- Email: [email protected]
Data processing methods in the measurements of magnetic penetration depth
Physica C 397 (2003) 80–85 www.elsevier.com/locate/physc
Data processing methods in the measurements of magnetic penetration depth R.F. Wang Department of Physics, Northern JiaoTong University, Beijing 100044, People’s Republic of China Received 9 December 2002; received in revised form 14 April 2003; accepted 24 April 2003
Abstract The data processing methods that are always used in the measurements of superconducting magnetic penetration depth k are discussed. The strong and weak points of each method are analyzed in detail. It is emphasized that the adjustable parameters in the comparison between theory and experiment always lead to some uncertainties in the conclusions. To eliminate these uncertainties, some effective data processing methods are recommended for different experiments. 2003 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.35.ha
1. Introduction The penetration depth k is a fundamental parameter of superconductivity. Its value is related to the density of superconducting electrons and its temperature dependence kðT Þ can give some important information about the symmetry of energy gap in superconductors (d- or s-wave) [1–4]. Therefore this quantity has received extensive attention for many years. But for k being a very small quantity (only about 100 nm), it is difficult to accurately measure its absolute value. Many methods, such as microwave techniques [5], can only measure the changes in kðT Þ with temperature
E-mail address: [email protected] (R.F. Wang).
but cannot determine its absolute values. The values of kð0Þ reported in many articles are usually values inferred by fitting experimental data to a theory of kðT Þ [6]. But from the same experimental data, different values are often obtained based upon different theories, which results in some uncertainties in the conclusions and some debates among different authors [7]. To eliminate these uncertainties and to draw a more confirmed conclusion, we should improve the data processing methods. This paper focuses on the problems in various data processing methods that are always used in the penetration depth measurements. To resolve these problems, we propose a new data processing method, with which one can draw some confirmed conclusions on the important properties of a superconductor even though the absolute value of kð0Þ is unknown.
0921-4534/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4534(03)01092-X
R.F. Wang / Physica C 397 (2003) 80–85
p ln½1 þ KðqÞ=q2 dq
81
2. A set of experimental results with k(T) measured accurately
keff ¼ R 1
We have accurately measured the absolute values of the penetration depth kðT Þ in a superconducting film using the two-coil mutual-inductance technique [8]. The details of the experiment have been described elsewhere [9]. Here, we only present the related experimental results and some necessary theoretical analysis in brief. The sample we used in the experiment was a Nb film that was magnetron-sputtered onto a silicon substrate 15 · 16 · 0.5 mm in size. The thickness d of the sample was about 70 nm, the residual resistivity q was 8.2 lX cm at T ¼ 10 K, the superconducting transition temperature TC was 8.83 K, and the measured value of the penetration depth k was 112 nm (3%) at T ¼ 2:0 K. The temperature dependence of kðT Þ for the sample is shown in Fig. 1. The experimental data lie above the dotted curve of the two-fluid model (kðtÞ ¼ kð0Þð1 t4 Þ1=2 , where t ¼ T =TC ), but are below the dashed curve of the London penetration depth kL ðT Þ [10]. This result can be explained in the context of BCS theory [6]. According to BCS theory, the penetration depth depends on two parameters: one is the coherence length n0 , and the other is the London penetration depth kL ðT Þ. If the surface scattering is taken as diffusion, the measured penetration depth keff ðT Þ can be calculated from the following expression:
with PippardÕs kernel KðqÞ that takes the electronic mean free path ‘ into account: ( ) 1 n 3 2 KðqÞ ¼ 2 0 ½ð1 þ ðqnÞ arctanðqnÞ qn kL n0 2ðqnÞ3
Fig. 1. A comparison between the theoretical predictions and the experimental results in the case that the absolute values of kðT Þ are measured accurately.
0
ð2:1Þ
ð2:2Þ 1 where n00 ¼ n0 =J ð0; T Þ, n1 ¼ n01 and 0 þ‘ , J ð0; T Þ has its usual meaning [6]. The solid line in Fig. 1 shows the calculated results with the bulk values of kL ð0Þ ¼ 35 nm, and n0 ¼ 43 nm. In the calculations, the electronic mean free path ‘ was obtained from the residual resistivity q and was about 4.1 nm for the sample. As can be seen in Fig. 1, the agreement between the experiment and theory is fair for the temperature dependence of k. So, in the case that the absolute values of kðT Þ can be measured with high accuracy, no adjustable parameter is needed in the comparison between experiment and theory, and the conclusions are also rather reliable.
3. Problems existing in several frequently used data processing methods As we know, many experiments can only measure the relative values of kðT Þ, i.e. DkðT Þ ¼ kðT Þ kð0Þ, but cannot determine their absolute values. So how to evaluate the absolute values of kðT Þ becomes a very important problem for these experiments. The usual way is to fit the experimental data to a particular theory of kðT Þ [2– 4,7,11,12], i.e. kðT Þ ¼ kfit ð0Þf ðtÞ; where kfit ð0Þ is taken as an adjustable parameter, and f ðtÞ ¼ 1 at t ¼ 0. If the experimental values of DkðT Þ agree with the predictions of the theory with kfit ð0Þ being taken a certain value, the fitted value kfit ð0Þ is always taken as the absolute value of kð0Þ for the sample. Usually, from the same experimental results, with the target theories different, the values of kfit ð0Þ determined by this way are different too. Here, to illustrate this problem in detail, we will repeat this method with the same experimental data in Fig. 1. Now, we assume that
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we had not exactly measured the absolute value of kð0Þ and have to adjust it to fit a particular theory. Of course, the relative values DkðT Þ between different experimental dots remain invariant in this process. Two theoretical models are always introduced as target functions: one is the two-fluid model represented by the dotted curve in Fig. 1, the other is the London limit model represented by the dashed line. In fact, they are two limits of the functions (1) and (2). The two-fluid model corresponds to the limit of n0 =kL ð0Þ ! 1, which is very close to the dash-dotted curve representing the theoretical results with n0 =kL ð0Þ ¼ 10; 000 in Fig. 1. So, this model applies to the type-I superconductor with n0 =kL ð0Þ 1. The London limit model corresponds to the contrary limit of n0 =kL ð0Þ ¼ 0, and applies to the type-II superconductor with n0 =kL ð0Þ 1. For an ordinary BCS superconductor, the ratio of n0 =kL ð0Þ lies between these two limits, and the temperature dependence of k should lie between these two curves. If the functions (1) and (2) are used as target functions, there are two parameters ðn0 ; kL ð0ÞÞ to be fitted, which is rather difficult. For convenience, many authors always choose one of these two limits as a target function to fit their experimental data, and at the same time, some uncertainties are inevitably introduced into their results. Figs. 2 and 3 show the fitted results with the two-fluid model and London limit model as target functions, respectively. Though the agreements between the experimental dots and the theoretical
Fig. 2. If kð0Þ is taken as an adjustable parameter, the same experimental data in Fig. 1 are re-plotted with the two-fluid model as a target function.
Fig. 3. If kð0Þ is taken as an adjustable parameter, the same experimental data in Fig. 1 are re-plotted with the London limit model as a target function.
models are satisfying, the fitted values of kð0Þ (148 and 80 nm, respectively) are quite different from the experimental value (112 nm). If, based upon these figures, we draw a conclusion that the temperature dependence of k in the Nb film obeys the two-fluid model or the London limit model with the absolute value of kð0Þ being 148 or 80 nm, obviously, such a conclusion is questionable. So this data analysis method is not very ideal, and some additional methods are needed to make the conclusion more reliable. There is another data processing method with the two-fluid model as a target function [13,14], which is to plot the kðT Þ or DkðT Þ against y ¼ ð1 t4 Þ1=2 . With this method, the same experimental data in Fig. 1 are re-plotted in Fig. 4, where all the experimental dots except those in the range of [1, 1.25] are located along a straight line. Even in the range of [1, 1.25], the deviation from
Fig. 4. Another data processing method with the two-fluid model as a target function.
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the straight line is very small and can be neglected. Some people may draw the conclusion that kðT Þ in the Nb film obeys the two-fluid model. But from Fig. 1, we know that it does not obey the two-fluid model at all. There are three reasons for this phenomenon. First, this method is only suitable for studying the temperature behavior of kðT Þ near TC . For y ¼ 1:25 corresponding to t ¼ 0:77, the deviation from the straight line in the range of [1, 1.25] means that in a quite large temperature range from 0 K to 0.77TC , the experimental dots cannot be described by the fitted straight line in Fig. 4. Second, the protraction of the straight line which is fitted to the measured data in the linear region in Fig. 4 does not pass through the origin of coordinates at y ¼ 0, which means that this straight line virtually represents a modified two1=2 fluid model [11] ðkðtÞ ¼ Að1 t4 Þ þ BÞ, but not the rigorous two-fluid model. Third, in fact, that the experimental dots near TC lie along a straight line in Fig. 4 is a universal characteristic for all the superconductors. For the superconducting transition is a second-order phase transition, the properties of superconductors can be described by G–L theory in the temperature range near TC . According to this theory, as T is near TC , the superconducting electron density nS is directly proportional to ð1 tÞ, i.e. nS / ð1 tÞ. In addition, for the Nb film, as T ! TC , the penetration depth k is much larger than the coherence length n0 and is related to nS by the London function k2 / n1 S . Combining these two functions, we can obtain [15] 1=2 1=2 k / ð1 tÞ . For y ¼ ð1 t4 Þ , y is also di1=2 rectly proportional to ð1 tÞ as t ! 1. So the straight line in Fig. 4 only means that the penetration depth kðtÞ diverges with proportion to y as t ! 1, but not that kðT Þ obeys the two-fluid model. So when we use this data processing method, care should be taken not to focus attention only on the linear part in the temperature range near TC but also not to neglect the temperature behavior of k in the low temperature range, which reflects many important properties of superconductors.
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4. A more sensitive and effective data processing method From the analysis above, we can see that a suitable data analysis method is important for us to draw a correct conclusion in the measurements of k. For the measurements where the absolute values of kðT Þ can be determined accurately, such as two-coil mutual-inductance techniques and muon spin rotation techniques, the data processing method shown in Fig. 1 is suitable. The conclusion drawn with this method is rather reliable, because no adjustable parameter is needed in the comparison between theory and experiment. For the measurements where only the changes in kðT Þ with temperature can be determined but the absolute values of kðT Þ cannot be measured directly from experiments, such as microwave techniques, etc., the method shown in Fig. 1 can no longer be used, because the value of kð0Þ is not known. In addition, the methods shown in Figs. 2 and 3 are somewhat unsatisfactory, for adjustable parameters have to be introduced into the comparison between theory and experiment, resulting in some uncertainties in the conclusions. So, we should search for another more sensitive data processing method with which the absolute value of kð0Þ is not needed. This request can be satisfied, if the data processing method used by Schawlow and Devlin [16] is modified to some extent. This new method is a modification of that shown in the above literature, which is to plot dk=dy vs. t instead of plotting dk=dy vs. y. (Here, to study the details of kðtÞ in the low temperature range, we use t instead of y as the horizontal axis.) For the superconducting transition being a second-order phase transition, the penetration depth kðT Þ should diverge with k / ð1 tÞ1=2 as the temperature tends to TC (i.e. t ! 1). Any correct theoretical model kth ðtÞ ¼ kth ð0Þf ðtÞ (where f ð0Þ ¼ 1 and f ð1Þ ¼ 1) should reflect this feature. 1=2 As we know, the function y ¼ ð1 t4 Þ also di1=2 verges with y / ð1 tÞ as t ! 1. So as the temperature T tends to TC both the experimental results Mkex =MyðtÞ and the theoretical predictions Mkth ðtÞ=MyðtÞ should tend to the constants Cex and Cth , respectively. As the temperature is lowered, Mkex =MyðtÞ and Mkth ðtÞ=MyðtÞ become
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temperature-dependent parameters. To eliminate adjustable parameters in the comparison between theory and experiment, we can divide Mkex =MyðtÞ and Mkth ðtÞ=MyðtÞ by Cex and Cth , respectively, i.e. to make both curves of Mkex =MyðtÞ and Mkth ðtÞ= MyðtÞ normalized at y 1 (i.e. t 1). Thus we can compare the theoretical predictions with the experimental results without needing adjustable parameters. In such comparisons, the differences of kðT Þ between different temperatures must be determined very accurately. The accuracy should be better than 0.1 nm, because the changes in kðT Þ with temperature are very small in the low temperature range. As mentioned above, we determined the values of kðT Þ with the accuracy of 3 nm in the experiment. This accuracy is acceptable for determining the absolute values of kðT Þ, but is unacceptable for determining the changes in kðT Þ between different temperatures, for the changes in kðT Þ between adjacent experimental dots in Fig. 1 are always less than 30 nm in the low temperature range. If we determine the values of DkðT Þ from these experimental results, the data will contain too much error to be used in this data processing method. To illustrate this method, we replace the experimental results with the calculated results for the Nb film in Fig. 1. Fig. 5 illustrates the details of this data analysis method, where the solid dots represent the calculations replacing the experimental results. We can see that both the two-fluid model (the solid straight line) and the London limit model (the dash-dotted curve) cannot well describe the ‘‘ex-
Fig. 5. A more sensitive data processing method without kð0Þ involved.
perimental’’ results (the solid dots). To explain the experimental results, we have to search for another suitable theoretical model. So, in this way, we can eliminate the arbitrariness in choosing theoretical models as in Figs. 2 and 3, and improve the reliability of the conclusions. This data processing method is especially suitable to those techniques by which the values of DkðT Þ can be measured very accurately, such as microwave techniques and ac susceptibility techniques. In Fig. 5, we can notice that both the curve representing the London limit model and the experimental dots representing the penetration depth of the Nb film tend to zero as t ! 0. This feature is related to the symmetry of the energy gap. For a BCS-type s-wave superconductor, the energy gap is homogenous and DkðT Þ goes to zero exponentially as kð0ÞT 1=2 eDð0Þ=kT as T ! 0, however, Dy ð¼ yðtÞ 1Þ goes to zero as a power law ð1=2Þt4 . So, it is inevitable that dk=dy tends to zero in the low temperature range, which is a common property for all s-wave superconductors. But, for some high temperature superconductors, the energy gap is anisotropic and DkðtÞ goes to zero as a power law tn as t ! 0, where n depends on whether the nodes with the energy gap are points or lines. If n < 4; dk=dy should diverge as t ! 0, but not tend to zero in Fig. 5, which provide us a confirmed way to test the symmetry of the energy gap in a superconductor. For an example [6], some experiments indicate the temperature dependence of kðtÞ in YBCO follows the dependence of ð1 t2 Þ1=2 . In 1=2 Fig. 5, we plotted the dependence of ð1 t2 Þ by the dashed curve, which obviously diverges as t ! 0. So, from this curve, we can conclude that the energy gap in YBCO is anisotropic and its pairing symmetry is different from that of a s-wave superconductor. But if one uses the data processing method described in Figs. 2 or 3, a contrary conclusion may be drawn. In the literature [16], the authors computed the absolute value of kð0Þ using other procedures, so they did not make the theoretical predictions and the experimental results normalized at y 1. But for some theoretical models (such as the two-fluid model), where the absolute value of kð0Þ cannot be calculated, the procedure of normalization is necessary. Comparing with that shown in Fig. 4, this
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method highlights the deviation of the experimental results from the two-fluid model and takes this deviation as important evidence to distinguish whether a theoretical model is feasible or not. On the contrary, this deviation is always neglected in Fig. 4. So this data processing method removes the shortcomings of those methods shown in Figs. 2– 4. and provides better sensitivity. Of course, though it is more sensitive, this method cannot provide the fitted value of kð0Þ, which is always important for us to study the properties of superconductors. To get this value, we can combine the methods shown in Fig. 5 and in Figs. 2 or 3. First of all, we can choose a suitable theoretical model with the method shown in Fig. 5, then get a fitted value of kð0Þ and a clear curve about kðT Þ with the method shown in Figs. 2 or 3.
5. Conclusion In summary, a suitable data processing method is very important to the penetration depth measurements. For the experiments where the absolute values of kðT Þ can be determined accurately, the method shown in Fig. 1 is suitable. This method can make a comparison between theory and experiment without any adjustable parameters, and the conclusions are rather reliable. For the experiments where only the changes in kðT Þ with temperature can be measured accurately, we should use the method shown in Fig. 5 to choose a correct theoretical model, then fit the experimental results to the chosen theoretical model to obtain the absolute value of kð0Þ with the method shown in Figs. 2 and 3.
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Acknowledgement We thank Professor S.P. Zhao for his helpful discussions about this work. References [1] C. Panagopoulos, J.R. Cooper, T. Xiang, G.B. Peacock, I. Gameson, P.P. Edwards, Phys. Rev. Lett. 79 (1997) 2320. [2] M.D. Lumsden, S.R. Dunsiger, J.E. Sonier, R.I. Miller, R.F. Kiefl, R. Jin, J. He, D. Mandrus, S.T. Bramwell, J.S. Gardner, Phys. Rev. Lett. 89 (2002) 147002. [3] F. Manzano, A. Carrington, N.E. Hussey, S. Lee, A. Yamamoto, S. Tajima, Phys. Rev. Lett. 88 (2002) 047002. [4] R. Prozorov, R.W. Giannetta, P. Fournier, R.L. Greene, Phys. Rev. Lett. 85 (2000) 3700. [5] A.B. Pippard, Proc. Royal Soc. A 216 (1953) 547. [6] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, Inc., New York, 1996 (Chapter 3). [7] A.F. Hebard, A.T. Fiory, D.R. Harshman, Phys. Rev. Lett. 62 (1989) 2885; R.L. Greene, L. Krusin-Elbaum, A.P. Malozemoff, Phys. Rev. Lett. 62 (1989) 2886. [8] A.T. Fiory, A.F. Hebard, P.M. Mankiewich, R.E. Howard, Appl. Phys. Lett. 52 (1988) 2165. [9] R.F. Wang, S.P. Zhao, G.H. Chen, Q.S. Yang, Appl. Phys. Lett. 75 (1999) 3865. [10] B. M€ uhlschegel, Z. Phys. 155 (1959) 313. [11] J.H. Classon, J.E. Eventts, R.E. Somekh, Z.H. Barber, Phys. Rev. B 44 (1991) 9605. [12] M.S. Pambianchi, L. Chen, S.M. Anlage, Phys. Rev. B 54 (1996) 3508. [13] D. Shoenberg, Superconductivity, Cambridge University Press, Cambridge, 1952 (Chapter 5). [14] J. Guimpel, F. de la Cruz, J. Murduck, I.K. Schuller, Phys. Rev. B 35 (1987) 3655. [15] P.G. De Gennes, Superconductivity of Metal and Alloys, Addison-Wesley Publishing Company, Inc., Boston, 1989, p. 180. [16] A.L. Schawlow, G.E. Devlin, Phys. Rev. 113 (1959) 120.
Data processing methods in the measurements of magnetic penetration depth R.F. Wang Department of Physics, Northern JiaoTong University, Beijing 100044, People’s Republic of China Received 9 December 2002; received in revised form 14 April 2003; accepted 24 April 2003
Abstract The data processing methods that are always used in the measurements of superconducting magnetic penetration depth k are discussed. The strong and weak points of each method are analyzed in detail. It is emphasized that the adjustable parameters in the comparison between theory and experiment always lead to some uncertainties in the conclusions. To eliminate these uncertainties, some effective data processing methods are recommended for different experiments. 2003 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.35.ha
1. Introduction The penetration depth k is a fundamental parameter of superconductivity. Its value is related to the density of superconducting electrons and its temperature dependence kðT Þ can give some important information about the symmetry of energy gap in superconductors (d- or s-wave) [1–4]. Therefore this quantity has received extensive attention for many years. But for k being a very small quantity (only about 100 nm), it is difficult to accurately measure its absolute value. Many methods, such as microwave techniques [5], can only measure the changes in kðT Þ with temperature
E-mail address: [email protected] (R.F. Wang).
but cannot determine its absolute values. The values of kð0Þ reported in many articles are usually values inferred by fitting experimental data to a theory of kðT Þ [6]. But from the same experimental data, different values are often obtained based upon different theories, which results in some uncertainties in the conclusions and some debates among different authors [7]. To eliminate these uncertainties and to draw a more confirmed conclusion, we should improve the data processing methods. This paper focuses on the problems in various data processing methods that are always used in the penetration depth measurements. To resolve these problems, we propose a new data processing method, with which one can draw some confirmed conclusions on the important properties of a superconductor even though the absolute value of kð0Þ is unknown.
0921-4534/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-4534(03)01092-X
R.F. Wang / Physica C 397 (2003) 80–85
p ln½1 þ KðqÞ=q2 dq
81
2. A set of experimental results with k(T) measured accurately
keff ¼ R 1
We have accurately measured the absolute values of the penetration depth kðT Þ in a superconducting film using the two-coil mutual-inductance technique [8]. The details of the experiment have been described elsewhere [9]. Here, we only present the related experimental results and some necessary theoretical analysis in brief. The sample we used in the experiment was a Nb film that was magnetron-sputtered onto a silicon substrate 15 · 16 · 0.5 mm in size. The thickness d of the sample was about 70 nm, the residual resistivity q was 8.2 lX cm at T ¼ 10 K, the superconducting transition temperature TC was 8.83 K, and the measured value of the penetration depth k was 112 nm (3%) at T ¼ 2:0 K. The temperature dependence of kðT Þ for the sample is shown in Fig. 1. The experimental data lie above the dotted curve of the two-fluid model (kðtÞ ¼ kð0Þð1 t4 Þ1=2 , where t ¼ T =TC ), but are below the dashed curve of the London penetration depth kL ðT Þ [10]. This result can be explained in the context of BCS theory [6]. According to BCS theory, the penetration depth depends on two parameters: one is the coherence length n0 , and the other is the London penetration depth kL ðT Þ. If the surface scattering is taken as diffusion, the measured penetration depth keff ðT Þ can be calculated from the following expression:
with PippardÕs kernel KðqÞ that takes the electronic mean free path ‘ into account: ( ) 1 n 3 2 KðqÞ ¼ 2 0 ½ð1 þ ðqnÞ arctanðqnÞ qn kL n0 2ðqnÞ3
Fig. 1. A comparison between the theoretical predictions and the experimental results in the case that the absolute values of kðT Þ are measured accurately.
0
ð2:1Þ
ð2:2Þ 1 where n00 ¼ n0 =J ð0; T Þ, n1 ¼ n01 and 0 þ‘ , J ð0; T Þ has its usual meaning [6]. The solid line in Fig. 1 shows the calculated results with the bulk values of kL ð0Þ ¼ 35 nm, and n0 ¼ 43 nm. In the calculations, the electronic mean free path ‘ was obtained from the residual resistivity q and was about 4.1 nm for the sample. As can be seen in Fig. 1, the agreement between the experiment and theory is fair for the temperature dependence of k. So, in the case that the absolute values of kðT Þ can be measured with high accuracy, no adjustable parameter is needed in the comparison between experiment and theory, and the conclusions are also rather reliable.
3. Problems existing in several frequently used data processing methods As we know, many experiments can only measure the relative values of kðT Þ, i.e. DkðT Þ ¼ kðT Þ kð0Þ, but cannot determine their absolute values. So how to evaluate the absolute values of kðT Þ becomes a very important problem for these experiments. The usual way is to fit the experimental data to a particular theory of kðT Þ [2– 4,7,11,12], i.e. kðT Þ ¼ kfit ð0Þf ðtÞ; where kfit ð0Þ is taken as an adjustable parameter, and f ðtÞ ¼ 1 at t ¼ 0. If the experimental values of DkðT Þ agree with the predictions of the theory with kfit ð0Þ being taken a certain value, the fitted value kfit ð0Þ is always taken as the absolute value of kð0Þ for the sample. Usually, from the same experimental results, with the target theories different, the values of kfit ð0Þ determined by this way are different too. Here, to illustrate this problem in detail, we will repeat this method with the same experimental data in Fig. 1. Now, we assume that
82
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we had not exactly measured the absolute value of kð0Þ and have to adjust it to fit a particular theory. Of course, the relative values DkðT Þ between different experimental dots remain invariant in this process. Two theoretical models are always introduced as target functions: one is the two-fluid model represented by the dotted curve in Fig. 1, the other is the London limit model represented by the dashed line. In fact, they are two limits of the functions (1) and (2). The two-fluid model corresponds to the limit of n0 =kL ð0Þ ! 1, which is very close to the dash-dotted curve representing the theoretical results with n0 =kL ð0Þ ¼ 10; 000 in Fig. 1. So, this model applies to the type-I superconductor with n0 =kL ð0Þ 1. The London limit model corresponds to the contrary limit of n0 =kL ð0Þ ¼ 0, and applies to the type-II superconductor with n0 =kL ð0Þ 1. For an ordinary BCS superconductor, the ratio of n0 =kL ð0Þ lies between these two limits, and the temperature dependence of k should lie between these two curves. If the functions (1) and (2) are used as target functions, there are two parameters ðn0 ; kL ð0ÞÞ to be fitted, which is rather difficult. For convenience, many authors always choose one of these two limits as a target function to fit their experimental data, and at the same time, some uncertainties are inevitably introduced into their results. Figs. 2 and 3 show the fitted results with the two-fluid model and London limit model as target functions, respectively. Though the agreements between the experimental dots and the theoretical
Fig. 2. If kð0Þ is taken as an adjustable parameter, the same experimental data in Fig. 1 are re-plotted with the two-fluid model as a target function.
Fig. 3. If kð0Þ is taken as an adjustable parameter, the same experimental data in Fig. 1 are re-plotted with the London limit model as a target function.
models are satisfying, the fitted values of kð0Þ (148 and 80 nm, respectively) are quite different from the experimental value (112 nm). If, based upon these figures, we draw a conclusion that the temperature dependence of k in the Nb film obeys the two-fluid model or the London limit model with the absolute value of kð0Þ being 148 or 80 nm, obviously, such a conclusion is questionable. So this data analysis method is not very ideal, and some additional methods are needed to make the conclusion more reliable. There is another data processing method with the two-fluid model as a target function [13,14], which is to plot the kðT Þ or DkðT Þ against y ¼ ð1 t4 Þ1=2 . With this method, the same experimental data in Fig. 1 are re-plotted in Fig. 4, where all the experimental dots except those in the range of [1, 1.25] are located along a straight line. Even in the range of [1, 1.25], the deviation from
Fig. 4. Another data processing method with the two-fluid model as a target function.
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the straight line is very small and can be neglected. Some people may draw the conclusion that kðT Þ in the Nb film obeys the two-fluid model. But from Fig. 1, we know that it does not obey the two-fluid model at all. There are three reasons for this phenomenon. First, this method is only suitable for studying the temperature behavior of kðT Þ near TC . For y ¼ 1:25 corresponding to t ¼ 0:77, the deviation from the straight line in the range of [1, 1.25] means that in a quite large temperature range from 0 K to 0.77TC , the experimental dots cannot be described by the fitted straight line in Fig. 4. Second, the protraction of the straight line which is fitted to the measured data in the linear region in Fig. 4 does not pass through the origin of coordinates at y ¼ 0, which means that this straight line virtually represents a modified two1=2 fluid model [11] ðkðtÞ ¼ Að1 t4 Þ þ BÞ, but not the rigorous two-fluid model. Third, in fact, that the experimental dots near TC lie along a straight line in Fig. 4 is a universal characteristic for all the superconductors. For the superconducting transition is a second-order phase transition, the properties of superconductors can be described by G–L theory in the temperature range near TC . According to this theory, as T is near TC , the superconducting electron density nS is directly proportional to ð1 tÞ, i.e. nS / ð1 tÞ. In addition, for the Nb film, as T ! TC , the penetration depth k is much larger than the coherence length n0 and is related to nS by the London function k2 / n1 S . Combining these two functions, we can obtain [15] 1=2 1=2 k / ð1 tÞ . For y ¼ ð1 t4 Þ , y is also di1=2 rectly proportional to ð1 tÞ as t ! 1. So the straight line in Fig. 4 only means that the penetration depth kðtÞ diverges with proportion to y as t ! 1, but not that kðT Þ obeys the two-fluid model. So when we use this data processing method, care should be taken not to focus attention only on the linear part in the temperature range near TC but also not to neglect the temperature behavior of k in the low temperature range, which reflects many important properties of superconductors.
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4. A more sensitive and effective data processing method From the analysis above, we can see that a suitable data analysis method is important for us to draw a correct conclusion in the measurements of k. For the measurements where the absolute values of kðT Þ can be determined accurately, such as two-coil mutual-inductance techniques and muon spin rotation techniques, the data processing method shown in Fig. 1 is suitable. The conclusion drawn with this method is rather reliable, because no adjustable parameter is needed in the comparison between theory and experiment. For the measurements where only the changes in kðT Þ with temperature can be determined but the absolute values of kðT Þ cannot be measured directly from experiments, such as microwave techniques, etc., the method shown in Fig. 1 can no longer be used, because the value of kð0Þ is not known. In addition, the methods shown in Figs. 2 and 3 are somewhat unsatisfactory, for adjustable parameters have to be introduced into the comparison between theory and experiment, resulting in some uncertainties in the conclusions. So, we should search for another more sensitive data processing method with which the absolute value of kð0Þ is not needed. This request can be satisfied, if the data processing method used by Schawlow and Devlin [16] is modified to some extent. This new method is a modification of that shown in the above literature, which is to plot dk=dy vs. t instead of plotting dk=dy vs. y. (Here, to study the details of kðtÞ in the low temperature range, we use t instead of y as the horizontal axis.) For the superconducting transition being a second-order phase transition, the penetration depth kðT Þ should diverge with k / ð1 tÞ1=2 as the temperature tends to TC (i.e. t ! 1). Any correct theoretical model kth ðtÞ ¼ kth ð0Þf ðtÞ (where f ð0Þ ¼ 1 and f ð1Þ ¼ 1) should reflect this feature. 1=2 As we know, the function y ¼ ð1 t4 Þ also di1=2 verges with y / ð1 tÞ as t ! 1. So as the temperature T tends to TC both the experimental results Mkex =MyðtÞ and the theoretical predictions Mkth ðtÞ=MyðtÞ should tend to the constants Cex and Cth , respectively. As the temperature is lowered, Mkex =MyðtÞ and Mkth ðtÞ=MyðtÞ become
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temperature-dependent parameters. To eliminate adjustable parameters in the comparison between theory and experiment, we can divide Mkex =MyðtÞ and Mkth ðtÞ=MyðtÞ by Cex and Cth , respectively, i.e. to make both curves of Mkex =MyðtÞ and Mkth ðtÞ= MyðtÞ normalized at y 1 (i.e. t 1). Thus we can compare the theoretical predictions with the experimental results without needing adjustable parameters. In such comparisons, the differences of kðT Þ between different temperatures must be determined very accurately. The accuracy should be better than 0.1 nm, because the changes in kðT Þ with temperature are very small in the low temperature range. As mentioned above, we determined the values of kðT Þ with the accuracy of 3 nm in the experiment. This accuracy is acceptable for determining the absolute values of kðT Þ, but is unacceptable for determining the changes in kðT Þ between different temperatures, for the changes in kðT Þ between adjacent experimental dots in Fig. 1 are always less than 30 nm in the low temperature range. If we determine the values of DkðT Þ from these experimental results, the data will contain too much error to be used in this data processing method. To illustrate this method, we replace the experimental results with the calculated results for the Nb film in Fig. 1. Fig. 5 illustrates the details of this data analysis method, where the solid dots represent the calculations replacing the experimental results. We can see that both the two-fluid model (the solid straight line) and the London limit model (the dash-dotted curve) cannot well describe the ‘‘ex-
Fig. 5. A more sensitive data processing method without kð0Þ involved.
perimental’’ results (the solid dots). To explain the experimental results, we have to search for another suitable theoretical model. So, in this way, we can eliminate the arbitrariness in choosing theoretical models as in Figs. 2 and 3, and improve the reliability of the conclusions. This data processing method is especially suitable to those techniques by which the values of DkðT Þ can be measured very accurately, such as microwave techniques and ac susceptibility techniques. In Fig. 5, we can notice that both the curve representing the London limit model and the experimental dots representing the penetration depth of the Nb film tend to zero as t ! 0. This feature is related to the symmetry of the energy gap. For a BCS-type s-wave superconductor, the energy gap is homogenous and DkðT Þ goes to zero exponentially as kð0ÞT 1=2 eDð0Þ=kT as T ! 0, however, Dy ð¼ yðtÞ 1Þ goes to zero as a power law ð1=2Þt4 . So, it is inevitable that dk=dy tends to zero in the low temperature range, which is a common property for all s-wave superconductors. But, for some high temperature superconductors, the energy gap is anisotropic and DkðtÞ goes to zero as a power law tn as t ! 0, where n depends on whether the nodes with the energy gap are points or lines. If n < 4; dk=dy should diverge as t ! 0, but not tend to zero in Fig. 5, which provide us a confirmed way to test the symmetry of the energy gap in a superconductor. For an example [6], some experiments indicate the temperature dependence of kðtÞ in YBCO follows the dependence of ð1 t2 Þ1=2 . In 1=2 Fig. 5, we plotted the dependence of ð1 t2 Þ by the dashed curve, which obviously diverges as t ! 0. So, from this curve, we can conclude that the energy gap in YBCO is anisotropic and its pairing symmetry is different from that of a s-wave superconductor. But if one uses the data processing method described in Figs. 2 or 3, a contrary conclusion may be drawn. In the literature [16], the authors computed the absolute value of kð0Þ using other procedures, so they did not make the theoretical predictions and the experimental results normalized at y 1. But for some theoretical models (such as the two-fluid model), where the absolute value of kð0Þ cannot be calculated, the procedure of normalization is necessary. Comparing with that shown in Fig. 4, this
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method highlights the deviation of the experimental results from the two-fluid model and takes this deviation as important evidence to distinguish whether a theoretical model is feasible or not. On the contrary, this deviation is always neglected in Fig. 4. So this data processing method removes the shortcomings of those methods shown in Figs. 2– 4. and provides better sensitivity. Of course, though it is more sensitive, this method cannot provide the fitted value of kð0Þ, which is always important for us to study the properties of superconductors. To get this value, we can combine the methods shown in Fig. 5 and in Figs. 2 or 3. First of all, we can choose a suitable theoretical model with the method shown in Fig. 5, then get a fitted value of kð0Þ and a clear curve about kðT Þ with the method shown in Figs. 2 or 3.
5. Conclusion In summary, a suitable data processing method is very important to the penetration depth measurements. For the experiments where the absolute values of kðT Þ can be determined accurately, the method shown in Fig. 1 is suitable. This method can make a comparison between theory and experiment without any adjustable parameters, and the conclusions are rather reliable. For the experiments where only the changes in kðT Þ with temperature can be measured accurately, we should use the method shown in Fig. 5 to choose a correct theoretical model, then fit the experimental results to the chosen theoretical model to obtain the absolute value of kð0Þ with the method shown in Figs. 2 and 3.
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Acknowledgement We thank Professor S.P. Zhao for his helpful discussions about this work. References [1] C. Panagopoulos, J.R. Cooper, T. Xiang, G.B. Peacock, I. Gameson, P.P. Edwards, Phys. Rev. Lett. 79 (1997) 2320. [2] M.D. Lumsden, S.R. Dunsiger, J.E. Sonier, R.I. Miller, R.F. Kiefl, R. Jin, J. He, D. Mandrus, S.T. Bramwell, J.S. Gardner, Phys. Rev. Lett. 89 (2002) 147002. [3] F. Manzano, A. Carrington, N.E. Hussey, S. Lee, A. Yamamoto, S. Tajima, Phys. Rev. Lett. 88 (2002) 047002. [4] R. Prozorov, R.W. Giannetta, P. Fournier, R.L. Greene, Phys. Rev. Lett. 85 (2000) 3700. [5] A.B. Pippard, Proc. Royal Soc. A 216 (1953) 547. [6] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, Inc., New York, 1996 (Chapter 3). [7] A.F. Hebard, A.T. Fiory, D.R. Harshman, Phys. Rev. Lett. 62 (1989) 2885; R.L. Greene, L. Krusin-Elbaum, A.P. Malozemoff, Phys. Rev. Lett. 62 (1989) 2886. [8] A.T. Fiory, A.F. Hebard, P.M. Mankiewich, R.E. Howard, Appl. Phys. Lett. 52 (1988) 2165. [9] R.F. Wang, S.P. Zhao, G.H. Chen, Q.S. Yang, Appl. Phys. Lett. 75 (1999) 3865. [10] B. M€ uhlschegel, Z. Phys. 155 (1959) 313. [11] J.H. Classon, J.E. Eventts, R.E. Somekh, Z.H. Barber, Phys. Rev. B 44 (1991) 9605. [12] M.S. Pambianchi, L. Chen, S.M. Anlage, Phys. Rev. B 54 (1996) 3508. [13] D. Shoenberg, Superconductivity, Cambridge University Press, Cambridge, 1952 (Chapter 5). [14] J. Guimpel, F. de la Cruz, J. Murduck, I.K. Schuller, Phys. Rev. B 35 (1987) 3655. [15] P.G. De Gennes, Superconductivity of Metal and Alloys, Addison-Wesley Publishing Company, Inc., Boston, 1989, p. 180. [16] A.L. Schawlow, G.E. Devlin, Phys. Rev. 113 (1959) 120.