Klein 2dtilt

8/6/2019 Klein 2dtilt

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A Two-dimensional Superconductor in a Tilted Magnetic Field - new states with

finite Cooper-pair momentum

U. Klein

Johannes Kepler Universitat Linz, Institut fur Theoretische Physik, A-4040 Linz, Austria(Dated: November 23, 2003)

Varying the angle between applied field and the conducting planes of a layered superconductor

in a small interval close to the plane-parallel field direction, a large number of superconducting stateswith unusual properties may be produced. For these states, the pair breaking effect of the magneticfield affects both the orbital and the spin degree of freedom. This leads to pair wave functionswith finite momentum, which are labeled by Landau quantum numbers 0 < n < . The stableorder parameter structure and magnetic field distribution for these states is found by minimizing thequasiclassical free energy near Hc2 including nonlinear terms. One finds states with coexisting line-like and point-like order parameter zeros and states with coexisting vortices and antivortices. Themagnetic response may be diamagnetic or paramagnetic depending on the position within the unitcell. The structure of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states at = 0 is reconsidered.The transition n of the paramagnetic vortex states to the FFLO-limit is analyzed and thephysical reason for the occupation of higher Landau levels is pointed out.

PACS numbers: 74.20Mn,74.25.Ha,74.70.Kn

Keywords: sup erconductivity; FFLO state; orbital pair breaking; paramagnetic pair breaking

I. INTRODUCTION

In this paper a theoretical study of a two-dimensional,clean-limit superconductor in a tilted magnetic field ispresented. Such systems exist in nature; several classes oflayered superconductors of high purity with conductingplanes of atomic thickness and nearly perfect decouplingof adjacent planes have been investigated in recent years.These include, among many others, the intercalated tran-sition metal dichalcogenide T aS2 (pyridine), the or-ganic superconductor (BEDT T T F)2Cu(N CS)2,and the magnetic field induced superconductor (BETS)2FeCl4.

Depending on the angle between applied field andconducting planes the nature of the pair-breaking mech-anism limiting the superconducting state can be contin-uously varied. For large the usual orbital pair-breakingmechanism dominates and the equilibrium state is theordinary vortex lattice. With decreasing , in a smallinterval close to the parallel direction, spin pair-breakingbecomes of a magnitude comparable to the orbital effectand both mechanisms must be taken into account. Forthe plane-parallel field direction, = 0, the orbital ef-fect vanishes completely and the superconducting state

is solely limited by paramagnetic pair breaking. The su-perconducting state expected in this limit is the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state1,2. The tilted-field arrangement, which allows to control externally therelative strength of both pair-breaking mechanisms, hasfirst been investigated by Bulaevskii3.

The upper critical field Hc2, where a second orderphase transition between the normal-conducting andthe superconducting state takes place, has been calcu-lated for arbitrary angle and temperature T = 0 byBulaevskii3. This treatment was generalized to arbitraryT by Shimahara and Rainer4 . The field Hc2 has a cusp-

like shape, considered both as a function of or T, withdifferent pieces of the curve belonging to different val-ues of the Landau quantum number n (n = 0, 1, . . .). Inthe orbital pair breaking regime, for large , one findsas expected n = 0. As is well known, this lowest valuen = 0 determines the (orbital) upper critical field of thefamiliar vortex state, both in the framework of Ginzburg-Landau(GL)- and microscopic theories of superconduc-tivity. With decreasing , higher-n segments of the criti-cal field curve appear close to the plane-parallel orienta-tion. For 0 one finds4 n and agreement withthe FFLO upper critical field. Thus, in this purely para-

magnetic limit, the stable state below Hc2 must be theFFLO state.

Paramagnetically-limited superconductivity differs infundamental aspects, such as Meissner effect and spin-polarization, from the behavior of the usual, orbitally-limited superconducting state. In the FFLO state pair-ing takes place between electrons with momentum and

spin values (k + q/2, ) and (k + q/2, ). This leadsto Cooper pairs with finite momentum hq and a spa-tially inhomogeneous superconducting order parametergiven by (r) = 0 exp(qr) (or by linear combinationsof such terms with the same absolute value of q). Thepair-breaking is entirely due to the Zeeman coupling be-tween the magnetic moment of the electrons and the

external magnetic field H. The general rule, for bulksuperconducting states, that gradient terms in the freeenergy must only be taken into account if a nontrivialvector potential is present breaks down for the FFLOstate.

At T = 0, the Cooper pair momentum of the FFLOstate is approximately given by hq = |pF pF|, where|pFpF| = H

2m/EF is the difference in Fermi mo-

mentum between spin-up and spin-down electrons. Withincreasing T the FFLO wave number q decreases and van-


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ishes at the tricritical point Ttri = 0.56 Tc. The FFLOstate is only stable for T < Ttri, where its upper criti-cal field HFFL O exceeds the Pauli limiting field HP ofthe homogeneous superconducting state5,6. At T = 0,HP = 0/

2, where 0 is the superconducting gap at

T = 0. The second order phase transition line HFFL O(T)depends on the shape of the Fermi surface. In this paperwe use a cylindrical Fermi surface appropriate for a two-

dimensional (2D) geometry. The corresponding criticalfield7 is given by HFFL O = 0 at T = 0.

Between the ordinary vortex state with n = 0 and theFFLO state with n a countable infinite numberof unconventional superconducting states, characterizedby Landau quantum numbers n = 1, 2, . . ., exist. Thetransition from the vortex state to the first of these, then = 1 state, occurs at an angle 1 given approximatelyby

sin 1 Horbc2

HP kBTc

mv2F, (1)

where Horbc2 and Hp are the pure orbital and param-agnetic upper critical fields, respectively. Since Hp

Horbc2 the experimental upper critical field for a three-dimensional sample is given by Horbc2 . Because 1 1(generally 1 will be of the order of magnitude of 1 De-gree), the perpendicular component H = Hsin for allof these states with n > 0 will be much smaller thanthe parallel component H = Hcos . Thus, these stateswill have some properties in common with the FFLOstate, namely strong paramagnetic-pair breaking, a spa-tially inhomogeneous order parameter, and Cooper pairswith finite velocity of the center of mass coordinate. De-spite this similarity with regard to general features, the

order parameter structure for the n > 0 states may becompletely different, even for large n, from the FFLOstate. The reason is, that a finite perpendicular compo-nent H, no matter how small, implies a new and ratherstringent topological constraint on the equilibrium struc-ture, namely the flux quantization condition. The sub-ject of the present paper is the detailed investigation ofthe structure of these n > 0 states, which might be re-ferred to either as FFLO precursor states or as param-agnetic vortex states, in the vicinity of the upper criticalfield Hc2. A theoretical treatment of these FFLO precur-sor states, reporting several essential results and an out-line of the calculation, has been published previously8.

This paper8

will be referred to as KRS in what follows.In the present paper many new results are reported andthe treatment is extended with regard to several points,including finite values of, the purely paramagnetic limit = 0, and the transition n .

It should be pointed out, that the physical origin of theLandau level quantization effects for Cooper pairs, con-sidered in the present paper, is very different from theLandau quantization effects for single electron states dis-cussed in a large number of publications by Tesanovic etal.9, Rajagopal et al.10, Norman et al.11 and others. Thelatter are mainly concerned with the relative-coordinate

degree of freedom of the two bound electrons constitut-ing a Cooper pair and lead to measurable consequencesonly outside the range of validity of the quasiclassical ap-proximation, at very low temperature T < (kBTc)

2/EFand/or high fields. In addition, a mechanism is re-quired to suppress the Zeeman effect, which is neglectedin the theoretical treatment and is not compatible withthe predicted phenomena. The question whether the

most dramatic consequences12 (reentrant superconduc-tivity) of this type of Landau quantization effects willbe observable, has been the subject of a controversialdiscussion13,14. In contrast, the present Landau levelquantization mechanism is a consequence of the Zeemaneffect, concerns the center of mass motion of the Cooperpairs, and can be described (as will be discussed shortly)by means of the quasiclassical theory of superconductiv-ity.

Restricting ourselves to the vicinity of the upper criti-cal field Hc2 we may use an expansion of the free energyin powers of the order parameter , keeping only a fi-nite number of terms. An analogous gradient-expansion,which would lead to a relatively simple GL-like theorywith a finite number of spatial derivatives of , does,unfortunately, not exist for the present problem. Suchan expansion may be performed for = 0, in the purelyparamagnetic limit, near the tricritical point Ttri, wherethe order parameter gradient is small because the char-acteristic length q1 of the FFLO state diverges at Ttri.However, for finite H a small characteristic length fororder parameter variations does not exist in the relevantrange of temperatures, and the spatial variation of must be taken into account exactly. One might stillhope that a GL theory with a finite number of deriva-tives, although not accurate, will be useful to predict thequalitative behavior of the superconducting states nearHc2 correctly; bearing in mind for example the results ofstandard GL for type II superconductivity. However, forthe mixed orbital-paramagnetic pair-breaking phenom-ena under discussion, there is not even a single pointon the temperature scale where a GL theory with a finitenumber of derivatives is valid. Such a theory is only validnear Tc where no FFLO state exists, or near Ttri in thevicinity of the paramagnetic limit, i.e. for extremelylarge n. The latter region is inaccessible both from anumerical and a experimental point of view. In this con-text, it should also be noted that the final equilibrium

structures do not show any continuity with regard to n.Fortunately, the present problem does not require solv-

ing the full set of Gorkovs equations because the simplerset of quasiclassical equations may be used instead, aspointed out by Bulaevskii3. The large parallel compo-nent H of the applied magnetic field, acting only on thespins of the electrons, is exactly taken into account by theZeeman term. Thus, with regard to this component noquestion, as to the validity of the quasiclassical approxi-mation, arises. The magnitude of the perpendicular com-ponent H, on the other hand, must obey the usual qua-siclassical condition hc < kBT, where c = eH/mc,


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or sin (ehH/mc) < kBT. Inserting the highest possiblefield H = HP in the latter relation, one finds that thequasiclassical approximation holds indeed for not too lowtemperatures, T /Tc > kBTC/EF, in the interesting rangeof tilt-angles < 1, where the new paramagnetic vortexstates appear.

In most papers on paramagnetic pair-breaking and theFFLO state the influence of orbital pair-breaking is com-

pletely neglected. This means, that the GL-parameter tends to infinity and that all spatial variations of the mag-netic field can be neglected. For three-dimensional super-conductors this approximation implies that the orbitalcritical field is much higher than the paramagnetic Pauli-limiting field. This is impossible to achieve15 for BCS-likesuperconductors, because the superconducing coherencelength cannot be smaller than an atomic distance. Itseems unlikely even for unconventional materials16 wheremany-body effects may lead to a strong renormalizationof the input parameters. For the present 2D situation,the suppression of the orbital pair-breaking effect is en-tirely due to geometrical reasons, and no restriction on

the value of is required in order to reach the purelyparamagnetic limit at parallel fields. Thus, keeping allterms in the quasiclassical free energy related to spatialvariations of the magnetic field, will allow us to studytype II superconductors with arbitrary or even type Imaterial. Large- superconductors show, however, still apractical advantage because of their larger critical angle1 [see Eq. (1)].

This paper is organized as follows. In section II Eilen-bergers quasiclassical equations generalized with regardto a Zeeman coupling term, as well as the correspondingfree energy functional, are reported. The expansion ofthe free energy near the upper critical field, for a general2D quasi-periodic state, is treated in section III. Twolimiting cases of the analytical results, the GL limit andthe structure of the ordinary vortex lattice, are reportedin appendices. The numerical results for the paramag-netic vortex states, at finite perpendicular field, are re-ported and discussed in section IV. The structure of theFFLO state, for the special case of vanishing perpen-dicular field, is reconsidered in the present quasiclassicalframework in section V. The non-trivial transition 0(or n ) to the purely paramagnetic limit is analyzedin section VI. An explanation for the increase in n, interms of the finite momentum of the Cooper pairs in theparamagnetic vortex states, is also reported in this sec-tion. The results are summarized in the final section VII.

II. QUASICLASSICAL EQUATIONS WITH

ZEEMAN TERM

We need a weak-coupling, clean-limit version of thequasiclassical theory17,18, which contains all terms re-lated to the coupling of the electrons spins to an exter-nal magnetic field. A general quasiclassical theory whichcovers Zeeman coupling has been published by Alexander

et.al19. The 4 4 Green s function matrix appearing inthis work may be considerably simplified for the presentsituation. Since we neglect spin-orbit coupling, the direc-

tion of the magnetic induction B in spin-space may be

chosen independently from the direction of B in ordinary

space; we adopt the usual choice of B being parallel tothe zdirection in spin space. Then, only six essentialGreens functions remain, which are denoted by

f(+) = f + f3 f() = f f3f+()

= f+ f+3 f+(+) = f+ + f+3g(+) = g + g3 g() = g g3.

Here, f, f+, g denote the Greens functions in the absenceof Zeeman coupling, and f3, f

+3 , g3 are the additional

Greens function components in the the zdirection ofspin space. The three equations for the left groupf(), f

+(+), g() are decoupled from the three equations

for the right group f(+), f+()

, g(+) and differ only by

a negative sign in front of the magnetic moment .

=h|e|/(2mc) of the electron. Also, for each group a sep-arate normalization condition g2(+) + f(+)f

+() = 1 and

g2() + f()f+(+)

= 1 respectively, exists. Therefore, it

is convenient to introduce Greensfunctions f, f+, g, de-fined by

f(r, k, s) = f()(r, k, ),

f+(r, k, s) = f+(+)

(r, k, ),

g(r, k, s) = g()(r, k, ),

which are functions of the spatial variable r, the quasi-

particle wave-number k, and the complex variable s = + B. The 2D variable r denotes positions in the con-ducting (x, y)-plane. The real variable takes the valuesof the Matsubara frequencies l = (2l+1)kBT; the Mat-subara index l will not always be written down explicitly.The second group of Greens functions f(+), f

+(), g(+)

may be expressed by similar relations in terms off, f+, gif s is replaced by s .

Using the Greens functions f, f+, g, the quasiclassicalequations with Zeeman coupling become formally similarto the quasiclassical equations without spin terms. Thenonlinear transport equations for f, f+ are given by

2s + hvF(k)r

f(r, k, s) = 2(r)g(r, k, s),

2s hvF(k)r

f+(r, k, s) = 2(r)g(r, k, s),

(2)

where the Greens function g is given by the normaliza-tion condition

g(r, k, s) =

1 f(r, k, s)f+(r, k, s)1/2

. (3)

Here, vF(k) denotes the Fermi velocity and r is

the gauge-invariant derivative defined by r = r


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(2e/hc) A. The order parameter and the vector po-

tential A must be determined selfconsistently.

The self-consistency equation for is given by

2kBTND

l=0

1

l

+ ln (T /Tc)(r) = kBTND

l=0

d2k f(r, k, s) + f(r, k, s ) , (4)

where ND is the cutoff index for the Matsubara sums.The self-consistency equation for A is Maxwells equation

r

B(r) + 4 M(r)

=

162ekBT NFc

NDl=0

d2k

4vF(k

)g(r, k, s),(5)

where NF is the normal-state density of states at theFermi level. The r.h.s. of Eq. (5) is the familiar (orbital)

London screening current while the magnetization M isa consequence of the magnetic moments of the electronsand is given by

M(r) =22NF B(r)

4kBT NFNDl=0

d2k

4g

B

B,

(6)

The first term on the r.h.s. of Eq. (6) is the normal

state spin polarization. The second term is is a spinpolarization due to quasiparticles in the superconductingstate.

The following symmetry relations hold for solutions ofEqs. (2,3,4,5)

g(r, k, s ) = g(r, k, s),f+(r, k,

s) = f(r,

k,

s),

g(r, k, s) = g(r, k, s),f(r, k, s) = f(r, k, s),f+(r, k, s) = f+(r, k, s),

(7)

which have been extensively used in the calculations de-scribed in the next sections.

The quasiclassical equations (2,4,5) may be derived asEuler-Lagrange equations of the Gibbs free energy func-tional G, which is given by

G =1

Fp

d3r

B2

8 2NF B2

B H

4+

NF

kBT

+l=

1

|l| + ln (T /Tc)

||2 kBT NF+

l=

d2k

4I(r, k, s)

.

(8)

The area of the sample is denoted by Fp and the

kdependent quantity I is given byI(r, k, s) = f

+ + f +

g l|l|

1

f

s +

hvF2

r

f +

1

f+

s hvF

2r

f+

.

An important reference state for the present problemis the purely paramagnetically limited homogeneous su-perconducting state, which is realized for our 2D super-conductor if the magnetic field is exactly parallel to theconducting planes. In this case, the vector potential and

the gradient terms in the transport equations may be

omitted. At T = 0 the free energy difference betweenthe superconducting and normal-conducting states maybe derived analytically. It is given by

Gs Gn = NF

2H2 20/2

, (9)

and vanishes at the Pauli critical field HP. For higherT the self-consistency equation for the gap must besolved numerically, yielding agreement with previousresults.7,20. Let us investigate the magnetic response inthis purely paramagnetic limit. It is neglected in mosttheoretical treatments, but is of particular interest if theinfluence of finite values of the GL-parameter is to be


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taken into account. To obtain the magnetization dueto the spins, the coupled self-consistency equations (4,5)have to be solved. Using dimensionless quantities definedin appendix A the gap equation takes the form

ln t tNDl=0

1

|

|2 + (l + B)

2+ c.c.

2

l

= 0,

(10)while Maxwells equation reduces to

B H = 2

B 2t ND

l=0

l + B||2 + (l + B)2

.(11)

Note that the orbital screening current [the r.h.s. ofEq. (5)] is completely absent for the plane-parallel fielddirection. At T = 0 the r.h.s. of Eq. (11) vanishes ex-actly. This means that the normal state spin polarization[first term on the r.h.s. of Eq. (11] is exactly canceled bythe spin polarization due to the superconducting quasi-

particles [second term on the r.h.s. of Eq. (11)]. Thenumerical solution shows that the quasiparticle polariza-tion decreases with increasing T and vanishes at = 0,where the magnetic behavior of the normal-conductingstate is recovered.

In the rest of this paper dimensionless quantities asintroduced by Eilenberger will be used. These quantitiesare listed in appendix A. Any exceptions will be men-tioned explicitly.

In the next sections the stable order parameter struc-ture of a 2D superconductor in the vicinity of the phaseboundary will be investigated. The phase boundaryHc2(T) itself is given by the highest solution of the

equation4

0 = ln t+t

0

ds1 eDs

sinh st

1

cos(Hs)eHs2/4Ln(Hs2/2)

,

(12)

where the integer n = 0, 1, 2, . . . is Landaus quantumnumber, D is the Debye frequency, and Ln is a Laguerrepolynomial21 of order n. A typical phase boundary isshown in Fig. 1. Each piece of the nonmonotonic Hc2curve is characterized by a single value of n. An infinitenumber of eigenstates n,k exists, belonging all to thesame, highly degenerate eigenvalue n. For the presentgauge, these are given by

n,k(r) =A(1)n

n!ekxe

H2

y kH

2

Hen

2H

y k

H

,

(13)

where k is a real number and Hen is a Hermitepolynomial21 of order n. The functions (13) are orthog-onal and normalized,

(n,k, m,l) = n,m(k l), (14)

8.00

8.50

9.00

9.50

10.00

10.50

11.00

11.50

12.00

Hc2

0.40 0.80 1.20 1.60 2.00

[deg]

0

1

2

3

FIG. 1: Phase boundary of the superconducting state att = 0.1 for tilt angles between 0.1 and 2.0 using a value

= 0.04 for the dimensionless magnetic moment of the elec-tron. The numbers 0, 1, 2, . . . are Landau quantum numberscharacterizing the individual pieces of the curve.

if the amplitude A in Equ. (13) is chosen according to

A =1

R0

HL2x

1/4, (15)

where Lx is the size of the system in xdirection andR0 is defined in appendix A. The gap, for the portion ofthe Hc2-curve characterized by n, is a linear combination

of all n,k belonging to this n. The harmonic oscillatoreigenfunctions (13) are extensively used in the theory ofthe quantum Hall22 effect and many other topics in thequantum theory of a charged particle in a magnetic field.

III. FREE ENERGY EXPANSION NEAR THE

UPPER CRITICAL FIELD

We assume that the transition between the supercon-ducting and normal-conducting states at the upper crit-ical field Hc2 will be of second order for arbitrary tilt-angle . Then, the order parameter , or more precisely

its amplitude may be used as a small parameter for ex-panding the free energy G in the vicinity ofHc2. We keepterms up to fourth order in and all orders in order pa-rameter derivatives and determine the energetically mostfavorable order parameter structure near Hc2. Similarcalculations for the ordinary vortex lattice, correspond-ing to the case of large of the present arrangement,have been performed by Eilenberger23 and by Rammerand Pesch24. No special assumptions on the order pa-rameter structure, such as the number of zeros per unitcell, will be made. We only assume that the order param-eter is quasi-periodic on a 2D lattice, with an arbitrary


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unit cell, characterized by the length of the two basis vec-tors and the angle between them. The free energy willbe minimized with respect to these unit cell parameters.

A. Order parameter

Let the unit vector a of our elementary cell be paral-lel to the xaxis, a = aex. The angle between a andthe second unit vector b is denoted by . To constructa quasi-periodic order parameter near Hc2, exactly thesame method as used by Abrikosov25, for the case n = 0,may be applied. The result is given by the following lin-ear combination of a subset of the basis functions (13)

n(r) = ACn

m=+m=

exp

b

am(m + 1) cos

exp

2

amx

hn (y mb sin ) , (16)

where

hn(z) =(1)n

n!e

B2

z2Hen

2Bz

.

This order parameter8,11,26 is not invariant under trans-

lations r r = r + na + mb but acquires phase factorsfor each elementary translation, which are uniquely de-fined within a fixed gauge. Surrounding a unit cell inanti-clockwise direction, these phase changes add up toa total factor of exp 2, i.e. each unit cell carries a sin-gle flux quantum 0 . We shall use this assumption ofa single flux quantum per unit cell, which is written as

Bab sin = 2 in the present units, throughout thispaper. Preliminary calculations27 show that states withtwo flux quanta per unit cell have higher free energy andcan be excluded. Also, a preference for multi-quanta vor-tices seems unlikely in the present situation, where thesingle flux quantum state is stable at large , while thetotal flux decreases to zero as 0.

The order parameter (16) describes a flux line latticewhere the Cooper pair states belong to arbitrary Landauquantum numbers n, depending on the tilt angle . Asis well known, the pairing states for the ordinary vortexstate belong to the lowest Landau level n = 0. Thepresent shift to higher Landau levels is, of course, related

to the large paramagnetic pair-breaking field H as willbe discussed in more detail in section VI.

The coefficient Cn in Eq. (16) may be expressed bythe spatial average of the square of the order parameter,using the relation

|n|2 = 1Fp

|Cn|2+M/2

m=M/2

1, (17)

where Fp is the area of the sample. The spatial aver-age over the unit cell area Fc = ab sin is defined in

appendix A. For later use, when performing the limit 0 in section VI, we assumed in Eq. (17) that thearea of the superconducting plane is finite and that thenumber of unit cells in one direction is M. At the end ofthe following calculation, Cn will be fixed according tothe requirement |n|2 = 1 and an infinitesimal ampli-tude will be attached in front of each power of n.

A useful quantity is the square of the order parameter

modulus, which may be written in the form

|n|2(r) =

l,j

(2n)l,j e Ql,jr. (18)

The Fourier coefficients (2n)l,j are given by

(2n)l,j = (1)lj elba cosexl,j/2Ln(xl,j ), (19)

where xl,j is defined by Eq. (B4). The order parametern is proportional to n but with an amplitude chosenaccording to |n|2 = 1. It is instructive to compareEq. (18) with the local magnetic field reported later insubsection IIIE.

B. General aspects of the expansion

A fourth order expansion of G requires first and thirdorder contributions in the Greens functions f, f+. Weuse the notation

f = f(1) + f(3), f+ = f+(1) + f+(3), (20)

where f(1) and f(3) are the contributions of order 1 and3 respectively. A consistent treatment of the magnetic

field terms28,29 requires a separation of B and A accord-ing to

B(r) = B + B1(r), A(r) = A(r) + A1(r), (21)

where B is the spatially constant magnetic induction,B1(r) is the rdependent deviation from B and A(r),A1(r) are the corresponding vector potentials. An evalu-ation of the magnetic field terms in G requires the lead-

ing order in B1(r), which is 2: B1 B(2)1 . The spatiallyconstant quantity Hc2B, where B = | B|, is small of or-der

2

. The whole expansion in will be done keeping

Bfixed; at the end of the calculation, the Gibbs free energyG will be minimized with respect to the order parameteramplitude and the induction B. The calculation canbe seen as an extension of Abrikosovs classical work25

to arbitrary temperatures below Tc.Let us choose the coordinate system in such a way

that the magnetic field lies in the (y, z)plane. Then,the induction B(r) (and the external field H) may besplit according to

B(r) = B(r)ey + B(r)ez, (22)


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in perpendicular and parallel components B, B. The

corresponding vector potentials are denoted by A, A.In order to fix the gauge we may employ here essentiallythe same method as used before in numerical calculationson the vortex lattice without Zeeman coupling29,30. The

gauge conditions which fix A1 are given by28

A1

r = 0,

d2r A1 = 0, A1 periodic. (23)

The vector potential A describing the average value B ofthe induction is chosen according to

A(r) =

Bz By

ex. (24)

The first term in Eq. (24) can be omitted in the gaugeinvariant derivatives of Eq. (2) since no zdependenceexists in our 2D system. Thus, the orbital pair-breakingcontribution in the transport equations consists of thesum of the second term B in Eq. (24) and therdependent part A1 (only the perpendicular compo-nent of A1 is relevant here). The (large) parallel com-

ponent B, on the other hand, enters the spin pair-

breaking term, which is proportional to B(r) = (B2(r) +

B2(r))1/2. Eqs. (23,24) fix the gauge, i.e. allow a

unique determination of A in terms of B. While ||2and B are periodic, i.e. invariant under translations be-

tween equivalent points in the 2D structure, and A areonly quasiperiodic, i.e. they differ by phase factors and

a change in gauge respectively. The phase factors arefixed within a given gauge and may be calculated usingEq. (24).

As a first step in the expansion of G, the Greens func-tion g is eliminated in favor of f, f+ by means of therelation

g = 1 f f+

2 f

2(f+)2

8+ . . . ,

which is valid for small . Second, the gradient termsin G may be eliminated with the help of the transport

equations (2). Then, the (dimensionless) Gibbs free en-ergy takes the form

G =1

Fp

d3r

2

B H2

2 B2+

ln t + 2

l=0

1

2l + 1

||2

t2

l=0

f+ + f +

1

4

f(f+)2 + f2f+

+ c.c.

,

(25)

where the bar denotes a Fermi surface average as definedin appendix A.

Inserting the expansions (20),(21) in the free en-ergy (25) and collecting terms of the same order in ,G takes the form

G = G + G(2) + G(4), (26)

where the terms G, G(2), and G(4) denote the free energycontributions of order 0, 2 and 4 respectively. Theterm G is given by

G = 2

B H2

2 B2. (27)

We will first simplify the quantities G(2) and G(4) andthen calculate the minimum of G with respect to theamplitude and the induction B.

C. Second order contribution

The second order contribution to the Gibbs free energyis given by

G(2) =1

A

d3r

ln t + 2

l=0

1

2l + 1

||2

t2

l=0

f+(1) + f(1) + c.c.

.

(28)

To calculate the lowest order Greens function only con-tributions of order 0, namely the spatially constant part

B =

B2 + B2

1/2of the induction and the lowest or-

der vector potential A, have to be taken into account inEq. (2). The resulting equation for f(1) is given by

[l + B + k(0)r ]f

1 = , (29)

where (0)r =

r + Byex. To proceed, we use well-

known methods31 and solve first the eigenvalue problem


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8

of the operator k(0)r . The solution is given by

k(0)r fk,p(r) = Ek,rfk,p(r), (30)

with the eigenvalues Ek,r = kp and the eigenfunctions

fk,p(r) = exp B2

xkx + ykyxky ykx Bxy

2+ pr

.

(31)

Using the completeness of this continuous set of eigen-functions the differential operator on the l.h.s. of Eq. (29)may be inverted and f(1) be represented in the form

f(1) =

d2p

42

d2r1

fk,p(r)fk,p

(r1)

l + B + kp(r1). (32)

Representing the denominator in Eq. (32) by means ofthe identity

1

r=

0

ds esr (33)

as an additional integral, both the pintegration and ther1integration may be performed analytically and thesolution of Eq. (29) takes the form

f(1)(k, s, r) =

0

du eus

exp

B

2

2uykx + u2kxky

(r uk).

(34)

The first order solution for f+ is given byf+(1)(k, s, r) = f(1)(k, s , r).

The evaluation of the remaining integrals may begreatly simplified by introducing the gap correlationfunction V(r1, r2). In the present gauge it is defined by

V(r1, r2) = (r1)(r2)exp

B2

(x1 x2)(y1 + y2)

.

(35)Of particular importance are the Fourier coefficientsVl,j (r), where r = r1 r2. The precise definition andcalculation of Vl,j (r) is reported in appendix (B).

All terms in Eq. (28) containing first order Greensfunctions may be expressed as integrals over a gap cor-relation function. The first of these takes the form

f+(1) =

0

du eusV(r + uk, r), (36)

while the corresponding expression for f(1) may bederived from (36) with the help of the symmetry rela-tions (7). To proceed, center of mass coordinates are in-

troduced and a Fourier expansion of VCM( R, r) with re-

gard to the variable R is performed, using the result (B3)

from appendix B. The remaining summations and inte-grations may be performed analytically21. Collecting allterms one obtains the final result for the second ordercontribution

G(2) =||2

ln t + t

0

ds1 eDs

sinh st

1

cos(Bs)eBs2/4Ln(Bs

2/2).(37)

While the order parameter expansion Eq. (16), whichentered the calculation of G(2), depends on the latticeparameters a,b,, this dependence is absent in the fi-nal result, Eq. (37). The quantity G(2), characterizingthe appearance of the superconducting instability, andnot the detailed structure below it, does only depend onthe eigenvalue n. The relation G(2) = 0 agrees with thelinearized gap equation (12) used to calculate Hc2.

The technique used here to calculate G(2) will be gener-alized in the next subsection to evaluate the fourth ordercontribution to the free energy.

D. Fourth order contribution

The free energy contribution of order 4 may be split,according to

G(4) = G(4)N + G

(4)M , (38)

in a nonmagnetic part G(4)N and a magnetic part G

(4)M . In

G(4)N the spatially constant induction

B and the corre-

sponding vector potential A(r) are used. The term G(4)M

collects all terms of order 4 where deviations B1(r)

2

(or the corresponding vector potential A1(r)) from the

average induction B are taken into account; for ,it becomes negigibly small.

The nonmagnetic part G(4)N is given by

G(4)N = G

(4)a + G

(4)b . (39)

The term G(4)a may be calculated using the solutionsf(1), f+(1) of order 1, already obtained in subsec-tion IIIC,

G(4)a =

t

8

ND

l=0

f(1)2f+(1) + f(1)f+(1)2 + c.c..(40)

The term G(4)b requires the nonmagnetic parts f

(3)N , f

+(3)N

of the third order Greens functions f(3), f+(3),

G(4)b =

t

2

NDl=0

f

(3)N + f

+(3)N + c.c.

. (41)

The magnetic part G(4)M is given by

G(4)M = G

(4)c + G

(4)d . (42)


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9

The term G(4)c is purely magnetic in origin, while the

term G(4)d contains the magnetic parts f

(3)M , f

+(3)M of the

third order Greens functions,

G(4)c = (2 2) B21, (43)

G(4)d =

t

2

ND

l=0 f

(3)M + f

+(3)M + c.c.. (44)

In a next step, the terms B1 and f(3) = f

(3)N +f

(3)M of or-

der 2 and 3 respectively, must be calculated. The samemethod used in subsection IIIC to calculate f(1), by in-verting the differential operator on the l.h.s. of Eq. (29),may be used here to obtain f(3). Using an operator nota-

tion for brevity, the sum of f(3)N and f

(3)M may be written

as

f(3) = [l + B + k(0)r ]

1D, (45)

D = 12

f(1)f+(1) P f(1). (46)

The first and second term in Eq. (46) gives f(3)N and f

(3)M

respectively. The term P is of order 2 and is given by

P =

BB B

(2)1 k A(2)1 . (47)

The magnetic contributions B1 = B(2)1 and

A(2)1 must be

determined by solving Maxwells equation (5). Expand-ing (5) one obtains two decoupled equations

1 2

2

B1 = 0 B, (48)

r 1 22

B1 + 0 B

= , (49)

for the parallel and perpendicular component B1 andB1 of B1. The quantities

0, , which are both of order2, are given by

0 =t

B

2

2

NDl=0

g, = 2t2

NDl=0

kg. (50)

The Greens function g in Eq. (50) may be replaced by

f(1)f+(1)/2 under the k-integral. Thus, 0, may becalculated by using the first order solutions f(1), f+(1)

as given by Eq. (34). The solution of Eqs. (48, 49)

is obtained by expanding the unknown variables B1,B1 and the parameters 0, , which are all invariantunder lattice translations, in Fourier series; the corre-

sponding Fourier coefficients are denoted by ( B1)l,m =(B1)l,mey, ( B1)l,m = (B1)l,mez and (

0)l,m, l,m.The explicit solutions will be reported at the end of thissubsection.

Given the second order contribution B1, the first term

G(4)c of f

(3)M [see Eq. (43)] can be evaluated. To calculate

the second term G(4)d one needs, in addition, the correc-

tion term A1 [see Eqs. (45)-47)]. Writing A1 = A1+ A1,

the Fourier coefficients of A1, A1 may be expressed29

in terms of the Fourier coefficients of the induction,

( A1)l,m =

Q 2l,m Ql,m,x(B1)l,mez,

( A1)l,m =

Q 2l,m(B1)l,m (Ql,m,yex Ql,m,xey) ,

(51)

using the gauge conditions defined by Eq. (23). Thequantities Ql,m,x, Ql,m,y in Eq. (51) are the x and y com-

ponents of the reciprocal lattice vector Ql,m defined inappendix B.

Each one of the four terms of order 4 in Eqs. (39, 42)may be represented as a multiple integral and Matsubarasum over the product of two gap correlation functions.

What remains to be done is to perform analytically asmany integrations as possible. The details of the calcula-

tion will be reported here for the first term G(4)a , defined

by Eq. (40); the evaluation of the other three terms issimilar.

Using the first order Greens functions (34) and thedefinition of the gap correlation function (35), the term

G(4)a takes the form

G(4)a = t8NDl=0

0

ds0

ds10

ds2 es(s+s1+s2)V(r s1k, r + s2k) V(r sk, r) + V(r, r + sk) + c.c..(52)

Expanding V in a Fourier series, the spatial average in Eq. (52) may be performed and G(4)a takes the form

G(4)a = t

2

NDl=0

1

2

20

dl,m

0

ds

0

ds1

0

ds2 el(s+s1+s2)Vl,m

(s1 + s2)k

Vl,m

sk

cos

Ql,ms

2k

cos

Ql,m

s1 + s22

k

cos

B (s + s1 + s2)

+ sin

Ql,m

s1 + s22

k

sin

B (s + s1 + s2)

.

where Vl,m is given by Eq. (B3) and the symmetry relations (7) have been used to rearrange the integrand. We


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10

introduce center of mass coordinates tS = s1 + s2, tR = s1 s2 in the s1, s2-plane. Replacing s1, s2 by the newvariables, the integration over tR may be performed and the double integral over s and tS becomes a product of two

independent, one-dimensional integrals. Performing this step, G(4)a takes the form

G(4)a = tNDl=0

1

2

20

dl,m

1

Ql,mk

0

ds els coss

2Ql,mk

Vl,m

sk

0 dtS e

ltS

sin tS

2Ql,m

k Vl,m tS k cos Bs cos BtS sin Bs sin BtS .

(53)

An attempt to simplify Eq. (53) further, by performingone of the remaining integrations analytically, was notsuccessful. At this point it seems already feasible to cal-culate the remaining integrals over s, and the sums overMatsubara and Fourier indices numerically. However, weprefer to proceed and calculate the remaining integralsby means of an asymptotic approximation.

Let us consider for definiteness the integral overs in Eq. (53). The integrand has its maximum at

s = 0. We analyze the behavior of the various fac-tors in the integrand as a function of s, and ne-glect the sdependence of the slowest varying factors.The characteristic lengths in sspace of the factorsexp(ls), cos

s Ql,mk

, Vl,m(sk) and cos

Bs

are

given by 1 = [(2l + 1)t]1, 2 = (B|rl,m|)1, 3 =

(nB)1/2 and 4 = (B)

1, where rlm = la + mb. Weconsider a range of inductions B 2. Choosing a

typical number = 0.1 for the dimensionless magneticmoment, our induction varies in the range B 0 this second term depends also (since a positive nis necessarily due to a finite ) on the spin pair-breakingeffect. The GL limit of the local induction is discussedin appendix C.

The validity of the asymptotic approximation used inthe derivation of Eqs. (54,61) is not restricted to low n,but sufficiently high temperatures, say t > 0.1, should beused. Clearly, if different states with very small free en-ergy differences are found, no conclusion as to the relativestability of these states can be drawn.

F. Extremal conditions

In thermodynamic equilibrium, the values of, B, Band the lattice parameters a, b, have to be chosen insuch a way that the free energy becomes minimal. To findthe equilibrium values of , B, B the extremal condi-tions

G

= 0,

G

B= 0,

G

B= 0, (64)

have to be solved near Hc2 . The question for the optimala, b, will be addressed in the next section.

Inserting the superconducting solution for in the freeenergy yields

G = G 14

G(2)

2G(4)

, (65)

where the coefficients G G2, G4 are defined by G =G + 2G(2) + 4G(4). Eq. (65) shows, that the stablelattice structure (see section IV) is determined by therequirement of minimal G4

To find the two-component macroscopic magnetiza-

tion relation between induction B, B and external fieldH, H, the above extremal conditions must be solved

for B, B. This cannot be done for arbitrary fieldsbut requires an appropriate expansion of the coefficientsfor small B Bc2,, B Bc2,. A lengthy butstraightforward calculation, generalizing Abrikosovs clas-sical work25 to the present situation, leads to the result

B = H + H +

B = H + H + .(66)


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12

The coefficients in this linear relation are given by

= 22

2

2 2 A /detM = 2

2A/detM

= 22

2

2 2 A /detM = 2

2 2

ABc2, + ABc2,

/detM

= 2 2 2 ABc2, + ABc2, /detM,where: detM = 2

2 2 2 2 2 A A .

The parameters A, . . . may be calculated for a givenlattice structure with the help of the relations

A =1

2G(4)

G(2)

B

2(67)

A =1

2G(4)

G(2)

B

2(68)

A =1

2G(4)G(2)

B

G(2)

B

, (69)

where the derivatives of G(2) have to be evaluated atB = Bc2 and the relation AA = A

2 may be shown

to be true.Eq. (66) constitutes the macroscopic relation between

induction and external field for a 2D superconductor in atilted magnetic field. It is, of course, strongly anisotropicand shows a coupling between the parallel and perpen-dicular field components. For H = 0, 0 and t 1,Eq. (66) should reduce to Abrikosovs GL solution25,32

for the magnetization of a triangular vortex lattice. Thisis indeed the case as shown in Appendix C.

For H = 0, 0, Eq. (66) describes the ordi-nary vortex lattice (near Hc2) for arbitrary tempera-tures. A numerical comparison with corresponding re-sults by Eilenberger23 and Rammer and Pesch24 has notbeen undertaken because a different (spherical) Fermisurface has been used in these works. However, the limitH = 0, 0 of the present theory will be checked inappendix D by calculating the critical value of separat-ing type II from type I superconductivity.

IV. RESULTS FOR FINITE PERPENDICULAR

FIELD

In this section we determine the stable order param-eter structures for the paramagnetic vortex states with1 n 4 in the vicinity of Hc2. The numerical pro-cedure to find the stable states is essentially the sameas in KRS8. First, the upper critical field Bc2 and thecorresponding quantum number n have to be found forgiven temperature t and tilt angle by solving the lin-earized gap equation (37). In a second step, the stablelattice structure, which minimizes the fourth order termG(4) = G(4)/4 [see Eq. (54], has to be determined. Be-cause of the flux quantization condition the minimum

with respect to only two parameters, which may be cho-sen as a/L and , must be found. In contrast to theordinary vortex lattice, where it is usually sufficient tocalculate only a few lattices of high symmetry (triangu-lar, quadratic) to find the stable state, the present situ-ation is characterized by a large number of local minimaof Eq. (54), corresponding to a large number of possiblelattices of rather irregular shape. Therefore, a graphi-

cal method was used to determine the stable state; thefree energy surface G(4)(a/L,) was plotted for the whole(a/L,)-plane and the global minimum was determinedby inspection. Basically, two material parameters, and , and two externally controlled parameters, t and, enter the theory. Numerical calculations have beenperformed for a single value of = 0.1, two differentreduced temperatures 0.2 and 0.5, four different values0.1, 1.0, 10, 100 of Eilenbergers parameter , and severalvalues of corresponding to different Landau quantumnumbers n. Some of the resulting order parameter andmagnetic field structures in the range n 4 will be re-ported here. These low-n pairing states are, of course,

the most important ones from an experimental point ofview.

For comparison we consider first, in appendix D, theordinary vortex lattice state with n = 0. This illustratesthe method and may also be used to check the accuracyof our asymptotic approximation. The equilibrium statefor low- type II superconductors is calculated and goodagreement with previous theories is found for not too lowtemperatures.

Considering now pairing states with n > 0, the num-ber of order parameter zeros per unit cell increasesclearly with increasing n. One finds8 two types of min-

ima of

G

(4)

(a/L,), isolated minima and line-like min-ima. The first type corresponds to ordinary 2D lat-tices, the second type, characterized in a contour plot[see Fig.1 of KRS8] by a line of constant a/L withG(4)(a/L, ) nearly independent of , corresponds toquasi-one-dimensional, or FFLO-like lattices (rows ofvortices and one-dimensional FFLO-like minima alter-nating). A convenient way to identify the type ofminimum and find its position on the a/Laxis, is toplot the projection of the G(4)(a/L,)surface on the(G(4),a/L)plane. An example for this perspective,where independent parts of the free energy surfaceshow up as lines, is given in Fig 2 for n = 7. The

coordinate of a 2D minimum cannot be read off fromsuch a plot and requires a second projection on the(a/L,)plane (such as Fig 11 or Fig.1 of KRS8). Thefree energy maps for other n > 0 states are in principlesimilar to Fig 2 but the different local minima show morepronounced differences for smaller n .

Let us start with the paramagnetic vortex state withn=1 and consider first the limit of large . As reported inKRS8, a quasi-one-dimensional state is found to be sta-ble in this case. Fig 3 shows the spatial variation of themodulus of the order parameter. One sees rows of vor-tices separated by a single, FFLO-like line of vanishing


13/24

13

0.40 0.80 1.20 1.60 2.00 2.40

a/L

0.40

0.60

0.80

1.00

1.20

G(4)

FIG. 2: Projection of the free energy G(4) on theG(4),a/Lplane. Using this perspective the independentparts of the free energy surface are displayed as lines. Inthe considered range of a/L one finds six local minima, corre-sponding to two FFLO-like and four two-dimensional lattices.The global minimum is at a/L 1.1 and corresponds to atwo-dimensional lattice. Parameters chosen in this plot aren = 7, t = 0.5, = 10, = 0.1, = 0.055.

order parameter. The unit cell of the structure shown inFig 3 is given by a/L = 1.0875, = 33deg. A shift ofthe vortex rows relative to each other leads to a latticewith the same a/L and a different , which has nearlythe same free energy (which is reasonable, since the inter-action between vortices from different rows is weak as aconsequence of the intervening FFLO domain wall). Thevortices are of the ordinary type, i.e. the phase of theorder parameter changes by +2 when surrounding thecenter.

It is of interest, to calculate the magnetic field belong-ing to this order parameter structure. We plot the par-allel and perpendicular components B1(r) and B1(r)

of the spatial varying part B1(r) of the magnetic fieldas given by Eqs. (61)-(63), omitting a common factor

t|n|2

/(2

2

). The field B1(r), which is entirelydue to the spin pair-breaking mechanism, is shown inFig 4. Due to its paramagnetic nature, the field B1(r)is expelled from regions of small (r). This behavior isexactly opposite to the usual orbital response, which im-plies an enhancement of the induction in regions of small||(r). As a consequence, the spatial variation of B1 isvery similar to that of ||2, shown in Fig 3.

The perpendicular field B1(r), shown in Fig 5, con-sists of a spin term proportional to 2, and a second termwhich depends [see Eq. (63)] not explicitly on . Theterm proportional to 2 is negligibly small and the total

0

1

2

3 0

0.5

1

1.5

2

-1

0

1

0

1

2

FIG. 3: Square of modulus of order parameter |1|2 as a

function of x/a, y/a in the range 0 < x/a < 2, 0 < y/a 0,the order parameter is not periodic but changes its phaseby certain factors under translations between equivalentlattice points. These phase factors are proportional tothe perpendicular induction [cf. Eq. (35)] and vanish for


16/24

16

0. Thus, the assumption of a periodic order param-eter for = 0 is reasonable (though not stringent). Itimplies, that all allowed wave vectors in the expansion of must be vectors of a reciprocal lattice.

A further slight simplification stems from the behaviorof the quasiclassical equations under the transformation

r r, k k, which implies that the order parametermust be either even or odd under a space inversion r

r. Thus, the order parameter may be written as aninfinite sum

(r) =

m

me Qmr, (70)

with coefficients defined by

m = ||I

i=1

ci (m,ni m,ni) . (71)

Here, a shorthand notation m is used for the two inte-

gers characterizing a 2D reciprocal lattice vector Qm [cf.the Fourier expansion at the beginning of appendix B].

The vectors actually entering the expansion are distin-guished by an index i, their total number is I, and the

two integers characterizing Qni are denoted by ni. Thecomplex numbers ci are the expansion coefficients; onemay set c1 = 1 since only the relative weight is impor-tant. It turns out, that the two solutions distinguished inEq. (71) by a sign are essentially equivalent, and only oneof them, say the even one, need be considered. Thus, theorder parameter becomes a linear combination of cosinefunctions.

All reciprocal lattice vectors used in Eq. (70) must beof the same length. Denoting this length by q(T), the

condition|

Qm|

= q(T) takes the forml

a

21

sin2 p+

j

b

21

sin2 p 2 lj

ab

cos p

sin2 p= 1, (72)

where the two integers l, j have been used here to rep-resent the double index m. The dimensionless quantitiesa, b are defined by a = q(T)ap/2, b = q(T)bp/2, whereap, bp, p denote the lattice parameters in the paramag-netic limit. IfI reciprocal lattice vectors exist, the latticeparameters ap, bp, p, fulfill I relations like Eq. (72) withI pairs of integers l1, j1, . . . lI, jI.

Using Eq. (70) the free energy expansion nearHF F L O(T), including terms of fourth order in the small

amplitude ||, may be performed by means of methodssimilar to section III. The result for the purely param-agnetic free energy Gp takes the form

Gp = Gp + G(2)

p + G(4)

p , (73)

where Gp = 2H2, and G(2)p and G(4)p are contributionsof order ||2 and ||4 respectively.

The second order term is given by

G(2)p = ||2I

i=1

|ci|2A, (74)

with the iindependent coefficient A defined by

A = 2

ln t + t

0

ds1 eDs

sinh st

1 cos(Bs)J0(sq)

.

The condition A = 0 determines the upper critical field;it may also be derived from Eq. (37), performing the limitn

.

The fourth order term is given by

G(4)p = ||4 I

i=1

|ci|4Ai +I

i=k

|ci|2|ck|2Bi,k+

Ii=k

(ci )

2 (ck)2 + c.c.

Ci,k

, (75)

G(4)

p depends on the lattice structure via the coefficientsAi, Bi,k, and Ci,k, which are defined by

Ai =t

2

NDl=0

20

d

2

Pni,ni,ni(k) + 2Pni,ni,ni(k)

Bi,k =t

2

NDl=0

20

d

22

Pni,nk,nk(k) + Pni,nk,nk(k)

Ci,k =t

2

NDl=0

20

d

2Pnk,ni,nk(k),

where

Pn1,n2,n3(k) =1

Nn1Nn2N

n3

+1

N+n1N+n2N

+n3

Nn = l + B + Qnk .

In contrast to Eq. (54) no approximations have been usedin deriving Eq. (75).

Using Eq. (72) all possible 2D lattices and wave vec-tors may be calculated numerically. The stable latticeat HFFL O(T) is then determined from the condition of

lowest G(4)p , taking also the LO state into considera-tion. It turns out, that it is energetically favorable atHFFL O (T) if all eigenfunctions in the order parameterexpansion (70) have equal weight, i.e. ci = 1 for all i.

The result of the numerical search for the lowest free

energy of periodic structures, characterized by maximalthree pairs of reciprocal wave vectors, is displayed inFig 8. The highest curve at a given temperature cor-responds to the stable lattice. For 0.22 < t < 0.56 theone-dimensional LO state is realized. For t < 0.22 2Dperiodic structures appear, namely the square state for0.05 < t < 0.22, and the hexagonal state for t < 0.05 (weuse here the notation of Shimahara39 for the 2D states).Besides the fact that the triangular state39 is absent,because it is neither even nor odd, the present resultsagree quantitatively with those of Shimahara39, obtainedwithin a different, but equivalent, formalism. Thus, more


17/24

17

complicated 2D periodic structures than those found al-ready in Ref39 do not exist in the considered range oftemperatures; the assumption of equal weight for differ-ent wave vectors [ci = 1 for all i in Eq. (71)] has alsobeen confirmed for these states.

0.00

0.10

0.20

0.30

0.40

0.50

Gp(4

) /A

0.00 0.05 0.10 0.15 0.20 0.25 0.30

T/Tc

LO

square

hexa

gonal

FIG. 8: The fourth order term G(4)p (minimized with respect

to ||) divided by A2 [see Eq. (74)] at HFFLO(T) for threedifferent periodic structures as a function of reduced temper-ature t = T /Tc. The part of the hexagonal curve which islower than the LO state is not visible, since the coefficientsci are determined automatically to yield the highest possible

solution for G(4)p /A2.

The temperature region below t = 0.01 has been inves-tigated recently by Mora and Combescot37. They founda series of states characterized by an even (total) num-ber 2N = 8, 10, . . . of different wave vectors, all enteringthe order parameter expansion with equal weight, andwith N increasing with decreasing temperature. Merg-ing these results with the present ones, one obtains a verysimple description of all of the FFLO states at the phaseboundary, namely an infinite number of states, each onebeing a linear combination of N = 1, 2, . . . cosine func-tions of equal weight and with N different, but equallyspaced, wave vectors.

Of course, it is also of interest to investigate the possi-

ble equilibrium structures in the region below the criticalfield. As a first step in this direction, preliminary cal-culations at 0.95HFFL O and 0.90HFFL O have been per-formed, using the fourth order expansion (73), which isnot valid near first order transition lines. The result issurprising and shows a revival of the LO state in the lowtemperature region.

In Fig. 9 the terms G(4)p /A2, for the three states dis-played in Fig 8, are plotted as a function of temperaturebelow the transition line, at 0.9HF F L O. The hexagonalstate (not visible) does not exist any more. The usualsquare state (characterized by ci = 1) is only stable in

0.00

0.10

0.20

0.30

0.40

0.50

0.60

Gp(4

) /A

0.00 0.05 0.10 0.15 0.20 0.25

T/Tc

square

LO

FIG. 9: The term G(4)p /A

2 for the LO state and the squarestate at 0.90HFFLO(T), as a function of reduced temperature

T /Tc. The hexagonal curve is lower than the LO state and isnot displayed in this figure.

a very small temperature interval 0.061 < t < 0.075.The LO state is now stable in a much larger interval0.075 < t < 0.56, as compared to Fig 8. It is also stablein a small temperature region below t = 0.061. But at

t 0.017 the factor G(4)p /A2 for the LO state has a sin-gularity and jumps from + to . This implies thatthe fourth order term (for the LO state) changes signand that a first order transition occurs somewhere in thevicinity of this singularity; higher order terms in the freeenergy would be required for a quantitative treatment.Between this singularity at t 0.017 and the lowest con-sidered temperature t = 0.01 the stable state is againcharacterized by a square unit cell. However, the orderparameter in this temperature range, 0.01 < t < 0.017,is given by a linear combination of plane wave states [seeEqs. (70),(71)] with a real coefficient c1 = 1 and an imag-inary coefficient c2 = . The usual order parameter struc-ture for the square lattice, which is characterized by tworeal weight factors of equal magnitude (c1 = c2 = 1), isnot equivalent to this case and has higher free energy.

The results below HFFL O (T) indicate, that the 2Dstates are only stable in a tiny interval near the phaseboundary, and that the one-dimensional LO state reap-pears inside the superconducting state. The square state- the one with the smallest N (N = 2) - has the largeststability region, as one would also expect from the freeenergy balance shown in Fig 8. We shall come back to thequestion of the stability of the 2D states in section VI,considering it from a different point of view. The struc-ture found below the singular point of the LO state (seeFig. 9) raises the question, if still other order parameterstructures, different from those found at the transition


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18

line, will appear near t = 0 deep in the superconductingstate. The present fourth order expansion is not reallyappropriate to answer this question.

VI. TRANSITION TO THE PURELY

PARAMAGNETIC REGIME

The limit n of the series of paramagnetic vortexstates, discussed in section IV, is now well known; for0.22 < t < 0.56 the one-dimensional LO state is real-ized, while 2D states of square or hexagonal type, pre-dicted by Shimahara39, appear at lower t. The region ofstill smaller t, below t = 0.01, which has been studiedby Mora and Combescot37, will not be considered here.The way, this limit is approached, is, however, unknown.Thus, we address ourselves in this section to the the ques-tion of how the one- or two-dimensional unit cell of theFFLO state develops from the unit cell of the paramag-netic vortex states if the Landau level index n tends toinfinity.

This limiting process is very interesting, because avast number of different states with different symmetryis passed through in a small interval of tilt angles . Theunit cell of the finite-n states is subject to the conditionthat it carries exactly a single quantum of flux of theperpendicular field B. Since B 0 as n , atleast one of the unit cell vectors must approach infinitelength - i.e. the dimension of the macroscopic sample - inthis limit. Thus, the n limiting process describesa transition from a microscopic (or mesoscopic) lengthscale to a macroscopic length scale.

The transition to the FFLO state has previously beeninvestigated by Shimahara and Rainer4 in the linear

regime. They found the important relation

q = limn

4eBn/hc, (76)

where q is the absolute value of the FFLO wave vector(here we changed to ordinary units). Eq. (76) has beenderived by identifying the asymptotic form of the Hermitepolynomials21 with the form of the LO order parameter.It implies that a relation

B n

, =hcq2

4e(77)

holds at large n. The validity of Eq. (76) may also be

checked numerically by comparing the numbers andq, which are both obtained from the upper critical fieldequation.

Relation (77) may be derived from basic physical prop-erties of the present system. The energy spectrum forplanar Cooper pairs in a perpendicular magnetic fieldB is the same as for electrons and is given by

En = h(n +1

2), =

eBmc

. (78)

Considering now the energy spectrum of Cooper pairsfor B = 0, one has to distinguish two cases. First, in

the common situation without a large spin-pair-breakingfield, all Cooper pairs occupy the lowest possible energyE = 0, which is the kinetic energy p2/4m taken at theCooper pair momentum p = 0. Second, if a large spin-pair-breaking field parallel to the conducting plane exists,the energy value to be occupied by the Cooper pairs,shifts to a finite value p2/4m, since the Cooper pairsacquire a finite momentum p due to the Fermi level shift

discussed in section I. Thus, in the latter case, which isof interest here, the Landau levels (78) must obey thecondition

En =he

mcB(n +

1

2)

B0

p2

4m(79)

for B 0. Ifp is replaced by the wave number q = p/h,Eq. (77) becomes equivalent to Eq. (79). The limitingbehavior expressed by Eq. (76) or Eq. (77) is thereforea direct consequence of Landaus result for the energyeigenvalues of a charged particle in a magnetic field.

Combining Eq. (77) with analytical results at T = 0,

the limiting behavior of the unit cell as n may beunderstood. Expressing the FFLO wave number in termsof the BCS coherence length 0 by means of the relationq = (2/)10 , and using the flux quantization conditionin the form

F(n)B(n) = 0, (80)

[with 0 = hc/2e and B defined by Eq. (77)] the areaF(n) of the unit cell for pairing in Landau level n is ap-proximately given by

F(n).

= 320n. (81)

Thus, the unit cell area diverges with the first power ofn. The behavior of the magnetic length L , which isdefined by the relation B = (0/)L

2, is given byL

.= 0n

1/2.Eq. (81) is not sufficient to determine the shape of the

unit cell in the limit of large n. However, a simple possi-bility to produce a one-dimensional periodic LO structurefor n is a divergence of one of the unit cell lengths,say b, of the form b n, while the second length a re-mains constant, i.e. a n0. The numerical results forthe states referred to in section IV as FFLO-like, orquasi-one-dimensional states show a behavior

a

L.

=

n, (82)

which is in agreement with this possibility. The numer-ical value of the constant is close to 2

2, which cor-

responds to a = 20 and to the lattice constant /q ofthe LO state. Thus, the LO state may be identified alsthe limiting case of the quasi-one-dimensional states ofsection IV for large n; the distance of the FFLO-lines isessentially independent of n, while the periodicity lengthb sin in the direction perpendicular to the lines tends


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19

to infinity (like b sin .

= n/q) for n . The one-dimensional FFLO unit cell is a substructure that de-velops inside the diverging unit cell of the paramagneticvortex states.

To complete the description of the transition to theLO state, the above lattice structure may be used inEq. (16) to perform the limit n of the order param-eter expansion n. We consider a 2D sample of finite

area Fp, which contains NaNb small unit cells of sizeFc = ab sin . The total area is given by Fp = LaLb sin with La = Naa and Lb = Nbb. For different n the sizeand shape ofFc may change while Fp remains, of course,unchanged. Adopting the above model for the behaviorof the unit cell as a function of n, we have n-independentnumbers Na and a, while b = b

(n) increases linearly with

n and Nb = N(n)b decreases consequently according to

N(n)b =

Lbb(n)

=2Lb sin

20

1

n. (83)

Thus, a largest possible Landau number n = nc exists,

which corresponds to b(nc) = Lb (or N(nc)

b = 1) and isgiven by

nc =2Lb sin

20. (84)

This cutoff nc agrees exactly, in the present model, withthe number of n = 1 unit cells fitting into a length Lb.As an additional consequence of the finite area of thesample, only a finite number of terms occur in the sumover m in Eq. (16). This number is fixed by the conditionthat the center positions ym = mb sin = mL2/a, lieinside the sample22. This leads to the condition

aLb sin 2L2

m < aLb sin 2L2

, (85)

which is in the limit n = nc only fulfilled for m = 0.Using the asymptotic expansion21 of the Laguerre poly-nomial Ln, for large and even n = 2j, and taking intoaccount only the term with m = 0 in the sum of Eq. (16),the order parameter takes the form

2j AC2jDj cos

8j

Ly

. (86)

The amplitude A [see Eq. (15)] is, in the present systemof (ordinary) units, given by

A =

2B0L2a

1/4.

The coefficient C2j is, in the limit n nc, simply givenby C2j = (Fp)

1/2 [see Eq. (17)], and the coefficient Djtakes the form

Dj =2j (j 1)!

(1)j

2(2j 1)! .

While these factors, A, Dj , C2j diverge for n , if thesample dimensions approach infinity, all singularities can-cel if n is replaced by the cutoff nc, and one obtains theexpected result, nc = cos qy, for the one-dimensionalperiodic order parameter structure in the purely param-agnetic limit.

The transition to the two-dimensional (square andhexagonal) periodic states found by Shimahara39 is more

involved than the transition to the LO state. Let us re-strict to the square state, which is the simplest of all 2Dstates, and is also most stable from a thermodynamicpoint of view.

For the square state, which is a linear combination oftwo LO states with orthogonal wave vectors, one wouldexpect a divergent behavior of both unit cell basis vec-tors of the type a n1/2, a = b n1/2. Consequently,choosing a square unit cell in the (exact) order parameterexpansion Eq. (18), one would expect to find a substruc-ture which becomes increasingly similar, with increasingn, to the structure of Shimaharas square state (line-likeorder parameter zeros, in the form of two sets of orthog-onal straight lines and circles). Numerical calculations,performed in the range n < 40 are, however, not in agree-ment with this expectation.

On the other hand, the mathematical limit of the orderparameter (18) yields in fact a 2D state with the period-icity of the FFLO wave vector and square symmetry, asshown in appendix E for a simplified model. The ex-planation for this apparent contradiction is provided bythe result [relation (E8) of appendix E], that the quan-tum number n for a square state must obey the conditionn = N2, where N is an integer. This is a general result,which has been derived using essentially only the behav-

ior a n1/2

for large n. The latter is a consequence ofthe flux quantization condition and the shape of the unitcell.

Of course, the relation n = N2 cannot be fulfilledexactly for finite numbers n, N (for a sample of finiteextension) since is an irrational number. The propermeaning of this relation is, that the sequence of stateswith quantum numbers n = int(N2), N = 1, 2, . . . rep-resents a sequence of approximations (of increased qual-ity) to the square state. Thus, the square state is thelimit of a sequence defined on a very small subset of theset of integer numbers.

This explains, why no systematic development of the

square state with increasing n has been observed in thenumerical calculations. The largest quantum number inthe considered range (n < 40), which fulfills the abovecondition is n = 28 (corresponding to N = 3). The orderparameter modulus for n = 28 is shown in Fig. 10. Itreveals, in fact, a certain similarity to the structure ofthe square state (at least more similarity than any otherstate in the considered range). The arrangement of iso-lated order parameter zeros in Fig. 10 shows a tendencytowards the formation of line-like zeros. Clearly, an ex-tremely high n and an extremely sharp definition of thetilt angle would be required to produce a really good ap-


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20

1.0 2.0 3.0 4.0

-0.00

0.20

0.40

0.60

0.80

1.00

y/a

-0.00 0.20 0.40 0.60 0.80 1.00

x/a

FIG. 10: Contour plot of the square of the order parametermodulus for Landau quantum number n = 28 and a unit cellwith parameters a = b, = /2.

proximation to the square state. The final conclusion ofthe present analysis for the square state, that extremerequirements with regard to the definition of the tilt an-gle must be fulfilled in order to produce it, will probablyhold for all other 2D states as well.

The above analysis of the formation of the FFLOstate(s) as limit(s) of the paramagnetic vortex states forn has been based on relation (79). In addition,relation (79) allows for an intuitive understanding of theunusual phenomenon of Cooper pairing at higher n, en-countered in the present configuration. The choice n = 0for the ordinary vortex state - in the absence of param-

agnetic pair-breaking - corresponds to the lowest energythe system can achieve for p = 0. For sufficient large Hand decreasing H, the Landau level spacing becomessmaller than the kinetic energy and the system has toperform a quantum jump from the n = 0 to the n = 1pairing state, in order to fulfill the requirement of givenenergy as close as possible within the available range ofdiscrete states [Inserting n = 1 in Eq. (79) determinesthe angle 1 as given by Eq. (1)]. For the same reason, aseries of successive transitions to superconducting statesof increasing n takes place with further decreasing H,until the FFLO state is finally reached at B = 0. TheFFLO state for n

may obviously be considered as

the continuum limit, or quasiclassical limit, of this se-ries of Cooper pair states, which starts with the ordinaryvortex state at n = 0.

VII. CONCLUSION

The paramagnetic vortex states studied here, appearin a small interval of tilt angles close to the parallel ori-entation. A common feature of all of these states is afinite momentum of the superconducting pair wave func-

tion, which is due to the large parallel component of theapplied magnetic field. In these new superconductingstates the Cooper pairs occupy quantized Landau levelswith nonzero quantum numbers n. The number n in-creases with decreasing tilt angle and tends to infinityfor the parallel orientation, where the FFLO state is re-alized. The unusual occupation of higher Landau levelsmay be understood in terms of the finite momentum of

the Cooper pairs.

The end points of the infinite series of Cooper-pairwave states occupying different n are the ordinary vor-tex state at n = 0 and the FFLO state at n = . Thedominant pair-breaking mechanism in the vortex state isthe orbital effect, while Cooper pairs can only by bro-ken by means of the spin effect in the FFLO state. Theequilibrium structure of the new states, which occupy thelevels 0 < n < , is very different from the structure ofthe FFLO state(s), despite the fact, that the difference intilt angles and phase boundaries may be small. Generallyspeaking, the equilibrium structures of the new states re-

flect the presence of both pair-breaking mechanisms; thefact that the local magnetic response may be diamag-netic or paramagnetic depending on the position in theunit cell may be understood in terms of this competition.A second unusual property, also closely related to the si-multaneous presence of both pair-breaking mechanisms,is the coexistence of vortices and antivortices in a singleunit cell.

The FFLO state has been predicted in 1964 and a largenumber of experimental and theoretical works dealingwith this effect have been published since then. A def-inite experimental verification has not been achieved by

now. However, recent experiments in the organic super-conductor (BEDTT T F)2Cu(NCS)2 and other lay-ered materials40,41,42,43 revealed remarkable agreement44

with theory, both with regard to the angular- and thetemperature-dependence of the upper critical field. Inthese phase boundary experiments, identification of theFFLO precursor states, studied in the present paper,seems possible if the tilt angle is defined with high pre-cision. To obtain a more direct evidence for all of theseunconventional states, including the FFLO limit, otherexperiments, such as measurements of the local densityof states by means of a scanning tunnelling microscopewould be useful.

Acknowledgments

I would like to thank D.Rainer, Bayreuth andH.Shimahara, Hiroshima for useful discussions and help-ful comments during the initial phase of this work.


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21

APPENDIX A: SYSTEM OF UNITS AND

NOTATION

In this appendix we use primes to distinguish Eilen-bergers dimensionless quantities, which will be used insections III-V, from ordinary ones. The primes will beomitted in sections III-V.

temperature: t = T /Tclength: r

= r/R0, R0 = hvF/2kBTc = 0.882 0, 0 isthe BCS coherence length.Fermi velocity: vF = vF/vFwave number: k

= kR0Matsubara frequencies:

l = l/kBTc = (2l + 1)t

order parameter:

= /kBTcmagnetic field: H

= H/H0, where H0 = hc/2eR20

vector potential: A

= A/A0, where A0 = hc/2eR0magnetic moment:

= /0 = kBTc/mv2F, where

0 = kBTc/H0. Note that the dimensionless magnetic

moment

agrees with the quasiclassical parameter.

Gibbs free energy: G

= G/[(kBTc)2 NFR30]

Eilenbergers parameter is related to the GL-parameter0 of a clean superconductor according to the relation

=718(3)

1/20 = 0.68370.

The symbol k denotes a dimensionless, 2D unit vector.

The Fermi-surface average of a kdependent quantitya(k) is denoted by a. For our cylindrical Fermi surfacethis average is simply an integral from 0 to 2 over theazimuth angle .

a =1

4d2k a(k) =

1

2

2

0

d a(k()).

Finally, the symbol a, defined by

a = 1Fc

unit cell

d2r a(r),

denotes a spatial average of a quantity a(r) over a unitcell of area Fc.

APPENDIX B: GAP CORRELATION

FUNCTION

It is convenient to express the gap correlation function,defined by Eq. (35) in terms of center of mass coordi-

nates R = (r1 + r2)/2, r = r1 r2, using the notationVCM( R, r) = V(r1, r2). The function V

CM( R, r) is in-

variant under center of mass translations R R+ la+jband may consequently be expanded in a Fourier series,

using reciprocal lattice vectors Ql,j = l Q1 + j Q2, l , j =0, 1, 2, . . ., with basis vectors

Q1 =2

a

1

1tan

, Q2 =

2

b

01

sin

.

The Fourier coefficients of VCM( R, r) are denoted byVl,j (r). The Fourier transform of Vl,j (r) with respect

to r is denoted by V(p)

l,j (p).

Using the behavior of the gap (r) under lattice trans-

lations r r + rl,j , where rl,j = la + jb, the importantrelation

Vl,j (r) = e

l(j+ ba cos)

V0,0(r + rj,l) (B1)may be proven. This relation, first reported by Delrieu45,shows that all Fourier coefficients are known if V0,0 isknown. A similar relation holds for the Fourier transformV(p)

l,j :

V(p)

l,j (p) = eprj,ll(j+

ba cos)V

(p)0,0 (p).

The functions Vl,j and V(p)

l,j , which are most useful forthe evaluation of the free energy, may be calculated byproceeding along the chain

VCM( R, r) V0,0(r) V(p)0,0 (p) V

(p)l,j (p) Vl,j (r),

where an arrow denotes either calculation of a Fouriercoefficient, or of a Fourier transform, or application ofDelrieus relation.

Using the order parameter expansion (16) and per-

forming the necessary manipulations, the result for V(p)

l,jis given by

V(p)

l,j (p) =4

B(1)n+lj|n|2e

p2

B Ln

2

Bp 2

enba cose

FC2 (pxQl,j,ypyQl,j,x),

(B2)

where Ql,j,x, Ql,j,y are the x and y components respec-

tively of the reciprocal lattice vector Ql,j . The final resultfor Vl,j is given by

Vl,j (r) = (1)lj|n|2

enba cose

B4

G2l,jLn

B

2G2l,j

, (B3)

B2

G2l,j =

FCr 2 + xQl,j,y yQl,j,x + xl,j ,

xl,j =

sin b

al2 +

a

bj2 2lj cos

. (B4)

The usefulness of the gap correlation function for pair-wave states with with arbitrary n is essentially based onthe translational invariance of the observable quantities|| and B.

APPENDIX C: THE GINZBURG-LANDAU

LIMIT

Let us first consider the upper critical field HGLc2 , whichis determined by G(2) = 0, for = 0, n = 0 (H = 0)


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22

and t 1. Solving this equation in this limit, one finds,using ordinary units,

HGLc2 = 1.2220

220(1 t). (C1)

Eq. (C1) differs from the usual GL result by a factor of3/2. This discrepancy is due to our use of a cylindrical

Fermi surface, instead of a spherical one, and can be elim-inated by replacing the GL parameter by 3GL/2 (thequantities used in Eilenberger units are derived assuminga spherical Fermi surface).

The magnetization relation (66) takes the followingform for H = 0, = 0, t 1:

B H = 4M = H Hc222/A 1 . (C2)

The coefficient A in (C2)is given by (68). The fourthorder free energy contribution (54) takes the form

G(4) = S(1)

4

l,m

f21 (xl,m) S(1)42

l,m

f21 (xl,m) , (C3)

where S(1) = 7(3)/8. The first sum in Eq. (C3) turnsout to agree with Abrikosovs geometrical factor A,

l,m

f21 (xl,m) = A, (C4)

as discussed in more detail in KRS8. Performing againthe above replacement of one arrives at Abrikosovswell-known result

4MH

Hc2

=1

(22GL 1)A. (C5)

Eq. (C4) remains also valid for n > 0. For the non-

magnetic terms in Eq. (54), the Matsubara sum S(1)l,m

may be considered as a low temperature correction tothe GL term (C4). The GL-limit of the local magneticfield B1 [see Eq. (63)] has also been calculated and hasbeen found to obey the correct GL relation32 betweenmagnetic field and square of order parameter. Here, thelow-temperature corrections are contained in the Mat-

subara sum S(2)l,m.

APPENDIX D: THE LIMIT OF THE ORDINARY

VORTEX LATTICE

It is of interest to investigate the limit of Eq. (54) cor-responding to the ordinary vortex lattice. We considera situation without paramagnetic pair-breaking, i.e. set = 0, = /2, and ask for the equilibrium structureof the vortex lattice and the critical value of separat-ing type I from type II superconductivity. To comparewith the usual notation, we use here the same scaling

2/3 of the GL parameter as in appendix C. Fig-ure 11 shows the free energy G(4) as a function of a/L, for = 1.46 ( = 1.5) at t = 0.5. The flat minimumof G(4) at a/L = 1.905, = 60 indicates that the stableconfiguration is, as expected, a triangular vortex lattice.No other local minimum of the free energy exists. Withdecreasing this minimum changes quickly into a maxi-mum; below

.= 1.36 the free energy has no minimum at

all which means that no spatially varying superconduct-ing state exists. The critical value of

.= 1.36 separating

type I from type II behavior at t = 0.5 agrees fairly wellwith the result of

.= 1.25 obtained by Kramer46 for

the phase boundary between typ II and type II/1 behav-ior. For lower temperature the agreement is worse; att = 0.2 the present theory gives = 2.5 while Kramerstheory46 gives = 1.7. Recall that the error inducedby the asymptotic approximation of subsection IIID in-creases with decreasing temperature.

0.507 0.508 0.509 0.510 0.511

40.00

50.00

60.00

70.00

80.00

[deg]

1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30

a/L

FIG. 11: Contour plot of the free energy G(4) as a function ofa/L and without paramagnetic pair-breaking. Parametersare = 1.5, t = 0.5, = 0, = /2 . The minimal value

G(4) = 0.5067 is at a/L = 1.905, = 60.

APPENDIX E: THE SQUARE LIMIT FOR A

MODEL ORDER PARAMETER

The square of the the order parameter modulus,Eq. (18), for a square lattice may be written in the form

|n|2(x,y,a) =l,j

Hl,j , (E1)

Hl,j = (1)lj e2 (l2+j2)Ln((l

2 +j2))e2a (lx+jy).

We are interested in the limiting behavior of Eq. (E1) for

n , a n 12 . In this limit, the quantity 2/a


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23

tends to zero and the double sum may be approximatedby a double integral. An appropriate tool to performsuch a calculation for infinite sums in a systematic wayis Poissons summation formula. Using a two-dimensionalversion, which is derived in exactly the same way as forsingle sums, Eq. (E1) may be written in the form

|n|2

(x,y,a) = a

22

mx

my

dk

x

dk

ye(mxk

x+myk

y)h(kx, x , ky, y , a),

(E2)

where h(kx, x , ky, y , a) is a function representing Hl,j .The problem here is the factor (1)lj [see Eq. (E1)],which must be represented by an infinite series of stepfunctions47.

Since we are more interested in the question if a limitwith the correct periodicity and symmetry exists, thanin the detailed functional form of this limit, we repre-sent the factor (1)lj approximately by the real part ofexpa2kxky/4, i.e. we use the function

h(kx, x , ky, y , a) = Hl,j

l= a

2kx, j=

a2

ky(E3)

to represent Hl,j . Using this model, the absolute valueof the r.h.s. of Eq. (E2) will be denoted by S(x,y,a)instead of |n|2(x,y,a). It takes the form

S(x,y,a) = a

2

2 mx

my

dkx

dky

h(kx, x mxa, ky, y mya, a)

,

(E4)

where h(kx, x , ky, y , a) is given by

h(kx, x , ky, y , a) = cos

a2

4kxky

e

a2

8 (k2

y+k2

y)

Ln

a2

4

k2y + k

2y

e(kxx+kyy).

In order to perform the integrations, the relation

Ln(x+y) =1

(1)n22nn!n

m=0

n

m

H2m(

x)H2n2m(

y)

may be used to rewrite the Laguerre polynomial in theintegrand as a sum of products depending on kx andky separately. Then, the integration over kx may beperformed21 and, after a simple shift of the integrationvariable, a second relation48

(2)nHn( x + y2

)Hn(x y

2) =

nm=0

(1)m

n

m

H2m(

x)H2n2m(

y)

may be used to calculate the sum over m. Performingthe integration over kx one obtains the final result

S(x,y,a) = n

22nn!

mx

my

e

a2x2Hn

2

ax

e

a2y2

Hn

2

a

ye 2a2 xy + (1)ne

2

a2xy,

(E5)

where the abbreviations x = x mxa, y = y mya havebeen used.

We are interested in the limiting value of Eq. (E5)for n . The asymptotic behavior of the Hermitepolynomials21 for large n implies

Hn

2

ax

cos

(2n + 1)1/2

2

ax

(2n + 1)1/22mx.(E6)

Since a n1/2, the factor in front of x in Eq. (E6) re-mains finite for n and defines the FFLO wave vectorq, i.e.

(2n + 1)1/2

2

a= q, for n . (E7)

Eq. (E7) implies a restriction on the possible quantumnumbers n of a square FFLO state. The fact that aninteger number N of wave lengths = 2/q must fit into

a length a, implies the condition

n = N2 (E8)

in the limit a . Condition (E8) is of a general na-ture and not a specific feature of our model. For quan-tum numbers n obeying Eq. (E8), all mx, mydependentphase factors in Eq. (E6) become multiples of 2 and maybe omitted. The limit of S(x,y,a) for a obtainedin this way is well-defined, i.e. independent of any cutoff,and is given by

limnS(x,y,a) | cos(qx) cos(qy)|. (E9)

A rotation of /4 transforms Eq. (E9) into the more fa-miliar form39 | cos(qx) + cos(qy)|. The expected cor-rect result for |n|2 is the square of the r.h.s. of Eq. (E9).Thus, the result of our model calculation differs from theexact result. A limiting state of the correct periodicityand symmetry has, however, been obtained.


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24

[email protected] P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964).

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