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Point-contact studies on the heavy-fermion superconductor UPt3
ELSEVIER
Physica B 206 & 207 (1995) 609-611
Point-contact studies on the heavy-fermion superconductor UPt 3 G. Goll a'*, Chr. Bruder b, H.v. L6hneysen a "Physikalisches lnstitut, Universitiit Karlsruhe, 76128 Karlsruhe, Germany blnstitut fiir Theoretische Festkdrperphysik, Universitiit Karlsruhe, 76128 Karlsruhe, Germany
Abstract The superconducting order parameter of the heavy-fermion superconductor UPt 3 has been investigated by Andreev reflection of charge carriers at the normal metal-superconductor interface between UPt 3 and Pt. The reflection mechanism leads to minima in the differential resistance dV/dI versus voltage V measured at low temperatures (T "~ T¢). Here we report on a comparison of the spectra with the theory. Using an isotropic order parameter A(k) = A0 the spectra exhibiting a double-minimum structure can be described only at low bias. Better agreement can be obtained by a fit with a two-dimensional order parameter together with a preferential forward direction for electron injection. The analysis favors a two-dimensional order parameter with an orbital part with a line of nodes in the basal plane and point nodes along the c-axis.
1. Introduction The existence of unconventional superconductivity in UPt 3 has been supported by various experiments. In particular, the complex (B, T) phase diagram was investigated extensively, revealing three superconducting phases in the Shubnikov state of UPt 3 [1]. The question of the order parameter (OP) of UPt 3 and its symmetry has been under intense theoretical scrutiny. Several scenarios invoking a symmetry-breaking field have been envisioned by analyzing the symmetryinvariant Ginzburg-Landau functions including a symmetry-breaking coupling term. The most convincing scenario is the two-dimensional (2D) representation either with even parity (E~g) [2[ or with odd parity (E2o) [3]. Other scenarios consider a one-dimensional (1D) representation A2u where the spin degeneracy leads to a splitting of the superconducting transition [4]. Also, two nearly degenerate 1D representations have been proposed [5]. Point-contact (PC) spectroscopy between a normal
* Corresponding author.
metal (N) and a superconductor (S) can be employed to obtain information about the superconducting energy gap and the OP via the mechanism of Andreev reflection (AR) at the NS interface [6]. This mechanism can help to identify the gap structure of anisotropic superconductors [7]. For UPt3, measurements of PC spectra of a single-crystalline sample have shown a strong gap anisotropy and an unusual temperature and magnetic-field dependence of the spectra [8,9]. Here we report on a comparison of different OP scenarios with the zero-field data.
2. Experimental Two samples of UPt 3 were investigated, a single crystal and a polycrystal which contained a few large grains. Both samples showed two sharp superconducting transitions. The samples were mounted inside the mixing chamber of a 3He/4He dilution refrigerator. Mechanical feed-throughs and a differential screw mechanism allowed establishing and changing of PC between UPt 3 and Pt as a normal-metal counterelectrode at low temperatures T. The differential resist-
0921-4526/95/$09.50 t~) 1995 Elsevier Science B.V. All rights reserved S S D I 0921-4526(94)00534-6
610
G. Goll et al. / Physica B 206 & 207 (1995) 609-611
ance d V / d l was measured down to T = 3 0 m K as a function of the applied voltage V. For further experimental details see Ref. [8]. The contact resistance could be changed in the range of a few tenths up to some 10~, but usually only contacts with a zero-bias resistance Ro < 5 ~ showed some structure in the d V / d l versus V curve due to superconductivity. Therefore, R 0 were chosen to be of the order of 1 ~. The typical value of 1 suggests that the contacts are in the ballistic regime for PC spectroscopy, for the contact diameter d = (16pl/ 37fRo) 1/2 ~ 600 ~, < l holds for such contacts. Here the Sharvin expression for R o was used,/9 is the resistivity and l the electron mean free path with p l = 2 × 10-15 f~_m2 for bulk UPt 3. From the T dependence of the resistance in the normal state RN and p, one can determine d and lpc, the mean free path in the contact region, independently using d = ( ( d p / d T ) / ( d R N / d T ) and/PC = (PZ/d)(RN(O) - (16pl)/(3~rd2)) -~ 110]. However, this determination yields values for d much larger than those calculated from the Maxwell or Sharvin resistance and 1 < d so that the PC should be in the so-called 'dirty limit' [9]. This difference probably arises from the fact that the resistivity in the contact region is much larger than in the bulk.
3. Results and discussion
Fig. l(a) shows the differential resistance d V / d l versus voltage V at T = 4 2 m K of a PC with R 0= 1.62 l~ between the single-crystalline UPt 3 and Pt with preferential current flow IIIc. Fig. l(b) shows a PC spectrum with R o = 1.09 I~ at T = 53 mK for the polycrystalline grain with 1 oblique at an angle of 20-30 ° off the c-axis as determined from Laue pictures. In both cases the normal-state background has been subtracted. This background, in particular the normalstate PC spectrum for low bias, was determined by measurements in the normal state either at T > Tc or in a magnetic field B >Be2. Both spectra show a double-minimum structure, which can be explained by A R of charge carriers at the NS interface. For energies e V < A an incident electron is reflected as a hole at the interface, resulting in a larger conductivity than for E ~>A when electrons can cross the interface and enter the superconductor as quasiparticle excitations. In the case of a contact with an ideal interface where the probability of A R is 1, and of ordinary reflection zero, the d V / d l is reduced by a factor of two for e V < A. Blonder et al. [6] calculated
i
I
i
I
~
I
t
I
'
, ,(aI , ,
(b)
"2bo
'' o V(p.V)
'200'
Fig. l. Differential resistance dVIdl versus voltage V after subtraction of the normal-state background for the single crystal (a) and the polycrystal (b). The lines represent fits for different order parameter symmetries (full line, E2u; longdashed line, Elg ; dashed line, isotropic OP).
the reflection and transmission coefficients for an swave superconductor (isotropic OP) with a non-ideal interface by introducing a phenomenological parameter Z which might arise from different Fermi velocities and/or a potential barrier at the interface. In this case d V / d l is enhanced at V ~ 0 and the characteristic double-minimum structure with minima at V . ~ A / e occurs. The double-minimum structure is also expected for an anisotropic superconductor [7]. In this work the spatial dependence of the OP near the interface was determined self-consistently for various non-ideal interfaces of d-wave superconductors and various directions of current flow. The non-ideality of the interface was taken into account by an interface potential parametrized by reflection coefficients. Using this formalism d V / d I was calculated for different OP symmetries assuming that the electrons are injected preferentially into forward direction with an angular
G. Goll et al. / Physica B 206 & 207 (1995) 609-611
spread described by a cone. A detailed description will be given elsewhere [11]. The result of these calculations is shown in Fig. 1(a) and (b). The full line represents a fit for a 2D-OP E~ e l ( k ) ~ Ik~l(kx + iky) 2 (similar to the E2u representation) and the long-dashed line is a fit for a 2D-OP EI~ A ( k ) - l k ~ l ( k x +iky) (similar to the Etg representation), while the dashed line is the well-known result for an isotropic OP A(k)~ Ao. In all three cases the theoretical expression for dV/dI has been scaled to fit the measured resistance change r = ( R N - R o ) / R o. The parameters of the s-wave fit are A0 = 29 i~eV, Z = 0.2 for Fig. 1(a) and A0 = 39 ixeV, Z -- 0.15 for Fig. l(b). The data can be fitted much better with both anisotropic OPs assuming a cone for the current flow with a maximum electron injection angle 0. In Fig. l(a) the parameters are el0 = 35 IxeV and Z = 0.23 for the E~ state and el0 = 54 IxeV and Z = 0.50 for the EII state, with 0 = 20° for both curves. With this 0 the EIt representation fits the data slightly better. However, the agreement for the E~ representation can be achieved by reducing 0. Thus, the precision of the data is not sufficient to distinguish between the two 2D representations. Also, T = 42 mK is not low enough. For even lower T, the difference between E~ and EII is expected to yield larger difference in dV/dI versus V [11]. While the data are described very well by a 2D-OP, an isotropic OP or an 1D-OP with lines of nodes (not shown) clearly cannot reproduce the data. For the 1D-OP, the curves are always much too wide. Fig. l(b) shows the data and corresponding fits for the polycrystalline grain where a preferential current flow of 25° off the c-axis was assumed. The fit parameters are 0 = 10°, elo = 26 IxeV and Z = 0.46 for the E~ state and 0 = 30°, el0 = 55 ~eV and Z = 0.54 for the E . state. Again the best agreement can be obtained with a 2D-OP, but for this current direction the E I state looks more 'isotropic' than the E H state. This can be explained by noting that while both 2D representations have the same topology of nodes, namely a line of nodes in the basal plane and point nodes at the poles, there is a difference in the excitation spectrum. In the E~ state the quasiparticle
611
spectrum opens linearly with a polar angle and quadratically in the E~I state. This difference is probably responsible for the larger difference between the two fits with a 2D-OP in Fig. l(b), While this does suggest that the EiI representation gives a better description of our data, a definite conclusion must await a more detailed study. Also, the above-mentioned problem of using r as an additional fit parameter must be solved. In conclusion, our results are in line with the anisotropy of the spectra and the occurrence of A R in the low-T-low-B phase only as reported earlier [8]. The above analysis supports the description of the unconventional superconductivity in UPt 3 by a 2D order parameter.
Acknowledgement This work was partly supported by Deutsche Forschungsgemeinschaft through SFB 195.
References [1] For a recent review, see: H.v. L6hneysen, Physica B 197 (1994) 551. [2] D.W. Hess, T.A. Tokuyasu and J.A. Sauls, J. Phys.: Condens. Matter 1 (1989) 8135. [3] J.A. Sauls, J. Low Temp. Phys. 95 (1994) 153. [4] K. Machida and M. Ozaki, Phys. Rev. Lett. 66 (1991) 3293; 67 (1991) 3732. [5] D.-C. Chen and A. Garg, Phys. Rev. Lett. 70 (1993) 1689. [6] G.E. Blonder, M. Tinkham and T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515. [7] Chr. Bruder, Phys. Rev. B 41 (1990) 4017. [8] G. Goll et al., Phys. Rev. Lett. 70 (1993) 2008. [9] H.v. L6hneysen and G. Goll, J. Low Temp. Phys. 95 (1994) 199. [10] A.I. Akimenko et al., Sov. J. Low Temp. Phys. 8 (1982) 130. [11] G. Goll, H.v. L6hneysen and Chr. Bruder, unpublished.
Physica B 206 & 207 (1995) 609-611
Point-contact studies on the heavy-fermion superconductor UPt 3 G. Goll a'*, Chr. Bruder b, H.v. L6hneysen a "Physikalisches lnstitut, Universitiit Karlsruhe, 76128 Karlsruhe, Germany blnstitut fiir Theoretische Festkdrperphysik, Universitiit Karlsruhe, 76128 Karlsruhe, Germany
Abstract The superconducting order parameter of the heavy-fermion superconductor UPt 3 has been investigated by Andreev reflection of charge carriers at the normal metal-superconductor interface between UPt 3 and Pt. The reflection mechanism leads to minima in the differential resistance dV/dI versus voltage V measured at low temperatures (T "~ T¢). Here we report on a comparison of the spectra with the theory. Using an isotropic order parameter A(k) = A0 the spectra exhibiting a double-minimum structure can be described only at low bias. Better agreement can be obtained by a fit with a two-dimensional order parameter together with a preferential forward direction for electron injection. The analysis favors a two-dimensional order parameter with an orbital part with a line of nodes in the basal plane and point nodes along the c-axis.
1. Introduction The existence of unconventional superconductivity in UPt 3 has been supported by various experiments. In particular, the complex (B, T) phase diagram was investigated extensively, revealing three superconducting phases in the Shubnikov state of UPt 3 [1]. The question of the order parameter (OP) of UPt 3 and its symmetry has been under intense theoretical scrutiny. Several scenarios invoking a symmetry-breaking field have been envisioned by analyzing the symmetryinvariant Ginzburg-Landau functions including a symmetry-breaking coupling term. The most convincing scenario is the two-dimensional (2D) representation either with even parity (E~g) [2[ or with odd parity (E2o) [3]. Other scenarios consider a one-dimensional (1D) representation A2u where the spin degeneracy leads to a splitting of the superconducting transition [4]. Also, two nearly degenerate 1D representations have been proposed [5]. Point-contact (PC) spectroscopy between a normal
* Corresponding author.
metal (N) and a superconductor (S) can be employed to obtain information about the superconducting energy gap and the OP via the mechanism of Andreev reflection (AR) at the NS interface [6]. This mechanism can help to identify the gap structure of anisotropic superconductors [7]. For UPt3, measurements of PC spectra of a single-crystalline sample have shown a strong gap anisotropy and an unusual temperature and magnetic-field dependence of the spectra [8,9]. Here we report on a comparison of different OP scenarios with the zero-field data.
2. Experimental Two samples of UPt 3 were investigated, a single crystal and a polycrystal which contained a few large grains. Both samples showed two sharp superconducting transitions. The samples were mounted inside the mixing chamber of a 3He/4He dilution refrigerator. Mechanical feed-throughs and a differential screw mechanism allowed establishing and changing of PC between UPt 3 and Pt as a normal-metal counterelectrode at low temperatures T. The differential resist-
0921-4526/95/$09.50 t~) 1995 Elsevier Science B.V. All rights reserved S S D I 0921-4526(94)00534-6
610
G. Goll et al. / Physica B 206 & 207 (1995) 609-611
ance d V / d l was measured down to T = 3 0 m K as a function of the applied voltage V. For further experimental details see Ref. [8]. The contact resistance could be changed in the range of a few tenths up to some 10~, but usually only contacts with a zero-bias resistance Ro < 5 ~ showed some structure in the d V / d l versus V curve due to superconductivity. Therefore, R 0 were chosen to be of the order of 1 ~. The typical value of 1 suggests that the contacts are in the ballistic regime for PC spectroscopy, for the contact diameter d = (16pl/ 37fRo) 1/2 ~ 600 ~, < l holds for such contacts. Here the Sharvin expression for R o was used,/9 is the resistivity and l the electron mean free path with p l = 2 × 10-15 f~_m2 for bulk UPt 3. From the T dependence of the resistance in the normal state RN and p, one can determine d and lpc, the mean free path in the contact region, independently using d = ( ( d p / d T ) / ( d R N / d T ) and/PC = (PZ/d)(RN(O) - (16pl)/(3~rd2)) -~ 110]. However, this determination yields values for d much larger than those calculated from the Maxwell or Sharvin resistance and 1 < d so that the PC should be in the so-called 'dirty limit' [9]. This difference probably arises from the fact that the resistivity in the contact region is much larger than in the bulk.
3. Results and discussion
Fig. l(a) shows the differential resistance d V / d l versus voltage V at T = 4 2 m K of a PC with R 0= 1.62 l~ between the single-crystalline UPt 3 and Pt with preferential current flow IIIc. Fig. l(b) shows a PC spectrum with R o = 1.09 I~ at T = 53 mK for the polycrystalline grain with 1 oblique at an angle of 20-30 ° off the c-axis as determined from Laue pictures. In both cases the normal-state background has been subtracted. This background, in particular the normalstate PC spectrum for low bias, was determined by measurements in the normal state either at T > Tc or in a magnetic field B >Be2. Both spectra show a double-minimum structure, which can be explained by A R of charge carriers at the NS interface. For energies e V < A an incident electron is reflected as a hole at the interface, resulting in a larger conductivity than for E ~>A when electrons can cross the interface and enter the superconductor as quasiparticle excitations. In the case of a contact with an ideal interface where the probability of A R is 1, and of ordinary reflection zero, the d V / d l is reduced by a factor of two for e V < A. Blonder et al. [6] calculated
i
I
i
I
~
I
t
I
'
, ,(aI , ,
(b)
"2bo
'' o V(p.V)
'200'
Fig. l. Differential resistance dVIdl versus voltage V after subtraction of the normal-state background for the single crystal (a) and the polycrystal (b). The lines represent fits for different order parameter symmetries (full line, E2u; longdashed line, Elg ; dashed line, isotropic OP).
the reflection and transmission coefficients for an swave superconductor (isotropic OP) with a non-ideal interface by introducing a phenomenological parameter Z which might arise from different Fermi velocities and/or a potential barrier at the interface. In this case d V / d l is enhanced at V ~ 0 and the characteristic double-minimum structure with minima at V . ~ A / e occurs. The double-minimum structure is also expected for an anisotropic superconductor [7]. In this work the spatial dependence of the OP near the interface was determined self-consistently for various non-ideal interfaces of d-wave superconductors and various directions of current flow. The non-ideality of the interface was taken into account by an interface potential parametrized by reflection coefficients. Using this formalism d V / d I was calculated for different OP symmetries assuming that the electrons are injected preferentially into forward direction with an angular
G. Goll et al. / Physica B 206 & 207 (1995) 609-611
spread described by a cone. A detailed description will be given elsewhere [11]. The result of these calculations is shown in Fig. 1(a) and (b). The full line represents a fit for a 2D-OP E~ e l ( k ) ~ Ik~l(kx + iky) 2 (similar to the E2u representation) and the long-dashed line is a fit for a 2D-OP EI~ A ( k ) - l k ~ l ( k x +iky) (similar to the Etg representation), while the dashed line is the well-known result for an isotropic OP A(k)~ Ao. In all three cases the theoretical expression for dV/dI has been scaled to fit the measured resistance change r = ( R N - R o ) / R o. The parameters of the s-wave fit are A0 = 29 i~eV, Z = 0.2 for Fig. 1(a) and A0 = 39 ixeV, Z -- 0.15 for Fig. l(b). The data can be fitted much better with both anisotropic OPs assuming a cone for the current flow with a maximum electron injection angle 0. In Fig. l(a) the parameters are el0 = 35 IxeV and Z = 0.23 for the E~ state and el0 = 54 IxeV and Z = 0.50 for the EII state, with 0 = 20° for both curves. With this 0 the EIt representation fits the data slightly better. However, the agreement for the E~ representation can be achieved by reducing 0. Thus, the precision of the data is not sufficient to distinguish between the two 2D representations. Also, T = 42 mK is not low enough. For even lower T, the difference between E~ and EII is expected to yield larger difference in dV/dI versus V [11]. While the data are described very well by a 2D-OP, an isotropic OP or an 1D-OP with lines of nodes (not shown) clearly cannot reproduce the data. For the 1D-OP, the curves are always much too wide. Fig. l(b) shows the data and corresponding fits for the polycrystalline grain where a preferential current flow of 25° off the c-axis was assumed. The fit parameters are 0 = 10°, elo = 26 IxeV and Z = 0.46 for the E~ state and 0 = 30°, el0 = 55 ~eV and Z = 0.54 for the E . state. Again the best agreement can be obtained with a 2D-OP, but for this current direction the E I state looks more 'isotropic' than the E H state. This can be explained by noting that while both 2D representations have the same topology of nodes, namely a line of nodes in the basal plane and point nodes at the poles, there is a difference in the excitation spectrum. In the E~ state the quasiparticle
611
spectrum opens linearly with a polar angle and quadratically in the E~I state. This difference is probably responsible for the larger difference between the two fits with a 2D-OP in Fig. l(b), While this does suggest that the EiI representation gives a better description of our data, a definite conclusion must await a more detailed study. Also, the above-mentioned problem of using r as an additional fit parameter must be solved. In conclusion, our results are in line with the anisotropy of the spectra and the occurrence of A R in the low-T-low-B phase only as reported earlier [8]. The above analysis supports the description of the unconventional superconductivity in UPt 3 by a 2D order parameter.
Acknowledgement This work was partly supported by Deutsche Forschungsgemeinschaft through SFB 195.
References [1] For a recent review, see: H.v. L6hneysen, Physica B 197 (1994) 551. [2] D.W. Hess, T.A. Tokuyasu and J.A. Sauls, J. Phys.: Condens. Matter 1 (1989) 8135. [3] J.A. Sauls, J. Low Temp. Phys. 95 (1994) 153. [4] K. Machida and M. Ozaki, Phys. Rev. Lett. 66 (1991) 3293; 67 (1991) 3732. [5] D.-C. Chen and A. Garg, Phys. Rev. Lett. 70 (1993) 1689. [6] G.E. Blonder, M. Tinkham and T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515. [7] Chr. Bruder, Phys. Rev. B 41 (1990) 4017. [8] G. Goll et al., Phys. Rev. Lett. 70 (1993) 2008. [9] H.v. L6hneysen and G. Goll, J. Low Temp. Phys. 95 (1994) 199. [10] A.I. Akimenko et al., Sov. J. Low Temp. Phys. 8 (1982) 130. [11] G. Goll, H.v. L6hneysen and Chr. Bruder, unpublished.