PHYSICAL REVIEW B VOLUME 48, NUMBER 5 1 AUGUST 1993-I

Phase diagram for a Josephson network in a magnetic field

Joseph P. Straley and George Michael BarnettDepartment of Physics and Astronomy, University ofKentucky, Lexington, Kentucky 40506-0055

{Received 25 January 1993; revised manuscript received 12 April 1993)

We have constructed the ground-state configurations for a square-lattice Josephson network in a mag-netic field H=p@o/qa, for all p and q in the range 1 ~p ~ q/2 and 2 & q 20, leading to a better under-standing of the zero-temperature phase diagram. We then compute the field dependence of the ground-state energy, helicity modulus, and critical current.

I. INTRODUCTION

A Josephson network is a model for an extreme type-IIsuperconductor, consisting of discrete superconductingelements ("grains") at lattice sites. It is assumed that thesuperconducting wave function is constant on each grain,and that its magnitude is fixed, and that only the phasevaries from site to site. The phases are Josephson cou-pled so that the network in a magnetic field is describedby the Hamiltonian

H = —Jg cos( tf';~—

P, + i j —X~ )

—J g cos(P; —P, , +,),

where P, is the phase of the superconducting wave func-tion on site (i,j). The frustration factor gi =2srfjrepresents the effect of the vector potential, wheref=a B/4& isothe magnetic flux per unit cell of the lat-tice, measured in terms of the Aux quantum No=ck/2e.

The phases also determine the supercurrents, so thatthe supercurrents fiowing from site (i,j) in the positivedirections are given by

d" sin( P;i—4, + i,. —X, ),

(P,"=8 sin( P, —(f, +, ),where c/=2eJ/fi is the critical current for a single bond.In what follows, currents will be quoted in unitsI=ficI/2e with limiting value J, to avoid introducingthese two closely related material constants. The condi-tion that the net supercurrent out of a site be zero isequivalent to BH/BP,"=0; thus the supercurrent is con-served at all sites in any configuration that is a localminimum of the energy. The average supercurrent perbond need not be zero; by choice of boundary conditionsits magnitude can be varied over a range comparable to8, by forcing P to have a net gradient. It is sometimesconvenient to make the imposed gradient explicit in theHamiltonian

where (5„,h ) is the "phase strain" that induces thecurrent; with this Hamiltonian the boundary conditionon P is that its average gradient should vanish, con-veniently accomplished by studying large periodic cells.We will refer to configurations that are local minima ofthe energy with respect to variation of P, ~

in the periodicboundary conditions with no phase strain as stableconfigurations.

Pioneering work on the determination of the groundstates of the Josephson network was done by Teitel andJayaprakash' (TJ). For rational f=p/q, the frustrationfactor is periodic (modulo 2') with period q. Then theground states can be sought on periodic cells, if theirwidth (the j coordinate) is a multiple of q. In most cases,the ground state is periodic on the q Xq cell however,we have found some exceptions. There are examples forwhich the current pattern requires a larger cell; these will

be noted below. There are also cases for which thecurrent pattern is periodic on q X q but the phase patternis not; in the examples we have found, P differs by rr atseparation q in one direction [this state is the lowest ener-

gy configuration for Eq. (3) that is periodic on q X q —butwe must put b,„=m /q, rather than 0]; this occurs for

f= 1/4, 1/6, 1/12, 1/14, 1/16, 3/10, 3/14, 3/16, 3/20,5/16, and 5/18. We found no example for which theperiodicity of the ground state was greater than 2q X2q.

It is readily seen that the value of BH/Bb, is propor-tional to the average current I"per bond, which is zero inthe stable configurations; we can further define the helici-ty modulus tensor, which has elements Y=N 'B H/BE and Y~~=N 'B H/BA . These mea-sure the ability of a phase strain to cause a supercurrent,and thus is closely related to the superAuid density. Thehelicity modulus is generally less than J, which is itsvalue at f=0.

For each site of the dual lattice (i.e., the center of anelemental square of the lattice) we can define the circula-tion

H= —

Juncos(4,

, —y, +, , —X,

—a„)

(3)

In the low-lying stable configurations with small f, thecirculation is approximately 2srf for dual s—ites in re-gions where P is slowly varying with position; but the netcirculation should be small (it is zero if we use periodicboundary conditions), and so there will also be vortices:

0163-1829/93/48{5)/3309{7)/$06.00 3309 1993 The American Physical Society

3310 JOSEPH P. STRALEY AND GEORGE MICHAEL BARNETT

dual sites with circulation 2~( 1 f—). In low-energyconfigurations, where the vortices are well defined, thefraction of dual sites having vortices is exactly f. The lo-cation of the vortices specifies the currents sufficientlythat they uniquely specify the low-lying stableconfigurations.

The energy of a configuration then depends on the lo-cation of the vortices: the vortices interact. At large sep-arations, the interaction resembles the two-dimensionalCoulomb potential

V(r) = —srYQi Qzlnr+const,

where the charges are Q =1 f for—the vortices, and f-for the dual sites not occupied by vortices, so that allconfigurations have zero net charge (this could also beviewed as Q= 1 for the vortices, with a neutralizingcharge f on e—very dual site). The helicity modulus hereplays a role similar to the reciprocal of a dielectric con-stant. As will be shown below, the ground-state energyfor the Josephson network problem for a wide range of fcan be closely approximated by the energy for the corre-sponding Coulomb lattice gas. This long-range interac-tion between vortices ensures that density of vortices isexactly the same as the applied magnetic fiux (the lowercritical field is zero ); this is quite different from the caseof particles interacting with a short-ranged potential,where the commensurate phases extend over a finiterange of chemical potential.

The ground-state configuration for the Coulomb gaswith neutralizing background is a triangular lattice,which can never be commensurate with the square-latticegeometry imposed in the Josephson problem. Commens-urate configurations can be made from triangular oneshaving integer unit-cell area by a combination of rotationand shear; the magnitude of the shear strain ~ to producea Bravais lattice having lattice vectors v, and v2 is

[u, +uz+iv, —v2i ]

—2,1

where 3 is the area of the unit cell. This will cost an en-

ergy per site

hU= —,'o.H,where 0. is the shear modulus.

For any one variable P, , it is easy to find that valuethat minimizes Eq. (1) with all other variables held fixed;iterating this over all sites many times quickly leads to alocal equilibrium configuration, which may be visualizedas a vortex configuration. Finding the ground state en-tails moving the vortices about, seeking lower-energy ar-rangements. We did this in part interactively, with a pro-gram that allowed us to edit the arrangement of phases,by copying the phases of the sites in one small cell (theoptimum size depends on the density; sizes from 3 X 3 to8X8 were used) onto another cell (adjusting the phasesby a constant so that the phase at one corner was un-changed). This is most successful if the number of vor-tices in the two cells is the same; then a direct replace-ment of the vortex arrangement frequently resulted. Wealso used a Monte Carlo version of the same program,

which attempted copying random cells to random targetpositions, and accepted the results if the energy de-creased, or increased by a less than some AE, which wasslowly decreased to zero in the course of the run. TheMonte Carlo program occasionally found a candidateground state; more frequently it was used to suggest theground-state configuration, which was then constructedinteractively. It should be understood that what we referto as "ground-state configurations" in what follows aremerely the lowest-energy configurations we could find.However, we attempted to validate these in several ways:where we could envision several competing structuresthat were almost equally e6'ective in spreading the vor-tices uniformly, we built them all; the Monte Carlo pro-gram was used for its ability to suggest competing struc-tures. All of our candidate ground states have inversionsymmetry of the current pattern, so that the net super-current is zero. As checks on the validity of our claim tohave found the ground state we will point to the internalconsistency of the classification scheme (which wasdeveloped in large part after the configuration library hadbeen constructed), and the systematic regularities thatwill appear in the graphs of ground-state energy and heli-city modulus vs f, and critical current vs 1/q (failure ofthese regularities have in several cases directed us to abetter configuration). However, we will concede that it ispossible that several of our configurations are not the trueground state.

II. GROUND-STATE CQNFIGURATIQNS

We have constructed the ground-state configurationsfor f=p/q with p and q in the ranges 1 ~p ~q/2 and2 ~q ~20, as well as some larger values, for periodicallycontinued cells of size q Xq, occasionally Zq X2q. Whenarranged in order of f, a classification scheme emerges.

The top level of the classification scheme are the Bra-vais states, which occur for p=1, and every q exceptq =7. The vortices form a Bavais lattice, having unit cellof area q. These are lattice approximations to the tri-angular arrangement of vortices that is formed in type-IIsuperconductors. The triangular pattern itself is notcommensurate with the square lattice but can be madecommensurate by application of a shear; in every case theground-state configuration was the one constructed bythe smallest shear as computed by Eq. (6). Some typicalBravais ground-state configurations are shown in Fig. 1.

One surprising discovery is the existence of a Bravaisground state for f=—'. [Fig. 2(a)]. For this configurationthe phase pattern is only periodic on 8 X 8, even thoughthe currents are periodic on 4X4; this is why it was notdiscovered previously. The ground state proposed by TJ[Fig. 2(b)] is rather more complicated. These twoconfigurations are constructed from the same four super-current values, and have exactly the same energy per site;thus they are both ground states. However, in what fol-lows the Fig. 2(a) configuration will prove to be more use-ful (because ground states for f close to —' can be inter-

preted in terms of it), and thus will hereafter be called theground state for this case.

Clearly the Bravais-lattice configurations are a vanish-

48 JOSEPHSON NETWORK IN A MAGNETIC FIELD 3311

I I I I

I(

y I

I ~l ~l )pl (g(I'' I'' I'' I'' I'

I ~l )~I (glI (tl (il (ol

II I.

I(

$ II

$ I) y I

Iy I

I Ill It( Ill It( I

I (ql (~ I )yl

I ~ I pl pl

~~

. ~ ) ~ ~&

~ ~g

~ ~, I ~ ~ ~

. I' . I', I . , I , ~ I . , I

~l . , ) . ~lI

I' I' ' I'

~~

~~

~ ~ ~

s'

~

FIG. 3. The ground state for f=4/17 (E= —1.37944J).Regions of the f=1/4 ground state [Fig. 2(a)] are clearlypresent; the domain walls can be interpreted to be narrow stripsof the f=1/5 ground state (Fig. 1).

FIG. 1. Some Bravais ground-state configurations. Theseare the cases f = (a) 1/2 (E= —1.414 21J), (b) 1/5(E= —1.447 21J), (c) 1/3 (E= —1.327 03J), and (d) 1/6(E= —1.473 92J). They are drawn on cells larger than q Xq torepresent more clearly the periodicity of the currents. The mag-nitude and direction of each supercurrent is represented by aline originating at the midpoint of the bond connecting the sitesat which the phase is defined. Vortices are present at dual siteshaving a large positive circulation; these are indicated by a dot.

ingly small subset of all possible f. The majority of theconfigurations in our sample belong to the secondclassification level, of what we will call interpolatingconfigurations. These consist of domains of one of theBravais configurations, separated by domain wall, whichfrequently can be viewed as being narrow slices of adifferent Bravais configuration. For example, Fig. 3shows the ground state for f=4/17, which lies between1/5 and 1/4. It is a panel of the Bravais 1/4configuration, interrupted by a narrow stripe, which canbe regarded as a quote from the 1/5 ground state. Thisdomain structure is similar to what is encountered in thetheory of incommensurate ordering; however, the factthat the domains and domain wall have differing densitiesof vortices (and thus different charge density in theCoulomb gas analog) implies that wide domains are un-stable.

The ground-state configuration for f=1/7 is not aBravais structure; it is instead an interpolation betweenthe f= 1/6 and 1/8 structures (the unit cell has area 14,and two vortices per unit cell and thus is more properlyrepresented as f=2/14; the current pattern is onlyperiodic on a 14X14 cell). This is not very surprising,since the Bravais candidate is not at all close to being tri-angular.

For f & 1/3, all ground states in the set studied areBravais or interpolating structures, although the casef=4/15 has to be interpreted in terms of bands of threestructures (1/3, 1/4, and 1/5), and in the cases 3/19 and3/20 the domain wall is irregular (Fig. 4).

The major class of interpolating configurations for1/3 &f & 1/2 have been previously discussed by Halsey.They consist of diagonal domains of the f= 1/2 checker-board configuration, separated by domain walls (ordomains) of the f= 1/3 structure (see Fig. 5 for an exam-ple). Since both of these component structures containdiagonal chains of vortices, this is the dominant featureof these configurations. The translational invariance of

I'

I'

I

, I

I

I'I

, I

I

', I'

. I'

~ ~

~ ~I

~ ~ ~

.I' ', '. I'~ ~I,I

, I . ~

I' . ' . I'I'

,I'

I)

0 I . ~I

0 I

, I

, ) . ~

I

, I ~

FIG. 2. The two ground states for f= 1/4. (a) The Bravaisstructure. (b) The structure found by Teitel and Jayaprakash.The two patterns are built from the same four supercurrentvalues and have the same energy per site (E= —1.447 21J). Al-though the current and phase pattern of (b) is periodic on a 4X4square; the phase pattern of 2(a) is only periodic on an 8X8square.

~ I ~

FICJ. 4. The ground state for f=3/19 (E=—1.495 33J).This pattern consists of stripe domains from the f= 1/6 pattern(Fig. 1); however, the domain wall is not itself a quotation fromsome other Bravais structure.

3312 JOSEPH P. STRALEY AND GEORGE MICHAEL BARNETT 48

I t I)

t II

t I

)t I t I

I t I ~

I, I, , I ~

I

t I t I)

t I ~ ~ ~

I

~ t &

(t l

l t I I

I I' ''II, I , , I

I '.I

, , I , , & , l , tI'. I

I ~ t &

tt I

&t I'. I

I ~

II

II ~ '

II '. I

I

FIG. 5. A Halsey state, for f=5/12 (E= —1.27688J). Thisconfiguration can be viewed as an interpolation between f=1/3and 1/2.

g i I . t I . i I i I 1 i & t I

I .I

t I . i I i I I ~ & t II . I

i Ig

It I .

Ii I . i I i I ~

. I . I

I ~ t « t I~ t I t I i I

I . I' . '

I '.I

'I

I ~ i &

l i Ig i I t I'. I . /

' . 'I . I

I t II

t I i I ~ i « t I~. I

'I

t '. I . I

I ~ i & i I ( i I ~

(i I i I

I

I t I I i & i Ig t I

Ii I

I . I

~ . . . I,

I

II, I -

II I ~ I ~

I

. I'.

I

FIG. 7. The ground state for f=6/17 (E= —1.27993J).The domain wall does not lie along the (1,1) diagonal.

the current pattern along the diagonal permitted Halseyto determine the energy per site for these configurationsexactly:

UH = —(2J/q )csc(m. /2q ), (8)

which (somewhat surprisingly) does not depend on p.We find that the Halsey state is the ground state for

many f in the region 1/3& f & 1/2; however, there aresome exceptions. For f ~ 7/16=0. 4385, the ground-state structures are the f=1/2 checkerboard patternwith a low concentration of missing vortices; These va-cancies again behave electively like charges and form aperiodic structure (Fig. 6). For 1/3& f &7/19=0. 3684,we found other types of domain walls (Fig. 7) and com-plex configurations (Fig. 8) that have lower energy thanthe Halsey states.

This new level to the classification scheme is readilyunderstood when it is realized that the vortices have along-range interaction. As f approaches any value 1/q,the interpolating configurations will become widedomains of the Bravais structure, separated by domainwalls carrying the excess vortices. In the continuum lim-it, this charged layer is unstable. In the vacancy crystal,

it has simply dissociated into free vacancies; in theconfiguration of Fig. 7, the inclination of the domain wallhas decreased the amplitude of the variations in the vor-tex density. We can anticipate similar phenomena nearevery Bravais structure.

The ground states for various p and q are classified inTable I. The classification scheme is adequate to describemost of the cases, although there are too few of the spe-cial configurations close to f= 1/3 to allow unambiguousgeneralization.

III. GROUND-STATE ENERGY AND COMPLIANCES

The ground-state energy for small f can be estimatedby a simple argument. In this limit the vortexconfiguration will be a nearly triangular arrangement ofvortices, and the shear energy can be ignored. Thecurrent distribution will be dominated by the closest vor-tex, and the current around any particular vortex will beapproximately (Y/J)8(l/r —1/R), where r is the dis-tance from the vortex and R =a /"&/m f is the radius overwhich the vortex dominates. In writing this we havemade the approximation that the arguments of the tri-

l I & I & l & I « I & lI Q ~ « i l t & t & i I t & i & i & t & t & t « t & i I

&i l t

I I & I & I & I « I & l I

't« t&lt

&'~ t &i i « iI & I & l & I « I & l

I Q i« t I i & ~ « t & ~ « t & t « t & t &

) i « i &

lt « t I i I

It« tl lt« tl tl &tl t« tl tl It« tl lt& lt& t& t& t&I & I & l & Q l & l I & l l I

i& t& i« t& t& il tI il ~ i& il t& i'& tI t& i&I « I & l & ~ l & l l & I & I

, ~I & I I I « I & l l lI i & ~ I t & t & i & i & i & i & t & t « t I ~ & '~ I ~ i &

&t I

i& i& tl i& tI i& ~ & i& t& i& t& i& t« tl tI tlI & I I & I & I & I « I & I

i I t I&i I Q i« i I i & t I

&i I

lt « i & i« i & i & t &

&i I

I & l & Q l & l I & I & I & I

~ « t & t ~

&i l + t &

&t & t &

lt &

&t &

lt

&t & i

1 I I & I & I & I

t& i& t& i& i& t& il tt ~ l ~ t& il t& t& tI tl ~ &

I & I « l & l l l & l

Ii Ii Ii &i lt« t&

~ t« t i t ~ ~l I

I i & i I t & t & t I i & i & i« i & i & i I ~ I t I ~ i &&

~ II 1 I & I & l « .

I & l & ~ lt& &tl tl t« tl tl &t& t& t& il &t& &t& ~ & ltl tl &tlLI & ~l I & l & l & & « l & l & Qt I

&i I t &

&t I Q t « t I t I t & t I t & t & t & t & t & t

I & l & ~ l & l l & I & l & lt& ~ & t& t& t& iI tl ~ t& t& iI t& tl tl t& i& t&

I & & I & l I & I & I

i& t& t& t« t& i« t& tI tl Q t& &il tI t& tI t&l « l & l I I & „ I

'~ &i&&i& ~ « i« t& &i& &i& t« t& tI &i&Q i&&i& tI t&l I & I I

I 'tl ~ I&tl t& i& il t& t& t& t& t& i& tI tl ~ i&

I « & I l & & li I t I i« i I i« t & t« t & i & i &

&i« t « i I i I

&~ l

~ l ~ t &&

t & t ~ t &

&t ~

lt &

&t &

&

't &

&t &

l't &

&~ &

&t &

&

't &&i

tl I l & I

il t& i& i& i& t& t& t& i« tl il tl t& tl ill & l & & I & l I I

~ I i I ~ i & t I i I i I i I ~ t & ~ IIi I'. I & '. I

i i I t I i I q i & i I ~ t & i I i I t & i II I —'- '-

I

~ i t ' t t ' ~ t t t ' i t ' e'. I ' '. I 1 . '. I '.I

' '.I

t&&tl ~ tl .

ltl tl

I ltl . , tl, t« tl . t& &tl . , tl, t&, tl

t« tl , tl ltl . . tl . t« tl&

tl l

~, il il . i& il . il ~

I'. I ' '. I

I i & t I ~ t I i & t I I t & i I ~ i I ~ i & t I. I

~'~ && tl ~ il il . 'tl ~ t&& ~ I ~ t

1 I t l '~ « t I 't ll

'~ l ~'~ l ~

i I&

t Il

t I i I i « i I&

t Il

t I t I

FIG. 6. The ground state for f=8/17 (E= —1.261 39J).The basic structure is the f= 1/2 checkerboard, from which afew vortices are missing. These vacancies have been markedwith circles for visual clarity.

FIG. 8. A candidate ground state for f= 5/14(E= —1.280 61J). This complicated structure was found by theMonte Carlo program.

JOSEPHSON NETWORK IN A MAGNETIC FIELD 3313

TABLE I. Bravais structures are indicated by single integers (3); interpolating structures by the component Bravais structures(4/5), except for the Halsey structures (H =2/3). Vacancy crystals are indicated V, and unclassifiable structures by S. Structures arelabeled in lowest common denominator form even though a larger cell may be required (e.g., 8/17 can only be constructed as 16/34).The duplicate structures that are not in lowest-common-denominator form are entered in a smaller font (e.g., p =2, q = 10 gives the2/5 structure).

p 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

10987654321 2 3

2 H3 3/46 6/8

2

H4

8

V3

4/59

2 Vh K

3/4 3/4s 5/610 11

2

H3

6

12

VH

3/44/56/713

2 Vh h

S 3

3/4 3/4/54/5 s

6/8 6/814 15

2

Hh

3/44

5/6

16

2

V v

H HS 3

3/4 3/44/5 4/s5/68/917 18

VHS

3/43/44/5—/69/10

19

2

Vh

S3/4

4

s—/8

10

20

U„,= Uo — flnf+ Cf,&Y(9)

where the last term corrects for overlap effects, discrete-ness, and failure of the small argument approximation; C

gonometric functions in Eq. (2) are small; in the same ap-proximation the energy density is —2+Y/r . Integrat-ing over the area for which a + r R and then dividingby ~R gives the energy density

has the character of a vortex core energy, although adifferent choice for the argument of the logarithm wouldaffect its absolute value (and even its sign). For small fwe should put U0= —2J and Y=J. The same derivationapplies near f=1/2 —except that now the ground stateis the checkerboard pattern of Fig. 1(a), f becomes(1/2 f ), the dens—ity of defects, and Y=&2J—giving

&Z~JU„,= —v'2J —

( —,' —f )ln( —,

' —f )+C'( —,' —f ) . (10)

I I I II

I I I II

I I I II

I l I II

I I I I—1

—1 2

—1.4

—1.6

—1.8

c i & I j i s i I & s s i I a & s i I s & ~

0 0. 1 0.2 0.3 0.4 0.5

FICr. 9. The ground-state energy per site, as a function of f.The two smooth curves drawn in are continuum approximationsappropriate for small f and f close to 1/2; the rigorous upperlimit provided by the Halsey states is also indicated. The lowercurve is the bound provided by the ground state of Harper' sequation (all f=p/q with 2p ~ q ~ 101 are represented in thiscurve).

The ground-state energy per site of our configurations isshown in Fig. 9. Three fitting curves have been added:for f ~

—,' we have drawn in Eq. (9) with C =0.34J; for

f ~ 0.43 we have drawn in Eq. (10) with C'= —0.66J (assuggested above, C' can be negative); and for1/3 ~f & 0.43 we have drawn in UH =4J/~, which is thelimit for large q of Halsey's result (8), and is a rigorousupper bound on the ground-state energy. The two coreenergies were chosen to make the fitting curves reason-ably match the data; these terms give a correction to Eqs.(9) and (10), which is only important when the argumentof the logarithm is no longer small —i.e., at the limit ofrelevance of the expression. It is seen that the largestpart of the f dependence of the ground-state energy is de-scribed by these simple functions, even though in thisfigure we have extrapolated out of the small argument re-gion.

A further generalization of Eq. (9) to the vicinity off=1/3 (where Uo= —4J/3 and Y=2J/3) indicates thatthe crystal of vacancies in the f= 1/3 structure will onlyexist in a narrow range f=0.33+0.01; to study this re-quires a prohibitively large cell. This mechanism ex-plains how the apparent discontinuities in U(f) near thesimple rational values will be filled in, so that U(f) is acontinuous function of the applied field, as has beendemonstrated by Vallat and Beck.

The Hamiltonian (1) can also be written in the form

M = —J Re pe 'Q,*Jg; + & 1

—J Re g Q,*J.f; J + &,lJ 1J

where P, =b, exp(iP, . ), and with 6 fixed at unity. If in-

3314 JOSEPH P. STRALEY AND GEORGE MICHAEL BARNETT 48

stead we minimize this function subject to the less restric-tive condition that

(12)

a new problem will result, which has been considered byAlexander; he showed that it reduces to Harper's equa-tion, ' which has been studied by Hofstadter. " Thisgives a lower bound on the ground state of Eq. (1),U~ —,'Js(f), where E(f) is the minimum eigenvalue ofHarper's equation (i.e., the bottom edge of Hofstadter'sbutterfly spectrum). This estimate is also entered on Fig.9. It gives exactly the right answer for f=1/2. Howev-er, in the continuum limit g will prefer to be small nearthe vortices (which are located at the nodes of this func-tion); the unity-modulus condition that turns Eq. (11)back into the original Eq. (1) cannot represent this verywell, and so the two problems are increasingly discrepantat small f.

For each ground-state configuration we computed thehelicity moduli. The two helicity moduli are different forvortex patterns which lack fourfold symmetry, althoughin practice these did not differ by a great deal. Figure 10shows the f dependence of the lower of the two helicitymoduli for the ground-state configurations. We note thatas in the case of the ground-state energy, the major partof the field dependence of the helicity modulus can be de-scribed by several simple trends. (The anomalously highvalue for f=4/15 suggests that in this case we have notyet found the ground state).

The ground states are constructed with zero super-current, which generally implies that the configurationminimizes the Hamiltonian (3) with zero phase strains b,

and 6 on a periodic cell of size 2q X2q. Minimum ener-

gy states with nonzero (I ) can be constructed quitesimply by changing the 6 adiabatically and relaxing theresulting configuration. We can then define a critical

current by slowly increasing the phase strains until arearrangement of the vortex pattern occurs (which is sig-naled by an abrupt decrease in supercurrent and energy).The critical current depends on the direction of the phasestrain (i.e., the relative sizes of 5 and 6 ). We studiedhe cases ~ =0, ~ =0, ~ =~, and ~„=—~ for eac

configuration. The values obtained for the criticalcurrent of a given configuration occasionally differed byas much as a factor of 2. The minimum (over these fourcases) value of the critical current was always attained foreither b, =0 or 5 =0.

The anisotropy of the critical current is particularlylarge for Halsey configurations, because the diagonalchains of vortices serve as a wall through which very lit-tle supercurrent can pass (the bonds bordering a vortexare already carrying close to the limiting current). It wasfound that by continually increasing 5 and 6 in tan-dem (and continually relaxing the resultingconfiguration), the rearrangements that take place at eachcritical current event can completely realign theconfiguration. In this way the f=5/13 Halsey state withstaircase going down to the right was converted into thestate with staircase going up. In an experimental situa-tion this would mean that application of a small voltagecould establish the presence of the maximum criticalcurrent alignment. '

The dependence of the minimum critical current on fwas found to be quite erratic (Fig. 11). This could be ex-pected from an argument by Teitel and Jayaprakash, "who noted that there are q equivalent ground states andclaimed that this implies I, = I/q (i.e., independent of p,where f=p/q). This argument has been criticized previ-ously. ' ' We find that the critical currents for the Hal-sey states are reasonably well described by the rule

0..3 I I I I I I I I ~I

I I ~ I II I I ~

II 1 I I

I I 1 II

I I I II

I I I ~I

I I ~ I I I I I I1

0.2

0.8

0.6

~0~ ~

e~0

+be~ ~

Q Il 0 ~ g+C

~r~4'

~ ~0.1

'g~

~ ~ ~0~e ~ ~ ~

4~ ~

~ ~ ~

~ ~

I I I ~ I I I I I I I I I I I ~ I I I I I0 80 0.1 0.2 0.3 0.4 0.5

FICx. 10. The helicity modulus as a function off The values.lie close to a smooth curve, excepting the non-Bravais lattice

f= 1/4 [Fig. 2(b)I, which is anomalously high; f=4/15, whichis slightly high; and f=6/17 and 7/19, which are too low.

I l I I I ~ I I I I ~ ~ I I I ~ I ~ I ~ I ~ I00 0.1 0.2 0.3 0.4 0.5

FIG. 11. The critical current, as a function of f.

48 JOSEPHSON NETWORK IN A MAGNETIC FIELD 3315

0. I I I ~1

1 ~ I ~I

I I I I I I I I II

1 I ~ I

I

C

0.2

f = n/(3n+1)

I I I I I ~ I I I

0.1 0.2 0.3 0.4 0.5

FIG. 12. The critical current, as a function of 1/q. The mag-netic fiux is represented by f=p /q with p and q relatively prime(even in the cases where a 2q X2q cell is required). Lines havebeen drawn to aid in identifying the special families f=1/q,f=n/(3n +1), and the Halsey states. The points correspond-ing to these cases have been entered as filled circles, and theremainder as crosses. Except for the cases 1/19, 1/16, 1/11, and1/9, these points are seen to lie close to their lines. (The case1/4 has been entered twice. The Bravais structure [Fig. 2(a)]gives the lower critical current. )

f=n l(3n +1) has I, =0.07J+0.3J/(3n +1); the seriesf=1/2 —1/q roughly corresponds to I, =O. OSJ+J/q.In the case of the series f= 1/q, the finite limit for I, canbe interpreted as representing a property of a single vor-tex, viewed as if it were a point particle: it is localizedwithin an elemental plaquette by a periodic potential, andthe current gives rise to a Lorentz force, which attemptsto push the vortex across the lattice; the critical currentcorresponds to the maximum slope of the pinning poten-tial. Similarly the large-q limit of I, will not be zero forsequences that approach a rational f, because asymptoti-cally we have the rational structure plus a few defects,which can be treated by the same argument. The zerolimit of I, for the Halsey states may reAect the fact thatmany of the f in this set belong to the Fibonacci sequenceof approximants to the golden ratio (3—&5)/2.

In Fig. 12 we have plotted the critical currents for allconfigurations against 1/q, we note that this does revealsome order that Fig. 11 fails to And, but that the onlyclear conclusion is that excluding the Halsey states, thecritical currents for this group seems to be bounded awayfrom zero.

Vallat and Beck have given an argument that purportsto show that the critical currents are bounded away fromzero. However, in this argument they have assumed thatthe phases in Eq. (3) are independent of the phase strains6; in reality well before their estimate of the criticalcurrent is reached, this set of phases will no longer be alocal minimum of the energy. The critical current dis-cussed here is determined not by the condition that thecurrent reaches a maximum value, but by the conditionthat the phase pattern becomes unstable against pertur-bations.

I, =0.8J/q; however, other families have nonzeroasymptotes in the q~ ~ limit: for example, the criticalcurrent for members of the series f=1/q is well de-scribed by I, =0. 1J+0.45J/q, and the series

ACKNOWI. KDGMKNTS

This work was supported by the National ScienceFoundation through Grant No. DMR90-03698. %'ethank E.B. Kolomeisky for some perceptive comments.

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~~The possibility of such realignment was suggested by Halseyin Ref. 6.S. Teitel and C. Jayaprakash, Phys. Rev. B 27, 598 (1983).

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