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Resistance scaling of point contacts between heavy-fermion superconductors and tungsten
PHYSICA[ Physica B 218 (1996) 169-172
ELSEVIER
Resistance scaling of point contacts between heavy-fermion superconductors and tungsten K. Gloos*, C. Geibel, R. Miiller-Reisener, C. Schank lnstitut J~r Festk6rperphysik, SFB 252, Technische Hochschule Darmstadt, Hochschulstr. 8, D-64289 Darmstadt, German),
Abstract We investigated point contacts between a tungsten tip and the heavy-fermion superconductors CeCuzSi z, UPt 3, URuzSi2, UNi2A13, and UPd2AI 3. A correlation between contact size, magnitude of the superconducting anomalies, and specific resistivity of the heavy-fermion superconductors suggests that these anomalies are mainly due to diffusive and thermal transport. The residual contact resistance can be described by a strongly enhanced Sharvin resistance, probably caused by both a badly conducting normal interface layer on the heavy-fermion side and strong ordinary reflection of ballistic electrons.
Point-contact spectroscopy has been used to study the superconducting (SC) state of the heavy-fermion (HF) compounds [-1-5]. These experiments can be roughly summarized as follows: (i) At low temperatures the voltage-dependent contact resistance follows approximately the temperature-dependent specific resistivity of the HF compound under investigation. (ii) The residual contact resistance is unusually high. (iii) The relative magnitude of the SC anomalies decreases with increasing residual contact resistance. These anomalies are interpreted by Andreev reflexion (AR), i.e. an electron incident on the normal-SC interface transforms into a Cooper pair and reflects a hole [6]. The BTK theory [7] is used to describe them. Goll et al. [4] reported that contacts with well-conducting UPt3 showed very small SC anomalies, of order 1% of the total resistance. UBel 3 represents the opposite limit. Its resistivity Pnr ~ 130 ~tf~cm near Tc implies an extremely small electronic mean free path. The SC anomalies of this compound seem to be dominated by thermal and diffusive transport [8]. Here we derive a correlation between the magnitude of the SC anomalies, the residual contact resistance, the * Corresponding author.
resistivity, and the contact size for the HFSC in contact with tungsten. According to this correlation, AR is unlikely to be the dominant process for the observed SC anomalies. For more details see Refs. I-8, 9]. Our analysis is based on the conventional model for metallic contacts. A ballistic point contact of radius a has Sharvin's resistance [10] RSH A = 2RK/(akv) 2, kv being the Fermi wave number and RK = h/e 2 = 25.8 kf2. Fenton [11] derived a reflection coefficient ~ = ( m n v - raM)2~ (mHv + raM)2 ~ 1 for ballistic electrons due to a mismatch between the effective electron masses mnF and mM of the HF compound and the simple metal (used as the normal counterelectrode), respectively. This enhances Sharvin's resistance by the factor 1/(1 - ~ ) > > 1 . Deutscher and Nozi+res [12] reported recently that the quasiparticle mass enhancement is cancelled for those contacts, i.e. ~ 0. Scattering processes in the bulk material on either side of the interface add Maxwell's resistance R M A x ( T ) = p(T )/2a, p ( T ) being the T-dependent (average) electrical resistivity in the contact region. Thus, the total contact resistance R(T) "~ RsH~/(I -- ~) + RMAx(T) [13]. If one of the electrodes becomes SC, the AR hole current reduces RSnA by a factor of two. Simultaneously, ordinary scattering processes in the SC freeze out, leaving
0921-4526/96/$15.00 ;~] 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 5 8 5 - 4
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K. Gloos et al. / Physica B 218 (1996) 169 172
Maxwell's contribution from the normal conductor [14]. Hence, the SC anomalies contain information about AR as well a s RMA x. Since the H F compounds have rather large and T-dependent resistivities in the 1-100 ~tf~cm range, RMAX plays an important role in this scenario. Our point contacts between polycrystalline tungsten tips and the HFSC CeCu2Si2, UPt3, URuzSi2, UNizAI3, and UPdzA13 were prepared with a spear-anvil-type of setup [15]. Table 1 summarizes the relevant data of the samples. We used the A coefficient of their electrical resistivity P n v ( T ) = Po + A T 2 to determine the absolute value of the bulk resistivity. Figs. l(a)-(c) show typical spectra of low-R contacts between tungsten and CeCu2Si2, URuzSi2, and UNizAI3, (high-R contacts did not show SC anomalies but zero-bias maxima even in the normal state). Above Tc the differential resistance depends weakly on T and on U. Below Tc pronounced maxima occur at voltages 2elUI down to the BCS energy gap 2Ao = 3.5kBTc (small maxima even appear at some contacts with UPt3). We discuss now in detail the properties of the CeCu2Si2-W contact of Fig. l(a). Identifying the T-dependent part of the contact resistance R ( U = 0) above Tc (Fig. 2) with the T-dependent part of RMax(T) results in a contact radius of a = 2.5 lam. At the SC transition the resistance drops by 6R = 73 gf~, approaching Ro = R ( T ~ O ) = 51 m~q. Such a behaviour has previously been attributed to the AR hole current [1 5]. Our alternative explanation takes into account that RuAx vanishes below T c. The resistance drop indicates a local resistivity of4a6R = 74 laf~ cm at most, about five times larger than the bulk P0. It is still a quite reasonable value because bulk SC in CeCu2Si2 with Tc = 0.6 K was found for po as high as 60 la~ cm [16]. This contact clearly demonstrates that R~AX contributes considerably to the ob-
served SC anomalies, but it is difficult to separate the effects of AR. We investigated 54 different CeCu2Si2-W contacts. Their systematic dependence 8 R oc 1/a like Maxwell's resistance, shown in Fig. 3(a), strongly supports our above interpretation. The scattering of the data points can have various reasons: (i) Maxwell's formula is valid for a circular orifice while the real contacts can have quite different shapes. (ii) The high pressure exerted by the tungsten tip can change th e physical properties in the contact region, see e.g. Ref. [17]. For example, the A coefficient (used to calculate the radius) is directly related to the large effective H F mass, depending strongly on pressure. (iii) The pressure gradient introduces disorder that increases the local resistivity and/or reduces Tc. For CeCu2Si2 the average 4 a 3 R ~- 40 laf~ cm from Fig. 3(a) exceeds the bulk po by a factor 2.5. This implies an only moderate distortion of the contact area, which stays SC. Fig. 3(b) shows the correlation between contact radius and residual Ro, described by a strongly enhanced Sharvin resistance Ro ~ 5 0 0 , RSnA vc l / a 2. This would correspond to an excessively large Fermi wavelength ,ea2/~o/2RK ~ 3 nm of the H F compounds (or tungsten). Two more realistic explanations have been proposed for such a big offset, a badly conducting normal interface layer on the H F side [2] and strong ordinary reflection of ballistic electrons due to a mismatch of the effective electron masses [9]. The point-contact spectra itself can be used to discriminate between the two possibilities of a high ordinary reflection and a normal-conducting interface layer. Reflection causes a step-like voltage drop at the contact. The interface layer thermalizes the electrons flowing through it. They have lost part of their energy when they arrive at the HFSC, shifting the SC anomalies to higher voltages. For CeCuzSi2 the
Table 1 Properties of the heavy-fermion superconductors Compound
CeCu2Si2 UBe~3 UPt3 URu2Si2 UNizAI3 UPdzA13
Tc
A
Po
(K)
(g~ cm/K 2)
(~t~ cm)
0.67 0.90 0,50 1,4 1~2 1.7
11 Linear 1.3 0.18 0.25 0.23
16 130 1.2 11.4 2.9 7.6
m*/m
380 260 180 140 70 65
vv
1
li
(km/s)
(nm)
(lam)
4.5 6.5 6.8 l0 11 12
3.5 0.43 90 7.3 87 33
2.1 3.1 3.3 4.8 5.3 5.8
The bulk T~ (midpoint) and the elastic mean free path 1 is derived from the specific resistivity, Po being the extrapolated normal-state value (for UBe~3 it is the extrapolated specific resistivity at To). The inelastic mean free path is 1~>~hvv/kBT (T = 0.1 K) [18]. Effective masses m*/m and Fermi velocities vv of the first four compounds are from Ref. [19], the effective masses of the others are from Ref. [20]. Information on the A coefficient can be found in Refs. [16, 21 23].
171
K. Gloos et al. /Physica B 218 (1996) 169 172
CeCu2Si2
.
.
.
i
.
.
.
.
, 1
0.35 0.30 ,-.--, 0.25 o.20
.~
.
CeCuzSi,_ - W
0.135
.
.
.
.
.
,
.
.
.
.
f
•
1
0.15
/
/ 0.10
o.15 O.lO 0.05
.-----,- 6 0 0 " c " - ' - T -
iiilIZf .
'2.00~"--
. ,~t. -
°
0.0
.
.
0.5
7-R,,J !
.
1.0
0.0
0.5
0.05
1.0
"1"2(K: )
°
x.3
.
(_x-¢~
Fig. 2. (a)Contact resistance R(U =0) versus T 2 of the
0.118 0,1 ,----
0.116
~
~
_
-
-
_
,.----
0.II0
Fig. 1. Typical spectra R = d U / d l versus U and T of contacts between tungsten and (a) CeCu2Si2, (b) URu2Si,, and (c) UNi2AI3. additional broadening of the spectra is so small that it cannot be caused by the action of a normal interface layer alone. The other H F S C reveal qualitatively similar results (Fig. 4). Over two orders of magnitude the average 4 a S R agrees remarkably well with the bulk Po, while the residual Ro is strongly enhanced by more than two orders of
1
10 a (pro)
0,1
I
10
Fig. 3. (a) Resistance drop 6R versus radius a of contacts between tungsten and CeCu2Siz. The solid line corresponds to 4aSR = 40 I.tf~cm. (b) Residual contact resistance Ro versus radius a. Solid lines represent (bottom) RSHA and (top) a fit R 0 = 350 kf~nm2/a 2.
magnitude. Fenton's prediction fits Ro reasonably for large m * / m , the deviations of UPd2A13 and UNi2AI3 may be explained by a rather thick normal interface layer. The good agreement between 8R and RMAX supports our method to derive the radius. It also suggests that the small SC anomalies of contacts with well-conducting UPt3 represent some upper bound for the AR signal for all H F S C in contact with tungsten. Since the SC anomalies of UPt3 itself seem to contain a certain contribution from RMAX, its AR hole current can be much smaller than the upper limit estimated from 8R. Maxwell's resistance, on the other hand, shows that the contact region really becomes SC. Electrons incident from tungsten must hit this SC region. With the
172
K. Gloos et al. / Physica B 218 (1996) 169-172
! O0
Acknowledgements
10
We thank F. Steglich for stimulating discussions. Part of this work was supported by the SFB 252 Dar-
t
mstadt/Frankfurt/Mainz. ¢,q
I
t
~
o.1 °'~
Fenton Sharvin
1
10 0,, (~cm)
100 50
100
500
0.01
m*/m
Fig. 4. Contacts between tungsten and the heavy-fermion superconductors: (a) 4a6R versus P0. The solid line shows Maxwell's resistance. (b) a2Ro/2RK ( = Fermi wavelength squared) versus effective mass m*/m of the heavy-fermion compounds. Solid lines show Rsn A and Fenton's prediction, respectively. The UBe13data are from Ref. [8].
conventional AR process most of the reflected holes flow back even in the presence of additional scattering centers. Because of the small H F Fermi energy ev, deviations from the ideal retro-reflection angle can be large (Ao/ ev ~ 5 mrad). But they should be small enough for the holes to return through these large contacts. This large size may enable two electrons to pass the constriction simultaneously and form a 'heavy' Cooper pair without AR. To conclude, our experimental results are inconsistent with the "conventional" transformation of "light" electrons into "heavy" quasi-particles or "heavy" Cooper pairs. Interpreting these point-contact spectra in terms of the B T K theory for AR is not meaningful. Local heating may be responsible for the shape of the spectra. Since both the elastic (l) and the inelastic (li) mean free path of the H F compounds are small (Table 1), the diffusion length ~ i / ~ 0.1-0.5 ~tm (T = 0.1 K) is comparable to the contact radius. The maxima in the differential resistance then mark the voltage at which (part of) the contact region is locally heated above Tc.
References [1] A. Nowack et al., Phys. Rev. B 36 (1987) 2436. [2] A. Nowack et al., Z. Phys. B 88 (1992) 295; A. Nowack et al., Z. Phys. B 97 (1995) 77. [3] K. Hasselbach, J.R. Kirtley and P. Lejay, Phys. Rev. B 46 (1992) 5826. [4] G. Goll et al., Phys. Rev. Lett. 70 (1993) 2008. [5] Y. De Wilde et al., Phys. Rev. Lett. 72 (1994) 2278. [6] A.F. Andreev, Soy. Phys. JETP 19 (1964) 1228. [7] G.E. Blonder, M. Tinkham and T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515. [8] K. Gloos et al., to be published. [9] K. Gloos et al., Physica B 206 & 207 (1995) 282. [10] Yu. V. Sharvin, Sov. Phys. JETP 21 (1965) 655. [11] E.W. Fenton, Solid State Commun. 54 (1985) 709. [12] G. Deutscher and P. Nozi&es, Phys. Rev. B 50 (1994) 13557. [13] A. Wexler, Proc. Phys. Soc. 89 (1966) 927; I.O. Kulik, Sov, J. Low Temp. Phys. 18 (1992) 450. [14] Yu. G. Naidyuk and O.E. Kvitnitskaya, Soy. J. Low Temp, Phys. 17 (1991) 439. [15] K. Gloos and F. Martin, Z. Phys. B, to be published. [16] U. Rauchschwalbe, Physica B 147 (1987) 1. [17] K. Gloos et al., Physica B 206 & 207 (1995) 279. [18] D. Wohlleben et al., Europhys. Lett. 3 (1987) 107. [19] N. Grewe and F. Steglich, in: Handbook on the Physics and Chemistry of Rare Earths, edited by K.A. Gschneidner Jr. and L. Eyring (North-Holland, Amsterdam, 1991), Vol. 14, p. 343. [20] C. Geibel et al., Z. Phys. B 83 (1991) 305; C. Geibel et al., Z. Phys. B 84 (1991) 1. [21] J. Hessert, Diploma Thesis, TH Darmstadt (1992), unpublished. [22] Y. Dalichaouch, M.C. de Andrade and M.B. Maple, Phys. Rev. B 46 (1992) 8671. [23] C. Wassilew, PhD Thesis, TH Darmstadt (1994), unpublished.
ELSEVIER
Resistance scaling of point contacts between heavy-fermion superconductors and tungsten K. Gloos*, C. Geibel, R. Miiller-Reisener, C. Schank lnstitut J~r Festk6rperphysik, SFB 252, Technische Hochschule Darmstadt, Hochschulstr. 8, D-64289 Darmstadt, German),
Abstract We investigated point contacts between a tungsten tip and the heavy-fermion superconductors CeCuzSi z, UPt 3, URuzSi2, UNi2A13, and UPd2AI 3. A correlation between contact size, magnitude of the superconducting anomalies, and specific resistivity of the heavy-fermion superconductors suggests that these anomalies are mainly due to diffusive and thermal transport. The residual contact resistance can be described by a strongly enhanced Sharvin resistance, probably caused by both a badly conducting normal interface layer on the heavy-fermion side and strong ordinary reflection of ballistic electrons.
Point-contact spectroscopy has been used to study the superconducting (SC) state of the heavy-fermion (HF) compounds [-1-5]. These experiments can be roughly summarized as follows: (i) At low temperatures the voltage-dependent contact resistance follows approximately the temperature-dependent specific resistivity of the HF compound under investigation. (ii) The residual contact resistance is unusually high. (iii) The relative magnitude of the SC anomalies decreases with increasing residual contact resistance. These anomalies are interpreted by Andreev reflexion (AR), i.e. an electron incident on the normal-SC interface transforms into a Cooper pair and reflects a hole [6]. The BTK theory [7] is used to describe them. Goll et al. [4] reported that contacts with well-conducting UPt3 showed very small SC anomalies, of order 1% of the total resistance. UBel 3 represents the opposite limit. Its resistivity Pnr ~ 130 ~tf~cm near Tc implies an extremely small electronic mean free path. The SC anomalies of this compound seem to be dominated by thermal and diffusive transport [8]. Here we derive a correlation between the magnitude of the SC anomalies, the residual contact resistance, the * Corresponding author.
resistivity, and the contact size for the HFSC in contact with tungsten. According to this correlation, AR is unlikely to be the dominant process for the observed SC anomalies. For more details see Refs. I-8, 9]. Our analysis is based on the conventional model for metallic contacts. A ballistic point contact of radius a has Sharvin's resistance [10] RSH A = 2RK/(akv) 2, kv being the Fermi wave number and RK = h/e 2 = 25.8 kf2. Fenton [11] derived a reflection coefficient ~ = ( m n v - raM)2~ (mHv + raM)2 ~ 1 for ballistic electrons due to a mismatch between the effective electron masses mnF and mM of the HF compound and the simple metal (used as the normal counterelectrode), respectively. This enhances Sharvin's resistance by the factor 1/(1 - ~ ) > > 1 . Deutscher and Nozi+res [12] reported recently that the quasiparticle mass enhancement is cancelled for those contacts, i.e. ~ 0. Scattering processes in the bulk material on either side of the interface add Maxwell's resistance R M A x ( T ) = p(T )/2a, p ( T ) being the T-dependent (average) electrical resistivity in the contact region. Thus, the total contact resistance R(T) "~ RsH~/(I -- ~) + RMAx(T) [13]. If one of the electrodes becomes SC, the AR hole current reduces RSnA by a factor of two. Simultaneously, ordinary scattering processes in the SC freeze out, leaving
0921-4526/96/$15.00 ;~] 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 5 8 5 - 4
170
K. Gloos et al. / Physica B 218 (1996) 169 172
Maxwell's contribution from the normal conductor [14]. Hence, the SC anomalies contain information about AR as well a s RMA x. Since the H F compounds have rather large and T-dependent resistivities in the 1-100 ~tf~cm range, RMAX plays an important role in this scenario. Our point contacts between polycrystalline tungsten tips and the HFSC CeCu2Si2, UPt3, URuzSi2, UNizAI3, and UPdzA13 were prepared with a spear-anvil-type of setup [15]. Table 1 summarizes the relevant data of the samples. We used the A coefficient of their electrical resistivity P n v ( T ) = Po + A T 2 to determine the absolute value of the bulk resistivity. Figs. l(a)-(c) show typical spectra of low-R contacts between tungsten and CeCu2Si2, URuzSi2, and UNizAI3, (high-R contacts did not show SC anomalies but zero-bias maxima even in the normal state). Above Tc the differential resistance depends weakly on T and on U. Below Tc pronounced maxima occur at voltages 2elUI down to the BCS energy gap 2Ao = 3.5kBTc (small maxima even appear at some contacts with UPt3). We discuss now in detail the properties of the CeCu2Si2-W contact of Fig. l(a). Identifying the T-dependent part of the contact resistance R ( U = 0) above Tc (Fig. 2) with the T-dependent part of RMax(T) results in a contact radius of a = 2.5 lam. At the SC transition the resistance drops by 6R = 73 gf~, approaching Ro = R ( T ~ O ) = 51 m~q. Such a behaviour has previously been attributed to the AR hole current [1 5]. Our alternative explanation takes into account that RuAx vanishes below T c. The resistance drop indicates a local resistivity of4a6R = 74 laf~ cm at most, about five times larger than the bulk P0. It is still a quite reasonable value because bulk SC in CeCu2Si2 with Tc = 0.6 K was found for po as high as 60 la~ cm [16]. This contact clearly demonstrates that R~AX contributes considerably to the ob-
served SC anomalies, but it is difficult to separate the effects of AR. We investigated 54 different CeCu2Si2-W contacts. Their systematic dependence 8 R oc 1/a like Maxwell's resistance, shown in Fig. 3(a), strongly supports our above interpretation. The scattering of the data points can have various reasons: (i) Maxwell's formula is valid for a circular orifice while the real contacts can have quite different shapes. (ii) The high pressure exerted by the tungsten tip can change th e physical properties in the contact region, see e.g. Ref. [17]. For example, the A coefficient (used to calculate the radius) is directly related to the large effective H F mass, depending strongly on pressure. (iii) The pressure gradient introduces disorder that increases the local resistivity and/or reduces Tc. For CeCu2Si2 the average 4 a 3 R ~- 40 laf~ cm from Fig. 3(a) exceeds the bulk po by a factor 2.5. This implies an only moderate distortion of the contact area, which stays SC. Fig. 3(b) shows the correlation between contact radius and residual Ro, described by a strongly enhanced Sharvin resistance Ro ~ 5 0 0 , RSnA vc l / a 2. This would correspond to an excessively large Fermi wavelength ,ea2/~o/2RK ~ 3 nm of the H F compounds (or tungsten). Two more realistic explanations have been proposed for such a big offset, a badly conducting normal interface layer on the H F side [2] and strong ordinary reflection of ballistic electrons due to a mismatch of the effective electron masses [9]. The point-contact spectra itself can be used to discriminate between the two possibilities of a high ordinary reflection and a normal-conducting interface layer. Reflection causes a step-like voltage drop at the contact. The interface layer thermalizes the electrons flowing through it. They have lost part of their energy when they arrive at the HFSC, shifting the SC anomalies to higher voltages. For CeCuzSi2 the
Table 1 Properties of the heavy-fermion superconductors Compound
CeCu2Si2 UBe~3 UPt3 URu2Si2 UNizAI3 UPdzA13
Tc
A
Po
(K)
(g~ cm/K 2)
(~t~ cm)
0.67 0.90 0,50 1,4 1~2 1.7
11 Linear 1.3 0.18 0.25 0.23
16 130 1.2 11.4 2.9 7.6
m*/m
380 260 180 140 70 65
vv
1
li
(km/s)
(nm)
(lam)
4.5 6.5 6.8 l0 11 12
3.5 0.43 90 7.3 87 33
2.1 3.1 3.3 4.8 5.3 5.8
The bulk T~ (midpoint) and the elastic mean free path 1 is derived from the specific resistivity, Po being the extrapolated normal-state value (for UBe~3 it is the extrapolated specific resistivity at To). The inelastic mean free path is 1~>~hvv/kBT (T = 0.1 K) [18]. Effective masses m*/m and Fermi velocities vv of the first four compounds are from Ref. [19], the effective masses of the others are from Ref. [20]. Information on the A coefficient can be found in Refs. [16, 21 23].
171
K. Gloos et al. /Physica B 218 (1996) 169 172
CeCu2Si2
.
.
.
i
.
.
.
.
, 1
0.35 0.30 ,-.--, 0.25 o.20
.~
.
CeCuzSi,_ - W
0.135
.
.
.
.
.
,
.
.
.
.
f
•
1
0.15
/
/ 0.10
o.15 O.lO 0.05
.-----,- 6 0 0 " c " - ' - T -
iiilIZf .
'2.00~"--
. ,~t. -
°
0.0
.
.
0.5
7-R,,J !
.
1.0
0.0
0.5
0.05
1.0
"1"2(K: )
°
x.3
.
(_x-¢~
Fig. 2. (a)Contact resistance R(U =0) versus T 2 of the
0.118 0,1 ,----
0.116
~
~
_
-
-
_
,.----
0.II0
Fig. 1. Typical spectra R = d U / d l versus U and T of contacts between tungsten and (a) CeCu2Si2, (b) URu2Si,, and (c) UNi2AI3. additional broadening of the spectra is so small that it cannot be caused by the action of a normal interface layer alone. The other H F S C reveal qualitatively similar results (Fig. 4). Over two orders of magnitude the average 4 a S R agrees remarkably well with the bulk Po, while the residual Ro is strongly enhanced by more than two orders of
1
10 a (pro)
0,1
I
10
Fig. 3. (a) Resistance drop 6R versus radius a of contacts between tungsten and CeCu2Siz. The solid line corresponds to 4aSR = 40 I.tf~cm. (b) Residual contact resistance Ro versus radius a. Solid lines represent (bottom) RSHA and (top) a fit R 0 = 350 kf~nm2/a 2.
magnitude. Fenton's prediction fits Ro reasonably for large m * / m , the deviations of UPd2A13 and UNi2AI3 may be explained by a rather thick normal interface layer. The good agreement between 8R and RMAX supports our method to derive the radius. It also suggests that the small SC anomalies of contacts with well-conducting UPt3 represent some upper bound for the AR signal for all H F S C in contact with tungsten. Since the SC anomalies of UPt3 itself seem to contain a certain contribution from RMAX, its AR hole current can be much smaller than the upper limit estimated from 8R. Maxwell's resistance, on the other hand, shows that the contact region really becomes SC. Electrons incident from tungsten must hit this SC region. With the
172
K. Gloos et al. / Physica B 218 (1996) 169-172
! O0
Acknowledgements
10
We thank F. Steglich for stimulating discussions. Part of this work was supported by the SFB 252 Dar-
t
mstadt/Frankfurt/Mainz. ¢,q
I
t
~
o.1 °'~
Fenton Sharvin
1
10 0,, (~cm)
100 50
100
500
0.01
m*/m
Fig. 4. Contacts between tungsten and the heavy-fermion superconductors: (a) 4a6R versus P0. The solid line shows Maxwell's resistance. (b) a2Ro/2RK ( = Fermi wavelength squared) versus effective mass m*/m of the heavy-fermion compounds. Solid lines show Rsn A and Fenton's prediction, respectively. The UBe13data are from Ref. [8].
conventional AR process most of the reflected holes flow back even in the presence of additional scattering centers. Because of the small H F Fermi energy ev, deviations from the ideal retro-reflection angle can be large (Ao/ ev ~ 5 mrad). But they should be small enough for the holes to return through these large contacts. This large size may enable two electrons to pass the constriction simultaneously and form a 'heavy' Cooper pair without AR. To conclude, our experimental results are inconsistent with the "conventional" transformation of "light" electrons into "heavy" quasi-particles or "heavy" Cooper pairs. Interpreting these point-contact spectra in terms of the B T K theory for AR is not meaningful. Local heating may be responsible for the shape of the spectra. Since both the elastic (l) and the inelastic (li) mean free path of the H F compounds are small (Table 1), the diffusion length ~ i / ~ 0.1-0.5 ~tm (T = 0.1 K) is comparable to the contact radius. The maxima in the differential resistance then mark the voltage at which (part of) the contact region is locally heated above Tc.
References [1] A. Nowack et al., Phys. Rev. B 36 (1987) 2436. [2] A. Nowack et al., Z. Phys. B 88 (1992) 295; A. Nowack et al., Z. Phys. B 97 (1995) 77. [3] K. Hasselbach, J.R. Kirtley and P. Lejay, Phys. Rev. B 46 (1992) 5826. [4] G. Goll et al., Phys. Rev. Lett. 70 (1993) 2008. [5] Y. De Wilde et al., Phys. Rev. Lett. 72 (1994) 2278. [6] A.F. Andreev, Soy. Phys. JETP 19 (1964) 1228. [7] G.E. Blonder, M. Tinkham and T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515. [8] K. Gloos et al., to be published. [9] K. Gloos et al., Physica B 206 & 207 (1995) 282. [10] Yu. V. Sharvin, Sov. Phys. JETP 21 (1965) 655. [11] E.W. Fenton, Solid State Commun. 54 (1985) 709. [12] G. Deutscher and P. Nozi&es, Phys. Rev. B 50 (1994) 13557. [13] A. Wexler, Proc. Phys. Soc. 89 (1966) 927; I.O. Kulik, Sov, J. Low Temp. Phys. 18 (1992) 450. [14] Yu. G. Naidyuk and O.E. Kvitnitskaya, Soy. J. Low Temp, Phys. 17 (1991) 439. [15] K. Gloos and F. Martin, Z. Phys. B, to be published. [16] U. Rauchschwalbe, Physica B 147 (1987) 1. [17] K. Gloos et al., Physica B 206 & 207 (1995) 279. [18] D. Wohlleben et al., Europhys. Lett. 3 (1987) 107. [19] N. Grewe and F. Steglich, in: Handbook on the Physics and Chemistry of Rare Earths, edited by K.A. Gschneidner Jr. and L. Eyring (North-Holland, Amsterdam, 1991), Vol. 14, p. 343. [20] C. Geibel et al., Z. Phys. B 83 (1991) 305; C. Geibel et al., Z. Phys. B 84 (1991) 1. [21] J. Hessert, Diploma Thesis, TH Darmstadt (1992), unpublished. [22] Y. Dalichaouch, M.C. de Andrade and M.B. Maple, Phys. Rev. B 46 (1992) 8671. [23] C. Wassilew, PhD Thesis, TH Darmstadt (1994), unpublished.