Short Wavelength Cutoff Effects in the AC Fluctuation Conductivity of Superconductors

21 Pages·0.6 MB·English

by  D. -N. Peligrad

Short Wavelength Cutoff Effects in the AC Fluctuation Conductivity of
Superconductors
D.-N. Peligrad and M. Mehring
2. Physikalisches Institut, Universit¨at Stuttgart, 70550 Stuttgart, Germany
3
0 A. Dulˇci´c
0 Department of Physics, Faculty of Science, University of Zagreb, POB 331, 10002 Zagreb, Croatia∗
2 (Dated: SubmittedFebruary 28, 2002; revised version November26, 2002)

n Theshortwavelengthcutoffhasbeenintroducedinthecalculation ofac fluctuationconductivity
a of superconductors. It is shown that a finite cutoff leads to a breakdown of the scaling property
J
in frequency and temperature. Also, it increases the phase φ of the complex conductivity

0 (tanφ = σ2/σ1) beyond π/4 at Tc. Detailed expressions containing all essential parameters
2 are derived for 3D isotropic and anisotropic fluctuation conductivity. In the 2D case we obtain
individual expressions for the fluctuation conductivity for each term in the sum over discrete
]
wavevectors perpendicular to the film plane. A comparison of the theory to the experimental
n
o microwave fluctuation conductivity is provided.
c
-
r PACSnumbers: 74.40.+k,74.25.Nf,74.76.Bz
p
u
s
. I. INTRODUCTION
t
a
m
FluctuationsoftheorderparameternearthecriticaltemperatureT aremuchlargerinhigh-T superconductorsthan
c c
- inclassicallowtemperaturesuperconductors. OneofthereasonsliesinthehigherthermalenergykBTcwhichprovides
d
theexcitations,andtheotherinaveryshortcoherencelengthswhichoccurinhigh-T cupratesuperconductors. With
n c
these properties, the region of critical fluctuations was estimated from the Ginzburg criterion to be of the order of
o
c 1K, or more, around Tc, which renders the critical region accessible to experimental investigations.1 Farther above
[ T , one expects to observe the transition from critical to noninteracting Gaussian fluctuations which are the lowest
c
order fluctuation corrections to the mean field theory.2
2
The layered structure of high-T superconductors requires some theoretical sophistication. One could treat these
v c
8 superconductorswithvariousmodelsfromthree-dimensional(3D)anisotropictocoupledlayersLawrence-Doniach,or
4 purely two-dimensional (2D) ones. Due to the temperature variation of the coherence lengths one could even expect
7 a dimensionalcrossoverin some systems. The fluctuation conductivity is alteredby dimensionality in various models
7 sothatadetailedcomparisonofmodelcalculationsandexperimentaldatacouldaddressthe dimensionalityproblem.
0
For the reasons stated above, the fluctuation conductivity in high-T superconductors was studied experimentally
c
2 by many authors.3,4,5,6,7,8,9,10,11,12,13,14 Most of them used dc resistivity measurements.3,4,5,9,10,11,12,13,14 The reports
0
were controversialin the conclusions about the dimensionality of the system, and the critical exponents. It has been
/
t shown that, in a wide temperature range above T , the fluctuation conductivity did not follow any of the single
a c
m exponent power laws predicted by scaling and mean-field theories.5 The data in the Gaussian regime could be fitted
by an expression derived within the Ginzburg-Landau (GL) theory with a short wavelength cutoff in the fluctuation
-
d spectrum. Recently,Silvaetal.13haveproventhattheGLapproachwithanappropriatechoiceofthecutoffparameter
n yields result which is identical to that of the microscopic Aslamazov-Larkin(AL) approach with reduced excitations
o of the short wavelength fluctuations.15 It has been further shown that the detailed temperature dependence of the
c
fluctuation conductivity was not universal, but sample dependent. In this respect, the GL approach has practical
:
v advantage since the cutoff parameter can be readily adjusted in fitting the experimental data. Silva et al.13 could fit
Xi very well the data on a number of thin films in the Gaussian region from Tc+1 K to Tc+25 K.
When critical fluctuations are studied, it becomes essential to know accurately the value of T . However, the
r c
a determination of Tc from dc resistivity measurements brings about some uncertainties. One should avoid the use
of unjustifiable definitions of T such as e. g.: (i) zero resistance temperature, (ii) midpoint of the transition, (iii)
c
maximum of the derivative dρ/dT, (iv) intersection of the tangent to the transition curve with the temperature
axis, etc. The correct value of the critical temperature can be determined as an additional fitting parameter in
the analysis of the fluctuation conductivity. Usually one assumes that a well defined power law holds in a given
narrow temperature range and then determine both, T and the critical exponent from the selected segment.10,12,14
c
However,the experimentaldata usually show analmostcontinuous changeofthe slope sothat the uncertaintyin the
determination of T is an unsolved problem. Besides, the effects of the cutoff have been neglected in the analysis of
c
the data close to T . Even though the values of the fluctuation conductivity near T are not much affected by the
c c
introduction of the cutoff, the slopes can be considerably changed,4 and the analysis may become uncertain.
2
Anumberofmicrowavestudieshavebeenreportedshowingclearsignsoffluctuationsinboth,therealandimaginary
parts of the ac conductivity.6,7,16,17,18 The real part σ of the complex conductivity (σ =σ iσ ) has a sharp peak
1 1 2
atT ,whichisnotobservedine.g. Nbasarepresentativeoflowtemperatureclassicalsuperco−nductors.19 Thesalient
c
feature of the ac case is that the fluctuation conductivity does not diverge at T because a finite frequency provides
c e
a limit to the observation of the critical slowing down near T . The real part σ has a maximum at T . It is also
c 1 c
important to note that σ and σ have individually different temperature and frequency dependences, even though
1 2
they result from the same underlying physics. Testing a given theoretical model becomes more stringent when two
curves have to be fitted with the same set of parameters.
The expressions for the ac fluctuation conductivity in the Gaussian regime have been deduced within the time
dependent Ginzburg-Landau (TDGL) theory by Schmidt.20 Using general physical arguments, Fisher, Fisher, and
Huse21 provided a formulation for the scaling of the complex ac conductivity as
σ(ω) ξz+2−D ±(ωξz) , (1)
∝ S
where ξ is the correlation length, z is the dynamical critical exponent, D is the dimensionality of the system, and
e e
±(ωξz)aresomecomplexscalingfunctionsaboveandbelowTc. Thisformofthefluctuationconductivitywasclaimed
Stoholdinboth,theGaussianandcriticalregimes. Dorsey22 hasdeducedthescalingfunctionsintheGaussianregime
aeboveTc,andverifiedthe previousresultsofSchmidt.20 Morerecently,WickhamandDorsey23 haveshownthateven
in the critical regime, where the quartic term in the GL free energy plays a role, the scaling functions preserve the
same form as in the Gaussian regime.
The above mentioned theoretical expressions of the ac fluctuation conductivity did not take into account the slow
variationapproximationwhichisrequiredforthevalidityoftheGinzburg-Landautheory.2 Itwasnotedlongtimeago
that the summation over the fluctuation modes had to be truncated at a wavevector which corresponded roughly to
theinverseoftheintrinsiccoherencelengthξ .24 Theimprovedtreatmentwithashortwavelegthcutoffwasappliedin
0
fluctuationdiamagnetism,25 anddc paraconductivityfaraboveT .26 Thisapproachwasalsoappliedindc fluctuation
c
conductivity of high-T superconductors where one encounters a large anisotropy.4,5,13 The introduction of the short
c
wavelength cutoff was found to be essential in fitting the theoretical expressions to the experimental data. In view
of the great potential of the microwave method described above, we find motivation to elaborate in this paper the
improved theory of ac fluctuation conductivity including the short wavelength cutoff. We find that the resulting
expressionscanbe writteninthe formofEq.(1). However,the cutoffintroduces abreakdownofthe scalingproperty
in the variable ωξz. Also, we find that the phase φ of the complex conductivity (tanφ = σ /σ ) evaluated at T
2 1 c
departs from the value π/4 when cutoff is introduced. Values of φ larger than π/4 were observed experimentally,7
but were attributed to an unusually large dynamic critical exponent. Also, deviation of the scaling in the variable
ωξz was observedalready at 2 K above T ,7 but no analysis was made considering the short wavelengthcutoff in the
c
fluctuationspectrum. Thepresenttheoryisdevelopedfordifferentdimensionalitieswhichfacilitatescomparisonwith
experimental data.
II. THE EFFECTS OF SHORT WAVELENGTH CUTOFF
Frequency dependent conductivity canbe calculated within the Kubo formalismfrom the currentcorrelationfunc-
tion. For the fluctuation conductivity one has to consider the current due to the fluctuations of the order parameter.
The resulting expression for the real part of the conductivity is27
2e~ 2 1 τ /2
σ1xx = m k T kx2 <|ψk|2 >2 1+(ωkτ /2)2 , (2)
(cid:18) (cid:19) B k k
X
where the current is assumed to be in the x-direction. ψ is the Fourier component of the order parameter, and
k
τ =τ /(1+ξ2k2) is the relaxationtime of the k-th component. The relaxationtime for the k=0 mode is givenby
k 0
π~ ξ(T) z
τ = , (3)
0
8k T ξ
B c (cid:18) 0 (cid:19)
where z is the dynamic critical exponent. An alternative approach is to calculate the response of the system to an
external field through the expectation value of the current operator averaged with respect to the noise.22 However,
the introduction of the short wavelength cutoff in this approach leads only to selfconsistent implicit expressions.28,29
Eq. (2) is obtained from the time dependent Ginzburg-Landau theory and represents the equivalent of the
Aslamazov-Larkin fluctuation conductivity obtained from microscopic calculations. In the following, we present the
results which take accountof the short wavelengthcutoff in this contribution to the ac fluctuation conductivity. The
3
other contributions such as Maki-Thomson (MT) and one-electron density of states (DOS) renormalization30,31,32
cannot be treated within the time dependent Ginzburg-Landau theory but require microscopic calculations. It has
been shown31 that MT anomalous contribution in high-T superconductors is almost temperature independent while
c
DOS contribution is strongly temperature dependent, and contains a number of parameters which have to be deter-
minedthroughacomplexfittingprocedureinanexperimentaldataanalysis.33Sincethethreetermsinthefluctuation
conductivity are additive it is important to have the Aslamazov-Larkin term corrected for short wavelength cutoff
which then allows to fit the MT and DOS contributions properly from the rest of the total experimental fluctuation
conductivity.
The sum in Eq. (2) can be evaluated by integration considering the appropriate dimensionality. In this section we
discuss the simplestcase ofan isotropic3D superconductor. The integrationin k-spaceneeds a cutoffsince the order
parameter cannot vary appreciably over distances which are shorter than some minimum wavelength. The cutoff in
k can be expressed as kmax = Λ/ξ , where Λ is a dimensionless cutoff parameter. Obviously, Λ would imply
nxo cutoff in the integratixon, wherea0s for Λ 1 one obtains the usually assumed cutoff at 1/ξ . I→n t∞he 3D isotropic
0
case, the same cutoff applies also to k and≈k so that for the 3D integration in the k-space one has to set the cutoff
y z
limit for the modulus kmax =√3Λ/ξ . With the change of variable q(T)=kξ(T) one obtains
0
e2 ξ(T) z−1 Q q4
σ3D,iso(ω,T,Λ)= dq , (4)
1 6π~ξ ξ (1+q2)[Ω2+(1+q2)2]
0 (cid:18) 0 (cid:19) Z0
where
ξ(T)
Q(T,Λ)=k ξ(T)=√3Λ (5)
cut
ξ
(cid:18) 0 (cid:19)
is the temperature dependent cutoff limit in the q-space, and
ωτ π ~ω ξ(T) z
0
Ω(ω,T)= = (6)
2 16k T ξ
B c (cid:18) 0 (cid:19)
is a dimensionless variable which depends on frequency and temperature as independent experimental variables.
For the dc case (ω =0), and no cutoff (Λ ), one finds from Eq. (4)
→∞
e2 ξ(T) z−1
σ3D,iso(T,Λ )= , (7)
dc →∞ 32~ξ ξ
0 (cid:18) 0 (cid:19)
which reduces to the well known Aslamazov-Larkinresult27 provided that relaxationaldynamics is assumed (z =2),
and ξ(T)/ξ is taken only in the Gaussian limit as 1/√ǫ. However, with a finite cutoff parameter Λ one obtains
0
5
e2 ξ(T) z−1 3Q2+1
σd3cD,iso(T,Λ)= 16π~ξ ξ arctan(Q)−Q(cid:18)(1+Q2)2(cid:19) . (8)
0 (cid:18) 0 (cid:19)
 
 
 
This result has been obtained by Hopfeng¨artner et. al.4 except that they used only the Gaussian limit 1/√ǫ for the
reducedcorrelationlengthξ(T)/ξ . Their analysishas shownthatthe cutoff playsnoroleexactly atT sinceQ
0 c
→∞
regardlessofΛ. However,atanytemperatureaboveT onegetsafiniteQandthevalueoftheconductivityislowered
c
with respect to the result given by Eq. (7). Their conclusion was that the Gaussian fluctuations with no cutoff yield
an overestimated fluctuation conductivity.
In this paper we are primarily interested in the ac case. Before integrating Eq. (4) with Ω = 0, we find the
6
corresponding expression for the imaginary part σ . We can apply Kramers-Kronig relations to each of the Fourier
2
components in Eq.(2), andcarry out the summation. This is equivalent to a calculationof the kernel for σ from
2 2
K
the kernel used in Eq. (4), namely
1
K
∞ ′
2Ω (Ω)
1 ′
K2(Ω)= π ΩK2 Ω′2 dΩ . (9)
Z0 −
With the kernel (Ω), the imaginarypartofthe fluctuation conductivity canbe calculatedfor any cutoff parameter
2
K
Λ
e2 ξ(T) z−1 Q Ωq4
σ3D,iso(ω,T,Λ)= dq . (10)
2 6π~ξ ξ (1+q2)2[Ω2+(1+q2)2]
0 (cid:18) 0 (cid:19) Z0
4
Finally, the complex fluctuation conductivity can be written in the form
e2 ξ(T) z−1
σ3D,iso(ω,T,Λ)= 3D,iso(ω,T,Λ)+i 3D,iso(ω,T,Λ) . (11)
32~ξ ξ S1 S2
0 (cid:18) 0 (cid:19)
(cid:2) (cid:3)
Theprefactorisequaeltothedc resultwithnocutoffeffectasinEq.(7). Thefunctions aregivenbythefollowing
1,2
S
expressions
S13D,iso(ω,T,Λ)= 3π1Ω2 P−(P+2 +2)L+2P+(P−2 −2)A+16 arctan(Q) , (12)
(cid:2) (cid:3)
S23D,iso(ω,T,Λ)= 3π1Ω2 2P−(P+2 +2)A−P+(P−2 −2)L−24Ω arctan(Q)+8Ω1+QQ2 , (13)
(cid:20) (cid:21)
where we used the following shorthand notations
P± =√2 Ω2+1 1 , (14)
±
q
p
2+Q2+(Q P−)2
L=ln − , (15)
2+Q2+(Q+P−)2
(cid:18) (cid:19)
2Q+P− 2Q P−
A=arctan +arctan − . (16)
P P
(cid:18) + (cid:19) (cid:18) + (cid:19)
It can be easily verified that the -functions given by Eq. (12) and Eq. (13) have proper limits. In the dc limit
1,2
(Ω 0), one finds that 0,Sand leads to the dc result of Eq. (8). One can also verify that the ac results
2 1
obta→inedpreviouslyby ScShm→idt20 andDSorsey22 canbe recoveredfromourEq.(12)andEq.(13) inthe limit Λ ,
→∞
i. e. when no cutoff is made.
The effects of the cutoff are not trivial in the ac case. It is essential to examine those effects in detail as they have
strongbearingontheanalysisoftheexperimentaldata. TheprefactorinEq.(11)dependsonlyontemperaturewhile
thecutoffparameterΛisfoundonlyinthe -functions. Therefore,theeffectsofthecutoffcanbestudiedthroughthe
S
-functions alone. We can look at the temperature and frequency dependences of these functions with and without
S
the cutoff. Fig. 1(a) shows a set of curves as functions of ξ(T)/ξ for three different frequencies. Far above T the
1 0 c
S
relaxationtime τ is so short that ωτ 1 for any of the chosenfrequencies. Therefore the response of the system is
0 0
like in the dc case. With no cutoff, ≪saturates to unity (dashed lines in Fig. 1(a)). This limit is required in order
1
S
that σ from Eq. (11) becomes equal to σ (Λ ) in Eq. (7). If a cutoff with a finite Λ is included, decays
1 dc 1
→ ∞ S
at higher temperatures (solid lines in Fig. 1(a)). The reduction of is more pronounced at smaller values of ξ(T)
1
S
since the integration in the q-space is terminated at a lower value Q = √3 Λ ξ(T)/ξ . At higher temperatures the
0
conductivity σ at any frequency behaves asymptotically as σ given by Eq. (8).
1 dc
At temperatures closer to T the relaxationtime τ increases as ξ(T)z with increasing correlationlength according
c 0
to Eq. (3) which is usually termed critical slowing down. When ωτ 1 for a given frequency, is sharply reduced
0 1
≈ S
and vanishes in the limit of T . With the diverging prefactor in Eq. (11) it can still yield a finite σ at T . It may
c 1 c
appearfromFig.1(a)thatcutoffmakesnoeffectwhenT isapproached,butweshowlaterthatanimportantfeature
c
still persists in σ .
1
Obviously, at lower operating frequencies one needs to approach T closer so that the critical slowing down could
c
reachtheconditionωτ 1. OnecanseefromFig.1(a)thatforfrequenciesbelow1GHzonewouldhavetoapproach
0
≈
T closerthan1 mK in orderto probe the criticalslowingdowninfluctuations. The higherthe frequency,the farther
c
above T is the temperature where the crossover ωτ 1 occurs. This feature expresses the scaling property of the
c 0
≈
conductivityinfrequency andtemperature variables. However,the scalingpropertyholdsstrictly onlyinthe absence
of the cutoff. Namely, if one sets Λ , the function depends only on the scaling variable Ω. Fig. 1(b) shows
1
→ ∞ S
the same set of curves as in Fig. 1(a), but plotted versus Ω. The three dashed curves from Fig. 1(a) coalesce into
one dashed curve in Fig. 1(b), thus showing the scaling property in the absence of cutoff. However, the full lines
representingthefunctions withafinitecutoffparameterΛdonotscalewiththe variableΩ. Thereasonisthatthe
1
S
function then depends also on Q, which itself is not a function of Ω. Namely, the cutoff in the q-space depends on
1
S
the properties of the sample, and on the temperature, but not on the frequency used in the experiment. Hence, the
5
cutoffbringsaboutabreakdownofthescalingpropertyinfrequencyandtemperature. Theeffectismorepronounced
at temperatures farther above T where the cutoff is stronger.
c
Thepropertiesofthefunction areshowninFig.2forthesamesetofthreemeasurementfrequenciesasinFig.1.
2
S
When plotted versus ξ(T)/ξ , the function exhibits a maximum at the point where the correspondingfunction
0 2 1
S S
showsthecharacteristiccrossoverduetoωτ 1asdiscussedabove. WhenT isapproached, tendstozero. When
0 c 2
≈ S
is multiplied with the diverging prefactor in Eq. (11), one finds a finite σ at T . Far above T , the function
2 2 c c 2
S S
vanishes, regardless of the cutoff. This is consistent with the behavior of . Namely, at high enough temperatures,
1
acquires asymptotically the dc value, as seen in Fig. 1. Obviously, thSe imaginary part of the conductivity must
1
vSanish when dc like limit is approached. The decrease of the function at higher temperatures is very rapid so
2
S
that the effects of the cutoff are unnoticeable on the linear scale. Only with the logarithmic scale used in the inset to
Fig. 2(a), one observes that the cutoff effects are present also in , though by a very small amount. Fig. 2(b) shows
2
S
the scaling property of with no cutoff and its breakdown when cutoff is included.
2
S
Wehavenotedabovethatboth, and tendtozerowhenT isapproached. Also,theeffectofcutoffisseentobe
1 2 c
S S
smallinthatlimit. Yet,thesefunctionsaremultipliedbythedivergingprefactorinEq.(11),andthenmayyieldfinite
σ and σ . A carefulanalysis is needed in orderto find the phaseφ ofthe complex conductivity (φ=arctan(σ /σ ))
1 2 2 1
atT . Fora3Disotropicsuperconductor,Dorsey22 haspredictedφ=π/4,i.e. σ =σ atT . Hisresultwasobtained
c 1 2 c
withnocutoffanditremainstobeseenifthispropertyispreservedevenwhenafinitecutoffismade. Fig.3(a)shows
and as functions of ξ(T)/ξ for 100 GHz frequency. The effects of cutoff on are noticeable only far above
1 2 0 2
S S S
T . Closer to T , the curves for calculated with, and without cutoff, are indistinguishable. In contrast, a finite
c c 2
S
cutoffreducesthe valuesof eveninthelimit ofT . As aresult,the finalcutoffparameterΛyieldsacrossingofthe
1 c
S
curves for and at some temperature slightly above T . It is better seen on an enlarged scale in Fig. 3(b). This
1 2 c
S S
is a surprising result which has bearing on the experimental observations. Due to the cutoff, the condition =
1 2
S S
(φ=π/4)isreachedatatemperatureslightlyaboveT . Sinceboth, and aremultipliedwiththesameprefactor
c 1 2
S S
in Eq. (11), one finds that the crossing of σ (T) and σ (T) does not occur at the peak of σ (T), but at a slightly
1 2 1
highertemperature. ExactlyatT ,σ ishigherthanσ sinceafinite cutoffparameterreducesσ ,butmakesnoeffect
c 2 1 1
on σ .
2
The observation that the cutoff brings about a reduction of σ at T is worth further investigation since it can be
1 c
measuredexperimentally. Fig.4(a)showsthe ratio / (equalto σ /σ ) attemperaturesapproachingT . Withno
2 1 2 1 c
S S
cutoff (dashed lines in Fig. 4(a)), this ratio reaches unity regardless of the frequency used. A finite cutoff parameter
(Λ = 0.5 in Fig. 4(a)) makes the ratio equal to unity at a temperature slightly above T , and in the limit of T the
c c
ratio saturates at some higher value. The saturation level is seen to be higher when a higher frequency is used.
One can find analytical expansions of the -functions in the limit of T (Ω ). The leading terms are
c
S →∞
3D,iso(W,Ω ) 4√2 C 1D 1 , (17)
S1,2 →∞ ≈ 3π ∓ 2 √Ω
(cid:18) (cid:19)
where we used the notation
C =arctan(1+√2W) arctan(1 √2W) , (18)
− −
1+√2W +W2
D =ln . (19)
1 √2W +W2!
−
The parameter W depends on the frequency ω and the cutoff parameter Λ
16k T
W =√3Λ B c . (20)
π ~ω
r
Both functions tend to zero in the limit of T (Ω ), but their ratio is finite and depends on the parameter W.
c
→ ∞
Fig. 4(b) shows the plot of / at T as the function of W. One can observe that for a given cutoff parameter Λ,
2 1 c
S S
the ratio / at T increases at higher frequencies (lower W). The limits at T in Fig. 4(a) represent only three
2 1 c c
S S
selected points on the curve for / in Fig. 4(b).
2 1
S S
In a given experiment, the ratio σ /σ at T can be directly determined from the experimental data so that the
2 1 c
corresponding value of the parameter W can be found uniquely from the curve of / in Fig. 4(b), and the cutoff
2 1
S S
parameter Λ is obtained using Eq. (20). We should note that Λ is a temperature independent parameter. It can be
determined by the above procedure from the experimental data at T , but it controls the cutoff at all temperatures.
c
OnemayobservefromEq.(17)thatinthelimitofT theleadingtermsintheexpansionsofthe -functionsbehave
c
as (ξ(T)/ξ )−z/2. Taking into account the prefactor in Eq. (11) one finds that σ and σ can hSave finite nonzero
0 1 2
6
values at T only if z = 2, i. e. for the purely relaxational dynamics. We have assumed this case in all the figures of
c
this section.
From the experimental data at T one can determine also the parameter ξ . Using Eq. (11) and the -functions in
c 0
S
Eq. (17), one obtains finite conductivities at T
c
σ3D,iso(ω,T ,Λ)= e2 2kBTc 3D,iso(W) , (21)
1,2 c 6~ξ π ~ω F1,2
0r
where
3D,iso(W)= 1 C 1D . (22)
F1,2 π ∓ 2
(cid:18) (cid:19)
As explained above, from the ratio of the experimental values σ /σ at T one can determine the parameter W, and
2 1 c
the values of canthen be calculatedfrom Eq.(22). The remaining unknownparameterξ canbe obtained using
1,2 0
F
Eq. (21) and either of the experimental values of σ (T ) or σ (T ).
1 c 2 c
Itisalsointerestingtolookattheplotsof (W)inFig.4(b). Onecanobservethat saturatestounityalready
1,2 2
F F
at small values of W 2. On the other hand, is smaller than unity at any finite value of W, in conformity with
1
≥ F
theratio( / )=( / )atT . AtthispointitisusefultofindtheexpectedrangeofthevaluesofW encountered
2 1 2 1 c
F F S S
in the experiments. For the microwave frequencies in the range 1-100 GHz, with Λ = 0.5, and T =100K, one finds
c
that W is in the range 9 - 90. According to Fig. 4(b), 1 in this range. This means that the cutoff makes no
2
F ≈
effect on , and only is reduced, in conformity with the calculated curves shown in Fig. 3.
2 1
S S
III. ANISOTROPY
Most high-T superconductors are anisotropic,some of them even having a high value of the anisotropy parameter
c
γ = ξ /ξ . Therefore, for practical purposes one needs adequate expressions for the ac fluctuation conductivity.
ab0 c0
The realpartof the fluctuation conductivity in the ab-planeis obtainedusing the Kubo formalismas in the isotropic
case. One obtains
2e~ 2 k2τ /2
σ3D,aniso = k T x 0 . (23)
1 m α B c (1+k2 ξ2 +k2ξ2)[Ω2+(1+k2 ξ2 +k2ξ2)2]
(cid:18) ab (cid:19) k ab ab c c ab ab c c
X
Taking k2 = k2 /2, and substituting the variables q = k ξ and q = k ξ , one can evaluate the sum in Eq. (23)
x ab ab ab ab c c c
by integration in the q -plane, and along the q -axis
ab c
e2 ξ(T) z−1 Qab Qc q3
σ3D,aniso(ω,T,Λ ,Λ )= ab dq dq , (24)
1 ab c 8π~ξ0c (cid:18) ξ0 (cid:19) Z0 Z−Qc (1+qa2b+qc2)[Ω2+(1+qa2b+qc2)2] ab c
where we allowed a cutoff Q (T) = √2Λ ξ (T)/ξ in the q -plane, and a possibly different cutoff Q (T) =
ab ab ab 0ab ab c
Λ ξ (T)/ξ along the q -axis. The dimensionless parameter Ω is the same as given by Eq. (6). We use the notation
c c 0c c
ξ(T)/ξ for both ξ (T)/ξ and ξ (T)/ξ .
0 ab 0ab c 0c
We may briefly examine the dc case (Ω=0). With no cutoff one obtains
e2 ξ(T) z−1
σ3D,aniso(T,Λ )= , (25)
dc ab,c →∞ 32~ξ ξ
0c (cid:18) 0 (cid:19)
which reduces to the Aslamazov-Larkin result for z = 2 (relaxational dynamics) and ξ(T)/ξ taken in the Gaussian
0
limit. Note that the fluctuation conductivity in the ab-plane depends on ξ . Finite cutoff parameters reduce the
0c
fluctuation conductivity when the temperature is increased above T
c
e2 ξ(T) z−1 Q2 Q
σ3D,aniso(T,Λ ,Λ )= arctan(Q ) ab c (26)
dc ab c 16π~ξ ξ c − 2(1+Q2 )(1+Q2 +Q2)
0c (cid:18) 0 (cid:19) (cid:20) ab ab c
2+3Q2 Q
ab arctan c .
−2(1+Q2ab)3/2 1+Q2ab!#
This expression has not been reported in the previous literature. The analysis of apdc fluctuation conductivity is
difficult because of the number of unknown parameters.
7
The ac fluctuation conductivity can be obtainedfrom the integralin Eq.(24) for the realpart while the imaginary
part is obtained by the procedure analogous to that of the isotropic case described in the preceding section
e2 ξ(T) z−1 Qab Qc Ωq3
σ3D,aniso(ω,T,Λ ,Λ )= ab dq dq . (27)
2 ab c 8π~ξ0c (cid:18) ξ0 (cid:19) Z0 Z−Qc (1+qa2b+qc2)2[Ω2+(1+qa2b+qc2)2] ab c
The full expression can again be written in the form
e2 ξ(T) z−1
σ3D,aniso(ω,T,Λ ,Λ )= 3D,aniso(ω,T,Λ ,Λ )+i 3D,aniso(ω,T,Λ ,Λ ) . (28)
ab c 32~ξ ξ S1 ab c S2 ab c
0c (cid:18) 0 (cid:19)
(cid:2) (cid:3)
The -fuenctions for the 3D anisotropic case are found to be
S
S13D,aniso(ω,T,Λab,Λc)= 3π1Ω2 2Qc(3+Q2c)L1+P−(P+2 +2)L2−T−(T+2 +2−Q2ab)L3+2P+(P−2 −2)A1 (29)
(cid:2)
2T (T2 2+Q2 )A 8 1+Q2 (2 Q2 )A +16A +12ΩQ A ,
− + −− ab 2− ab − ab 3 4 c 5
q (cid:21)
S23D,aniso(ω,T,Λab,Λc)= 3π1Ω2 4Qc(3+Q2c)A5+2P−(P+2 +2)A1−2T−(T+2 +2−Q2ab)A2−P+(P−2 −2)L2 (30)
(cid:2) 2+Q2
+T (T2 2+Q2 )L +12Ω ab A 24ΩA 6ΩQ L ,
+ −− ab 3 1+Q2 3− 4− c 1#
ab
where we used the shorthand notations for P± as in Eq. (14), and tphe following
T± =√2 (1+Q2 )2+Ω2 (1+Q2 ) , (31)
ab ± ab
r
q
(1+Q2)2[Ω2+(1+Q2 +Q2)2]
L =ln c ab c , (32)
1 (1+Q2 +Q2)2[Ω2+(1+Q2)2]
(cid:18) ab c c (cid:19)
L =ln 2+Q2c +(Qc−P−)2 , (33)
2 (cid:18)2+Q2c +(Qc+P−)2)(cid:19)
L3 =ln(cid:18)22((11++QQ2a2abb))++QQ2c2c ++((QQcc−+TT−−))22(cid:19) , (34)
2Qc+P− 2Qc P−
A1 =arctan +arctan − , (35)
P P
(cid:18) + (cid:19) (cid:18) + (cid:19)
2Qc+T− 2Qc T−
A2 =arctan +arctan − , (36)
T T
(cid:18) + (cid:19) (cid:18) + (cid:19)
Q
c
A =arctan , (37)
3
1+Q2 !
ab
p
A =arctan(Q ) , (38)
4 c
1+Q2 +Q2 1+Q2
A =arctan ab c arctan c . (39)
5
Ω − Ω
(cid:18) (cid:19) (cid:18) (cid:19)
8
Theeffectsofcutoffaresimilarasthosedescribedatlengthintheprecedingsectionforthesimplercaseof3Disotropic
superconductors. In this section we discuss only the modifications in the limit T T where the relevantparameters
c
→
can be determined. The -functions can be expanded in the limit of T (Ω ), and the leading terms are
c
S →∞
3D,aniso(W ,W ,Ω ) 4√2 C U 1+W4 1W2 C + 3 W C +√2W3 C (40)
S1 ab c →∞ ≈ 3π 1− + ab− 2 ab 2 √2 c 3 ab 4
(cid:20) (cid:18)q (cid:19)
1 1 1 1 1
−2D1+ 2U− 1+Wa4b+ 2Wa2b D2+ 2√2Wc3D3 √Ω ,
(cid:18)q (cid:19) (cid:21)
S23D,aniso(Wab,Wc,Ω→∞)≈ 43√π2 C1−U− 1+Wa4b+ 12Wa2b C2+ √12Wc3C3+ √32WabC4 (41)
(cid:20) (cid:18)q (cid:19)
1 1 1 3 1
+ D U 1+W4 W2 D W D ,
2 1− 2 + ab− 2 ab 2− 2√2 c 3 √Ω
(cid:18)q (cid:19) (cid:21)
where we used the following shorthand notations
U± = 1+W4 W2 , (42)
ab± ab
r
q
C =arctan 1+√2W arctan(1 √2W ) , (43)
1 c c
− −
(cid:16) (cid:17)
U−+√2Wc U− √2Wc
C =arctan arctan − , (44)
2
U+ !− U+ !
C =arctan W2 +W2 arctan W2 , (45)
3 ab c − c
(cid:0) (cid:1) (cid:0) (cid:1)
W
c
C =arctan , (46)
4
W
(cid:18) ab(cid:19)
1+√2W +W2
D =ln c c , (47)
1
1−√2Wc+Wc2!
D =ln 1+Wa4b+√2WcU−+Wc2 , (48)
2
p1+Wa4b−√2WcU−+Wc2!
p
W4[1+(W2 +W2)2]
D =ln c ab c . (49)
3 (W2 +W2)2(1+W4)
(cid:18) ab c c (cid:19)
The cutoff parameters appear in
16k T
W =√2Λ B c , (50)
ab ab π ~ω
r
16k T
B c
W =Λ . (51)
c c π ~ω
r
9
We note that in the anisotropic case, the -functions behave also as 1/√Ω when T T . As already discussed in
c
S →
the previous section, this implies that finite nonzero σ (T ) and σ (T ) can be obtained only for z = 2 (relaxational
1 c 2 c
model). Since the available experimental data in anisotropic high-T superconductors6,7,8 show finite nonzero σ (T )
c 1 c
and σ (T ), we can adopt z =2 in the remainder of this section.
2 c
In analogy to the 3D isotropic case described in the preceding section, one may define the functions
3D,aniso(W ,W )= 3√Ω 3D,aniso(W ,W ,Ω ) (52)
F1,2 ab c 4√2 S1,2 ab c →∞
so that the conductivities at T are given by
c
σ3D,aniso(ω,T ,Λ ,Λ )= e2 2kBTc 3D,aniso(W ,W ) . (53)
1,2 c ab c 6~ξ π ~ω F1,2 ab c
0cr
The ratio of experimental values σ /σ at T does not define uniquely the cutoff parameters Λ and Λ . It puts,
2 1 c ab c
however, a constraint on their choice. Fig. 5(a) shows the plot of / given by Eq. (52) as a function of two
2 1
F F
variables, W and W . It is evident that a fixed value of / defines a simple curve of the possible choices of
ab c 2 1
F F
(W ,W ). Fig. 5(b) shows a selection of such curves for / =1.05, 1.1, 1.15, and 1.2. The dashed line marks the
ab c 2 1
F F
condition Λ =Λ (W =√2W ). Experimentally,one has to probe the possible choices for (W , W ) and look at
ab c ab c ab c
the fits of the theoretical curves to the experimental data at T >T . The parameter ξ in Eq. (28) can be obtained
c 0c
once the choice for (W , W ) is made.
ab c
Note thatin practicalapplications ofthe abovetheory one needs measurementswhere the microwavecurrentflows
onlyintheab-plane. Particularlysuitableforthispurposearethemeasurementsinwhichthesuperconductingsample
is placed in the antinode of the microwave electric field E in the cavity.34,35
ω
IV. 2D FLUCTUATIONS
Superconductingtransitiondoesnotoccurinastrictly2Dsystem. However,ifthesampleisaverythinfilmsothat
its thickness is much smaller than the correlation length, the fluctuations will be restricted within the film thickness
d in one direction, and develop freely only in the plane of the film. Using the formalism described in the preceding
sections, we find that the fluctuation conductivity is given by
e2 ξ(T) 2 Q q3
2D
σ (ω,T,Λ)= dq , (54)
1 4~d ξ (1+q2+q2)[Ω2+(1+q2+q2)2]
(cid:18) 0 (cid:19) Xqn Z0 n n
e2 ξ(T) 2 Q Ωq3
2D
σ (ω,T,Λ)= dq , (55)
2 4~d ξ (1+q2+q2)2[Ω2+(1+q2+q2)2]
(cid:18) 0 (cid:19) Xqn Z0 n n
e2 ξ(T) 2
2D 2D 2D
σ (ω,T,Λ)= (Ω,Q,q )+i (Ω,Q,q ) , (56)
16~d ξ S1 n S2 n
(cid:18) 0 (cid:19) Xqn (cid:2) (cid:3)
e
where
ξ ξ(T)
0
q =nπ , (57)
n
d ξ
(cid:18) (cid:19)(cid:18) 0 (cid:19)
and Q = √2 Λ (ξ(T)/ξ ). The prefactor is the Aslamazov-Larkin result for the 2D case with no cutoff, and the
0
Gaussian form 1/ǫ replaced by the more general expression (ξ(T)/ξ )2.
0
The -functions are given by
S
1 1+Q2+q2 1+q2
2D(Ω,Q,q )= 2 arctan n 2 arctan n (58)
S1 n Ω Ω − Ω
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
1+q2 (1+q2)2[Ω2+(1+Q2+q2)2]
+ n ln n n ,
Ω (1+Q2+q2)2[Ω2+(1+q2)2]
(cid:18) (cid:19) (cid:18) n n (cid:19)(cid:21)
10
1 2(1+q2) 1+Q2+q2 1+q2
2D(Ω,Q,q )= n arctan n arctan n (59)
S2 n Ω Ω Ω − Ω
(cid:20) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(1+q2)2[Ω2+(1+Q2+q2)2] Q2
ln n n 2 .
− (1+Q2+q2)2[Ω2+(1+q2)2] − 1+Q2+q2
(cid:18) n n (cid:19) n(cid:21)
The summation overq in Eq.(56) has to carriedout until the factor nπ(ξ /d) reaches some cutoff value Λ, which
n 0
is of the orderof unity. If the film thicknessis large(d ξ ), one has to sum upto a highn-value. In suchcases,the
0
summation is well approximated by an integration, an≫d one retrieves the 3D case of the preceding section. The 2D
character is better displayed when the film thickness is comparable to ξ . Then, only a few terms have to be taken
0
into account. In the extreme case of d<ξ , only the n=0 term is found below the cutoff limit.
0
The zero frequency limit (Ω 0) yields
→
e2 ξ(T) 2 Q4
2D
σ (T,Λ)= . (60)
dc 16~d ξ (1+Q2+q2)2(1+q2)
(cid:18) 0 (cid:19) Xqn n n
The n=0 term yields the previous result of Hopfeng¨artner et al.4 and Gauzzi et al.5
In the limit of T (Ω ) one obtains
c
→∞
e2 k T W4 1+(W2+W2)2
σ2D(T )= B c 2 arctan(W2+W2) 2 arctan(W2)+W2 ln n n , (61)
1 c π~d ~ω n " n − n n (1+(cid:2)Wn4)(W2+Wn2)2(cid:3)#
X
e2 k T W4 1+(W2+W2)2
σ2D(T )= B c 2W2 arctan(W2+W2) 2W2 arctan(W2) ln n n , (62)
2 c π~d ~ω n " n n − n n − (1+(cid:2)Wn4)(W2+Wn2)2(cid:3)#
X
where we used the notation
16k T
W =√2Λ B c , (63)
π ~ω
r
ξ 16k T
0 B c
W =nπ . (64)
n d π ~ω
(cid:18) (cid:19)r
Onemayobservethatforn=0therealpartoftheconductivityisfinite,buttheimaginarypartdiverges. Itisdueto
the logarithmic term in Eq. (62). This is an unphysical result. It may indicate that the n=0 term is not physically
acceptable, or that the 2D model should not be applied exactly at T .
c
V. COMPARISON WITH EXPERIMENT
The relevance of the theoretical expressions derived in the preceding sections can be demonstrated by comparison
ofthecalculatedandexperimentallymeasuredacfluctuationconductivity. Asanexamplewepresenthereananalysis
of the data in Bi2Sr2Ca2Cu3O10−δ thin film. The experimental results of the complex conductivity measured at
9.5 GHz are shown in Fig. 6(a). The main features are the same as reported previously in single crystals of high-
T superconductors.6,18 We have to note that in our measurement the thin film was positioned in the center of an
c
elliptical microwave cavity resonating in TE mode, and oriented in such a way that the electric field E was in
e 111 ω
the ab-plane. Thus the in-plane conductivity was measured and the application of the theoretical expressions of the
preceding sections is appropriate. Other experimental details have been reported previously.35,36,37 In this section
we are interested in the fluctuation conductivity near T which is shown on an enlarged scale in Fig. 6(b). The real
c
part of the conductivity has a maximum when the coherence length diverges. Since the critical temperature of a
phase transition is characterized by the divergence of the coherence length, we use the maximum of σ in Fig. 6(b)
1
to determine T =84.04 K. One can also observe in Fig. 6(b) that the imaginary part of the conductivity crosses the
c
real part at a temperature slightly aboveT . This is a direct experimental evidence of the short wavelengthcutoff as
c
discussed in Section II. The experimental values of σ and σ at T can be used in the evaluation of the parameters
1 2 c
which enter the theoretical expressions of the preceding sections.