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The electrical resistivity of Na below 1 K
Volume 108A, number 3
PHYSICS LETTERS
25 March 1985
T H E E L E C T R I C A L R E S I S T I V I T Y OF Na B E L O W 1 K ~ Z.-Z. YU 1, j. BASS and W.P. P R A T T Jr. Department of Physics and Astronomy, Michigan State University, East Lansing, M148824, USA
Received 24 December 1984
High precision ( - 0.1 ppm) measurements of the electrical resistivity of high purity Na, prepared in He gas, show strong deviations below 1 K from the simple T 2 behavior expected for electron-electron scattering.
The standard theory [ 1] of the temperature dependent electrical resistivity p(T) predicts that when the temperature is so low that electron-phonon scattering becomes negligible, p(T) should be dominated by electron-electron scattering, for which p(T) ~ T 2 . Indeed, Levy et al. [2] measured p(T) down to 1.1 K for a Na sample held at room temperature for varying times, and they reported that below about 2.2 K their data were consistent with the form p(T) = A T 2. They attributed this behavior to electron-electron scattering. Their directly measured coefficients A were nearly constant at about 1.9 f ~ m/K 2, but they interpreted their data in terms of a model of Kaveh and Wiser [3] in which A varies with the relative contributions to the residual resistivity from dislocations and impurities. In this letter we extend measurements of p(T) for Na down to 0.1 K. Below 1.1 K, our data display strong deviations from T 2 behavior. These deviations, as well as variations in the magnitude of the data for samples obtained from different sources, suggest that there remains considerable uncertainty about the intrinsic electron--electron scattering term in p(T) for Na, both concerning the magnitude of any such term and whether this magnitude varies significantly with changes in relative dislocation and impurity concentrations. Work supported in part by the U.S. National Science Foundation Low Temperature Physics Program via Grants DMR 83-03206 and DMR 83-05389. Present address: Department of Physics, Syracuse University, Syracuse, NY, USA. 164
Details of our experimental apparatus and techniques have been given elsewhere [4]. The only points we note here are that we measured samples of Na obtained from two different sources, and that measurements were made concurrently on two, essentially identical, free-hanging samples. To focus attention on deviations from simple T 2 behavior, it is convenient to plot the quantity (l/T) × dp/dT, which should yield a horizontal straight line for simple electron---electron scattering. Fig. 1 shows a plot of (po/PT) d p / d T versus T from 3.6 K to 0.1 K for four samples of thick Na (d = 1.0 mm) obtained from two different sources (below 3.6 K, the ratio P0/P is essentially unity). The open circles indicate data for sample Na 1, made from Na with RRR =R(300 K)/R(0 K) ~ 4800, kindly supplied by Professor J.C. Garland of Ohio State University. A few data points from its "twin" sample Na 2 are shown as filled circles. The open and filled triangles indicate data from sampies Na 3 and Na 4 with RRR ~ 400 obtained from the Callery Chemical Company. Fig. 1 also contains, for comparison, dotted curves representing data of Levy et al. [2] which extend down to approximately 1.1 K. As noted above, they reported that their data displayed T 2 behavior below about 2.2 K. From 3.6 K down to about 2 K, our data decrease with decreasing temperature as expected for electron - p h o n o n scattering. Below 2 K, the data can be interpreted in either of two alternative ways: (a) That between about 1.9 K and 1.2 K, the data follow a horizontal straight line as expected for electron-electron scattering to within experimental uncertainty of 1-2%, 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 108A, number 3
PHYSICS LETTERS
25 March 1985
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Fig. 1. (po/PT) dp/dT versus Tfor Na samples from 0.1 K to 3.6 K. The dotted lines represent p = / ~ fits to Na data from ref. [2] up to 2.2 K, above which the lines curve upward to indicate the onset of electron-phonon scattering.
and then below 1.2 K the data rise above this line due to some additional scattering mechanism; or (b) That the additional mechanism takes over from electronphonon scattering in the vicinity of 1.5 K, so that the apparent horizontal region between 1.9 and 1.2 K arises fortuitously from the overlap of the effects of electron-phonon scattering and this additional mechanism. In this second case, any contribution from electronelectron scattering would be too small to be reliably isolated. To investigate the form of the additional contribution to p(T) below 1.2 K, we plot in fig. 2 the quantity (P4.2K/P) dp/dT versus T for our Na samples along with similar data for Rb, Li, and K samples [5,6]. The dotted and dashed lines in this figure indicate T 2 resistivities inferred from flat portions of curves plotted as in fig. 1.
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Fig. 2. (p4.2K/p) dp/dT versus T for Rb, Li, K, and Na sampies from 0.07 K to 1.4 K. The Na data are the same as in fig. 1. The K data are from ref. [5] and the Li and Rb data are from ref. [6]. The dashed and dotted lines indicate the 7~ resistivities inferred from the flat portions of curves plotted as in fig. 1.
Our Na data show departures from a T2 dependence similar to those seen in Li and Rb. These departures peak at very different temperatures for the Na sampies obtained from the two different sources. In K, similar peaks can be produced by means of plastic deformation [7]. Those peaks have been attributed to the introduction of defects such as dislocations which can scatter electrons inelastically. As in the case of Li [6], the martensitic transformation which Na undergoes at about 35 K provides a plausible generation mechanism for such defects. 165
Volume 108A, number 3
PHYSICS LETTERS
We conclude that below 1.2 K the resistivity o f Na exhibits anomalous behavior quite different from the simple T 2 form expected for e l e c t r o n - e l e c t r o n scattering. Both the magnitude of this anomaly, and the relatively high temperature to which it extends, make it difficult to isolate reliably any "intrinsic" electron - e l e c t r o n contribution to the resistivity o f Na for a given sample. It is, therefore, even more difficult to reliably infer extrapolated behavior for electron-electron scattering in either high dislocation or low dislocation limits as was a t t e m p t e d b y Levy et al. [2]. Studies o f Na samples plastically deformed at 4.2 K are planned to see whether the anomalous behavior is due to the presence o f dislocations and, if so, whether we can reliably isolate any e l e c t r o n - e l e c t r o n contribution to p(T).
166
25 March 1985
References [1] J.M. Ziman, Electrons and phonons (Oxford Univ. Press, London, 1960). [2] B. Levy, M. Sinvani and A.J. Greenfield, Phys. Rev. Lett. 43 (1979) 1822. [3] M. Kaveh and N. Wiser, J. Phys. F10 (1980) L37. [4] D.L. Edmunds, W.P. Pratt Jr. and J.A. Rowlands, Rev. Sci. Instrum. 51 (1980) 1516; W.P. Pratt Jr., Can. J. Phys. 60 (1982) 703. [5] C.W. Lee, M.L. Haerle, V. Heinen, J. Bass, W.P. Pratt Jr., J.A. Rowlands and P.A. Schroeder, Phys. Rev. B25 (1982) 1411. [6] Z.Z. Yu, M.L. Haerle, J.W. Zwart, J. Bass, W.P. Pratt Jr. and P.A. Sehroeder, Phys. Lett. 97A (1983) 61. [7] M.L. Haerle, W.P. Pratt Jr. and P.A. Schroeder, J. Phys. F13 (1983) L243.
PHYSICS LETTERS
25 March 1985
T H E E L E C T R I C A L R E S I S T I V I T Y OF Na B E L O W 1 K ~ Z.-Z. YU 1, j. BASS and W.P. P R A T T Jr. Department of Physics and Astronomy, Michigan State University, East Lansing, M148824, USA
Received 24 December 1984
High precision ( - 0.1 ppm) measurements of the electrical resistivity of high purity Na, prepared in He gas, show strong deviations below 1 K from the simple T 2 behavior expected for electron-electron scattering.
The standard theory [ 1] of the temperature dependent electrical resistivity p(T) predicts that when the temperature is so low that electron-phonon scattering becomes negligible, p(T) should be dominated by electron-electron scattering, for which p(T) ~ T 2 . Indeed, Levy et al. [2] measured p(T) down to 1.1 K for a Na sample held at room temperature for varying times, and they reported that below about 2.2 K their data were consistent with the form p(T) = A T 2. They attributed this behavior to electron-electron scattering. Their directly measured coefficients A were nearly constant at about 1.9 f ~ m/K 2, but they interpreted their data in terms of a model of Kaveh and Wiser [3] in which A varies with the relative contributions to the residual resistivity from dislocations and impurities. In this letter we extend measurements of p(T) for Na down to 0.1 K. Below 1.1 K, our data display strong deviations from T 2 behavior. These deviations, as well as variations in the magnitude of the data for samples obtained from different sources, suggest that there remains considerable uncertainty about the intrinsic electron--electron scattering term in p(T) for Na, both concerning the magnitude of any such term and whether this magnitude varies significantly with changes in relative dislocation and impurity concentrations. Work supported in part by the U.S. National Science Foundation Low Temperature Physics Program via Grants DMR 83-03206 and DMR 83-05389. Present address: Department of Physics, Syracuse University, Syracuse, NY, USA. 164
Details of our experimental apparatus and techniques have been given elsewhere [4]. The only points we note here are that we measured samples of Na obtained from two different sources, and that measurements were made concurrently on two, essentially identical, free-hanging samples. To focus attention on deviations from simple T 2 behavior, it is convenient to plot the quantity (l/T) × dp/dT, which should yield a horizontal straight line for simple electron---electron scattering. Fig. 1 shows a plot of (po/PT) d p / d T versus T from 3.6 K to 0.1 K for four samples of thick Na (d = 1.0 mm) obtained from two different sources (below 3.6 K, the ratio P0/P is essentially unity). The open circles indicate data for sample Na 1, made from Na with RRR =R(300 K)/R(0 K) ~ 4800, kindly supplied by Professor J.C. Garland of Ohio State University. A few data points from its "twin" sample Na 2 are shown as filled circles. The open and filled triangles indicate data from sampies Na 3 and Na 4 with RRR ~ 400 obtained from the Callery Chemical Company. Fig. 1 also contains, for comparison, dotted curves representing data of Levy et al. [2] which extend down to approximately 1.1 K. As noted above, they reported that their data displayed T 2 behavior below about 2.2 K. From 3.6 K down to about 2 K, our data decrease with decreasing temperature as expected for electron - p h o n o n scattering. Below 2 K, the data can be interpreted in either of two alternative ways: (a) That between about 1.9 K and 1.2 K, the data follow a horizontal straight line as expected for electron-electron scattering to within experimental uncertainty of 1-2%, 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 108A, number 3
PHYSICS LETTERS
25 March 1985
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Fig. 1. (po/PT) dp/dT versus Tfor Na samples from 0.1 K to 3.6 K. The dotted lines represent p = / ~ fits to Na data from ref. [2] up to 2.2 K, above which the lines curve upward to indicate the onset of electron-phonon scattering.
and then below 1.2 K the data rise above this line due to some additional scattering mechanism; or (b) That the additional mechanism takes over from electronphonon scattering in the vicinity of 1.5 K, so that the apparent horizontal region between 1.9 and 1.2 K arises fortuitously from the overlap of the effects of electron-phonon scattering and this additional mechanism. In this second case, any contribution from electronelectron scattering would be too small to be reliably isolated. To investigate the form of the additional contribution to p(T) below 1.2 K, we plot in fig. 2 the quantity (P4.2K/P) dp/dT versus T for our Na samples along with similar data for Rb, Li, and K samples [5,6]. The dotted and dashed lines in this figure indicate T 2 resistivities inferred from flat portions of curves plotted as in fig. 1.
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Fig. 2. (p4.2K/p) dp/dT versus T for Rb, Li, K, and Na sampies from 0.07 K to 1.4 K. The Na data are the same as in fig. 1. The K data are from ref. [5] and the Li and Rb data are from ref. [6]. The dashed and dotted lines indicate the 7~ resistivities inferred from the flat portions of curves plotted as in fig. 1.
Our Na data show departures from a T2 dependence similar to those seen in Li and Rb. These departures peak at very different temperatures for the Na sampies obtained from the two different sources. In K, similar peaks can be produced by means of plastic deformation [7]. Those peaks have been attributed to the introduction of defects such as dislocations which can scatter electrons inelastically. As in the case of Li [6], the martensitic transformation which Na undergoes at about 35 K provides a plausible generation mechanism for such defects. 165
Volume 108A, number 3
PHYSICS LETTERS
We conclude that below 1.2 K the resistivity o f Na exhibits anomalous behavior quite different from the simple T 2 form expected for e l e c t r o n - e l e c t r o n scattering. Both the magnitude of this anomaly, and the relatively high temperature to which it extends, make it difficult to isolate reliably any "intrinsic" electron - e l e c t r o n contribution to the resistivity o f Na for a given sample. It is, therefore, even more difficult to reliably infer extrapolated behavior for electron-electron scattering in either high dislocation or low dislocation limits as was a t t e m p t e d b y Levy et al. [2]. Studies o f Na samples plastically deformed at 4.2 K are planned to see whether the anomalous behavior is due to the presence o f dislocations and, if so, whether we can reliably isolate any e l e c t r o n - e l e c t r o n contribution to p(T).
166
25 March 1985
References [1] J.M. Ziman, Electrons and phonons (Oxford Univ. Press, London, 1960). [2] B. Levy, M. Sinvani and A.J. Greenfield, Phys. Rev. Lett. 43 (1979) 1822. [3] M. Kaveh and N. Wiser, J. Phys. F10 (1980) L37. [4] D.L. Edmunds, W.P. Pratt Jr. and J.A. Rowlands, Rev. Sci. Instrum. 51 (1980) 1516; W.P. Pratt Jr., Can. J. Phys. 60 (1982) 703. [5] C.W. Lee, M.L. Haerle, V. Heinen, J. Bass, W.P. Pratt Jr., J.A. Rowlands and P.A. Schroeder, Phys. Rev. B25 (1982) 1411. [6] Z.Z. Yu, M.L. Haerle, J.W. Zwart, J. Bass, W.P. Pratt Jr. and P.A. Sehroeder, Phys. Lett. 97A (1983) 61. [7] M.L. Haerle, W.P. Pratt Jr. and P.A. Schroeder, J. Phys. F13 (1983) L243.