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Thin film microcalorimeter for heat capacity measurements from 1.5 to 800 K

, D. W. Denlinger, E. N. Abarra, Kimberly Allen, P. W. Rooney, M. T. Messer, S. K. Watson, and F. Hellman Department of Physics, University of California at San Diego, La Jolla, CaIifomia 92093

(Received 18 November 1993; accepted for publication 13 January 1994)

A new microcalorimeter for measuring heat capacity of thin films in the range 1.5-800 K is described. Semiconductor processing techniques are used to create a device with an amorphous silicon nitride membrane as the sample substrate, a Pt thin film resistor for temperatures greater than 40 K, and either a thin film amorphous Nb-Si or a novel boron-doped polycrystalline silicon thermometer for lower temperatures. The addenda of the device, including substrate, is 4X 10m6 J/K at room temperature and 2~ 10e9 J/K at 4.3 K, approximately two orders of magnitude less than any existing calorimeter used for measuring thin films. The device is capable of measuring the heat capacity of thin film samples as small as a few micrograms.

I I I. INTRODUCTION

Many materials of current fundamental or technolog- ical interest can only be made, or are primarily made, in thin flhn form. Examples include multilayers, many amorphous materials, and ultrathin films of reduced dimension- ality. In the study of these materials, measurement of the specific heat can provide information on the electron and phonon densities of states, magnetic interactions, and structural or electronic phase transitions. Specific heat measurements can also discriminate between supercon- ducting transitions which are characteristic of bulk properties or of a minority second phase. Magnetic ordering can be detected in zero applied magnetic field. Specific heat measurements of thin films have, however, been limited to relatively thick films at low temperatures. We describe here the development of a microcalorimeter capable of measuring the thermodynamic properties of thin films between 1.5 and 800 K.

As technology has progressed, so has the ability to fabricate calorimeters capable of measuring smaller masses over a wider temperature range (a good review of small sample calorimetry is given by Stewart’). Commercially available differential scanning calorimeters are currently capable of measuring samples as small as a few mg from 77 K to well above room temperature. Differential scanning calorimetry is primarily used for measuring the heat capacity of bulk samples but thin film samples can also be measured if enough material can be prepared and if the film can be removed from the substrate, which would oth- erwise overwhelm the small sample signal. This is usually a difficult task, almost inevitably exposing the sample to air and limiting the types of samples and substrates which can be used.

All techniques for measuring thin film samples face two fundamental problems. First, the substrate, thermometer, and heater used for the measurement all possess significant heat capacity which must be subtracted from the total measured heat capacity. Second, for most experi- ments, it is impossible to make the thermal link small enough to permit an adiabatic measurement of the heat

capacity, as is usually done for bulk samples, due to the need for electrical leads.

The first problem is addressed by using thin film heat- ers and thermometers. At low temperatures, thin insulating substrates with high Debye temperatures (e.g., sapphire) may be used; this technique has been used by Kenny and Richards,2 for example, to allow measurements of monolayers of He. It is not, however, useful above about 100 K, where even a high Debye temperature material begins to have an appreciable heat capacity since the substrate is inevitably many times the thickness of the thin film sample.

The second problem has been addressed through various measurement techniques. In 1968 Sullivan and Seidel’ developed an ac method capable of measuring bulk samples of 100 mg at low temperatures (down to 1.4 K). In 1972 Bachmann et aL4 reported on a calorimeter which used a relaxation method to measure bulk samples as small as 1 mg from 1 to 35 K. About the same time, Greene et tiL5 developed an apparatus which measured thick films at low temperatures. Specific heat measurement over a wide temperature range (4-380 K) for bulk samples (>20 mg) was achieved in 1980 by Griffing and Shivashankar’j who employed a GaAsP light emitting diode (LED) as the temperature sensor. A very small sample calorimeter was developed by Graebner7 in 1989 in which bulk samples as small as a few micrograms are mounted directly on a Chromel-Constantan thermocouple; a similar technique has been used by Geer et al.* to measure specific heat of self-supporting single layers of liquid crystals in which a thermocouple can be embedded. This technique permits measurements of extremely small samples in particular cases but does not eliminate the substrate problem for thin films. Earlier work on thin film calorimeters had been done by Early, Hellman, Marshall, and Geballe’ in 1981 using doped epitaxial Si thermometers on a sapphire substrate (and a relaxation method patterned after Bachmann et ai. ) to measure samples smaller than 1 mg at low temperatures.

Attempts have been made” to use semiconductor pro-

946 Rev. Sci. Instrum. 65 (4), April 1994 0034-6746/94/65(4)/946/14/66.60 @ 1994 American Institute of Physics

cessing technology to fabricate devices with thin film leads and submicron thick substrates in order to reduce the addenda contribution below that of previous devices which use relatively massive wires and sample platforms, but the delicate substrates proved to be an insurmountable problem. Recent advances in membrane technology have made it possible to fabricate strong, thin silicon nitride membranes which form thesample platform for our microcalorimeters. We use a 180 nm amorphous Si-N membrane together with thin film metal leads and thin film thermometers to create a calorimeter with two orders of magnitude less addenda than existing thin film calorimeters. The low thermal conductivity of the Si-N provides the necessary thermal isolation of the sample from its environment. Our total addenda heat capacity (including substrate) is approximately 4~ low6 J/K at room temperature and drops to below 1 X10-” J/K below 2 K. This enables us to measure, for example, Al samples with mass as small as 3 pg. Our devices use a thin film Pt thermometer and either thin film amorphous Nb-Si or boron-doped polycrystalline silicon thermometers which, together, cover the range from 1.5 to 800 K. All materials used in the construction of the microcalorimeters are metallurgically stable over the temperature range of interest and exhibit good thermal cycling. The upper temperature limit is set at present by the use of Au in the leads which becomes mobile at approximately 800-900 K, replacement of the Au with a different material would likely allow the devices to be useful to close to 1200 K, the processing temperature of the nitride and polycrystalline silicon.

We use the relaxation technique described by Bach- mann et al. However, the microcalorimeters could easily be used with other techniques such as the ac method” or the sweep method.4’11

In Sec. II we describe the apparatus, focusing in particular on the fabrication of the microcalorimeters, the development of the doped polycrystalline silicon low- temperature thermometer, the device mounting, and the electronics. In Sec. III we discuss the thermal relaxation method for measuring specific heat and give details of our experimental method, and in Sec. IV we show results.

II. APPARATUS

A. Microcalorimeter construction

The processing steps to prepare the microcalorimeters and samples are shown in Fig. 1. We start with a (lOO)- oriented Si wafer which provides a relatively high-thermal- conductivity “frame” to anchor to the copper block used for the specific heat measurements. A thin ( 180 run) amorphous silicon nitride layer is deposited on both sides of the wafer using a Si-rich low-stress low pressure chemical vapor deposition (LPCVD) process developed by Sekimoto et al. I2 and refined at the Berkeley Microfabrication Laboratory.13 The Si-N layer on the back of the wafer is used as a mask to deilne a 0.5 X 0.5 cm2 square area at the center of the device within which the Si is etched away in KOH. The nitride on the front side becomes a 0.5 cm square membrane at the center of a 1 cm square Si frame

[Figs. 1 (a) and 1 (b)]. We make the devices on 2 in. wafers and are able to fit nine devices on a wafer. The rest of the processing is done on the membrane, necessitating careful handling.

We use platinum for the high-temperature thermometer (40-800 K) and for the sample heater. Pt is sputtered on the membrane with a 3 nm Ti binding layer and wet etched to make the pattern shown in Fig. 1 (c). The thermometers are measured by an ac bridge technique described below which uses a thermometer on the sample and a matching thermometer on the Si frame as two arms of a resistance bridge. The thickness and path lengths are chosen to give a room temperature resistance of approximately 1200 n for all three resistors (sample and matching thermometers and heater). Our Pt films have a resistivity of about 23 psi cm, which is quite high due to a large residual resistivity po. l4 This does not seriously affect the sensitivity of the bridge since’ dp/dT is independent of po; the value of p, and hence R, only enters as $( fi) due to the need to limit 12R self-heating (see Ref. 15).

The next processing step is to fabricate the electrical leads connecting the heater and thermometers (both the Pt thermometer and the Nb-Si low-temperature thermometers described below) on the membrane to the Si frame [see Fig. 1 (d)]. There are several design requirements: the leads should have small electrical resistances ( < 1% of the heater and thermometer resistance), they should have a relatively low thermal conductance to keep the sample thermally isolated, and the thermal conductivity should be a weakly varying, preferably linear, function of temperature. We have chosen an alloy of Au-Pd (93 at. % Au) which meets these criteria.16,‘7 It is metallurgically stable up to at least 500 “C! and has no phase transitions in the temperature range of interest.‘* The alloy is coevaporated onto the membrane from a single thermal evaporation boati using a 3 nm Ti binding layer. This bilayer is photolithographically patterned and wet etched to provide both the leads and the contact pads on the Si frame [see Fig. 1 (d)]. The high percentage of gold facilitates the subsequent bonding of electrical leads from the cryostat to the contact pads on the Si frame.

At lower temperatures, a more sensitive thermometer than the Pt is needed, and, because of the demands for large dR/dT, no single thermometer is adequate over the entire low-temperature range desired (1.5-40 K). There- fore we use two thermometers of identical composition with different geometries, giving resistances that differ by about a factor of 10, a technique developed by Early.”

In order to keep the effective sensitivity of the device thermometers constant with temperature, they need to have a resistivity which depends on temperature as Te2.15 Amorphous Nb-Si is known to go through the metal- insulator transition at approximately 12 at. % Nb.” On the insulating side of this transition the resistivity is thermally activated, while (barely) on the metallic side it is proportional to T- . 1’2 21 By choosing a composition between, the desired dependence can be approximately reached. Figure 2 shows the temperature dependence of a

Rev. Sci. Instrum., Vol. 65, No. 4, April 1994 Thin film calorimeter 947

U Silicon q Si-N

1 cm

Nb-Si thermometer containing approximately 10 Nb, which we have found to be ideal.

at. %

This material is prepared by electron beam coevaporation on the amorphous nitride membrane by using a copper

(e)

(f)

(9)

-T .25 cm

FIG. 1. Microcalorimeter at various stages of fabrication. (a)-(b) ( lOO)- oriented silicon wafer with 180 nm low-stress amorphous Si-N is etched in KOH to leave a free-standing membrane 0.5 x0.5 cm2 supported by a Si frame 1 X 1 cm’. (c) Pt is sputtered and patterned to form a high- temperature sample thermometer, matching thermometer, and a heater. (d) Au-Pd leads and contacts on the frame are prepared by thermal evaporation and photohthographic patterning. (e) Nb-Si is codeposited by electron beam coevaporation; this is patterned into low-temperature thermometers using a copper lift-off technique. (f) The 0.25 x0.25 cm” sample is deposited through a shadow mask onto the back side of the membrane. (g) Profile of finished device showing the shadow mask in place (not to scale: typical film thicknesses are 100-500 nm and the shadow mask is approximately 25 pm from the membrane).

0 Silicon q Si-N E3 Sample

E Platinum u Nb-Si

lift-off technique to form two pairs of strips as shown in Fig. 1 (e). The higher-temperature thermometer has a longer, narrower path than the lower-temperature thermometer. We use a sample thermometer and a matching

946 Rev. Sci. Instrum., Vol. 65, No. 4, April 1994 Thin film calorimeter

1.4 I

A

- 1.4 5. 1.2

& 1.0 - 0.8 .G -4

0.6 0.4

-2 0.2 e 0.0

A A A A 0.0 ' ' ' ' ' ' ' ' ' ' 1 p ' ' +

0 20 40 60 80 Temperature (K)

FIG. 2. Resistivity of coevaporated Nb-Si with approximately 10 at. % Nb. Inset shows resistivity vs l/T*, the desired temperature dependence (see Ref. 15).

thermometer on the Si frame as was done with the Pt thermometers. We have found that we can use a substrate temperature of at least 200 “C for this deposition, allowing us to thermally cycle the completed devices up to at least this temperature (for sample preparation, for example) with no degradation of properties on returning to low temperatures.22 Ifit is necessary to cycle the devices to still higher temperatures, we use the polycrystalline silicon thermometers which are described in the next section. It is important to note that our measurement technique (the relaxation method, to be described below) does not depend on the thermometers maintaining any sort of calibration on thermal cycling, only that they maintain the necessary sensitivity to temperature.

The last step is to deposit the sample at the center of the back side of the membrane which electrically insulates it from the thermometers. We have successfully made samples of a wide variety of materials by sputtering and evaporation. The 0.25 cmX0.25 cm sample can be patterned photolithographically by spinning a layer of photoresist on the sample at the bottom of the etch pit and focusing the projection aligner down at this level. Alternatively, we have developed Si evaporation masks [see Fig. 1 (f)] which fit inside the membrane cavity and define the sample area. Semiconductor processing techniques allow us to create masks with clean square edges, and by using Si for the mask, the sides of the mask automatically have the same 55” slope of the etch pit which is inherent in the anisotropic KOH etching process of ( lOO)-oriented Si wafers. Use of a shadow mask is essential if the devices are to be used for in situ sample preparation and measurements.

The robustness of the devices may be of concern for some applications, but we find they stand up to routine careful handling and have even been successfully trans- ported by mail.

B. Polycrystallinb silicon thermometers

We have developed a new boron-doped polycrystalline silicon (polysilicon) low-temperature thermometer which is metallurgically stable up to at least 1100 K, making it ideal if we wish to make low-temperature specific heat measurements on a sample which must be prepared at high temperatures. However, due to the complicated processing steps required to incorporate the polysilicon into our membrane device, we prefer to use the amorphous Nb-Si thermometers described above.

Doped epitaxial Si is a widely used low-temperature thermometer, 1*9p20123 but one which is incompatible with the microcalorimeter processing because it cannot be prepared on an amorphous substrate (the nitride membrane). Conduction in doped polysilicon has been studied at temperatures above 77 K (Refs. 24-32) and the material has been used as a thermometer at high temperatures,33 but the presence of grain boundaries makes extrapolation to low temperatures difficult. To the best of our knowledge, the resistivity of doped polysilicon has never been measured below 77 K. We have found that heavily doped polysilicon makes an excellent low-temperature resistance thermometer, with a thermally activated temperature dependence. Boron was chosen as the dopant as it does not chemically segregate to grain boundaries.24

The polysilicon thermometer that we have developed has two levels of doping: the active temperature-sensing portion has a concentration of about 1.6 x 1Or9 cm-s while a doping of 2 x 102’ cmw3 is used to provide lower resistance leads and to make an ohmic contact with Au-Pd pads on the frame. By using polysilicon itself as the material for the leads, instead of the Au-Pd, we avoid the pos- sibility of exothermic reactions taking place on the membrane should we use the devices at high temperatures. (Reactions between Au and Si, for example, occur not far above room temperature). As with the Nb-Si thermometers described above, we use two thermometers with different geometries of the actively doped area to span the temperature region from 1.5 to 40 K [see Fig. 3 (a)]. A plot of the resistance of the two thermometers is shown in Fig. 3 (b). An insert shows the temperature dependence of the resistivity plotted as log(p) vs 1/T1’4; the data are consis- tent with thermally activated variable range hopping.34

Polysilicon is thus a very satisfactory thermometer, but for our particular devices, owing to the fabrication of the membrane, its use leads to more complicated processing steps. The difficulty lies in the fact that the polysilicon thermometer must be made prior to etching the silicon away and forming the membrane since the membranes cannot stand up to the polysilicon processing. This means that an approximately 100 nm thick polysilicon layer must be protected while the 330 pm thick Si wafer is etched in KOH. We have found that the LPCVD Si-N is the only mask which stands up against this KOH step. We therefore must deposit the nitride for the membrane in two layers, sandwiching the polysilicon thermometer between two 90 nm thick films of nitride.

We start by depositing the first 90 nm nitride t?lm. We then deposit approximately 110 nm of polysilicon in an

Rev. Sci. Instrum., Vol. 65, No. 4, April 1994 Thin fi lm calorimeter 949

1.5

s

0 20 40 60 80

ON Temperature (K)

PIG. 3. (a) Diagram of two polysilicon low-temperature thermometers with different active area geometries. Leads are heavily doped polysilicon, while the active area (at the center of each strip) is masked from the heavy doping and receives a lighter dose. Photolithographically delined SiOa masks (shaded) define a straight (narrow path) and serpentine (wide path) active area. (b) Resistance of the active area of the polysilicon low-temperature thermometers (lead resistance has been subtracted) with boron concentration of 1.25~ lOI cmd3 and polysilicon leads with boron concentration of 3 x low cmm3. Inset shows log(p) vs 1/T”4 for the active area, the temperature dependence expected for variable range hopping (see Ref. 34).

LPCVD furnace at 605 “C and auneal it for 1 h at 1000 “C. The wafers are then commercially ion implanted with the first of two implant? and annealed at 900 “C for 30 min to activate the doping. We use boron at an energy of 20 keV which puts the peak of the implant distribution near the middle of the layer.36 Boron concentrations between 1.25~ 1019 and 2.0X 1019 crn3 (1.38~ 1014 and 2.2~ 1014 cmm2 area dose) were successfully used for this hrst implant which creates the active temperature sensing areas. In the next step, 450 nm of SiOZ is deposited, again in an LPCVD furnace, and patterned to act as a mask for several small areas against a second high dose implant which creates the leads (2 x 102’ cmm3; 2.2 x 1015 cme2 area dose of B at 20 keV, also annealed at 900 “C for 30 min) . The leads develop a relatively low and temperature-independent resistivity (about 750 $I cm) with any concentration above 1 x 10”’ cmw3. The temperature dependence of the lead and active area resistances are shown in Fig. 3 (b) .

The Si02 mask is a relatively tall and narrow structure and can lead to problems with nitride coverage if the second nitride layer is deposited on top of it. We therefore remove the Si02 after the heavy implant. The polysilicon is then patterned into the strips shown in Fig. 3(a) after which the second nitride layer is deposited. The low-stress

nitride process is prone to developing particulates which are incorporated in the film and can lead to pinholes. This sets a lower limit of approximately 50 nm on the thickness of nitride which makes a sufficient mask in KOH, 90 nm is our usual choice.

After the Si wafer is etched in KOH to form the membrane, electrical contact to the polysilicon thermometers must be made on the Si frame. We use a plasma etcher to etch small holes in the second nitride layer. Care must be taken not to etch too far, as once the nitride is etched away, the polysilicon etches very rapidly. The wafer is dipped in HF to remove the silicon oxide immediately prior to the Ti/Au-Pd evaporation which forms the contacts for the polysilicon thermometers. Using this technique we have found that the metal makes good ohmic contact to the polysilicon leads.

C. Microcalorimeter mounting

The Si frame of the microcalorimeter is attached to a removable copper mount with either Apiezon II or N grease37 and pressed down using phosphor-bronze pins. Electrical connections are made by bonding 0.001 in. gold wires from the Au-Pd contact pads on the Si frame to gold pads (cut from ceramic integrated circuit mounts) epoxied on the copper mount. The copper mount (with the sample) is pressed onto a Cu block in a cryostat by screws, again using Apiezon H or N to ensure thermal contact. Copper wires soldered onto the other end of the gold pads connect to the wires in the cryostat sample chamber via gold pins. Commercially calibrated Ge (Ref. 38) and Pt (Ref. 39) thermometers are pressed into holes in the Cu block close to the sample mount attachment point, using copper-filled conducting grease. Two more calibrated thermometers are mounted on top of the sample chamber for routine calibration checks.

This whole assembly is inside a relatively standard immersion-type vacuum cryostat with a variable heat link to the bath.40 For measurements below 77 K, the cryostat is evacuated, sealed, precooled with LN2, and immersed in LHe. For all measurements above 77 K (including those to date above room temperature), the cryostat is immersed in liquid nitrogen and is continuously pumped with a turbo pump. In the latter case, the pressure at the top of the cryostat is typically around 6x 10Y7 Torr, and is lower at the sample space due to cryopumping by the LN2-cooled walls.

D. Electronics

An ac bridge technique is used in which the bridge is balanced when the sample and block are at To and the small temperature rise ST of the sample relative to the block is detected as an off-null signal by a lock-in amplifier. The sample thermometer R, and the matching thermometer on the Si frame, R, (see Fig. 1) serve as the arms of the bridge. A schematic of the essential components of the bridge is shown in Fig. 4. Having the matching resistor on the device itself makes the wiring and stray capacitances of the two arms more symmetric than the usual approach of

950 Rev. Sci. Instrum., Vol. 65, No. 4, April 1994 Thin fi lm calorimeter

COARSE ADJ.

L- +=

FROM OSC r

FIG. 4. A schematic of the ac bridge. R, is the sample thermometer; R, is the matching thermometer on the Si frame. R, is used to measure the current in the bridge. The bridge is balanced by changing RY which varies the voltages at op-amps 3 and 4. Transformers Tl and T2 are used for isolation. The oscillator signal is amplified by a programmable amplifier 1, and buffered by op-amp 2. The phase shifters needed to balance the reactive component of the bridge for the high resistance thermometers at low temperatures are the RC circuit elements next to op-amps 3 and 4.

an external matching resistor. For the Pt thermometers, R, is typically 1000-1600 fi and is within 100 fi of R,. For the low-temperature thermometers (both Nb-Si and doped polysilicon), R, varies from 10 to 300 kfi as a function of temperature. Referring again to Fig. 4, RA is a resistor allowing the measurement of the current through the bridge for thermometer resistance measurements, and RB is an identical resistor added to keep the bridge symmetric.

puter and communication between equipment is made through a GPIB interface.

A Philips PM5 110 oscillator provides the excitation voltage for the bridge and the reference for. the Stanford Research SR530 lock-in amplifier. The bridge is balanced by changing R,, which varies the voltages to op-amps 3 and 4 (which are on a single chip), thereby changing Vi and Vz, the voltages across the sample and matching thermometers, respectively. The magnitudes of the bridge excitation voltages Vi and I’, are a compromise between minimizing self-heating and making measurements of the off-null voltage possible at a lock-in gain which results in a full scale range of at least 2 ,!.iV. Irl and V2 are typically 10-13 mV for the Pt thermometer, giving a self-heating of <2 mK at room temperature, about 4 mK at 80 K, and < 10 mK around 40 K. For the low-temperature thermometers, larger excitation voltages can be used; at every temperature, self-heating is kept to less than 0.03% of To.41 RV consists of two helipots connected to dc motors for automated coarse and fine adjustments. The resistor- capacitor (RC) components next to the op-amps serve to balance the reactive part of the bridge.

The frequency used for the bridge when using the Pt thermometer is 733 Hz, which is fast enough for the time constants encountered above 40 K (see Table I). At low temperatures, the faster relaxation time constants (as little as a few ms) necessitate a frequency of 1500 Hz or greater.

The block temperature iS read and controlled by a Quantum Design 401802 Digital R/G Bridge. Stability is approximately It2 mK at 80 K and better than f4 mK above room temperature using the Pt block thermometer, and 0.002%-0.005% at lower temperatures with the Ge thermometer.

TABLE I. Addenda and time constants for microcalorimeters with Au sample.

Sample thickness T CT-- c h addenda Ti”tc 4 (lo-” m) 6) (PJAQ @J/K) (ms) bs)

1000 1 7.2 x 1o-5 5 x 1o-5 0.03 1 0.55 1000 4.3 1.9 x 10-3 2 x 10-3 0.22 2.7 loo0 10 2.7 x lo-’ 1 x lo-* 1.4 22 1000 40 0.68 0.18 4.4 150 2OuO 100 2.6 1.2 6.9 380 2000 300 3.1 4.0 11 240

10000 900 18 6 7 95

The center tap of the bridge provides the input to the lock-in amplifier. The output* of the lock-in amplifier is recorded by a 1 MHz, 1Zbit RC electronics computer os- cilloscope, allowing averaging of many relaxation sweeps. Data acquisition is automated by a pc-compatible 386 com-

‘Specific heat of metal layers based on Ref. 42. hMeasured values except 1 and 900 K which were extrapolated from measured values.

“ri,,=&/2K where d=0.125 cm, the distance to the edge and K is the thermal diffusivity of the bilayer consisting of the gold sample layer and the nitride membrane under it. lY= CKnittni*+ K.4utAu)/

~~~~~~~~~~~~~~~~~~~~~~~ ) where fi are the thicknesses, ar the thermal con- ductivifies, q the specific heats, and pi the densities of the layers, determined as discussed in the text.

dMeasured external relaxation time constant; values for the thicknesses shown were calculated by scaling the data from a 307 nm gold sample.


2OK 20K

I in cryostat

To computer

FIG. 5. A schematic diagram of the filtered switch for the current to the sample heater. The switch is a MOSFET 2N7OC0.

Current to the sample heater is supplied by a Keithley Model 2243 programmable current source and is turned on and off using a metal-oxide-semiconductor field-effect tran- sistor (MOSFET) 2N7000 switch. There is significant capacitance between the Pt thermometer and the heater through the sample (which acts like a ground plane), and between their respective leads and pads on the frame (via the semiconducting Si frame). The capacitance via the sample is approximately 40-80 pF and that via the Si frame is approximately 400 pF at room temperature and decreases with decreasing temperature as the Si becomes more insulating. Due to these capacitances, the sample heater with its relatively high voltage is capacitively cou- pled to the low-voltage bridge thermometers. An abrupt change in the heater voltage introduces a high frequency spike into the bridge and overloads the lock-in amplifier. To avoid the high frequencies associated with an abrupt transition, we filter the trigger to the heater switch (see Fig. 5) in order to turn the heater off more slowly than a step function. This filter time constant has to be much less than the relaxation time constant being measured, but long enough to minimize interference and prevent the lock-in from overloading. A Computer Boards CIO-D1024 card provides the trigger and also controls the relays via a rib- bon cable.

Work at low temperatures (below 10 K) is complicated by noise and capacitance problems. Typical resistances of the polysilicon thermometers at 4 K are greater than 100 kfi, and typical stray capacitances are 0.1 nF. For an excitation frequency of 1500 Hz, wR,C is approximately 0.1, introducing attenuations into measurements of R, and AR,. This problem necessitates the use of two frequencies in the measurement. For measuring the resistance R, and AR, (see section on thermal conductance) we use 100 Hz, and for measuring the relaxation curves, where the attenuation of the signal is irrelevant because only the determination of Q- matters, we use frequencies of 1500 Hz or greater.

III. EXPERIMENTAL METHOD

We use the relaxation method4 to measure heat capacity, although we note that the microcalorimeters may be


used with any of the small sample calorimetry techniques. The microcalorimeter with its thin film sample [see Figs. 1 (f)-1 (g)] is pressed against a copper block at temperature Ta. A small dc current (typically 5-300 PA) is supplied to the thin film sample heater, causing the temperature of the sample and the addenda (the amorphous Si-N membrane under the sample, the thin film sample thermometers, and the thin film heater itself) to rise to a temperature T,+ST. 6T is determined by the thermal link between the sample and the outside world, Ker, which includes conduction through the Au-Pd leads .and the amorphous Si-N “border” and, at higher temperatures, radiation. We then turn off the sample heater and measure the relaxation of the sample temperature back to the block temperature. The temperature decays exponentially

T- T,+ST exp( -UT),

where r= C/K,, and C is the sum of the sample heat capacity and the contributions from the various addenda.

The determination of the sample specific heat using the relaxation method thus involves four separate measurements: the time constant of the relaxation 7, the thermal link Ker, the heat capacity of the addenda which must be subtracted from the total C, and the sample volume or weight which is used to convert heat capacity into specific heat. In this section, we discuss the range of validity for this technique and how each of these measurements are implemented for the microcalorimeters.

A. Internal and external time constants

The bridge is balanced with the temperature of the block (and sample) stable at a temperature To. The sample heater is turned on, changing R T and driving the bridge off balance. The current to the sample heater is chosen such that 6T<0.003To to ensure that the values of C and K change by less than 1% within the temperature interval.” At time t=O, the sample heater is turned off and the off- null signal of the bridge (proportional to ST) is recorded up to tz2.5~. Several sweeps (25 to 225) are taken, averaged, and fitted to an exponential. A rounding of the signal near t=O is introduced by the lock-in filter time constant and by the filtered heater switch (see Fig. 5), both of which are set to <~/lo. Fitting is done from t=O.lT (away from the rounding) to 1.17. Figure 6 shows an example of a measured relaxation curve out to 77. For To above 40 K, we find no detectable drifts in the base line, allowing the base line to be recorded before and after the set of sweeps and averaged. Below 40 K, the noise in the base line is greater (due to the larger resistances of the low-temperature thermometers and the shorter time constants). We therefore record and subtract the base line after each sweep.

A simple exponential decay is dependent on the condition that internal time constants Tint (through the sample and between the sample, the heater, and the thermometer) are much faster than the external time constant 7. Physi- cally, the sample must be isothermal during the relaxation to the block temperature and must be at the same temperature as the thermometers. The internal time constants are

Time (ms)

0 500 1000 1500 2000 Time (ms)

FIG. 6. A typical relaxation curve (averaged over 100 sweeps) with r==252.3 ms at room temperature. This data was taken with a sampling time of 2 ms, sample heater power of 30 mW, lock-in amplifier full scale range of 5 pV, and integration time of 10 ms. Inset shows the log plot of the same relaxation curve. Note the linearity down to 77 indicating a single time constant.

controlled by the thermal diffusivities of the various components and by the distances involved. The time constant associated with thermal diffusion through the thickness of the sample/membrane/thermometers is extremely fast since the sample, the thermometers, and the heater are evaporated directly onto the 180 nm thick membrane. The internal time constant associated with lateral thermal diffusion is slower due to the much longer distances involved. To ensure that this time constant is sufficiently fast re- quires that the thermal conductance of the sample be large compared to K,,. This requirement is met when the sample is conducting or when a parallel conducting layer such as gold or aluminum is deposited either under or over the sample. An “empty” device therefore cannot be measured because 7int is not (7.

The lateral internal time constant is estimated based on thermal diffusion through a bilayer or trilayer consisting of the sample, the nitride membrane, and any parallel conducting layer. The thermal diffusivity of a bilayer is given by

K= Kltl +Kz tz w1t1+ P2 c2 t2 ’

where ti are the thicknesses, Ki the thermal conductivities, cf the specific heats, and pi the densities of the layers. The time constant q,,, is equal to d2/2K where d=O. 125 cm, the distance from the center of the sample to the edge. The density, thermal conductivity, and specific heat for the conducting layer are based on literature values’7p42 assuming a mean free path limited by an approximate grain size of 100 nm. The density, thermal conductivity, and specific heat for the Si-N membrane are based on measurements made on this material at room temperature by Mastrangelo et a1.43 The values at other temperatures were calculated by scaling the values for vitreous silica42,44 to the room

10 15 20 25 30 Time (ms)

FIG. 7. Relaxation curves (r=7.35 ms) at 10 K using a boron-doped polycrystalline Si resistance thermometer. This figure shows the relaxation curves for three different sample heater currents giving three different powers: 30.02, 37.99, and 46.90 nW. The largest current gives 6V =12.2 pV, which corresponds to 6T=25 mK. The inset shows the logarithm of all three curves, scaled by Z”, which is proportional to the relative power input. Note that they lie on top of each other, indicating not only a single time constant but the same time constant and thermal conductance. The high frequency oscillation is the incompletely filtered 2f component of the bridge excitation.

temperature nitride values. Table I shows estimates of 7int at various temperatures together with the measured external relaxation time constant 7 for a typical microcalorimeter with a 100-200 nm Au sample. Typical Caddenda are also shown and will be discussed below. We note that these estimates of 7int are worst case estimates since we neglect the Pt thermometer and heater which have good thermal conductivity and will decrease the internal time constant, and since we believe from measurements of the addenda that the room temperature value (and hence the values at all temperatures) for c&d& may be overestimated (discussed below). Faster internal time constants are obtained by using Al for a conducting layer instead of Au due to both higher thermal conductivity and lower specific heat.

Experimental data for the relaxation shows one time constant from 0.17 up to 7~, beyond which the signal is dominated by noise (see Figs. 6 and 7) .45 This observation supports the conclusion of Table I that, with an appropriate choice of conducting layer, CTint<T. Figure 7 shows relaxation curves of 153 nm (2.6 lug) Al taken at 10 K for several values of sample heater current. The curves scale with the power (as shown in the inset), as expected, showing that the thermal conductance and the heat capacity are not changing appreciably within the temperature throw ST and giving further credibility to the simple time constant analysis.46

B. Thermal conductance

After measuring the time constant 7, we measure the heat link Ker between the sample and the block. The current source is used to provide electrical power P to the sample heater, producing a steady-state temperature rise


AT (typically larger than 6T used for the determination of 7). In steady-state conditions, the power in (P) equals the power out. Therefore,

P=K,,dAT+a~[(To+AT)‘-~] I

=: (Kco,d+4~cAT;)AT=Ke~AT,

where Kcond is the thermal conductance of the Au/Pd leads and the Si-N membrane border, u is the Stefan-Boltzmann constant, E and A are, respectively, the emissivity and surface area of the sample/membrane/thermometers. The contribution to the power from radiation assumes an isothermal environment held at To. This is ensured by a copper heat shield surrounding the block. Note that we do not need to calculate any of these quantities; K,, is found by measuring P and AT. Typical values used for AT are O.O2T,; the averaging of K,, over AT is removed by inter- polation once we know the functional form of the temperature dependence of KeE.

To measure AT, we calibrate the sample thermometers during each measurement using the commercially calibrated block thermometers. This process is somewhat more involved than the techniques described in Refs. 1, 3, 4, and 20 because the matching resistor R, on the Si frame also depends on temperature. With the heater turned off and both the sample and block at To, VI, and V, (see Fig. 4) are adjusted to balance the bridge ( Vs=O) . The current I through the bridge is determined by the voltage V, across RA . When power P is input, RT changes to R,+ AR,, and the bridge is driven off balance with an off-null signal A Vc . The temperature rise AT=AR,(dRT/dT)-‘. For the closely matched case RTzRM, VlzV,, and AR, is approximately given by 2AV,,/I (the exact solution is given in Ref. 47 ) . To determine dR r/dT, R T is measured at each block temperature To. The voltage V, across R, is measured and the current is simply I= VA/RA, so that R,=R,VR/V,. The voltages V,, V,, V,, V,, and AV, are all measured with the lock-in amplifier, using low-noise relays to switch measurements between them. R, measurements are made to better than 0.5%. Measurements are carried out over some temperature range with r, P/AR=, and R, versus temperature stored in a file. R,(T) is fit to a spline and the derivative dRT/dT calculated. AR, is then converted to AT, which together with r and P/ART, yields K,,(T) and C(T). Typical dR/dT for the Pt thermometers is 2.5 R/K and does not vary greatly for a given thermometer from 40 to 360 K. For the polysilicon or the Nb-Si thermometers, dR/dT is a strongly varying function of temperature (see Figs. 2 and 3).

C. Addenda

Because it is not possible with our microcalorimeters to measure an empty device, the most accurate method for determining the heat capacity addenda is to use a differential technique. First we deposit a conducting layer and measure the heat capacity, which becomes the addenda for the subsequent layer. We then deposit the sample, remea- sure the heat capacity, and take the difference. The first (conducting) layer must be thick enough to ensure a re-

954 Rev. Sci. Instrum., Vol. 65, No. 4, April 1994

laxation curve with a single time constant. 250 nm of Al or Au is sufficient for measurements around room temperature. Thinner layers are sufficient at low temperatures where the specific heat and the thermal conductivity of the metal samples become much larger than that of the silicon nitride.

For some samples, it is desirable to deposit them directly on the Si-N membrane, rather than on an Al or Au film. In this case, if a conducting layer is needed, it must be deposited on top of the sample and the differential technique is not possible. We will show in the Results section that the addenda varies very little between different devices and hence an approximate value can be used if high accuracy is not needed or if only the temperature dependence through a phase transition is wanted. To improve on the accuracy of the estimated addenda, the usual technique used, for example, in Refs. 1, 3,4, and 20 is to calculate the addenda based on measurements of specific heat for the various contributions and determinations of the mass of each contribution (e.g., the mass of the sapphire substrate and of the wires used for electrical leads). In the ideal case, a sample with substrate and thermometers with infinite thermal conductivity is attached to the block by leads with no heat capacity and finite thermal conductivity KL. In this case, r= C/KL, where C= C,+C,. C, is the sample heat capacity and C, is the combined heat capacity of the substrate (i.e., the membrane beneath the sample), the thermometers, and the heater. A correction can be made to include the effect of finite lead heat capacity CL. Specifi- cally, c=c*+c,+ 1/3CL to within 1% when CL/( C,+ C,) < 0. 1.4 This still gives a single time constant r= C/K,. For the microcalorimeters, the Au-Pd leads and the Si-N border around the sample contribute to the total thermal conductance and hence to the heat capacity. The heat capacity C, of the Si-N border is relatively large, so that we must consider (CL+ C,)/( C,+ C,) which may be nearer the value of 0.7-1.0, making the expansion which led to the correction 1/3CL invalid. We performed a finite element heat flow calculation to determine the effect of this large heat capacity and found that l/3 (CL+ C’) still gave an adequate approximation to the addenda heat capacity (within a few percent for the sample and membrane thicknesses described in this article). This calculation will be described below.

The high heat capacity of the Si-N border also led us to consider the potential problem of heat flow from the border back to the sample, which would give rise to mul- tiple time constants. [Note, however, that the observation of a single time constant experimentally (Figs. 6 and 7) gives confidence that heat flow back from the border is not a problem for the devices and conducting layers we use.] In other words, can the Si-N border act as a heat reservoir, storing heat and eventually reintroducing it back to the sample? A complication in analyzing the Si-N border is that it constitutes a two-dimensional heat flow problem. The square nitride membrane border can be reduced to an effective one-dimensional problem by replacing it in the calculation with nitride “leads” which are the width of the sample, correcting for the missing corners by slightly in-

Thin film calorimeter

creasing the conductivity and specific heat. Using a simple graphical method in determining the shape factor,48 the conductance increases by approximately 20% when the corners are included. To test the importance of heat flow- ing back to the sample, and to determine what contribution the Si-N border makes to the heat capacity addenda [i.e., whether using l/3( C,+C,) is appropriate], a one- dimensional problem with two different parallel conducting paths was analyzed. We solved the time-dependent heat flow equation for a sample with a given heat capacity C, and thermal conductance K, connected to the block by two leads, one with high conductivity but low heat capacity (representing the metallic leads) and one with low conductivity but greater heat capacity (representing the Si-N border) using finite differences4’ We examined the relaxation curve generated in this simulation and determined whether it fit a simple exponential form with a time constant r= C/K, where C= C,+ l/3 (CL + Cb) . ( C’ in this simulation includes the heat capacity of the sample, the heater, the thermometers, and the portion of the membrane under the sample.)

We found from this calculation that the total heat capacity which should be included as addenda depends on the ratios CJC, and C/C, and on the ratios KJK, and KJKb, where the subscript b refers to the Si-N border, L to the Au-Pd leads, and C and K to the extensive quantities of heat capacity and thermal conductance, respeo tively. The danger therefore is that even our differential technique for determining the addenda might lead to an error as the ratios CJC, and CJC, change with the addition of a second sample layer. We have found, however, that the corrections to the simple l/3( C,+C,> addenda contribution are quite small (less than 3% ) for all cases relevant to the microcalorimeters, that is, when r&r. For example, at room temperature, CJC, is approximately 1.6 for a 250 nm Al layer, CJC, is approximately 18, KJK, is approximately 11, and KJKb is approximately 25. With these numbers, the calculation shows a single relaxation time constant with r= C/K, where K= KL+Kb and C= C,+ l/3 ( CL+ C,), with C, determined to an accuracy of 3%. For this case, the ratio of thermal conductance through the Au-Pd leads to that through the Si-N border KJKb is 2.3. If KL/Kb increases above 3, deviations from one time constant appear in the calculation. For larger values of C/C,, i.e., thicker films or lower temperatures, the accuracy improves to better than l%, and in fact the limit on the ratio of KL/Kb for observation of a single time constant can be relaxed. In practice, these limits are the same as those that ensure q&r.

D. Sample volume determination

The sample area is known to high accuracy as it is photolithographically determined either by etching or by deposition through the evaporation masks shown in Fig. 1 (g). The sample thickness is determined by measuring neighboring samples from the same deposition, using either Rutherford backscattering or a profilometer technique.

50 1”‘,““,“,,,“‘,

40 -

30 -

20 -

10 -

0 ltll’(*I~‘illl’ll~~ 0 100 200 300 400

Temperature (K)

FIG. 8. Typical measured thermal link KeR. K,, can be written as K,,=a+b~7’+cT~. The linear term (shown in the figure with circles) is from conduction through the Au-Pd leads and S-N border and the cubic term from radiation with c=4aui where A is the sample/membrane/ thermometers surface area , ~,2(0.25 cm)* and (T is the Stefan-Boltzmann constant. For this particular device and sample (aluminum), the emissivity E is found from the fit to be -0.1.

IV. RESULTS

A. Thermal conductance

The effective thermal link Ker= Kand + 4~x54 T3 can be written as a+ bT+cT3, where a and b are constants related to the thermal conductance of the leads and c=4mA. Typical measurements shown in Fig. 8 show that indeed K,, has this temperature dependence, which allows us to extract an approximate value for the emissivity (assuming A=0.125 cm2, typical values determined for E are 0.1, a reasonable number). The T3 (radiation) contribution to K,, is extremely small until we reach temperatures above 200 K.

B. Addenda

We measured a series of samples (Au, Cu, and Al) of different thicknesses (200 to 1000 nm). The heat capacity of the Au, Cu, or Al (calculated from literature values42) was subtracted from the total measured heat capacity. This process is shown graphically in Figs. 9 (a>-9( c) . The addenda for a number of microcalorimeters from different processing batches are shown in Fig. 10; results are re- markably similar to each other. Au and Al have very different Debye temperatures and hence their specific heats have very different temperature dependencies in the temperature interval studied. The similarity of the addenda for such different thicknesses of different types of samples supports the reliability of the measurement. The small differences that are seen are due to differences in the nitride and metallization thicknesses, and can be corrected for since these thicknesses are readily measured on each device. (In fact, these thickness variations have been much reduced in recent batches, as is evidenced by the similarity of batch B and C in the figure).


G r; 0.8 2

total measured heat copocity +H+w++

A++++@+

++++ +++

++++ +f’ literature value for Al - +if -. -

l - -5z-..

f //--- - --,-.-’

C’ _._’ ,,0 _.- / .’ &dendo = Ctotol-CA, / / ,/

.A’ .fl ,

0.0 t --+-I' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 1 0 100 200 300 400

(4 Temperature (K)

-5’ 1.3 ‘23 u n F

/ + / + /’

+ r”

:~:~/, , (, I-jI:~~~:~:__‘_i . 30 40 50 60 70 80

04 Temperature (K)

total measured heat capacity,,’

f,’ /

/ /’ literature

/ ’ value for Au

+/ /

/ I

addenda =

0 ICI 20 30 40 50

(4 Temperature (K)

FIG. 9. Measurement of the heat capacity of microcalorimeter plus aluminum or gold sample. (a) 278 nm (4.7 pg) of Al on 199 nm nitride membrane from 80 to 370 K; includes addenda data from (b) and (c); (b) 345 nm (5.8 pg) of Al on 165 nm nitride from 40 to 70 K; (c) 307 nm (37 pg) of Au on 199 mn nitride from 4 to 40 K. The literature values for the specific heat for Au and Al are from Ref. 42. The sample areas were defined photolithographically; the sample thicknesses were from a crystal monitor. Subtracting the heat capacity of the Au or Al from the total measured heat capacity gives the addenda shown on each figure.

Figure 11 shows calculations of the various contributions to the addenda as a function of temperature, compared to a typical measured value. The largest contribution at higher temperatures comes from the nitride membrane, both the portion under the sample and the border outside the sample area (which contributes l/3 of its total heat

5p 8”,” I1,1111,1111j

Q : 14 - -3 - v- - x3 - .e ki : Q2 - s - %I- 2 -

--Batch A -‘.“. Batch B - -Batch C

01, 3 ’ 1 ” b 3-t ’ * ’ * I ” 9 t 81 0 100 200 300 400

Temperature (K)

FIG. 10. Addenda for six different devices from three different processing batches determined as shown in Fig. 9. Note the similarity especially in the B and C devices. Batch A has about 165 nm nitride thickness while B and C have about 199 nm. The thicknesses and areas of the metallic leads and thermometers are also slightly different for these three batches.

capacity). The thermometers, heater, and Au-Pd (or heavily doped polysilicon) leads contribute about 12% of the total addenda above 77 K, and increasing amounts at lower temperatures. The heat capacity of the nitride membrane is calculated from the specific heat for vitreous silica (from Refs. 42 and 44) since the room temperature value agrees with the single measurement for amorphous silicon- rich silicon nitride (0.7 J/g K, from Ref. 43). The calculated total addenda exceeds the measured values by a significant amount at all temperatures; since the nitride membrane is the dominant contribution, the most likely

///

CD0 _.C’

@“‘* y,H.y~H’~ji$kme

“Jf,./ ‘. sample

: 1 I

0 0 100 200 300 400

Temperature (K)

FIG. 11. Comparison of measured and calculated addenda. (0) are typical addenda, determined as shown in Fig. 9. (- -) represents the total addenda calculated for the metallic portions of the microcalorimeters. (-.-a ) represents the heat capacity of the nitride membrane directly beneath the sample, assuming the nitride specific heat to be the same as that of vitreous silica. (-) shows the calculated total addenda. This calculation exceeds the measurement while scaling the nitride specific heat to a room temperature value of 0.5 J/g K fits our data better.


2.0x1 o-” 1” ,

s? --'I.5 -

Au sample+Au layer+addenda

2 x

2 g 1.0 - a

s $J 0.5 -

I2

o.o- r ' * t ' a ‘ * t ' * ‘ t " q ' 0 100 200 300 400

(4 Temperature (K)

2.5x10-5~ 5 ' ' ' 1 I ' 8 ' 1 ' ' 4 ' 1 ' ' ' ' 4

c 2.0 - ;3 =-, 1.5 - .A=! ii Ek 1.0 - cl 5 0.5 -

t+ +++c2 ,t+ 1 ,+++++iu samD!etAi layer j

faddenda r

0.01, * B I ' * * * t " I ' 1 ' p t t .-i 0 100 200 300 400

(b) Temperature (K)

FIG. 12. Total heat capacity of a microcalorimeter with the first (conducting) layer (0 ), with the iirst and second (sample) layer ( + ), and the difference between these (A). (a) The first layer is 290 nm (35 pg) of Au and the second (sample) layer is 390 *40 nm (47 S= 5 pg) of Au. The difference of the measurements fits the literature value for 425 nm (51 ,ug) of Au. (b) First layer is 595 nm (10 pg) of Al and second layer is 470+50 nm (57*6 pg) of Au. The difference data fits the literature value for 532 mn (64 pg) of Au. Literature values are from Ref. 42.

explanation is an overestimate of its specific heat. At low temperatures (below 40 K), the difficulties as-

sociated with attenuation from using high bridge frequencies (necessitated by fast time constants) and with the large temperature dependence of dR/dT have delayed the completion of measurements to the same accuracy that we have achieved above 40 K. Values of KeE are currently uncertain to approximately 10% with corresponding un- certainties in the total heat capacity C and hence Caddenda. We have solved the attenuation problem by using two measurement frequencies (as previously discussed), but we have not yet completed calibration runs to the accuracy shown in Figs. 9(a), 10, and 12. A relaxation curve taken at 10 K was shown in Fig. 7, demonstrating that the technique works. Figure 9(c) shows preliminary results of measurements from 4.3 to 40 K with a sample of Au. Al samples have also been measured. For the purposes of determining addenda, a few measurements were made near 4.2 K where the total heat capacity for a 307 nm (37 pg) Au sample plus addenda was found to be 9 X lo-’ J/K.

The literature value for gold is 7X 10e9 J/K; the addenda is therefore approximately 2 x 10M9 J/K. At lower temperatures, we have extrapolated the addenda, based on literature values for a-SiOz,44 to give a value of approximately 1 X lo-*’ J/K at 1.5 K. These lowest temperature measurements are currently in progress.

At present we do not have an apparatus capable of making specific heat measurements above 360 K, but microcalorimeters with samples, Pt thermometers and heat- ers, and Au/Pd leads on them have been cycled in a furnace up to 800 K without damage. The Pt thermometers have been measured up to 800 K; their temperature sensitivity does not change. We do not, therefore, anticipate any problems with using these devices up to at least this temperature.

C. Calibration samples

A variety of samples have been prepared and measured using these devices. As discussed in the section on processing, samples have been prepared by evaporating or sputtering onto the back side of the membrane, then spinning on photoresist at the bottom of the etch pit and photolithographically defining the sample area. As a simpler alterna- tive, we use evaporation masks [see Fig. 1 (g)] which fit down into the pit, approximately 25 pm away from the membrane. This close fit is needed to avoid shadowing effects, particularly when coevaporating materials. For samples where the differential method is used, the first (highly conducting) layer is made with a slightly larger area than the second (sample) layer to ensure that the sample lies completely within the highly conducting area defined by the first layer.

Figures 9 and 10 show measurements which were made on single layers of either Au or Al on several devices from different batches at temperatures ranging from 4 to 360 K. Sample thicknesses ranged from 150 to 600 nm.

For a better test of the microcalorimeters, we measured a series of gold samples deposited as the second layer onto previously measured microcalorimeters with various types of first layers. Figures 12(a) and 12 (b) show the results of the first measurement, the second measurement, and the difference between the two which should be the heat capacity of the gold sample. The thickness (and hence the volume and weight) of the gold layer is uncertain to approximately 10% due to the sample preparation technique used. This measured heat capacity is shown compared with the literature values for bulk gold,42 with the sample volume as the single adjustable parameter. In each case, the difference between the two measurements fits the literature value for Au to the accuracy of the film thickness. In particular, the temperature dependence is well fit.

V. CONCLUSION

We have developed a thin tllm microcalorimeter which is capable of measuring the specific heat of thin film samples as small as a few micrograms over a wide temperature range. The contribution of the addenda for these devices is more than two orders of magnitude less than addenda for

Rev. Sci. Inetrum., Vol. 65, No. 4, April 1994 Thin film calorimeter 957

any comparable system capable of measuring thin films. Measurements to date have been made from 4.3 to 360 K; the thermometers have a sensitivity which allows their use down to 1.5 K. Devices have been thermally cycled to 800 K and the temperature dependence of the Pt thermometers successfully measured, we anticipate no problems extend- ing specific heat measurements up to 800 K.

ACKNOWLEDGMENTS

We would like to thank J. Birmingham, E. D. Dahl- berg, R. C. Dynes, T. H. Geballe, A. M. Goldman, P. L. Richards, and G. R. Stewart for useful discussions con- cerning specific heat measurements and for assistance in writing this manuscript. For numerous resistivity measurements, we would like to thank Erica Perry Lewis, Dennis Palmer, and Nancy H. Trissel. For assistance in the development of the microcalorimeter, we would like to thank R. S. Muller, R. M. White, and the staff and students at the UC Berkeley Microlab: Katalin Voros, Bob Hamilton, Kris Pister, Debra Hebert, Dave Hebert, James Bustillo, William Flounders, Leslie Field, and Phil1 Guillory. We would also like to thank George Kassabian and Alan White for their help in developing the electronics for the experiment and Arnold Krause and Andy Pommer for help in machining. This work has been supported by the National Science Foundation (Grant Nos. DMR 9 l-09004 and DMR 88-10374).

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15To keep sensitivity constant with temperature T, the ac bridge offset voltage SF, should be constant. This voltage can be written as SV,= fL#WdT)&i,,,,, . I,,, is determined by limiting the self-heating of the thermometer ST, = ILR/K to 0.0003 T. Koc T in the region of interest, say K=aT. Then ST, =I2 R/aT=O.O003T or I,,,,,= ~O.O003u/RT. Therefore SV0=5 0.0003a/RT(dR/dT)6T,,,,I,. We ‘? would like to keep GTsnmple small to limit the averaging over the (temperature dependent) specific heat to approximately 1%. Quantitatively we want AC/C=[C( T0+6T,,,,, ) - C( T,)]/C( To) ~0.01, where T, is the block temperature. We assume we are in a region where C(T) =flT3 (the limitation on ST-,,, when C is linear in T is even less stringent). Using C(T,+ST,,,,,) =C( T, ) +GT,,,,,,dC/dT we fmd AC/C=36T,,,,,, /T(O.Ol, or ST,,,,,,<O.O03T. Then the bridge

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43C H. Mastrangelo, Y.-C. Tai, and R. S. Muller, Sensors Actuators A 23, 856 (1990).

44R. C. Zeller and R. 0. Pohl, Phys. Rev. B 4, 2029 (1971). 45All devices which meet the criteria discussed in the text show a single

time constant, although not all devices were checked to >7r at all temperatures to the accuracy shown in Fig. 6.

46Further confidence is given by the accuracy of the measurements of the heat capacity of standard materials like Au and Al (see Results section).

958 Rev. Sci. h-strum., Vol. 85, No. 4, April 1994 Thin film calorimeter

J7The resistance change AR, can be solved exactly. Note that V, and V, stay constant and it is the current that changes when R, changes on turning on the sample heater. The off-null voltage AV, = V, -Z’(R,+RT+ART), where I’ is the current when the bridge is off-balance. Rearranging, ART= ( VI -A V,)/Z’- VI/Z, where we have used the condition at balance V,=Z(R,+ RT). The current I’ is related

to Z, i.e., Z’=Z[l +ART/(R,+RT+RB +R,)]-’ Substituting I’ into the expression for ART yields ART= (-AVdZ)[l- (l+ V,/V,)-’ -tAVd(V,+ Vdl-‘, which reduces to AR, z -2AVdZ when ?~~Z v,.

4sF Kreith Principles of Heat Transfer (Harper and Row, New York, 198;).


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