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3. High Technology - Vol. 32
Fluctuation Phenomena in High Temperature Superconductors edited by
Marcel Ausloos SUPRAS, Institute of Physics, University of Lidge, Liäge, Belgium
and
Andrei A. Varlamov "Forum": INFN Laboratory for the Theory of Condensed Matter, Department of Physics, University of Florence, Florence, Italy
Springer-Science+Business Media, B.V.
Proceedings of the NATO Advanced Research Workshop on Fluctuation Phenomena in High Critical Temperature Superconducting Ceramics Trieste, Italy 5—9 August, 1996
A C L P . Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6331-9 ISBN 978-94-011-5536-6 (eBook) DOI 10.1007/978-94-011-5536-6
Printed on acid-free paper
All Rights Reserved © 1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997 Softcover reprint of the hardcover 1st edition 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photo­ copying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
PART I. Near the Ginzburg-Landau Temperature 1
A. A. Varlamov and M. Ausloos, Fluctuation phenomena in superconductors 3
o. Rapp, J. Axn1ls, Yu. Eltsev and W. Holm, C-axis transport
in YBa2Cu307_B
T. Plackowski, Analysis of the specific heat jump at Tc for
5mB~Cu3~_B
S. K. Patapis, Fluctuation conductivity in a-b plane and along c-axis
in films of YBaCuO
superconducting transition
W. Lang, G. Heine, W. Liebich, X. L. Wang and X. Z. Wang,
Magnetoresistance in HTSC far above T c : fluctuations versus
normal-state contributions
P. Konsin, B. Sorkin and M. Ausloos, Electric field effects in high-Tc superconductors
M. Houssa, H. Bougrine and M. Ausloos, Effect of fluctuations on the
thermal conductivity of High-T c superconductors
I. G. Gorlova, S. G. Zybtsev, V. Va. Pokrovskii and V. N. Timofeev,
Fluctuation phenomena in BSCCO (2212) whiskers
43
53
61
73
81
91
101
113
VI
F. Federici and A. A. Varlamov, The fluctuation induced pseudogap in the
infrared optical conductivity of high temperature superconductors 121
A. Carrington, F. Bouquet, C. Marcenat, D. Colson, V. Viallet and A. Tyler,
Specific heat studies oflow dimensional high-T c superconductors 131
P. Carretta, A. Rigamonti, A. A. Varlamov and D. Livanov,
d-pairing in high temperature superconductors: pro and contra from the
fluctuation phenomena analysis
J. Booth, Dong-Ho Wu and S. M. Anlage, Measurements of the frequency
dependent microwave fluctuation conductivity of cuprate thin film
superconductors
A. Barone and A. A. Varlamov, Fluctuation phenomena in tunnel
and Josephson junctions
G. Balestrino, E. Milani and A. A. Varlamov, The role of density of
141
151
179
states fluctuations on the c-axis resistivity of High Tc superconductors 201
PART II. Near the Vortex-lattice Melting Transition 217
N. K. Wilkin and H. J. Jensen, The effect of disorder on melting and
decoupling transitions in layered superconductors 219
Tao Chen and S. Teitel, Vortex line fluctuations and phase transitions in
type IT superconductors 231
D. Stroud and R. Sasik, Flux lattice melting in the lowest Landau level
approximation: results in three dimensions 239
VII
E. Silva, R. Fastampa, M. Giura, D. Neri and S. Sarti, Fluctuational
contribution to the resistivity in YBa2Cu307_o in magnetic fields 251
P. Pureur and R. Menegotto Costa, Low field fluctuation
magnetoconductivity in Bi2Sr2CaCu208 and YB~Cu307 :
gaussian, critical, and LLL scatings
A. V. Nikulov, Fluctuation effects in mixed state of type II
superconductor
superconductors
Ginzburg-Landau theory
S. A. Ktitorov and E. S. Babaev, Fluctuations in the lattice
Ginzburg-Landau model
in strong magnetic field
G. Carneiro, Vortex fluctuations in vortex liquids
A. Buzdin and V. Dorin, Magnetic field crossover between 2D and 3D
regimes of gaussian fluctuations in layered superconductors
259
271
279
293
301
311
323
335
PART III. A Little Bit More Theoretical 343
H. Won, K. MaId and Y. Sun, Aspects of the d-wave superconductivity 345
T. Schneider and J. M. Singer, Universal critical quantum properties of
cuprate superconductors 361
B. N. Narozhny, Theory of superconducting fluctuations in the strong
coupling model 369
R. A. Klemm, Fluctuation phenomena in layered superconductors 377
B. L. GyOrffy, G. Litak and K. I. WysokiIlski, Anderson theorem and
spatial fluctuations in the gap of disordered superconductors
L. Capriotti, A. Cuccoli, V. Tognetti, R. Vaia and P. Verrucchi,
Berezinskii-Kosterlitz-Thouless transition in two-dimensional XX:Z
385
easy-plane quantum Heisenberg magnets 397
J. F. Annett, Pairing symmetry and pairing interactions in the cuprates 405
C. Attanasio, C. Coccorese, L. Maritato, L. Mercaldo and M. Salvato,
Superconducting properties of Nb/CuMn multilayers 415
INDEX 425
PREFACE
These Proceedings of a NATO-ARW (HTECH ARW 96 00 52) held at the
International Center for Theoretical Physics, Trieste, Italy from Aug 5 till Aug 9,
1996 resulted from many discussions between various workers, concerning the need for
a gathering of all (if possible) who were concerned about the subject of
superconductivity fluctuations in High critical Temperature Superconductors (HTS). It
appeared to many that the Skocpol-Tinkham work of 1975 had to be revitalized in view
of the discovery of the new superconducting ceramics and the enormous amount of
work having already taken place. The study of HTS is one of the most prominent
research subject in solid state sciences. The understanding of the role of fluctuations is
also thought to be necessary before technological applications since the fluctuations
may destroy the superconducting state.
The workshop discussions have touched upon (i) Superconducting fluctuations in the
vicinity of the critical transition, (ii) Superconductivity fluctuations near the
percolation transition, and (iii) Fluctuations of the vortex lattice at the lattice melting
temperature. These topics served as initiators for a very great amount of discussions
with many comments from the audience. More than forty "long lectures" and two
"poster sessions" were held. Private discussions going unrecorded but obviously took
place at many locations : lecture halls, staircases, cafetaria, bedrooms, bars, beach, ...
Arguments openly repol ted for the first time were often quite sharp ones
The effect of fluctuations on static properties, and on electrical and thermal transport,
properties in the ab- plane and along the c-axis, the effect of magnetic field on
fluctuation phenomena in specifically layered compounds were intensely debated
subjects. Whether the d-wave order parameter is definitely the main HTS key feature is
still under debate. Nevertheless, it seems that the fluctuation contribution above T c
away from the critical temperature should take into account a density of states effect
beside the Aslamazov-Larkin and the Maki-Thompson contribution. The c-axis data
on the paraconductivity and the subsequent analysis seems convincing in that respect.
IX
x
The effect of a magnetic field is still unclear. The transport property behaviors near the
Kosterlitz-Thouless transition have been much discussed for various sytems. It is still
hard to say which of the quasi particle scattering or vortex motion is the key
dissipation mechanism in the mixed state and what their respective role is at the
transitions.
Specific fluctuations of the vortex lattice in HTCS as compared to conventional
superconductors were very intensely debated. Quite interesting considerations were
reported for the first time, either from Monte-Carlo simulations or theoretical work. It
appeared that the considerations do not necessarily describe the overall experimental
situations well enough nor give some clearly phrased insight on the processes at hand.
Therefore much work has still to be done on that question. Nevertheless the
complicated (H, T) phase diagram should emerge from all those considerations and
expectations which were presented and discussed. However the type of (first or second
order) transition, the critical exponents, the line(s) or point(s) singularities are major
raised questions for the future and will obviously attract much attention as seen from
the debates which went on. Whether the lowest Landau level approximation or the
vortex motion dissipation is the key theoretical ingredient is still opened. Maybe they
do not describe the same region of the (H, T) plane.
The Proceedings are more or less arbitrarily divided into three parts : one is reserved for
papers mainly considering the Landau-Ginzburg transition, sometimes in presence of a
magnetic field, and including the percolation region. The second part is mainly
concerned with the vortex structure behavior, whether fluctuating or not between
various phases. The third part contains more theoretical papers which could also have
found their place in Part I or Part II ... but a ternary structure is always more appealing.
The various presentations were allowed some space for these proceedings. Due to
various committrnents not all could be inserted in due time. There is more theory than
experiment reported in the following, but often theoreticians took great care in
describing their work with respect to available experimental data.
Xl
The editors had to choose some order of presentation. That was very hard to do.
Sometimes there is some overlap, and why should one paper reporting some
consideration be located, printed and read before another? There is no strict rule. We
decided to choose an order "counter-alphabetically" in each Part. In so-doing the paper
by the editor comes first, and nobody should be upset; that is a privilege. Notice that
the editors would have likely chosen the alphabetical order for presentation, if the order
of authors had been reversed on the first paper! Never mind! In so doing, the second
paper of the first Part appears to be following well the first one, and so on, and the
first Part ends with a very interesting work as well. The first Part has thus nice
boundaries. The same is fortunately true for the second and third Part. Moreover in
choosing that order and the three part structure, the last paper was becoming a fine
more or less conclusion paper. However the editors made then one exception, and
interchanged the last and next to the last paper. That is only reflecting the friendly and
open schedule that we had during the meeting. In fact, in so doing, the last paper of
these proceedings does not directly concern High T c Superconductors.
The editors and organizers (including Dr. Tesanovic) are very pleased to acknowledge
the financial support of NATO for this ARW. The financial matters were dealt
extremely well. We have a deep appreciation for the help and cooperation of the NATO
Scientific Affair Division staff and leaders, who allowed us much freedom in the
organizational and scientific matters (within NATO rules of course). The format of this
NATO-ARW was proven to be as successful as that used in other NATO-ARW. Post
workshop collaborations are already taking place among groups which had no previous
direct connexions.
Moreover, we emphasize here our expression of the deepest appreciation for the help
and cooperation the members of the ICTP, Trieste gave us. We should mention Dr.
Yu Lu, Dr. Hilda Cerdeira, Mrs. Marialuisa Viani, and Mr. M. Michelcich. Special
thanks go to our family and to our scientific coworkers as well for their understanding
now and then.
PART I.
FLUCTUATION PHENOMENA IN SUPERCONDUCTORS
A.A.VARLAMOV
Istituto Nazionale di Fisica della Materia Laboratorio "Forum" Dipartimento di Fisica, Universitd di Firenze L.E.Fermi, 2 Firenze 50125, Italy and Moscow Institute of Steel and Alloys Leninsky prospect 4, 117936 Moscow, Russia
AND
M. AUSLOOS
SUPRAS Institute of Physics, B5 Universite de Liege, B-4000 Liege, Belgium
1. Introduction
During the first half of this century after the prominent discovery done by Kamerlingh-Onnes, the problem of fluctuations, smearing the supercon­ ducting transition, had not even been taken into account: in bulk samples of traditional superconductors the critical temperature Tc sharply divides the superconducting and the normal phases indeed. It is worth mention­ ing that such a behavior of the physical characteristics of superconductors is in perfect agreement both with the Ginzburg-Landau phenomenological theory (1950) and the BCS microscopic theory of superconductivity (1957) [1 J. Nevertheless, in the same time it was well known that thermodynamical fluctuations play an important role in the description of many other phase transitions [2J, so-called second order phase transitions, (like the "-point in liquid helium) often strongly smeared out by order parameter thermal fluctuations.
The characteristics of high temperature and organic superconductors, low dimensional and amorphous superconducting systems studied today, differ strongly from the traditional superconductors included in textbooks.
3
M. Ausloos and A.A. Varlamol' (eds.), Fluctuation Phenomena in High Temperature Superconductors. 3-41. © 1997 Kluwer Academic Publishers.
4
The transition points turn out to be much more smeared out here. The appearance of nonequilibrium Cooper pairs in a non-equilibrium thermody­ namic state (superconducting fluctuations) above the critical temperature leads to the rise of precursor effects of the superconducting phase already in the normal phase, often far away from Te. The electrical conductivity, the heat capacity, the diamagnetic susceptibility, the sound attenuation, the thermoelectric power, the thermal conductivity, etc. may considerably vary in the vicinity of the transition temperature due to contributions from these fluctuating states.
What is the principal difference between conventional and unconven­ tional superconductors, and in general, what determines the role and the strength of fluctuations in the vicinity of the superconducting transition? How widely turns out to be smeared the transition point in already de­ signed superconducting devices? How to separate the fluctuation contribu­ tions from the other ones? What microscopic information can be extracted from the analysis of the fluctuation corrections in different physical char­ acteristics of superconductors?
These questions, side by side with many others, find their answers in the theory of fluctuation phenomena in superconductors. This chapter of superconductivity phenomena was developed in the last 25 years by the combined efforts of many theoreticians and experimentalists.
The first numerical estimation of the fluctuation contribution to the heat capacity of superconductors in the vicinity of Tc was done by Ginzburg in 1960 [3]. In that paper he showed that superconducting fluctuations increase the heat capacity already above the transition temperature. In this way the fluctuations smear the jump in the heat capacity which, in accordance to the phenomenological Ginzburg-Landau theory of second order phase transitions (see for instance [1]), takes place at the transition point itself. The range of temperatures where the fluctuation correction to the heat capacity of a bulk and clean conventional superconductor is important was estimated by Ginzburg as being equal to
(1 )
where a is the interatomic distance, tF is the Fermi energy, ~o is the coher­ ence length 1. It is easy to see that this range of values turns out to be many orders of magnitude less than what is verifiable with experiments having
1 In the theory of phase transitions now the relative width of fluctuation region is called the Ginzburg-Levanyuk parameter Gi(D) ; its value really depends on the spa"", dimensionality D and on the impurity concentration,
5
some temperature instability. That is why, during a long time, the fluctua­ tion phenomena in superconductors had been considered to be inaccessible for experimental studies.
The formulation of the microscopic theory of bulk superconductivity, the theory of the second kind superconductors, the search of high Tc su­ perconductivity attracted some attention towards dirty systems, while the properties of superconducting films and filaments began to be studied as well. In 1968, in the paper of Aslamazov and Larkin [4], that is well known now, a consistent microscopic theory of fluctuations in the normal phase of a superconductor in the vicinity of the critical temperature was formu­ lated. The microscopic approach confirmed the Ginzburg's evaluation [3] for the width of the fluctuation region in a bulk clean superconductor. But much more interesting results were found in [4] for dirty and low dimen­ sional superconducting systems. The power of the ratio (al ~o), which enters in (1), drastically decreases as the effective dimensionality of the electron spectrum diminishes. Another possibility to increase the strength of the fluctuation effects is to decrease the coherence length. That really happens in dirty superconductors because of the diffusive character of the electron scattering. It means that the fluctuation phenomena can be more easily observable in amorphous materials with reduced dimensionality, like films and whiskers, where both facts mentioned above take place. High tempera­ ture superconductors present a special interest in this sense, because their electron spectrum is extremely anisotropic and their coherence length is very small. As a result the temperature range in which the fluctuations are important may be measured by tens of degrees[5].
The manifestation of superconducting fluctuations above the critical temperature may be- conveniently demonstrated in the case of the electrical conductivity. In a first approximation it may be reduced to three different effects. The first one, a direct effect, consists in the appearance of nonequi­ librium Cooper pairs with a characteristic lifetime TG£ rv (hiT - Tc) in the vicinity of the transition. In spite of their finite lifetime, some definite number of such pairs (of course depending on the closeness to Tc ) is al­ ways present in the unit volume of the normal phase (below Tc they are in excess in comparison with the equilibrium value). Their presence gives rise, for instance, to the appearance of the precursor of the Meissner-Ochsenfeld anomalous diamagnetism in the normal phase, which is manifested by the anomalous increase of the diamagnetic susceptibility at the edge of the transition. As far as the electrical conductivity is concerned, one can say that above Te , because of the presence of nonequilibrium Cooper pairs, a new, nondissipative, channel of charge transfer is opened. Such direct fluc­ tuation contribution to the conductivity is called the para conductivity or the Aslamazov-Larkin contribution [4].
6
t f 0 ,7
-z.O Tl,C -1/1
fiT
Figure 1. The normalized correction oN (f) /e2 ,c to the single-particle density of states vs the energy f in units of Tc for a two-dimensional sample in the case of a clean super­ conductor above Te. r;;i assumes the values O.02T" O.04Te and O.06T,. In the inset the behaviour of fO (T) vs r;;l is shown.
Another consequence of the appearance of fluctuating Cooper pairs above Tc is the decrease of the one-electron density of states at the Fermi leveL If some electrons are involved in the pairing they cannot simultane­ ously participate in the charge transfer and in the heat capacity as one­ particle excitations. Nevertheless, the total number of the electron states cannot be changed by the Cooper interaction, and only a redistribution of the levels is possible along the energy axis [6, 7]{Fig.l). One can speak about the opening of a fluctuational pseudogap in the Fermi level in connection with what has been said.
The decrease of the one-electron density of states at the Fermi level leads to a reduction of the normal metal conductivity. This, indirect, fluctuation correction to the conductivity is called the density of states contribution and it appears side by side with the paraconductivity. It has an opposite (negative) sign and turns out to be much less singular in T - Tc in compar­ ison with the Aslamazov-Larkin contribution, so that in the vicinity of Tc it can be usually omitted. Nevertheless, in many cases [8,9, 10, 11, 12, 13], far from Tc or when, because of some special reasons, the main corrections are suppressed, the density of states contribution becomes of the greatest importance. Namely such "exotic" situation takes place in tunnel structures and in the modern problem of the c-axis resistivity component of strongly anisotropic high temperature superconductors, where the electron motion between the conducting planes has a tunneling character [8, 9, 14].
Finally we have to mention the third, purely quantum, fluctuation COIl­
tribution, generated by the coherent elastic scattering of electrons formillg
7
Cooper pair, on impurities. It is the so called anomalous Maki-Thompson contribution [15, 16] which often turns out to be very important in conduc­ tivity [13J and in other transport phenomena at the edge of the transition. Its temperature singularity is similar to the paraconductivity one, but this contribution turns out to be extremely sensitive to electron phase breaking processes (like spin-flip scattering, proximity effect, etc.).
Below we present an introduction in the form of a brief review on fluc­ tuation phenomena in superconductors.lt is geared to newcomers in the field.
2. The description of fluctuations in the Ginzburg-Landau func­ tional formalism.
In the study of the superconducting transition thermodynamics near the critical temperature, we. start from the well known functional for the free energy in the Ginzburg-Landau form:
(2)
+ (ii - V x 1)2 (V. 1)2 (ii. V x 1)] -----+ + .
8~ 8~ 4~
This functional takes into account the first terms of the free energy expansion over the order parameter and its space derivatives (1L112 deter­ mines the concentration of Cooper pairs). Such an approximation is valid from both sides of the transition point between the normal and supercon­ ducting phases. The expansion (2) is accomplished in the presence of an external magnetic field ii. The vector potential A describes the distribution of the magnetic field in the volume of the sample, moreover its calibration is chosen to have a gauge-invariant form(2}. The magnetic contribution to the energy of the superconductor in an external field is taken into account through the last two terms in (2). The Band C coefficients, in the vicinity of the transition, may be assumed as positive constants, the coefficient it has to be positive at T > Teo and negative at T < Teo. Such a definition of A provides for the existence of the free energy minimum at L1 i' 0, for temperatures below the critical one. The microscopic theory gives for these coefficients the explicit expressions:
8
(3)
where
V F 2
{ 2 [ ( 1 1) ( 1 ) ] rt~ ( 1) } 1] - - Tt 'IjJ - + - 'IjJ - - --'ljJI - - 3 ~ 2 41rTeOTtr 2 41rTeo 2
(4)
in clean case (TT ~ 1) (5)
in dirty case (TT <t:: 1).
Here ((;v) is the Riemann zeta-function, and 'IjJ(;v) is the Euler digamma­ function.
The expansion (2) may be written both above and below Teo. However, a smooth variation of the order parameter as a function of the coordinates is essential for its validity, i.e. V D,( r) has to be small. This condition imposes a restriction on the value of the external magnetic field if and as a matter of fact requires H <t:: He2 ( 0).
Minimizing the free energy functional (2) one can find the equilibrium value of L\(T) and the appropriate value of the free energy F(T). Therefore all thermodynamical characteristics of a superconductor in the vicinity of the transition temperature can be determined. In the simple H = 0 case, when one can easily see that
if T > Teo;
In agreement with (6) the heat capacity presents a jump
(7)
9
as the temperature goes through the critical point.
In the general case the minimization of the functional (2) over the func­ tions ~(T) and A( i) results in the so called Ginzburg-Landau equations, well known in the theory of superconductivity, and from which the equilib­ rium configuration expressions ~o( T) and Ao( T) can be found. The problem of the description of the superconductivity thermodynamics can be solved in this way in a general form in the vicinity of the transition and for suf­ ficiently weak magnetic fields. However, the thermodynamical fluctuations which become particularly important at the edge of the transition drop out from such considerations in the scheme proposed here above.
The full description of the phase transition can be done through the exact calculation of the partition function which is determined by the func­ tional integral carried out over all possible configurations of the complex function ~(T):
(8)
The equilibrium functions ~o( T) and Ao( T), found from the Ginzburg­ Landau equations, correspond to the saddle point in the integral (8).
It is worth mentioning that, generally speaking, one should carry out the integration in (8) over all A( T) field configurations too. This procedure corresponds to taking into account the fluctuations of the electromagnetic field as well. However, such fluctuations are not specific to the superCOll­ ducting transition point and here we shall omit them, thus assuming the absence of electromagnetic field fluctuations 2 •
If one knows the exact partition function Z, it is easy to find the free energy of a superconductor including the fluctuation contribution:
F = -TlnZ. (9)
Of course, in the general case the path integral (8) cannot be carried out and some simplifying assumptions have to be made before its computation. Some of them will be demonstrated below as an example of the calculation of the fluctuation contributions to the thermodynamical characteristics of a superconductor.
2The calculation of the contribution of the electromagnetic field fluctuations leads to some slight decrease in the superconducting transition temperature.
10
3. Fluctuation correction to the heat capacity.
Let us begin with the calculation of the fluctuation contribution to the heat capacity in the normal phase of a superconductor. We restrict ourselves to the region of temperatures where this correction is still small. We also omit the high order terms in A( r') and its derivatives. For simplicity, from the beginning, we shall assume if = 0 and 1 = 0, such that in the Ginzburg­ Landau functional (2) only the first and third term are kept. So far as we consider the system above the transition temperature, A( r} has the meaning of a fluctuating order parameter: its mean value is equal to zero. It depends on the space variables even in the absence of a magnetic field. After expanding the partition function in a Fourier series one can find:
(10)
It
The integration in the last expression is carried out over all complex Fourier components of the order parameter
(11 )
and the product over k is related to the Fourier sum which takes place in the exponent of (10) and is carried out over all
- 211" - 211" - 211" - k = -nxi + -nyj + -nzl,
Lx Ly Lz
where Lx,y,z are the sample dimensions in appropriate directions; i,), f are unit vectors along the axes; nx,y,z are integer numbers; V is the volume of the sample.
The fluctuation contribution to the free energy F = - TIn Z is found after integrating over d2 Af to be:
(12)
The appropriate correction to the heat capacity of a superconductor at
temperatures above the critical one may thus be expressed as
II
(13)
Only the most singular term in C 1 , € <{::: 1, is kept in this formula. The result of the following summation over k strongly depends on the
linear sizes of the sample, i.e. on its effective dimensionality. As it is clear from (13), the scale with which one has to compare these sizes is determined by the value (1]/ €)~, which, following the Ginzburg-Landau theory, coincides with the temperature dependent effective size of the Cooper pair ~(T). Thus, if all dimensions of the sample considerably exceed the value (1]/ €) ~ one can integrate over (21rt3 L"LyLzdk"dkydkz, instead of summing over n", ny, n z • The fluctuation correction to the heat capacity of the sample turns out to be equal to
(14)
In the same way one can easy find the general formula for the fluctuation correction to the heat capacity of a superconductor in its normal phase
(15)
where
for a film of section S; (16)
for.a bulk sample of volume V.
In the case of small superconducting particles or granules, with char­ acteristic sizes R ::; ~(T) we can speak about "zero-dimensional" samples and the appropriate fluctuation contribution to the heat capacity is
(II)
From the formula given above it is easy to see that the role of the fluctuations increases when the effective dimensionality of the sample or the
12
electron mean free path decreases. Let us warn that in the above formula (16) the energy is measured in Kelvins with kB = 1. Practically, the results so obtained should be multiplied by the Boltzmann constant kB = 1.38 . 10-16erg/K.
From the expressions of the fluctuation correction to the heat capacity quoted above, one can reproduce the already cited Ginzburg-Levanyuk pa­ rameter which gives an estimate for the width of the critical fluctuations region (the region where the fluctuation contribution begins to be of the same order or even to exceed the normal value of the heat capacity or of some other thermodynamical or kinetic characteristic under consideration):
6.Te B2Te2 -- f'V--
Te C3a . (18)
From formula (10), one can find the fluctuation contribution to the heat capacity at temperatures below Te too. For this purpose let us restrict our­ selves to the region of temperatures not very close to Te , where fluctuations are sufficiently weak. In this case the order parameter can be presented as the sum of the equilibrium (6.0) and fluctuation part (6.1):
(19)
As already mentioned here above the value 16.012 = -( aiEI/ B) is determined from the Ginzburg-Landau equation; in the language of path integral ap­ proach (6) it is its saddle point. Keeping in this expression the terms of the second order over 6. 1 , one can easily find
z = f].. J dR6. 1 d8'6. 1 exp { - ; [27 + 1]PR26.1 + 1]P8'2.6.d } . (20) k
Carrying out this integral one can see that the fluctuation correction to the heat capacity is proportional to that one calculated above for the normal phase of a superconductor
(21)
but for temperatures below Teo Hence, in the framework of the Ornstein­ Zernike fluctuation theory which is like a mean field theory [2J we find that the heat capacity of the superconductor tends to infinity. ,X-point with a finite jump of the heat capacity when the temperature goes to the critical point from both sides of the transition.
13
Strictly speaking, these results do not permit us to discuss this diver­ gence at the critical point itself. The calculations are in principle valid only in that region of temperatures where the fluctuation correction is small. The criterium of applicability of the mean field theory is evident. Below To the zero-order term should dominate over the fluctuation corrections in the expansion of the heat capacity: this zero order term is the magnitude of the heat capacity jump (7). Therefore it would be more reasonable to assume that the theory proposed here above is valid up to temperatures at which the fluctuation corrections (20) are small in comparison with the value of the heat capacity jump:
(22)
The next fluctuation corrections to the heat capacity are also positive. They diverge at the transition point and they change only the power law with which the heat capacity tends asymptotically to infinity. Therefore the behavior of the heat <:apacity in the critical region oftemperatures cannot be strictly studied inthe framework of the present theory. Other theoretical ideas, for instance the renormalization group theory [17], should be brought forward.
It is worth mentioning that, because of the large value of the coherence length, whence of the Cooper pair size, which drastically exceeds the in­ teratomic distance, the fluctuation correction to the heat capacity, occurs in the immediate vicinity of the transition temperature, and is relatively small. The critical region determined from the condition (22), for clean bulk superconductors may be estimated in 3D to be
a Gi rv (~O)4 rv 10-8 -7- 10-16
where a rv 10-scm is the interatomic distance and ~o rv 10- 6 -7- 10- 4cm is the coherence length.
However, the fluctuation effect increases for small effective sample di­ mensionalityand small electron mean free path. For instance, the fluctu­ ation heat capacity ofa small superconducting granular system is readily accessible for experimental studies of such effects.
4. Fluctuation correction to the diamagnetic susceptibility.
In contrast to the case of the heat capacity, the role of fluctuations in the diamagnetic susceptibility, at temperatures above To turns out to be much more important. Both facts that the susceptibility of a normal metal is extremely small and the superconductor is an ideal diamagnetic suggest
14
that a noticeable role can be played by the superconducting fluctuations in the diamagnetism of the normal phase at the edge of the transition. It is easy to qualitatively estimate the expected value of this effect [18]. Above Te , a Cooper pair will appear and decay as a result of the thermodynam­ ical fluctuations. The characteristic size of the fluctuating Cooper pair is determined by the Ginzburg-Landau coherence length
(23)
The density of such nonequilibrium Cooper pairs is determined by the average square of the order parameter 3 n "-'< 1~12 >. Hence their energy may be estimated as
~2 4 n 1 12 3
E = m.~GL2 < ~ > 31r~GL (24)
and this energy has to be of the order of the thermal energy kBT. This condition gives
(25)
A qualitative understanding of the phenomenon of the diamagnetic sus­ ceptibility increase may be obtained from the well-known Langevin expres­ sion for the atomic susceptibility:
X "-' -----,--­ mc2
(26)
Referring to Cooper pairs which can be imagined as a set of two-particle rotating system it is clear that < r2 > has the meaning of ~bL' where n and m have been defined above. Thus
(27)
T 1 diverges as ( e ) as the temperature tends to the critical one.
T - Te
3It is worth noting that inspite of the condition < /:i >= 0, because of the presence of fluctuations, we can have < /:i 2 ># 0 even above Tc.
15
Next we derive the exact expression for X/I in a weak magnetic field starting from the Ginzburg-Landau functional for the free energy (2). How­ ever let us discuss first the evident restrictions on the magnetic field and the temperature ranges which should be assumed in this approach. In [19, 20] the attention was focussed on the fact, that the Ginzburg-Landau free­ energy functional is valid only for long wavelength fluctuations, whereas it turns out that fluctuations of all wavelengths contribute to the magneti­ zation at all but very small values of T - Tc (where T - Tc <t: Tc). The presence of a magnetic field gives rise to a nonuniformity of the system. We can take it into account by considering the first terms of the expansion over the gradient of ~ only if the coherence length ~o is much less than the magnetic length Jhe/eB. This condition can be rewritten as
he ~o B<t:-rv-
e~o 2 ~o 2 (28)
where ~o = 2.05 . 10-70e . em2 is the quantum magnetic flux. The latest value in (28) gives an estimate for Bc2 in the case of a type
II superconductor. Hence, from the very beginning we shall restrict our consideration to the case T - Tc <t: Tc and B --+ 0 and follow the Schmid and Schmidt theory [18, 21]. Then the exact (in the framework of the GL-functional approach) Prange [22] theory will be discussed for the case T - Tc <t: Tc and B <t: B c2 ' The extension of the theory to the case of strong fields (B f'V Bc2 ) and arbitrary temperatures needs a more sophisticate diagrammatic approach.
Let us begin, as usual, from the functional (2). Above To where fluc­ tuation effects are comparatively small, the average magnetic field in the metal, jj, may be assumed to be equal to the external field and in (2) one can omit the last three terms. Because of the smallness of 1~12, which we have assumed, the fourth power term may be omitted as well.
The expression (29) can be rewritten in a more convenient form after an expansion of the order parameter over the basis of eigenfunctions {rPnk ( r)} of a free electron in a permanent magnetic field:
(30) n,k
J FGLdY = a ~ {€ + 11 [p + iJ (n + ~)]} iLln,ki2 (31) n,k
where k is the momentum along the direction of the magnetic field; if is the transversal momentum; Llnk = J ¢:k( r )Ll( i)¢nk( i)dY and iJ = (4eH Inc). Substituting this expression in (2) and carrying out the integra­ tion over the order parameter configurations one can find
Z = J 11 d2 Llnk exp ( - k;T J FGLdY) n,k
(32)
where
FGL = -TlnZ = -T2: ln _7rkB~ (33) n,k a{€ + 11[k2 + B(n + ~)]}
Taking into account that the number of the single particle states with definite quantum numbers n and k is (2eB 127rnc) multiplied by the crossec­ tion ofthe sample in the plane perpendicular to the magnetic field direction, we obtain the following expression for the free energy:
(34)
where Y is the volume of the sample. Firstly [18, 21] this formula was eval­ uated in a very simple way by the application of the Poisson transformation [23] of the sum in (33) into an integral
Here F(O) is the free energy of the sample in the absence of the magnetic field. The integral over J; can be carried out integrating twice by parts and, for sufficiently weak field B ~ if!o/ ~GL \ the integral may be approximated by
100 h2iJ 1 1 dx( .... ) = --( )2 22 •
o 2m 21rS h k -+al£1 2m
17
(36)
The last summation and the integration over k are trivial and one can find
(0) 1 (e 2 2 F = F + -V -) kBT . ~GLB • 121r he
Using the definition
1 {PF X = -V 8B2·
one can make oneself sure about the validity of the above estimate: •
1 ( e )2 Xjl = -- - kBT· ~GL 61r he
(37)
(38)
(39)
Note that this formula predicts a nontrivial increase of Xjl for clean metals. The usual statement that fluctuations are most important in dirty super­ conductors with a short electronic mean free path does not hold in this particular case because ~GL turns out to be now in the numerator of the expression (39). Inserting the value of ~GL for a clean metal
~GL = 7((3)
one obtains
The expression in curly brackets is the free electrons Landau diamag­ netic susceptibility XL. Hence, the fluctuation contribution turned out to be of the same order of magnitude as XL and it is possible to distinguish them through the temperature dependence of Xj/.
In the same way, for a film of thickness d <s:: ~GL in a perpendicular magnetic field in [18], was found to be
18
(42)
However, the "ointment" we have described above has a quite large "fly" in it. The first problem consists in the fact that with, an applied magnetic field, the critical temperature becomes to be dependent on B: Tc = Tc{B). This fact was ignored in the theory of [18, 21J. The result of their theory may be considered as the zero-field limit for the susceptibility in the rigorous definition of this property only. Indeed even in small fields the experimental curves depart from (41 )-( 42). In fact, in the first experimental test of the prediction of [18, 21J theory, Gollub et al. [24J pointed out that since experiments are done in a finite magnetic field which is lower than the nucleation temperature of the transition, a generalization of (41 )-( 42) is required. This would at least involve [T - Tc2(B)Jl/2 instead of (T - Tc)1/2, where Tc2 is the nucleation temperature.
This fact was automatically taken into account in the subsequent mod­ ernization of the theory by Prange [22J. Starting from the same quadratic part of the usual Ginzburg-Landau functional for the free energy he pro­ posed an exact solution for the experimentally observed value of the fluc­ tuation magnetization. Below we will follow his considerations [22J.
Let us transform the sum over n in (34) into an integral not by means of the Poisson formula, but in an exact way, introducing the appropriate 6-function in the integral. If G(y) = [yJ - y, where [yJ is the integer part of y, the derivative G'(y) will have 6-functional contributions in the points y = n + ~ and we can rewrite (34) in the following form:
VkBT roo f dk [ '( X) ] F = -~ Jo dx 271"£(k,x) G iJ + 1 (43)
where
(44)
For the magnetization one obtains
M = - (~~) _ VkBT 4e roo dx J dk -!-£(k, x )G" (:) 471" he Jo 271" B2 B
Carrying out the differentiation and the integral over k one can find
_ M = VkBT (4e)3/2 fCr) ~ 4~ he
where the parameter is
19
(46)
(47)
The zero-field limit of (49) is given by the limit, -+ 00 and it reproduces the result of Schmidt and Schmid: [18,21] f cr) = (1/24,1/2) (Fig.2). The field dependence now enters through the parameter ,. It is clear that it is not enough to change only Teo with Te2 (B) [22]. So one of the impor­ tant discrepancies between this exact result and the result of [18, 21] for the fluctuation part of the magnetization consists in the field dependence. Indeed, according to Prange,
M JHT = fCr) (49)
1 T - Teo ,=- . 2 Teo - Te2 (H)
Hence, because of the presence of the magnetic field, the transition shifts from the temperature T = Teo, where, = 0, to the temperature Tc2 (H).
1 (( 1) ( 1)-1/2) where, = -"2 f , ~ -"2 = 4 , + "2 .
Thus the magnetization diverges as 4 (T - Te2 )(H)r 1 / 2 at the point Te2 (H), but at T = Teo, cr = 0) remains finite, f(O) = 0.09, and increas('~ as IIi rather than H as it follows from (49).
20
0.16
0.14
0.12
0.02 -O.S 0.0 O.S 1.0 1.S 2.0 2.S 3.0
Y Figure 2. Function f( "Y). Also plotted is the form used for the first time by Gollub et al. in their experimental results . .It was heuristically obtained by modifying Schmid's result by simply shifting the transition temperature.
If we study the magnetization as a function of H, assuming the tem­ perature to be constant, the magnetization diverges with the field in the vicinity of HC2(T) as (H - He2(T)r 1/ 2.
The effect of fluctuations on the magnetization of a superconductor in the normal phase has been experimentally studied many times. E.g. Gollub et al. [24] measured the magnetic moment of bulk cylindrical samples as a function of temperature, at fixed values of the magnetic field with a SQUID magnetometer. The fluctuation part of the magnetization was isolated tak­ ing into account its explicit temperature dependence, since the normal state diamagnetism is not dependent on temperature. The absolute magnitude of the fluctuation contribution was established using the results at sufficiently high temperatures, where the fluctuation effects are negligible. Gollub et al. found that the "fluctuation tails" of the diamagnetic magnetization in lead samples extend far from Te and they can be detectable even at twice
Te· Gollub et al. measured the magnetic moment at T = Teo as a function
of the external magnetic field. According to Prange, the ratio ~' III
HTco this case has not to depend on the value of the magnetic field, and it has to be a universal constant
M' = -O.323A-.- 3 / 2 k IJiT '1'0 B yn cO
(50)
o __ t. , ! ,. , ,! ' t" ., .1
0 ·01 0· 1 I
x [n 2 I • Pb J6 o Nb 100 ... In-80Mt 13
... In-16 0,1, n 130
21
1
Figure 3. Universal behavior of scaled fluctuation magnetization at Teo versus scaled field for a number of materials. The broken curve is the clean-limit microscopic theory.
High n.!dl
'"' Q
x
~ I
5·0 5· S 65
Figure 4. Measured fluctuation magnetization above Te versus temperature and field of a bulk cylindrical sample of indium. At large fields the magnetization is suppressed.
A quite satisfactory agreement with this statement was found in the experiment [24], in the region of weak magnetic fields. But when increasing
the field the ratio ;' diminishes (Fig.3). The deviation from Prange HTco
formula itself is not surprising. We mentioned above that this result was obtained in the framework of the Ginzburg-Landau functional for the free energy and the limitation H <t: Hc2 was assumed. But in the experiment the considerable deviations from (50) come from some characteristics of every
22
metal field H. rv 210 He2(0). We can see an analogous behaviour of the
temperature dependence of M' for different values of the magnetic field in FigA for indium. The fluctuation part of the magnetization increases as the temperature is lowered, tending to diverge at Te2( H) < Teo. Since indium is a type I superconductor, the temperature Tc2 plays the role of a supercooling limit, so that a first-order jump for the Meissner state takes place well before Te2 . But even for low fields M' increases with H less than linearly as was expected from [18, 21].
However for larger fields M' begins to decrease with the increasing of H because the magnetic field suppresses the fluctuations more strongly in comparison to what we found in the framework of the GL theory. This is why the systematic deviation from Prange's "universal curve" was found for the temperature dependence of M' for fixed values of the field.
In order to better understand the reasons of such a disagreement be­ tween the theory [22] arld the experiment one has to formulate the micro­ scopic theory of fluctuations along the lines of the diagrammatic technique and to apply this more general approach to the problem discussed above. This was done in [25].
5. Time-dependent Ginzburg Landau equation. Paracondudiv­ ity.
Let us now discuss the superconducting fluctuation effects on the transport properties of a superconductor above the critical temperature [26, 27].
As we mentioned in the Introduction, the appearence of the fluctuating Cooper pairs above Te leads to the opening of a new channel for the charge transfer. This phenomenon is called paraconductivity4.
This fluctuation contribution was firstly found through the diagram­ matic approach in a microscopic theory [4]. Nevertheless, as proposed by Abrikosov in [1], this result can be reproduced in a way analogous to that of the heat capacity correction (Sect. 3) as we do in this section.
In order to find the paraconductivity value, a time-dependent gener­ alization of the Ginzburg-Landau equations is required. This is associated with the fact that the electric field can be defined as if = -c-1oA/ at, where A is the vector potential; but in this case, 1 has to be regarded as being dependent on time. It may also be assumed that if = -V</> and A = 0, but, as it will been shown below, the scalar potential</> is contained in the equation for the order parameter ~, introduced in section 2, in combination
4This term may have different origins. First of ali, evidently, paraconductivity is anal­ ogous to paramagnetism and means excess conductivity. Another possible origin is an incorrect onomatopoeic translation from the russian "paraprovodimost" that means pair conductivity (this story belongs to L. Aslamazov).
23
with 0.6 lot. In other words, the electric field in superconductors necessarily leads to nonstationary phenomena. The London equation o( AJ) I ot = E, where A = mlnee2 [1], also corresponds to this.
The general nonstationary BCS equations are very complicated, even in the limit of slow time and space variations of the field and the order parameter. It is for this reason that we shall not give their derivation here; instead, we write the model equation for the vicinity of To which in gen­ eral correctly reflects the qualitative aspects of the behaviour of the order parameter and in some cases turns out to be exact.
Let us keep in mind the Ginzburg-Landau functional formalism we in­ troduced in section 2. If a departure from equilibrium is assumed, then it is no more possible to derive the Ginzburg-Landau equation as in [1] from the condition that the variational derivative of the free energy is zero. At the same time, in the non-equilibrium case .6. depends on time. Nevertheless, at small deviations from equilibrium it is natural to assume that 0 tl I 8t is proportional to the variational derivative of the free energy 5F I 5tl *.
However, the gauge invariance requires that 8tll8t should be included in the equation in the following way:
otl .e)..A - + 2z-'f'u ot n (51)
where cfJ is the scalar potential ofthe electric field. The complete condition of gauge invariance requires, in fact, the invariance of the electric and magnetic fields
E -101 ---- ~ V<jJ c 8t
(52 ) -H VxA
upon variation of the potentials 1 --+ 1 + V Il, and cfJ --+ <jJ ~ C-181L / ot, where Il is a scalar. It is not difficult to see that when i is included in the
equation in the combination [-inV- (2e/c) 1] tl and <jJ in the combina­
tion [01 ot + 2i (eln) cfJ] tl, the gauge trasformation is compensated by the variation of the phase of the function tl. Thus we have
(53)
This microscopic derivation shows that, in the paraconductivity problem, (53) gives an exact answer if the correct choice of the constant e is made.
24
The constant 0 can be determined from the following considerations. On the right hand side of (53), see (2), there is a term -CV2.6.. On the whole this equation resembles the diffusion equation that can be generally written as
(54)
where n is the electron density and D is the diffusion coefficient [1]. The role of the diffusion coefficient D is played by C/O. On the other
hand, in presence of impurities, the Ginzburg-Landau equation may be written in the form [1]
7rDh (V _ 2ie 1)2 tP + (Tc ~ T) tP _ (7(;3~) ItPI2 tP = 0 (55) 8Tc he Tc 87r Tc
where we have put
.6. = i n~;:T tP.
It is easy to see that in this equation the coefficient 7r Dh/8Tc plays the same role as the coefficient C / a in a pure superconductor. Writing
(56)
(57)
This value coincides with the value obtained in the BeS microscopic theory. We are interested in the fluctuation correction to .6. that arises under
the action of a constant electric field. It is assumed to be small and therefore should be proportional to E in the first order correction. But since E does not depend on time, the same can be said on the correction to .6.. In view of this, the derivative o.6./ot may be omitted from (53). Assuming .6. to be small, we retain the linear terms in .6.. Substituting the Ginzburg-Landau functional into (53) yields
(58)
25
We have chosen the gauge at which A = 0 and E = - V ¢. Since E is homogeneous, it follows that ¢ = -E· r. Recall that E = (T - Te) Te.
In the absence of an electric field !:1 undergoes equilibrium fluctuations. The probability of fluctuations is proportional to the partition function Z we have introduced in Sect. 3.
W <X exp [- ~ ~ (ad p'/4m) 1"',1'1 ' (59)
where !:1p = J !:1( T)e-ipi/1idV. Hence the average equilibrium fluctuation
1!:1~of is given by
J 1!:1i'l exp [-~ (a€ + p2/4m) l!:1pI2] d ILlpl2
Jexp [-~(a€+p2/4m)l!:1pI2] dl!:1pl2
(60)
In the presence of a weak electric field we have !:1p = !:1~0) + Ll~). Since the momentum representation of the quantity rLl( T) is iii (8 I 8i) !:1p, WE'
obtain in the first approximation in E:
8!:1 (0) ( p2 ) 2()eE 8; + a€ + 4m !:1~1) = 0, (61)
from which it follows that
!:1~1) = -2 Be E8!:1(O). P a€ + p2 14m 8p (62)
The average electric current in the Ginzburg-Landau theory is
26
If we substitute (1~~0)12) into this formula in accordance with (60), we
obtain that; is zero. In the next approximation we have
J
(64)
-2- L...J E- D..- . m _ a€ + p2 /4m op ( p )
p
( 65)
The change from summation to integration is carried out according to the following rules:
p
1 J dp S 27rli
three - dimensional case,
(66)
thin wire, cro888ection S ~ e. Taking the integrals in (65) for all three cases and substituting e ac­
cording to (57), we obtain the values of the paraconductivity:
(1"=
1 e2
16lid€ 7r
three - dimensional case,
film, thickness d ~ ~,
{
--- 7r TtT P6
27
Figu.re 5. Resistance vs rec;Iuced temperature T fTc curves for the three film samples [29] utilized for fluctuation measurements. The resistance are normalized to their values at T = 1.33T,.
The order of magnitude is easily checked to be : u' f'V (e2 H,o"') C l/2 for the three dimensional case, u' f'V (e 2 / d",) C 1 for the two dimensional case, and u' f'V (e 2 ~o/ S",) c a/2 for the one dimensional case.
The two-dimensional case is the most interesting one since formulae (67) do not contain any characteristics of the film, except its thickness and Te. Moreover, as has already been pointed out, the fluctuations in a film are stronger than in a three-dimensional specimen; this is also seen from formula (67). Finally, such films are easy to prepare. Experimental data [28] have completely confirmed the formula (67) for this case. In Fig.5 and Fig.6 one can observe the smearing out of the resistive transition, due to the fluctuations, in a Bi2Sr2CaCu20S+6 thin film and the fitting of the excess conductivity with the Aslamazov-Larkin theory and its extension taking into account the short wavelength fluctuations [29]
Formulae (67) contain an infinite divergence as T -+ Te , which is an evidence that they are not valid in the immediate vicinity of Te.The range of validity is set up by the requirement that the fluctuation correction must be much less than the normal conductivity (j = nee2Ttr 1m.
Another source of error is the neglect of the interaction of electrons with fluctuating Cooper pairs. This effect enhances the conductivity, but cannot be interpreted so simply as in the above-described paraconductivity of fluc­ tuational Cooper pairs. The corresponding "anomalous Maki-Thompson conductivity" [15, 16], proves to be small in many cases 5, in particular
"The Maki-Thompson conductivity disappears if, for example, the superconductor contains magnetic impurities.
28
... ~.
\ .. ~
- 8 . 0 -"',a ... . e -1.0 -4 .0 -' . 0 -z.o -I.Q 0.0
in c
Figure 6. Normalized excess conductivity J(€) = (16hd/e2)AO' vs € = In.(T/Tc) in a In.-ln. scale. The solid line represents the extended theory by Balestrino et al. [29], and the dashed line represents the Aslamazov-Larkin theory.
under experimental conditions [28], which makes it possible to obtain a confirmation of formulae (67).
In this respect, let us point out the results of ref. [13] where the electrical resistivity of a well oxygenated polycrystalline Y Ba2CUa07_6 sample was studied over a large temperature range with the same precision as that necessary in the vicinity of a critical temperature when one extracts critical exponents. The polycrystalline sample had a critical temperature at Te = 90.5K as determined by the infexion point in p{T). It has been shown that a logarithmic behavior exists without any doubt at high temperature, hidden in the linear temperature regime of p{T) above Te. After substraction of the linear background in order to get !1p, a log-log plot of d(!1p}/dT in terms of € = (T - Te)/Te, shows that the Maki-Thompson regime extends to ca. 2Te •
It has been claimed that those values correspond to the temperature at which "Cooper pairs" could start breaking apart and therefore at which such pairs, whence superconducting fluctuations, start to occur. Nowadays the picture is clearly said to correspond to "a gap opening" in the density of states of such materials but it is clearly due to the existence of such Cooper pair fluctuations. It can also be pointed out that in Bi2Sr2CanCUn+108+y sytems such a logarithmic term is hardly seen. This is markedly due to the preexistence of the gap in the density of states in the CU02 planes and the narrower transition interval in temperature, - even though the systems should be more two dimensionallike. This fact is a signature of the d-wave order parameter symmetry resulting in the gap anisotropy.
29
6. The effect of superconducting fluctuations on the one-electron density of states.
As it was aheady mentioned, the possible appearance of nonequilibrium Cooper pairing above Te leads to the redistribution of the one-electron states around the Fermi level. A semi-phenomenological study of the fluc­ tuation effects on the density of states of a dirty superconducting material was first carried out while analysing the tunneling experiments of granular Al in the fluctuating regime just above Te [30]. The second metallic elec­ trode was in the superconducting regime and its gap gave a bias voltage around which a structure, associated with the superconducting fluctuations of AI, appeared. The measured density of states has a drop at the Fermi level6 , reaches its normal value at some energy Eo(T) and shows a maximum at the energy value equal to several times Eo, decreasing again towards its normal value at higher energies (Fig.l). The characteristic energy Eo was found to be of the order of the inverse of the Ginzburg-Landau relaxation time TGL introduced above.
The presence of the depression at E = 0 and of a peak at E f'V (hjTGd in the density of states above Te are the precursor effects of the appearance of the superconductive gap in the quasiparticle spectrum at temperatures below Te.
The calculation of the fluctuation contribution to the one-electron den­ sity of states is a nontrivial problem and cannot be carried out in the framework of the phenomenological Ginzburg-Landau theory. It can be solved with the diagrammatic technique calculating the fluctuation cor­ rection to the one-electron temperature Green function with its subsequent analytical continuation on the axis of real energies [6, 7]. We omit here the details of the cumbersome calculations and present only the results obtained from the first order perturbation theory for fluctuations. They are valid near the transition temperature, in the so-called classical region, where the deviations from the classical behaviour are small. The theoreti­ cal results reproduce the main features of the experimental behaviour cited above. The strength of the depression at the Fermi level is proportional to different powers of TGL, depending on the effective dimensionality of the electronic spectrum and the character of the electron motion (diffusive or ballistic). Namely, in a dirty superconductor for the most important cases of dimensions D = 3,2 one can find the value of the relative correction to the density of states at the Fermi level [6]:
6Here we count the energy E from the Fermi level, where we assume E = O.
30
(69)
where D is a diffusion coefficient. At large energies E ~ Tai the density of states recovers its normal value, according to the same laws (69) but with the substitution TGL ~ E-1
7. The peak in c-axis resistance of HTCS.
Among all unconventional properties of the high temperature superconduc­ tors, the transport properties are the most puzzling ones. In contrast to the in-plane resistivity (which is almost linear in temperature in clean samples) the transverse resistivity' at first grows moderately with decreasing temper­ ature and then rises precipitously at lower temperatures. Such a behaviour was observed in all high Tc compounds [14, 31, 32,. 33, 34, 35, 36] and even in conventional layered superconductors [37]. As noted by Anderson and Zou [38], the difference in temperature dependence between the transverse and in-plane resistivities is very difficult to explain in a conventional Fermi liquid theory.
We identify here at least one physical source for this difference: the suppression of the positive paraconductivity along the c-direction by the square of the interlayer transparency together with the growth of the normal resistance related to the fluctuation depression of the density of states at the Fermi level as discussed here above.
Let us start the discussion from the qualitative understanding of the effect of the transverse resistance of the fluctuation growth. Finally we will present the result of the exact diagram calculations. Below we assume the electron spectrum of a layered metal in the form of a corrugated cylinder:
~(p) = E(p) - Ep = vp(iPlli- pp) + W cos(P.La) (70)
where PII and P.L are the projections of the electron momentum in the layer plane and in the perpendicular direction respectively; a is the interlayer distance and W is the hopping integral, the square of which is propor­ tional to the probability P1 of the electron hopping from one layer to the neighboring one. This spectrum permits to find all possible types of di­ mensional crossovers in the problem under discussion, but for qualitative understanding we restrict ourselves to the region of temperatures not too close to Te , above the Lawrence-Doniach[39] crossover point (but of course
31
T - Tc 1 . d) h h fl . . . ak 1 . £ = <t: IS suppose ,were t e uctuatlOn paIrIng t es p ace In Tc
each plane separately only. The longitudinal paraconductivity in the plane is determined by the
well known formula [4]:
(71)
To modify this result for the c-axis paraconductivity one has to take into account the hopping character of the electron motion in this direction. One can easily see that the probability of the coherent hopping of the two
electrons, during the virtual Cooper pair lifetime TeL f'V kB(TIi_ Tc)' can
be calculated as the conditional probability of these two events:
(72)
and the transverse paraconductivity may be estimated as
AL P. AL p2 1 0' .l f'V 2' 0'11 f'V 1 2"'
£ (73)
It is easy to see that the temperature singularity of O'tL turns out to be even stronger than that one in O'tL because of the hopping character of the electron motion (the critical exponent two in the conductivity is relative to the zero-dimensional band motion), but, in the case of strongly anisotropic layered superconductor O'tL , is considerably suppressed by the square of the small probability of interplane electron hopping which enters into the prefactor.
Namely, this suppression determines the necessity of taking into account the DOS contribution to the transverse conductivity, which is less singular in temperature but, in contrast to the paraconductivity, manifests itself at the first, not the second, order in the "interlayer transparency". One can easily estimate it as follows.
The density of Cooper pairs is determined by the average value of the square of the order parameter modulus:
(74)
32
where 1] is the gradient term coefficient of the GL theory. The decrease of the electron density is evidently estimated by the same number. It is im­ portant to point out that, in contrast to the paraconductivity, the negative correction to the one-electron transverse conductivity is obviously propor­ tional only to the first order in Pl :
(75)
Excluding temporarily from consideration the anomalous Mald-Thompson contribution, one can say that the shape of the temperature dependence of the transverse resistance is determined by the competition of two oppo­ site sign contributions: the paraconductivity, that is strongly dependent on temperature but is somewhat reduced by the square of the barrier trans­ parency ("" w4 ) and by the first order in barrier transparency ("" w2 ) DOS contribution which depends on the reduced temperature only logarithmi­ cally. Namely these circumstances determine the possible competition be­ tween the contributions of different orders in transparency and lead to the formation of the maximum with the following temperature dependence:
(76)
(here a and f3 are coefficients which will be presented below). The exact evaluation of the diagram expressions for all contributions to
the transverse fluctuation conductivity of a layered superconductor for an arbitrary impurity concentration, [8, 9] up to the first order in perturba­ tion theory (including Maki-Thompson and other corrections) leads to the general formula:
- -rkln + + e2a [ ( 2 )2 [(£+r)1/2_£1/ 2J2 161] £1/2 + (£ + r )112 4[£( € + r )]1/2
(77)
+ ( £ + f3 + r ) k[( )1/2 1/2]2] [€(£ + r)]1/2 + [f3(f3 + r)]1/2 -1 - € + r - £ .
Here r is the reduced Lawrence-Doniach crossover temperature, f3 _11"_ is the dimensionless phasebreaking parameter, k and k are some 8TeT</> functions of the impurity concentration [9]. In the simple case of 2D fluc­ tuations and strong pair breaking, this formula reproduces the result of previous qualitative considerations (76).
34
tance temperature are shifted towards much lower values. The full theoret­ ical treatment of the magnetic field effect on the fluctuation conductivity of layered superconductors above Te has been given in [9], where the AL, DOS, regular and anomalous Maki-Thompson fluctuation contributions to the c-axis conductivity have been considered in details.
A useful analogy for the analysis of the c-axis resistivity in HTCS may arise in the discussion of the separation of different contributions in amor­ phous films [9, 41]. In these systems, in the weak localization regime, the total conductivity is interpreted as the sum of several contributions (lo­ calization corrections, corrections originating from the Coulomb electron­ electron interaction, the Cooper channel etc.). The crucial point is that the temperature and the magnetic field dependences of these contributions are very different, and this fact allows to separate and to identify each of them, extracting relevant informations concerning the microscopic parameters of these systems.
Here we apply a somewhat similar approach to the analysis of the anomalous negative c-axis magnetoresistance which has been recently ob­ served in strong magnetic fields parallel to the c-axis on BSCCO single crystals [42]. From the reported data we can see that the effect becomes significant below approximately 120K and its magnitude increases dramat­ ically as the temperature goes down to 95K. From [9] one can find the following expression for the fluctuation c-axis magneto conductivity close to Te , in the presence of not too strong magnetic fields:
(78)
where B is the magnetic field (all quantities are measured in c.g.s units). The temperature dependent factor I(T) is :
I(T)
+
10 r2 {3( £ + r /2) - 8k [~ + ~ (1 + ~) 1 + [£(£+ r)]3/2 £(£+r) r 2 k
(79)
[13(13 + r)]3/2 [(£(£ + r))1/2 + (13(13 + r))1/2]
where 10 is given by:
33
800 ........
r~ t:i , ,",. 'r;; 200 ~ 600 tJ I~ 10' no
~ Tomp''''''''' (1{)
Temperature (K)
Figure 1. Fit of the temperature dependence of the transverse resistance of an under­ doped BSCCO c-axis oriented film with the results of the fluctuation theory. The inset shows the details of the fit in the temperature range between Tc and llOK.
The quantitative agreement of this theory with the experimental data was shortly thereafter proved [14, 32, 40] by fits on resistivity peaks of BSCCO and YBCO samples, having good metallic behaviour far from the transition, and therefore showing a relatively small peak in the c-axis resistivity(Fig.7).
In these experiments the carrier concentration and the anisotropy of a Bi2Sr2CaCu208+z film grown on a misaligned substrate were changed by reducing and oxidizing the samples through annealing treatments. Since the AL contribution is more heavily dependent on the interlayer coupling than the DOS one, a more pronounced peak is expected for materials with higher anisotropy. The carrier density also affects the magnitude of the peak, since a higher carrier concentration means a lower fluctuation contribution and a higher normal-state conductivity. The evolution of the resistivity peak under redox treatments confirmed these predictions.
However, for strongly oxygen deficient Y B a2 C Ua 0 z (x := 6.4 - 6.8), the increase of the c-axis resistivity begins far from Tc and the peak has such a large magnitude [32] that it cannot be due to fluctuation effects only: in this case the effect is probably due to some insulating behavior.
8. The effect of a magnetic field on the c-axis resistivity. Anoma­ lous magnetoresistance.
The behaviour of the resistivity peak under an external c-axis oriented mag­ netic field is also very interesting [31]: its magnitude strongly increases with the field intensity while the position of the maximum and the zero resis-
35
(1/I(z) is the digammafunction and kB is the Boltzmann's constant). The first term in (79) represents the AL contribution, the second is the sum of DOS and regular MT contributions and the third is the anomalous MT one. Their different temperature dependences allow one to separate them and therefore to extract the values of the physical parameters which are involved.
To compare (78) with the experimental data of [42], we fitted them using as adjustable parameters Vp, T and the phase pair-breaking lifetime Ttl>. The values of the interlayer spadng a ~ 10-7 em and of the hopping integral w ~ 40K have been taken from literature data [40], since they are not likely to vary strongly from sample to sample (at least for BSCCO samples with metallic behaviour far from Tc), while Pc{H, T) and Tc ~ 85 K have been deduced from [42], to use as few as possible adjustable parameters. We stress, anyway, that all these parameters are not phenomenological constants, but they have a well defined physical meaning, allowing an a posteriori analysis about the consistency of the obtained values.
The results of the fit performed using (78) for the magnetoresistance curves are shown in (Fig.8). The curves measured at T = 95 K and T = 100 K have been fitted simultaneously (Le. using the same values of the fitting parameters for both curves in order to put several constraints upon them), while the curve at T = 105 K (and the curves measured at higher temper­ atures) have not been considered in the fit because the theory is valid only in the limit E ~ 1, while at 105K we have already € = 0.21. The theory cannot therefore be quantitatively used at these temperatures. However, the theoretical curve at 105K has been drawn in (Fig.8) using the values of VF, T and Ttl> found by fitting the curves at 95K and lOOK in order to show that, even at higher temperatures, the calculated temperature dependence of the transverse magnetoresistance is qualitatively in agreement with the experimental data. The values of the fitting parameters extracted from the
6 kB kB fit are VF = 3.1 x 10 cm/s, r;TcT = 0.11 and r;TcTtI> = 0.96. Reliable
values for the errors on these parameters cannot be calculated, partially because of their strong correlation, but we estimate them not to be negli- gible with respect to the parameter values themselves. The ratio Tef> ~ 10
T is in good agreement with the expected one [8], while the values of VF and T are on the lower side of the literature data. However, we believe that these values are quite reasonable even though there are large errors on the parameter best values, the temperature dependences of Ttl> and T have been neglected in the small temperature range considered, there are difficulties in the experimental evaluation of the c-axis resistivity in single crystals of layered superconductors, and the approximations made in the choice of Te , a and w can be debated upon.
36
0 <"'I a ....-< ,-..... -2 0 II e; -4 - a.V ...... a...V -6 - <I
-8 0 2 4 6 8 10 12 14
H (T)
Figure 8. Fit of transverse magnetoresistance data of a BSCCO crystal with the fluc- tuation theory results. .
While the field dependence of the magneto conductivity is simply B2, its behaviour with temperature, given by (79), is much more interesting. In Fig.9 we plot (79), using the above found values for the fitting parameters. It can be seen that the theory predicts the existence of a temperature T .. at which there is an inversion of the sign of magneto conductivity (provided, of course, that the calculated T .. lies in the region of applicability E(T .. ) < 1 of the theory). The physical origin of this change of sign is the same as for the appearance of the peak discussed above: relatively far from Tc the AL negative magneto conductivity is suppressed by its dependence on the square of the barrier transparency. The positive DOS contribution domi­ nates, while very close to Te the very singular temperature dependence of the negative AL contribution (I'V c 4 ) makes it to prevail on the less sin­ gular DOS contribution (I'V E- 2), in spite of the linear dependence on the transparency of the latter. Unfortunately the temperature range of the data in [42] does not allow to check this prediction (in our simulation T .. is about 88K). Nevertheless, the data of c-axis magnetoresistance of YBCO, in the immediate vicinity of Te , show an effect of positive sign, rapidly decreasing as temperature is increased [43], providing indirect support of the temper­ ature dependence of magnetoresistivity calculated in (79) for layered su­ perconductors. This fluctuation magnetoresistivitycontribution should also depend, through the parameter r, on the sample anisotropy (and therefore oxygen content), a higher effect being expected for the more anisotropic samples.
Finally, it is worth mentioning that the speculation proposed here above, concerning the change in sign of the fluctuation correction, is general for
37
1
"" ~ ('d '-"
£ '-' b
Temperature (K)
Figure 9. Calculated from the fluctuation theory temperature dependence of the mag­ netoresistivity of BSCCO. .
the c-axis characteristics and is valid for the magnetization as well. In the experiments with BSeeO single crystals and films the temperature dependences of this value often exhibit a fixed point in the fluctuation region above Tc , for different magnetic fields, in agreement with the idea proposed above.
9. The thermoelectric power and the thermal conductivity.
Let us briefly discuss the superconducting fluctuation effects on transport properties of a superconductor in presence of a thermal gradient above the critical temperature[26]. The shape of the electrical resistance versus tem­ perature curve in the vicinity of a superconductivity transition depends on various contributions. The same is true for the thermoelectric power and the thermal conductivity. However more than in the case of the elec­ trical transport those latter properties also are sensitive to the amount of superconducting materials (among other things, like impurities or grain boudaries or twin structures, ... ) in the sample. Therefore the data analysis is not so simple, due to the lack of clear normal background terms to be substracted off at first. Moreover the finite thermal gradient imposes that it is hardly possible to closely approach the true critical point [44].
While for the resistivity the normal state background in high Tc super­ conductors is usually admitted to be linear. The normal background of the thermoelectric power (TEP) is already quite controversial for normal sys­ tems [45], the more so ir high Tc superconductors. Even the sign of the TEP in the normal state is nd so immediately explained, because the behavior
38
can be due to many diffusion mechanisms for carriers occupying different bands as found in several superconducting cuprates [46]. In that respect some work taking into account a multiband structure, like in semi-metals, should be here mentioned [47]. Warn however that no trivial substraction procedure of contributions to TEP can be made in order to obtain some relevant term, because Matthiesen's rule does not apply to TEP indeed [45]; this is not often well known and is a source of erroneous interpreta­ tions. The TEP coefficient cannot be written indeed as being proportional to some relaxation time, but rather to some logarithmic dependence of the lattter. Thus a linear superposition cannot be made, but has rather to be performed through the scattering cross sections themselves [45]. This severe criticism should be repeated here since many authors superpose the normal state contribution S." and the fluctuation part (or the "excess TEP" ) Sfl such that S = S",+Sfl' This is alas a standard approach.
On the other hand, beside the paper of Houssa et al.[48], see also these proceedings [49], there seems to be only one paper reporting experimental data attempting to extract some fluctuation contribution from the thermal conductivity (ThC) [50].
The theory of superconducting fluctuation effects in TEP can be found in various works. From a theoretical point of view, direct and indirect con­ tributions can also be distinguished as for the paraconductivity. Maki [51J was the first to calculate some contribution to the excess TEP above Tc. He predicted a cusp in 3D and a logarithmic divergence in 2D. A prefactor included the pair breaking parameter. However fluctuation contributions were not all equally treated in the different factors and terms of the excess TEP expansion, and the calculation was made for the dirty limit and with other not necessarily reliable approximations. The most divergent contribu­ tion to the diffusion TEP was the opposite of that required to fit the data on YBCO single crystals. Maki [52] recently rederived a new expression for the excess TEP relating together the conductivity, the electron phonon coupling constant., the electron density of states at €F and the interlayer distance. The direct Aslamazov-Larkin contribution was investigated in pa­ pers [53, 55J (see also the contribution of D.Livanov in the present volume) [54J. Very recently the contribution to the excess TEP arising from the density of states fluctuations was found to be of the first importance in the problem under discussion [56]. Finally, Ausloos et al. reinvestigated the low dimensional (1D and 2D) thermoelectric power of high temperature super­ conductors in the fluctuation regime through a calculation of the so-called electrothermal conductivity, i.e. S / p [57].
Some report appeared in [26J discussing available investigations at that time. Different behaviors were observed, like 2D or/and 3D fluctuation regimes, with the appropriate AL exponents and within the ranges limited
39
by the VL crossover temperature [53J. Some 5/6 exponent characterizing a percolation path could be found even above TGL in polycrystalline ma­ terials. It has also been claimed that some anisotropy was observed in the basal plane of untwinned YBCO samples.
The theory of AL-like fluctuations above Tc in the ThC was made by Varlamov and Livanov[53J. By using the appropriate normal phonon back­ ground [58J and the basic electronic contribution derived from a fit of the re­ sistivity within a Wiedemann-Franz law assumption, Houssa et. al have ex­ tracted the behavior of fluctuations in ThC for different polycrystalline ma­ terials [48J. made possible following some precise experimental data taken with a technique allowing simultaneous TEP and ThC measurements [59J. Only the Gaussian fluctuation region could be probed and above Tc only [49]. Nevertheless the data analysis leads to a precise measurement of the interlayer barrier from the knowledge of the crossover temperature (TLD = 83.1K, to be compared with Tc = 79.5K) in a Bi2Sr1.8Ca1.2CU208+y' temperature could not be found in a magnetically textured poly crystalline DyBa2Cu807_z due to the more three dimensional nature of the com­ pound. Interestingly these analyses lead to some knowledge of the relax­ ation time Tt1' (ca. 9.1 X 10-18 and 2.1 X 10-18 s near Tc respectively) and to the corresponding mean free path (ca. 53.4 nm and 17.4 nm respectively). found in the literature near Tc when measured through other means. This is likely due to the neglect of impurity scattering in the Varlamov-Livanov theory.
In conclusion, it seems very relevant to suggest that further investiga­ tions on theory and experimental work on properties involving a thermal gradient should take into account the new discussions geared towards the electrical resistivity behavior both in the ab-planes and along the c-axis direction. To measure and to discuss such properties along the latter di­ rection are certainly huge challenges. considered since superconductivity fluctuations seem to have an interesting signature in those effects. See the "obvious" superconductivity fluctuations in the Nernst effect of various high critical temperature superconductors [60,61, 62J.
10. Acknowledgments
We are grateful to A. A. Abrikosov, G. Balestrino, A. Barone, P. Clippe, R. Cloots, C. Di Castro, F.Federici, M. Houssa, A. Larkin, D. Livanov, E. Milani, M. Pekala, V. Tognetti, R. Vaglio, N. Vandewalle and 1. Yu for many useful discussions concerning the material presented in these notes. This work is also part of the ARC (94-99/174) grant from the Ministery of Higher Education through the Research Council of the University of Liege. We thank NATO very much for financial support during this ARW.
40
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