Carbon resistance thermometers and variable-range hopping

Carbon resistance thermometers and variable-range hopping* N.S. Sullivan and C M . Edwards Department of Physics, University of Florida, Gainesville, FL 3261 1, USA

Received 15 February 1985 The observed temperature dependence of a wide variety of carbon resistance thermometers is compared with the simple models of variable-range hopping. It is shown that for some resistors at very low temperatures, the ~-power law (for hopping in 2 D) provides a slightly more accurate fit than the ¼-power law (for hopping in 3 D). In other cases, the two power laws are indistinguishable within the accuracy of the data. Keywords: thermometers; resistors; low temperature studies

Carbon radio resistances such as those manufactured by Allen-Bradley ~ or Speer 2 have been used as low temperature thermometers in almost every cryogenic laboratory in the world since 1936 when they were introduced by Giaque 3. While there are empirical formulae for their resistance values, R(T), as a function of temperaturd ,4-8, e g as illustrated by the Clement-Quinnell formula ~ log10 R

+

K/(logloR )

=

2/o +

a/T

(1)

only recently has attention been devoted to the underlying physical origin of the dependences R(T) which do show some qualitative universal behaviouro Anderson et al 9 and Sanchez et al 1° have reported that some carbon composition resistors vary as T -1/4 at low temperatures with R = Roexp(AT -1/4) which can be attributed to variable-range hopping We have compared four very different types of carbon resistance thermometers and find that their conductivities can be described just as well, and in some cases slightly more accurately, by a T -1/3 variation which is given by Motfs result" for electrical conductivity, (r(r), due to variable-range hopping in two dimensions rather than three dimensions. Mott's model '~:2 was originally invoked to describe conduction by a degenerate electron gas in a highly disordered medium He considered the d.~ conduction when the Fermi energy lies in the range of energies where the electron states are localized. This is expected to be the case for resistances made by compacting fine graphite particles for which the conduction could be expected to be dominated by electrons on the surface of the microscopic grains of the material The surface of the grains is characterized by defects such as broken and dangling bonds which modify the localization and enhance the conductivity due to variable range hopping. At low temperatures the dominant mechanism for electrical conductivity is thermally activated hopping of electrons in *Work supportedin part bythe NationalScienceFoundationu Low Temperature PhysicsGrant DMR-8304322 0011-2275/86/040211-04 $03.00 © 1986 Butterworth Et Co (Publishers) Ltd

states near the Fermi energy. The probability, v, of hopping a distance R is given by the product of (1) the overlap integral of the electron wave functions fz = exp(-2aR) and

(2)

and (2) the Boltzman factor

f2 = exp(-W/kB T)

(3)

where W is the activation energy for hopping W ~ 1/(density of states)

(4)

In two dimensions, for large values of R, the average spacing at the Fermi level is

AW = 1/[4rrR2N(EF)]

(5)

Hence v(R) ~ exp [-2aR - 1/(47rRZNFkBT)]

(6)

This is optimized by a particular value of R, namely R Sopt = 1/(41rOtNFkaT) The conductivity is governed by

1/R(T) ~/)opt

~

exp(-3ORopt)

(7)

This T -i/3 law is specific to two-dimensional hopping The Mott T-1/4 law 11 which explains the d.c. conductivity of amorphous semiconductors 13-Is applies to 3-dimensional hopping. An elegant demonstration of the

Cryogenics 1 9 8 6 Vol 26 April

21 1

Carbon resistance t h e r m o m e t e r s : N.S. Sullivan and C M . Edwards I00 K

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Figure 2 Comparison of the temperature dependence of an AllenBradley radio resistor (10 N, 1 / 8 W) with the T -1In laws ~ , 9 . 6 1 9 T -1/3 - 1 0 . 3 0 1 ; . . . . , 1 3 . 1 7 T - 1 / 4 - 1 3 . 8 5 5 . These refer to ~ - and ¼-power laws, respectively

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T -~/3 law has been obtained by Pepper et aL '6'~? for 2-dimensional conduction in MOS and MNOS transistors. In Figures 1-4 we compare a T -1/3 variation ( ) and a T-~/4 variation ( . . . . ) with the calibration of three standard low temperature carbon resistors: (1) a 220f~ (1/2W) Speer resistor, (2) a 10N (I/8W) AllenBradley resistor, (3) a Matsushita radio resistor t8, and(4) a typical home-made thick film resistoP. The difference betwen the fits to a 1/3-power law and a l/a-power law are very small and in only two cases, that of the Matsushita resistor(Figure3) and the 100~ Speer resistor(Figure5), is there any evidence that one exponent provides a better fit than the other. In both Figures 3-5 we find that V3-power law is slightly more accurate. In the comparison of the two power laws with the reported data for the other resistors we cannot distinguish between the two exponents. The range of resistor values explored in Figure I is somewhat larger than that of Reference 9 for the same type of resistor. The different fits obtained from a least-squares analysis are summarized in Table 1. The variance~ ~r2, is appreciably less tor the V3-exponent for the 100II Speer, the Matsushita and the Koppetski resistances. It is interesting to note that in addition to a description of the qualitative temperature dependenc~ the variable-range hopping model can also explain why different types of resistances show the characteristic

212

Cryogenics 1986 Vol 26 April

04 LLI (J Z

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3

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temperature

dependence of

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Matsushita radio resistor(330 E~ 1/8 W) with the T-I/" law~ The data points are taken from Reference 16. , 4 . 0 4 0 T -1/3 - 5.214; , 6 . 8 5 5 7 "-1/4 - 8 . 7 5 9 . These refer to ~ - and tA-power laws, respectively. Note the departure from the low temperature law for temperatures T > 1 K

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Temperature dependence of a carbon glass thermometer (Lake Shore model CGP,-1-2000, Reference 18). , 7 . 8 5 5 7 "-1/3 + 1.807 (T -1/3 law); . . . . , 7 . 8 0 8 T "1/4 + 1.025 (T-~/4 law). These lines do not fit the calibration data for this type of resistor ILl

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exponential increase of R over different temperature ranges. We expect from this model that

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Figure 5 Comparison of the ~ - and N-power laws for the calibration of a Speer resistor ( 1 0 0 f~, 1 / 2 W) reported in Reference 9. , 1 . 2 8 2 T -~/3 - 2 . 2 5 6 ; . . . . , 1 . 9 8 3 T -1/4 - 2 . 9 7 1

Table 1

) 1/0+a)

(8)

where A is a constant; at is the overlap factor, d the dimension for the hopping; a n d N F the density of states at E F which depends on the material. These parameters can be modified by choosing different preparations, e.g by varying the particular size to change ~ and by pyrolizing to reduce the number of free electrons and, thus, N F. For completeness, we note that Sheng and Klafter ~°, using a very different theoretical approach to that of Mott and co-workersH, 12, find that logR ~ T a with at varying from 1/(l + at) at low temperatures to at > 1/2 at high

Least squares fit to the temperature dependence log R(7) = A T -wn + B a

Resistor tape

~-Iaw A

Speer (220 fi, 1 / 2 W) Allen-Bradley (10 fl, 1 / 8 W) Matsushita ( 3 0 0 D,, 1 / 8 W) Koppetski (Reference 17) Speer ( 1 0 0 El, 1/2 W) Carbon glass (Lake Shore CGR-1-20OO)

1.714 9.619 4.040 1.027 1.282 7.855

B -2.025 -10.301

-5.214

+1.460 -2.256 +1.807

¼-law o2

A

B

02

0.036 0.162 0.065 0.0062 0.026 0.0189

2.693 1.317 6.855 1.010 1.983 7.808

-3.074 -13.855 -8.759 +I .362 -2.971 +1.025

0.024 0.204 0.233 0.0065 0.064 0.0285

aR is in kfl, 02 is the variance (arbitrary units)

Cryogenics

1986

Vol 26 April

213

Carbon resistance thermometers: N.S. Sullivan and C.M. Edwards temperatures The temperature at which the crossover to the stronger power law occurs depends on the grain size. The calibrations we are discussing refer to the low temperature regime where variable range hopping is relevant We also compared the temperature dependence of a carbon glass thermometer with the T-'/3 and T-~/4 laws and found that the behaviour of this type of resistance thermometer cannot be adequately described by such power laws As shown in Figure 6, the calibrations* indicate a stronger temperature dependence, and this may be due to the very different origin of the charge carriers in the glass material compared to the carbon resistor. In conclusion, we stress that the accuracy of the calibration of most carbon radio resistors at low temperatures does not allow one to distinguish between 1/3 and 1//4power laws, and extrapolation using either power law may lead to errors at very low temperatures In some cases, a 1/3power law does provide a slightly more accurate description over a wider temperature range.

References 1

Clement, J.R. and Quinnel, E.H. Rev Sci Instr (1952) 23 213

2 3 4 5 6

7 8 9 10 11 12 13 14 15 16 17 18

*The calibrations were carried out by Lake Shore Cryogenics, Westerville, OH 43081, USA

214

Cryogenics 1986 Vol 26 April

19 20

Black,,W.C., Roach, W.IL and Wheatlcy, J.C. Rev Sci Inst (1964) 35 587 Giaque, W.F. lnd Eng Chem (Ind) (1936) 28 743 Hudson, R.P. in Experimental Cryophysics (Eds, Hoare, EE., Jackson, LC. and Kurti, N.) Butterworths, London, UK(1961) 9.5 Scott, R.B. Cryogenic Engineering D. Van Nostrand, London` UK (1967) 5.23 White, G.K. Experimental Techniques in Low Temperature Physics Clarendon Press, Oxford, UK (1979) Hudson, R.P., Marshak, I-L, Soulen, Jr., R.J. and Utton, D.B. J Low Temp Phys (1975) 20 1 Lounasmaa, O.V. Experimental Principles and Methods Below IK Academic Press, New York, USA (1974) Anderson, b.C., Anderson, J.I-L and Zaitlin, M.P. Rev Sci Instr (1976) 47 407 Sanchez, J., Benoit, b. and FIouqnet, J. Rev Sci Instr (1977) 48 1090 Mott, N.F.JNon-CrystSolids (1968) 1, 1;ContempPhys (1969) 10 1251 Mott, N.F. and Davis, E.b. ElectronicProcessesin Non-Crystalline Materials Clarendon Press, Oxford, UK (1979) Walley, P.A. and Jonscher, b.lC Thin Solid Films (1967) 1 367 Clark, A.H.PhysRev (1967) 154, 750;JNoncrystSolid (1970) 2 5265 Chopra, ILL. and Bhal, S.K. Phys Rev (1970) BI 2545-56 Pepper, M., Pollitt, S. and Adkins, C.J. JPhys C Solid State(1974) 7 I_243 Pepper, M., Pollitt, S., Adkins, C.J. and Oakley, R.E. PhysLett (1974) 47 71 Kobaysi, S., Shinoharn, M. and Ono, K. Cryogenics (1976) 16 597 Koppetzki, N. Cryogenics (1983) 23 559 Sheng, P. and Kiafter, J. Phys Rev (1983) B27 2583