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Temperature dependence of the conductivity of a silicon inversion layer at low temperatures
~
Solid State Communications, Vol.58,No.8, pp.511-514, 1986. Printed in Great Britain.
0038-1098/86 $3.00 + .00 Pergamon Journals Ltd.
TEMPERATURE DEPENDENCE OF THE CONDUCTIVITY OF A SILICON INVERSION LAYER AT LOW TEMPERATURES R.P. Smith and P.J. Stiles Physics Department, Brown University, Providence, RI
02912 USA
(Received December 6, 1985 by J. Tauc)
The temperature dependence of the conductivity of a high-mobility silicon MOSFET is reported for temperatures from 0.2 to 23 K and electron densities from I to 20 x iOllcm-2. The results show a strong increase in conductivity with decreasing temperature. However, this rate of increase was observed to rapidly diminish outside of a restricted range of temperature for a given density. The decreasing temperature dependence at the lower temperatures is attributed to the saturation of screening as the effects of broadening become comparable to thermal effects. The diminishing temperature dependence at the higher temperatures puts limits on the applicability of recent calculations for the conductivity.
i. Introduction
number of samples for ns>1012cm -2. Kawaguchi and KawaJi 12 measured the properties of lower mobility samples (~<16,000 cm2/V-s) for temperatures between about 2 K and 50 K. Both groups observed an approximately linear temperature dependence with some decrease in the temperature dependence at low temperatures. A recent paper by Dorozhkin and Dolgopolov 4 reported on the temperature dependence of the conductivity for 3xlOll
We measured t h e temperature dependent conductivity for a high mobility n-type silicon MOSFET for a wide range of low temperatures and electron densities. The data shows large increases in conductivity for decreasing temperatures, and as recent theories I-3 predict the relative increases become larger as the density decreases. While it has been understood for some time that this can be explained qualitatively by temperature dependent corrections to the polarizability I, the data does not agree with some recent calculations for the entire range of temperatures2, 3 and shows that the simple linear relationship between conductivity and temperature with a slope independent of density exhibited in experimental work by Dorozhkin and Dolgopolov 4 probably does not apply for densities and temperatures outside of the limited ranges covered in their work. A saturation of the temperature dependence is observed at temperatures T<0.OSEF/k B (where E F is the Fermi energy and k B is Boltzmann's constant). This saturation can be attributed to a smoothing of the structure in the polarizaability at 2kp due to impurity interactions as shown by Das Sarma and Vinter5, 7, Ando 6, and Das Sarma 8. However, this prediction is somewhat speculative since no detailed calculations of the mobility including this effect for high mobility samples have been done. There are a number of papers on the conductivity at relatively high electron densities 9-]2 (i.e., electron density n s greater than 1011 cm-2). The temperature dependence of the conductivity in samples with large values of N i, the ionized impurity density, was measured and analyzed by Hartstein at. al. 9-I0 and concentrated primarily on high values of n s. Other workers have also recently investigated the conductivity at low temperatures. Cham and Wheeler II reported on systematic studies for a 511
512
Vol. 58, No. 8
SILICON INVERSION LAYER AT LOW TEMPERATURES
should itself be renormalized self-consistently due to the electron-lmpurlty Interaction 5-8, resulting in smearing of the polarizability function even at T=O K. More detailed calculations of the effect of the renormallzation by Ando 6 and Das Sarma 8 show that the temperature dependence of ~he conductivity implied by Maldague's work ~ will weaken below some temperature related to this impurityinduced broadening. Calculations for the temperature dependent mobility have been made that incorporate some of the above ideas. Stern 17 did a numerical calculation using the temperature dependence of the screening and numerically self-conslstent wave functions 17 for the z dependence that successfully described experimental results for relatively high values of n s. He also pointed out that different energy dependences of the various scattering mechanisms make it necessary to add the effects of those mechanisms within the calculations instead of simply applying Matthiessen's rule i/~ = l/b+ 1 / ~ r a f t e r considering the various effects separately. This work was recently extended by Stern and Das Sarma 2 to lower values of n s. A very recent analytical expansion of Maldague's I temperature corrections to the screening and its effect on the conductivity has been given by Gold and Dolgopolov 3. The results are remarkably simple and appear to explain the recent experimental results of Dorozhkin and Dolgopolov 4 even though they do not incorporate the impurity broadening in their calculations.
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Experimental
Results
The conductivity was measured in a silicon MOSFET at temperatures between 0.2 K and 20 K and at electron concentrations of I to 25 x 10 11 cm -2. The sample, a Hall geometry device with an aluminum gate and an oxide thickness of about 1,000~, had a peak mobility as high or higher than samples used in previous studies (p~ 24,000 cm2/V-s at the time of this work) and is the same sample used in the fractional quantum Hall effect work reported by Furneaux st. al. 18 ; their studies and high field work done at the time of this work shows that the inversion layer is homogeneous. The measurements were taken with a four point probe method at a frequency of about 145 Hz. The current was taken from a function generator in series with a I0 M~ resistor; the potential was measured from probes along one edge of the sample with a differential amplifier whose output was connected to a lock-in amplifier. The data was taken in two low temperature apparatus--the 0.2 to 23 K data in a dilution refrigerator and 1.3 to 4.Z K data (not shown) in pumped He 4. The number of electrons in the inversion layer was determined by Shubnikov-de Haas measurements at a number of different magnetic fields and by transconduetance techniques at 77 ° K and was found to be independent of magnetic field and temperature. The threshold voltage of 0.4 volts indicates that the number N~ of positive ions at the interface is in the neighborhood of 2 x 1010cm-2. Figure one shows the conductivity and the mobility for temperatures ranging from 0.2 to ZO K. It can be seen that the mobility
20
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Figure 1. (a) The conductivity and (b) mobility versus electron density for 0.2 K < T < 23 K. The densities are accurate to about 2 x iO I0. increases with decreasing temperature, and that this effect is much stronger for low densities until the density becomes very low (ns~1011 cm-2). The peak values of the mobility are approximately linear in n s (with the possible exception of the 23 K curve), but there is no correspondingly ~imple dependence on temperature. Figure two shows the conductivity as a function of temperature for a number of electron populations. All of the curves show an inflection point near a temperature T = (O.I)EF/k B. These linear regions occur at approximately the same values of kBTIE P as the wider and more linear regions reported by Dorozhkin and Dolgopolov 4. The temperature dependence in our data becomes much weaker for temperatures above or below the inflection points. For the higher densities, the resistivity increases monotonically and at an increasing rate with increasing temperature as with most previously reported data. However, at densities below 5xlOL1cm -2 the second derivative of the resistivity with respect to temperature becomes negative at higher temperatures. For the lowest
SILICON INVERSION LAYER AT LOW TEMPERATURES
Vol. 58, No. 8
ns= 215 xlOH/cm2
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T (°K)
Figure 2. The conductivity plotted as a function of temperature. The different plots are for different densities. The error bars at the top and middle show the range of estimated error in the data for those curves; curves between them have intermediate amounts of error, and curves below have less error. Data was taken for temperatures of 0.2, 0.5, 1.0, 1.4, 2.6, 3.5, 4.2(2), 6.4, 8.0, 10.9, 15.9, and 23 K. The dashed curve gives T = F/K B and the dotted curve T = ( 0 . 1 ) ~ / ~ . density reported, the resistivity itself decreases slightly with increasing temperature for 8K
5]3
their theory does not consider the terms of higher order than T 3/2 in the expansion of o. We assumed that the higher order terms were dominated by a positive T 2 term in fitting the data. The error incurred by trying to determine daldT in our data is quite large because of the short interval of T over which the slope is large. The experimentally determined values of the linear coefficient C(n,a) for the temperature d e p e n d e n c e are approximately 20~ smaller than predicted, which is comparable to experimental error. This discrepancy could be due to the fact that t h e low temperature saturation is not considered in the theory. C(n,a) as a function of ns has a broad maximum in the vicinity of ns=TxlO£1cm -2, in qualitative agreement with the theory. The only significantly larger deviation occurs for the smallest value of ns reported, for which our data has a maximum amount of uncertainty. Recent numerical calculations by Stern and Das Sarma for temperatures below 5 K very nearly predict the correct values of de/dT for ns=3xlOllcm -2. However, they slightly overestimate the slope for higher n s. It is possible that the smoothing of the polarizability function by impurity interactions and surface roughness accounts for the discrepancies for large n s. Also, the calculations are sensitive to certain sample parameters, such as the ionized impurity density, the locations of the impurities, and the typical surface roughness dimensions, which were not precisely determined for the sample reported on here. Also shown in figure two is the temperature at which the broadening parameter ? is equal to the thermal energy, where F=h/2z. The impurity broadening work5-8 indicates that the temperature dependent part of the polarizability should saturate with decreasing temperature when the thermal energy kBT is approximately equal to ? for broadening parameters approximately equal to (0.2)EF; the temperature d e p e n d e n t part of the conductivity will saturate at higher values of kBT/E F for smaller values of V/EF and lower values of k B T / ~ for larger values of ?/EF. The deviation of F/k B away from the saturation region at the lowest densities is in the correct direction to be explained by this effect, but the deviation appears to be too large. At high densities FlkB appears to deviate to a smaller value of T than the apparent onset of saturation; this deviation is not in the predicted direction for the impurity broadening models, but more extensive high-temperature data should be taken to verify this. The scattering is dominated by surface roughness effects at the higher densities, and the impurity broadening models may not be quantitatively applicable. Weak localization causes the conductivity to decrease with decreasing temperature 20 but according to our current understanding this term can only account for less than I0~ of the downward deviation with decreasing temperatures for the range of data shown. It cannot account for the deviation from the linear behavior observed at low temperatures. 4. Conclusion Conductivity measurements were performed on a silicon MOSFET over a wide range of low
514
SILICON INVERSION LAYER AT LOW TEMPERATURES
temperatures and densities corresponding to the regime where the temperature dependence of the screening of impurities is expected to be strongest. It is shown that the conductivity is approximately linear for a small region for kBT-(O.1)E F. Outside of thls region the temperature dependence becomes progressively weaker. More detailed work for different samples and conditions will follow.
Vol. 58, No. 8
helpful discussions regarding their theories and for sending us unpublished work. We would also like to thank M. Graf for allowing us to do the work in his dilution refrigerator and Dale Syphers and John Furneaux for their help. We would especially like to thank R.G. Wheeler for supplying us with the sample and helpful conversations. Thls work was performed under NSF grant DMR-8314397.
Acknowledgements--We would like to thank F. Stern, S. Das Sarma, and A. Gold for their REFERENCE ] P.F. Maldague, Surface Science 73, 133 (1978). 2 F. Stern and S. Das Sarma, Solid State Electronics 28, 211 (1985) and private communication. 3 A. Gold and V.T. Dolgopolov, Journal of Physics 18, L463 (1985) and A.Gold and V.T. Dolgopolov, to be published in Physical Review B. 4 S.I. Dorozhkin and V.T. Dolgopolov, JETP Letters 40, 1019 (1985). 5 S. Das Sarma and B. Vinter, Physical Review B 24, 549 (1981). 6 T. Ando, Journal of the Physical Society of Japan 51, 3215 (1982). 7 S. Das Sarma and B. Vinter, Surface Science
113, 176 (1982). 8 S. Das Sarma, Physical Review Letters 50, 211 (1983). 9 A. Hartstein, T.H. Ning and A.B. Fowler, Surface Science 58, 178 (1976). i0 A. Hartstein, A.B. Fowler and M. Albert, Surface Science 98, 181 (1980). ii K.M. Cham and R.G. Wheeler, Physical Review Letters 44, 1472 (1980).
12 y. Kawaguchl and S. KawaJi, Surface Science 98, 211 (1980), and Solid State Comm. 36, 257 (1980),
13 F. Stern and W.E. Howard, Physical Review 163, 816 (1967). 14 y. Matsumoto and Y. Uemura, Japan Journal of Applied Physics, Supplement 2, Part 2, 367 (1974). 15 T. Ando, Journal of the Physical Society of Japan 43, 1616 (1977). 16 F. 3tern, Physical Review Letters 18, 546 (1967). 17 F. Stern, Physical Review Letters 44, 1469 (1980). 18 J.E. Furneaux, D. Syphers, J.S. Brooks, G.M. Schmiedeshoff, R.G. Wheeler, and P.J. Stiles, to be published in the proceedings of the Sixth International Conference on the Electronic Properties of Two-Dimensional Systems. 19 A. Gold, private communication. 20 For a good review, see P.A. Lee and T.V. Ramakrishnan, Reviews of Modern Physics 57, 287 (1985).
Solid State Communications, Vol.58,No.8, pp.511-514, 1986. Printed in Great Britain.
0038-1098/86 $3.00 + .00 Pergamon Journals Ltd.
TEMPERATURE DEPENDENCE OF THE CONDUCTIVITY OF A SILICON INVERSION LAYER AT LOW TEMPERATURES R.P. Smith and P.J. Stiles Physics Department, Brown University, Providence, RI
02912 USA
(Received December 6, 1985 by J. Tauc)
The temperature dependence of the conductivity of a high-mobility silicon MOSFET is reported for temperatures from 0.2 to 23 K and electron densities from I to 20 x iOllcm-2. The results show a strong increase in conductivity with decreasing temperature. However, this rate of increase was observed to rapidly diminish outside of a restricted range of temperature for a given density. The decreasing temperature dependence at the lower temperatures is attributed to the saturation of screening as the effects of broadening become comparable to thermal effects. The diminishing temperature dependence at the higher temperatures puts limits on the applicability of recent calculations for the conductivity.
i. Introduction
number of samples for ns>1012cm -2. Kawaguchi and KawaJi 12 measured the properties of lower mobility samples (~<16,000 cm2/V-s) for temperatures between about 2 K and 50 K. Both groups observed an approximately linear temperature dependence with some decrease in the temperature dependence at low temperatures. A recent paper by Dorozhkin and Dolgopolov 4 reported on the temperature dependence of the conductivity for 3xlOll
We measured t h e temperature dependent conductivity for a high mobility n-type silicon MOSFET for a wide range of low temperatures and electron densities. The data shows large increases in conductivity for decreasing temperatures, and as recent theories I-3 predict the relative increases become larger as the density decreases. While it has been understood for some time that this can be explained qualitatively by temperature dependent corrections to the polarizability I, the data does not agree with some recent calculations for the entire range of temperatures2, 3 and shows that the simple linear relationship between conductivity and temperature with a slope independent of density exhibited in experimental work by Dorozhkin and Dolgopolov 4 probably does not apply for densities and temperatures outside of the limited ranges covered in their work. A saturation of the temperature dependence is observed at temperatures T<0.OSEF/k B (where E F is the Fermi energy and k B is Boltzmann's constant). This saturation can be attributed to a smoothing of the structure in the polarizaability at 2kp due to impurity interactions as shown by Das Sarma and Vinter5, 7, Ando 6, and Das Sarma 8. However, this prediction is somewhat speculative since no detailed calculations of the mobility including this effect for high mobility samples have been done. There are a number of papers on the conductivity at relatively high electron densities 9-]2 (i.e., electron density n s greater than 1011 cm-2). The temperature dependence of the conductivity in samples with large values of N i, the ionized impurity density, was measured and analyzed by Hartstein at. al. 9-I0 and concentrated primarily on high values of n s. Other workers have also recently investigated the conductivity at low temperatures. Cham and Wheeler II reported on systematic studies for a 511
512
Vol. 58, No. 8
SILICON INVERSION LAYER AT LOW TEMPERATURES
should itself be renormalized self-consistently due to the electron-lmpurlty Interaction 5-8, resulting in smearing of the polarizability function even at T=O K. More detailed calculations of the effect of the renormallzation by Ando 6 and Das Sarma 8 show that the temperature dependence of ~he conductivity implied by Maldague's work ~ will weaken below some temperature related to this impurityinduced broadening. Calculations for the temperature dependent mobility have been made that incorporate some of the above ideas. Stern 17 did a numerical calculation using the temperature dependence of the screening and numerically self-conslstent wave functions 17 for the z dependence that successfully described experimental results for relatively high values of n s. He also pointed out that different energy dependences of the various scattering mechanisms make it necessary to add the effects of those mechanisms within the calculations instead of simply applying Matthiessen's rule i/~ = l/b+ 1 / ~ r a f t e r considering the various effects separately. This work was recently extended by Stern and Das Sarma 2 to lower values of n s. A very recent analytical expansion of Maldague's I temperature corrections to the screening and its effect on the conductivity has been given by Gold and Dolgopolov 3. The results are remarkably simple and appear to explain the recent experimental results of Dorozhkin and Dolgopolov 4 even though they do not incorporate the impurity broadening in their calculations.
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Experimental
Results
The conductivity was measured in a silicon MOSFET at temperatures between 0.2 K and 20 K and at electron concentrations of I to 25 x 10 11 cm -2. The sample, a Hall geometry device with an aluminum gate and an oxide thickness of about 1,000~, had a peak mobility as high or higher than samples used in previous studies (p~ 24,000 cm2/V-s at the time of this work) and is the same sample used in the fractional quantum Hall effect work reported by Furneaux st. al. 18 ; their studies and high field work done at the time of this work shows that the inversion layer is homogeneous. The measurements were taken with a four point probe method at a frequency of about 145 Hz. The current was taken from a function generator in series with a I0 M~ resistor; the potential was measured from probes along one edge of the sample with a differential amplifier whose output was connected to a lock-in amplifier. The data was taken in two low temperature apparatus--the 0.2 to 23 K data in a dilution refrigerator and 1.3 to 4.Z K data (not shown) in pumped He 4. The number of electrons in the inversion layer was determined by Shubnikov-de Haas measurements at a number of different magnetic fields and by transconduetance techniques at 77 ° K and was found to be independent of magnetic field and temperature. The threshold voltage of 0.4 volts indicates that the number N~ of positive ions at the interface is in the neighborhood of 2 x 1010cm-2. Figure one shows the conductivity and the mobility for temperatures ranging from 0.2 to ZO K. It can be seen that the mobility
20
24
28
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28
ns(lOUcm -2) 25 / ! /
20
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Figure 1. (a) The conductivity and (b) mobility versus electron density for 0.2 K < T < 23 K. The densities are accurate to about 2 x iO I0. increases with decreasing temperature, and that this effect is much stronger for low densities until the density becomes very low (ns~1011 cm-2). The peak values of the mobility are approximately linear in n s (with the possible exception of the 23 K curve), but there is no correspondingly ~imple dependence on temperature. Figure two shows the conductivity as a function of temperature for a number of electron populations. All of the curves show an inflection point near a temperature T = (O.I)EF/k B. These linear regions occur at approximately the same values of kBTIE P as the wider and more linear regions reported by Dorozhkin and Dolgopolov 4. The temperature dependence in our data becomes much weaker for temperatures above or below the inflection points. For the higher densities, the resistivity increases monotonically and at an increasing rate with increasing temperature as with most previously reported data. However, at densities below 5xlOL1cm -2 the second derivative of the resistivity with respect to temperature becomes negative at higher temperatures. For the lowest
SILICON INVERSION LAYER AT LOW TEMPERATURES
Vol. 58, No. 8
ns= 215 xlOH/cm2
4 -
13.3
z
~
C" ,?, o
2
,~-.! . ~5 ~ , 0
5
I0
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.
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20
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T (°K)
Figure 2. The conductivity plotted as a function of temperature. The different plots are for different densities. The error bars at the top and middle show the range of estimated error in the data for those curves; curves between them have intermediate amounts of error, and curves below have less error. Data was taken for temperatures of 0.2, 0.5, 1.0, 1.4, 2.6, 3.5, 4.2(2), 6.4, 8.0, 10.9, 15.9, and 23 K. The dashed curve gives T = F/K B and the dotted curve T = ( 0 . 1 ) ~ / ~ . density reported, the resistivity itself decreases slightly with increasing temperature for 8K
5]3
their theory does not consider the terms of higher order than T 3/2 in the expansion of o. We assumed that the higher order terms were dominated by a positive T 2 term in fitting the data. The error incurred by trying to determine daldT in our data is quite large because of the short interval of T over which the slope is large. The experimentally determined values of the linear coefficient C(n,a) for the temperature d e p e n d e n c e are approximately 20~ smaller than predicted, which is comparable to experimental error. This discrepancy could be due to the fact that t h e low temperature saturation is not considered in the theory. C(n,a) as a function of ns has a broad maximum in the vicinity of ns=TxlO£1cm -2, in qualitative agreement with the theory. The only significantly larger deviation occurs for the smallest value of ns reported, for which our data has a maximum amount of uncertainty. Recent numerical calculations by Stern and Das Sarma for temperatures below 5 K very nearly predict the correct values of de/dT for ns=3xlOllcm -2. However, they slightly overestimate the slope for higher n s. It is possible that the smoothing of the polarizability function by impurity interactions and surface roughness accounts for the discrepancies for large n s. Also, the calculations are sensitive to certain sample parameters, such as the ionized impurity density, the locations of the impurities, and the typical surface roughness dimensions, which were not precisely determined for the sample reported on here. Also shown in figure two is the temperature at which the broadening parameter ? is equal to the thermal energy, where F=h/2z. The impurity broadening work5-8 indicates that the temperature dependent part of the polarizability should saturate with decreasing temperature when the thermal energy kBT is approximately equal to ? for broadening parameters approximately equal to (0.2)EF; the temperature d e p e n d e n t part of the conductivity will saturate at higher values of kBT/E F for smaller values of V/EF and lower values of k B T / ~ for larger values of ?/EF. The deviation of F/k B away from the saturation region at the lowest densities is in the correct direction to be explained by this effect, but the deviation appears to be too large. At high densities FlkB appears to deviate to a smaller value of T than the apparent onset of saturation; this deviation is not in the predicted direction for the impurity broadening models, but more extensive high-temperature data should be taken to verify this. The scattering is dominated by surface roughness effects at the higher densities, and the impurity broadening models may not be quantitatively applicable. Weak localization causes the conductivity to decrease with decreasing temperature 20 but according to our current understanding this term can only account for less than I0~ of the downward deviation with decreasing temperatures for the range of data shown. It cannot account for the deviation from the linear behavior observed at low temperatures. 4. Conclusion Conductivity measurements were performed on a silicon MOSFET over a wide range of low
514
SILICON INVERSION LAYER AT LOW TEMPERATURES
temperatures and densities corresponding to the regime where the temperature dependence of the screening of impurities is expected to be strongest. It is shown that the conductivity is approximately linear for a small region for kBT-(O.1)E F. Outside of thls region the temperature dependence becomes progressively weaker. More detailed work for different samples and conditions will follow.
Vol. 58, No. 8
helpful discussions regarding their theories and for sending us unpublished work. We would also like to thank M. Graf for allowing us to do the work in his dilution refrigerator and Dale Syphers and John Furneaux for their help. We would especially like to thank R.G. Wheeler for supplying us with the sample and helpful conversations. Thls work was performed under NSF grant DMR-8314397.
Acknowledgements--We would like to thank F. Stern, S. Das Sarma, and A. Gold for their REFERENCE ] P.F. Maldague, Surface Science 73, 133 (1978). 2 F. Stern and S. Das Sarma, Solid State Electronics 28, 211 (1985) and private communication. 3 A. Gold and V.T. Dolgopolov, Journal of Physics 18, L463 (1985) and A.Gold and V.T. Dolgopolov, to be published in Physical Review B. 4 S.I. Dorozhkin and V.T. Dolgopolov, JETP Letters 40, 1019 (1985). 5 S. Das Sarma and B. Vinter, Physical Review B 24, 549 (1981). 6 T. Ando, Journal of the Physical Society of Japan 51, 3215 (1982). 7 S. Das Sarma and B. Vinter, Surface Science
113, 176 (1982). 8 S. Das Sarma, Physical Review Letters 50, 211 (1983). 9 A. Hartstein, T.H. Ning and A.B. Fowler, Surface Science 58, 178 (1976). i0 A. Hartstein, A.B. Fowler and M. Albert, Surface Science 98, 181 (1980). ii K.M. Cham and R.G. Wheeler, Physical Review Letters 44, 1472 (1980).
12 y. Kawaguchl and S. KawaJi, Surface Science 98, 211 (1980), and Solid State Comm. 36, 257 (1980),
13 F. Stern and W.E. Howard, Physical Review 163, 816 (1967). 14 y. Matsumoto and Y. Uemura, Japan Journal of Applied Physics, Supplement 2, Part 2, 367 (1974). 15 T. Ando, Journal of the Physical Society of Japan 43, 1616 (1977). 16 F. 3tern, Physical Review Letters 18, 546 (1967). 17 F. Stern, Physical Review Letters 44, 1469 (1980). 18 J.E. Furneaux, D. Syphers, J.S. Brooks, G.M. Schmiedeshoff, R.G. Wheeler, and P.J. Stiles, to be published in the proceedings of the Sixth International Conference on the Electronic Properties of Two-Dimensional Systems. 19 A. Gold, private communication. 20 For a good review, see P.A. Lee and T.V. Ramakrishnan, Reviews of Modern Physics 57, 287 (1985).