Thermal diffusivity measurements on small samples

Advanced Energy Conversion.

Vol. 2, pp. 45-51.

Pergamon Press, 1962. Printed in Great Britain

THERMAL DIFFUSIVITY MEASUREMENTS ON SMALL SAMPLES* A. B.

TIMBERLAKE,~P. W. DAVIS~and T. S. SHILLIDAY~

Summary--A method is described for the measurements of thermal diffusivities of small samples of semiconducting materials. Results of measurements on InSb and GaAs are given. It is shown that in some cases the simple theory of Angstrom's method must be modified to include some other mechanism of energy transfer, possibly that of ambipolar diffusion. INTRODUCTION THE precise measurement of thermal conductivity is always a rather difficult process. When the additional restrictions of high temperatures and small samples are imposed, the difficulties are aggravated. These additional restrictions are sometimes encountered in studies of the high-temperature thermoelectric properties of materials. Samples of homogeneous composition are often small, and data are needed on the thermal and electrical properties over a wide range of temperature. A method is presented here which has enabled measurements of thermal diffusivity of InSb and GaAs to be made over the temperature range 23 °C to 600°C, using samples of dimensions as small as 1 cm × 1 cm × 0-06 cm. The method employed for the measurement of thermal diffusivities is one of several recently explored by Becker [1] and is an adaptation of Angstrom's original diffusivity method. In essence, Angstrom's method is that of imposing a cyclic temperature variation at one end of a sample and then measuring, at various known distances along its length, the amplitude attentuation or phase variation of the resulting temperature wave propagated down the sample. In the method here described, a chopped beam of radiation from an incandescent lamp is focused on one face of a sample having dimensions of about 1 cm × 1 cm × 1 mm thick. If the radiation is properly filtered, all of the incident energy is absorbed at the front face of the sample, producing the desired cyclic temperature variation. This temperature variation is then propagated through the sample as a wave which is damped exponentially. The degree of damping, which is measured at the back surface of the sample, is dependent on the material parameters. The general equation describing the temperature wave in the medium has been derived by Carslaw and Jaeger [2]. Their expression, adapted to account for the experimental conditions which apply here, is

Oac(a, where

Oac(a,

t ) = (EP~e ~¢1/2/tol/2k)exp(--a~/to/2 r ) c o s ( t o t -

\

ax/to/2r-- :]

(1)

t) = a.c. portion of the temperature rise above ambient at a distance " a " into the medium, °C, = emissivity of irradiated surface,

* Supported by Contract NObs-77034. Battelle Memorial Institute, Columbus, Ohio. 45

46

A.B. TIMBERLAKE,P. W. DAVIS,and T. S. SmLLIDAY to ---- chopping frequency, radians per sec, K = thermal diffusivity, cmZ/sec, k ---- thermal conductivity, W/cm °C, P0 =c = amplitude of a.c. portion of incident radiation flux, W/cm 2.

Considering only the r.m.s, magnitude, 0r.m.s., of the temperature wave, manipulation of equation (1), yields

ln(col/2 8r.m.s. ) ~--- --aa,/oJ/2 K + const

(2)

If measurements are made over a range of chopping frequencies and ln(oY2 8r.m.s.) is plotted against o~1/9.,a straight line is obtained which has slope --a/,~/2K. Since V = a0, where Vis the thermoelectric voltage measured, and where the Seebeck coefficient a can be assumed a constant over the very small temperature range considered, a plot ofln(oj1/2 V,.m.s.) versus ~ol/2 gives an identical result. Thus, the thermal diffusivity can be determined from

where m is the slope of a plot of log10 (Vf 1/2) versusf 1/2 a n d f i s the chopping frequency in c/s. E X P E R I M E N T A L DETAILS The optical arrangement employed in impinging a cyclic source of heat on the front face of the sample is shown in Fig. 1. Evacuable chamber .

M,

@

--{

M, L -Projection lamp MrSphericol mirror of f = 14 cm C -Chopper M2-Spherical mirror of f= 14cm W-Window S -Sample

FIG. 1. Optical diagram of thermal diffusivityapparatus. The source lamp is a 1000 W projection bulb whose image is first focused in the plane of the slotted chopper wheel and again, reduced in linear dimensions by a factor of two, on the sample which is contained in an evacuable chamber. The chopper wheel is driven by a synchronous motor powered by an audio oscillator through a 20 W amplifier. The chopping frequency range is 20-1200 c/s. In order to measure the cyclic temperature variation at the back face of the sample, use is made of the Seebeck voltage generated by the sample itself. This is accomplished by contacting one tungsten lead to the edge of the sample and another to the back face of the sample at

Thermal DiffusivityMeasurementson SmallSamples

47

a point opposite to the position of the source image on the front face. Since the source image on the sample face is kept considerably smaller than the sample, the edge contact is far enough away from the heated area that it does not receive any cyclic temperature signal at the frequencies employed. The signal at the back surface is, however (assuming proper frequency and sample thickness), still appreciable, being damped to about 5 per cent of the front surface value. There is, therefore, a resultant Seebeck voltage arising from the two metal semiconductor contacts.

~ Preamplifier~aWove nalyzer

I

0.1~ ~,l,Geoforme r,, tronsformer I osci Audi laotor I

FIG. 2. Electroniccircuitryfor thermal diffusivitymeasurements. This voltage is amplified using the circuit shown in Fig. 2. The signal is first fed into a geoformer transformer and then to a preamplifier from which it is read on a wave analyzer. Arrangements are made to calibrate the signal absolutely by putting a known voltage from an audio oscillator across a resistance in series with the sample itself. With the sample in the dark, the oscillator output is set to match the thermally-generated sample signal and the voltage read. Thus, any dependence of the gain on frequency or amplitude is eliminated. The signal obtained is therefore proportional to the cyclic temperature amplitude at the back surface. Knowledge of this signal as a function of chopping frequency together with the sample thickness permit determination of the thermal diffusivity of the sample with the aid of equation (3). RESULTS InSb

Intrinsic single-crystal InSb was measured over a range of temperatures, 23°C-500°C. The form of the data obtained is shown in Fig. 3. It is to be noted that the magnitude of the signal does not always decrease with increasing temperature. There are a number of factors which could affect the signal amplitude, such as contact resistance and light intensity. The important parameter is the slope of the line from which the thermal diffusivity is calculated. The values of thermal diffusivity measured by the described method have been reduced to thermal conductivity by the relation k = K cp where c and p are specific heat and density respectively. The density used in these calculations was 5.78 g/cm8 [3]. The specific heat from 100°K to 773°K is shown in Fig. 4. The circles are values reported by Gul'tyaev and Petrov

48

A . B . TIMBERLAKE,P. W. DAVIS, and T. S. SHILLIDAY

[4]; the solid dots are results of measurements made in this laboratory by the method of Ginnings and Corrucini [5]. The thermal conductivities calculated from these data, as well as those of other experimenters, are shown in Fig. 5 [6, 7, 8]. The results attributed to Kanai and Nil are calculated from their diffusivity values using the same specific heat and density as employed for the data from this laboratory. The results obtained are in the high part of the range reported by others. These high values are believed to result from energy transport by processes which can occur when a semiconductor is illuminated by photons of energy greater than Eg, the gap energy. G~irtner has

>

=o u_

\1oo °

o '~.

2 g

3000

i0-~ 4'0

-

5'0

6"0

7.0 I

(Chopping Frequency)'~, sec-'lz FIG. 3. Typical experimental data from measurements on an InSb sample at various temperatures. Sample thickness 0"143 cm for T = 25°C, 100°C, 200°C, and 0'119 cm for T ~ 200°C, 300°C, 400°C, and 500°C.

recently treated this effect, the photothermal effect, in considerable detail [9]. He has shown that a semiconductor under illumination may exhibit an anomalous high thermal conductivity, the magnitude of which depends on the energy flux on the surface of the material, the front and back surface recombination velocities, diffusion constant, energy gap, and lifetime of injected carriers. A correction of our results based on G~irtner's work has not yet been made. However, an approximate correction has been made by considering only ambipolar diffusion of photoinjected electron-hole pairs. The initial mechanism of absorption of photons incident on the sample is the creation of electron-hole pairs.* These pairs recombine after diffusing a mean * The glass window in the evacuable chamber filtered out all radiation of longer wavelength than the absorption edge of the sample.

Thermal Diffusivity Measurements

on

Small Samples

49

distance L, the ambipolar diffusion length, where they a r e a s s u m e d to give up their energy to the lattice. Thus there may be considered a region of length L behind the illuminated surface in which an energy Eg per incident photon is transported without attenuation. The measured thermal diffusivity must then be corrected by a factor [ ( a - L)/a] 2, since the sample is effectively shortened by a distance L. The results obtained when this correction is applied to the measured values of thermal diffusivity are shown by the triangles in Fig. 5. The ambipolar diffusion length used to make the correction was 40/~ for the entire temperature range [10]. It can be seen that the corrected values are in very good agreement with those of the other experimenters.

1.4 1.3 1.2 I.I 1.0

/

/..<,

0.9

0.8 Q)

,,2,

E

L)

0.7

•

Measurements made at Battelle o Gultyoev and Petrov

0.6

v (j

0.5

0.4 0.3 0.2 0,1

00

I00

200

300

400

500

600

700

800

Tern perature,"K FIG. 4. Specific heat of InSb.

GaAs The thermal diffusivity of GaAs has been measured from 23 °C to 600°C. With this material it is necessary to filter out all radiation of wavelength longer than the absorption edge at 0.8 t~, i.e., radiation which would not be absorbed at the front surface of the sample. Otherwise, two signals are generated simultaneously at the back surface: One arising from the radiation transmitted through the sample, and absorbed at the back contact, the other from the radiation absorbed at the front and conducted to the back. Two methods were tried to confine the absorption to the front surface. In one case the radiation was filtered using a Corning No. 9788 filter in conjunction with a 1 in. water cell. The other method was to coat the front surface of the sample with a thin layer of Aquadag. The filter was used with a sample 0.107 cm thick. The Aquadag coating was used with the

50

A. B. TIMBERLAKE,P. W. DAVIS, and T. S. SHILLIDAY

500

O~Z

400

Temperature'C ~

300

I

I

IO0

I

I

0 o

0.20 T Od9 t~

~

o.m 0.17

0.16 >, •~ 0"15

.' / I "°f /

0,14 ~0'13 U 0'12

i

:-" ~

/

O'll O.lO F O~C

/

007 OOC I'0

1"5

20

2"5

3~0

3'5

I000 ~- r ,°K" FIG. 5. Thermal conductivity of ~Sb. ( (. . . . . (- . . . . (...... (C)) From (A) From

) Bowers et aL 90 per cent confidence. -) Stuckes, I0 I~ c m -a, n-type. ) Stuckes, 2 x 1015 cm -a, p-type. ) Kanai and Nii, 101~ cm -a, n-type. BMI diffusivity measurements. BMI diffusivity measurements corrected for 40 # ambipolar diffus{on length.

Temperature,°C 500 350 250 150 600 4003OO 200 I

i

I

I

I

I

IO0

I

23 I

I

1

0"45

0(

n ' o~ "i"

0-40

×

E 0.35 u

• (~ "-I

0.30 o.z5

×

"~ 0.20 ~

i

0.I 5

ID ,- (NO I---

X×O il~ ~ -

I

I

Legend • X Z~ []

0"05

Thickness: 0 ' 0 5 4 9 cm Thickness: 0 ' 1 0 7 cm Abrahams, et al. Bate, static h e a t - f l o w measurement at Battelle J

1-00

1"50

2"00 I000 T, ° K

J

2.50

I

3"00

3"50

FIG. 6. Thermal conductivity of GaAs (n = 4" 3 x 1017 cm-3).

Thermal Diffusivity Measurements on Small Samples

51

same sample lapped to 0.0549 cm thickness. There was no essential difference in the values obtained by the two methods. The results are shown in Fig. 6 in terms of thermal conductivities. They were calculated from the measured diffusivities using a specific heat of 0.347 J/g per deg and density of 5.31 g/cm 3. The specific heat was measured in this laboratory and was constant over the temperature range of the diffusivity measurements. The conductivity shows the expected linear dependence on reciprocal temperature. The only other data available for comparison are those of Abrahams, Rosi, and Braunstein [12] and of Bate [13] of this laboratory. These were room temperature values made using static heat flow methods. Abrahams, et al., measured 0.37 W/cm per deg for the lattice conductivity and Bate obtained values of 0.38 and 0.42 W/cm per deg for the total conductivity of two specimens. All of these values are slightly lower than the room temperature values given in Fig. 6. CONCLUSIONS The technique employed seems to be a satisfactory method for the measurement of thermal diffusivities of small samples of elevated temperatures. It must be kept in mind, however, that the essence of the method is that of introducing thermal energy at the front surface of the saple by the use of electromagnetic radiation. As has been indicated above, precautions must be taken to assure that the energy introduced is absorbed at the surface and is promptly converted to thermal energy. When the photothermal effect is corrected for or eliminated by proper sample preparation, the method described has several attractive features. Measurements may be made on samples several millimeters in length and width by several tenths of m m thick. The usual problem encountered at elevated temperatures, radiation loss, is accounted for in the theory. An independent check of the results may be obtained (see equation 1) by comparing the phase and amplitude of the temperature waves at the front and back surfaces simultaneously. Acknowledgements---We are indebted to O. J. Mengali and R. T. Bate for many helpful discussions, and to E. A. Eldridge for specific heat measurements. We are especially greateful to J. H. Becker of the National Bureau of Standards for the advice and information he very generously gave.

REFERENCES [1] J. H. BECKER,J. Appl. Phys., 31, 612 (1960). [2] H. S. CARSLAWand J. C. JAEGER,Conduction of Heat in Solids (lst Ed.), p. 56, Oxford University Press (1947). [3] L. H. DEVAtrXand F. A. P I z ~ o , Phys. Rev. 102, 85 (1956). [4] P. V. GUL'TVAEVand A. V. P~-raov, Soy. Phys. Solid State 1, 330 (1959). [5] D. C. GrNNINGSand R. J. ComtuCtNI, Res. Nat. Bur. Stand. 38, 583 (1947). [6] R. BOWEaS,R. W. UP.~, J. E. BAtraLEand A. J. ConrqISrI,J. Appl. Phys. 30, 930 (1959). [7] A. D. SxtrCKEs,Phys. Rev. 107, 427 (1957). [8] Y. KANAtand R. NH, J. Phys. Chem. Solids 8, 338 (1958). [9] W. W. G.~tTNER,Phys. Rev. 122, 419 (1961). [10] D. G. AVERYand D. P. JENKIrqS,J. Electron. 1, 145 (1955). [11] P. W. DAVIS,A. B. TmmL'RLAKEand T. S. SmLUDAY.J. Appl. Phys. 33, 765 (1962). [12] M. S. AaRAI-IAMS,R. BRAUNS~ and F. D. RosI, J. Phys. Chem. Solids 10, 190 (1959). [13] R. BATE. Unpublished data.