Sliding Abrikosov lattice in a superconductor with a regular array of artificial pinning centers: AC conductivity and criticality at small frequencies

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Sliding Abrikosov lattice in a superconductor with a regular array of artiﬁcial

pinning centers: AC conductivity and criticality at small frequencies

T. Maniv

a

_{, B. Rosenstein}

b

_{, I. Shapiro}

c

_{, B.Ya. Shapiro}

_c,*

_{, R.F. Hung}

b a

Schulich Faculty of Chemistry, Technion-IIT, 32000 Haifa, Israel

b

Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, ROC

c

Department of Physics, Institute of Superconductivity, Bar-Ilan University, 52900 Ramat-Gan, Israel

a r t i c l e

i n f o

Article history:

Available online 4 March 2010 Keywords:

Sliding vortex lattice Periodic pinning array

Time-dependent Ginzburg–Landau theory

a b s t r a c t

Dynamics of the flux lattice in the mixed state of strongly type-II superconductor near the upper critical field subjected to AC field and interacting with a periodic array of short range pinning centers is consid-ered. The superconductor in a magnetic field in the absence of thermal fluctuations on is described by the time-dependent Ginzburg–Landau equations. For a special case of the d-function shaped pinning centers and for pinning array commensurate with the Abrikosov lattice (so that vortices outnumber pinning cen-ters) an analytic expression or the AC conductivity is obtained. It is found that below a certain critical pin-ning strength and for sufficiently low frequencies there exists a sliding Abrikosov lattice, which vibrates nearly uniformly despite interactions with the pinning centers. At very small frequencies the conductivity diverges at the critical pinning strength.

The great interest in the problem of magnetic flux pinning in type-II superconductors stems from its relevance to technological applications as well as with its implications to the general problem of complex nonlinear dynamics with tunable parameters. An important challenge in applications of type-II superconductors is in achieving optimal critical currents under given magnetic fields. This requires preventing depinning of Abrikosov vortices during formation of the resistive state under the applied current. Recently there have been advances in the study of vortex pinning by fabri-cating periodic arrays of pinning sites where each pinning site may be either magnetic or normal inclusion effectively trapping vortices. Pinning arrays with triangular, square, and rectangular geometries have been fabricated using either microholes or blind holes arrays of magnetic dots and periodic array of columnar de-fects[1]. The resulting critical current is enhanced when vortex lat-tice is commensurate with the periodic array of pinning sites. In addition this system is a convenient experimental tool to study the general problem of interacting periodic system moving in peri-odical potential like dislocations in crystals or charge density waves. Theory of dynamics of the pinned vortex matter by a ran-dom distribution of pins is very complicated. However in the ab-sence of significant thermal fluctuations, the problem simplifies considerably. It was studied theoretically, mostly in 2D systems, using either numerical methods within a model of interacting points-like particles representing vortices subject to pinning

potential and driving force[2]or within the framework of elasticity theory, in which the vortex matter is treated as an elastic manifold subject to both the pinning stress and a driving force[3].

Theory of the Abrikosov lattice subjected to an AC field and periodic pinning is simpler, but so far has been treated either numerically using molecular dynamics approach or by means of the elastic manifold approach in London approximation. On the other hand this approach completely ignores the contribution of the vortex cores essentially important when the distances be-tween vortices and artificial pinning sites are not much larger than the size of the coherence length. In fact there is still no an analytical theory describing AC properties of a type-II supercon-ductor with periodic pinning array subjected to a strong magnetic field. Here we present a theory of AC conductivity in the time-dependent Ginzburg–Landau (TDGL) approximation describing superconductor in a strong magnetic field. In the absence of ther-mal fluctuations, an exact solution for the linear response in the case of a d-function model for the periodical array of the pinning centers in which it is commensurate with the Abrikosov lattice (vortices outnumber pinning centers) is obtained for the first time.

Let us consider a type-II superconductor under a constant exter-nal magnetic ﬁeld H parallel to a system of pinning centers direc-ted along z axis and carrying electric current along the y axis, see Fig. 1.

The simplest relaxation dynamics of a superconductor in the presence of electric ﬁeld is described by TDGL[4] and Maxwell equations (in the dimensionless variables)

*Corresponding author.

E-mail address:[email protected](B.Ya. Shapiro).

Physica C 470 (2010) 744–746

Contents lists available atScienceDirect

Physica C

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fGL¼ wH _ w ahwwþ 1 2ðw wÞ2; @tw¼ H _ w ahwþ ww2 i

U

w; ð1Þ H _ p¼ D2 2 h 2þ U0 X a dðr raÞ; ah¼ 1 th h 2 u0; th¼ T Tc ; ð2Þ H _ ¼ H _ p u0; U0¼

pw

2

_e

Tc ; j ¼

rU

þi 2½w _{Dw c:c:;}

r

j ¼ 0 ð3Þ

Here D are the covariant derivatives,

w

is the order parameter and

Uthe electric potential, h the magnetic ﬁeld in units of Hc2, fGLis GL

energy density, w the radius of the single pinning potential and u0= 2

p

U0nph is dimensionless pinning strength. In our units

r

n=

q

n= 1.

This set of the equations can be simpliﬁed near the coexistence line where ah 1 and for small electric ﬁelds. Expanding the order

parameter in ﬁrst order in the form

w

(r, t) = u(r) + h(r, t) one obtains @th¼ H _ hþ i

U

h; E ¼

rU

; E ¼ j i½h_{D/ hðD/Þ} ð4Þ here /ðrÞ ¼ ah bA 1=2

u

0ðrÞ þ Oða 3=2

h Þ where

u

0ðrÞ is the Lowest Landau

level order parameter. Deﬁning the retarded Green function by

ð@tþ H _

ÞGðr; r0_;_{t t}0_{Þ ¼ dðr r}0_;_{t t}0_Þ _ð5Þ

one obtains the following relation between the current density and the electric ﬁeld

Eðr; tÞ ¼ jðtÞ Z t t0_¼1 jðt0_Þ Z r0 Gðr; r0_;_{t t}0_Þy0

_uðr

0_ÞðD y/Þþ c:c: ð6Þ where E = j + O(ah).

For a uniform AC density j(t) = j0cos

x

t one obtains after

aver-age over volume of the sample for complex conductivity

qðxÞ ¼ lim

T!1 1 Tj0 Z T 0

dteixthEðr; tÞir ð7Þ

Performing integrations one obtains

qðxÞ ¼

1

1þrsðxÞ 1

r

sðxÞ

r

sðxÞ ¼ R_r0y0h/ðr0ÞðDy/ÞGðr; r0;

xÞ þ /

ðr0ÞðDy/ÞGðr; r0;xÞir ð8Þ

Eq.(1)allows to relate the dynamic conductivity in the supercon-ductor with the Green function (GF) of the quantum mechanical Hamiltonian bHpof a charged particle in magnetic ﬁeld in the

pres-ence of periodic potential. Representing Green function in the inte-gral form, one obtains the Dyson equation

Gðr; r0_;

_{xÞ ¼ G}

clðr; r0;

xÞ U

0 X

a

Gclðr; ra;

xÞG

clðra;r0;

xÞ

ð9Þ

where Gclðr; r0;

x

Þ is the Green function of the clean superconductor

Gclðr; r0;tÞ ¼ e ih 2ðxy0yx0Þg clðr; r0;tÞ gclðr; r0;tÞ ¼ CðtÞe r2 2gðtÞ CðtÞ ¼ h 4pe ht 2sinh1 ht 2

gðtÞ ¼

2 htanh ht 2 ð10Þ

In particular at pinning points r = rb, assuming commensurability

with the vortex lattice, one obtains

Gðra;r0;

xÞ ¼

X

a M1

abðxÞGclðrb;r0;

xÞ

ð11Þ

where a symmetric matrix Mba(

x

) is deﬁned by

M1 abðxÞ ¼S1BZ R qBZeiqðrb raÞ

_P

q;x

P

q;x¼_1þU₀np1Gclðq;xÞ ð12Þ

Substituting it into the expression for full GF with arbitrary posi-tions one obtains

Gðr; r0_;

_{xÞ ¼ G}

clðr; r0;

xÞ

_SU0 BZ P a;b R qeiqðr arbÞ

_P

q;xKðr; ra;rb; ;r0;

xÞ

Kðr; ra;rb;r0;

xÞ ¼ G

clðr; ra;

xÞG

clðrb;r0;

xÞ

ð13Þ

To determine the operator GF for operator bH one has to subtract the constant u0from bHp. In the

x

x

þ iu0Þ

ðXÞ

HereW(X) is digamma function and Kmaxis maximal vector of the

reciprocal pinning centers’ lattice.

Let us consider some limiting cases important for experiment.

Fig. 1. The hexagonal Abrikosov vortex lattice (distribution of the superconducting density |w(r)|2

) and pinning centers. Zeroes of order parameter fall on the locations of the columnar defects (red squares), so that vortices outnumber the pins. Vectors d1and d2are lattice vectors of pinning array. Distance between nearest neighbors of

the Abrikosov lattice is aD. (For interpretation of the references to colour in this

ﬁgure legend, the reader is referred to the web version of this article.)

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(i) No pinning U0¼ 0

r

FF¼ ah bA 1 ixþ h ð16Þ

If

x

= 0 when Eq.(18)gives a well known Bardeen–Stephen re-sult for ﬂux ﬂow conductivity.

(ii) Criticality near the critical pinning strength h = u0for small

frequency. This means that 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. In this case the real part of the conductivity diverges. Near this line the conductivity reads (seeFigs. 2 and 3):

r

sðx! 0Þ ah 2pnpbAh 1 Uc0 U0 ð17Þ where Uc

0¼ ð2

p

npÞ1. Therefore the pinning strength is only factor

determining the transition into the pinned state. The critical value is independent of the magnetic induction.

(iii) AC conductivity at the critical line (u = uc)

In this case the conductivity at small frequencies

x

h has only the imaginary part

r

s ah bA

1

ix ð18Þ

(iv) AC for subcritical pinning strength In this case u 1 and AC conductivity reads

r

s

r

FF¼ 1 þ 0:6uh ðixþ hÞ ix ixhþ u

H

1 þ ix h

where the second term in this expression describes pinning correc-tion to usual Bardeen–Stephen conductivity.

In summary, we developed the theory of AC conductivity for a superconductor with periodic pinning array in the Ginzburg–Lan-dau approximation and predicted that above some pinning strength, the AC conductivity in the limit of small frequency shows typical for ideal superconductor behavior.

Acknowledgements

We acknowledge support from the Israel Science Foundation Grant No. 425/07. Work of B.R. was supported by NSC of ROC Grant #972112M009048 and MOE ATU Program.

References

[1] Q.H. Chen et al., Phys. Rev. B 73 (2006) 014506.

[2] A.B. Kolton, D. Dominguez, N. Gronbech-Jensen, Phys. Rev. Lett. 86 (2001) 4112. [3] Y.Q.H. Chen, C. Carballeira, T. Nishio, B.Y. Zhu, V.V. Moshchalkov, Phys. Rev. B 78

(2008) 172507.

[4] B. Rosenstein, D.P. Li, Rev. Mod. Phys. 82 (2010) 109. Fig. 2. Real part of conductivity atx?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 value the conductivity diverges.

Fig. 3. Real and imaginary parts of conductivity atx?0 as function of the pinning strength u for magnetic ﬁeld h = 0.95. When the pinning strength approaches the critical u the conductivity diverges at small frequencies.