THE ELECTRICAL CONDUCTIVITY OF SILICON BETWEEN 500 C AND 1200 C

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1 R949 Philips Res. Repts 31, ,1976 THE ELECTRICAL CONDUCTIVITY OF SILICON BETWEEN 500 C AND 1200 C M. C. H. M. WOUTERS*) **), H. M. EIJKMAN*) and L. J. van RUYVEN*) Abstract The electrical conductivity of p- and n-type dislocation-free silicon has been measured in the temperature range 500 C to 1200 o C, using the Van der Pauw method. The results were compared with the extrapolated values obtained from existing theories. An empirical formula for the effective intrinsic carrier concentration in this temperature range has been evolved using a computer approximation for the data combined with the standard expressions for the mobilities and their temperature dependence. 1. Introduction The electrical conductivity of silicon at elevated temperatures is one of the important parameters in the determination of the amount of coupling between R.F. heating coils and silicon. Actual measurements of the conductivity have only been done at temperatures below 700 DC and, therefore, some investigators 1.2) use extrapolated values at higher temperatures. Although the qualitative theory of electrical transport at high temperatures is quite well understood 3), the inaccuracies in the individual contributions of the various effects make the ultimate value for the conductivity meaningless. For example, the conductivity of silicon at 1100DC, a temperature commonly used in chemical vapour deposition, estimated from the formulae published by different authors 3.4.5) can vary by as much as a factor of two. 2. Theory At low temperatures, i.e., the extrinsic conductivity range, Hall measurements and conductivity-mobility measurements give information about the separate parameters of the transport equation. The conductivity in the intrinsic range is due to the contributions of an equal number of electrons and holes with different mobilities : In which (1 is the specific conductivity, (U- 1 cm-i), q the electron charge (I) *) Semiconductor Development Laboratory N.V. PhiIips' Gloeilampenfabrieken, Nijmegen, The Netherlands. **) In partial fulfillment for a degree, College of Advanced Technology, Dordrecht, The Netherlands.

2 THE ELECTRICAL CONDUCTIVITY OF SILICON BETWEEN 580 C AND 1200 C O- 19 C, III the intrinsic concentration of electrons or holes (cm="), and /-ln and /-lp the mobility of electrons and holes (cm 2 /V s) respectively. With the exception of the electron charge the parameters in eq. (1) are dependent on temperature. The concentration of electrons III is controlled by the excitation process across the bandgap Eg, in thermal equilibrium: Ne and N; are the effective densities of states for the conduction and valence band, respectively, Eg depends on T. The effective densities of states, under the approximation of a temperatureindependent effective mass m*, obey the following relations in which Me represents the number of equivalent conduction band minima. (2) (3a) For silicon (4a) N; = t+, (3b) Ne = Tt, (4b) where Tis expressed in K, and N; and Ne in cm=". In the low-temperature range the dependence of Eg on temperature has been obtained by Varshni 6) from transport measurements, and more recently by Bludau et al. 7) from the shift of the exciton line with temperature. At very high temperatures an additional correction!le on the magnitude of Eg has to be applied because of the electrostatic interaction between the negative electrons and positive holes 8): and!le = 7.10' nit. r+ ev. (5) The temperature dependence of the electron and hole mobilities is determined by the scattering mechanism operating in the temperature range under consideration, which for intrinsic silicon is phonon scattering. The temperature dependence of the mobilities due to acoustical phonons has been measured frequently in the low-temperature range;' viz.

3 280 M. C. H. M. WOUTERS. H. M. EIJKMAN AND L. J. VAN RUYVEN One may expect that at high temperatures the optical phonons, in addition to the acoustical phonons, contribute to the scattering process. The carrier scattering due to optical phonons has been estimated 9), however, and appears to be negligible; as is also the case for electron-hole scattering. 3. Experimental results and discussion We have measured the specific conductivity of silicon in the temperature range from 550 oe to 1200 oe using the standard Van der Pauw technique. The samples were circular silicon slices with a diameter of 50.8 mm (2 inch) and a thickness of approximately 300 [Lmprepared from dislocation-free silicon, cut perpendicular to the (111) direction. Both p- and n-type slices of high and low room-temperature resistivities have been studied. The apparatus consisted of an electrical furnace "Multiple Unit" type HDH 2718, designed for temperatures from 500 oe to 1300 oe, and a quartz holder shown in fig. I on which the samples were clamped between 4 tungsten wires touching the slice at 4 positions on its circumference. The temperature was measured with a thermo-couple at a position approximately 10 mm from the slice, and it was calibrated for this particular configuration. The reproducibility of the measurements was within 5 % over the full range and the accuracy amounts to 7 %. Fig.!. Detail of the quartz-slice holder for high temperature van der Pauw measurements.

4 THE ELECTRICAL CONDUCTIVITY OF SILICON BETWEEN 500 C AND c '900 _ r t-c) _10 3 ft (K-l) Fig. 2. Th e logarithm of the conductivity in ohm - 1cm-1 versus loooit in K - 1, the dashed lines give the extrapolated calculation of Morin and Maita 5), the fullline gives the computer approximation to our measurements as indicated by x. Fig. 2 gives the results for different samples with the logarithm of the conductivity plotted against 1000/T in K-l. For convenience the temperatures have been indicated in centigrades as well. Conductivity data at the high-temperature end and at the low-temperature end are already available. Glasov et al. 10) have studied the change in conductivity for the transition from solid to liquid silicon near the melting point (1420 "C), It is not clear whether the samples used were single crystals; a few of their measurements were taken in the range 1200 C to l400 C and agree within 10 % with our data. The results at the low-temperature end of our data can be compared with those of Morin and Maita 5). The data from the lightly doped samples agree within the experimental error, indicating that the use of dislocation-free silicon has no observable effects on the electrical conductivity' in this temperature range. Morin and Maita 5) obtained from their measurements a nearly linear relation between log a and T-l. This can easily be seen from (1) with: q = (C), nl- 2 = T3 exp (-Eg*/kT) (cm- 6 ), Eg* = nit Tt (ev), /-lp = T-2.3 (cm 2 fv s), /-ln = T-2.6 (cm2fv s). (6) (7) (8) (9)

5 282 M. C. H. M. WOUTERS. H. M. EIJKMAN AND L. J. VAN RUYVEN The second term in E g *, defined in (5), is the correction resulting from the Coulomb interaction between charge carriers of opposite sign, which becomes effective in the temperature range above 400 C 8). A graphical representation of the above formulae in the temperature range of interest is given in fig. 2 by the dashed line. Although Morin and Maita limit the validity of their formulae to the temperature range below 800 C it is interesting to extrapolate this formula to the temperatures of our measurements. At 1000 "C the deviation from our measurements is 18 %, at c 22 % and at 1400 C 40 %. Starting from (1) and retaining the same functional dependence on T as in (6), we have constructed a modified formula for n[> based on a computer approximation to our data, and the standard expressions (8) and (9) for the mobilities, and Varshni's 6) numerical expression for Eg, given in (10). The resulting approximation to the conductivity is presented in fig. 2 by the full line. Thus, using /hp = T- 2 3 (8) /hn = T- 2 6 (9) T 2 E= O- 4 g T+, 1108 (10) the result is III = Tt exp (-Eg/2kT). (11) This empirical expression describes the measurements with an accuracy of better than 5 % over the temperature range 800 K to 1400 K, and agree within 10% with the measurements of Morin and Maita 5) in the low-temperature range. In table I the measured conductivities are presented at temperatures from 600 C to c.

6 THE ELECTRICAL CONDUCTlVITY OF SILICON BETWEEN SOO c AND 1200 C 283 TABLE I The intrinsic conductivity of silicon at temperatures from 600 oe to 1400 oe. resulting from a computer approximation to the measurements temperature COC) conductivity (0-1 cm- 1 ) l Nijmegen, June 1976 REFERENCES 1) L. J. Giling and J. Bloem, J. Crystal Growth 31,317, ) S. E. Mayer and D. E. Shea, J. Electrochem. Soc. 111, 550, ) K. Seeger, Semiconductor Physics, Springer-Verlag, Wien, ) E. H. Putley and W. H. Mitchell, Proceedings ofthe Physical Society, A72, 193, ) F. J. Morin and J. P. Maita, Phys, Rev. 96, 28, ) Y. P. Varshni, Physica 34, 149, ) W. B1udau, A. Onton and W. Heinke, J. appl. Phys. 45,1846,1974. B) W. J. Moore, Physical chemistry, Prentice Hall Inc., New York, ) F. J. Morin and J. P. Maita, Phys. Rev. 94, 1525, ) V. M. Glazov, S. N. Chizhevskaya and N. N. Glagoleva, Liquid semiconductors, Plenum Press, New York, 1969.