- Email: [email protected]
Angular dependence of resistive transition in strongly anisotropic superconductors under magnetic field
Physica C 201 (1992) 386-390 North-Holland
Angular dependence of resistive transition in strongly anisotropic superconductors under magnetic field Ryusuke Ikeda Department of Physics, Kyoto University, Kyoto 606, Japan Received 13 July 1992
Resistivity and specific heat for arbitrary field and current configurations of strongly anisotropic high-T¢ superconductors are studied according to the renormalized fluctuation theory and by specifically considering the 2D limit of the layered model and the anisotropic 3D GL model. It is pointed out that the theory can naturally explain phenomena peculiar to strongly anisotropic materials, such as the independence of the resistivity in arbitrary field configuration from the in-plane field component, and socalled Lorentz force-free dissipation and field independence observed in resistivity measurements in the H_L c k I configuration of BSCCO.
1. Introduction
There has been a growing interest in various extraordinary features appearing in transport and thermodynamic data for the mixed state of high-To superconductors. In the field configuration Hllc-axis, the resistivity data show remarkable broadening phenomena for both cases of HZI and Hill (I is the applied current), which are at present understood quite well according to the order parameter fluctuation theory [ 1-3 ]. In other field configurations, one encounters various interesting phenomena in resistivity data carefully measured for strongly an±sotropic high-To superconductors such as BSCCO. For instance, several authors pointed out [4-6 ] that the in-plane (I±c) resistivity of BSCCO is, even in the specific case H ± c [ 5 ], essentially independent of the in-plane component Hb, of the applied field. Recently, such H,-independence has also been found inthe out-of-plane (Illc) resistivity of LSCO single crystals [ 7 ]. Further, the in-plane resistivity in H i c of BSCCO is independent of the relative angle between H and i (namely, Lorentz force-free) [4,5]. This iS in contrast to that in 90 K YBCO [8], in which case the H ± I resistivity is larger than the Hill one in the Ohmic regime. Clearly, these phenomena should be explained within a single theoretical framework.
In this paper, we point out that such striking features observed in resistivity data for strongly anisotropic materials can naturally be understood, as well as in the HIIccase [ 1-3 ], according to the fluctuation theory [ 1 ] for strongly type-II superconductors. Interestingly, we find that the Lorentz force-free behavior in H Z c can be obtained in the strongly anisotropic limit even if the discrete layer structure is neglected.
2. Calculations
Our analysis is based upon, as in previous works [ 1,9 ], the Lawrence-Doniach model in the extreme type-II limit where fluctuations in the magnetic field are entirely neglected:
+'2 =~x,r(-iO=- ~oAa)¥~(r)Z+q(~)2l~u,(r) (i+ ] )s
-~',+,(r) exp(i~ox
0921-4534/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
f
dZA~(r))12 ]
is
b
}
-4-~l~(r) l4 ,
(1)
R. Ikeda/Angulardependenceofresistivetransition where X, Y, Z denote the directions of the main crystal axis, and r the in-plane coordinates (see ref. [ 1 ] on other notations). Throughout this paper, the applied field H is assumed to be in the X-Z plane, and the angle between H and 2~is expressed by 0. We primarily focus on the qualitative results that the theory gives. Below, the 2D limit of eq. (1) and the anisotropic 3D model (s--,0 limit of eqs. ( 1 ) ) will be discussed separately.
2.1. The 2D limit First we will briefly explain the derivation of the formulae in the 2D limit, where the mass anisotropy r/, proportional to the Josephson coupling strength in the Z-direction, is vanishingly small. We can always choose the gauge A = - H ( A " cos 0 - 2 sin 0) Y. For any r/#0, it is quite difficult to study the renormalization of fluctuations in the layered system ( 1 ) with 0 # 0. Here, we merely focus on the leading contributions to physical quantities in the q--,0 limit and study the situation in which the Josephson coupling is negligible in renormalizing the fluctuations. Then, the hamiltonian (1) (under the present gauge) is equivalent to a sum of the 2D one in each layer under a field H I = H cos 0 perpendicular to the layers: ocf= ~ , . . It means that the nonvanishing physical quantities in the 2D limit, such as the in-plane fluctuation conductivity a± and the (normalized) specific heat C/AC, are obtained in the 0 # 0 case by, in the corresponding formulae for the Hll Z case [ 1,9 ], taking the q-,0 limit and replacing h=2nH~2/Oo by h cos 0. In contrast to them, the out-of-plane fluctuation conductivity G vanishes when the r/-, 0 limit is strictly taken, since the supercurrent in the Zdirection
JZ = iaq ~f ~oo( ~t*~, +, exp( i ~o HSY sin O) - c.c.) (2)
is explicitly multiplied by q. Nevertheless, one can see that the a~ formula in the ~/--.0 limit is precisely given, up to lowest order in q, by taking the 2D limit used above only in renormalizing the fluctuations. In fact, the current-current correlation function appearing in the Kubo formula for az is in the present case given by
387
Z (JZ(r)jZ(r')> t,J
, (4natl~ 2~2
=~,~)
2
~ <~t~(r')~(r)q4+~(r')q/*+j(r) >
2
=2(2~aq~O] E <~t(r')~,(r)> \ SOo ] , X ( ~+1 (r')~*+x (r) > ,
(3)
where we used the fact that, in the 2D limit, there are no inter-layer correlations between the order parameters in neighboring layers. Note that no vertex corrections [ 1 ] appear in eq. (3). This can be regarded as a quantitative justification for fitting results to the HIIIIIcBSCCO data given in ref. [ 2 ]. The az formula is easily obtained by means of eq. (3) and along the same line as in ref. [1]. Summarizing the abovementioned results, we give the formulae for a . , az and C/AC in the 2D limit: 82
a k ~ 4 - ~ s n=O ~ (n+l)
x (~__~+--# t +ll e2
{
-4+ 1) '
# n -[- #~t
22~
"~
ks/
C'/AC~- ks H
-
Oo
1 oHcos0 2 .=o#~ '
1 0#;
A---C0o--Scos 0 ,=o E #~ Oe •
(4)
For the renormalized mass parameters #~, we will use, as in refs. [ 1,9 ], the high-field expressions which are given by #~, = e + ( 2 n + 1 ) ~ ~2H cos 0 eo +#;(x'+ln(1 +x')),
_(
2x'
Og'o l+x'-ln(l+x')+ l---~x'] Oe x'= - kB H cos 0/#; 2 . - AC OoS
'
(5)
In eq. (5), we assumed that the contributions of higher Landau level modes to the mass renormalizations can be dropped out, since they can be considered to be absorbed into a slight renormalization of TcH [ 1 ]. A similar treatment for high-energy flue-
388
R. Ikeda/Angular dependenceof resistive transition
tuations was done in ref. [ 10 ] in deriving the zerofield formula (6) given below. Therefore, we expect these expressions (4) and (5), broadly speaking, to be useful even in the H±->0 limit, where all Landau levels are not negligible. Consequently, in this limit, tr'~ and C ' / A C reduce to their 2D gaussian formulae in zero field, respectively. If one would like to include the effect of the vortex unbinding transition, which occurs in the q--,0 limit, in t r ± ( H l - , 0 ) , it should be understood to reduce to the Halperin-Nelson interpolation formula [ l 0 ]
a'z
H±~0
e 2
~/p~_¢]2
1F.
(6)
= auN - 1~s/7 ~ [ smh N/ ~ _l '
where ~ - 1 +T/TKT (TKT is the true transition temperature in ~/=0), ec~-kB/ACns~, and fl is a constant of order unity [ I 1 ]. The corresponding limiting form of trz becomes n±~o
__~
1
+~o2 =~x, Y I - i O n - Oo a) ¢/I +~2<1 --iO:--
Az ~ulz + ~ I~ul4 ,
(9)
1
t,77 ' 2 r/,
Next, we consider the anisotropic 3D GL model
(7)
This is the leading term in r/of
2:4(~)
2.2. The anisotropic 31_) case
2~ A "~ E
/72 1 ~"
eE {~o~ E E
result is nothing but the statement noted by Kes et al. [6]. However, in deriving the H . - > 0 results above, the q ~ 0 limit was assumed at the outset for any 0, and thus, it is not clear at this stage if a calculation performed independently in the H ± Z (0 = 90 ° C) case leads to essentially the same result in the strongly anisotropic limit. In addition, the apparently different behavior seen in YBCO data [ 8 ] should be understood on the same footing. Below, we show that these requirements are indeed realized in the anisotropic 3D case.
(8,
which was used in fitting zero-field data in refs. [ 1,2 ]. Evidently, the formulae (4) indicate that the outof-plane conductivity G, as well as the in-plane conductivity tr±, is independent of the in-plane component of the field in strongly anisotropic systems. This result, consistent with recent LSCO data [ 7 ], does not seem to be understood according to other ideas. For instance, it is unreasonable to assume the presence of inter-layer phase slippage events in order to explain the large Ohmic dissipation observed in the configuration HIIIIIc of BSCCO [12 ]. Actually, since each Cu-O layer has macroscopic size, the plausible object which may be excited by the current is a vortex ring confined between neighboring layers, which, however, should not lead to any Ohmic dissipation [ 13 ]. The above result of tr'~ in H± -o0 suggests the field independence and the so-called Lorentz force-free behavior in H_I_Z. Actually, this apparently trivial
where the layer structure is neglected, and the sample anisotropy is taken into account as the ratio of the coherence lengths. In the fluctuation theory from higher temperature, it is always most useful to represent ~u in terms of complete and orthogonal basis functions fully diagonalizing the quadratic part of the hamiltonian. In the present case, such a basis set is given, under the gauge A = - H ( c o s 0 ~ ' - sin 0 2~) Y, by
--n D+
Unpq(r)=Ao x f ~ , n ! xex
- ~ (y+ ~ (p+~q))2+i(pX+qg)
,
(10) where ) ~ = y - t ( X cos O - Z sin 0), y=~,Y, g = Z cos O + X sin O, Ao is a normalization constant, and /5+ = - i ( G + o t O z ) + h y - O ; ,
(ll)
or=y-3( 1 - q ) sin 0cos 0, 74 = cosE0+ t/sinE0.
(12)
R. lkeda/Angulardependenceof resistivetransition Note that, in 0~0, 90 °, the ortho-normalized basis functions (10) are not Landau level eigen functions but rather their superpositions. Their completeness is readily shown in the same way as for the usual Landau level basis by noting the relation
389
where #. = ~ ( 2 n + 1 )h72+flo
u,m(r)u*m(r') P,q
0-7
-
g X
1
3
-l
Z Uo,~(r)u*,~(r'), P,q
where Ln (x) is the Laguerre polynomial. Therefore, following the procedures in previous works, it is not difficult to see that calculating transport and thermodynamic quantities for 0 ~ 0 and the 3D anisotropic case is merely an extension of the 3D limit ( s ~ 0 ) of results in refs. [ 1,9]. We have only to perform the following two modifications: 1 ) The normalized magnetic field h = 2X~o2H/Oo in the nil c formulae is replaced by h72 everywhere it appears in them. 2) The coefficient ~2 of the q2 term in the quadratic (oc 10,12) terms in HIIc is replaced by ~2~7-4. Consequently, we obtain the following formulae for conductivities aa (a=X, Y, Z) in the directions of the main axis of the crystal and the specific heat C/AC: ~x = (COS20~± + ?/sin20~, )Y-4, O'y = ~3_ ,
az = q (cos/0OI, + q sin200± ) y - 4 ,
(13)
e2
-
5" ( n + l ) 8h~o¢. =% 1
X
+ ~n+l
~x/~5+#,+ l '
e2
Oil- 32h~---~by2 n=0 ~ fi~-3/2,
kahy 2 11~3/2 0flo C/AC= 8naC~o2~oc ~ o ~* '
(14)
kB
h
y21f1312 .
g = 2nAC #o2#o<
(15)
In obtaining eq. ( 15 ), we used a fact commented on in ref. [ 16 ] of ref. [ 1 ]. Note that, in a r and C/AC, the field and angular dependences appear only in the form of h72. This functional form, usually resulting from the mean field theory, explains why the (incorrect) Hc2(0) estimated from the in-plane resistivity Pv( ~-a 71 ) in early experimental studies for 90 K YBCO precisely agreed [ 14 ] with the angular dependence of the upper critical field in the mean field theory. Therefore, we expect that the estimation of the anisotropy q of less anisotropic materials like 90 K YBCO from the angular and field dependence of constant Pr (or C/AC) is a quantitatively reliable procedure. Further, the fact that the normalized magnetic field h is always accompanied by the angledependent factor 72, even in other quantities ax and az, suggests that the strongly anisotropic limit (q--. 0) ofeq. (13) is qualitatively the same as in the 2D limit ofeq. ( 1 ) mentioned above. In fact, in this limit, y2 is approximated by cos 0 except in the vicinity of 0=90 °, and the second terms of ax and az will be unimportant over the temperature range of our interest. Consequently, the sin20 contributions unrelated to the field h are entirely lost in eq. ( 13 ), and one finds that, for instance, ax becomes equivalent to at, although they are usually different from each other. Finally, we will consider the specific case 0= 90 ° (H_L2) of ax and e r in eq. (13). As already mentioned, ax and ay are not equivalent to each other in general. In fact, in 3D-like systems such as 90 K YBCO, deviations between them evidently appear in H_L c [ 8 ]. It was pointed out in ref. [15 ] that theoretical H_L c resistivity curves calculated in terms of
R. Ikeda/Angular dependenceof resistive transition
390
trx and try of eq. ( 13 ) reproduce the typical features seen in the 90 K YBCO data of ref. [ 8 ]. W h e n taking the r/--,0 limit in trx and try of this 0 = 9 0 ° case, we can find interesting results: since, in 0 = 9 0 ° , hy2=hx/-~ tends to vanish in q ~ 0 , the n - s u m m a t i o n s can be replaced by the corresponding integrals if the field strength H is not too large. Then, trx and tryprecisely coincide with each other and become H~O
e±
e2
1
- 32h~cx~oo"
(16)
Even in this case, the renormalization terms for/io, as in the 2D limit, can be regarded as being absorbed into a renormalization of T¢, indicating that the implicit field dependence in eq. (16) is also negligible [16]. Therefore, even if the layer structure is neglected, the H ± Z resistivity in the strongly an±sotropic limit shows the Lorentz force-free behavior a n d field i n d e p e n d e n c e on Ohmic regime, which qualitatively agree with observations [4,5] in BSCCO.
3. C o n c l u s i o n
In conclusion, we showed that, by studying the 2D limit of the L a w r e n c e - D o n i a c h model and the anisotropic 3D G L model, the i n d e p e n d e n c e of the resistivity from the in-plane field c o m p o n e n t and the Lorentz force-free behavior in H ± c, characteristic of strongly anisotropic systems, can naturally be understood according to the order parameter fluct u a t i o n theory. The agreement of our results based on the anisotropic 3D G L model with experimental observations in the O h m i c regime of the H i c configuration suggests that the origin of such peculiar p h e n o m e n a is not the discrete layer structure but rather the strong anisotropy of the materials.
Acknowledgements
The author would like to thank Y. Iye a n d K. Kadowaki for informative conversations and N. Motohira for sending him their resistivity data.
References
[1] R. Ikeda, T. Ohm± and T. Tsuneto, J. Phys. Soc. Jpn. 61 (1991) 1051. [2] R. Ikeda, T. Ohm± and T. Tsuneto, Phys. Rev. Lett. 67 (1991) 3874. [3]R. Ikeda, T. Ohm± and T. Tsuneto, in: Mechanisms of Superconductivity, ed. Y. Muto (JJAP series 7, 1992) p. 354. [4] Y. lye, T. Tamegaiand S. Nakamura, PhysicaC 174 ( 1991 ) 227. [5] K. Kadowaki, Physica C 185-189 ( 1991 ) 1811. [6] P.H. Kes, J. Aarts, V.M. Vinokur and C.J. van der Beek, Phys. Rev. Lett. 64 (1990) 1063. [ 7 ] N. Motohira et al., private communication. [8] W.K. Kwok, U. Welp, G.W. Crabtree, K.G. Vandervoort, R. Hulscher and J.Z. Liu, Phys. Rev. Len. 64 (1990) 966. [9 ] R. Ikeda and T. Tsuneto, J. Phys. Soc. Jpn. 61 ( 1991 ) 1337. [10] B.I. Halperin and D.R. Nelson, J. Low Temp. Phys. 36 (1979) 599. [ 11 ] We assumed here a possible existence of Hc~ for the component H± to be negligible. Indeed, this will be reasonable, because the energy of a singlepoint vortex in a multilayered system is proportional to In R, leading to vanishing H¢~, where R is the system size in directions parallel to the layers; see, for instance, A. Buzdin and D. Feinberg,J. Phys. (Paris) 51 (1990) 1971. [ 12] G. Briceno, M.F. Crommie and A. Zettl, Phys. Rev. Lett. 66 (1991) 2164. [ 13] See, e.g., D.S. Fisher, M.P.A. Fisher and D.A. Huse, Phys. Rev. B 43 (1991) 130. [ 14] See, for instance, Y. lye et al., Physica C 153-155 (1988) 26. [ 15] R. Ikeda, preprint (1992). [ 16] Of course, the Landau level-splittingin this anisotropic 3D case is no longer negligibleat sufficientlylow temperatures, leading to a slight field dependence of the resistivity.
Angular dependence of resistive transition in strongly anisotropic superconductors under magnetic field Ryusuke Ikeda Department of Physics, Kyoto University, Kyoto 606, Japan Received 13 July 1992
Resistivity and specific heat for arbitrary field and current configurations of strongly anisotropic high-T¢ superconductors are studied according to the renormalized fluctuation theory and by specifically considering the 2D limit of the layered model and the anisotropic 3D GL model. It is pointed out that the theory can naturally explain phenomena peculiar to strongly anisotropic materials, such as the independence of the resistivity in arbitrary field configuration from the in-plane field component, and socalled Lorentz force-free dissipation and field independence observed in resistivity measurements in the H_L c k I configuration of BSCCO.
1. Introduction
There has been a growing interest in various extraordinary features appearing in transport and thermodynamic data for the mixed state of high-To superconductors. In the field configuration Hllc-axis, the resistivity data show remarkable broadening phenomena for both cases of HZI and Hill (I is the applied current), which are at present understood quite well according to the order parameter fluctuation theory [ 1-3 ]. In other field configurations, one encounters various interesting phenomena in resistivity data carefully measured for strongly an±sotropic high-To superconductors such as BSCCO. For instance, several authors pointed out [4-6 ] that the in-plane (I±c) resistivity of BSCCO is, even in the specific case H ± c [ 5 ], essentially independent of the in-plane component Hb, of the applied field. Recently, such H,-independence has also been found inthe out-of-plane (Illc) resistivity of LSCO single crystals [ 7 ]. Further, the in-plane resistivity in H i c of BSCCO is independent of the relative angle between H and i (namely, Lorentz force-free) [4,5]. This iS in contrast to that in 90 K YBCO [8], in which case the H ± I resistivity is larger than the Hill one in the Ohmic regime. Clearly, these phenomena should be explained within a single theoretical framework.
In this paper, we point out that such striking features observed in resistivity data for strongly anisotropic materials can naturally be understood, as well as in the HIIccase [ 1-3 ], according to the fluctuation theory [ 1 ] for strongly type-II superconductors. Interestingly, we find that the Lorentz force-free behavior in H Z c can be obtained in the strongly anisotropic limit even if the discrete layer structure is neglected.
2. Calculations
Our analysis is based upon, as in previous works [ 1,9 ], the Lawrence-Doniach model in the extreme type-II limit where fluctuations in the magnetic field are entirely neglected:
+'2 =~x,r(-iO=- ~oAa)¥~(r)Z+q(~)2l~u,(r) (i+ ] )s
-~',+,(r) exp(i~ox
0921-4534/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
f
dZA~(r))12 ]
is
b
}
-4-~l~(r) l4 ,
(1)
R. Ikeda/Angulardependenceofresistivetransition where X, Y, Z denote the directions of the main crystal axis, and r the in-plane coordinates (see ref. [ 1 ] on other notations). Throughout this paper, the applied field H is assumed to be in the X-Z plane, and the angle between H and 2~is expressed by 0. We primarily focus on the qualitative results that the theory gives. Below, the 2D limit of eq. (1) and the anisotropic 3D model (s--,0 limit of eqs. ( 1 ) ) will be discussed separately.
2.1. The 2D limit First we will briefly explain the derivation of the formulae in the 2D limit, where the mass anisotropy r/, proportional to the Josephson coupling strength in the Z-direction, is vanishingly small. We can always choose the gauge A = - H ( A " cos 0 - 2 sin 0) Y. For any r/#0, it is quite difficult to study the renormalization of fluctuations in the layered system ( 1 ) with 0 # 0. Here, we merely focus on the leading contributions to physical quantities in the q--,0 limit and study the situation in which the Josephson coupling is negligible in renormalizing the fluctuations. Then, the hamiltonian (1) (under the present gauge) is equivalent to a sum of the 2D one in each layer under a field H I = H cos 0 perpendicular to the layers: ocf= ~ , . . It means that the nonvanishing physical quantities in the 2D limit, such as the in-plane fluctuation conductivity a± and the (normalized) specific heat C/AC, are obtained in the 0 # 0 case by, in the corresponding formulae for the Hll Z case [ 1,9 ], taking the q-,0 limit and replacing h=2nH~2/Oo by h cos 0. In contrast to them, the out-of-plane fluctuation conductivity G vanishes when the r/-, 0 limit is strictly taken, since the supercurrent in the Zdirection
JZ = iaq ~f ~oo( ~t*~, +, exp( i ~o HSY sin O) - c.c.) (2)
is explicitly multiplied by q. Nevertheless, one can see that the a~ formula in the ~/--.0 limit is precisely given, up to lowest order in q, by taking the 2D limit used above only in renormalizing the fluctuations. In fact, the current-current correlation function appearing in the Kubo formula for az is in the present case given by
387
Z (JZ(r)jZ(r')> t,J
, (4natl~ 2~2
=~,~)
2
~ <~t~(r')~(r)q4+~(r')q/*+j(r) >
2
=2(2~aq~O] E <~t(r')~,(r)> \ SOo ] , X ( ~+1 (r')~*+x (r) > ,
(3)
where we used the fact that, in the 2D limit, there are no inter-layer correlations between the order parameters in neighboring layers. Note that no vertex corrections [ 1 ] appear in eq. (3). This can be regarded as a quantitative justification for fitting results to the HIIIIIcBSCCO data given in ref. [ 2 ]. The az formula is easily obtained by means of eq. (3) and along the same line as in ref. [1]. Summarizing the abovementioned results, we give the formulae for a . , az and C/AC in the 2D limit: 82
a k ~ 4 - ~ s n=O ~ (n+l)
x (~__~+--# t +ll e2
{
-4+ 1) '
# n -[- #~t
22~
"~
ks/
C'/AC~- ks H
-
Oo
1 oHcos0 2 .=o#~ '
1 0#;
A---C0o--Scos 0 ,=o E #~ Oe •
(4)
For the renormalized mass parameters #~, we will use, as in refs. [ 1,9 ], the high-field expressions which are given by #~, = e + ( 2 n + 1 ) ~ ~2H cos 0 eo +#;(x'+ln(1 +x')),
_(
2x'
Og'o l+x'-ln(l+x')+ l---~x'] Oe x'= - kB H cos 0/#; 2 . - AC OoS
'
(5)
In eq. (5), we assumed that the contributions of higher Landau level modes to the mass renormalizations can be dropped out, since they can be considered to be absorbed into a slight renormalization of TcH [ 1 ]. A similar treatment for high-energy flue-
388
R. Ikeda/Angular dependenceof resistive transition
tuations was done in ref. [ 10 ] in deriving the zerofield formula (6) given below. Therefore, we expect these expressions (4) and (5), broadly speaking, to be useful even in the H±->0 limit, where all Landau levels are not negligible. Consequently, in this limit, tr'~ and C ' / A C reduce to their 2D gaussian formulae in zero field, respectively. If one would like to include the effect of the vortex unbinding transition, which occurs in the q--,0 limit, in t r ± ( H l - , 0 ) , it should be understood to reduce to the Halperin-Nelson interpolation formula [ l 0 ]
a'z
H±~0
e 2
~/p~_¢]2
1F.
(6)
= auN - 1~s/7 ~ [ smh N/ ~ _l '
where ~ - 1 +T/TKT (TKT is the true transition temperature in ~/=0), ec~-kB/ACns~, and fl is a constant of order unity [ I 1 ]. The corresponding limiting form of trz becomes n±~o
__~
1
+~o2 =~x, Y I - i O n - Oo a) ¢/I +~2<1 --iO:--
Az ~ulz + ~ I~ul4 ,
(9)
1
t,77 ' 2 r/,
Next, we consider the anisotropic 3D GL model
(7)
This is the leading term in r/of
2:4(~)
2.2. The anisotropic 31_) case
2~ A "~ E
/72 1 ~"
eE {~o~ E E
result is nothing but the statement noted by Kes et al. [6]. However, in deriving the H . - > 0 results above, the q ~ 0 limit was assumed at the outset for any 0, and thus, it is not clear at this stage if a calculation performed independently in the H ± Z (0 = 90 ° C) case leads to essentially the same result in the strongly anisotropic limit. In addition, the apparently different behavior seen in YBCO data [ 8 ] should be understood on the same footing. Below, we show that these requirements are indeed realized in the anisotropic 3D case.
(8,
which was used in fitting zero-field data in refs. [ 1,2 ]. Evidently, the formulae (4) indicate that the outof-plane conductivity G, as well as the in-plane conductivity tr±, is independent of the in-plane component of the field in strongly anisotropic systems. This result, consistent with recent LSCO data [ 7 ], does not seem to be understood according to other ideas. For instance, it is unreasonable to assume the presence of inter-layer phase slippage events in order to explain the large Ohmic dissipation observed in the configuration HIIIIIc of BSCCO [12 ]. Actually, since each Cu-O layer has macroscopic size, the plausible object which may be excited by the current is a vortex ring confined between neighboring layers, which, however, should not lead to any Ohmic dissipation [ 13 ]. The above result of tr'~ in H± -o0 suggests the field independence and the so-called Lorentz force-free behavior in H_I_Z. Actually, this apparently trivial
where the layer structure is neglected, and the sample anisotropy is taken into account as the ratio of the coherence lengths. In the fluctuation theory from higher temperature, it is always most useful to represent ~u in terms of complete and orthogonal basis functions fully diagonalizing the quadratic part of the hamiltonian. In the present case, such a basis set is given, under the gauge A = - H ( c o s 0 ~ ' - sin 0 2~) Y, by
--n D+
Unpq(r)=Ao x f ~ , n ! xex
- ~ (y+ ~ (p+~q))2+i(pX+qg)
,
(10) where ) ~ = y - t ( X cos O - Z sin 0), y=~,Y, g = Z cos O + X sin O, Ao is a normalization constant, and /5+ = - i ( G + o t O z ) + h y - O ; ,
(ll)
or=y-3( 1 - q ) sin 0cos 0, 74 = cosE0+ t/sinE0.
(12)
R. lkeda/Angulardependenceof resistivetransition Note that, in 0~0, 90 °, the ortho-normalized basis functions (10) are not Landau level eigen functions but rather their superpositions. Their completeness is readily shown in the same way as for the usual Landau level basis by noting the relation
389
where #. = ~ ( 2 n + 1 )h72+flo
u,m(r)u*m(r') P,q
0-7
-
g X
1
3
-l
Z Uo,~(r)u*,~(r'), P,q
where Ln (x) is the Laguerre polynomial. Therefore, following the procedures in previous works, it is not difficult to see that calculating transport and thermodynamic quantities for 0 ~ 0 and the 3D anisotropic case is merely an extension of the 3D limit ( s ~ 0 ) of results in refs. [ 1,9]. We have only to perform the following two modifications: 1 ) The normalized magnetic field h = 2X~o2H/Oo in the nil c formulae is replaced by h72 everywhere it appears in them. 2) The coefficient ~2 of the q2 term in the quadratic (oc 10,12) terms in HIIc is replaced by ~2~7-4. Consequently, we obtain the following formulae for conductivities aa (a=X, Y, Z) in the directions of the main axis of the crystal and the specific heat C/AC: ~x = (COS20~± + ?/sin20~, )Y-4, O'y = ~3_ ,
az = q (cos/0OI, + q sin200± ) y - 4 ,
(13)
e2
-
5" ( n + l ) 8h~o¢. =% 1
X
+ ~n+l
~x/~5+#,+ l '
e2
Oil- 32h~---~by2 n=0 ~ fi~-3/2,
kahy 2 11~3/2 0flo C/AC= 8naC~o2~oc ~ o ~* '
(14)
kB
h
y21f1312 .
g = 2nAC #o2#o<
(15)
In obtaining eq. ( 15 ), we used a fact commented on in ref. [ 16 ] of ref. [ 1 ]. Note that, in a r and C/AC, the field and angular dependences appear only in the form of h72. This functional form, usually resulting from the mean field theory, explains why the (incorrect) Hc2(0) estimated from the in-plane resistivity Pv( ~-a 71 ) in early experimental studies for 90 K YBCO precisely agreed [ 14 ] with the angular dependence of the upper critical field in the mean field theory. Therefore, we expect that the estimation of the anisotropy q of less anisotropic materials like 90 K YBCO from the angular and field dependence of constant Pr (or C/AC) is a quantitatively reliable procedure. Further, the fact that the normalized magnetic field h is always accompanied by the angledependent factor 72, even in other quantities ax and az, suggests that the strongly anisotropic limit (q--. 0) ofeq. (13) is qualitatively the same as in the 2D limit ofeq. ( 1 ) mentioned above. In fact, in this limit, y2 is approximated by cos 0 except in the vicinity of 0=90 °, and the second terms of ax and az will be unimportant over the temperature range of our interest. Consequently, the sin20 contributions unrelated to the field h are entirely lost in eq. ( 13 ), and one finds that, for instance, ax becomes equivalent to at, although they are usually different from each other. Finally, we will consider the specific case 0= 90 ° (H_L2) of ax and e r in eq. (13). As already mentioned, ax and ay are not equivalent to each other in general. In fact, in 3D-like systems such as 90 K YBCO, deviations between them evidently appear in H_L c [ 8 ]. It was pointed out in ref. [15 ] that theoretical H_L c resistivity curves calculated in terms of
R. Ikeda/Angular dependenceof resistive transition
390
trx and try of eq. ( 13 ) reproduce the typical features seen in the 90 K YBCO data of ref. [ 8 ]. W h e n taking the r/--,0 limit in trx and try of this 0 = 9 0 ° case, we can find interesting results: since, in 0 = 9 0 ° , hy2=hx/-~ tends to vanish in q ~ 0 , the n - s u m m a t i o n s can be replaced by the corresponding integrals if the field strength H is not too large. Then, trx and tryprecisely coincide with each other and become H~O
e±
e2
1
- 32h~cx~oo"
(16)
Even in this case, the renormalization terms for/io, as in the 2D limit, can be regarded as being absorbed into a renormalization of T¢, indicating that the implicit field dependence in eq. (16) is also negligible [16]. Therefore, even if the layer structure is neglected, the H ± Z resistivity in the strongly an±sotropic limit shows the Lorentz force-free behavior a n d field i n d e p e n d e n c e on Ohmic regime, which qualitatively agree with observations [4,5] in BSCCO.
3. C o n c l u s i o n
In conclusion, we showed that, by studying the 2D limit of the L a w r e n c e - D o n i a c h model and the anisotropic 3D G L model, the i n d e p e n d e n c e of the resistivity from the in-plane field c o m p o n e n t and the Lorentz force-free behavior in H ± c, characteristic of strongly anisotropic systems, can naturally be understood according to the order parameter fluct u a t i o n theory. The agreement of our results based on the anisotropic 3D G L model with experimental observations in the O h m i c regime of the H i c configuration suggests that the origin of such peculiar p h e n o m e n a is not the discrete layer structure but rather the strong anisotropy of the materials.
Acknowledgements
The author would like to thank Y. Iye a n d K. Kadowaki for informative conversations and N. Motohira for sending him their resistivity data.
References
[1] R. Ikeda, T. Ohm± and T. Tsuneto, J. Phys. Soc. Jpn. 61 (1991) 1051. [2] R. Ikeda, T. Ohm± and T. Tsuneto, Phys. Rev. Lett. 67 (1991) 3874. [3]R. Ikeda, T. Ohm± and T. Tsuneto, in: Mechanisms of Superconductivity, ed. Y. Muto (JJAP series 7, 1992) p. 354. [4] Y. lye, T. Tamegaiand S. Nakamura, PhysicaC 174 ( 1991 ) 227. [5] K. Kadowaki, Physica C 185-189 ( 1991 ) 1811. [6] P.H. Kes, J. Aarts, V.M. Vinokur and C.J. van der Beek, Phys. Rev. Lett. 64 (1990) 1063. [ 7 ] N. Motohira et al., private communication. [8] W.K. Kwok, U. Welp, G.W. Crabtree, K.G. Vandervoort, R. Hulscher and J.Z. Liu, Phys. Rev. Len. 64 (1990) 966. [9 ] R. Ikeda and T. Tsuneto, J. Phys. Soc. Jpn. 61 ( 1991 ) 1337. [10] B.I. Halperin and D.R. Nelson, J. Low Temp. Phys. 36 (1979) 599. [ 11 ] We assumed here a possible existence of Hc~ for the component H± to be negligible. Indeed, this will be reasonable, because the energy of a singlepoint vortex in a multilayered system is proportional to In R, leading to vanishing H¢~, where R is the system size in directions parallel to the layers; see, for instance, A. Buzdin and D. Feinberg,J. Phys. (Paris) 51 (1990) 1971. [ 12] G. Briceno, M.F. Crommie and A. Zettl, Phys. Rev. Lett. 66 (1991) 2164. [ 13] See, e.g., D.S. Fisher, M.P.A. Fisher and D.A. Huse, Phys. Rev. B 43 (1991) 130. [ 14] See, for instance, Y. lye et al., Physica C 153-155 (1988) 26. [ 15] R. Ikeda, preprint (1992). [ 16] Of course, the Landau level-splittingin this anisotropic 3D case is no longer negligibleat sufficientlylow temperatures, leading to a slight field dependence of the resistivity.