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Pairing symmetry of Tl-2201 from measurements of temperature and magnetic field dependencies of the anisotropic penetration depth
Physica C 335 Ž2000. 134–138 www.elsevier.nlrlocaterphysc
Pairing symmetry of Tl-2201 from measurements of temperature and magnetic field dependencies of the anisotropic penetration depth Y.T. Wang, A.M. Hermann ) Department of Physics, UniÕersity of Colorado, SuperconductiÕity Laboratories Campus, Box 390, Boulder, CO 80309-0390, USA
Abstract We report a complete set of measurements of the anisotropic penetration depth of single crystals of Tl-2201 at varying doping levels Žand corresponding critical temperatures.. Using low field Žbelow Hc1 . dc magnetization data, we solve the London equation for the anisotropic specimen geometry and obtain both l ab and l c as a function of temperature and magnetic field. Values of the penetration depth were calculated by a self-consistent condition, and we have found that both temperature and field dependences are linear for samples with 20 K O Tc O 70 K. These dependencies give strong evidence for d-wave pairing with nodes in the superconducting gap, a not-surprising finding for this simplest of HTSC tetragonal cuprates. The data for optimally doped crystals with Tc s 90 K show, however, a quadratic temperature dependence. This latter feature is difficult to reconcile with known theories of s- or d-wave pairing, and this issue poses a central question yet to be answered. q 2000 Elsevier Science B.V. All rights reserved. Keywords: TI-2201; Magnetic field dependencies; Anisotropic penetration depth
1. Introduction The nature of the pairing symmetry in high-temperature superconducting materials is very important, particularly the issue of s-wave vs. d-wave. Measurement of the temperature andror magnetic field dependent penetration depth at low temperature is a useful probe of the energy gap at the Fermi surface and of the superfluid electrodynamics to reveal this important symmetry. For the temperature depen-
)
Corresponding author. Tel.: q1-303-492-2661; fax: q1-303278-8664. E-mail address: [email protected] ŽA.M. Hermann..
dence, D l, defined as lŽT . y lŽ0., was calculated by Mattis and Bardeen w1x and Turneaure et al. w2x in the case of local limit, i.e. the superconducting coherence length much smaller than the London penetration depth. For s-wave ŽBCS. pairing states excited by a finite energy D l is given by
D lŽ T .
lŽ 0.
f 3.33
Tc
ž / T
1r2
yD
exp
ž / T
,
Ž 1.
where D is the energy gap at zero temperature. Therefore, the exponential dependent penetration depth is the one of the properties in s-wave superconductor. For d-wave symmetry with nodes on the
0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 0 0 . 0 0 1 5 8 - 1
Y.T. Wang, A.M. Hermannr Physica C 335 (2000) 134–138
Fermi surface, D lŽT . is calculated to be power law of temperature, T p . For instance, Annett et al. w3x have shown that all the possible non-s-wave pairing states with tetragonal or orthorhombic symmetry and a Fermi surface with spherical or cylindrical topology have line nodes in the gap that result in a linear temperature dependence in the penetration depth, D lŽT . ; T. However, an impurity scattering can change the temperature dependence from T to T 2 w4,5x. For the dependence of the penetration depth on magnetic field in a d-wave superconductor, according to Yip and Sauls w6x and Xu et al. w7x, the variation of l as a function of magnetic field H is linear in H even at T s 0 K: D lŽ H ,T .
lŽ H s 0.
fa ŽT .
ž
H H0 Ž T .
/
,
Ž 2.
where D lŽ H,T . s lŽ H,T . y lŽ0,T ., H0 ŽT . is a characteristic field of the order of the thermodynamic critical field Hc ŽT ., and a ŽT . is a temperature-dependent coefficient. At T s 0 K, a s 0.6667 and 0.4714 for the field parallel and perpendicular to the node direction, respectively. This result is very different from that for an s-wave superconductor w11x, for which D l changes with H 2 in the Meissner state: D lŽ H ,T .
lŽ H s 0.
fb ŽT .
ž
H H0 Ž T .
2
/
.
Ž 3.
At low temperatures, nonlinear corrections to the London equations cause the quadratic field dependence since the supercurrent density does not scale exactly with the velocity of the superfluid. The main idea is that the supercurrent causes pair-breaking and then decreases the superfluid density as a function of increasing magnetic field, which is proportional to the superfluid velocity. In recent years, measurements of l have provided strong evidence for unconventional pairing in the high Tc superconductors for which there are line nodes in the energy gap. Hardy et al. w8x used a superconducting resonator technique to measure the change of penetration depth of YBCO single crystal and showed a strong linear term extending from 3 to
135
25 K, which is highly suggestive of line nodes in a clean d-wave superconductor. Sonier et al. w9x, using muon-spin-rotation Ž m SR. technique, displayed an unconventional pairing symmetry by showing a linear dependence of 1rl a b in temperature for YBCO single crystal. Maeda et al. w10,12x measured the change of penetration depth of YBCO-123, BSCCO2212 and TBCCO-2212 and obtained a linear increase of l with field H at low temperature with an almost temperature-independent gradient to reveal evidence that the order parameter of the double layered cuprates has nodes in k-space. Most studies have concentrated on YBCO w8,13,14x and only a few have examined highly anisotropic materials such as BSCCO w15–17x or TBCCO w18–21x, the latter being especially important by virtue of its tetragonal structure. All of these samples showed either linear or quadratic dependence of temperature. Sonier et al. w22x showed linear dependence of both temperature and magnetic field in the same sample of YBCO single crystal. In our report, Tl-2201 was chosen because of its simple structure with tetragonal crystal symmetry and a single CuO 2 plane per formula unit. We have studied Tl-2201 at low temperatures and low fields to reveal its near ground state nature, and we chose direct dc-magnetization measurements in the Meissner state. Both the temperature and magnetic field dependence of the penetration depth were calculated for the same Tl-2201 single crystal with different transition temperatures adjusted by oxygen annealing to investigate systematically its pairing symmetry.
2. Experimental Tl-2201 single crystals were grown by the twostep self-flux technique. Details of crystal growth were described by Duan et al. w23x. The as-grown crystals have Tc of ; 90 K, which can be lowered by annealing. Crystals with Tc in the 70 K range were produced by placing as grown crystals in a furnace heated to ; 3308C in air with subsequent immediate furnace shut down. The crystals were heated to ; 3508C again for 30 min to reduce Tc to ; 44 K and then were heated at 4508C for 4 h in air and followed by fast cooling to reach Tc of about 20
Y.T. Wang, A.M. Hermannr Physica C 335 (2000) 134–138
136
K. Most of the data reported here were taken on one crystal, but many features of the data were verified on other crystals. All the magnetic measurements were performed using a SQUID magnetometer. The sample was mounted in a sample rod, which was rotated for the applied field perpendicular or parallel to the c-axis. In order to study the penetration depth at low temperature and low magnetic field, all the dc-magnetization measurements were conducted in the Meissner state in which the applied field was below Hc1. The lower critical field was estimated from the first point of the initial magnetization curve, which deviated from linearity in H w27x. Corresponding penetration depths were calculated by solving the London equation in the Meissner state, H - Hc1. In the case of H I c , the magnetic moment m was solved to give
ms
I1
BaV 4p Ž 1 y N .
R
ž / ž / l R
I0
l
2l y1 ,
Ž 5.
R
where N is the demagnetizing factor, V is the volume of sample, I0 and I1 are modified Bessel functions. In this configuration, the demagnetizing factor is so large that it cannot be ignored. The initial slope of the magnetization curve, M Ž H ., was measured to calculate N in Eq. Ž5. above as 1
Ha
4p
M
ž /ž /
Ns1q
.
Ž 6.
The penetration depth, l a b can then be obtained by solving this transcendental equation using the experimental data of mŽT .. When the applied field is perpendicular to c-axis, we obtain 1y
mŽ T .
s
I2
a1
m0
tanh
a1 I2
q
I1
a2
tanh
a2 I1
,
Ž 8.
where m0 s y
a1 s
BaV
(ž
4p
, 2
p 2
I1
/
q1 ,
a2 s
(ž
2
p 2
I2
/
q1 ,
and I1 s
2 lab d
,
I2 s
2 lc L
.
In this field orientation, the demagnetizing factor N is close to zero. l a b was determined from measurements with the H parallel to the c-axis as described above. Using these values of l a b and solving Eq. Ž8. with experimental mŽT . data for the field perpendicular to the c-axis, l c was calculated.
3. Results and discussion The theoretical predictions and our experimental results are summarized in Table 1. The temperature dependence of penetration depth with field parallel to c-axis, D lŽT ., displayed a strong linear term at low T with slopes of 14, 19, 24.5 ArK for sample with Tc s 70, 44 and 20 K, respectively, while in the case of Tc s 90 K, l ab is seen to obey a T 2 behavior at low temperature as shown in Fig. 1. The l a b data for Tc O 70 K strongly suggest an unconventional pairing state. Similar results were observed in the case of field perpendicular to c-axis as shown in Table 1 and Fig. 2. The slopes of D l c as a function of T are 12, 9 and 7 ArK for Tc s 70, 44 and 20 K, respectively. The slope increases as Tc decreases in H I c case, while it decreases in H I a b case. The slopes of D l a b are in the same range of that in YBCO system, which span from 4.8 to 20 ArK w24x, and similar to the value of 13 ArK in the Tl-2201 single crystal of Tc s 78 K measured by Broun et al. w19x, who also found a linear temperature dependence. The values of l a b Ž0. derived from our data are 168, 172, 212 and 332 nm and the corresponding values of l c Ž0. are 2112, 2555, 4385 and 8408 nm for Tc s 90, 70, 44 and 20 K, respectively. As Tc is reduced by raising the carrier concentration, 1rl2 Ž0.4 drops, a behavior which has been widely discussed and is known as the Uemura relation w25,26x. The value of l a b Ž0. for Tc s 70 K agrees with the data from Broun et al. w19x, for which a Tl-2201 single crystal with Tc of 78 K was measured by surface impedance measurements and l a b Ž0. s 165 nm was obtained.
Y.T. Wang, A.M. Hermannr Physica C 335 (2000) 134–138
137
Table 1 Summary of theoretical predictions and our experimental results
lŽT .
Theoretical predictions
Experimental results
For s-wave: D lŽT .rlŽ0. f 3.33ŽTcrT .1r 2 expŽyDrT .
Not observed
For d x 2 – y 2-wave: D lŽT . ; T ; T 2 with impurity scattering
D l ; T observed in l a b and l c for sample with Tc s 20, 44 and 70 K Tc s 20 K Tc s 44 K Tc s 70 K Slope ŽArK. 24.5 19 14 l a b ŽT s 0. Žnm. 332 212 172 l c ŽT . Slope ŽArK. 7 9 12 l c ŽT s 0. Žnm. 8408 4385 2555 Dl ; T 2 observed in l a b and l c for sample with Tc s 90 K
l a b ŽT .
lŽ H . For s-wave: DlŽT .rlŽ0. f b ŽT .Ž HrH0 ŽT .. 2 For d x 2 – y 2-wave: D lŽT .rlŽ0. f a ŽT .Ž HrH0 ŽT ..
Not observed Observed in both l a b and l c for sample with all Tc ’s Ž20, 44, 70 and 90 K. Tc s 20 K Tc s 44 K l a b Ž H . Slope ŽArG. 6 7.2 l c Ž H . Slope ŽArG. 6 8
Tc s 70 K 9 11
Tc s 90 K 9.5 12
For the field dependence of the penetration depths, in both field orientations, H I c and H I a b , D lŽ H .’s are linearly dependent on the applied field for the sample at all Tc ’s. The slopes in the H I c case are 6, 7.2, 9 and 9.5 ArG for sample of Tc s 20, 44, 70 and 90 K, while the corresponding values are 6, 8, 11 and 12 ArG in the case of H I a b . The slopes increase as Tc increases in both cases and are much larger than those in the YBCO system, which are
about 3–3.5 ArGw10,12x. In order to rule out the anisotropic s-wave possibility, a wide range of energy gap, D, in Eq. Ž1. from 0.01 to 1000 was selected to fit our data, but no converging fit was found Ža reasonable value of D is 1.76Tc w28x.. Both temperature and field dependencies of in-plane and out-of-plane penetration depths suggest a d-wave symmetry.
Fig. 1. Temperature dependence of the in-plane penetration depth of the sample with Tc s90 K.
Fig. 2. Temperature dependence of the out-of-plane penetration depth of the sample with Tc s90 K.
138
Y.T. Wang, A.M. Hermannr Physica C 335 (2000) 134–138
References w1x D.C. Mattis, J. Bardeen, Phys. Rev. 111 Ž1958. 412. w2x J.P. Turneaure, J. Halbritter, H.A. Schwettman, J. Supercond. 4 Ž1991. 341. w3x J.F. Annett, N. Goldenfeld, S.R. Renn, Phys. Rev. B 43 Ž1991. 2778. w4x F. Gross, B.S. Chandrasekhar, D. EinZel, K. Andres, P.J. Hirschfeld, H.R. Ott, J. Beuers, Z. Fisk, J.L. Smith, Z. Phys. B 64 Ž1986. 175. w5x M. Prohammer, J.P. Carbotte, Phys. Rev. B 43 Ž1991. 5370. w6x S.K. Yip, J. Sauls, Phys. Rev. Lett. 69 Ž1992. 2264. w7x D. Xu, S.K. Yip, J.A. Sauls, Phys. Rev. B 51 Ž1995. 16233. w8x W.N. Hardy et al., Phys. Rev. Lett. 70 Ž1993. 3999. w9x J.E. Sonier et al., Phys. Rev. Lett. 72 Ž1994. 744. w10x A. Maeda et al., J. Phys. Soc. Jpn. 65 Ž1996. 3638. w11x J.E. Sonier et al., Phys. Rev. B 55 Ž1997. 11789. w12x A. Maeda et al., Physica C 263 Ž1996. 438. w13x J.F. Annet, N. Goldenfeld, S.R. Renn, D.M. Ginsberg ŽEd.., Physical Properties of High Temperature Superconductors World Scientific, Singapore, 1990.
w14x A. Porch, Physica C 214 Ž1993. 350. w15x J.R. Cooper, L. Forro, B. Keszei, Nature ŽLondon. 343 Ž1990. 444. w16x A. Maeda et al., Phys. Rev. B 46 Ž1992. 14234. w17x A. Maeda et al., Phys. Rev. Lett. 74 Ž1995. 1202. w18x L.J. Chen, J.T. Lue, Solid State Commun. 105 Ž1998. 155. w19x D.M. Broun, D.C. Morgan, R.J. Ormeno, S.F. Lee, A.W. Tyler, A.P. Mackenzie, J.R. Waldram, Phys. Rev. B 56 Ž1997. R11443. w20x G. Brandstatter, F.M. Sauerzopf, H.W. Weber, F. Ladenberger, E. Schwarzmann, Physica C 235–240 Ž1994. 1845. w21x R. Thompson, J.G. Ossandon, R.S. Yang, M. Paranthaman, J. Brynestad, Phys. Rev. B 48 Ž1993. 14031. w22x J.E. Sonier et al., Hyperfine Interact. 105 Ž1997. 161. w23x H.M. Duan, T.S. Kaplan, B. Dlugosch, A.M. Hermann, Physica C 203 Ž1992. 257. w24x C. Panagopoulos, J.R. Cooper, T. Xiang, Phys. Rev. B 57 Ž1998. 13422. w25x Y.J. Uemura et al., Phys. Rev. Lett. 66 Ž1991. 2665. w26x J.L. Tallon et al., Phys. Rev. Lett. 74 Ž1995. 1008. w27x A. Umezawa et al., Phys. Rev. B 38 Ž1988. 2843. w28x B. Muhlschlegel, Z. Phys. 155 Ž1959. 313.
Pairing symmetry of Tl-2201 from measurements of temperature and magnetic field dependencies of the anisotropic penetration depth Y.T. Wang, A.M. Hermann ) Department of Physics, UniÕersity of Colorado, SuperconductiÕity Laboratories Campus, Box 390, Boulder, CO 80309-0390, USA
Abstract We report a complete set of measurements of the anisotropic penetration depth of single crystals of Tl-2201 at varying doping levels Žand corresponding critical temperatures.. Using low field Žbelow Hc1 . dc magnetization data, we solve the London equation for the anisotropic specimen geometry and obtain both l ab and l c as a function of temperature and magnetic field. Values of the penetration depth were calculated by a self-consistent condition, and we have found that both temperature and field dependences are linear for samples with 20 K O Tc O 70 K. These dependencies give strong evidence for d-wave pairing with nodes in the superconducting gap, a not-surprising finding for this simplest of HTSC tetragonal cuprates. The data for optimally doped crystals with Tc s 90 K show, however, a quadratic temperature dependence. This latter feature is difficult to reconcile with known theories of s- or d-wave pairing, and this issue poses a central question yet to be answered. q 2000 Elsevier Science B.V. All rights reserved. Keywords: TI-2201; Magnetic field dependencies; Anisotropic penetration depth
1. Introduction The nature of the pairing symmetry in high-temperature superconducting materials is very important, particularly the issue of s-wave vs. d-wave. Measurement of the temperature andror magnetic field dependent penetration depth at low temperature is a useful probe of the energy gap at the Fermi surface and of the superfluid electrodynamics to reveal this important symmetry. For the temperature depen-
)
Corresponding author. Tel.: q1-303-492-2661; fax: q1-303278-8664. E-mail address: [email protected] ŽA.M. Hermann..
dence, D l, defined as lŽT . y lŽ0., was calculated by Mattis and Bardeen w1x and Turneaure et al. w2x in the case of local limit, i.e. the superconducting coherence length much smaller than the London penetration depth. For s-wave ŽBCS. pairing states excited by a finite energy D l is given by
D lŽ T .
lŽ 0.
f 3.33
Tc
ž / T
1r2
yD
exp
ž / T
,
Ž 1.
where D is the energy gap at zero temperature. Therefore, the exponential dependent penetration depth is the one of the properties in s-wave superconductor. For d-wave symmetry with nodes on the
0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 0 0 . 0 0 1 5 8 - 1
Y.T. Wang, A.M. Hermannr Physica C 335 (2000) 134–138
Fermi surface, D lŽT . is calculated to be power law of temperature, T p . For instance, Annett et al. w3x have shown that all the possible non-s-wave pairing states with tetragonal or orthorhombic symmetry and a Fermi surface with spherical or cylindrical topology have line nodes in the gap that result in a linear temperature dependence in the penetration depth, D lŽT . ; T. However, an impurity scattering can change the temperature dependence from T to T 2 w4,5x. For the dependence of the penetration depth on magnetic field in a d-wave superconductor, according to Yip and Sauls w6x and Xu et al. w7x, the variation of l as a function of magnetic field H is linear in H even at T s 0 K: D lŽ H ,T .
lŽ H s 0.
fa ŽT .
ž
H H0 Ž T .
/
,
Ž 2.
where D lŽ H,T . s lŽ H,T . y lŽ0,T ., H0 ŽT . is a characteristic field of the order of the thermodynamic critical field Hc ŽT ., and a ŽT . is a temperature-dependent coefficient. At T s 0 K, a s 0.6667 and 0.4714 for the field parallel and perpendicular to the node direction, respectively. This result is very different from that for an s-wave superconductor w11x, for which D l changes with H 2 in the Meissner state: D lŽ H ,T .
lŽ H s 0.
fb ŽT .
ž
H H0 Ž T .
2
/
.
Ž 3.
At low temperatures, nonlinear corrections to the London equations cause the quadratic field dependence since the supercurrent density does not scale exactly with the velocity of the superfluid. The main idea is that the supercurrent causes pair-breaking and then decreases the superfluid density as a function of increasing magnetic field, which is proportional to the superfluid velocity. In recent years, measurements of l have provided strong evidence for unconventional pairing in the high Tc superconductors for which there are line nodes in the energy gap. Hardy et al. w8x used a superconducting resonator technique to measure the change of penetration depth of YBCO single crystal and showed a strong linear term extending from 3 to
135
25 K, which is highly suggestive of line nodes in a clean d-wave superconductor. Sonier et al. w9x, using muon-spin-rotation Ž m SR. technique, displayed an unconventional pairing symmetry by showing a linear dependence of 1rl a b in temperature for YBCO single crystal. Maeda et al. w10,12x measured the change of penetration depth of YBCO-123, BSCCO2212 and TBCCO-2212 and obtained a linear increase of l with field H at low temperature with an almost temperature-independent gradient to reveal evidence that the order parameter of the double layered cuprates has nodes in k-space. Most studies have concentrated on YBCO w8,13,14x and only a few have examined highly anisotropic materials such as BSCCO w15–17x or TBCCO w18–21x, the latter being especially important by virtue of its tetragonal structure. All of these samples showed either linear or quadratic dependence of temperature. Sonier et al. w22x showed linear dependence of both temperature and magnetic field in the same sample of YBCO single crystal. In our report, Tl-2201 was chosen because of its simple structure with tetragonal crystal symmetry and a single CuO 2 plane per formula unit. We have studied Tl-2201 at low temperatures and low fields to reveal its near ground state nature, and we chose direct dc-magnetization measurements in the Meissner state. Both the temperature and magnetic field dependence of the penetration depth were calculated for the same Tl-2201 single crystal with different transition temperatures adjusted by oxygen annealing to investigate systematically its pairing symmetry.
2. Experimental Tl-2201 single crystals were grown by the twostep self-flux technique. Details of crystal growth were described by Duan et al. w23x. The as-grown crystals have Tc of ; 90 K, which can be lowered by annealing. Crystals with Tc in the 70 K range were produced by placing as grown crystals in a furnace heated to ; 3308C in air with subsequent immediate furnace shut down. The crystals were heated to ; 3508C again for 30 min to reduce Tc to ; 44 K and then were heated at 4508C for 4 h in air and followed by fast cooling to reach Tc of about 20
Y.T. Wang, A.M. Hermannr Physica C 335 (2000) 134–138
136
K. Most of the data reported here were taken on one crystal, but many features of the data were verified on other crystals. All the magnetic measurements were performed using a SQUID magnetometer. The sample was mounted in a sample rod, which was rotated for the applied field perpendicular or parallel to the c-axis. In order to study the penetration depth at low temperature and low magnetic field, all the dc-magnetization measurements were conducted in the Meissner state in which the applied field was below Hc1. The lower critical field was estimated from the first point of the initial magnetization curve, which deviated from linearity in H w27x. Corresponding penetration depths were calculated by solving the London equation in the Meissner state, H - Hc1. In the case of H I c , the magnetic moment m was solved to give
ms
I1
BaV 4p Ž 1 y N .
R
ž / ž / l R
I0
l
2l y1 ,
Ž 5.
R
where N is the demagnetizing factor, V is the volume of sample, I0 and I1 are modified Bessel functions. In this configuration, the demagnetizing factor is so large that it cannot be ignored. The initial slope of the magnetization curve, M Ž H ., was measured to calculate N in Eq. Ž5. above as 1
Ha
4p
M
ž /ž /
Ns1q
.
Ž 6.
The penetration depth, l a b can then be obtained by solving this transcendental equation using the experimental data of mŽT .. When the applied field is perpendicular to c-axis, we obtain 1y
mŽ T .
s
I2
a1
m0
tanh
a1 I2
q
I1
a2
tanh
a2 I1
,
Ž 8.
where m0 s y
a1 s
BaV
(ž
4p
, 2
p 2
I1
/
q1 ,
a2 s
(ž
2
p 2
I2
/
q1 ,
and I1 s
2 lab d
,
I2 s
2 lc L
.
In this field orientation, the demagnetizing factor N is close to zero. l a b was determined from measurements with the H parallel to the c-axis as described above. Using these values of l a b and solving Eq. Ž8. with experimental mŽT . data for the field perpendicular to the c-axis, l c was calculated.
3. Results and discussion The theoretical predictions and our experimental results are summarized in Table 1. The temperature dependence of penetration depth with field parallel to c-axis, D lŽT ., displayed a strong linear term at low T with slopes of 14, 19, 24.5 ArK for sample with Tc s 70, 44 and 20 K, respectively, while in the case of Tc s 90 K, l ab is seen to obey a T 2 behavior at low temperature as shown in Fig. 1. The l a b data for Tc O 70 K strongly suggest an unconventional pairing state. Similar results were observed in the case of field perpendicular to c-axis as shown in Table 1 and Fig. 2. The slopes of D l c as a function of T are 12, 9 and 7 ArK for Tc s 70, 44 and 20 K, respectively. The slope increases as Tc decreases in H I c case, while it decreases in H I a b case. The slopes of D l a b are in the same range of that in YBCO system, which span from 4.8 to 20 ArK w24x, and similar to the value of 13 ArK in the Tl-2201 single crystal of Tc s 78 K measured by Broun et al. w19x, who also found a linear temperature dependence. The values of l a b Ž0. derived from our data are 168, 172, 212 and 332 nm and the corresponding values of l c Ž0. are 2112, 2555, 4385 and 8408 nm for Tc s 90, 70, 44 and 20 K, respectively. As Tc is reduced by raising the carrier concentration, 1rl2 Ž0.4 drops, a behavior which has been widely discussed and is known as the Uemura relation w25,26x. The value of l a b Ž0. for Tc s 70 K agrees with the data from Broun et al. w19x, for which a Tl-2201 single crystal with Tc of 78 K was measured by surface impedance measurements and l a b Ž0. s 165 nm was obtained.
Y.T. Wang, A.M. Hermannr Physica C 335 (2000) 134–138
137
Table 1 Summary of theoretical predictions and our experimental results
lŽT .
Theoretical predictions
Experimental results
For s-wave: D lŽT .rlŽ0. f 3.33ŽTcrT .1r 2 expŽyDrT .
Not observed
For d x 2 – y 2-wave: D lŽT . ; T ; T 2 with impurity scattering
D l ; T observed in l a b and l c for sample with Tc s 20, 44 and 70 K Tc s 20 K Tc s 44 K Tc s 70 K Slope ŽArK. 24.5 19 14 l a b ŽT s 0. Žnm. 332 212 172 l c ŽT . Slope ŽArK. 7 9 12 l c ŽT s 0. Žnm. 8408 4385 2555 Dl ; T 2 observed in l a b and l c for sample with Tc s 90 K
l a b ŽT .
lŽ H . For s-wave: DlŽT .rlŽ0. f b ŽT .Ž HrH0 ŽT .. 2 For d x 2 – y 2-wave: D lŽT .rlŽ0. f a ŽT .Ž HrH0 ŽT ..
Not observed Observed in both l a b and l c for sample with all Tc ’s Ž20, 44, 70 and 90 K. Tc s 20 K Tc s 44 K l a b Ž H . Slope ŽArG. 6 7.2 l c Ž H . Slope ŽArG. 6 8
Tc s 70 K 9 11
Tc s 90 K 9.5 12
For the field dependence of the penetration depths, in both field orientations, H I c and H I a b , D lŽ H .’s are linearly dependent on the applied field for the sample at all Tc ’s. The slopes in the H I c case are 6, 7.2, 9 and 9.5 ArG for sample of Tc s 20, 44, 70 and 90 K, while the corresponding values are 6, 8, 11 and 12 ArG in the case of H I a b . The slopes increase as Tc increases in both cases and are much larger than those in the YBCO system, which are
about 3–3.5 ArGw10,12x. In order to rule out the anisotropic s-wave possibility, a wide range of energy gap, D, in Eq. Ž1. from 0.01 to 1000 was selected to fit our data, but no converging fit was found Ža reasonable value of D is 1.76Tc w28x.. Both temperature and field dependencies of in-plane and out-of-plane penetration depths suggest a d-wave symmetry.
Fig. 1. Temperature dependence of the in-plane penetration depth of the sample with Tc s90 K.
Fig. 2. Temperature dependence of the out-of-plane penetration depth of the sample with Tc s90 K.
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Y.T. Wang, A.M. Hermannr Physica C 335 (2000) 134–138
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