D,m)/k2D,m5~a! 1 ~empty membrane!, ~b! 2 ~e.g.,Cr only!, ~c! 10, ~d! 50, ~e! 100 ~standard experimentaconditions with Cu conduction layer!. For ~a!–~c!, 5%contours are shown; for~d! and ~e! 2% contours areshown. For these figures,k2D,Pt50.11mW/K was used,the value estimated at 20.3 K.

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1k2D,m)/k2D,m usingk2D,Pt50.110mW/K ~the value at 20.3K!. (k2D,s1k2D,m)/k2D,m is the ratio of the 2D thermal conductivity in the central sample area which consists of samand membrane (5k2D,s1k2D,m) to the 2D thermal conduc


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tivity of the membrane border area. The highest temperais found near the center of the sample area, at the samheater. Twenty equally spaced contours are shown for raof 1, 2, and 10; 50 are shown for ratios of 50 and 100~inorder to display the small gradient still seen in the samarea!. The isothermal contours do not depend onT0 or onP,


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but they do require thatk2D,s and k2D,m be independent otemperature over the temperature intervalDT, which there-fore must be small compared toT0 . When the thermal conductivity ratio is small, e.g., 2, the 0.25 cm30.25 cm centralsample area is far from isothermal; the temperature drfrom its maximum value at the heater to approximately 7of this at the position of thermometer T2 and 65% atposition of T1. When this ratio is large, e.g., 100, the 0.cm30.25 cm sample area is isothermal to within 2%, andtemperature gradient is nearly entirely found in the bormembrane surrounding the sample area. It is importannote that these % are relative toDTmax at the heater positionwhich is of order 0.01* T0 , and hence, reflect very smaoverall temperature differences.

The effect of the Pt leads was studied by repeatingabove simulations while varying k2D,Pt over a range from 0to 0.18mW/K. With increasingk2D,Pt, DT decreases for constantP, as expected due to this increasing thermal link. Tsimulations show thatP/DT increases linearly over thirange, as would be expected from a simple addition of pallel thermal conduction paths~membrane1Pt leads!. ~At 20K, radiation is negligible, sok is due to thermal conductionthrough the membrane and the Pt leads.! Comparisons of theshapes of the isotherms calculated with and without theleads to thermometer T2 and as a function ofk2D,Pt showvery little difference, although they slightly increase the thmal homogeneity around T2. Including the T1 Pt leawould similarly slightly improve the thermal homogeneitbut added significant calculation time and hence wereout.

As the sample thermal conductivityk2D,s decreases, thesimulatedDT at T1 and T2 increases~for the sameP!, eventhough bothk2D,m and k2D,Pt are unchanged, because tsample area is no longer isothermal. In the standard reation method measurement, with a high intemal thermal cductivity layer such as Cu,P/DT is the thermal linkk ~W/K!connecting the sample to the environment, used to obtainheat capacityc5textk. Figure 3 shows the calculated depedence ofP/$DT2% on k2D,s ~with k2D,m fixed at 0.190mW/Kand P50.576mW), including the effects of the Pt leadshown in Fig. 1~b!. DT2 begins to increase andP/$DT2% todecrease below the high ratio limit fork2D,s,100* k2D,m ; itis reduced by 20% byk2D,s54* k2D,m .

Although the detailed temperature profiles cannot berectly measured with the present design of the microcaloreters, the temperatures measured by thermometers T1 anprovide experimental verification of results shown in Figsand 3, and allow us to extract parameters of the devices,as the thermal conductivity of the membrane and thesample layer, and an approximate value for the Cu usedthermal conducting layer. The experimentally measuredDTandP/DT for thermometers T1 and T2 are reported in TaI for power P50.576mW dissipated in the heater. With thCu thermal conduction layer~measurement 3!, the tempera-ture difference between T1 and T2 is small, of order 1consistent with Fig. 2~e!, suggesting a thermal conductivitratio of approximately 100. In measurements 1 and 2, wout the Cu layer,DT is seen in Table I to be significantllarger than it is in measurement 3, and is dependent on t


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mometer, consistent with Fig. 2. For the empty membra~measurement 1!, because thermometer T1 extends over sua large area, covering several isotherms,DT1 is not mean-ingful and is not reported. For the device with Cr samponly ~measurement 2!, the difference in temperaturesDT1and DT2 is significant and consistent with the isothermshown in Fig. 2~b!, which show T2 at approximately the 20%isotherm and the more distributed thermometer T1 extendfrom approximately the 35% to the 45% isotherms.

From the measuredDT shown in Table I, we are able todetermine the thermal conductivity of the membranek2D,m at20.3 K through comparison with numerical simulations,cluding the effects of the Pt leads (k2D,Pt50.110mW/K). Wevaried k2D,m so that the calculated temperatureDT2 at theposition ~0;20.1! equals the experimental values givenTable I for measurements 1~bare membrane! and 3 ~mem-brane plus Cu conduction layer in sample area!, where thethermal conditions and the measuredDT ~for the sameP! arequite different. Two independent values ofk2D,m are therebydetermined. The values arek2D,m50.194 and 0.186mW/Krespectively, close to each other, as expected from the re

FIG. 3. Simulated values ofP/DT with DT determined at~0;20.1!, theposition of thermometer T2, as a function of the sample 2D thermal cductivity k2D,s for k2D,m50.190mW/K ~found from matching the measureDT in Table I!, using P50.576mW and the calculatedk2D,Pt

50.11mW/K. For high intemal thermal conductivity~large k2D,s /k2D,m),P/DT5k, the thermal link which includes thermal conduction through bothe membrane and the Pt leads. The contribution of the Pt leads is appmately 8% of the totalk.

TABLE I. Measured temperatures at thermometers T1 and T2 for poP50.576mW dissipated in the heater with the Si frame held at temperatT0520.3 K for an empty membrane, a membrane with a 1035 Å Cr layethe sample area, and a membrane with both Cr and Cu conduction laysample area. With the Cu conduction layer,P/DT5k, the thermal linkconnecting the sample to the environment used to obtain the totalcapacityctot5textk.

Measurement ~1! Empty membrane ~2! Cr only ~3! Cr1Cu

DT2 ~K! 0.437 0.372 0.279P/DT2 (mW/K) 1.32 1.55 2.07DT1 ~K! ~Not well defined! 0.291 0.275P/DT1 (mW/K) ~Not well defined! 1.98 2.10


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ducibility of these micromachined devices. The small diffence between these values, measured on two differentvices, are attributed to slightly different thickness aperhaps composition of thea-Si–N membrane. They are ithe range of variation reported in other work.4,12The averageof these valuesk2D,m50.190mW/K gives the thermal con-ductivity of thea-Si–N membranekm510.6 mW/K cm at 20K, consistent with other measurements of low strea-Si–N.15,20 In the following calculations, we will use thevaluek2D,m50.194mW/K when measurements made on tempty microcalorimeter are under consideration and 0.mW/K for the Cr or Cr1Cu microcalorimeter.

Next, we consider measurement 2. Here, the 2D therconductivity of the Cr is added to that of the membrane~andthe Pt in the places shown in Fig. 1, usingk2D,Pt

50.110mW/K) in the central 0.25 cm30.25 cm samplearea. Usingk2D,m50.186mW/K found above, we variedk2D,s to fit DT2, giving (k2D,s1k2D,m)50.429mW/K, andk2D,s50.243mW/K. The thermal conductivity of the Cr filmkCr is then calculated fromkCr5k2D,s /tCr523.5 mW/K cmfor tCr51035 Å. This gives (k2D,s1k2D,m)/k2D,m52.3.From the isotherms in Fig. 2~b!, we estimate the expectevalue ofDT150.30 K, very close to the measured value0.29 K ~note again that the distributed nature of T1 makea less good thermometer when the sample area is not isomal!.

For a high ratio of internal to external thermal condutivity ( k2D,s1k2D,m)/k2D,m , we calculate the 2D geometricafactor relating the thermal linkk ~[measuredP/DT) to thelocal, microscopic parameterk2D,m . In the limit of thermalconduction solely through the 2D membrane, i.e., wk2D,Pt50, we find thatk510.33k2D,m510.33kmtm . As dis-cussed above, the simulations show that the Pt leads conute linearly to the thermal linkk. Thus,

k2D,m5ktm5S P

DT2kPtD 1

10.33, ~2!

wherekPt5k2D,Pt w/l is the thermal link of the Pt leads caculated from their lengths and widths.

The 2D geometrical number 10.33 can be qualitativunderstood by considering a 1D model of the membranehaving four equal ‘‘legs,’’ each with widthw50.25 cm andlength l 50.125 cm, thereforew/ l * 458, plus a contributionfrom the square area corners which was estimated gracally in our original work as an additional 20%, or coueven more naively be considered as1

2 of 4 squares, i.e., 2giving a naive geometry factor of;10.4

This analysis also allows us to determine the thermconductivitykf of a film of thicknesst f deposited on top ofthe whole membrane including the border area, with a thCu thermal conduction layer in the central sample area oIn this configuration (k2D,s1k2D,m)/k2D,m@1, hence, thethermal conductivitykf of the film is found from the follow-ing expression:

kf5F S P

DT2kPtD 1

10.332k2D,mG 1

t f, ~3!


-e-

s

6

al

fiter-

-

ib-

yas

hi-

l

ky.

where kPt is the thermal conductance of the Pt leads (kPt

5kPtt/@Lw#, andt, L, andw are the Pt thickness, and lengand combined width of the lithographically defined Pt lead!.We have used this technique to measure the thermal contivity of various thin film samples over a wide temperaturange, including effects of radiation losses which becoappreciable above 100 K.15

In order to extend these results to the entire temperarange of measurement, k2D,m(T) and the ratiok2D,s /k2D,m(T) are needed, wherek2D,s is the conductivityof the Cu thermal conduction layer deposited in the samarea. To obtain these, we made careful measurementsdevice with a thick Cu conduction layer to ensure a lar(k2D,s1k2D,m)/k2D,m) ratio and used Eq.~2!. The contribu-tion of the Pt leads,kPt, was calculated from their geometrand measured resistivity using the Wiedemann–Franz rtion ~we note that the Pt contribution tok is less than 20% aall temperatures.15 The thermal conductivity of the Cu conduction layerkCu was estimated from the measured resistity of the thermally evaporated Cu used for our conductlayers using the Wiedemann-Franz relation. An improvedtermination of the ratiok2D,Cu/k2D,m can be made in thefuture by directly measuring the thermal conductivityk(T)of Pt and Cu; the estimate shown here however is goodwithin a factor of 2, which is sufficient for present purpose

Figure 4 shows the experimentally-determinedkCu, km ,and the ratio ofkCu/km for our devices as a function otemperature. Since the thickness of the Cu and the membare approximately the same,k2D,Cu/k2D,m'kCu/km.100 atall temperatures. Based on this, the systematic error madusing Eq.~3! to determine the thermal conductivity of a filmplaced across the entire membrane using a 2000 Å Cu t

FIG. 4. Approximate experimental ratio of thermal conductivity of CukCu

used for thermal conduction layer and low stress LPCVD Si–NkSiNx5km

used for membrane vsT. Inset shows values~*100 for Si–N!. kCu is esti-mated from measurements of electrical resistivity of Cu samples preparthe same time as the thermal conduction layer using Wiedemann-FkSiNx

is determined from the measuredk5P/DT for a device with a Cuthermal conduction layer to give good intemal thermal conductivity and wdefinedDT. Following Eq.~2!, kPt has been calculated and subtracted frothe measuredk, and the resultingkSiNx

has been divided by the geometrifactor 10.33 and multiplied by membrane thicknesstSiNx

51800 Å to getkSiNx.


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mal conduction layer in the sample area is to underestimthe real thermal conductivity of the membrane or samplethe fractional change shown in Fig. 3; fork2D,Cu/k2D,m

.100, this is,1.5% ~5the % decrease inP/DT for T2 atfinite k2D,Cu/k2D,m compared to the extremely higk2D,Cu/k2D,m limit ! which is within the uncertainty in filmthickness for most cases. It is also possible to use the slated P/DT values shown in Fig. 3 to make a correctionextracting the geometric factor connecting a givenP/DT ~forT1 or T2! with k2D,m , at least in the limit where the samparea is nearly isothermal@large but not infinite (k2D,s

1k2D,m)/k2D,m], since Fig. 3 is at constantk2D,m . For ex-ample, at a ratio of 100, a common lower limit for the dvices reached through use of a Cu conduction layer, the ctour lines of Fig. 2 and the simulation data in Fig. 3 shothat the geometrical factor is reduced from 10.33 to 10.0T2 ~the same 1.5% correction!.

B. Time dependence: PÄ0

In this section, we turn to the analysis of the relaxatiDT(t). Experimentally, the relaxation is measuredswitching off ~at timet50) the current flowing in the heateand recording the relaxation of the off-null voltage acrothermometers T1 and T2, using an ac bridge describeRef. 4. In the high ratio limit (k2D,s1k2D,m)/k2D,m@1 pro-duced by a parallel thermal conduction layer such as1800 Å of Cu used here in measurement 3, we have prously shownexperimentallythat a single time constant iseen, out to 7text.

4

We start by comparing simulation and experiment oftime dependence for the empty calorimeter in order to extthe specific heat of thea-Si–N membrane at 20.3 K, thtemperature chosen for absolute calibration purposes.then turn to simulations of the relaxation response ofmembrane with a sample of various thermal conductivity aspecific heat. We show that even for a relatively low ratiosample to membrane thermal conductivity, fits to a sintime constant can be used to extract the sample heat capand we derive the systematic errors introduced bymethod as a function of this ratio. We also derive the conbutions to the total heat capacity of the membrane borderthe Pt leads. We then study the differential method direcsimulating~a! an initial sample layer~e.g., of Cu!, followedby a second identical layer, as in a calibration measuremand ~b! with the second layer being a more normal samwith high heat capacity but relatively low thermal conductity. The systematic error in this differential method is showto be less than in the single layers, and is derived as a fution of the ratio of internal to external thermal conductiviand sample to membrane specific heat. This systematic eis shown to be less than 2% under standard operating cotions of a Cu conduction layer with thickness equal to thathe membrane. Finally, a method for an iterative correctfor this systematic error is discussed.

1. Empty calorimeter: Membrane heat capacity

Figure 5 shows the simulated~line! and experimenta~circles! temperature relaxation for anempty calorimeter,with no thermal conduction layer, using thermometer T2


tey

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T0520.3 K. The inset shows the semilog plot of the data asimulation. In the simulation,k2D,m50.194mW/K was takenfrom the comparison between the steady state simulationthe measurements made on the same device, as descearlier, andk2D,Pt50.11mW/K, and c2D,Pt50.69mJ/K cm2.The parameterc2D,m was then varied and the calculated rlaxation compared with the measured temperature relaxaFigure 6 shows the sum of the square error~SE! as a functionof the parameterc2D,m calculated from timet52.4 to t590 ms, which corresponds to 95% of the relaxation. ThSE points were fitted with a fourth-order polynomial whominimum was found forc2D,m50.2669mJ/K cm2, givingCm50.0148 J/K cm350.0051 J/K g at 20.3 K, using a den

FIG. 5. Normalized temperature relaxation of thermometer T2 measurethe empty microcalorimeter atT0520.3 K ~circles!. Solid line is the resultof the numerically computed relaxation forc2D,m50.2669mJ/K cm2, whichgave the lowest square error for these data~using k2D,m50.186mW/K,k2D,Pt50.11mW/K, and c2D,Pt50.69mJ/K cm2). Inset shows data on asemilog plot.

FIG. 6. Variation of the square error~difference between experimentallmeasured and simulated! as a function of the parameterc2D,m for the emptycalorimeter. The sum is made for times from 2.4 to 90 ms which excluthe first five points. In these fits,k2D,m50.186mW/K, k2D,Pt50.11mW/K,and c2D,Pt50.69mJ/K cm2. Inset: residue plot of the fit usingc2D,m

50.2669mJ/K cm2.


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sity of 2.9 g/cm3 for low stress low-pressure chemical vapdeposition~LPCVD! a-Si–N from Refs. 21 and 22. This isignificantly smaller than the specific heat of silicon oxide20 K. The solid line in Fig. 5 shows the relaxation calculatusing this value; the difference between this calculationexperiment is shown in the inset of Fig. 6.

We will use the 20.3 K value found here forc2D,m for thesimulations discussed below, but in order to extend thesults to the entire temperature range of measuremc2D,m(T) and c2D,Cu/c2D,m(T) is needed. One approacwould be to measure and simulate the relaxation curvesthe bare membrane at all temperatures. Instead, by usihigh thermal conductivity layer such as Cu forc2D,s , mea-surements of the addenda heat capacity~membrane with Cuconduction layer! were analyzed using results derived in tnext section to give the membrane heat capacity from msurements of P/DT and t from a simple fit toe(2t/t). Withthe Cu conduction layer, to an absolute accuracy of a few~discussed in the next section!, the total heat capacityctot

5tP/DT can be written as a sum of the heat capacity ofCu plus the membrane including a contribution from tmembrane border~to be numerically analyzed in the nesection! plus the Pt and Nb–Si heaters and thermomet~The total contribution of Pt and Nb–Si is less than 15%ctot at all temperatures!.23 The specific heat of Cu can bdirectly measured by adding a second~sample! layer of Cu,or from the literature values for Cu,24 allowing extraction ofc2D,m to ;10% accuracy, which is more than sufficientmake an estimate ofc2D,s /c2D,m needed in the followingsections. Using this procedure,c2D,m has been determined a

FIG. 7. C (J/mol K) for 1800-Å-thick low-stress LPCVD Si–N membran~density 2.9 g/cm3 and 21 g/mol were used! and c2D,Cu/c2D,m ~shown ininset! vs T for 1800-Å-thick Cu conduction layer based on literature valufor Cu ~see Ref. 24!. C for Si–N calculated from measurements ofctot fordevices with Cu conduction layers fromctot5As(CCutCu1Cmtm)1XAbCmtm1cPt1cNb–Si with X50.24 ~as shown in Fig. 9!. Ab is the areaof the border portion of the membrane. Pt leads and Pt thermometerNb–Si thermometer contributions were calculated and subtracted fromctot

but are small~,15% total; see Ref. 23!. tm is measured by ellipsometry,tCu

is measured by Dektak profilometry on a neighboring sample and islargest source of uncertainty. In Ref. 23, several different devices withferent conduction layers~Cu and Al! were compared and shown to give thsameCm .


t

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e-nt,

ora

a-

%

e

s.f

a function of temperature, and is shown in Fig. 7 along wc2D,s /c2D,m vs T for c2D,s calculated for an 1800-Å-thick Cuconduction layer. The specific heat derived by this meanthe low stress a-Si–N is discussed in a separapublication;23 we note here only that the molar specific heis low at all temperatures, indicative of a high Debye teperatureuD ~for crystalline Si3N4 , uD is 1130 K,25 so a highvalue for the amorphous material is not surprising!.

2. Calorimeter with sample

To model the standard experimental relaxation methtime-dependent relaxation curves were simulated formembrane with sample layers of various specific heatthermal conductivity. For sample layers where a single exnential decay provides a reasonably good description ofresults, the quality of the fit and the contributions of tmembrane border and the Pt leads are obtained and discu~Sec. III B 2 a!. In Sec. III B 2 b, weconsider the differentialmethod in which the sample heat capacity is determinfrom the difference between two measurements: the mbrane with a first sample layer~typically, Cu!, which pro-vides the addenda, and the same membrane with thelayer and an added sample layer. The second sample lwas considered to have either similar properties to thelayer~as in a calibration measurement where both are Cu!, orsignificantly lower thermal conductivity as in a standameasurement of a sample of interest. The membrane speheat and thermal conductivity were held constant~their val-ues at 20.3 K were used! but we reiterate that it is the ratioof k2D,s /k2D,m andc2D,s /c2D,m that are important.

a. Single sample layer and contributions of membraborder and Pt leads In the first set of simulations, we toothe specific heat of the sample as fixed~values for a typicalmetal were used! and studied the effect of changing its themal conductivity and thickness~which changes bothk2D,s

and c2D,s , and more importantly k2D,s /k2D,m andc2D,s /c2D,m). The temperature at the position of thermomeT2 ~0;20.1! was calculated for 150 equally spaced timeThe initial conditions att50 were set using the calculatestatic distribution of temperatures as described in the precing section. In order to more clearly observe the dependeof the parameters obtained on the ratio ofk2D,s /k2D,m , weperformed this set of simulations primarily without thecontributions ~i.e., with k2D,Pt5c2D,Pt50); tests of the Ptcontribution will be discussed below.

The values used for the calculations arek2D,m

50.2mW/K, c2D,m50.3mJ/K cm2, and sample specific heac2D,s from 0.1 to 300 mJ/K cm2 ~giving a range ofc2D,s /c2D,m from 0.3 to 1000; a typical metal at 20.3 Kmight have C50.055 J/K cm3; this then corresponds tothicknesses from 180 Å to 45mm!. The 2D sample thermaconductivity k2D,s was allowed to range over five decadefrom equal to the membrane thermal conductivity to 15

greater. The steady-state results fork5P/DT are shownabove in Fig. 3 and are not affected byc2D,s .

The time-dependent temperature relaxation of T2 wasto a single exponentiale(2t/t) between 100% and 0.3% of thinitial t50 value. The quality of this fit as well as the valufor t were determined. The total specific heat from the sim

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lation was then determined fromctot5tP/DT, as in the ex-periments. Figure 8 shows the results for two sets of samwith c2D,s comparable toc2D,m the limit most commonlyused for these devices as a function ofk2D,s/k2D,m.

A single exponential decay is found to be a good dscription of the relaxation for high internal thermal condutivity k2D,s /k2D,m , for c2D,s comparable to or greater thac2D,m , as expected based on simple models for the heatin the standard semiadiabatic limit. Whenk2D,s /k2D,m dropsbelow 10, there starts to be significant deviation fromsingle exponential decay, but even as low as 2, the relaxacan still be fit with a single exponential, albeit less well@asmeasured by the higher normalized residual, shown in8~c!#. Qualitatively, the relaxationDT(t) in this limit can beviewed as a simple exponential with a time delay. The Figinset shows the extreme example ofk2D,s /k2D,m50; here,the delay is obvious, yet the data can be fit to a simexponential which gives a result within a factor of 2 of texact fit derived above~the systematic error, to be discussbelow, is large in this case!. We explored the use of differen

FIG. 8. Simulation of experimental parameterst and ctot as a function ofintemal to extemal thermal conductivity ratio fork2D,m50.2mW/K andc2D,m50.3mJ/K cm2. ~a! t from fit to DTe2t/t calculated at position~0;20.1! ~thermometer T2!; ~b! total specific heatctot5tk, with k5P/DTfrom Fig. 3, ~c! residual5% deviation of simulated relaxation data fromsingle exponential decay. Lower curve in each:c2D,s50.5mJ/K cm2 ~typicalmetal 1000 Å thick at 20 K! andk2D,s52,20,200,...,200 000* k2D,m ; uppercurve is membrane with sample layer twice the thickness of the first thfore c2D,s51.0mJ/K cm2 and k2D,s54,40,400,...,400 000* k2D,m . Arrowshows differential technique for deriving sample heat capacity~differencebetween the two measurements givescs,sim). Similar results found for layersc2D,s51.1mJ/K cm2, c2D,s52.2mJ/K cm2. Pt contributions not includedhere (c2D,Pt5k2D,Pt50) in order to display the dependence onk2D,s /k2D,m .


es

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5

e

types of fits, but found that the single exponential fit, eventhe low k2D,s /k2D,m limit, yielded data that were the mosinterpretable in terms of the desired heat capacity. Figurshows a generalization of Fig. 8~c!; normalized residuals areplotted as a function ofc2D,s /c2D,m for a high, a moderateand a relatively low value ofk2D,s /k2D,m (53106, 100, and10!. A single exponential decay is only exact in the limithigh c2D,s /c2D,m and k2D,s /k2D,m . For lower c2D,s /c2D,m ,even whenk2D,s /k2D,m is very large, deviations from pureexponentials are seen@normalized residuals of 1%–2%, asFig. 8~c!#, attributable to the finite thermal diffusivity of themembrane itself.

Figures 8~a! and 8~b! show that for highk2D,s , valuesfor t and ctot are independent ofk2D,s , hence, independent othe ratio k2D,s /k2D,m , as expected. They begin to deviasignificantly whenk2D,s /k2D,m drops below 100, for allc2D,s

studied. The deviation int and ctot is 10%–15% atk2D,s /k2D,m52, the lowest ratio studied.

The heat capacityctot contains contributions from thesample layer (c2D,sAs), the Pt heater, the Pt leads, and tmembrane both directly under the sample area (c2D,mAs) andin the border area, whereAs is the area of the sample~0.25cm30.25 cm! and Ab is the area of the membrane bord@~0.5 cm30.5 cm!2~0.25 cm30.25 cm!#. The Pt heater inthe sample area contributes its full heat capacity (c2D,PtAPt).The contributions of the Pt leads and of the membrane bder are less straightforward. In 1D, the contribution of tleads to the total specific heat has been solved analyticalthe limit ks@km andcs.cm; approximately 1/3 of their totaspecific heat contributes. However, even in 1D, this conbution is not exactly 1/3 but depends on the ratio ofcs to cm

~see, e.g., Ref. 26, and references therein!. In 2D, the analy-sis is less straightforward. We have, therefore, performedmerical simulations to determine the contribution of the 2membrane border toctot as a function of the ratioc2D,s /c2D,m

andk2D,s /k2D,m .For this set of simulations, we did not include the

leads as their 1D nature would complicate the interpretaof the 2D membrane contribution~i.e., we setk2D,Pt5c2D,Pt

50). We, therefore, define the fractional contribution of t

e-

FIG. 9. Deviation of simulated relaxation from single exponentialDTe2t/t.Relaxation calculated at position~0;20.1! ~thermometer T2!. Normalizedresidual defined as the sum on all points of ABS@~simulated data-fit!/fit#/number of data points.


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membrane border X from ctot5As(c2D,s1c2D,m)1XAbc2D,m . As before,ctot5tP/DT. Figure 10 showsX as afunction of c2D,s /c2D,m for variousk2D,s /k2D,m ~from 10 to53106). For high internal thermal conductivity, the averavalue is;0.24, somewhat less than the 1D value of 1/3would be expected for a square 2D membrane.@Qualita-tively, the corners do not contribute much to the specheat; the isotherms in Fig. 2 show their temperature toclose to the block temperatureT0 ; also, qualitatively, treat-ing the membrane border as four 1D leads, each 0.125long and 0.25 cm wide, and taking 1/3 of this, gives a cotribution close to~within 11% of! the exact solution.# Forvalues ofk2D,s /k2D,m5100– 400~similar to what is experi-mentally generally used, see Fig. 4!, and c2D,s /c2D,m

50.1– 10 ~a likely range!, the fractional contributionX isslightly lower, but still remains close to 24%.

In order to determine the contribution of the Pt leadsctot , we repeated this simulation first assumingc2D,Pt50,then c2D,Pt5331027 J/K cm2 in the largek2D,s /k2D,m limit(k2D,s5231022 W/K, k2D,m5231027 W/K), and rela-tively large c2D,s /c2D,m limit ( c2D,s5331026 J/K cm2,c2D,m5331027 J/K cm2). From the difference betweenctot

obtained in each case, we determined that 29.4% of thleads in the membrane border area contribute to the tspecific heat, compared to the purely 1D case of 33%.difference can be attributed to the fact that the leads hsome width~hence, are not strictly 1D! and are thermallyanchored to the membrane. The value 29.4% varies61%with different choices forc2D,s andk2D,s .

b. Differential method: Difference between two layerDeriving the sample heat capacity directly fromctot is prob-lematic whenc2D,s /c2D,m andk2D,s /k2D,m are not both verylarge, due both to deviations from a pure exponential~shownin Fig. 9! and the dependence of border contributionX onthese ratios~seen in Fig. 10!. The relaxation method usinthe membrane-based devices is most commonly perforin a differential mode, in which a device is measured withCu conduction layer only, giving the addenda, and thensame device~or its equivalent, given the reproducibility othe Si microfabrication processes! is measured with a sampllayer added. Conceptually, this can be viewed as taking

FIG. 10. Fractional contributionX of membrane border as a function ok2D,s /k2D,m and c2D,s /c2D,m . Membrane border areaAb50.530.5 cm2

20.2530.25 cm2 and contributesXAb to ctot .


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difference between the two sets of data shown, for examin Fig. 8~b!, where data are shown for two different thicknesses of a typical metal. However, care must be takenthis due to possible changes in 2D thermal conductivityadding the second layer. If the second layer is a typisample with low thermal conductivity, thenk2D,s changesvery little. If, however, the second layer has the same therconductivity as the first layer~as in Fig. 8 or in a calibrationmeasurement where both layers are Cu!, thenboth k2D,s andc2D,s double on adding the second layer, this can be seethe data shown in Fig. 8~b! where the values ofctot for vari-ousk2D,s /k2D,m are offset along thex axis from each other.

Two significant effects enter into the systematic erromade by the differential method.~1! In the limit of moderatevalues ofk2D,s /k2D,m (,400), the relaxation is less well fiby a single exponential and deviations are seen fromtheoretical values ofctot . These deviations, however, arethe same sign for both the first and second layer@see Fig.8~b!# so that thedifference~the sample heat capacity! is sig-nificantly closer to the correct theoretical value than eithlayer individually.~2! The contribution of the membrane boder ~and Pt leads! can change due to the changing ratioc2D,s /c2D,m , as seen in Fig. 10, leading to a systematic erin the differential value of the sample heat capacity eventhe limit of extremely highk2D,s /k2D,m . This second effectcan have either sign, as bothc2D,s /c2D,m and k2D,s /k2D,m

change on adding the sample layer, effectively movingtween different curves in Fig. 10. Both of these effects wbe discussed below.

Figures 8~a! and 8~b! showed the results forc2D,s

50.5mJ/K cm2 (ts5900 Å) and c2D,s51.0mJ/K cm2 (ts

51800 Å), for a range ofk2D,s /k2D,m . This can be thoughtof as an experiment with a first layer of 900 Å of a mewith some thermal conductivity followed by adding a secolayer of the same metal, also 900 Å, and measuring againin a differential experiment, we take the difference betwethe first and second layers, as shown by the diagonal arin Fig. 8~b!. ~Note that the second combined layer has higk2D,s than the first layer, hence, the diagonal nature of tline!. Figures 11~a! and 11~b! show the simulated additionasample heat capacity cs,sim5ctot(second layer)2ctot(first layer) and the error (cs,sim2cs,theory)/cs,theory ~as a%! as a function of the ratio of thermal conductivitiek2D,s /k2D,m . cs,sim is thus the value obtained from the timdecays analyzed as in experiments andcs,theory is the inputvalue c2D,sAs where As50.25 cm30.25 cm. This ‘‘experi-ment’’ was also performed for thicknesses of 200012000 Åand showed very similar results.

Figures 8~a! and 8~b! showed significant~.10%! devia-tions of the values oft and ctot from the largek2D,s /k2D,m

limit as k2D,s /k2D,m drops below 10. However, even at thlowest ratio studied (k2D,s /k2D,m52), the deviation in thesamplespecific heat~the differencebetween the two measurements in each ‘‘experiment’’! is only a few %, signifi-cantly less than the deviation in each quantity separately,the final result forcs,sim in the differential method is lessaffected by a lowk2D,s /k2D,m ratio than the intermediateresults fort, k, andctot . This can be seen visually in the facthat the two curves in Fig. 8~b! are somewhat parallel to eac


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other, causing the deviation in thedifferenceto be substan-tially smaller than the individual curve’s deviation. We noalso that for lowk2D,s /k2D,m , cs,sim is dependent on whethek2D,s /k2D,m changes on adding the second layer, even thothe additional heat capacitycs,theory is the same, as can bseen in Fig. 8~b! considering the diagonal arrow discussabove compared to a vertical arrow for constantk2D,s /k2D,m .

Even in the extremely highk2D,s /k2D,m ratio limit, theheat capacity obtained for the difference between theand second layers~the ‘‘sample’’! deviates from the correc~input! value cs,theory by 1%, as shown in Figs. 11~a! and11~b!. This is because the contributionX from the membraneborder toctot ~shown in Fig. 10! depends on the ratio ofc2D,s

to c2D,m , which changes~by a factor of approximately 2! onadding the ‘‘sample.’’ Physically, this error results from thchange in the fractional membrane border contributionctot5tP/DT on changing the ratioc2D,s /c2D,m by adding thesample. Since this contribution to the addenda changesdifferential method does not correctly account for the adenda. Whenk2D,s /k2D,m remains constant~as in the stan-dard measurement with a Cu conduction first layer ansample layer with low thermal conductivity! the changeinmembrane contributionX with increasingc2D,s is alwaysnegative, causing a systematicunderestimateof the samplespecific heat in the differential method,@proportional to theslope dX/d(c2D,s /c2D,m), for the appropriatek2D,s /k2D,m

FIG. 11. Simulated results of differential method:~a! sample specific heafrom difference between curves shown in Fig. 8:cs,sim5ctot(1)2ctot(2), asshown with arrow in Fig. 8~b!. Line indicates input value ofcs[cs,theory

5@c2D,s(2)2c2D,s(1)#* 0.25 cm30.25 cm50.5mJ/K cm2* 0.0625 cm2. ~b!% error in sample specific heat5(cs,theory2cs,sim)/cs,theory.


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curve in Fig. 10#. For largek2D,s /k2D,m , this change is smalbut not zero. For lower values ofk2D,s /k2D,m , the membranecontribution change is more significant. In the most genecase of a differential method, bothk2D,s /k2D,m andc2D,s /c2D,m varies between the two layers. In that case,can be deduced from Fig. 10, the error in the sample speheat resulting from the dependence ofX on(k2D,s /k2D,m ,c2D,s /c2D,m) can be negative, positive, or nuldepending on whetherX increases or decreases. This is win Fig. 11~a!, wherek2D,s /k2D,m andc2D,s /c2D,m doubled onadding the second layer,cs,sim is larger than thecs,theory fork2D,s /k2D,m,100 and smaller fork2D,s /k2D,m.100.

To determine the error made under standard experimtal conditions, we simulated the full differential methoddetermining sample heat capacity~steady-stateP/DT andtime relaxation fit to a single exponentiale(2t/t) for eachlayer! for a wide range ofc2D,s first and second layers~from0.1 to 300mJ/K cm2, with intervals equally divided on a logscale, i.e., 1, 1.7, 3, 10,...! and c2D,m50.3mJ/K cm2. Forthese simulations, we used values ofk2D,s /k2D,m from 10 to53106 (k2D,s actually varied, withk2D,m50.194mW/K)and assumed thatk2D,s did not change significantly on adding the second~sample! layer, which was assumed to be olower thermal conductivity and/or thinner than the first~ad-denda! layer. For these simulations, Pt contributions wereincluded (k2D,Pt5c2D,Pt50).

In Fig. 12, we plot the fractional systematic error

FIG. 12. Percent error incs determined by the differential method vs thratio of heat capacities of sample to membrane for various ratios of samto membrane thermal conductivity. Data assumes constantk2D,s /k2D,m ~de-termined experimentally by the Cu conduction layer which dominates mk2D,s) and changingc2D,s due to the added sample. In these simulatioc2D,s ranged from 0.1 to 300mJ/K cm2 ~with intervals equally divided on alog scale, i.e., 1, 1.7, 3, 10,...! and c2D,m50.3mJ/K cm2, k2D,m

50.195mW/K, k2D,Pt5c2D,Pt50. The % error in cs is defined as100* (cs,theory2cs,sim)/cs,theory. cs,theory is the theoretical value of heat capacity (c2D,sAs) entered into the simulation.cs,sim5@ctot(1)2ctot(2)# wherectot

5tk for a first ~second! layer with t obtained from a simple exponential fito the relaxation data for each layer andk5P/DT ~the same for both lay-ers!. Data points are shown plotted atc2D,s /c2D,m5mean value of layers 1and 2 for each point. For a copper conduction layer with thickness equthe membrane as addenda, the systematic error is below 2% for all posadded samples since neitherk2D,s /k2D,m nor c2D,s /c2D,m can be less than thevalues for Cu alone~.100 and.2, respectively!.


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cs,sim for various ratiosk2D,s /k2D,m between 10 and 53106. As before, we define cs,sim5@ctot(layer 2)2ctot(layer 1)# wherectot[tP/DT for each layer, and the error is defined as (cs,theory2cs,sim)/cs,theory. This systematicerror takes into account errors produced both by the detions from a single time constant caused by low interthermal conductivity and the shifts in fractional contributioof the membrane border on increasing the sample heapacity in the standard difference method technique. Inhigh ratio limit (53106), this systematic error varies between 3% and 0% for values ofc2D,s /c2D,m ranging from 0.5to 10. Considering more physical thermal conductivity ratik2D,s /k2D,m5100– 400, the derivative remains small fphysically relevant values ofc2D,s /c2D,m ranging from 0.5 to10, hence, the error remains below a few %. In the limitlargec2D,s /c2D,m (.10), where Fig. 10 showed the derivtive dX/d(c2D,s /c2D,m) is high even for relatively high intemal thermal conductivityk2D,s /k2D,m5100– 400, the frac-tional contribution of the membrane will change significanon adding the sample~e.g., X goes from 20% to 10% onadding a sample which hasc2D,s /c2D,m5100). However, theerror introduced into the determination ofcs by the differen-tial method is nonetheless very small~as seen in Fig. 12! asthe total contribution of the membrane is small comparedthe sample.

Thus, the relaxation method is only exact in the limitlarge k2D,s /k2D,m and largec2D,s /c2D,m . The errors intro-duced as either is reduced, however, partially cancel inusual differential method of measurement in which tsample heat capacity is determined from the differencetween the heat capacity of a device with a conduction laand a similar device with conduction layer plus sample. Uder our standard operating conditions ofk2D,s /k2D,m.100,andc2D,s /c2D,m between 1 and 10~Cu conduction layer plussample!, the error is 2% or less, and would, therefore, likebe dominated by other errors in the measurement, sucuncertainty in sample thickness. Even for relatively lowtios of k2D,s /k2D,m ~e.g. 10!, the error is below 7% for allreasonable values ofc2D,s /c2D,m , largely because of thecompensating effects of the differential method.

In all cases where the systematic error is large, detions from single exponential decay are also seen, i.e.,normalized residual becomes larger than the backgronoise of the simulation that is controlled by the absoltolerance@see Figs. 8~c! and 9#. Figure 9 shows the deviatiofrom a pure exponential as a function ofk2D,s /k2D,m andc2D,s /c2D,m ; these data nearly perfectly parallel the depedence of systematic errors shown in Fig. 12. However,deviation is not as large as perhaps might be expected~e.g.,even data for the bare membrane shown in Fig. 5 couldqualitatively and mistakenly viewed as exponential!, andcould be masked experimentally by noise; it is, therefoimportant to verify that the membrane-based devicesused in limits where systematic errors are calculated tosmall, based on estimates ofk2D,s /k2D,m andc2D,s /c2D,m .

C. Analysis of thin Cr film sample

In this section, we discuss further simulations of the mcrocalorimeters in a limit of low internal thermal conducti


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ity. Specifically, we analyze the time-dependent relaxatsimulated and measured for a thin Cr layer only~no Cu con-duction layer!, and show that the precise geometry of tmicromachined device allows us to extract the specific hof the Cr, even though a single time constant relaxation isobserved. We are able to demonstrate these results usinexisting calorimeters; an improved thermometry deswould make this type of analysis a powerful tool.

For this simulation, the 2D thermal conductivity of thCr derived previously is added to that of the membrane~andthe Pt in the places shown in Fig. 1! in the central 0.25cm30.25 cm sample area, giving (k2D,s1k2D,m)/k2D,m

52.3, well below the desired limit of 100. Figure 13 showthe experimentally measured time-dependent relaxationthermometer T2. The data are shown normalized to thvalue att50. ~As previously discussed, thermometer T1less reliable in limits of low sample thermal conductivity duto its physically distributed nature!. As anticipated, thesedata do not fit a single exponential decay~semilog plotshown as an inset!. The temperature relaxation data depeon bothc2D,s(5c2D,Cr) andc2D,m , as well as the values ok2D,s(5k2D,Cr) andk2D,m found from the steady-state condtions previously discussed.

We performed an analysis of the data shown in Fig.using k2D,s50.243mW/K, k2D,m50.186mW/K, c2D,m

50.2669mJ/K cm2, k2D,Pt50.110mW/K, and c2D,Pt

50.69mJ/K cm2. The relaxation of thermometer T2 is caculated for 150 times between 0 and 120 ms for differvalues ofc2D,s and the square error calculated for each fro4 to 120 ms, as shown in Fig. 14. The same fitting procedas was used for the bare membrane was used here to exc2D,s50.257mJ/K cm2 ~shown in Fig. 14!. The residue plotobtained with this value is shown in the inset of Fig. 1Calculations of the temperature relaxation at different poi

FIG. 13. Normalized relaxation of the relative temperatureDT measuredwith thermometer T2 at base temperatureT0520.3 K for a device with 1035Å Cr layer only. The inset shows the semilog plot of the data, show~small! deviations from a single exponential decay, as expected for thisthermal conductivity sample. Solid line is the result of the numerical simlation of relaxation at position~0;20.1! ~T2! for c2D,s50.5239mJ/K cm2,which gave the lowest error~using c2D,m50.2669mJ/K cm2, k2D,m

50.186mW/K, k2D,s50.243mW/K, k2D,Pt50.110mW/K, and c2D,Pt

50.69mJ/K cm2).


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in the sample area, in particular, two positions within tlarge area thermometer T1, with thec2D andk2D parametersfound here, were also made, but only qualitatively matchthe data for T1, which contain a complicated averaging dto variations in both its temperature and its temperature ssitivity ~a result of small compositional gradients in the NbSi!. Finally, the specific heat of Cr is calculated fromc2D,s

50.257mJ/K cm2, which gives C53.455 mJ/K g5180 mJ/K mol~a densityr57.19 g/cm3 and the molar mass52.0 g/mol have been used!.

IV. DISCUSSION

Under conditions similar to our standard experimenoperating conditions, with an 1800-Å-thicka-Si–N mem-brane, an 1800 Å Cu thermal conduction layer, and a samwith heat capacitycs , the simulations confirm the usual relaxation method fitting with a single exponentialtext, andconfirm the differential determination ofcs from textP/DT2ca , with DT measured on either T1 or T2 andca measuredusing the same relaxation method (5text8 P/DT) for the Culayer alone.~We note thatca can be measured on the samdevice before depositing the sample or on an equivalentvice!. The absolute accuracy is better than 2%, dependenthe ratio of sample heat capacity to addenda heat capaThis small systematic error~,2%! is shown in Figs. 12 andin 11~b! for an 1800 Å membrane and 1800 Å Cu conductilayer, with the second sample layer being either Cu orother material,k2D,s /k2D,m.100 andc2D,s /c2D,m.2. It isimportant to note that this absolute systematic error of,2%for any sample of any thickness refers only to the systemerrors discussed in this article, not the many other expmental errors inherent in measuring a thin film sample sas uncertainty in sample thickness and noise, both of whwill introduce errors that depend on sample thickness.

FIG. 14. Variation of the square error in T2 as a function of the paramc2D,s for calorimeter with Cr film only. The sum is made for times from 4120 ms, which excludes the first five points. Inset: residue plot of theusing c2D,s50.5239mJ/K cm2, which gave the lowest square error~withc2D,m50.2669mJ/K cm2, k2D,m50.194mW/K, k2D,s50.243mW/K, c2D,Pt

50.69mJ/K cm2, andk2D,Pt50.11mW/K).


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Figure 12 can be used to provide a first-order iteratcorrection to the experimentally determined heat capabased on the measured values ofc2D,s and c2D,m ; this isuseful if other uncertainties such as film thickness are bethan 2%, or if it is desired to eliminate the Cu layer suchfor a sample of moderately high thermal conductivity. Inhigh precision experiment,c2D,s for a real sample would beestimated from a traditional difference method: the sampestimated 2D heat capacity is determined fro@ctot(addenda1sample)2ctot(addenda)#/As . ~Here, addenda1sample refers to a device measured with Cu conduclayer1sample; addenda to a device with conduction laonly!. This is then added to the Cu 2D heat capacityc2D,Cu togive c2D,s . The resulting estimate for the totalc2D,s /c2D,m

then gives a % correction toc2D,s based on Fig. 12; this %correction would then be used to correct the sample hcapacity itself, since the Cu conduction layer contributionc2D,s did not change between the two measurements~thesystematic error comes from the changing value of mebrane contributionX!. When a Cu layer is used,k2D,s /k2D,m

exceeds 100 for both addenda and addenda plus samplethe difference in Fig. 12 betweenk2D,s /k2D,m5100 and 53106 is small, hence, the increased value ofk2D,s /k2D,m onadding the sample is likely to be not important. If no Clayer is used, the changing value ofk2D,s /k2D,m would needto be considered, and the errors and their corrections ik,ctot , and changing membrane contributionX calculated di-rectly from Figs. 3, 8, and 10.

The availability of two thermometers on opposite sidof the sample area with respect to the heater~which is asym-metric! gives the possibility of improving the accuracy of thspecific heat measurement by making simulations of datwas done here for the Cr film. We note also that the quatative simulation technique in this thin film limit could bimproved by using a simpler design of the thermometry ccuits on the microcalorimeter, in particular, matching the gometries of T1 and T2 to the isothermal contour lines.addition, a quantitative study of the thermal properties ofPt films would reduce the uncertainty in the thermal paraeters used to describe the leads and heater.

We briefly comment on the values obtained in this woThe thermal conductivity of low stressa-Si–N is signifi-cantly larger than that of vitreous silica and silica-basglasses at 20 K, but is consistent with other values repofor low stressa-Si–N.15,20,27 We have also found that thplateau commonly observed in the thermal conductivityinsulating glasses is at 30–50 K ina-Si–N, while it is near10 K in silica glasses, and the specific heat~in J/g K! ofa-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 sptered film. Measurement of the resistivity of Cr films depoited under similar conditions show a residual resistivity ofmV cm, which gives L5rk/T51.7331028 W V K21, inrough agreement with the Wiedemann–Franz relationL52.4531028 W V K21. The specific heat of the Cr filmmeasured in this workCCr5180 mJ/K mol627 mJ/K mol.The error onCCr is estimated at 15%, taking 5% errors oncm

andcs . The same Cr film was measured with the single tim

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constant technique, using an 1800-Å-thick Cu film as a thmal conduction layer.16 In this previous measurement, wfoundCCr5234 mJ/K mol670 mJ/K mol at 20 K; the contri-bution of the Cr film at 20 K was only 10% of the total hecapacity so that the error incCr was of order 30%. Using thenumerical technique described here, with no Cu layer,cCr isfound with less uncertainty as it represents 50% ofctot . Wenote that this numerical technique is especially adaptedmeasurement of the low-temperature specific heat of suconducting films and particles for two reasons:~1! the ther-mal conductivity of metals is lowered in the superconductstate and~2! using a metallic thermal conduction layer wouintroduce spurious effects because of the proximity effeThe technique can be straightforwardly adapted to measment of the specific heat of submicron particles depositedthe membrane as the local thermal parameters can betrolled in the calculation.

ACKNOWLEDGMENTS

The authors thank B. Woodfield for valuable discussioconcerning calorimetry and the NSF, DOE, and Swiss Ntional Fund for Scientific Research for support.

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