Electrodynamics of the Josephson vortex lattice in high-temperature superconductors

A. E. KoshelevMaterials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

�Received 31 May 2007; published 31 August 2007�

We studied the response of the Josephson vortex lattice in layered superconductors to the high-frequencyc-axis electric field. We found a simple relation connecting the dynamic dielectric constant with the perturba-tion of the superconducting phase, induced by oscillating electric field. Numerically solving equations for theoscillating phases, we computed the frequency dependences of the loss function at different magnetic fields,including regions of both dilute and dense Josephson vortex lattices. The overall behavior is mainly determinedby the c-axis and in-plane dissipation parameters, which are inversely proportional to the anisotropy. The casesof weak and strong dissipations are realized in Bi2Sr2CaCu2Ox and underdoped YBa2Cu3Ox, respectively. Themain feature of the response is the Josephson-plasma-resonance peak. In the weak-dissipation case, additionalsatellites appear in the dilute regime in the higher-frequency region due to the excitation of the plasma modeswith the wave vectors set by the lattice structure. In the dense-lattice limit, the plasma peak moves to a higherfrequency, and its intensity rapidly decreases, in agreement with experiment and analytical theory. The behav-ior of the loss function at low frequencies is well described by the phenomenological theory of vortex oscil-lations. In the case of very strong in-plane dissipation, an additional peak in the loss function appears below theplasma frequency. Such peak has been observed experimentally in underdoped YBa2Cu3Ox. It is caused by thefrequency dependence of the in-plane contribution to losses rather than a definite mode of phase oscillations.

DOI: 10.1103/PhysRevB.76.054525 PACS number�s�: 74.25.Nf, 74.25.Op, 74.25.Gz, 74.20.�z

I. INTRODUCTION

The Josephson plasma resonance �JPR�1–3 is one of themost prominent manifestations of the intrinsic Josephson ef-fect in layered superconductors.4,5 It has been established asa valuable tool to study the intrinsic properties ofsuperconductors,6,7 and it was extensively used to probe dif-ferent states of vortex matter.8–14 The value of the JPR fre-quency depends on the anisotropy factor, and it rangeswidely for different compounds, from several hundred giga-hertz in Bi2Sr2CaCu2Ox �BSCCO� to several terahertz in un-derdoped YBa2Cu3Ox �YBCO�.

An interesting issue is the influence of the magnetic fieldapplied along the layer direction on the high-frequency re-sponse and, in particular, on the JPR. Such field forms thelattice of Josephson vortices �JVL�. The anisotropy factor �and the interlayer periodicity s set the important field scale,Bcr=�0 / �2��s2�. At B�Bcr the Josephson vortices are wellseparated and form a dilute lattice. When the magnetic fieldexceeds Bcr, the Josephson vortices homogeneously fill alllayers �dense-lattice regime�.15–17 The crossover field rangesfrom �0.5 T for BSCCO to �10 T for underdoped YBCO.The JVL state is characterized by a very rich spectrum ofdynamic properties. In particular, the transport properties ofthe JVL in BSCCO have been extensively studied by severalexperimental groups.18–20

The high-frequency response in the magnetic fields alongthe layers has been studied experimentally in BSCCO usingthe microwave absorption in cavity resonators,21 inLa2−xSrxCuO4,7 and in underdoped YBCO22,23 by the infra-red reflection spectroscopy. A detailed comparison betweenthe behaviors of the high-frequency response in the in-planemagnetic field for these compounds has been made recentlyby LaForge et al.22 In BSCCO, two resonance absorptionpeaks have been observed.21 The upper-resonance frequency

increases with the field and approaches the JPR frequency atsmall fields, while the lower-resonance frequency decreaseswith the field and approaches approximately half of the JPRfrequency at small fields. In the underdoped YBCO,22,23 theJPR peak in the loss function rapidly loses its intensity withthe increasing field, while the resonance frequency eitherdoes not move or slowly increases with the field. In addition,another wide peak emerges at a smaller frequency and itsintensity increases with increasing field.

Several theoretical approaches have been used to describethe response of the JVL state to the oscillating electric field.A phenomenological vortex-oscillation theory has been pro-posed by Tachiki et al.2 This theory describes the response ofthe JVL at small frequencies and fields in terms of phenom-enological vortex parameters, mass, viscosity coefficient, andpinning spring constant. This approach is expected to work atfrequencies much smaller then the JPR frequency; i.e., itcannot be used to describe the JPR peak itself.

On the other hand, the plasmon spectrum at high magneticfields, in the dense-lattice limit, has been calculated by Bu-laevskii et al.24 It was found that the plasma mode at a zerowave vector increases proportionally with the magnetic field,�p�B�=�p�0�B / �2Bcr�. This prediction describes very wellthe behavior of the high-frequency mode in BSCCO.21 Amore quantitative numerical study of the JVL plasma modesin the dense-lattice regime has been done by Koyama.25 Hereproduced the mode linearly growing with field and alsofound additional modes at smaller frequencies.

In this paper, we develop a quantitative description of thehigh-frequency response for a homogenous layered super-conductor valid for whole range of frequencies and fields.We relate the dynamic dielectric constant via simple aver-ages of the oscillating phases. The high-frequency responseis mainly determined by the c-axis and in-plane dissipationparameters, which are inversely proportional to the aniso-

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tropy. Analytical results for the dynamic dielectric constantand loss function can be derived in limiting cases, at smallfields and frequencies �vortex-oscillation regime� and at highfields, in the dense-lattice regime. Numerically solving dy-namic equations for the oscillating phases, we studied appli-cability limits of the approximate analytical results and in-vestigated the influence of the dissipation parameters on theshape of the loss function. Computing the oscillating phasesfor different vortex lattices, we traced the field evolution ofthe loss function with increasing magnetic field for the casesof weak and strong dissipations, typical for BSCCO and un-derdoped YBCO, respectively.

II. DYNAMIC PHASE EQUATIONS, DIELECTRICCONSTANT, AND LOSS FUNCTION

The equations describing phase dynamics in layered su-perconductors can be derived from Maxwell’s equations ex-pressing fields and currents in terms of the gauge invariantphase difference between the layers �n=�n+1−�n− �2�s /�0�Az. These equations have been presented in sev-eral different forms.26 We will use them in the form ofcoupled equations for the phase differences and magneticfield when charging effects can be neglected �see, e.g., Refs.27 and 28�,

c�0

2�csjJ

��n

�t+ sin �n +

1

�p2

�2�n

�t2 −c

4�jJ

�Bn

�x=

1

4�jJ

�Dz

�t,

�1a�

�4�ab

c2

�

�t+

1

ab2 �� �0

2�s

��n

�x− Bn� = −

�n2Bn

s2 , �1b�

where the magnetic field is along the y axis, ab and c arethe components of the quasiparticle conductivity, ab and care the components of the London penetration depth, jJ=c�0 / �8�2sc

2� is the Josephson current density, �p

=c /��cc is the plasma frequency, Dz is the external electricfield, and �n

2Bn�Bn+1+Bn−1−2Bn. Neglecting chargingeffects,29 the local electric field is connected with the phasedifference by the Josephson relation

Ez ��0

2�cs

��n

�t. �2�

The average magnetic induction inside the superconductor,By, fixes the average phase gradient ��n

�0� /�x=2�sBy /�0.To facilitate analysis, we use a standard transformation to

the reduced variables,

x/J → x, t�p → t, hn = 2��s2Bn/�0,

with J=�s, and introduce the dimensionless parameters

l =ab

s, �c =

4�c

�c�p, �ab =

4�abab2 �p

c2 .

It is important to note that both damping parameters �c and�ab roughly scale inversely proportional to the anisotropyfactor �, meaning that the effective damping is stronger in

less anisotropic materials. In particular, the cases of weakand strong dissipations, are realized in BSCCO and under-doped YBCO, respectively. Due to the d-wave pairing in thehigh-temperature superconductors, both dissipation param-eters do not vanish at T→0. Another important feature of thehigh-temperature superconductors is that the in-plane dissi-pation is typically much stronger than the c-axis dissipation,�ab �c.

31 This is a consequence of a rapid decrease of thein-plane scattering rate with decreasing temperature, whichmanifests itself as a large peak in the temperature depen-dence of the in-plane quasiparticle conductivity.32,33

Assuming an oscillating external field and using a com-plex presentation, Dz�t�=Dz exp�−i�t�, we obtain for smalloscillations

�− i�c�̃ + Cn�x� − �̃2��n − l2�hn

�x=

i�

4�jJDz, �3a�

��n

�x− hn +

l2

1 − i�ab�̃�n

2hn = 0, �3b�

where �̃=� /�p, Cn�x��cos��n�0��x��, and the static phases,

�n�0��x�, are determined by the following reduced equations:

�2�n�0�

�x2 + �−1

l2 + �n�sin �n�0� = 0, �4�

with the addition condition ��n�0� /�x=h=2��s2By /�0. We

introduce the reduced oscillating phase �n,

�n = �n

�pDz

4�jJ,

for which we can derive from Eqs. �3a� and �3b� the follow-ing reduced equation:

−�2�n

�x2 + � 1

l2 −1

1 − i�ab�̃�n

2��Cn�x� − �̃2 − i�c�̃��n = −i�̃

l2 .

�5�

From the Josephson relation �2�, we find Ez= �−i�̃ /�c��̄Dz,meaning that the dynamic dielectric constant is connected

with the average reduced oscillating phase �̄ by a simplerelation

�c��̃� = − �c/�i�̃�̄� . �6�

Consider the case of a zero magnetic field first. In thiscase, �n

�0��x�=0 and the oscillating phase is given by

�n = �̄ =− i�̃

1 − �̃2 − i�c�̃. �7�

In this case, Eq. �6� gives well-known results for the dy-namic dielectric constant, �c0����Dz /Ez, and the loss func-tion, L0���=Im�−1/�c0���� at the zero magnetic field, whichwe present in real units,

�c0��� = �c −�c�p

2

�2 +4�ic

�, �8�

A. E. KOSHELEV PHYSICAL REVIEW B 76, 054525 �2007�

054525-2

L0��� =4��3c/�c

2

��2 − �p2�2 + �4��c/�c�2 . �9�

The zero-field loss function has a peak at the Josephsonplasma frequency with width determined only by the c-axisquasiparticle conductivity.

Consider now the case of a finite magnetic field applied inthe layer direction. Such magnetic field generates the latticeof the Josephson vortices, which is described by the staticphase distribution, �n

�0��x�, obeying Eq. �4�. The oscillatingphase, in turn, is fully determined by this phase via the spa-tial distribution of the average cosine, Cn�x�. Averaging Eq.�4�, we obtain an obvious identity sin �n

�0�=0, indicatingthat there is no average current in the ground state. In thefield range By �0 /2�abc, the term 1/ l2 can be neglected.In this limit, it is more convenient to operate with the in-plane phases �n

�0��x�, defined by the relation �n�0�=�n+1

�0�

−�n�0�+hx, which obey the following equation:

�2�n�0�

�x2 + sin��n+1�0� − �n

�0� + hx� − sin��n�0� − �n−1

�0� + hx� = 0.

�10�

Due to the translational invariance, every solution �n�0��x�

generates a continuous family of solutions �n�0��x−u� corre-

sponding to the lattice shifts u. In particular, the phasechange for a small displacement is given by ��n=−u��n

�0� /�x. Taking the x derivative of Eq. �10�, we find that��n

�0� /�x obeys the following equation:

�3�n�0�

�x3 + Cn

��n+1�0�

�x+ Cn−1

��n−1�0�

�x− �Cn + Cn−1�

��n�0�

�x

= − �nCnh , �11�

with Cn=cos��n+1�0� −�n

�0�+hx� and �nCn=Cn−Cn−1. The con-dition � sin �n

�0� /�x=0 gives another useful identity for��n

�0� /�x,

Cn� ��n+1�0�

�x−

��n�0�

�x� = − Ch . �12�

We now proceed with the analysis of the dynamic phaseequation �Eq. �5��. Averaging this equation, we obtain thefollowing relation:

�i�c� + �2��̄ − Cn�x��n = i� .

Splitting �n�x� into the average and oscillating-in-space

parts, �n�x�= �̄�1+wn�x��, with wn�x�=0, we obtain

�̄ =i�̃

�2 − C − Cn�x�wn + i�c�̃.

Using relation �6�, we obtain the following result for thedielectric constant:

�c

�c��̃�=

�̃2

�̃2 − C − Cn�x�wn + i�c�̃. �13�

Therefore, the dynamic dielectric constant is fully deter-mined by the two simple averaged quantities, the static av-erage cosine, C, and the average including the spatial distri-bution of the oscillating phase, Cn�x�wn.

Introducing again the in-plane oscillating phases, wn=�n+1−�n, and neglecting terms of the order of 1 / l2, wederive the following equation:

�2�n

�x2 +1

1 − i�ab�̃�n�Cn�x� − �̃2 − i�c�̃��n�n = −

�nCn�x�1 − i�ab�̃

.

�14�

The oscillating phase has the same symmetry properties asthe vortex lattice. Comparing this equation with Eq. �11�, wesee that in the limit �→0, the solution is given by

�n →1

h

��n�0�

�x. �15�

This solution corresponds to the homogeneous lattice shift.As follows from Eq. �12�, the combination C+Cn�x�wn=C+Cn�x���n+1−�n�, which determines the dielectric constant�Eq. �13��, vanishes in this limit. This property is a conse-quence of the translational invariance, and it is only true inthe absence of pinning of the vortex lattice.

In summary, to find the dynamic dielectric constant at agiven field, one has to find first the static phase from Eq.�10�, assuming a definite vortex-lattice structure, and com-pute the average cosine, C= cos��n+1

�0� −�n�0�+hx�. After that,

one has to solve the dynamic equation �Eq. �14�� and com-pute the average Cn�x�wn=Cn�x���n+1−�n�. These two aver-ages completely determine �c��� via Eq. �13�. In the follow-ing sections, we will consider regimes in which analyticalsolutions are possible, the high-field limit and the vortex-oscillation regime at small frequencies. Then, we will presentthe results of the numerical analysis in the full field andfrequency range in the cases of weak dissipation �BSCCO�and large dissipation �underdoped YBCO�.

III. HIGH-FIELD REGIME

In this section, we consider the high-field regime, h 1,corresponding to the dense-lattice limit in which all inter-layer junctions are homogeneously filled with vortices. Thisregime allows for the full analytical description using anexpansion with respect to the Josephson currents. In particu-lar, the spectrum of the plasma modes and their damping inthis limit have been found by Bulaevskii et al.24 We use thesame approach to derive the dynamic dielectric constant.

The static phase solution at high fields, describing thetriangular lattice, is given by

�n�0� �

�n�n − 1�2

+2

h2 sin�hx + �n� . �16�

Using this distribution, we compute the average cosine

ELECTRODYNAMICS OF THE JOSEPHSON VORTEX… PHYSICAL REVIEW B 76, 054525 �2007�

054525-3

C � cos hx + �n −4

h2 sin�hx + �n�� �2

h2 . �17�

At high fields we can neglect rapidly oscillating Cn�x� in theright hand side of Eq. �14�. This allows us to obtain thesolution

�n = −Cn�x�/2

�̃2 − �1 − i�ab�̃�h2/4 + i�c�̃, �18�

compute the average

Cn�x���n+1 − �n� =1/2

�̃2 − �1 − i�ab�̃�h2/4 + i�c�̃,

and, finally, obtain the high-field limit of �c��� from Eq.�13�,

�c��̃��c

� 1 +i�c

�̃−

2

h2�̃2 −1/�2�̃2�

�̃2 + i�c�̃ − �1 − i�ab�̃�h2/4.

�19�

In the low-damping regime �c ,�ab�1, the loss function hasa peak at �̃=h /2 �in real units of �=�p��s2By /�0� corre-sponding to the homogeneous plasma mode.24 Such lineargrowth of the plasma frequency with field has been indeedobserved experimentally in underdoped BSCCO by Kakeyaet al.21

To verify the accuracy of the high-field approximation�Eq. �19��, we compare in Fig. 1 the loss function obtainedfrom this formula with an accurate numerical solution fortwo field values, h=2 and 4 in the case of weak dissipation,�c=0.01 and �ab=0.1. One can see that the high-field for-mula accurately describes the low-frequency region alreadyat h=2, but it overestimates the peak frequency. At h=4, thehigh-frequency approximation is already undistinguishablefrom the exact result.

IV. VORTEX-OSCILLATION REGIME AT SMALL FIELDSAND FREQUENCIES

In this section, we consider the phenomenological theoryof vortex oscillations, which describes the response of thevortex lattice at small frequencies, ���p. For the Abrikosovvortex lattice, such a theory was developed by Coffey andClem.30 The dynamic dielectric constant for the Josephsonvortex lattice has been derived using a similar approach byTachiki et al.2

Consider a superconductor in the vortex state carrying acsupercurrent js�exp�−i�t� along the c axis. The ac electricfield consists of the London term and the contribution fromthe vortex oscillations,

Ez = −4�c

2

c2 i�js −By

ci�u . �20�

The vortex-oscillation amplitude u can be found from theequation

�− �J�2 − i�J� + K�u =

�0

cjs, �21�

where �J is the linear mass of the Josephson vortex,34 �J isits viscosity coefficient,35,36 and K is the spring constant dueto pinning �it is neglected in the rest of the paper�. In Appen-dix A, we present formulas for �J and �J in terms of super-conductor parameters. Finding the oscillating amplitude, weobtain

Ez = − �4�c2 +

By�0

− �J�2 − i�J� + K

� i�js

c2 . �22�

A finite electric field also generates the quasiparticle currentjn=c,n���Ez. Therefore, the total conductivity c�By ,��= �js+ jn� /Ez is given by

c�By,�� = c,n��� −�c�p

2

4�i��1 +

1

− �J�2 − i�J� + K

By�0

4�c2�−1

.

�23�

The interesting feature of the real part of conductivity,

c,1�By,�� = Re�c,n����

+�J�c�p

2By�0/�4�c�2

�K + By�0/�4�c2� − �J�

2�2 + ��J��2 ,

�24�

is a resonance peak at the frequency �r

=��K+By�0 / �4�c2�� /�J. As one can see from Eq. �22�, this

resonance is a result of compensation of the London andvortex contributions to the oscillating electric field at ���r. Using the result for the vortex linear mass �J fromAppendix A and neglecting pinning, the formula for�r can be rewritten more transparently as �r

�0.84�p�2��s2By /�0. At �→0, Eq. �24� gives a known

result for the dc flux-flow conductivity.36

Using the well-known relation between the dynamic di-electric constant and conductivity, �c���=�c−4�c��� / i�,we obtain2

0

1

2

3

0 0.5 1 1.5 2 2.5 3

h=2, numericalh=2, theoryh=4, numericalh=4,theory

LossFunction

�

�c=0.01, �ab=0.1

x5

FIG. 1. �Color online� Comparison between the approximatehigh-field limit of the loss function based on Eq. �19� and exactnumerical solution for two field values, h=2 and 4, in the case ofweak dissipation, �c=0.01 and �ab=0.1. For clarity, the verticalscale for the h=4 plots is magnified five times.

A. E. KOSHELEV PHYSICAL REVIEW B 76, 054525 �2007�

054525-4

�c�By,�� = �c −4�c,n���

i�

−�c�p

2

�2 �1 +1

− �J�2 − i�J� + K

By�0

4�c2�−1

.

�25�

Using again formulas for �J and �J from Appendix A andneglecting pinning, we rewrite this result in reduced vari-ables in the form convenient for comparison with numericalcalculations,

�c�h,�̃��c

= 1 +i�c

�̃−

1/�̃2

1 − 2�h/�Cc��̃2 + i�c�̃� + Cabi�ab�̃�,

�26�

with Cc�9.0 and Cab�2.4. Remember that this formula isvalid only at small fields h�1 and frequencies �̃�1.

V. FIELD EVOLUTION OF THE LOSS FUNCTION

In the full range of frequencies and fields, the dynamicdielectric function can only be computed numerically. At thefirst step, one has to find the static phase distribution fromEq. �10�, assuming a definite vortex lattice structure. Toprobe general trends, we limit ourselves here only by simplealigned lattices. At a fixed magnetic field, such a lattice isfully defined by the number of layers separating the layersfilled with vortices, Nz �see sketch of the aligned lattice withNz=2 in the inset of Fig. 3�. At small fields, such lattices arerealized in ground states in the vicinity of two sets of com-mensurate fields, B1�Nz�=�3�0 / �2Nz

2�s2� and B2�Nz�=�0 / �2�3Nz

2�s2� �h1�Nz�=��3/Nz2 and h2�Nz�=� / ��3Nz

2� inreduced units�. At the intermediate field values, the groundstate is given by misaligned lattices.16,17

At the first stage, we solved the static phase equations�Eq. �10�� for fixed h and Nz. This solution has been used asan input for the dynamic phase equations �Eq. �14��. Finally,the oscillating phase determines the dynamic dielectric con-

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

�ab=0.40.61.02.04.0

LossFunction

�

h=0.5, Nz=2, �c=0.1

Role of in-plane dissipation

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

�c=0.1

0.20.30.40.5

LossFunction

�

h=0.5, Nz=2, �

ab=0.4

Role of c-axis dissipation

FIG. 2. �Color online� Influ-ence of the dissipation parameterson the loss-function shape in thedilute-lattice regime at h=0.5 andNz=2. In-plane dissipation �leftplot� does not influence much theJPR peak, but strongly influencesthe shape below the peak.

0

1

2

3

4

Numerical SolutionVortex model

LossFunction

�c=0.1, �ab=1.0, h=0.5, Nz=2

Nz=2

0

1

2

3Numerical solutionVortex model

LossFunction

�c=0.1, �ab=4.0,h=0.5, Nz=2

0

0.02

0.04

0.06

0.08

0 0.2 0.4 0.6 0.8 1

Numerical solutionVortex model

LossFunction

�

�c=0.01, �

ab=0.1, h=0.12, N

z=6

FIG. 3. �Color online� Comparison between the vortex-oscillation model and exact numerical calculation for the loss func-tion at several representative values of the dissipation parameters,field, and Nz �shown in the plots�. One can see that this modeltypically describes the high-frequency response roughly up to halfof the plasma frequency. The inset in the upper plot illustrates thealigned lattice with Nz=2.

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stant �Eq. �13�� and the reduced loss function L��̃�=−Im��c /�c��̃��.

We start with the discussion of several general propertiesof the high-frequency response. To illustrate the roles of twodifferent dissipation channels, we show in Fig. 2 series of thefrequency dependencies of the loss function when only onedamping constant is changed for fixed field h=0.5 and Nz=2 corresponding to a dilute lattice. We can see that thewidth of the JPR peak is mainly determined by the c-axisdissipation, while the in-plane dissipation has no visible in-fluence on the loss function near the JPR peak. On the otherhand, the in-plane dissipation strongly influences the shapeof the loss function below the peak. In particular, at a verystrong in-plane dissipation, a peaklike feature appears belowthe JPR frequency which looks like an oscillation mode.

In Fig. 3, we compare the behavior of the numericallycomputed loss function at low frequencies with predictionsof the vortex-oscillation model described in Sec. IV. We cansee that this model typically accurately describes the behav-ior at frequencies smaller then half the plasma frequency. Atlow dissipation, the loss function has an additional peak atsmall frequencies. This peak disappears with increasing dis-sipation. We have to emphasize that these plots are made foran ideal homogeneous system. In a real layered supercon-ductor, pinning of the Josephson vortices by inhomogeneitieswill lead to the resonance peak at the pinning frequency.

A. Small dissipation „Bi2Sr2CaCu2Ox…

To illustrate a theoretically expected behavior of the high-frequency response in BSCCO, we made calculations with

small dissipation parameters, typical for this compound, �c=0.01 and �ab=0.1. Figure 4 shows the evolution of the lossfunction with increasing magnetic field in the frequencyrange near the plasma resonance. For such small dissipation,the loss function at zero field has a very sharp peak at theplasma frequency. In the dilute-lattice regime at small fields,this peak decreases in amplitude and is displaced to slightlylower frequencies.

The most interesting feature of the dilute regime is theappearance of the satellite peaks in the high-frequency part.The strongest satellite is observed at Nz=2 near h�0.45. Aphysical origin of these satellites is clear. In the vortex latticestate, the homogeneous oscillations are coupled to theplasma modes with the wave vectors set by the lattice.Therefore, the location of the satellites is determined by theplasma spectrum. For reference, we present the plasma fre-quencies at the reciprocal-lattice vectors in Appendix B. Atsmall dissipations, the shape of the loss function is not rig-idly fixed by the value of the magnetic field; it is also sensi-tive to the lattice structure. This is illustrated in Fig. 5, wherethe loss function is plotted at fixed h for two values of Nz, forwhich the lattice energies are close. One can see that the lossfunction is almost Nz independent at frequencies below theJPR peak, indicating that in this frequency range the vorticescontribute independently to the response. However, abovethe peak, the shape of the loss function is very sensitive toNz. This means that the high-frequency response can poten-tially be used to probe the lattice structure.

In the dense-lattice limit, the plasma peak moves to ahigher frequency and its intensity rapidly decreases, as pre-dicted by the analytical theory in Sec. III. This behavior is

0.8 1.0 1.2 1.40

10

20

30

40

50

60

70

80

90

100

Nz=1,h=2

Nz=1,h=1.4Nz=2,h=1Nz=2,h=0.8 Nz=3,h=0.5Nz=2,h=0.45

Nz=3,h=0.4Nz=4,h=0.3Nz=3,h=0.2

Lossfunction

�

�c=0.01, �ab=0.1

h=0Nz=4,h=0.1

0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4 h=0Nz= 4, h=0.1Nz= 3, h=0.2Nz= 4, h=0.3

Nz= 2, h=0.4Nz= 2, h=0.45Nz= 3, h=0.5Nz= 2, h=0.8Nz= 2, h=1.0Nz= 1, h=1.4Nz= 1, h=2.0

�

FIG. 4. �Color online� Series of the frequency-dependent loss functions near JPR frequency at different fields for �c=0.01 and �ab

=0.1. The inset shows the low-frequency region for the same set of parameters.

A. E. KOSHELEV PHYSICAL REVIEW B 76, 054525 �2007�

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also consistent with experiment.21 The transition to the denselattice is very distinct. While for Nz=2 and h=1 the peak isstill located very close to the zero-field JPR frequency and itsamplitude is comparable with the zero-field peak, at veryclose field h=1.4 in the dense-lattice regime, Nz=1, the peakis noticeably shifted to a higher frequency, its amplitude con-siderably dropped, and the width increased.

An additional broad peak exists at low frequencies. Itsintensity is much smaller than the JPR peak. The evolutionof this peak with increasing magnetic field is shown in theinset of Fig. 4. One can see that the intensity of this peakmonotonically increases with the magnetic field. This peak iswell described by the vortex-oscillation model.

We did not find any intrinsic resonances in the loss func-tions at frequencies smaller than the JPR frequency, meaningthat there are no modes coupled to homogeneous oscillationsin this frequency range for a homogeneous superconductor.37

This suggests that the resonance feature observed in under-doped BSCCO at �0.5�p in Ref. 21 probably has an extrin-sic origin; e.g., it may be caused by the pinning of the Jo-sephson vortices.

B. Large dissipation (underdoped YBa2Cu3Ox)

In this subsection we discuss the high-frequency responseof the Josephson vortex lattice in the case of large effectivedissipation, which is realized, e.g., for the underdopedYBCO.22,23 Figure 6 shows the frequency dependences of theloss functions for the dissipation parameters �c=0.32 and�ab=6.0. For comparison, we also present the loss functionsat different in-plane fields for an underdoped YBCO samplefrom Ref. 22. One can see that two experimental features are

well reproduced by the theory: decreasing intensity of themain peak with increasing field and the appearance of thesatellite peak in the low-frequency part of the line. The ori-gin of this satellite peak is distinctly different from the high-frequency satellites found for small dissipation. Figure 2demonstrates that this peak only appears for sufficientlystrong in-plane dissipations. As this peak is absent in thelow-dissipation limit, it does not correspond to any specificmode of phase oscillations. It is caused by the decrease of therelative contribution of the in-plane dissipation channel withincreasing frequency. To verify this interpretation, we plot inFig. 7 the loss function for Nz=2 and h=1, together with therelative contribution of in-plane dissipation to losses, Rab

=abEx2 / �cEz

2+abEx2�, which can be rewritten in reduced

coordinates,

Rab =�ab���n/�x�2

�c�1 + ��n+1 − �n�2� + �ab���n/�x�2. �27�

One can see that Rab decreases with increasing frequency andalmost vanishes above the plasma resonance. The small-frequency peak roughly corresponds to the frequency whereRab drops to one-half. The decrease of the in-plane dissipa-tion also leads to the peak in the real part of the opticalconductivity 1, also shown in Fig. 7. Note that the peak in1 corresponds to the dip in the loss function.

At high fields, in the dense-lattice limit, only a singlebroad dissipation peak remains. In contrast to the low-dissipation case, this peak is displaced to lower frequencieswith increasing magnetic field. This behavior is not verifiedexperimentally yet; data are only available for field range h�1.22,23

0

20

40

60

0.9 0.95 1 1.05 1.1

Nz=3Nz=5

LossFunction

�

h=0.2

0.95 1 1.05 1.1 1.15

Nz=2Nz=3

�

h=0.4FIG. 5. �Color online� Plots of the loss func-

tion at the same h but with different Nz for �c

=0.01 and �ab=0.1. This figure illustrates thesensitivity of the loss-function shape to the vortexlattice structure �value of Nz� above the JPR peak.

0

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

h=00.2, Nz=50.5, Nz=31.0, Nz=21.3, Nz=12.0, Nz=1

�

�c=0.32,�ab=6.0

0.2, Nz=50.5, Nz=31.0, Nz=21.3, Nz=12.0, Nz=1

0 20 40 60 80 100 1200.00

0.02

0.04

0.06

0.08

0.10

0.12

Tc = 65 K

YBa2Cu3O6.75

T = 8 KE || cH || CuO2Lo

ssFunction

Frequency (cm-1)

H=0TH=2TH=4TH=8T

b)a)

FIG. 6. �Color online� �a� Se-ries of the frequency-dependentloss functions at different fieldsfor �c=0.32 and �ab=6.0. Forcomparison, plot �b� shows theevolution of the loss function withincreasing in-plane field for un-derdoped YBCO from Ref. 22.

ELECTRODYNAMICS OF THE JOSEPHSON VORTEX… PHYSICAL REVIEW B 76, 054525 �2007�

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VI. SUMMARY

In summary, we developed a comprehensive descriptionof the high-frequency response of the Josephson vortex lat-tice in layered superconductors. We found the general rela-tion �13� connecting the dynamic dielectric constant with theaverages containing the static and oscillating phases. Ana-lytical formulas for the dynamic dielectric constant and lossfunction were derived for the high-field regime and thevortex-oscillation regime at low frequencies. Numericallysolving equations for the oscillating phases, we explored theevolution of the loss function with increasing magnetic fieldfor the cases of weak dissipation describing BSCCO andstrong dissipation describing underdoped YBCO. Severalfeatures were found. The most interesting feature in theweak-dissipation case is the high-frequency satellites in thedilute-lattice regime, which appear due to the excitation ofplasma modes at the wave vectors of the reciprocal lattice. Inthe strong-dissipation limit, we reproduced the additionalpeak in the loss function below the JPR peak experimentallyobserved in underdoped YBCO.22,23 We established that this

peak appears due to the frequency dependence of the in-plane contribution to losses.

ACKNOWLEDGMENTS

This work was motivated by illuminating discussions ofexperimental data with D. Basov, A. LaForge, and I. Kakeya.This work was supported by the U.S. DOE, Office of Sci-ence, under Contract No. DE-AC02-06CH11357.

APPENDIX A: JOSEPHSON-VORTEX MASSAND VISCOSITY

The viscosity of an isolated Josephson vortex has beenconsidered in Refs. 35 and 36. In the case when the dissipa-tion caused both c-axis and in-plane quasiparticle transport,the viscosity coefficient is given by

�J =1

�s2� �0

2�c�2�Ccc + Cab

ab

�2 �=

�c�p�02

��4�cs�2��Cc�c + Cab�ab� , �A1�

where the numerical constants Cc and Cab are determined bythe phase distribution of an isolated Josephson vortex, �n

�0�,

Cc = �n=−�

� �−�

�

du ���n+1�0� − �n

�0���u

�2

� 9.0,

Cab = �n=−�

� �−�

�

du� �2�n�0�

�u2 �2

� 2.4.

The mass of the Josephson vortex has been considered inRef. 34. This mass is determined by the kinetic energy Ek,which, for a moving Josephson vortex, can be presented as

Ek =� d3r�cEz

2

8�= s�

n� d2r

�c

8�� �0

2�cs�2

�̇n2.

For a slowly moving vortex with velocity v, the phase dif-ference is determined by its static distribution, �n�r , t�=�n

�0��x−vt�, and we obtain

Ek = Lyv2

2s�

n� dx

�c

4�� �0

2�cs�2�d�n

�0�

dx�2

.

As, by definition, Ek=Ly�Jv2 /2, we obtain the following re-sult for the linear vortex mass:

�J =��0

2

��4�c�2s�

n=−�

� �−�

�

dx�d�n

dx�2

=Cc��0

2

���4�cs�2 . �A2�

The reduced combination in the dynamic dielectric constant�Eq. �25�� can be represented as

1

�J�2 + i�J�

B�0

4�C2 =

2�H

CC��̃2 + i�C�̃� + Cabi�ab�̃.

0

0.5

1

1.5

2L

�c=0.32, �ab=6.0, Nz=2, h=1

00.51

1.52

2.53

� 1

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Rab

�FIG. 7. �Color online� The frequency dependences of the loss

function, real part of conductivity, and relative contribution of in-plane dissipation �Eq. �27�� for the same damping parameters as inFig. 6, Nz=2, and h=1. Rapid decreasing of in-plane inhomogene-ity of the oscillations with increasing frequency can also be seenfrom the visualization of oscillating patterns of the local electricfields at different frequencies �Ref. 38�.

A. E. KOSHELEV PHYSICAL REVIEW B 76, 054525 �2007�

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APPENDIX B: PLASMA FREQUENCIES ATRECIPROCAL-LATTICE VECTORS: EXPECTED

LOCATION OF SATELLITES

The satellite peaks are expected approximately at theplasma frequencies for the wave vectors of the reciprocallattice. In the reduced units, the plasma spectrum at the zeromagnetic field and without the charging effects is given by

�p2�k,q� = 1 +

k2

1/l2 + 2�1 − cos q�, �B1�

where k and q are the wave-vector components along andperpendicular to the layers, respectively. The reciprocal vec-tors of the vortex lattice are given by

�k�n,m�,q�n,m�� = �2�n/a,2��m − n/2�/Nz� ,

where n and m are the integer indices and a is the vortexseparation, a=2� /hNz. Neglecting the small factor 1 / l2, weobtain for the satellite frequencies

�s2�n,m� � 1 +

�hNzn�2/2

1 − cos�2��m − n/2�/Nz�. �B2�

The most intense satellite is expected for the basic reciprocalvector with indices �n ,m�= �1,0�

�s2�1,0� = 1 +

�hNz�2/2

1 − cos��/Nz�.

In particular, for Nz=2, this gives �s2�1,0�=1+2h2.

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25� found the mode whose behavior strongly resembles the be-

havior of the experimental low-frequency peak in Ref. 21. How-ever, this mode is the antiphase mode in which the phases os-cillate with the phase shift � in neghboring layers. At highfields, the frequency of this mode can be computed analytically,�p���=�p

�2/h. As the average oscillating phase is zero in thismode, it does not contribute to the response to thehomogeneous-in-space oscillating electric field in a homoge-neous superconductor. Coupling of this mode to homogeneousoscillations can be mediated, e.g., by correlated disorder.

38 See EPAPS Document No. E-PRBMDO-76-051729 for anima-tions of oscillating patterns of local electric field at differentfrequencies for �c=0.32, �ab=6.0, Nz=2, and h=1. For moreinformation on EPAPS, see http://www.aip.org/pubservs/epaps.html.

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