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Andreev reflections on the heavy-fermion superconductor UPt3
PHYSICA
Physica B 194-196 (1994) 2073-2074 North-Holland
A n d r e e v reflections on the heavy-fermion s u p e r c o n d u c t o r UPt3 Y. De Wilde a,b , J. Hell a , A.G.M. Jansen a , P. Wyder a , R. Deltour b and L. Taillefer c ~Grenoble High Magnetic Field Laboratory, Max-Planck-Institut fiir Festk6rperforschung, B.P. 166, F-38042 Grenoble Cedex 9, France bUniversit@ Libre de Bruxelles, Service de Physique des Solides, C.P. 233, Boulevard du Triomphe, 1050 Bruxelles, Belgium ~Department of Physics, McGill University, Montreal H3A2TS, Canada The superconducting heavy fermion system UPt3 has been studied by means of point contacts adjusted with a bimorph stepping motor working in the mixing chamber of a dilution refrigerator (T > 30 mK). The measured differential resistance curves of Ag/UPt3 point contacts are compared with the usual isotropic BTK model describing the Andreev-reflection process of quasiparticles in normal~superconductor point contacts. For this heavy-fermion superconductor, the experimental data do not fit at all with the predictions of this model. An adaptation of the BTK theory to the case of anisotropic superconductors has been performed and demonstrates that our data can be interpreted in terms of the hybrid symmetry Ak = Ao]cz(k~ + iky), if one assumes a preferential injection o{ the electrons in the forward direction.
There exist ..several indirect indications that heavy fermion ..systems (HFS) could have a superconducting gap A which vanishes at points or lines at the Fermi surface [1]. Here we report about normal-superconductor (NS) point-contact (PC) measurements performed on UPt3 that contain information on the symmetry of the superconducting state. The problem of electronic transport in NS junctions can be described in the BTK formalism [2]. When a voltage V is applied on the junction, electrons injected through a PC with an excess energy E = eV > A can cross the NS interface via quasiparticle excitations. Below the forbidden energy g'ap, the traversing electron condenses into the superconducting state by forming a Cooper pair with another electron from the normal metal. This so-called Andreev-reflection process (retroreflection of a quasi particle) leads to a smaller PC resistance for eV < A. In terms of the Andreev-reflection probability A and the normal reflection probability B the differential resistance Rn~ of the NS junction normalized to the normal state value R,,~ is expressed by (valid at T = 0 )
R.,(E) R,,
_
i..
1-
Z ~
I , , -- 1 + A ( E , A ) -- B ( E , A)"
(1)
Here Z is a phenomenological parameter account0921-4526/94/$07.00 © 1994
SSDI 0921-4526(93)1669-D
-
ing for all the processes responsible for a transmission coefficient of the metallic PC less than unity. In the most simple case of a spherical Fermi surface with an anisotropic energy gap A(]c), the BTK formula (Eq.1) generalizes to
n,,(Z)
n..
_
I,,
_
(2)
I.,(E) Z ~ 1 - O-4--U~
1 + N f dk]c. fiP(]~)[A(E, A ( k ) ) - B ( E , A(]¢))] ' where the integral in the denominator is taken over all the directions in k-space of electrons arriving to the NS interface, fi and ]c are unit vectors, respectively in real space (perpendicular to the contact surface) and in momentum space. N = ( f dk]~. fiR(it)) -1 is a normalization factor and P(k) the angular distribution function of the injected electrons. The PC were adjusted inside the mixingchamber of a dilution cryostat of the nontoploading type by means of a bimorph steppingmotor [3]. Figure 1 presents R ( V ) curves obtained when pressing a sharp Ag-tip along the caxis of a grain of a freshly cleaved UPt3 polycrystal. The curves were recorded at different tem-
Elsevier Science B.V. All rights reserved
2074
1,30
I I I I [ I I , ~ , I I I
' ' I ' ' ' I ' ' ' I ' ' ' I j/.,~exp.
1.15 r~
1.10
4,5
f,,----hybrid ~polar ~f-axial ' ~hybrid f /.,_btk
1.05
1.00 z,
t I i ~ , I L i , I , L-200 0 200 V[~V]
Figure 1. Differential resistance R/R,o of a A g / U P t a PC. T h e curves correspond to different t e m p e r a t u r e s ranging from T = 4 0 m K (lower curve) to T = 5 0 0 m K (upper curve). T h e zero bias P C resistance R0 = 1.4~. p e r a t u r e s ranging from T = 4 0 m K to T = 500mK ( ~ critical t e m p e r a t u r e ) . A pronounced structure is visible around V = 0 m V which disappears in the n o r m a l state of [lPl3. In figure 2 we present the normalized characteristic R,+,/Rn,~. T h e triangular structure found does not fit at all with what. is expected for isotropic superconductors. A t t e m p t s to fit this t r i a n g u l a r shape with Eq.2 have been (tone for the three simplest, anisotropic s y m m e t r i e s of At/c) [4]: A(k) = A0cosOal (polar), At/c) =
A0 sin 0k(axial ) and ~X(k) : A0k~(k~ + ~i:~) (h:'brid). T h e hybrid s y m m e t r y is often proposed to be the one corresponding to the s y m m e t r y of U P t a [4, 5], As shown in figure 2, with these s y m m e t r i e s no a g r e e m e n t can be found between theory and e x p e r i m e n t for an isotropic angular distribution, P ( k ) = const. Once a preferential injection of (he electrons is introduced in Eq.2 i.]le hybrid s y m m e t r y gives the best description with respect to the two others and a good agreement is found for P(O) = e -(°/°*), with 0* = 20 ° . Here P(O) is the angular distribution flmction (in polar coordinates) corresponding to a preferential injection along the c-axis(0 = 0). For a slightly reduced transmission (Z = .3) of the NS junction. a characteristic with two m i n i m a at V = :1:22/A.' is found, in good a g r e e m e n t with other experimental results on U P t a [5].
J
I
I
-200
,
,
I
I
i
i
0
,
I
200
i
+ i
I
400
v[t,v] Figure 2. C o m p a r i s o n of the voltage dependence of the e x p e r i m e n t a l R ~ , / R , m curve (exp.) with c o m p u t e d curves for different s y m m e t r i e s for A0 = 75#V and Z = 0(full lines) or Z = .3 (dashed line). Curve 0: B T K model [2]. Curves l-a: isotropic injection, P(O) = eonst. Curves 4,5: forward injection, P(O) = e -(°/°°)~, with 0* = 20 °. To conclude, for UPt3 the a n o m a l o u s Andreevreflection structure in the NS point-contact characteristics can be interpreted in t e r m s of hybrid s y m m e t r y for the s u p e r c o n d u c t i n g gap. T h e preferential current distribution, assumed in our model, could also result from the integration in Eq.2 over the anisotropic Fermi surface of UPta. REFERENCES
1. 2. 3. 4. 5.
P. Fulde, a. Keller, and G. Zwicknagl, Solid State Phys. 41 (1988) 1. G . E . Blonder,M. T i n k h a m , and T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515. J. Heft, Y. De Wilde, A. G. M. Jansen, P. Wyder, W. Grill, Rev. Sci. I n s t r u m . , in press. P. J. Ilirschfeld, P. W61fle, and D. Einzel, Phys. Rev. B 37 (1988) 83. G. Goll, H. v. LShneysen, I.K. Yanson, and L. Taillefer, Phys. Rev. Lett. 70 (1993) 2008.
Physica B 194-196 (1994) 2073-2074 North-Holland
A n d r e e v reflections on the heavy-fermion s u p e r c o n d u c t o r UPt3 Y. De Wilde a,b , J. Hell a , A.G.M. Jansen a , P. Wyder a , R. Deltour b and L. Taillefer c ~Grenoble High Magnetic Field Laboratory, Max-Planck-Institut fiir Festk6rperforschung, B.P. 166, F-38042 Grenoble Cedex 9, France bUniversit@ Libre de Bruxelles, Service de Physique des Solides, C.P. 233, Boulevard du Triomphe, 1050 Bruxelles, Belgium ~Department of Physics, McGill University, Montreal H3A2TS, Canada The superconducting heavy fermion system UPt3 has been studied by means of point contacts adjusted with a bimorph stepping motor working in the mixing chamber of a dilution refrigerator (T > 30 mK). The measured differential resistance curves of Ag/UPt3 point contacts are compared with the usual isotropic BTK model describing the Andreev-reflection process of quasiparticles in normal~superconductor point contacts. For this heavy-fermion superconductor, the experimental data do not fit at all with the predictions of this model. An adaptation of the BTK theory to the case of anisotropic superconductors has been performed and demonstrates that our data can be interpreted in terms of the hybrid symmetry Ak = Ao]cz(k~ + iky), if one assumes a preferential injection o{ the electrons in the forward direction.
There exist ..several indirect indications that heavy fermion ..systems (HFS) could have a superconducting gap A which vanishes at points or lines at the Fermi surface [1]. Here we report about normal-superconductor (NS) point-contact (PC) measurements performed on UPt3 that contain information on the symmetry of the superconducting state. The problem of electronic transport in NS junctions can be described in the BTK formalism [2]. When a voltage V is applied on the junction, electrons injected through a PC with an excess energy E = eV > A can cross the NS interface via quasiparticle excitations. Below the forbidden energy g'ap, the traversing electron condenses into the superconducting state by forming a Cooper pair with another electron from the normal metal. This so-called Andreev-reflection process (retroreflection of a quasi particle) leads to a smaller PC resistance for eV < A. In terms of the Andreev-reflection probability A and the normal reflection probability B the differential resistance Rn~ of the NS junction normalized to the normal state value R,,~ is expressed by (valid at T = 0 )
R.,(E) R,,
_
i..
1-
Z ~
I , , -- 1 + A ( E , A ) -- B ( E , A)"
(1)
Here Z is a phenomenological parameter account0921-4526/94/$07.00 © 1994
SSDI 0921-4526(93)1669-D
-
ing for all the processes responsible for a transmission coefficient of the metallic PC less than unity. In the most simple case of a spherical Fermi surface with an anisotropic energy gap A(]c), the BTK formula (Eq.1) generalizes to
n,,(Z)
n..
_
I,,
_
(2)
I.,(E) Z ~ 1 - O-4--U~
1 + N f dk]c. fiP(]~)[A(E, A ( k ) ) - B ( E , A(]¢))] ' where the integral in the denominator is taken over all the directions in k-space of electrons arriving to the NS interface, fi and ]c are unit vectors, respectively in real space (perpendicular to the contact surface) and in momentum space. N = ( f dk]~. fiR(it)) -1 is a normalization factor and P(k) the angular distribution function of the injected electrons. The PC were adjusted inside the mixingchamber of a dilution cryostat of the nontoploading type by means of a bimorph steppingmotor [3]. Figure 1 presents R ( V ) curves obtained when pressing a sharp Ag-tip along the caxis of a grain of a freshly cleaved UPt3 polycrystal. The curves were recorded at different tem-
Elsevier Science B.V. All rights reserved
2074
1,30
I I I I [ I I , ~ , I I I
' ' I ' ' ' I ' ' ' I ' ' ' I j/.,~exp.
1.15 r~
1.10
4,5
f,,----hybrid ~polar ~f-axial ' ~hybrid f /.,_btk
1.05
1.00 z,
t I i ~ , I L i , I , L-200 0 200 V[~V]
Figure 1. Differential resistance R/R,o of a A g / U P t a PC. T h e curves correspond to different t e m p e r a t u r e s ranging from T = 4 0 m K (lower curve) to T = 5 0 0 m K (upper curve). T h e zero bias P C resistance R0 = 1.4~. p e r a t u r e s ranging from T = 4 0 m K to T = 500mK ( ~ critical t e m p e r a t u r e ) . A pronounced structure is visible around V = 0 m V which disappears in the n o r m a l state of [lPl3. In figure 2 we present the normalized characteristic R,+,/Rn,~. T h e triangular structure found does not fit at all with what. is expected for isotropic superconductors. A t t e m p t s to fit this t r i a n g u l a r shape with Eq.2 have been (tone for the three simplest, anisotropic s y m m e t r i e s of At/c) [4]: A(k) = A0cosOal (polar), At/c) =
A0 sin 0k(axial ) and ~X(k) : A0k~(k~ + ~i:~) (h:'brid). T h e hybrid s y m m e t r y is often proposed to be the one corresponding to the s y m m e t r y of U P t a [4, 5], As shown in figure 2, with these s y m m e t r i e s no a g r e e m e n t can be found between theory and e x p e r i m e n t for an isotropic angular distribution, P ( k ) = const. Once a preferential injection of (he electrons is introduced in Eq.2 i.]le hybrid s y m m e t r y gives the best description with respect to the two others and a good agreement is found for P(O) = e -(°/°*), with 0* = 20 ° . Here P(O) is the angular distribution flmction (in polar coordinates) corresponding to a preferential injection along the c-axis(0 = 0). For a slightly reduced transmission (Z = .3) of the NS junction. a characteristic with two m i n i m a at V = :1:22/A.' is found, in good a g r e e m e n t with other experimental results on U P t a [5].
J
I
I
-200
,
,
I
I
i
i
0
,
I
200
i
+ i
I
400
v[t,v] Figure 2. C o m p a r i s o n of the voltage dependence of the e x p e r i m e n t a l R ~ , / R , m curve (exp.) with c o m p u t e d curves for different s y m m e t r i e s for A0 = 75#V and Z = 0(full lines) or Z = .3 (dashed line). Curve 0: B T K model [2]. Curves l-a: isotropic injection, P(O) = eonst. Curves 4,5: forward injection, P(O) = e -(°/°°)~, with 0* = 20 °. To conclude, for UPt3 the a n o m a l o u s Andreevreflection structure in the NS point-contact characteristics can be interpreted in t e r m s of hybrid s y m m e t r y for the s u p e r c o n d u c t i n g gap. T h e preferential current distribution, assumed in our model, could also result from the integration in Eq.2 over the anisotropic Fermi surface of UPta. REFERENCES
1. 2. 3. 4. 5.
P. Fulde, a. Keller, and G. Zwicknagl, Solid State Phys. 41 (1988) 1. G . E . Blonder,M. T i n k h a m , and T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515. J. Heft, Y. De Wilde, A. G. M. Jansen, P. Wyder, W. Grill, Rev. Sci. I n s t r u m . , in press. P. J. Ilirschfeld, P. W61fle, and D. Einzel, Phys. Rev. B 37 (1988) 83. G. Goll, H. v. LShneysen, I.K. Yanson, and L. Taillefer, Phys. Rev. Lett. 70 (1993) 2008.