complex impedance of a two-dimensional josephson junction array

ELSEVIER Physica C 277 (1997) 161-169

PIU

Complex impedance of a two-dimensional Josephson junction array A n n a Jonsson *, Pe t t e r M i n n h a g e n

Department of Theoretical Physics, Umed University, 901 87 Umed, Sweden

Received 20 January 1997; revised 11 February 1997

Abstract

Simulations on the two-dimensional XY model are performed. The results are compared to experiments on a Josephson junction array for the case of a triangular lattice well below the Kosterfitz-Thouless temperature in a weak perpendicular magnetic field. Comparisons between experiments, existing theories and the simulations suggest that the characteristic features of the complex impedance seen in the experiments are due to vortex correlations induced by the presence of vortices with opposite vorticity.

© 1997 Elsevier Science B.V.

PACS: 74.40+k; 74.60Ge; 75,10Hk; 75.40Mg Keywords: Josephson junction array; XY model; Vortex; Dynamical response; Simulations

1. Introduct ion

A two-dimensional (21)) Josephson junction array (JJA) in the absence of a perpendicular magnetic field undergoes a Kosterlitz-Thouless (KT) transition to a superconducting state at a critical temperature TS:T [ 1,2]. This transition is driven by the unbind- ing of thermally created vortex-antivortex pairs [ 1,2]. These thermally created vortex pairs dominate both the static and dynamic properties close to the transition [2]. Adding a small perpendicular magnetic field perpendicular to the JJA destroys the superconducting state at a much lower temperature [ 2].

The question we are addressing here is the dynamical response of a JJA in a small magnetic field and a temperature which is well below TI~T (defined in the

* Corresponding author. Fax: +46 90 169556; e-mail: [email protected].

absence of the magnetic field) but which is still well above the superconducting transition in the presence of the magnetic field. This question is inspired by the experimental findings of Thtron et eL1. [ 3 ]. They found a very interesting anomalous dynamical response. Two main types of explanations have been proposed for these experimental findings [3-8] . One is based on the observation that the response has characteristic features very similar to the response just above the KT transition for the case when no magnetic field is present [4,8,9]. This suggests that the response is due to bound vortex-antivortex pairs to the extent that the hand-waving arguments for the bound vortex pair response behind the phenomenological description proposed by Minnhagen [2] (MP description) are basi- cally sound. This hand-waving link between the vortex pair response and the MP description has recently been put on a firmer footing by Capezzali et al. [7].

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162 A. Jonsson, P. Minnhagen / Physica C 277 (1997) 161-169

The trouble with this type of explanation is that thermally excited vortex-antivortex pairs are expected to be very scarce at the temperature corresponding to the experimental findings in Ref. [ 3 ]. The other explanation invokes the coupling between the vortices and the spin-waves [5,6].

In order to get some further insight into this issue we have performed simulations on a 2D XY model on a triangular lattice where we have chosen the parameters to correspond as closely as possible to the experimental conditions in Ref. [ 3 ]. Our main inference from these simulations is that it is the correlations between vortices of different vorticity, which give the characteristics of the dynamical response. We also find that the estimate of the vortex-spin-wave coupling by Korshunov [6] is at variance with the simulations and we suggest a possible explanation for this.

The content of this article is as follows: In Sec- tion 2 we review the experimental findings by Thtron et al. [ 3 ]. Next we in Section 3 review the results of Korshunov's vortex-spin-wave coupling theory [6]. The predictions from this theory can be very directly compared to our simulations. Section 4 describes our simulations and Section 5 contains the conclusions.

2. Experimental results

The experimental results of Thtron et al. [ 3 ] are for a 2D triangular JJA of SNS type with superconducting islands of Pb on top of the normal metal Cu. The array undergoes a KT transition at TKT ~ 3.70 K and the measurements are done at about 3.27 K. The complex sheet inductance Z(w) was measured and the result was expressed in terms of the Coulomb gas dielectric function ~((0) which is related to the inductance by Z((0) -- ioJLke((0), where Lk is the "bare" sheet kinetic inductance [2]. The measurements were made at a fixed o) as a function of the perpendicular magnetic field. The magnetic field was expressed in terms of the frustration f , i.e. the number of flux quanta per elementary triangle of the array. The range of f was f < 0.05. The results were expressed in terms of l / E ( ( 0 , f ) as a function of (0/f. The results for the frequencies 10 and 1.2 kHz are shown in Fig. 1.

Two important things about the data shown in Fig. 1 are that the data for the two frequencies as a function of f to good approximation fall on a single curve and

0.6

0.4

0.2

0.0

/ ~ 0 w i 1 i i i

1 2 ,o/2,~y [10% -1]

Fig. 1. The dielectric function 1 / ~ ( w , f ) measured by Th~ron et al. [3]. The real part is plotted against o~/2zrf. The open circles correspond to the frequency 10 kHz. The open diamonds correspond to a mixture of 10 and 1.2 kHz data, which means that there is a good overlap of the data for these two frequencies in this region [ 3 ]. The data fall to excellent agreement on a straight line through the origin for small o~/2¢rf (full line), The inset shows R e [1 /~ ] (open circles) and I Im[1 /*] l (solid circles) over a larger region. The broken curves in the inset are guides to the eye.

that the real part is proportional to ~ = (0/f, for small ~. This latter fact is very clearly seen in Fig. 1. In the present context it is of particular interest that the real part is linear in ~. It implies that the effective vortex mobility vanishes as 1/I In (01, for small (0 [3]. A constant mobility in the limit of vanishing (0 would instead imply a real part proportional to (02, for small (0 [31.

One suggestion for the linear o~ dependence comes from a phenomenology by Minnhagen (MP) [2]. A crude hand-waving argument for the response of bound vortex pairs leads to the response function [2]

Re = (0+w0 ' (1)

Im = , (2) 7"/" (02 _ (02

where (00 is a characteristic frequency. This response function contains the feature Re [ l / e ] cx (0, for small (0, which in turn suggests that this feature can be associated with the response from the vortex-antivortex pairs [2]. This link between the MP form of the response and the vortex-antivortex pairs has recently been put on a firmer footing by Capezzali et al. [7]. As mentioned above, a problem with this vortex- antivortex pair explanation is that it presumes the

A. Jonsson, P. Minnhagen / Physica C 277 (1997) 161-169 163

existence of both type of vortices. The data shown in Fig. 1 are taken well below the KT transition and consequently the vortex-antivortex pairs are expected to be very few. The vast majority of vortices present is induced by the external magnetic field. Another possible explanation which does not presume the presence of vortex-antivortex pairs has been suggested by Korshunov [6]. This explanation is based on the coupling between a vortex and the spin waves and the results of this theory is reviewed in Section 3.

3. Korshunov's theory

A 2D JJA is believed to be very well represented by a 2D XY model and the attempts to understand the experiments by Thtron et al. [3] are based on this representation.

The XY model is defined on a 2D lattice, where each lattice point i is associated with a phase angle Oi. In the present case it is a triangular lattice. The lattice points interact with a nearest-neighbor coupling given by the Hamiltonian

n = ~ t l(6~r = o; - o r ) , (3 ) (i j)

where

U(&) = 2J[ 1 - cos2P2(~b/2) 1. (4)

Here (i j ) denotes sum over nearest neighbor pairs and -~r < ~b < ¢r. For p = 1 the nearest-neighbor interaction U(~b) reduces to U(~b) = J( 1 -cos(~b) ) and this form of the interaction is usually the one taken to represent a JJA. J is the Josephson coupling energy which for a JJA depends on the temperature T. The measurements by Thtron et al. [3] in Fig. 1 are for T = 3.27 K and k B T / J ( T ) = 0.5 whereas kBTKT/J(TKT) = 1.5 and the large difference between these two ratios suggests that the vortex fluctuations are very rare at T = 3.27 K. The point with the generalization to p > 1 in Eq. (4) is that on the one hand it does not change the theoretical arguments, but on the other hand increases the number of vortex-antivortex pairs [ 10,8]. Chang- ing the p-value thus provides an additional possibility of testing predictions against simulations [ 8 ].

One way of introducing dynamics into the model is through the Langevin equation

dOi(t) FOH = - ~ / / + r/i(t) , (5)

where/" is a constant which determines the relaxation and rli(t) is a fluctuating noise associated with each lattice point such that

(rli( t )r l j ( t ') ) = 2FTt irS( t - t ') , (6)

where T is the temperature (in energy units, Boltz- mann's constant kB = 1). This dynamics is of time- dependent Ginzburg-Landau (TDGL) type and it is this type of dynamics to which the theory of Kor- shunov applies [6]. Another way of introducing the dynamics is through the resistor shunted Josephson junction model (RSJ). These two models of the dynamics give, according to Capezzali et al. [7 ], a qualitatively similar result with respect to vortex-antivortex pairs. However, the vortex-spin-wave coupling is quite different and only TDGL gives the logarithmic vanishing of the vortex mobility [6]. Microscopically the TDGL dynamics can be motivated for an SNS array, where current leak from the superconducting grain to the metallic substrate, whereas an SIS junction would have local current conservation and as a consequence would be better described by the RSJ model [ 6 ]. How- ever, from a more phenomenological point of view the situation is less clear because it is found that the TDGL dynamics seem to describe experiments on 2D superconducting films very well [ 8].

The analysis by Korshunov [6] of the 2D XY model with TDGL dynamics starts from the Euclidean action

1 _ 2 dtf dt'(Or(' t'Or(")) y ~ f dt J (1 - cos(Oj( t ) - O j , ( t ) ) ) . +

07') (7)

A crucial step in the treatment is to replace the cosine potential with a harmonic one, i.e.

J( 1 - c o s ( 0 r ( t ) - 0j, ( t ) )

.~ __Jeff (Oj(t) - Oj,( t) - mjr,) 2 (8) 2

where the integer variable mjj, = O, 1 (mjj, = - m r , j ) keeps track of the position of a vortex and Jeff is an effective coupling constant which should be chosen


to make the harmonic approximation as good as possible. Korshunov's treatment for a single vortex presumes that one can approximate Jeff with J without any qualitative changes of the result [6]. The result for the vortex mobility/.,(to) of a single vortex obtained by Korshunov in case of 2D XY model on a triangular lattice with TDGL dynamics is [6]

/.t(to) = - i toG(to), (9)

with

i,rto ( J e f f F 8 ~ ~r21tol G-'(to) =-TCcS--'Fln \ Itolv5 / + 2V'~-----F' (10)

for [to[ << JefrF, and

2~r2 ( - i to + JeeeF), G - I ( t o ) = - ~ -- (11)

for I tol >> J=fer. From Eqs. (9) and (10) one concludes that the mo-

bility of a single vortex,/z(to), vanishes as - 1 / I n Itol for small Itol due to the coupling to the spin-waves degrees of freedom (the spin-waves are described by the 0-variables in the harmonic approximation given by Eq. (8)) .

The experiment by Th6ron [3] is for a perpendicular magnetic field corresponding to a frustration f = number of vortices per elementary triangle. This gives rise to a density of free vortices nF which in the sim- plest estimate is just f divided by the area of an elementary triangle. This density of free vortices con- tributes to the dielectric function E(to) and in terms of the single vortex mobility/z(to) the contribution is given by [ 11 ]

e(to) = 1 + c n F ~ (to) , (12) - - l to

where C is a constant (the explicit value for the 2D XY model is C = 2~rT/T c°, see Section 4 for the defini- tion of T c° ). From Eqs. (9), ( 10 ) and (12) one hence concludes that Re [1/,(to)] cx to. Consequently Ko- rshunov's vortex-spin-wave coupling offers an explanation for the linear to dependence in Fig. 1 which does not require any vortices in addition to the ones associated with the applied magnetic field.

4. Simulations

Our simulations are for the 2D XY model defined by Eqs. (3) and (4) together with the dynamics given by Eqs. (5) and (6). The procedure is to integrate Eq. (5) by discretizing time into small enough steps subject to a random noise defined by Eq. (6). We use periodic boundary conditions and a two-valued random noise r/i = + ~ . The number of lattice points L 2 is L 2 = 64, 96, and 128. The lattice constant is taken to be unity so that the total area [2 is/2 = v/3L2/2. Details on this type of calculation can be found in Ref. [ 8].

The output from the simulations is given by the time correlation function H( t ) defined by

H(t ) - I ( F ( t ) F ( O ) ) , (13)

where

O F( t ) ---- Z -~ i j U(dpi j ( t ) )eij " e x ' (14)

(i j)

and the sum is over all nearest-neighbor pair links (i j) . The unit vector from i to j is eij and ex is a fixed unit vector pointing in the direction of one of three basic directions of the links of the lattice.

The frequency dependent dielectric function is given by [ 8 ]

2~'toT°3 / -~ T2 dt s in to tH( t ) , (15)

0

and o o

= ia dt c o s t o t H ( t ) , (16)

0

where T c° is the Coulomb gas temperature given by [ 12]

/ , 0 3 = ] T

In the present case we have a finite frustration f . This frustration is introduced by changing the argument $/j of U(dpij) in Eq. (3) into ~b/j ---* dpij -


Aue x • e U, with the vector potential given by A 0 = 2~'(m x + m J ) f , where m x consecutively numbers the rows of lattice points parallel to ex [ 8].

We will discuss the results of our simulations in terms of the Coulomb gas charge analogy of the 2D XY model [12] 1. The Coulomb gas charge, An( l ) , associated with an elementary triangle I is in this analogy given by [ 12]

v~TC~ ( E U ' I - f (18) An(1) ~ ~ \(U)EI /

where the sum is over the links which make up the triangle I (and U' denotes the derivative of U with respect to its argument). The average Coulomb gas charge density is then defined as

n = l ( ~ l A n ( 1 ) l ) . (19,

The Coulomb gas temperature T c6 defined by Eq. (17) is the effective temperature variable for the Coulomb gas charges [2,12]. In the vortex-Coulomb gas analogy the vortices corresponds to the Coulomb gas particles [2] . Consequently n is also a measure of the vortex density 2

Fig. 2 shows n as a function of TcG for f = 0 and f = 1/64. The two vertical lines denote the KT temperature cT~ and cT~/3, respectively. The experiment by Th6ron et al. [ 3] was for f < 0.05 so that f = 1/64 ~ 0.016 is representative for the experiment. The experimental temperature T/J was about 1/3 of TKT/J. For the XY model this means that the corresponding T c6 is about cT~/3. Consequently the Coulomb gas temperature c 7 ~ / 3 is representative for the experiment. As seen from Fig. 2 the additional vortex density naaa = n - 4 f x / ~ 3 at cT~/3 is very small, i.e. the vast majority o f vortices present for f - 1/64 corresponds to the frustration.

] The connection given in Ref. [ 12] gives the optimal Coulomb gas analogy of the 2D XY model.

2 n differs slightly from the conventional vortex density based on the angular difference around an elementary triangle. However, n turns out to be a more fundamental quantity for the 2D X¥ model (see Ref. [ 8 ] ) . 3 Since f is per elementary triangle the corresponding density is

4f/v~.

0.15

0.1

0.05

0 0

0.1 _, ' /

, 0 ~ . , / / , 0 1/3 0.6 / / i Tc°/T:° , / /

0.1 0.2 0.3 0.4 TcG

Fig. 2. The density nA(~ charge per elementary triangle) as a function of T cO for f = 0 (lower full curve) and f = 1/64 (upper full curve). The two broken vertical lines denote c2c2~ and ~c c/3, respectively. The inset shows the additional vortex density nA - f

for f = 0 (open circle), f = 1/64 (open diamond), f = 1/8 (open square), and f = 1/64 with p = 2.5 (cross). As seen from the inset the additional vortex density nA is small at 7"~c °/3 and f = 1/64. It increases with increasing f and increasing p.

Fig. 3 shows the frequency dependent dielectric function 1/e(~o) as a function of o~ obtained for parameters nominally representative for the experiment by Th&on et al [3] , i.e. f = 1/64 and T c ° = c7"~/3. Contrary to the experimental results given in Fig. 1, the simulations show no anomaly: the response function is perfectly Drude like and quite consistent with

0 i 0 0.5 1

Fig. 3. The dielectric function 1 /¢ (o~) obtained from the simulations for f = 1/64 and T cG = T~c o/3. The open circles correspond to Re [ 1/¢] and the solid circles to Ilm [ 1/~]1. The broken curves are a fit to the Drude form given by Eqs. (20) and (21). The inset shows that the leading behavior of Re [ 1/~] for small co is proportion~ tO off.

166 A. Jonsson, P. Minnhagen/Physica C 277 (1997) 161-169

a normal finite mobility/z(o~ = 0) = po. Inserting a constant po into Eq. (12) gives

= ~2 + ~o 2 (2o)

and

Im eo 2 +o-~ (21)

with

OrO = CnFlzO = 2q'rTnFp.,o /-co (22)

Eqs. (20) and (21) are of the conventional Drude form. As seen from Fig. 3 the conventional Drude response form gives a perfect fit to the simulation data, The Drude form predicts that Re [ 1/E(o~) ] should be proportional to w2 for small ~o (compare Eq. (20)) . This w2-dependence is also seen in the simulation data (inset in Fig. 3). This is in contrast to the linear w dependence inferred from the experimental data in Fig. 1.

The quantity o0 in Eq. (22) involves the density of free vortices nF. If we assume that the only vortices present are to good approximation the ones associated with the frustration and furthermore assume that these induced vortices are free then we get the approximate relation

4 nF = f---~_ • (23)

V3

0.4

0.3

0.2

0.1

0 0

T S / °-

0.1 0.2 0.3 0.4 0.5 f T / T cG

Fig. 4. or0 defined by the Drude form given by F_.qs. (20) and (21) as a function of frustration f . o'o is plotted against fT/T cO (open circles). The data fall on a straight line through the origin (full drawn line). The mobility/zo is determined by the slope of the line, The inset show/1,0 determined for each individual data point.

Consequently Eqs. (22) and (23) predicts that o'0 cx f where the proportionality constant is

87rT/z0

v37~" In order to test this prediction we have simulated a se- quence o f f keeping T c° fixed at T~ G/3. In each case a Drude response was found and o'0 was determined. In Fig. 4 we have plotted the result as o'0 against f T / T c° which according to Eqs. (22) and (23) should just be a straight line. This prediction is borne out to excellent degree as seen from Fig. 4. From the slope of this line we can determine the mobility go. Such a deter- mination is shown in the inset. The value found is to good approximation/zo = 0.0658F. The conclusion from the simulations is hence that the data is perfectly consistent with the Drude form given by Eqs. (20) - (22), There is no sign of any anomaly and the inferred mobility is just a conventional finite constant.

This result is in contradiction to the experimental results shown in Fig. 1 and furthermore to the single vortex theory by Korshunov [6]. The discrepancy between the experiments by Th4ron et al. [ 3] and the present simulations will be discussed below. Our conclusion is that the experiments are indeed consistent with a 2D X Y model but with parameters which are somewhat different from the nominal parameters 4 inferred in Ref. [3]. The theory by Korshunov is on the other hand for precisely the same model as we are simulating. Consequently the discrepancy has to be sought in the link between Korshunov's single vortex calculation and the present finite frustration simulation.

We wilt here only offer a suggestion as to where the discrepancy with Korshunov's calculation [6] might arise: In general when replacing the cosine potential with a harmonic one, as in Eq. (8), there is the question of the proper value for Jeer, a quantity which is proportional to the helicity modulus or the superfluid density [2]. Should it be the bare value (as assumed in Ref. [6] for the single vortex case), some partially renormalized value, or should it be the fully renormalized value? Here we note that the problem at hand

4 This either suggests that the mapping between the JJA model and the XY model with TDGL dynamics includes a non-trivial mapping of effective parameters, or that the experimental determi- nation of the Coulomb gas temperature from the sheet inductance in Ref. l 3 ] could differ from the one determined and used in the simulations.


is for finite frustration f so that the helicity modulus renormalized with respect to the vortices due to the finite frustration is zero. Thus if we calculate the mobility of one particular vortex in the background of all the others it seems natural to include this background vortex renormalization into Jeer. This changes Jeff to zero which in turn suggests that the limit given by Eq. (10) does not exist and that the proper result for small oJ is G -I oc -i~o, as is indicated by Eq. (11). Inserting this into Eq. (9) gives the conventional result bt(w = 0) = constant, in agreement with the present simulations.

The other proposed explanation for the experiment by Th6ron et al. [ 3 ] invokes vortex-antivortex correlations [2,4,7-9]. The discrepancy between the experiment and the present simulations may then be due to the mapping between the actual JJA array and the 2D XY model, i.e. although the XY model per se gives a good representation of the physics of the JJA, the nominal parameter values deduced from Ref. [ 3 ] might not be the relevant ones. In fact we note that if the vortex- antivortex correlations are responsible for the anomalous mobility observed in the experiments then the corresponding effective parameters for the XY model must be such that enough vortex-antivortex correlations are present. The reason why the anomaly does not show up in the present simulations should accord- ingly be due to the fact that too few vortex-antivortex correlations are present for the parameters used in the simulations.

In order to investigate this possibility further we first note that there are three ways of increasing the density of antivortices (vortices with vorticity opposite to the ones corresponding to the applied frustration): increasing the p-value in Eq. (4), increasing f , and increasing TcG [8]. As a first test we simulate the 2D XY model for the same values of f (= 1/64) and TcG (= ~T~'G/3) as in Fig. 3 but for p = 2.5 instead of the nominal p = 1. As seen from the inset of Fig. 2 the additional vortex density nadd is increased. As demonstrated in Fig. 5 the response is no longer of the Drude type but is now well described by the MP form of Eqs. ( 1 ) and (2). The conclusion is then that by increasing the density of vortex-antivortex pairs the response changes into the MP form. This is consistent with the vortex-antivortex correlation explanation for the anomalous dynamics proposed in Refs. [ 2,4,7,8 ]. In order to show that what matters here is the increase

0.5 ~a

o -

o

i i j ,~ ~l~"ll"q~ ~ t~ Q" I~" ° IIH~ ID ID ID 0 Q I l i tD 'Q 'Q t

/ I

t t ¢1

L [ I

O0 1 2 3 o.,'

Fig. 5. The dielectric function I/~(oJ) obtained from the simulations for f = 1/64 and T Co = T~c G/3 with p = 2.5 instead of 1. The open circles correspond to Re [ l / e ] and the solid circles to lira [ l / c ] I. The broken curves are a fit to the MP form given by Eqs. (1) and (2).

of vortex-antivortex correlations per se and not by what method it is increased, in Fig. 6 we demonstrate that the same change into the MP form is achieved by changing f to f = 1/8 keeping TcG and p at same vales as in Fig. 3, i.e. ~c6/3 and 1, respectively. As seen from the inset in Fig. 2 nadd is again increased. Thus our conclusion is that the anomalous response from the experiments by Th6ron et al. [ 3 ] shown in Fig. 1 is indeed due to vortex-antivortex correlations and that a JJA has qualitatively the same physics as a 2D XY model. However, at the same time we conclude that the mapping between the JJA in Ref. [3]

~ 0.5

0 . .~ . 0 -0." c~

e ~

¢ _

p~6

. o . O - • ..O - ' O

O ~ t Q ~ O Q ~ O

I I

O0 1 2 (,d

Fig. 6. The dieleclric function 1/~(to) obtained from the simulations for f = 1/8 and T cc = T~c G/3 with p = 1. The open circles correspond to R e [ I / e l and the solid circles to Ilm l 1/e] ]. The broken curves are a fit to the MP form given by Eqs. ( 1 ) and (2).


1.0

0.5

0.0

0.2 f , , - -ll

' , , , . 00/": I1 X - ' o ~ 0 0.5 1.0 [

I I I

5 10 15 ~/2~rl [I0% -x]

Fig. 7. The dielectric function l/e(o~,f) measured by Th6ron et at. I3] and plotted against co/2~rf. The open circles correspond to Re[l/el and the solid circles to [imI1/e]l. The full curve corresponds to the imaginary part obtained from the Krarners-Kronig relation. There is a clear disagreement between the Kramers-Kronig predicted full curve and the data (solid circles) for the imaginary part. The broken curve is a fit to the Drude form of Eq. (20) for the largest values of oo/f (¢o/f > 107 s-I). The inset shows that this fit (broken curve) fails for the small ~o/2~rf where the real part is proportional to ¢o/f as expected from the MP form of the response.

and a 2D XY model appears less straightforward than the one proposed in Ref. [3], i.e. the JJA in Fig. 1 ipso facto seem to have more vortex-antivortex correlations than what is suggested by the straightforward mapping to the 2D XY model.

The measurements shown in Fig. 1 are plotted against the variable go = o) / f and were measured for a few fixed frequencies with f as the variable. It was found that the data as a function of go approximately fall on a single curve. Consequently the data in Fig. 1 should at least approximately also represent the data for fixed f and variable o). One way of testing the degree of this approximation is to note that as a function of o) the real and imaginary part of the response are connected by a Kramers-Kronig relation. In Fig. 7 we have started from the real part of the response and calculated the imaginary part (full curve in Fig. 7). As seen in the figure the full curve deviates from the data. This reflects the fact that go is only an approximate scaling variable. The deviation from the oJ/f scaling can be studied in more detail within the 2D XY model [8]. In the MP region the response is given by Eqs. (1) and (2) with w0 <x n [8]. Conse- quently the correct scaling variable is win instead of

o) / f [8]. This may partly explain the deviation from the Kramers-Kronig relation in Fig. 7. However, in the simulations there is yet another effect: As seen from Figs. 3 and 6 the response goes from Drude like to MP like when f is increased and T cC is constant. The response is to very good approximation of form [ 8 ]

o) 2 + ~oo)0 + o'~ (24)

For small f the ratio O'o/o)o is large and Eq. (24) turns into the Drude form Eq. (20). In this small f - limit we have a Drude response and a o)/f-scaling as is apparent from Fig. 4. However, as f is increased the ratio o'o/wo decreases and finally the MP response of Fxl. ( 1 ) is recovered for large f (compare Fig. 6) with a oJ/n-scaling which corresponds to an approximate w/f-scaling [8] 5. This means that, for fixed o) and with f as the variable, the response goes from Drude like for large go = w / f to MP like for small go. The part corresponding to the largest go of the data by Thdron et al. [3] can be very well fitted to the Drude form (broken curve in Fig. 7). However, the part corresponding to smallest go clearly has the linear frequency behavior contained in the MP form (as shown in the inset). We conclude that also the crossover from Drude to MP with increasing frustration f found in the 2D XY model seems consistent with the data of Thdron at al [ 3 ].

5. Conclusions

The 2i3 XY model on a triangular lattice with TDGL dynamics was simulated for parameter values nominally corresponding to the experiment by Th6ron et al. [3]. The response obtained was perfectly nor- real and Drude like in contrast both to the non-Drude like behavior found in the experiments and theoret- ically derived from a vortex-spin-wave coupling by Korshunov [ 6]. Since Korshunov's calculation [ 6] is for the same model as we are simulating something appears to be lacking in the link between the theoretical treatment and the present simulations and we

5 The ratio ~r0/oJ0 is a good way of quantifying the degree of Drude and MP behavior. For example the data in Fig. 3 corresponds to almost pure Drude with tro/wo ~-, oo, Figs. 5 and 6 correspond to almost pure MP with ~ro/wo ~ 0.18 and 0.15, respectively.


made a suggestion in terms of the effective coupling constant which might vanish.

We believe that the difference between the simulations and the experiments is due to that the effective parameters for the XY model corresponding to the experiment for the JJA are different from the nominal ones proposed in Ref. [ 3 ]. In order to make this more likely we showed that by changing the parameters into a range where more vortex-antivortex correlations are present we find a MP response which is very similar to the response found in the experiments. Thus at least in the case of the 2D XY model the MP form of the response is due to vortex-antivortex correlations in agreement with earlier suggestions [2,4,7-9]. This leads us to believe that the very similar response in the experiments have the same origin. We also showed that the response in the simulations goes from Drude like to MP like with increasing frustration f and that such a feature also appears to be consistent with the experiment of Th~ron et al. [ 3 ].

Consequently the comparison between our simulations of a 2D XY model and experiments on a JJA leads us to believe that the anomalous response in both cases is due to vortex correlations induced by the presence of vortices with opposite vorticity. However, this also implies that the precise mapping of effective parameters between the experimental measured JJA and the 2D XY model needs to be better understood.

Acknowledgements

This work was supported by the Swedish Natural Research Council through contracts E-EG 04040-323 and F-FU 04040-322.

References

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complex impedance of a two-dimensional josephson junction array

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