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Electrical resistivity and hall effect of single crystal HgBa2Ca3Cu4O10+σ
PHYSICA ELSEVIER
Physica C 266 (1996) 104-108
Electrical resistivity and Hall effect of single crystal HgBa 2ca 3Cu4O 1 0 + J. LShle a,,, j. Karpinski a, A. Morawski b, p. Wachter a a Laboratoriwn fur Festkfrperphysik, ETH-Zi~rich, 8093 Zi~rich, Switzerland b High Pressure Research Centre "Unipress' ', Polish Academy of Science, 01142 Warsaw, Poland Received 21 December 1995; revised manuscript received 23 April 1996
Abstract We report the first measurements of the in-plane transport properties of single crystals of the four-layer superconductor HgBa2CaaCU4Olo+s with a transition temperature of 130 K. In the normal state the temperature dependence of the electrical resistivity and the Hall coefficient indicates that these crystals are underdoped. For the cotangent of the Hall angle we found a T 2 dependence. In the mixed state, the following form holds for a universal correlation between Hall resistivity PH and electrical resistivity p: On --- P', with n ~ 1.75.
1. Introduction The recently discovered homologous series HgBa2Ca~_ jCu~O2~+2÷~ with n --- 1, 2, 3 . . . o f f e r s the highest superconducting transition temperature (Tc) known to date [1,2]. Electrical transport measurements on single crystals of these compounds are of high interest in order to elucidate the reasons for the similarities and differences between the Hg family and other cuprate superconductors. These measurements should be carried out on single crystals, because in polycrystalline samples the effect of grain boundaries can drastically adulterate the results. In this paper we report for the first time on the transport properties of the four-layer HgBa2Ca3Cu4Oi0+s single crystals.
" Corresponding author. Fax: +41 1 633 10 77.
2. Experimental method Single crystals of HgBa2Ca3CU4Ol0+S were grown by a high pressure synthesis technique as described elsewhere [3]. They are untwirmed and have a tetragonal crystal structure. The crystals selected for the measurements were sized between 0.2 and 0.3 mm along the edge of the square and a c-axis dimension between 10 and 20 ixm. For measuring the electrical in-plane resistivity and the in-plane Hall effect electrical contacts with diameters of around 40 ixm were attached to the crystals with conducting silver epoxy (Epo-tec H20 silver paste) and 25 txm gold wire. In general four wires were pasted on the edges of the crystal in such a way that a quasi-two-dimensional current could flow along the ab-plane. A 5 minute heat treatment at 500°C in flowing 02 produced contact resistances of around 1 fl.
0921-4534/96/$15.00 Copyright © 1996 Elsevier Science B.V. All fights reserved PH S 0 9 2 1 - 4 5 3 4 ( 9 6 ) 0 0 3 3 1 - 0
J. LShle et aL / Physica C 266 (1996) 104-108
The resistance measurements were done by a standard four-point low-frequency ac (33 Hz) method. For the magnetoresistance and Hall effect measurements a magnetic field was applied parallel to the e-axis. For the Hall effect measurements the same frequency was used and for each measurement point the crystal was rotated by 180°, so that the small unbalanced resistivity signal could be subtracted.
105
180
.
.
.
.
.
160
---o---
0T 0.25 T
140
~
1 T
120
- . -
4T
/p ". ,~ ,~~"
2T 100
,
, ~
6T
80
3. Experimental results and discussion
Q.
60
3.1. Resistivity Fig. 1 shows a typical resistivity curve of a single crystal of HgBa2Ca3Cu40~0+s with an onset transition temperature of 130 K. For temperatures below 320 K the curve shows a non-linear temperature behaviour, which is typical for underdoped cuprates [4-6]. For temperatures higher than 320 K the shape of p(T) approaches linearity. Some authors interpreted the start of the deviation from linearity as an indication of the opening of a pseudo-gap in the spin fluctuation spectrum [5,7,8]. In this particular material this characteristic temperature seems to be twice as high as in YBa2Cu307_ x and YBa2Cu4Os, which means, if the above interpretation is applied, that the strength of the spin interaction and the size of the pseudo-gap is much larger. This can eventually explain the high transition temperatures of the Hg family. Similar resistivity results have been recently
600
....
sooI
, ....
, ....
, ....
, ....
, ....
,.
4OO o~300
100
~
,=",="~ ~ :
~"~'
0 J''' ......... ' .... ' ......... ~' 100 150 200 250 300 350 40( T [K] Fig. 1. Resistivity versus temperature for a HgBa2Ca3Cu4Olo+8 single crystal with an onset temperature of 130 K. p(T) is linear above 320 K. The inset is a plot of ap/~T versus temperature.
t
20
t ooO°°...'2 : :,
oooooOg
40
+ + +x~xxX~$vvvvgA~
I
I
60
80
0 o~
i
100
J
~9 I
120
I
140
160
T[K] Fig. 2. Temperature dependence of the resistivity for seven selected values of the magnetic field.
published on the three-layer HgBa2Ca3Cu4010+8 crystals [5,9]. Although, according to the authors of Ref. [9] these are multiphase crystals framed by a majority lattice of HgBa2Ca2Cu 308 + 8.
3.2. Magnetoresistance The temperature dependence of the resistivity is shown in Fig. 2 for seven different values of the magnetic field. In zero magnetic field the transition is sharp, where as in higher fields the transition broadens. The resistive behaviour below Tc can be explained for small temperatures in the framework of Thermally Acticated Flux Flow (TAFF) and for temperatures close to Tc by Flux Flow (FF). The resistive transition in a magnetic field can be used to determine the irreversibility line H *(T). This line devides the H-T-phase diagram in two regions of irreversible (low H, T) and reversible magnetic behaviour. Above this line no lossless current transport is possible due to flux motion. In this region of the H-T-phase diagram the superconducting material is
106
J. l_~hle et aL / Physica C 266 (1996) 104-108 . . . .
I
. . . .
I
. . . .
I
. . . .
I
0.3
10
"
ooO ° oo
o
/' o o
(a)
,
(b)
o
(c) (d)
• %
° oo° oo oo o o°
0.2
oo oo oo 0o
..r
6
.
•
•
1 O0
E
i °,
4
o
.
I
. . . .
150
l
.
.
i
.
200
f
. . . .
I
250
i
300
T[K] Fig. 4. InverseHallcoefficientas a functionof temperature.
2
linear behaviour, which can be fitted by the following expression:
k cl
ooO o°°
~o
a,
v
,5
o
,oo
T [K] Fig. 3. Irreversible field B* versus temperature for (a) HgBa2Ca3Cu40=o+8 single crystals, (b) YBa2Cu3OT_ ~ [11], (c) YBa2Cu408 [12] and (d) Bi2Sr2CaCu20 x [13].
not useful for technical applications. The upper-limit criterion for the determination of the irreversible temperature T * ( H ) was 10 -9 ~ cm. The irreversibility line B *(T)= Fo H *(T) is plotted in Fig. 3. We found nearly the same values as for the three-layer HgBa2Ca2Cu308+ ~ ceramics [10]. For comparison the irreversibility lines of YBa2Cu307_~ [11], YBa2Cu408 [12] and Bi2Sr2CaCu20 x [13] are added in Fig. 3.
cot(0H) = A T 2 + B.
(1)
A similar behaviour is found for underdoped YBa2Cu307_ x single crystals [4,14], YBa2Cu40 s single crystals [7] and for HgBazCaCu206+ s ceramics [15]. The T2-dependence of cot(0 n) can be explained by two different models. Within the 2D Luttinger liquid model, first proposed by Anderson [16], the separation of charge and spin leads to the existence of two different electronic relaxation times. On the one hand, the transport electronic relaxation time Tio~ T -1, due to scattering between holons and spinons, which determines the longitudinal resistivity; on the other hand, the transverse relaxation time, Ttr.= T -2 determined by scattering between spinons alone. The theory predicts that the Hall angle is only a function of Ttr., cot(0 H) = (tocTt~.)-1 , where t% is I
3.3. Hall effect
I
200
o o o o o
At temperatures above the transition temperature the Hall resistivity is linear in the magnetic field, therefore it is possible to define a Hall coefficient. In Fig. 4 the inverse Hall coefficients 1 / R H are plotted. For higher temperatures the inverse Hall coefficients vary linearly with temperature. Using the measured data of the resistivity and the Hall effect, we computed the values of cot(0H)= p / ( R r d x o H ) with i~oH = 6 T as a function of T 2 as it is shown in Fig. 5. We observed in the high temperature regime a
° oo°
150 0 o° o o
loo
o o 0 o
8
o o ooo o°
5O 0
o o
oo
I
2
,
I
4
i
I
6
,
I
i
8
T z [104 K 21 Fig. 5. T z dependence of the cotangens of the Hall angle.
10
J. l_~hle et al./ Physica C 266 (1996) 104-108
0
,
,
,
,
,
--o--- 130 K
6
--
00K
Luo et al. [22] found in YBa2Cu307 films such a behaviour with n = 1.7. Budhani [21] derived n = 1.85 from measurements on TI2Ba2Ca2Cu30 m films and Samoilov [23] deduced n = 2 from monocrystalline Bi2Sr2CaCu20 x. Looking for such a scaling behaviour in our samples we plotted log(pn) versus log(p) for different magnetic fields. Fig. 7 shows the resulting curves. In the TAFF regime we found the same universal power law dependence for all magnetic fields. The curve can be fitted in this regime with the parameters m = 0.04 ( ~ cm) -°'75 and n = 1.75. The explanation for such a scaling law was given by the theory of Vinokur et al. [24] in terms of weak pinning of the vortex liquid. The authors wrote the equation for the forces acting on the individual flux line in the following form:
/o
----o---100 K 90 K 80K --x-- 70K
8
//
o/° ./°
/° ,S o /o/ / /°,o
/
/0/° =
/
/o/ / / / / °/° /" / / / /° / / /° /o / / /°/ //-
"r Q.
/o
2
o.j-,-.,,-,;< o/O / /
107
/"
(~q + ~ ) v +(x(v X n) = * o ( j × I
I
I
I
I
I
0
2
4
6
8
10
n),
(3)
were ~q represents the viscosity coefficient, -y is a damping term, originating from the average pinning force acting on a flux line, a is the Hall drag coefficient, v is the flux line velocity, ~0 is the flux
B['r] Fig. 6. Magnetic-field dependence of the Hall resistivity for different temperatures below Tc.
10 the cyclotron frequency. More recently, an other model for the T 2 temperature dependence of the Hall angle has been proposed. Levin and Quader [17] propose a subband phenomenological model, which predicts the existence of two types of quasi particles with different relaxation rates and a T 2 dependence for cot(0n). Furthermore, we investigated the Hall effect below T~ in the TAFF and FF regime. For temperatures below Tc the definition of a Hall coefficient is useless, because the Hall resistivity Pn is not linear with increasing field. Fig. 6 shows the magnetic-field dependence of the Hall resistivity at different temperatures below Tc. The Hall resistivity does not undergo a sign change in contrast to other cuprate superconductors [18-21]. This may be a result of strong pinning centres in the crystals [21]. Some authors have found in the TAFF regime a correlation between the Hall and the magneto resistance of the following form:
p. = rap".
(2)
" lOT - 8T o
6T
,
4T
E
A "-%7~i"~ "~"~.o ~ Fo, . J ,
/
0.1
o*
-r
/
0.01
o
I o
ol
0.001 0.1
. . . . . . . .
I
1
.
,
,
,,,,,I
10 p [p.ocm]
,
,
......
I
.
,
100
Fig. 7. Hall resistivity versus electrical resistivity for different magnetic fields.
108
J. L~Me et aL /Physica C 266 (1996) 104-108
quantum, and n is the unit vector in the direction of the magnetic field. The term on the right-hand side is the Lorentz force acting on the flux line due to the presence of the transport current j. Vinocur and coworkers solved this equation for v, and calculated with this solution the electric field E = ( H × v)(iXo/C) induced by flux motion. The use of Ohm's law led to the following relationship between p and PH:
p. = p2(a/I n I)(c2/~000).
(4)
Comparison with Eq. (2) leads to m = (or/ I H I)(c2/ix0@0 ), if n = 2 was supposed. In general p and ot are functions of the temperature. In the TAFF regime, at low temperatures, p changes rapidly in a narrow temperature interval, thus the dominant temperature dependence factor will be p2. In the FF regime the temperature dependence of a is not negligible. In this regime a determines the sign of P H and et will be the dominant temperature dependence factor. In the TAFF regime we find the same m coefficient for all magnetic fields, which means that ct is proportional to the magnetic field. This result is consistent with the Bardeen-Stephen theory
[25]. 4. Summary In conclusion it can be said that the growth of untwinned reasonably large single crystals of the Hg cuprate family enable for the first time measurements of electrical transport properties as a function of temperature and magnetic field. Similar measurements on single crystals of other members of the Hg family are in progress.
Acknowledgements The authors wish to thank F. Hulliger for critical reading of the manuscript.
References [1] S.N. Putilin, E.V. Antipov, O. Chmaissen and M. Marezio, Nature 362 (1993) 226. [2] A. Schilling, M. Cantoni, J.D. Guo and H.R. Ott, Nature 363 (1993) 56. [3] J. Karpinski, H. Schwer, I. Mangelschots, K. Conder, A. Morawski, T. Lada and A. Paszewin, Physica C 234 (1994) 10. [4] P. Xiong, G. Xiao and X.D. Wu, Phys. Rev. B 47 (1993) 5516. [5] A. Cartington, D. Colson, Y. Dumont, C. Ayache, A. Bertinotti and J.F. Marucco, Physica C 234 (1994) I. [6] A.S. Alexandrov, A.M. Bratkovsky and N.F. Mott, Phys. Rev. Lett. 72 (1994) 1734. [7] B. Bucher, P. St¢iner, J. Karpinski, E. Kaldis and P. Wachter, Phys. Rev. Lett. 70 (1993) 2012. [8] T. lto, K. Takenaka and S. Uchida, Phys. Rev. Lett. 70 (1993) 3995. [9] D. Colson, A. Bertinotti, J. Hammatm, J.F. Marucco and A. Pinatel, Physica C 233 (1994) 231. [10] A. Schilling, O. Jeandupeux, J.D. Guo and H.R. Ott, Physica C 216 (1993) 6. [11] A. Schilling, H.R. Ott and Th. Wolf, Phys. Rev. B 46 (1992) 14253. [12] J. L0hle and P. Wachter, unpublished results. [13] J. LShle, K. Mattenberger, O. Vogt and P. Wachter, J. Appl. Phys. 76 (1994) 7446. [14] J.M. Harris, Y.F. Yan and N.P. Ong, Phys. Rev. B 46 (1992) 14293. [15] J.M. Harris, H. Wu, N.P. Ong, R.L. Meng and C.W. Chu, Phys. Rev. B 50 (1994) 3246. [16] P.W. Anderson, Phys. Rev. Lett. 67 (1991) 2092. [17] G.A. Levin and K.F. Quader, Phys. Rev. B 46 (1992) 5872. [18] S.J. Hagen, C.J. Lobb, R.L. Greene, M.G. Forrester and J.H. Kang, Phys. Rev. B 41 (1990) 11630. [19] Y. lye, S. Nukamura and T. Tamegai, Physica C 159 (1989) 616. [20] J. Schoenes, E. Kaldis and J. Karpinski, Phys. Rev. B 48 (1993) 16869. [21] R.C. Budhani, S.H. Liou and Z.X. Cai, Phys. Rev. Lett. 71 (1993) 621. [22] J. Luo, T.P. Orlando, J.M. Graybeal, X.D. Wu and R. Muenchhausen, Phys. Rev. Lett. 68 (1992) 690. [23] A.V. Samoilov, Phys. Rev. Lett. 71 (1993) 617. [24] V.M. Vinokur, V.B. Geshkenbein, M.V. Feigel'man and G. Blatter, Phys. Rev. Lett. 71 (1993) 1242. [25] J. Bardeen and M.J. Stephen, Phys. Rev. 140 (1965) A1196.
Physica C 266 (1996) 104-108
Electrical resistivity and Hall effect of single crystal HgBa 2ca 3Cu4O 1 0 + J. LShle a,,, j. Karpinski a, A. Morawski b, p. Wachter a a Laboratoriwn fur Festkfrperphysik, ETH-Zi~rich, 8093 Zi~rich, Switzerland b High Pressure Research Centre "Unipress' ', Polish Academy of Science, 01142 Warsaw, Poland Received 21 December 1995; revised manuscript received 23 April 1996
Abstract We report the first measurements of the in-plane transport properties of single crystals of the four-layer superconductor HgBa2CaaCU4Olo+s with a transition temperature of 130 K. In the normal state the temperature dependence of the electrical resistivity and the Hall coefficient indicates that these crystals are underdoped. For the cotangent of the Hall angle we found a T 2 dependence. In the mixed state, the following form holds for a universal correlation between Hall resistivity PH and electrical resistivity p: On --- P', with n ~ 1.75.
1. Introduction The recently discovered homologous series HgBa2Ca~_ jCu~O2~+2÷~ with n --- 1, 2, 3 . . . o f f e r s the highest superconducting transition temperature (Tc) known to date [1,2]. Electrical transport measurements on single crystals of these compounds are of high interest in order to elucidate the reasons for the similarities and differences between the Hg family and other cuprate superconductors. These measurements should be carried out on single crystals, because in polycrystalline samples the effect of grain boundaries can drastically adulterate the results. In this paper we report for the first time on the transport properties of the four-layer HgBa2Ca3Cu4Oi0+s single crystals.
" Corresponding author. Fax: +41 1 633 10 77.
2. Experimental method Single crystals of HgBa2Ca3CU4Ol0+S were grown by a high pressure synthesis technique as described elsewhere [3]. They are untwirmed and have a tetragonal crystal structure. The crystals selected for the measurements were sized between 0.2 and 0.3 mm along the edge of the square and a c-axis dimension between 10 and 20 ixm. For measuring the electrical in-plane resistivity and the in-plane Hall effect electrical contacts with diameters of around 40 ixm were attached to the crystals with conducting silver epoxy (Epo-tec H20 silver paste) and 25 txm gold wire. In general four wires were pasted on the edges of the crystal in such a way that a quasi-two-dimensional current could flow along the ab-plane. A 5 minute heat treatment at 500°C in flowing 02 produced contact resistances of around 1 fl.
0921-4534/96/$15.00 Copyright © 1996 Elsevier Science B.V. All fights reserved PH S 0 9 2 1 - 4 5 3 4 ( 9 6 ) 0 0 3 3 1 - 0
J. LShle et aL / Physica C 266 (1996) 104-108
The resistance measurements were done by a standard four-point low-frequency ac (33 Hz) method. For the magnetoresistance and Hall effect measurements a magnetic field was applied parallel to the e-axis. For the Hall effect measurements the same frequency was used and for each measurement point the crystal was rotated by 180°, so that the small unbalanced resistivity signal could be subtracted.
105
180
.
.
.
.
.
160
---o---
0T 0.25 T
140
~
1 T
120
- . -
4T
/p ". ,~ ,~~"
2T 100
,
, ~
6T
80
3. Experimental results and discussion
Q.
60
3.1. Resistivity Fig. 1 shows a typical resistivity curve of a single crystal of HgBa2Ca3Cu40~0+s with an onset transition temperature of 130 K. For temperatures below 320 K the curve shows a non-linear temperature behaviour, which is typical for underdoped cuprates [4-6]. For temperatures higher than 320 K the shape of p(T) approaches linearity. Some authors interpreted the start of the deviation from linearity as an indication of the opening of a pseudo-gap in the spin fluctuation spectrum [5,7,8]. In this particular material this characteristic temperature seems to be twice as high as in YBa2Cu307_ x and YBa2Cu4Os, which means, if the above interpretation is applied, that the strength of the spin interaction and the size of the pseudo-gap is much larger. This can eventually explain the high transition temperatures of the Hg family. Similar resistivity results have been recently
600
....
sooI
, ....
, ....
, ....
, ....
, ....
,.
4OO o~300
100
~
,=",="~ ~ :
~"~'
0 J''' ......... ' .... ' ......... ~' 100 150 200 250 300 350 40( T [K] Fig. 1. Resistivity versus temperature for a HgBa2Ca3Cu4Olo+8 single crystal with an onset temperature of 130 K. p(T) is linear above 320 K. The inset is a plot of ap/~T versus temperature.
t
20
t ooO°°...'2 : :,
oooooOg
40
+ + +x~xxX~$vvvvgA~
I
I
60
80
0 o~
i
100
J
~9 I
120
I
140
160
T[K] Fig. 2. Temperature dependence of the resistivity for seven selected values of the magnetic field.
published on the three-layer HgBa2Ca3Cu4010+8 crystals [5,9]. Although, according to the authors of Ref. [9] these are multiphase crystals framed by a majority lattice of HgBa2Ca2Cu 308 + 8.
3.2. Magnetoresistance The temperature dependence of the resistivity is shown in Fig. 2 for seven different values of the magnetic field. In zero magnetic field the transition is sharp, where as in higher fields the transition broadens. The resistive behaviour below Tc can be explained for small temperatures in the framework of Thermally Acticated Flux Flow (TAFF) and for temperatures close to Tc by Flux Flow (FF). The resistive transition in a magnetic field can be used to determine the irreversibility line H *(T). This line devides the H-T-phase diagram in two regions of irreversible (low H, T) and reversible magnetic behaviour. Above this line no lossless current transport is possible due to flux motion. In this region of the H-T-phase diagram the superconducting material is
106
J. l_~hle et aL / Physica C 266 (1996) 104-108 . . . .
I
. . . .
I
. . . .
I
. . . .
I
0.3
10
"
ooO ° oo
o
/' o o
(a)
,
(b)
o
(c) (d)
• %
° oo° oo oo o o°
0.2
oo oo oo 0o
..r
6
.
•
•
1 O0
E
i °,
4
o
.
I
. . . .
150
l
.
.
i
.
200
f
. . . .
I
250
i
300
T[K] Fig. 4. InverseHallcoefficientas a functionof temperature.
2
linear behaviour, which can be fitted by the following expression:
k cl
ooO o°°
~o
a,
v
,5
o
,oo
T [K] Fig. 3. Irreversible field B* versus temperature for (a) HgBa2Ca3Cu40=o+8 single crystals, (b) YBa2Cu3OT_ ~ [11], (c) YBa2Cu408 [12] and (d) Bi2Sr2CaCu20 x [13].
not useful for technical applications. The upper-limit criterion for the determination of the irreversible temperature T * ( H ) was 10 -9 ~ cm. The irreversibility line B *(T)= Fo H *(T) is plotted in Fig. 3. We found nearly the same values as for the three-layer HgBa2Ca2Cu308+ ~ ceramics [10]. For comparison the irreversibility lines of YBa2Cu307_~ [11], YBa2Cu408 [12] and Bi2Sr2CaCu20 x [13] are added in Fig. 3.
cot(0H) = A T 2 + B.
(1)
A similar behaviour is found for underdoped YBa2Cu307_ x single crystals [4,14], YBa2Cu40 s single crystals [7] and for HgBazCaCu206+ s ceramics [15]. The T2-dependence of cot(0 n) can be explained by two different models. Within the 2D Luttinger liquid model, first proposed by Anderson [16], the separation of charge and spin leads to the existence of two different electronic relaxation times. On the one hand, the transport electronic relaxation time Tio~ T -1, due to scattering between holons and spinons, which determines the longitudinal resistivity; on the other hand, the transverse relaxation time, Ttr.= T -2 determined by scattering between spinons alone. The theory predicts that the Hall angle is only a function of Ttr., cot(0 H) = (tocTt~.)-1 , where t% is I
3.3. Hall effect
I
200
o o o o o
At temperatures above the transition temperature the Hall resistivity is linear in the magnetic field, therefore it is possible to define a Hall coefficient. In Fig. 4 the inverse Hall coefficients 1 / R H are plotted. For higher temperatures the inverse Hall coefficients vary linearly with temperature. Using the measured data of the resistivity and the Hall effect, we computed the values of cot(0H)= p / ( R r d x o H ) with i~oH = 6 T as a function of T 2 as it is shown in Fig. 5. We observed in the high temperature regime a
° oo°
150 0 o° o o
loo
o o 0 o
8
o o ooo o°
5O 0
o o
oo
I
2
,
I
4
i
I
6
,
I
i
8
T z [104 K 21 Fig. 5. T z dependence of the cotangens of the Hall angle.
10
J. l_~hle et al./ Physica C 266 (1996) 104-108
0
,
,
,
,
,
--o--- 130 K
6
--
00K
Luo et al. [22] found in YBa2Cu307 films such a behaviour with n = 1.7. Budhani [21] derived n = 1.85 from measurements on TI2Ba2Ca2Cu30 m films and Samoilov [23] deduced n = 2 from monocrystalline Bi2Sr2CaCu20 x. Looking for such a scaling behaviour in our samples we plotted log(pn) versus log(p) for different magnetic fields. Fig. 7 shows the resulting curves. In the TAFF regime we found the same universal power law dependence for all magnetic fields. The curve can be fitted in this regime with the parameters m = 0.04 ( ~ cm) -°'75 and n = 1.75. The explanation for such a scaling law was given by the theory of Vinokur et al. [24] in terms of weak pinning of the vortex liquid. The authors wrote the equation for the forces acting on the individual flux line in the following form:
/o
----o---100 K 90 K 80K --x-- 70K
8
//
o/° ./°
/° ,S o /o/ / /°,o
/
/0/° =
/
/o/ / / / / °/° /" / / / /° / / /° /o / / /°/ //-
"r Q.
/o
2
o.j-,-.,,-,;< o/O / /
107
/"
(~q + ~ ) v +(x(v X n) = * o ( j × I
I
I
I
I
I
0
2
4
6
8
10
n),
(3)
were ~q represents the viscosity coefficient, -y is a damping term, originating from the average pinning force acting on a flux line, a is the Hall drag coefficient, v is the flux line velocity, ~0 is the flux
B['r] Fig. 6. Magnetic-field dependence of the Hall resistivity for different temperatures below Tc.
10 the cyclotron frequency. More recently, an other model for the T 2 temperature dependence of the Hall angle has been proposed. Levin and Quader [17] propose a subband phenomenological model, which predicts the existence of two types of quasi particles with different relaxation rates and a T 2 dependence for cot(0n). Furthermore, we investigated the Hall effect below T~ in the TAFF and FF regime. For temperatures below Tc the definition of a Hall coefficient is useless, because the Hall resistivity Pn is not linear with increasing field. Fig. 6 shows the magnetic-field dependence of the Hall resistivity at different temperatures below Tc. The Hall resistivity does not undergo a sign change in contrast to other cuprate superconductors [18-21]. This may be a result of strong pinning centres in the crystals [21]. Some authors have found in the TAFF regime a correlation between the Hall and the magneto resistance of the following form:
p. = rap".
(2)
" lOT - 8T o
6T
,
4T
E
A "-%7~i"~ "~"~.o ~ Fo, . J ,
/
0.1
o*
-r
/
0.01
o
I o
ol
0.001 0.1
. . . . . . . .
I
1
.
,
,
,,,,,I
10 p [p.ocm]
,
,
......
I
.
,
100
Fig. 7. Hall resistivity versus electrical resistivity for different magnetic fields.
108
J. L~Me et aL /Physica C 266 (1996) 104-108
quantum, and n is the unit vector in the direction of the magnetic field. The term on the right-hand side is the Lorentz force acting on the flux line due to the presence of the transport current j. Vinocur and coworkers solved this equation for v, and calculated with this solution the electric field E = ( H × v)(iXo/C) induced by flux motion. The use of Ohm's law led to the following relationship between p and PH:
p. = p2(a/I n I)(c2/~000).
(4)
Comparison with Eq. (2) leads to m = (or/ I H I)(c2/ix0@0 ), if n = 2 was supposed. In general p and ot are functions of the temperature. In the TAFF regime, at low temperatures, p changes rapidly in a narrow temperature interval, thus the dominant temperature dependence factor will be p2. In the FF regime the temperature dependence of a is not negligible. In this regime a determines the sign of P H and et will be the dominant temperature dependence factor. In the TAFF regime we find the same m coefficient for all magnetic fields, which means that ct is proportional to the magnetic field. This result is consistent with the Bardeen-Stephen theory
[25]. 4. Summary In conclusion it can be said that the growth of untwinned reasonably large single crystals of the Hg cuprate family enable for the first time measurements of electrical transport properties as a function of temperature and magnetic field. Similar measurements on single crystals of other members of the Hg family are in progress.
Acknowledgements The authors wish to thank F. Hulliger for critical reading of the manuscript.
References [1] S.N. Putilin, E.V. Antipov, O. Chmaissen and M. Marezio, Nature 362 (1993) 226. [2] A. Schilling, M. Cantoni, J.D. Guo and H.R. Ott, Nature 363 (1993) 56. [3] J. Karpinski, H. Schwer, I. Mangelschots, K. Conder, A. Morawski, T. Lada and A. Paszewin, Physica C 234 (1994) 10. [4] P. Xiong, G. Xiao and X.D. Wu, Phys. Rev. B 47 (1993) 5516. [5] A. Cartington, D. Colson, Y. Dumont, C. Ayache, A. Bertinotti and J.F. Marucco, Physica C 234 (1994) I. [6] A.S. Alexandrov, A.M. Bratkovsky and N.F. Mott, Phys. Rev. Lett. 72 (1994) 1734. [7] B. Bucher, P. St¢iner, J. Karpinski, E. Kaldis and P. Wachter, Phys. Rev. Lett. 70 (1993) 2012. [8] T. lto, K. Takenaka and S. Uchida, Phys. Rev. Lett. 70 (1993) 3995. [9] D. Colson, A. Bertinotti, J. Hammatm, J.F. Marucco and A. Pinatel, Physica C 233 (1994) 231. [10] A. Schilling, O. Jeandupeux, J.D. Guo and H.R. Ott, Physica C 216 (1993) 6. [11] A. Schilling, H.R. Ott and Th. Wolf, Phys. Rev. B 46 (1992) 14253. [12] J. L0hle and P. Wachter, unpublished results. [13] J. LShle, K. Mattenberger, O. Vogt and P. Wachter, J. Appl. Phys. 76 (1994) 7446. [14] J.M. Harris, Y.F. Yan and N.P. Ong, Phys. Rev. B 46 (1992) 14293. [15] J.M. Harris, H. Wu, N.P. Ong, R.L. Meng and C.W. Chu, Phys. Rev. B 50 (1994) 3246. [16] P.W. Anderson, Phys. Rev. Lett. 67 (1991) 2092. [17] G.A. Levin and K.F. Quader, Phys. Rev. B 46 (1992) 5872. [18] S.J. Hagen, C.J. Lobb, R.L. Greene, M.G. Forrester and J.H. Kang, Phys. Rev. B 41 (1990) 11630. [19] Y. lye, S. Nukamura and T. Tamegai, Physica C 159 (1989) 616. [20] J. Schoenes, E. Kaldis and J. Karpinski, Phys. Rev. B 48 (1993) 16869. [21] R.C. Budhani, S.H. Liou and Z.X. Cai, Phys. Rev. Lett. 71 (1993) 621. [22] J. Luo, T.P. Orlando, J.M. Graybeal, X.D. Wu and R. Muenchhausen, Phys. Rev. Lett. 68 (1992) 690. [23] A.V. Samoilov, Phys. Rev. Lett. 71 (1993) 617. [24] V.M. Vinokur, V.B. Geshkenbein, M.V. Feigel'man and G. Blatter, Phys. Rev. Lett. 71 (1993) 1242. [25] J. Bardeen and M.J. Stephen, Phys. Rev. 140 (1965) A1196.