Sliding Abrikosov vortex lattice in the presence of a regular array of columnar pinning centers: ac conductivity and criticality near the transition to a pinned state

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Sliding Abrikosov vortex lattice in the presence of a regular array of columnar pinning centers:

ac conductivity and criticality near the transition to a pinned state

T. Maniv,1 _{B. Rosenstein,}2_{I. Shapiro,}3_{and B. Ya. Shapiro}3 1_{Schulich Faculty of Chemistry, Technion-IIT, 32000 Haifa, Israel}

2_{Department of Electrophysics, National Chiao Tung University, 30010 Hsinchu, Taiwan, Republic of China} 3_{Department of Physics, Institute of Superconductivity, Bar-Ilan University, 52900 Ramat-Gan, Israel}

共Received 20 May 2009; revised manuscript received 6 August 2009; published 19 October 2009兲

The dynamics of the flux lattice in the mixed state of strongly type-II superconductor near the upper critical field Hc2共T兲 subjected to ac field and interacting with a periodic array of short-range pinning centers

共nano-solid兲 is considered. The superconductor in a magnetic field in the absence of thermal fluctuations on the mesoscopic scale is described by the time-dependent Ginzburg-Landau equations. An exact expression for the ac resistivity in the case of a␦-function model for the pinning centers in which the nanosolid is commensurate with the Abrikosov lattice共vortices outnumber pinning centers兲 is obtained. It is found that below a certain critical pinning strength uc and sufficiently low frequencies there exists a sliding Abrikosov lattice, which

moves nearly uniformly despite interactions with the pinning centers. At small frequencies the conductivity diverges as共u−u_c兲−1_{, whereas the ac conductivity on the depinning line diverges as i}_␻−1_{. This sliding lattice}

behavior, which does not exists in the single vortex-pinning regime, becomes possible due to strong interac-tions between vortices when they outnumber the columnar defects. Physically it is caused by “liberation” of the temporarily trapped vortices by their freely moving neighbors.

DOI:10.1103/PhysRevB.80.134512 PACS number共s兲: 74.20.De, 74.25.Qt, 74.78.Fk

I. INTRODUCTION

The great interest in the problem of magnetic-flux pinning in type-II superconductors is associated with its relevance to technological applications of superconductivity as well as with its implications to the general problem of complex glass dynamics with tunable parameters. An important challenge in applications of type-II superconductors is in achieving op-timal critical currents under given magnetic fields. This re-quires preventing depinning of Abrikosov vortices during formation of the resistive state under the applied current. Random pointlike pinning centers naturally appear due to imperfections of lattice structure or chemical disorder. Pin-ning can be artificially enhanced by ion irradiation,1_{in which} case one obtains a random array of columnar defects. Theory of dynamics of the pinned vortex matter by a random distri-bution of pins is very complicated.2–4 _{However in the} ab-sence of significant thermal fluctuations on the mesoscopic scale the problem simplifies considerably. It was studied theoretically mostly in two-dimensional 共2D兲 systems using either numerical methods within a model of interacting points such as particles representing vortices subject to pin-ning potential and driving force5 _{or within the elasticity} theory3 _{in which the vortex matter is treated as an elastic} manifold subject to both pinning stress and driving force. Elastic manifold under and ac electromagnetic radiation was investigated in Ref. 6.

Recently there have been advances in the study of vortex pinning by fabricating periodic arrays of pinning sites where each pinning site may be either magnetic or normal nanopat-tern inclusion effectively trapping vortices. Pinning arrays with triangular, square, and rectangular geometries have been fabricated using either microholes or blind holes,7_{arrays of} magnetic dots,8_{and periodic array of columnar defects.}9_The resulting critical current is enhanced when vortex lattice is

commensurate with the periodic array of pinning sites.8 _In addition this system is a convenient experimental tool to study the general problem of interacting periodic system moving in periodical potential such as dislocations in crys-tals or charge-density waves.10 _{Theory of the Abrikosov} lat-tice subjected to an ac field and periodic pinning is simpler but so far has been treated either numerically using molecular-dynamics approach11 or by means of the elastic manifold approach.12_{In the one-dimensional situation of the} vortex transport in narrow channels a sliding vortex phase was studied numerically13 _{following the Frenkel-Kontorova} model approach.14

Theoretically a basic question concerns the importance of correlation of defect centers for pinning of highly correlated vortex matter. It is sometimes believed that pinning of vortex lines in type-II superconductors is analogous to localization of correlated electrons by impurities in metals and semicon-ductors. According to this line of thought the vortices are mapped onto quantum particles, rather than considered as classical line—such as objects. If this were the case, a peri-odic array would not be able to trap the vortex lattice. How-ever the vortices are topological solitons of the essentially nonlinear Ginzburg-Landau equations and behave like clas-sical objects. Consequently both the random pinning and highly correlated pinning are expected to result in a roughly similar critical current and other characteristics of the pinned state especially when the density of columnar defects is much smaller than the density of vortices. We therefore con-centrate on a simpler problem of a periodic array of short-range pins 共to be named “nanosolid”兲 and employ the time-dependent Ginzburg-Landau共TDGL兲 equations for the order parameter⌿ 共Ref.15兲 to describe the dynamics of the vortex

matter in a magnetic field.

Considering the electric current applied to the system as a perturbation, a linear-response theory is used allowing the

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calculation of the ac resistivity. In the absence of thermal fluctuations on the mesoscopic scale, an exact solution for the linear response in the case of a␦-function model for the pinning centers in which the nanosolid is commensurate with the Abrikosov lattice共vortices outnumber pinning centers兲 is obtained. In the strong magnetic field regime near Hc2共T兲

investigated here, two features emerge as compared to the low-field regime, commonly studied previously by the Lon-don approach. The number of vortices关described by a set of zeros of the order parameter field ⌿共x,y兲兴 significantly ex-ceeds the number of columnar pins. In addition the custom-ary London approach is inapplicable since the distances be-tween vortices are not much larger than the size of the vortex cores. Interactions between vortices in this regime are sig-nificant and neighbors of the trapped vortices can liberate them from the potential trap resulting in a significant de-crease in the critical current.

We find, that below a certain critical pinning strength uc,

Abrikosov lattice of vortices is moving coherently despite interactions with the pinning centers, thus forming a “sliding Abrikosov lattice state.” The dependence of the ac conduc-tivity on pinning strength u, magnetic induction and fre-quency in this phase is calculated. In particular, at small frequencies conductivity as a function of u behaves as 共u − uc兲−1, demonstrating an ideal metal behavior, i␻−1, as

func-tion of frequency at u = uc.

The rest of the paper is organized as follows. In the next section the model is presented. In Sec. III a general linear-response theory of a strongly type-II superconductor under a magnetic field H to external alternating electric current is constructed to leading order in the small expansion param-eter ah=关1−T/Tc− H/Hc2共0兲兴/2. This parameter describes

deviation of H from the mean-field upper critical field

H_c2共T兲. The condensate part of conductivity is presented via

Green’s function共GF兲 of quantum charged particle subjected to both magnetic field and periodic potential. The exact Green’s function for the corresponding linearized TDGL equations in the presence of periodic array of short-range potentials 共commensurate with the vortex lattice兲 is calcu-lated using the inversion of the Lippmann-Schwinger equations16_{in Sec.}_IV_{. The results for the ac conductivity are} presented and analyzed in Secs.VandVI. Criticality at low frequencies is discussed in Sec.VI. Technical details are rel-egated to Appendices.

II. MODEL

Let us consider a type-II superconductor under a constant external magnetic field H parallel to a system of columnar defects directed along z axis and carrying electric current along the y axis, see Fig.1. The columnar defects are located at points ra 关2D vectors r=共x,y兲 will be denoted by bold

letters兴 and are assumed to be short range 共on the scale of the coherence length␰兲.

A. Basic equations

The static magnetic properties of the superconductor are described by the GL Gibbs energy as a functional of the order parameter ⌿ and vector potential A,

FGL关⌿,A兴 =

冕

dzdr

冋

ប 2 2mcⴱ 兩⳵z⌿兩2+ ប2 2mⴱ兩D⌿兩 2_{− a}

_⬘

_{共r兲兩⌿兩}2 +b

⬘

2兩⌿兩 4₊ 1 8␲共B − H兲 2

册

_. _共1兲 Here D⬅⵱−i2_⌽␲

0A, denotes the covariant derivative and ⌽0

=hc_eⴱ, eⴱ= 2兩e兩 is the unit of flux, B=⵱⫻A is the magnetic

induction. We chose the vector potentials in the form inde-pendent of time, Ax= − 1 2By, Ay= 1 2Bx. 共2兲

Assuming that the ratio␬⬅␭/␰Ⰷ1, where ␭ is the penetra-tion depth, the magnetizapenetra-tion M is by the factor 1/␬2smaller than the field and consequently for magnetic fields few times larger than Hc1, B⬇H. Thermal fluctuations on the

mesos-copic scale are ignored. The temperature is taken into ac-count only on the mean-field level, namely, coefficients of the Ginzburg-Landau energy depend on temperature T.

When the columnar defects are absent共the “clean” case兲,

a

⬘

共r兲=␣共Tc− T兲 is uniform and the free energy is minimized

by a hexagonal Abrikosov lattice of vortices with cores lo-cated at rn₁,n₂= n1a1+ n2a2 with a1= a⌬共1/2,

冑

3/2兲, a2 = a_⌬共1,0兲, where the lattice spacing is a_⌬= 21/23−1/4

冑

⌽0/B. The columnar defects are represented by an inhomogeneous coefficient

a

⬘

共r兲 =␣关Tc− T + V共r兲兴, 共3兲

where V consists of “potentials” around pinning centers ra,

V共r兲 = Tc

兺

a

U共r − ra兲. 共4兲

On microscopic scale the potential arises, for example, from deviation of local charge carriers density ne共r兲 from that of

the uniform sample, n0, sometimes represented as2

FIG. 1.共Color兲 Hexagonal Abrikosov vortex lattice 共distribution of the superfluid density兩␺共r兲兩2_{兲 and pinning centers. Zeros of order}

parameter fall on the locations of the columnar defects共red squares兲 so that vortices outnumber the pins. Vectors d1and d2are lattice

vectors of the lattice of pinning sites. Distance between nearest neighbors of the Abrikosov lattice is a_⌬.

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V共r兲 = Tc

⳵ln Tc

⳵ln ne

␦ne共r兲

n₀ . 共5兲

We assume that the defects are thin on the scale of coherence length at certain temperature, which is quite large for low-Tc

superconductors compared to the size of damage of ions or electrons used in irradiation experiments or nanosize antidots with effective radius w and strength␧⬎0 considered as phe-nomenological parameters within the Ginzburg-Landau ap-proach. As will be shown below, the only solvable configu-ration corresponds to a hexagonal periodic array of very thin pinning points located at ra= n1d1+ n2d2commensurate with the static Abrikosov lattice so that d1= s1a1, d2= s2a2 with

s₁, s₂integers. The density of pinning centers is proportional to the fractional filling factor f = s1s2, namely,

Np

LxLy

= 2

f

冑

3a_⌬2, 共6兲

for hexagonal lattice. In particular, it means that magnetic field cannot be too small B⬎⌽0Np

LxLy.

The simplest relaxation dynamics of a superconductor in the presence of electric field is described by TDGL equation15 ប2_␥ 2mⴱDt⌿ = − ␦ ␦⌿ⴱFGL, 共7兲 where Dt⬅_⳵t⳵− i eⴱ

ប⌽ and the electric field is E=−⵱⌽−1c

⳵ ⳵tA. The vortices are moving along the x direction, see Fig. 1. Maxwell equations are

ⵜ ⫻ B =4␲

c J, J = Jn+ Js. 共8兲

Superconducting component of the current density is

Js=

ieⴱប

2mⴱ共⌿

ⴱ_D_{⌿ − ⌿D⌿}ⴱ_兲, _共9兲

while the normal electron component of the current density is Jn=␴nE.

Neglecting the time dependence of the electric charge 关screened on the Thomas-Fermi length, which is smaller than

␰共Ref.15兲兴, the charge conservation law in a superconductor

reads

ⵜ · J = 0. 共10兲

In our case of large␬ and the magnetic field not far from

Hc2共T兲, the charge conservation Eq. 共10兲 implies

homogene-ity of the current denshomogene-ity. Indeed taking curl of the Maxwell Eq. 共8兲, one obtains

ⵜ ⫻ J = ⵜ2_M_{⬃ O共}_␬−2_兲. _共11兲 Consequently the ac current density is uniform and is ori-ented along the y axis, J共t兲=J0cos共␻t兲.

B. Dimensionless units

The system is invariant under translations in the field di-rections, so we use a 2D dimensionless energy density f_GL:

FGL= LzHc2

2 _/共8_␲␬2_兲兰drfGL_{. In this paper} _␰₌_ប/共2mⴱ_␣_T

c兲1/2

will be used as a unit of length, r→r/␰, while Hc2

=⌽0/共2␲␰2_{兲 is the unit of magnetic field, h⬅B/H}

c2. The

scaled order parameter is defined by ␺= 2−1/2⌿/⌿0, where ⌿0=共␣Tc/b

⬘

兲1/2, so that the dimensionless energy density is

can be written in the following form

fGL=␺ⴱHˆ␺− ah␺ⴱ␺+

1 2共␺

ⴱ_␺_兲2_. _共12兲 The dimensionless parameter,

ah= 1 − t − h 2 − u0, t = T Tc , 共13兲

has a physical meaning of “distance” from the static normal-mixed-state boundary in the H − T space and the constant shift u0 reflects an average pinning effect. It will be deter-mined by exploiting the bifurcation 共or transition兲 point expansion17,18 _{around a}

h= 0 at which the order parameter

vanishes. Defining a Hamiltonian for the case without pin-ning, Hˆcl= − 1 2D 2₋h 2 共14兲

and a dimensionless pinning potential U共r兲 one can represent the linear operator in Eq. 共12兲 in the following form

Hˆ = Hˆcl− u0−

兺

a

U共r − ra兲 ⬅ Hˆp− u0. 共15兲 It should have the property17,18 that its lowest eigenvalue is zero. It is well known that the lowest eigenstate of the Lan-dau Hamiltonian Hclis degenerate, however the presence of

the pinning potential in Eq. 共15兲 partially lifts the Landau

degeneracy. The lowest energy corresponds to the lowest Landau-level 共LLL兲 state in which zeros of the Abrikosov wave function fall on the pining center sites ra, see Fig.1

and expression in Appendix A.

In analogy to the coherence length, one can define a char-acteristic time scale. In the superconducting phase that is a typical “relaxation” time is tGL=␥␰2/2 and unit of electric field, EGL= Hc2␰/共ctGL兲, E=E/EGL. In these units the

time-dependent equation takes a form

−⳵t␺= Hˆ␺− ah␺+␺ⴱ␺2− i⌽␺. 共16兲

The current density is scaled as J = cHc2/共2␲␰␬2兲j, in

par-ticular,

js=

i

2关␺

ⴱ_D_␺₋_␺_共D_␺_兲ⴱ_兴 _共17兲 being the dimensionless supercurrent density. The conductiv-ity will be given in units of

␴GL=

c2tGL 2␲␭2=

c2␥

4␲␬2. 共18兲

This unit is close to the normal-state conductivity ␴n in

low-Tc superconducting metals in the dirty limit, for which

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relating the two: ␴n= k␴GL. The corresponding equation for the dimensionless electric field, takes a form

kE = j − js. 共19兲

From now on we will take for simplicity k = 1. The model will be solved using expansions in electric field and ah but

exactly in pinning strength in the next section.

III. LINEAR-RESPONSE THEORY AND EXPANSION IN POWERS OF SUPERFLUID DENSITY

(PARAMETER a_h)

A. Linear response of the system to electric current

The set of Eqs.共16兲 and 共19兲 can be solved using

expan-sion in both the total current, which can be viewed as “ex-ternal,” and the parameter ahdefined in Eq.共13兲. To the first

order in j the order parameter is represented as follows:

␺共r,t兲 =␾共r兲 +␪共r,t兲, 共20兲 where ␾ is the static mean-field solution specified below. Substituting Eq.共20兲 into TDGL, Eq. 共16兲, one obtains

−⳵t共␾+␪兲 = Hˆ共␾+␪兲 − ah共␾+␪兲 + 共␾+␪兲ⴱ共␾+␪兲2− i⌽␾.

共21兲 To this order in j one observes that the correction to the order parameter,␪, satisfies

⳵t␪=共− Hˆ + ah− 2␾ⴱ␾兲␪−␪ⴱ␾2+ i⌽␾. 共22兲

Similarly substituting Eq. 共20兲 into the expression for y

component of the supercurrent Eq. 共17兲, Eq. 共19兲 takes a

form

E = j − i关␪ⴱ_D

y␾−␪共Dy␾兲ⴱ兴. 共23兲

This equation expresses the linear response since the correc-tion␪in turn depends on the current via the electric potential ⌽ appearing in Eq. 共22兲. The equation Eq. 共22兲 can be solved

using expansion in the small parameter ah.

B. Expansion in ah

It is important to note that by exploiting the ultimate lo-calized form, i.e., proportional to a sum of delta functions, for the pinning potential U共r兲 in Eq. 共4兲, the static

configu-ration, ␾共r兲, of the order parameter to leading order in ah,

can be calculated exactly. In fact, the definition of the param-eter ahin Eq.共13兲 already took this fact into account. Indeed,

the solution at small ahwithout pinning is well known17,18

␾共r兲 =

冉

ah

␤A

冊

1/2

␸0共r兲 + O共ah

3/2_兲 _共24兲 with the functions␸N共r兲 constituting a basis of orbitals with

Landau-level index N = 0 , 1 , . . ., defined in symmetric gauge in Appendix A. The next to leading order in ah provides

normalization and fixes certain linear combination of the LLL functions so that␸0共r兲 is a hexagonal lattice. Yet with-out pinning this solution is highly degenerate since one can shift or rotate the lattice as a whole.

When the commensurate pinning potential is added, the degeneracy is lifted and the only configuration of the mini-mal energy is the one with vortex cores located exactly at the sites of the columnar pins. Energy of all the other configu-rations are higher. The lowest eigenvalue of the linear opera-tor Hˆp, Eq. 共15兲, 关determining the Hc2共T兲 line兴 is shifted by

u0. This justifies the dependence on the pinning strength in the definition of ah, Eq.共13兲. It is therefore concluded that

such a pinning does not change the shape of the configura-tion in the leading order but does suppress the value of the amplitude of the order parameter and the critical field.

Now we return to the linear-response relation, namely, to expansion of physical quantities to the first order in j, Eq. 共23兲. It is clear in view of Eq. 共24兲, that the correction to the

order parameter ␪ is of order ah

1/2 _{共thus allowing to neglect}

ah3/2and higher order terms兲, and obeys

⳵t␪= − Hˆ␪+ i⌽␾. 共25兲

For the ac transport along the y direction

⌽共r,t兲 = −

冕

0

y

inhomogeneous correction to the electric field in Eq. 共30兲

共induced by the supercurrent js兲 is however responsible for

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C. ac resistivity

For a homogeneous ac current density j共t兲= j0cos共␻t兲,

Eq. 共30兲, averaged over volume of the sample, gives the

following expression for complex resistivity

␳共␻兲 =1

T

冕

_t=0

T

e−i␻t具E典r/j0, 共31兲

where in the end the large time limit T→⬁ should be taken. Performing integration over y

⬙

one obtains共recalling that in our dimensionless units␴n=␳n= 1兲

␳共␻兲 = 1 −␴s共␻兲 ⬇

1 1 +␴s共␻兲

, 共32兲

with the condensate contribution to conductivity

␴s共␻兲 = −

冕

r⬘

y

⬘

具␾共r

⬘

兲关Dy␾共r兲兴ⴱG共r,r

⬘

,␻兲

,␻兲.

It is important to note that since the real part of the resis-tivity is positive, in view of Eq. 共32兲, our theory strictly

speaking is valid only when the condensate contribution to conductivity is smaller than␴n. Equation 共33兲 allows to

re-late the dynamic conductivity in the superconductor with the GF of a quantum-mechanical Hamiltonian Hˆp of a charged

particle in magnetic field in the presence of a periodic poten-tial defined in Eq. 共15兲. This will also allow calculation of

the lowest-energy eigenvalue determining the shift u. The next section and Appendix B deal with this problem.

IV. GREEN’S FUNCTION FOR A SYSTEM WITH

PERIODIC␦-PINNING ARRAY

To find the GF we approximate the potential by an array of delta functions U共r兲 = − U0

兺

a ␦共r − ra兲, 共34兲 where U0= ␲w2␧ ␰2_T c , 共35兲

and ␧ is the pinning energy. The delta function represents sufficiently localized defects on the scale of coherence length. First we review the comprehensively studied clean case.

A. Retarded Green’s function Gclfor a system without

pinning potential

The GF with added potential U共r兲, Eq. 共34兲, obeys

关⳵t+ Hˆp兴G共r,r

⬘

,t − t

⬘

兲 =␦共r − r

⬘

兲␦共t − t

⬘

兲, 共39兲

where one chooses to divide the operator Hˆpof Eq.共15兲 into

− t

⬘

兲.

共40兲 For the Fourier transform G共r,r

⬘

,␻兲.

共42兲 In particular, at pinning points r = rbone gets关using Eq. 共37兲兴

G共rb,r

⬘

,␻兲 = Gcl共rb,r

⬘

,␻兲 − U0

兺

a

e+ih/2ra⫻rb_g

cl

⫻共rb− ra,␻兲G共ra,r

⬘

,␻兲. 共43兲

Here we assumed commensurability with vortex lattice for pins in zeros so there will be no phase factor on the right-hand side due to flux quantization,

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ra⫻ rb=

2␲f

h 共na1nb2− nb1na2兲. 共44兲

If f = s1s2is even the phase in Eq.共43兲 disappears while for an odd s₁s₂one still can solve the pinning problem by divid-ing the pinndivid-ing sites into two sublattices. We continue here with the even case. Under this conditions, translation sym-metry allows to solve the set of linear equations

兺

_a Mba共␻兲G共ra,r

⬘

,␻兲 = Gcl共rb,r

⬘

,␻兲共45兲

with a symmetric matrix M defined by

Mba共␻兲 =␦ab+ U0gcl共rb− ra,␻兲, 共46兲

Using Fourier transform

gcl共ra,␻兲 = 1 共2␲兲2

冕

k g_k,cl_␻exp共ik · ra兲 = 1 共2␲兲2

冕

q_苸BZ g ˜_q,_␻exp共iq · ra兲. 共47兲

The function gcl共ra,␻兲 is more conveniently written via

g

˜_q,_␻=

兺

K

g_{共q+K兲,␻}cl , 共48兲 where K are reciprocal pinning centers lattice points

K = m1e1+ m2e2, e1= 2␲ s1a

冉

0,

_冑

2 3

冊

, e2= 2␲ s2a

冉

1,−

_冑

1 3

冊

. 共49兲 Here a = a_⌬/␰= 2⫻3−1/4

冑

␲/h is the dimensionless Abrikosov lattice spacing.

The result reads formally

G共ra,r

⬘

,␻兲 =

兺

b

Mab

−1_共_␻_兲G

cl共rb,r

⬘

,␻兲. 共50兲

The Kronecker delta function appearing in Eq.共46兲 has the

following integral representation,

␦ab=

1

S_BZ

冕

q_苸BZ

eiq·共rb−ra兲_, 共51兲

where the area of the Brillouin zone 共BZ兲 of the pinning centers lattice is SBZ=共2␲兲2␳p= 2 31/2f

冉

2␲ a

冊

2 =2␲h f . 共52兲

Using this representation the inverse matrix reads

Mba−1共␻兲 =

1

SBZ

冕

q苸BZ

eiq·共rb−ra兲⌸

q,␻, 共53兲

where the polarization kernel is defined as ⌸q,␻=

1 1 + U0␳p˜gq,␻

. 共54兲

As a result Eq.共50兲 becomes explicit,

G共ra,r

⬘

,␻兲 = 1 S_BZ

兺

b

冕

q苸BZ eiq·共rb−ra兲⌸ q,␻Gcl共rb,r

⬘

,␻兲. 共55兲 Substituting this into the expression of the full GF with arbitrary positions, Eq.共42兲, one obtains

C. Shift of the mean-field transition line

Position of the mean-field transition line is determined by the lowest eigenvalue of the operator Hˆp, Eq.共15兲. This can

be obtained from poles of the resolvent of the operator Hˆp

which is simply related to the GF. In particular, the ground state is determined by the large-time asymptotic.

The resolvent of Hˆp is defined as

R共␻兲 =

冕

r

G共r,r,␻兲. 共57兲

Substituting Eq.共56兲 one obtains

R共␻兲 =

冕

r Gcl共r,r,␻兲 − U0 SBZ

兺

a,b

R共␻兲 =

兺

Nk 1 i␻+ Nh− U0 h 2␲⌸q=0,␻

兺

a

兺

冊

. 共62兲

For the magnetic field considered to obey h⬎␳p, it

origi-nates from the LLL contribution to⌸q=0,␻,

R共␻兲 = h 2␲ 1 i␻

冢

1 − 2␲U0␳ph 1 i␻ 1 + 2␲U0␳ph 1 i␻

冣

= h 2␲ 1 i␻+ u₀, 共63兲 where u₀= 2␲U₀␳ph⬅ uh 共64兲

is the lowest eigenvalue of Hˆp. One observes that it is

pro-portional both to the pinning strength and magnetic field. This should be contrasted with the shift of the ground-state energy in the absence of the magnetic field for the same potential which is finite: u0= u. Our derivation is valid only for magnetic fields h⬎␳p to satisfy the commensurability

condition and therefore the limit h→0 cannot be taken. Physically one notes that the system takes advantage of zeros of the wave function created by magnetic field to avoid the increase in ground-state energy due to repelling delta poten-tial barriers.

D. Final expression of the Green’s function of Hˆ

To determine the operator Hˆ of the bifurcation point ex-pansion, Eq. 共15兲, one subtracts the constant u0 of Eq.共64兲

from Hˆp: Hˆ =Hˆp− u0. In the␻space such a transformation is equivalent to a shift of frequency by the imaginary number

iu0 in the GF

G共r,r

⬘

,␻兲 → G共r,r

⬘

⬘

兲Gcl共rb,r

⬘

,␻+ iu0兲. 共71兲

The first part is the flux flow conductivity ␴I=␴FF in the

absence of pinning while the second term vanishes in this limit. We therefore first calculate␴FF.

A. Flux flow ac conductivity in the clean system

Substituting the clean 共u=0兲 retarded GF and using the Abrikosov wave function expressed in Appendix A in terms of the Landau harmonics ␸N共r兲, the expression of Eq. 共68兲

includes

y

⬘

␸0共r

⬘

兲 =

_冑

1

⫻

冓

冋

y − ␲ ah共2l + 1兲

册

共Dy␸0兲 ⴱ

(8)

⫻exp

冋

−i␲ a 共2l + 1兲共x − iy兲 − h 2y共ix + y兲

册

冔

r , 共73兲 where C共t兲 and␩共t兲 are given in Eq. 共38兲. Substituting Dy␸0 from Appendix A, the average over r is performed

␴FF共␻兲 = 2␲ ah ␤A

冕

t=0 ⬁ dte−i␻tC共t兲 2/␩共t兲 − h 关2/␩共t兲 + h兴2= ah ␤A 1 i␻+ h, 共74兲 which also can be directly computed using the Landau-level basis.

To make the resulting expression more transparent physi-cally it may be rewritten in terms of dimensional parameters 共recalling that this is just the superconducting component兲

␴FF共␻兲 = c2␥ 8␲␬2_␤ A H_c2共T兲 − B i␻t_GLH_c2+ B, 共75兲

where Hc2共T兲=Hc2共1−T/Tc兲. There exists a well-defined

limit ␻→0 which coincides with the dc conductivity. For

␻= 0 the result ␴FF= c2␥ 8␲␬2␤A Hc2共T兲 − B B =␴n关Hc2共T兲/B − 1兴 共76兲

is well known15_{and consistent with the Bardeen-Stephen law} derived in the London limit 共well-separated vortex cores兲. Indeed, adding the normal component of the conductivity one obtains

␴=␴n+␴FF=␴nHc2共T兲/B. 共77兲

The dependence of the flux-flow conductivity on frequency is very weak for␻Ⰶt_GL−1⬃1013 _{Hz, even for low-T}

c

materi-als. This is not the case in the presence of strong pinning when electromagnetic shaking even at low frequencies leads to depinning and hence strong increase in resistivity.

B. ac conductivity in the presence of pinning

In the presence of pinning the first contribution to the conductivity, Eq.共68兲 is renormalized into

␴I共␻兲 = ah ␤A 1 i␻+ h − u0 . 共78兲

Now we consider the second contribution to conductivity Eq. 共67兲. The Gaussian integration over r in the first integral of

Eq. 共70兲 gives ⌺a 1_共_␻_{兲 =}4␲3 1/8

冑

LxLy

冉

ah ␤A

冊

1/2 _h i␻+ h − u0

兺

m e−i␲m2/2 ⫻

冋

ya− ␲共2m + 1兲 ah

册

⫻exp

再

i␲m 2 − ␲2_{共2m + 1兲}2 2a2h +

冋

␲共2m + 1兲 a − hya 2

册

共ya− ixa兲

冎

= −

冉

hah 2␤A

冊

1/2_␸ 1 ⴱ_共r a兲

冑

LxLy 1 i␻+ h − u0 . 共79兲

Similarly integration over r

⬘

results in

⌺b2共␻兲 = − h⌺b1ⴱ共−␻兲, 共80兲 so that ␴II共␻兲 = ah ␤A 兩␸1共r = 0兲兩2 LxLySBZ U0 共i␻+ h − u0兲2

冕

q

兺

a,b eiq·共rb−ra兲⌸ q,␻+iu0. 共81兲 Performing the double lattice sum,

兺

a,b eiq·共rb−ra兲_{= N} p 2

兺

K ␦q,K, 共82兲

and recalling that q is restricted to the first BZ, the resulting reciprocal-lattice sum reduces to the single term with q = K = 0 so that ␴II共␻兲 = ah ␤A U0␳p兩␸1共0兲兩2 共i␻+ h − u0兲2⌸q=0,␻+iu. 共83兲

An explicit expression for the polarization kernel, Eq. 共54兲,

can be obtained by using the results derived in Appendix B, i.e.,

冕

冊

册

, 共86兲 where we have used ␸1共r=0兲=1.373共1+i兲h1/2 from Appen-dix A and Eq.共52兲.

A large positive denominator suppresses the second con-tribution to the conductivity␴II, Eq.共81兲, due to “screening”

of the pinning potential by excitations with high LL quantum numbers N. In the presence of thermal共or quantum兲 fluctua-tions 共e.g., close to the vortex lattice melting point兲, when

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the energy scale of the vortex lattice shear fluctuations19 be-comes comparable to the pinning energy scale u, this de-nominator can be reduced significantly resulting in large en-hancement of the pinning conductivity.

Combining the two contribution of the previous section, one expresses the ac conductivity at frequency␻as a func-tion of two dimensionless parameters u = 2␲U0␳p 共pinning

strength兲 and h, ␴s共␻兲 = ah ␤A 1 i␻+ h⌬ ⫻

冦

1 + 0.6u

冉

i␻ h +⌬

冊

冋

i␻ i␻− uh+ u⌰

冉

i ␻ h +⌬

冊

册

冧

, 共87兲 where ⌬ = 1 − u 共88兲

is the distance from the “pinning-depinning line.” It is im-portant to note that for⌬⬍0 the linear-response approxima-tion is invalid. Physically this means that the flux flow state does not exists and the I-V curve becomes nonlinear. The reason is that there exists共assuming no thermal fluctuations on the mesoscopic scale兲 a nonzero critical 共threshold兲 cur-rent. Vortices are effectively pinned and as a consequence electric field cannot penetrate the superconductor.

In Fig.2the dependence in the␻→0 limit on the pinning strength u for magnetic fields in the h = 0.85– 0.99 range is presented. One observes that when the pinning strength ap-proaches the critical value, u = 1, the conductivity diverges. In Fig. 3 the dependence of the real关dissipation, Figs.3共a兲 and3共c兲兴 and the imaginary 关inductive, Figs. 3共b兲 and3共d兲兴 parts of the ac conductivity as function of u at two values of magnetic fields close to Hc2共T兲关h=0.95 in Fig. 3共a兲and h

= 0.99 in Fig.3共b兲兴 is shown.

Returning to physical units, one obtains

␴共␻兲 = c 2_␥ 2␬2_␤ A Hc2共T兲 − B i␥␻⌽0+ 4␲B共1 − u兲

冦

1 + 0.6u

冉

i␥⌽0␻ 4␲B + 1 − u

冊

冋

i␥⌽0␻ i␥⌽0␻− 4␲uB+ u⌰

冉

i ␥⌽0␻ 4␲B + 1 − u

冊

册

冧

, 共89兲

where Hc2共T兲=Hc2共1−T/Tc兲 within linear approximation for

the coefficient a

⬘

of the Ginzburg-Landau energy, Eq. 共3兲.

Note that, as mentioned above, use of the ah expansion

re-stricts the range of frequencies since the correction to resis-tivity, Eq.共32兲, should not exceed␳n. In the next section we

analyze several simple cases which explain the main features of the conductivity shown in Figs.2and3and, in particular, the transition to the pinned state.

VI. CRITICAL BEHAVIOR NEAR THE DEPINNING LINE A. Criticality for small frequencies

When pinning is present the limit␻→0 in the expression for the ac conductivity is nonzero and exhibits criticality features at the continuous transition

h = u0. 共90兲

Approaching the line u0= h −⌬ for small ⌬ the first contribu-tion to the conductivity, Eq.共87兲, diverges

␴共␻→ 0兲 = ah ␤A 1 ⌬+ ah ␤A 1 ⌬2 0.6 ⌰共⌬兲⬇ ah ␤A 1.6 ⌬ . 共91兲 The depinning line, Eq.共90兲, determines the critical

pin-ning strength since according to Eq.共64兲

h = u0= 2␲U0 c ␳ph→ U0 c = 1 2␲␳p . 共92兲

which, in terms of dimensional parameters, reads 2␲2Npw2␧

LxLyTc

= 1. 共93兲

Therefore the pinning strength is only factor determining the transition into the pinned state. The critical value is indepen-dent of the magnetic induction. To conclude the conductivity diverges as a power ⌬−␯共z−1兲 with critical exponent ␯共z−1兲 = 1, hence superconductivity is recovered. This means that

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 pinning strength u Σs ΣGL Ω 0 HHc20.99 HHc20.95 HHc20.9 HHc20.85

FIG. 2.共Color兲 Conductivity at␻→0 as function of the pinning strength u for magnetic field in the h = 0.85– 0.99 range. When the pinning strength approaches the critical, u = u_c, the conductivity diverges.

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the vortex lattice is pinned and electric field cannot penetrate the superconductor despite persistent current flow in it at least when the current is not large. This is consistent with results in the London limit for both a 2D system with point-like defects and a three-dimensional system with columnar defects2_{共in the absence of thermal fluctuations on the} meso-scopic scale兲.

B. Dependence of the ac conductivity on frequency on the depinning line

It can be readily seen by analyzing Eqs.共78兲 and 共86兲 that

on the depinning line ⌬=0 the ac conductivity of Eq. 共87兲

simplifies to ␴共␻兲 =ah ␤A 1 i␻

冦

1 + 0.6 i␻ h

冋

i␻ i␻− h+⌰

冉

i ␻ h

冊

册

冧

, 共94兲

and for␻Ⰶh leads to the same dynamic critical exponent

␴共␻兲 = ah

␤A

1.6

i␻. 共95兲

C. Subcritical pinning strength

When uⰆ1 the expression 共87兲 can be simplified by

ex-pansion in this parameter共⌬=1兲,

␴共␻兲 = ah ␤A 1 i␻+ h ⫻

冦

1 + 0.6u

冉

i␻ h + 1

冊

冋

i␻ i␻− uh+ u⌰

冉

i ␻ h + 1

冊

册

冧

. 共96兲 Let us consider two cases. The simplest case is for very large frequencies or very small fields, ␻Ⰷh. In this case the con-ductivity simplifies significantly

␴共␻兲 = ah

␤A

1

i␻. 共97兲

It is largely inductive and pinning does not influence it. In the opposite case of very large fields or relatively small fre-quencies, hⰇ␻, considered next disorder is important.

In the opposite case of very large fields or relatively small frequencies,␻Ⰶh, Eq. 共96兲 in this case simplifies into

␴共␻兲 = ah ␤A 1 h i␻−关⌰共1兲 + 0.6兴u2h i␻−⌰共1兲u2h . 共98兲

There are two cases. For␻Ⰶhu2_{the system becomes purely} dissipative 0.88 0.90 0.92 0.94 0.96 0.98 1.00 0 2 4 6 8 10 pinning strength u Re Σ ΣGL Ω2102_t GL Ω102_t GL Ω4103_t GL Ω2104_t GL 0.88 0.90 0.92 0.94 0.96 0.98 1.00 0 1 2 3 4 5 6 7 pinning strength u Im Σ ΣGL Ω2102_t GL Ω102_t GL Ω4103_t GL Ω2104_t GL 0.980 0.985 0.990 0.995 1.000 0 2 4 6 8 10 pinning strength u Re Σ ΣGL Ω4103_t GL Ω2103_t GL Ω103_t GL Ω2104_t GL 0.980 0.985 0.990 0.995 1.000 0 2 4 6 8 10 pinning strength u Im Σ ΣGL Ω4103_t GL Ω2103_t GL Ω103_t GL Ω2104_t GL (a) (b) (c) (d)

FIG. 3. 共Color兲 ac conductivity for various frequencies as function of the pinning strength u for magnetic field close to Hc2: h = 0.95 in

(11)

␴共␻兲 =ah ␤A 1 h ⌰共1兲 + 0.6 ⌰共1兲 . 共99兲

In the intermediate case hu2_Ⰶ_␻_Ⰶh

␴共␻兲 = ah

␤A

1

h. 共100兲

VII. SUMMARY

To summarize we have extended the theory of electro-magnetic response of a type-II superconductor in electro-magnetic field with a periodic array of pinning centers beyond the London approximation 共employed commonly in the litera-ture兲 using the Ginzburg-Landau approach. The ac conduc-tivity as a function of magnetic field and frequency, see Eq. 共89兲 and Figs.2 and3, is calculated analytically and can be applied to an experimentally important regime of strong magnetic fields when the London approximation is inappli-cable. This is the main result of the present paper. The ana-lytic solution was obtained by the Lippmann-Schwinger method not far from the Hc2共T兲 line for a 2D Abrikosov

lattice of vortices is commensurate with the short-range pinning array.

It is predicted that in strong magnetic fields the short-range pinning can effectively influence the vortex dynamics due to long-range correlations of the superconducting order parameter. In particular, a magnetic field independent critical pinning strength U₀c=␲w 2_␧ Tc = l 2 2␲, 共101兲

was found at which the conductivity at low frequencies di-verges as a power2_␴_⬀共U0_{− U}

0

c_兲−␯共z−1兲_{with the critical} expo-nent␯共z−1兲=1. Here l is the distance between pinning sites of共energy兲 depth ␧ and width w. For columnar defects due to irradiation ␧⬃2Tc, w⬃␰共T兲, making this condition l

⬍2␲␰共T兲. At criticality the conductivity diverges Using an accepted 2D value of ␯= 1,2 _{this implies z = 2. The ac} con-ductivity on the depinning line diverges as i␻−1_{. Below U}

0

c

and sufficiently low frequencies there exists a sliding Abri-kosov lattice, which moves nearly uniformly. This sliding lattice behavior, which does not exists in the single vortex-pinning regime, becomes possible due to strong interactions between vortices when they outnumber the columnar defects. Physically it is caused by “liberation” of the temporarily trapped vortices by their freely moving neighbors.

Finally, let us qualitatively contrast the results of two phe-nomenological approaches to the problem, the TDGL method adopted in the present paper and the London ap-proximation approach used in previous numerical simulation.11,20 _{The former is valid for fields much larger} than Hc1 while the later is valid far below Hc2. In strongly

type-II superconductors the applicability ranges might over-lap. The difference is more pronounced in the case of short-range pinning considered in the present paper. Indeed this type of pinning is very ineffective within the London ap-proach since it assumes that a vortex is a “pointlike” 2D

particle and consequently its velocity tracks the pinning po-tential landscape only at pinning centers. In contrast, within the GL approach the mixed state is described by a wave function. It senses pinning potential even when the size of the pin is smaller than ␰共T兲. Formally it is very similar to influence of the deltalike potential on wave function in quan-tum mechanics, as we have used while calculating Green’s functions in Sec.III.

ACKNOWLEDGMENTS

We appreciate the useful discussion with E. Zeldov, L. Feigel, D. P. Li, Yu. Galperin, I. Aranson, Y. Yeshurun, and A. Shaulov. B.S. and I.S. acknowledge support from the Is-rael Scientific Foundation 共Grant No. 499/07兲. The work of B.R. was supported by NSC of R.O.C. Grant No. 972112M009048 and the MOE ATU program. B.R. acknowl-edges the hospitality and support at the Physics Department of Bar Ilan University during sabbatical leave. T.M. is in-debted to the Department of Electrophysics, National Chiao Tung University, Hsinchu Taiwan, for supporting this re-search during a visit to Hsinchu Taiwan and acknowledges the support of the Israel Science Foundation Grant No. 425/ 07, and of the Posnansky Research fund in superconductiv-ity.

APPENDIX A: ABRIKOSOV FUNCTIONS COMMENSURATE WITH A

HEXAGONAL ARRAY OF PINNING CENTERS

The first two Landau harmonics in dimensionless units are

␸0= 31/8

冑

h

兺

l ei␲l2/2exp

再

i

冋

−h 2xy + ␲共2l + 1兲 a

冉

x − a 4

−h 2xy +␲共2l + 1兲 a

冉

x − a 4

冊

册

− h 2

冋

y − ␲共2l + 1兲 ah

册

2

冎

, 共A2兲

where the dimensionless Abrikosov lattice spacing a = a_⌬/␰. In particular, at origin ␸0共0兲=0 while

␸1共0兲 = − 33/8

冑

_␲_h_/2

兺

l 共2l + 1兲 ⫻exp

再

␲ 4

冋

i共2l 2_{− 2l − 1兲 −}

冑

3 2 共2l + 1兲 2

册

冎

_. We also need the covariant derivative of the LLL,

Dy␸0=

冉

⳵ ⳵y+ i h 2x

冊

␸0= −

冑

h 2␸1. 共A3兲

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APPENDIX B: FOURIER TRANSFORM OF THE GAUGE INVARIANT PART OF THE GREEN’S FUNCTION

OF THE CLEAN SYSTEM

In the t space the inverse Fourier transform with respect to position is gQ cl_{共t兲 =}

冕

r gcl共r,t兲exp共− iQ · r兲 =

冕

r C共t兲exp

冋

− r 2 2␩共t兲

册

exp共− iQ · r兲 = 2␲␩共t兲C共t兲exp

冋

−␩共t兲Q 2 2

册

, 共B1兲

where Q = q + K, with q belonging to the first Brillouin zone and K runs over the reciprocal lattice of the pinning centers. This is transformed into the ␻ space g_Q,cl_␻=兰_te−i␻tg_Qcl共t兲. The

quantity appearing in the expression for polarization kernel, Eq. 共54兲, has a form

g ˜_q,_␻= 2␲

兺

K

冕

t e−i␻t␩共t兲C共t兲exp

冋

− ␩共t兲共q + K兲 2 2

册

. 共B2兲 It is useful to divide the integrand into the well-known LLL part,21 共which diverges when␻= 0兲 and the rest as 共higher Landau levels, HLL兲 g ˜_q,_␻= SBZ„gHLL共Q兲 + gLLL共Q兲…, 共B3兲 gHLL_{= 2}

冕

t=0 ⬁ e−i␻t

兺

K

冦

exp

冋

−Q 2 h tanh

冉

ht 2

冊

册

1 + exp共− ht兲 − exp

冉

− Q2 h

冊

冧

, 共B4兲 gLLL= 2

兺

K exp

冉

−Q 2 h

冊

冉

−K 2 h

冊

= 2␲h SBZ 1 i␻. 共B6兲 Similarly, the HLL part becomes

gHLL=

冕

t=0 ⬁ e−i␻t f 2␲h

冕

K

再

eht/2 cosh共ht/2兲 ⫻exp

冋

−K 2 h tanh

冉

ht 2

冊

册

− 2 exp

冉

− K2 h

冊

冎

, 共B7兲 which reduces to gHLL= f 2␲h

冕

t=0 ⬁ e−i␻t

冕

0 K_max2 dK2 ⫻

冦

exp

冋

−K 2 h tanh

冉

ht 2

冊

册

e−ht+ 1 − exp

冉

− K2 h

冊

冧

共B8兲 =− f 2␲

冕

_t=0 ⬁

at the K_max2 →⬁ limit directly

g₂HLL= f

2␲h

冋

−␥E−␺

冉

i

␻

h + 1

冊

册

. 共B12兲

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