Eugene V. Slobodzian^a, Charles W. Smith^a*, Paul J. Dolan, Jr.^b

a Department of Physics and Astronomy, University of Maine, 5709 Bennett Hall, Orono, ME 04469 USA

b Department of Physics, Northeastern Illinois University, 5500 N. St. Louis Ave., Chicago, IL 60625 USA

PACS: 74.80.Fp, 81.40.Rs, 81.40.Ef

Keywords: Point contact, Andreev reflection, Normal-metal/superconductor interface, Inelastic scattering

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ABSTRACT

Charge transport measurements, as a function of temperature, for normal metal / superconductor point contacts of silver on annealed niobium, on work hardened niobium and on niobium containing interstitial hydrogen, are compared to detailed predictions of the BTK theory, modified to include inelastic scattering. An analysis procedure for independently extracting the elastic scattering parameter, Z, and the inelastic scattering parameter, G, from transport data, is demonstrated. Results indicate that, to a good approximation, both Z and G are temperature independent and uncorrelated.

1. Introduction

An ideal electrical point contact is dominated by ballistic transport, in that the charge carrier mean free path is greater than the length scale of the confining geometry. For a point contact between a normal metal and a superconductor, a peak in the differential resistance at zero bias voltage, resulting from Andreev reflection of the charge carriers at the contact interface, dominates the electrical characteristics. For the case where both the normal side and the superconductor side of the point contact are metallic, the shape and temperature dependence of the Andreev peak is well described by the BTK theory [1], from the metallic limit to the tunneling limit. In this theory, the mismatch in transport properties at or near the contact interface is modeled as elastic scattering, using a repulsive delta function barrier of height H and described by a unitless and temperature independent barrier strength parameter Z=H/?v_F, where v_F is the Fermi velocity. For cases where the Fermi velocity mismatch between the two sides of the contact is significant, an effective barrier strength parameter, Z_eff is used. Z_eff is defined as

(1)

where r is the ratio of the two Fermi velocities. Z_eff defined in this way is invariant under interchange of the Fermi velocities. The BTK theory is thus a one parameter model (Z) for charge transport in normal metal / superconductor point contacts.

Studies of metal oxide superconductors [2] show that, in addition to elastic scattering, inelastic scattering of the charge carriers plays an important role. A recent modification to the BTK theory [3] explicitly incorporates inelastic scattering into the Bogoliubov-deGennes formalism for the quasiparticle states of the system. A linear approach is taken by adding to the field term, a negative (positive) scattering term, weighted inversely by a finite quasiparticle lifetime, t , for the electron-like (hole-like) charge carriers. The quasiparticle lifetime is defined as , where G is a temperature independent inelastic scattering parameter that scales t into energy. As a consequence of this modification, the Bogoliubov-deGennes coherence factors are imaginary over the entire energy range, the quasiparticle density of states depends explicitly on t , the point contact electrical characteristics, as a function of bias voltage, are broadened and the temperature dependence of the Andreev peak is weakened. This is intuitively what one might expect, since inelastic scattering would shorten the quasiparticle lifetime and lead to suppression of various features in the transport spectrum.

In this paper we illustrate the effects of inelastic scattering on charge transport in normal metal / superconductor point contacts of silver on niobium, by systematically shortening the quasiparticle lifetime by introducing either interstitial hydrogen or lattice defects and dislocations into the niobium. We will demonstrate an analysis procedure for independently extracting the values for the elastic scattering parameter and the inelastic scattering parameter from charge transport data. We show that the extracted values for Z and G are unique, in that one and only one pair of values will fit the entire data set over the full temperature range. And finally we are able to illustrate some of the limits underlying the assumptions in the modified BTK theory.

2. Sample System and Preparation

Although the issues of inelastic scattering in normal metal / superconductor point contacts are acute for metal oxide superconductors, there are several reasons not to employ these materials as the model system for understanding the parameterization and limitations of the extended BTK theory. These include sample preparation challenges like granularity in the bulk and loss of stoichiometry at metallic contacts, and intrinsic directional gap properties as in d-wave materials, for which the measurement itself may take uncontrollable or unknown real space or momentum space averages. [4]

Our approach was to begin with a normal metal / superconductor combination with minimal Fermi velocity mismatch and to systematically shorten the quasiparticle lifetime by introducing sources of inelastic scattering into an otherwise ideal metallic superconductor. Of the many bimetallic systems from which to choose, the silver / niobium system provides an excellent Fermi velocity match (1.39x10⁸ cm/s, 1.37x10⁸ cm/s, respectively) while pairing a well understood normal metal with an easy-to-modify superconductor. Inelastic scattering was introduced into the niobium using two entirely different techniques: work hardening to produce lattice defects and dislocations, and thermal diffusion or electromigration of hydrogen into interstitial lattice sites. Data for all three cases will be shown, in addition to that for pure vacuum annealed niobium.

Source material for the superconducting side of the contact was 99.995% pure vacuum annealed niobium sheet, 0.10 mm thick. Small rectangles, 4.0 mm by 2.0 mm were formed and one of the 4.0 mm edges carefully micro-lapped to knife-edge sharpness. This forms the wedge in the wire/wedge point contact geometry described in the next section. To introduce lattice defects and dislocations, a 4.0 mm by 8.0 mm rectangle of niobium sheet was bent to failure by repeated 90^o flexing at its center. A knife edge was micro-lapped along both 4.0 mm edges. The flexed-to-failure edge always showed inelastic scattering and decrease of the critical temperature by 0.10 to 0.20 Kelvin. The undisturbed edge was used as a control. No further quantification of the mechanically modified niobium was made other than to note that x-ray measurements showed broadening of the diffraction peaks of the flexed-to-failure edge.

It is well known that niobium can accommodate large quantities of hydrogen on the tetrahedral interstitial sites of its bcc lattice. Its unusually large diffusion coefficient for hydrogen, even at room temperature, assures a uniform distribution throughout the bulk. Hydrogen can be introduced directly by surface penetration of the gas phase or by using a protonic electrolyte. Both techniques were employed in this study.

An ultrahigh vacuum chamber capable of a base pressure of 2x10^-9 torr was loaded with several prepared (sharpened) niobium wedges. The wedges were raised to a temperature of 325 K in hydrogen at 2.0 atmospheres pressure. One group was dosed for 1.0 hour and a second group for 2.0 hours. For this treatment, under ideal conditions, an atomic ratio of H/Nb of approximately 30% is predicted [5]. Mass change in the samples indicates ratios of 15%5% and 20%5% for the 1.0 hour and 2.0 hour samples, respectively. These smaller than ideal values were probably the result of a barrier potential at the surface of the niobium. We chose not to chemically or mechanically clean the surface of the wedges prior to hydrogenation, out of concern of introducing additional sources of inelastic scattering over which we had little or no control. The measured H/Nb ratios are consistent with the observed depression of the critical temperature in the samples of about 0.15 K. As will be shown in the results section, the longer exposure time leads to a greater degree of inelastic scattering.

Hydrogen was introduced by a second technique. Prepared vacuum annealed niobium wedges were placed in a slightly acidic (hydrochloric acid) distilled water solution and attached to the negative electrode of a constant current dc power supply. Niobium sheet was also used for the positive counter-electrode. A current of thirty milliamperes for 1.0 hour was sufficient to produce a mass change and critical temperature depression comparable to that produced in the 20%5% H/Nb ratio thermal diffusion samples.

3. Experimental Techniques

The wire/wedge geometry was used for the point contacts in this study and although described and illustrated elsewhere [6], will be briefly reviewed in this section. A wedge shaped superconductor sample (usually an edge sharpened sheet on the order of a few millimeters on a side) is placed in a miniature electronically insulated vise mounted in a dip-probe cryostat. A fine normal metal wire is crossed over the sharp edge of the wedge and held against it under spring tension. The location where the wire touches the wedge is the normal metal / superconductor point contact and is a few millimeters from a cryogenic thermometer incorporated in the miniature vise.

A pivot arm linkage attached to the end of the normal metal wire and an externally controllable adjustment screw permit the contact to be made at low temperature. The system is both mechanically and thermally very stable. The contact resistance and hence the contact area of a gold on gold wire /wedge varies less than a few parts in 5000 over the temperature range from 4.2 to 25.0 K.

The differential resistance of a point contact, as a function of bias voltage is measured using a standard lock-in amplifier technique. A typical data set consists of digitized dV/dI versus V curves containing 1024 points. dV/dI versus V curves are taken for 15 to 20 different temperature values, from 4.2 K to the critical temperature, T_c, over a bias range of nominally 20.0 millivolts. A precision standard resistance box is used to calibrate the differential resistance axis.

4. Analysis

A goal within the BTK theory, as well as within the modified BTK theory, was to derive an expression for the transport current for a normal metal / superconductor point contact that can be compared in a direct way with data of the type shown in Fig. 1. The expression for the total current through the contact is

(2)

where f(E) is the Fermi distribution function, R_N is the point contact resistance just above T_c, and A(E) and B(E) are the probabilities of Andreev reflection and ordinary reflection, respectively. In the modified BTK theory A(E) and B(E) contain both the elastic scattering parameter , Z, and the inelastic scattering parameter, G. As expected, the modified BTK theory rigorously produces the BTK theory for G=0.

To facilitate a direct comparison of theory and experiment, and to illustrate a method to uniquely determine Z and G from experimental data, it is easier to work with the normalized conductance at zero bias voltage rather than with the transport current. The normalized conductance at zero bias voltage Y(Z, G, T) is given by

(3)

Fig. 2 shows Y(Z, G, T) plotted against a reduced temperature scale from t ( ? T/T_C) = 0 to 1.0. Each panel shows seven curves, one each for Z = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 and 10.0, top to bottom respectively. The upper panel shows the BTK theory results, i.e. G=0. The center panel, G=0.25, and the lower panel, G=0.45, demonstrate that the effect of inelastic scattering on the transport of charge through point contacts is to suppress Andreev reflection and collapse the normalized conductance toward 1.0, the normal / normal contact limit. One can also see that any particular curve of Y versus t is unique for a particular Z, G pair, that is, as G increases the curves monotonically collapse toward Y=1, without intersecting. We can exploit this uniqueness as a means to determine Z and G for a particular point contact, in the following way. Fig. 3 shows Y at t=0.50 plotted against dY/dt at t=0.50, for steps in Z and G of 0.05. This Z-G surface is the entire parameter space for the modified BTK theory with the top edge of the surface being the BTK theory itself, i.e. G=0. Analysis begins with experimental data for dV/dI versus bias voltage V, for several temperatures, i.e. data of the type shown in Fig. 1. Y is determined using equation (3) (i.e. R_N/R₀(T) ) for each temperature in the dV/dI vs. V data set. Y vs. t is plotted and the value of Y at t=0.50 and dY/dt at t=0.50 is graphically determined. Using these values, Z and G are estimated from Fig. 3. These estimates are used in equation (3) and a curve is drawn through the Y vs. t data. In practice a condition of goodness of fit is chosen and this process is iterated on a computer. Convergence is rapid and Z and G can usually be determined to within a few percent. The reduced temperature evaluation point of t=0.50 is, of course, arbitrary and a Z-G surface, like the one in Fig. 3, can be generated for any reduced temperature value. However, as can be seen from Fig. 2, if the reduced temperature value is chosen substantially less than t=0.50 all curves have a slope near zero, while on the other hand if the reduced temperature value is chosen substantially larger than t=0.50, all curves have a value of Y approaching 1.0. The value of t=0.50 is a practical choice, which spreads the Z-G surface over a useful range in Y and dY/dt. Once values for Z and G are determined, one can use equation (2) to plot I_NS, or its derivative, as a function of bias voltage and temperature. It should be emphasized that this analysis procedure extracts a unique Z,G pair for the entire data set and underscores the fact that point contact measurements must be made as a function of temperature to enable a rigorous comparison with theory. In other words, one I vs. V (or dV/dI vs. V) curve does not contain all the information needed for a comparison to theory. We also note that although Z is unitless, as defined in the BTK theory, G , as defined in the modified BTK theory, has the units of energy. In the above presentation and the remainder of this paper values for G are normalized by eD₀, where D₀ is the BCS gap parameter at t = 0.0, i.e. G /D₀ is unitless. And finally, we emphasize that the modified BTK theory and how it is used here has no free parameters: T_C, D₀, Z and G are all extracted from the experiment. The theory either does or does not describe the point contact.

5. Results and Discussion

Fig. 1 shows the differential resistance as a function of bias voltage, at several temperature values, for a point contact of niobium with interstitial hydrogen. Fig. 4 shows the zero bias conductance ratio Y(Z, G , t) versus the reduced temperature, t, derived from the data in Fig. 1. Note: each curve in Fig. 1 corresponds to a point in Fig. 4 and thus the entire data set is contained in this figure. The solid line in Fig. 4 is the theoretical description for this data, using equation (3) and the modified BTK theory, where the values Z=0.54 and G =0.27 have been determined using the analysis technique discussed in the previous section.

Table 1 lists the results for eleven different point contacts of silver on niobium with interstitial hydrogen introduced by thermal diffusion. Table 2 lists the results for four point contacts for which hydrogen was introduced electrolytically. The Z,G values for these fifteen point contacts are plotted in Fig. 5 on the Z-G surface. For the thermal diffusion introduced hydrogen, one can see by comparison of the filled squares (one hour exposure) to the open circles (two hour exposure), that increased exposure to hydrogen resulted in increased inelastic scattering, i.e. larger G values, as expected. This data also shows a slight tendency of larger values of Z to correlate with larger values of G , within each exposure group. The data for the electrolytically introduced hydrogen (open triangles) falls among the data for thermal diffusion, indicating that once hydrogen is introduced, point contacts behave similarly, regardless of the hydrogenation technique employed.

Table 3 lists the results for nine work hardened niobium point contacts and three pure vacuum annealed niobium point contacts. The Z,G values for these contacts are plotted in Fig. 6 on the Z-G surface. The data for pure vacuum annealed niobium, filled circles, are along the top edge of the surface, i.e., G ~0, as discussed in the previous section. The data for the nine work hardened niobium contacts (open circles) covers an unusually wide range in Z (from 0.10 to 1.50) as well as a wide range in G (0.19 to 0.98) with little correlation between Z and G . The modified BTK theory assumes elastic scattering and inelastic scattering are independent and to a good approximation this experimental study supports that assumption. The correlation coefficient for the twenty-four tabulated point contacts with inelastic scattering is 0.03, indicating essentially no trend between Z and G values. However, since each sample preparation technique affects the surface and interior of the niobium differently, it would be an oversimplification to presume the correlation between Z and G within each sample preparation group is also 0.03. In fact, within the thermal diffusion group, a weak correlation of 0.46 is observed, and within the work hardened group the correlation is 0.33. These two groups taken together show little correlation between Z and G at 0.15. However, the electromigration group is different. It shows a negative correlation coefficient (-0.70) as might be expected, since a smaller Z value indicates a cleaner surface and therefor a lower threshold to hydrogen penetration. Thus, for this preparation technique, the smaller Z values correlate with the larger G values.

6. Limitations on Z and G

The elastic scattering parameter, Z, in the BTK theory sets the strength of the interaction at the normal metal / superconductor interface and is assumed to be temperature independent. In principle, Z values can range from zero in the ideal metallic microconstriction limit to ten or greater in the tunneling limit (see, for example, calculations for Z=50.0, Fig. 6, BTK). The role of Z, as the strength parameter for elastic scattering, becomes clear in the T ® T_C limit of the BTK theory. In this limit the normal metal / normal metal point contact, modeled as a repulsive d -function barrier, is recovered, with its transmission coefficient given by (1+Z²)^-1 and corresponding reflection coefficient, Z²(1+Z²)^-1. Experimental studies of normal metal / superconductor point contacts of Cu/Nb [7] support both the efficacy of the broad range of allowed values for Z and its temperature independence. Goodness of fit to dV/dI versus V data and Y(Z, t) versus t data have been illustrated and discussed in the literature. [8]

Constraints on the BTK theory regarding applicability to heavy fermion systems and metal oxide superconductors are not a limit on Z per se, but systemic to the assumptions in the theory regarding properties of simple Fermi surfaces and non-phononic transitions between allowed particle states in the normal metal and allowed quasiparticle states in the superconductor.

The incorporation of inelastic scattering into the Bogoliubov-deGennes framework is carried out by inserting a term linear in t^-1, where t is the lifetime of quasiparticles between interactions. t is assumed to be temperature independent. As a consequence of this linear approach the Bogoliubov coherence factors become imaginary over the entire energy range, the quasiparticle density of states depends explicitly on G (= ?/t ) and the temperature dependence of the Andreev peak at zero bias voltage is weakened, but Z and G remain independent of each other.

Limits on G may come about in at least two different ways: breakdown of the linear approximation and variation of G with temperature. In this study, G values (G /D₀) were observed to range from 0.00 to 1.50. A G value of 1.00 for niobium corresponds to a lifetime of t=2.2x10^-13 s, already smaller by about a factor of 10 than known scattering and recombination lifetimes for quasiparticles in niobium and about a factor of 3 smaller than phonon interactions.[9] However, for data with G <1.0, agreement is remarkably close to expected lifetimes, given that the modified BTK model does not in fact postulate a specific mechanism for t . If we look at the quality of fit for data for small G values (see Fig. 4, which is typical) it is on the whole better than data for larger G values, as illustrated in Fig. 7, for G =1.49. This figure is typical of several point contacts we have measured with G >1.0. The data drops below the theoretical prediction, especially in the range 0.75<t<1.00, i.e. near T_C. It is difficult to say if this is indicative of a breakdown in the linear approximation or if we are detecting a temperature dependence in G . Since the poorer fits are always observed for point contacts with large values of G , the authors feel it may reflect a limitation of the linear approximation. However, a temperature dependence of G can not be ruled out altogether, since when the poorer fits do occur, it is always near T_C, where scattering and recombination lifetimes are known to change rapidly with temperature. These two effects may not be mutually exclusive.

7. Conclusions

This paper presents a detailed experimental study of normal metal / superconductor point contacts of silver on niobium. Inelastic scattering has been enhanced in the niobium by two different techniques: hydrogenation to produce interstitial hydrogen throughout the bcc lattice and work hardening to introduce lattice defects and dislocations. The modified BTK theory has been used to describe charge transport data for these contacts. A new analysis method is demonstrated that can uniquely extract a value for the elastic scattering strength, Z, and a single value of the inelastic scattering parameter, G , which can be used to characterize the entire charge transport data set. In practice both Z and G are observed to be essentially temperature independent and at most only weakly correlated. For large values of the inelastic scattering parameter, i.e. G > 1.0, the goodness of fit to the data begins to degrade, especially near the critical temperature.

REFERENCES

[1] G.E. Blonder, M. Tinkham and T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515.

[2] P.W. Anderson, Science 256 (1992) 1562.

[3] A. Plecenik, M. Grajcar, S. Benacka, P. Siedel and A. Pfuch, Phys. Rev. B 49 (1994) 10016.

[4] Chr. Bruder, Phys. Rev. B 41 (1990) 4017.

Fig. 1. Differential resistance versus bias voltage for a point contact of silver on hydrogen diffused niobium. Temperature in K, top to bottom, 9.46, 9.09, 8.81, 8.63, 8.35, 8.35, 8.08, 7.53, 6.98, 6.61, 6.33, 5.97, 5.51, 5.05 and 4.59. Data curves are shifted 0.10 Ohm for clarity. Data for HD070, see Table 1.

Fig. 2. Normalized conductance Y(Z, G , t) versus reduced temperature, t. Top panel G =0.00, center panel G =0.25 and bottom panel G =0.45. Each panel shows curves for seven values of Z. Top to bottom, Z=0.0, 0.2, 0.4, 0.6, 0.8, 1.0 and 10.0.

Fig. 3. Y at t=0.50 versus dY/dt at t=0.50. This Z-G surface is the parameter space for the modified BTK theory.

Fig. 4. Normalized conductance versus reduced temperature for the data set shown in Fig. 1. The curve is for Z=0.54 and G =0.27; contact HD070, see Table 1.

Fig. 5. The Z-G surface showing point contacts with interstitial hydrogen. One hour diffusion time, filled squares; two hour diffusion time, open circles; electromigration, open triangles.

Fig. 6. The Z-G surface showing point contacts for vacuum annealed niobium, filled circles and work hardened niobium, open circles.

Fig. 7. Normalized conductance versus reduced temperature for a hydrogen diffused point contact. The curve is for Z=0.29 and G =1.49; contact HD052, see Table 1.

TABLE 1. Point contacts of silver on niobium, with interstitial hydrogen, introduced by thermal diffusion.
 

Contact Identifier	Y	DY/dt	Z	G	RN(?)	TC(K)	Exposure (hours)
HD018	1.610	-0.236	0.22	0.22	4.46	9.16	1
HD034	1.060	-0.050	0.30	1.50	4.26	9.00	2
HD052	1.060	-0.064	0.29	1.49	12.90	9.15	2a
HD070	1.150	-0.156	0.54	0.27	1.06	9.18	1b
HD085	1.280	-0.022	0.42	0.23	1.92	9.20	1
HD104	1.412	-0.085	0.33	0.18	0.97	9.20	1
HD157	1.140	-0.073	0.39	0.70	8.48	9.13	1
HD171	1.373	-0.145	0.31	0.26	1.63	9.12	2
HD191	1.134	-0.069	0.39	0.73	10.05	9.12	2
HD207	1.294	-0.126	0.33	0.36	4.15	9.12	2
HD223	1.224	-0.095	0.36	0.46	7.36	9.00	2

 

1. Data for contact HD052 is shown in Fig. 7.
2. Data for contact HD070 is shown in Fig. 1 and Fig. 4.

TABLE 2. Point contacts of sliver on niobium, with interstitial hydrogen, introduced electrolytically.
 

Contact Identifier	Y	dY/dt	Z	G	RN(?)	TC(K)
HE0212	1.17	-0.10	0.35	0.64	11.3	9.10
HE1325	1.18	-0.15	0.22	0.74	2.47	9.11
HE2637	1.07	-0.06	0.22	1.49	6.46	9.10
HE3851	1.35	-0.12	0.33	0.26	5.51	9.10

 

TABLE 3. Point contacts of silver on work hardened niobium (WH identifier) and vacuum annealed niobium (VA identifier).
 

Contact Identifier	Y	dY/dt	Z	G	RN(?)	TC(K)
WH008	0.87	0.21	1.50	0.98	2.29	9.11
WH017	0.66	0.75	1.29	0.19	1.43	9.10
WH033	1.51	-0.35	0.10	0.27	1.55	9.05
WH064	0.87	0.40	0.98	0.39	1.49	9.10
WH077	1.10	0.07	0.56	0.48	1.32	9.10
WH096	1.09	-0.03	0.47	0.79	12.04	9.10
WH115	1.22	-0.02	0.42	0.36	4.40	9.00
WH157	1.38	-0.07	0.35	0.20	2.88	9.05
WH176	1.09	-0.01	0.50	0.70	8.03	9.07
VA089	1.23	0.52	0.53	0.00	4.14	9.20
VA149	1.55	0.19	0.36	0.00	1.29	9.20
VA280	0.94	0.74	0.73	0.03	3.59	9.20