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Structural transitions of the vortex lattice in anisotropic superconductors and fingering instability of electron drople...

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STR UCTURAL TRANSITIONS OF THE V OR TEX LA TTICE IN ANISOTR OPIC SUPER CONDUCTORS AND FINGERING INST ABILITY OF ELECTR ON DR OPLETS IN AN INHOMOGENEOUS MA GNETIC FIELD By ALEXIOS KLIR ONOMOS A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2003

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A CKNO WLEDGMENTS First and foremost I w ould lik e to thank Professor Alan T. Dorsey for his kind and motiv ating men torship. His dedication, abilit y and encouragemen t ha v e b een inspiring in dicult times. I w ould also lik e to extend m y gratitude to m y immediate family and to m y close and dear friends for their supp ort and tolerance. ii

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T ABLE OF CONTENTS A CKNO WLEDGMENTS . . . . . . . . . . . . ii ABSTRA CT . . . . . . . . . . . . . . . v CHAPTERS 1 SUPER CONDUCTIVITY . . . . . . . . . . . 1 1.1 Con v en tional Sup erconductivit y . . . . . . . . . . . 2 1.2 Ginzburg-Landau Theory . . . . . . . . . . . . . 7 1.3 London Mo del . . . . . . . . . . . . . . . . 13 1.4 Anisotrop y and Structural Phase T ransitions . . . . . . . 15 1.5 Elasticit y and Melting of the V ortex Lattice . . . . . . . 18 2 GINZBUR G-LAND A U THEOR Y . . . . . . . . . 20 2.1 Anisotropic Ginzburg-Landau Theory . . . . . . . . . 20 2.2 Virial Theorem . . . . . . . . . . . . . . . 25 3 MEAN FIELD THEOR Y . . . . . . . . . . . 28 3.1 Solution of the First Ginzburg-Landau Equation . . . . . . 28 3.1.1 Landau Gauge . . . . . . . . . . . . . 29 3.1.2 Symmetric Gauge . . . . . . . . . . . . . 31 3.2 Structure of the V ortex Lattice and Minimization of the F ree Energy 35 4 ELASTICITY THEOR Y F OR THE V OR TEX LA TTICE . . . . 39 5 STR UCTURAL PHASE TRANSITIONS . . . . . . . 47 5.1 Nondisp ersiv e Elastic Mo duli . . . . . . . . . . . 47 5.2 Fluctuations . . . . . . . . . . . . . . . . 50 6 FINGERING INST ABILITY OF ELECTR ON DR OPLETS . . . 56 6.1 Man y-b o dy W a v efunction . . . . . . . . . . . . 58 6.2 Conformal Mapping Metho d . . . . . . . . . . . . 61 6.3 Mon te Carlo Sim ulation . . . . . . . . . . . . . 64 7 CONCLUSION . . . . . . . . . . . . . 75 A EXTENSION OF THE VIRIAL THEOREM . . . . . . 77 A.1 A Useful Iden tit y . . . . . . . . . . . . . . . 79 A.2 Generalized Abrik oso v Iden tities . . . . . . . . . . . 79 A.3 F ree Energy Magnetization, Gibbs Energy . . . . . . . . 81 iii

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B CALCULA TION OF f A h t ( 2 X )Tj/T1_168 11.9552 Tf0.0194 Tc 11.9894 0 Td( 2 Y ) 2 t i hj t j 2 t ( 2 X )Tj/T1_168 11.9552 Tf0.0195 Tc 11.8693 0 Td( 2 Y ) 2 t i . . 83 C MOMENTS . . . . . . . . . . . . . . 86 C.1 Conformal Map z = f ( w ) = aw + bw )Tj/T1_170 7.9701 Tf12.4874 Tc 6.5992 0 Td(M . . . . . . . . 86 C.1.1 Exterior Momen ts . . . . . . . . . . . . . 87 C.1.2 In terior Momen ts . . . . . . . . . . . . . 88 C.2 Conformal Map z = f ( w ) = aw + P M k =1 b k w )Tj/T1_170 7.9701 Tf8.0414 Tc 6.5992 0 Td(k . . . . . . . 90 C.3 Conformal Map z = f ( w ) = r w + Q= ( w )Tj/T1_169 11.9552 Tf0 Tc 11.9894 0 Td(w 0 ) + Q= ( w + w 0 ) . . . 90 C.4 Connection with Magnetic Field Inhomogeneit y . . . . . . 91 C.4.1 Exterior Momen ts . . . . . . . . . . . . . 92 C.4.2 P oten tial V ( z ). . . . . . . . . . . . . . 93 D CODE F OR THE NUMERICAL CALCULA TIONS . . . . . 97 D.1 driv er1.m . . . . . . . . . . . . . . . . . 98 D.2 energy .m . . . . . . . . . . . . . . . . . 99 D.3 mo duli.m . . . . . . . . . . . . . . . . . 109 D.4 driv er2.m . . . . . . . . . . . . . . . . . 120 D.5 csquash.m . . . . . . . . . . . . . . . . 121 E MONTE CARLO CODE . . . . . . . . . . . 132 REFERENCES . . . . . . . . . . . . . . 160 BIOGRAPHICAL SKETCH . . . . . . . . . . . 167 iv

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Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y STR UCTURAL TRANSITIONS OF THE V OR TEX LA TTICE IN ANISOTR OPIC SUPER CONDUCTORS AND FINGERING INST ABILITY OF ELECTR ON DR OPLETS IN AN INHOMOGENEOUS MA GNETIC FIELD By Alexios Klironomos Ma y 2003 Chairman: Alan T. Dorsey Ma jor Departmen t: Ph ysics I presen t a deriv ation of the nondisp ersiv e elastic mo duli for the v ortex lattice within the anisotropic Ginzburg-Landau mo del. I deriv e an extension of the virial theorem for sup erconductivit y for anisotropic sup erconductors, with the anisotrop y arising from s d mixing or an anisotropic F ermi surface. The structural transition from rhom bic to square v ortex lattice is studied within this mo del along with the eects of thermal ructuations on the structural transition. The reen tran t transition from square to rhom bic v ortex lattice for high elds and the instabilit y with resp ect to rigid rotations of the v ortex lattice, predicted b y calculations within the nonlo cal London mo del, are also presen t in the anisotropic Ginzburg-Landau mo del. I also study the ngering of an electron droplet in a sp ecial Quan tum Hall regime, where electrostatic forces are w eak. P erforming Mon te Carlo sim ulations I study the gro wth and ngering of the electron droplet in an inhomogeneous magnetic eld as the n um b er of electrons is increased. I expand on recen t theoretical results and nd excellen t agreemen t b et w een m y sim ulations and the theoretical predictions. v

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CHAPTER 1 SUPER CONDUCTIVITY The fascinating phenomenon of sup erconductivit y although disco v ered in 1911 b y H. Kamerlingh Onnes, has in the last t w o decades b egun to pla y a v ery imp ortan t role in ev eryda y life through tec hnology This tec hnological rev olution w as ev en more apparen t to scien tists and la ymen alik e after the disco v ery of high temp erature sup erconductors (HTSCs) b y Bednorz and M uller in 1986. The theory of con v en tional sup erconductivit y is w ell understo o d to da y There exists an accepted microscopic theory the so called BCS theory whic h explains adequately the b eha vior of con v en tional or isotropic, lo w temp erature sup erconductors lik e Pb, Al, Nb, Nb 3 Sn. Ho w ev er, HTSCs ha v e b een pro v en dicult b easts to tame. No microscopic theory with the degree of success of BCS in describing the relev an t phenomena exists as of to da y There are man y phenomenological mo dels that describ e the phenomena in b oth kinds of sup erconductors with v arious degrees of success. One v ery complicated and extremely imp ortan t issue for tec hnological applications is the morphology and prop erties of the v ortex lattice in t yp e-I I sup erconductors. The v ortex lattice is an arrangemen t of magnetic lamen ts inside the bulk of the sup erconductor whic h en ter under certain conditions when the sup erconductor is placed inside a magnetic eld. The distinction b et w een t yp e-I and t yp e-I I sup erconductors will b e claried later in this in tro ductory surv ey In this w ork I will use one particular phenomenological mo del, the GinzburgLandau theory to study the elastic prop erties of the v ortex lattice in anisotropic sup erconductors|in other w ords, the resp onse of the v ortex lattice to deformations. I also address the issue of structural phase transitions of the v ortex lattice and the eects of thermal ructuations. 1

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2 The surv ey of sup erconductivit y that follo ws will attempt to dene and explain the basic concepts required and th us pro vide the theoretical bac kground for the understanding of the main b o dy of the researc h that is rep orted here. F or a more detailed in tro duction to sup erconductivit y the reader can refer to v arious sources on the sub ject [1{3]. 1.1 Con v en tional Sup erconductivit y The t w o distinctiv e phenomena asso ciated with sup erconductivit y are the disapp earance of resistance at some critical temp erature T c (p erfect conductivit y) and the expulsion of magnetic elds from the bulk of a metal once it en ters the sup erconducting phase (p erfect diamagnetism or Meissner eect). Con v ersely the application of a sucien tly strong magnetic eld is found to destro y sup erconductivit y There exists a classication sc heme for sup erconductors according to their b eha vior when an external magnetic eld is applied. F or t yp e-I sup erconductors, the magnetic eld cannot en ter the bulk of the material un til the so-called thermo dynamic critical eld H c ( T ) is applied. The thermo dynamic critical magnetic eld is related to the free energy dierence b et w een the normal and the sup erconducting phases. F or t yp e-I I sup erconductors the magnetic eld p enetration b egins at a stronger eld, called H c 2 ( T ), and p ersists do wn to a smaller critical eld, H c 1 ( T ), with H c 2 > H c > H c 1 The c haracteristic b eha vior of eac h t yp e of sup erconductor when a magnetic eld is applied is illustrated in gure (1.1). T yp e-I I sup erconductors and esp ecially the in termediate phase, also called the mixed phase or Sh ubnik o v phase, are of in terest to me. In the mixed phase the magnetic eld en ters the bulk of the material in the form of rux tub es, eac h carrying a quan tum of magnetic rux 0 = hc 2 e = 2 : 07 10 7 gauss-cm 2 : (1.1)

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3 B Type II Type I 0 H PSfrag replacemen ts H c 1 H c H c 2 Figure 1.1: Flux p enetration b eha vior for t yp e-I and t yp e-I I sup erconductors with the same thermo dynamic critical eld H c F or most of the region b et w een the t w o critical elds H c 1 ( T ) and H c 2 ( T ), the rux tub es form a regular arra y that in a sense resem bles the crystal lattice of ordinary crystals. The ma jor dierence is that the v ortex lattice is essen tially t w o dimensional as the rux tub es can b e considered as straigh t lines whic h, ho w ev er, can b end and deform. The existence of magnetic elds inside the sup erconductor and sup erconductivit y ma y at rst seem to b e t w o m utually exclusiv e ev en ts. Ho w ev er, the magnetic tub es ha v e a v ortex of sup ercurren t swirling around them whic h screens the magnetic eld from the rest of the bulk. Sup erconductivit y is completely destro y ed in the core of the magnetic rux tub e where the magnetic eld attains its largest v alue. The in teractions b et w een the v ortex lines are repulsiv e. This in teraction, balanced b y the external magnetic eld pressure, stabilizes the v ortex lattice. Alexei

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4 Abrik oso v, in a seminal w ork [4], rst studied and predicted the existence of the v ortex lattice in con v en tional, t yp e-I I sup erconductors. There are t w o v ery imp ortan t length scales asso ciated with sup erconductivit y The p enetration length determines the electromagnetic resp onse of the sup erconductor pro viding a measure of the spatial exten t of the magnetic eld in the bulk of the material. The coherence length pro vides a measure of the size of the normal core. The coherence length, is also the c haracteristic lengthscale asso ciated with v ariations of the sup erconducting phase in more general situations. It can b e sho wn that H c 1 0 = 4 2 and H c 2 0 = 2 2 In addition, = = is a dimensionless quan tit y appro ximately indep enden t of temp erature, called the Ginzburg-Landau parameter. F or a t ypical elemen tal sup erconductor 500 A and 3000 A whic h giv es a Ginzburg-Landau parameter 1. The v alue of the Ginzburg-Landau parameter facilitates the c haracterization of a material as t yp e-I or t yp e-I I. Actually = 1 = p 2 serv es as the b oundary v alue b et w een the t w o regimes. F or smaller v alues one has a t yp e-I sup erconductor while for larger v alues a t yp e-I I. T ypical HTSCs are extreme t yp e-I I sup erconductors with of the order of 10 2 There are sev eral metho ds that one can use to observ e and study the static prop erties and the morphology of the v ortex lattice. Small angle neutron scattering (SANS) [5{14], Bitter decoration [15, 16], scanning tunneling microscop y (STM) [17, 18], magneto-optical metho ds [19] and m uon spin rotation metho ds ( SR) [20] are the ones widely used. Of all these metho ds, small angle neutron scattering stands out as the most useful to mak e bulk measuremen ts as the ma jorit y of the men tioned metho ds are limited to imaging the rux lines at the surface of the sup erconductor. 1 In short, the SANS metho d tak es adv an tage of the w eak in teraction of neutrons with mat1 One can prob e the bulk of the sup erconductor using SR. Ho w ev er, sophisticated mo deling is necessary to in terpret the data.

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5 ter through short range n uclear forces and dip ole in teractions with spatially v arying magnetic elds. The neutrons are scattered from the bulk of the material rev ealing information ab out crystal structure, the v ortex lattice and magnetic ordering. More sp ecically from the diraction pattern one can deduce the symmetry of the v ortex lattice. Measuring the scattered in tensit y enables one to determine the c haracteristic length scales and and the eld prole around the rux lines, whic h in turn allo ws the determination of the critical elds. Also, from the width of the diraction p eaks one can estimate the v ortex lattice correlation lengths. The existence of this new sup erconducting phase and the phase transition whic h accompanies it leads naturally to the idea of the order parameter. Although prop osed b y Ginzburg and Landau [21] b efore the disco v ery of the BCS theory the order parameter can b e though t of as the w a v efunction for the cen ter of mass motion of the Co op er pairs. I will dev ote a separate section to the study of the Ginzburg-Landau theory F or t yp e-I I sup erconductors the mean eld H { T phase diagram is rather simple, as illustrated in gure (1.2). The c haracterization \mean eld" implies that ructuation eects are not considered, whic h is a drastic simplication. Although this dissertation deals with mainly static prop erties of the v ortex lattice, I will address here some of the dynamic prop erties of the v ortex system to un v eil the complexit y that emerges once one c ho oses to depart from the simple (but nonetheless imp ortan t) approac h of mean eld theory F or an excellen t extensiv e review the in terested reader migh t refer to the classic w ork of Blatter and others [22]. F or a more p edagogical in tro duction t w o other nice reviews are a v ailable [23, 24] When an external curren t densit y is applied to the v ortex system, the rux lines will start to mo v e under the inruence of the Loren tz force F = j B =c In a homogeneous system the only resistance to the motion is pro vided b y the friction force, whic h is prop ortional to the steady state v elo cit y v of the v ortex system. The

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6 0 T H Mixed Phase Meissner Phase Normal Metal PSfrag replacemen ts H c 1 ( T ) H c 2 ( T ) T c Figure 1.2: The H { T phase diagram of a t yp e-I I sup erconductor. A t high elds and temp eratures the material is in the normal metallic phase. A t lo w temp eratures and elds the material is in the Meissner phase while for in termediate temp eratures and elds the material is in the mixed phase. steady state v elo cit y will then b e: v = j B =c where is a generalized friction co ecien t. Ho w ev er, as the rux tub es mo v e, an electric eld E = B v =c app ears parallel to the external curren t densit y With the curren t densit y j and the electric E in parallel there is p o w er dissipated in the system whic h implies that the fundamen tal prop ert y of dissipation-free curren t ro w in a sup erconductor is lost. Disorder comes to the rescue, pinning the rux lines and restoring dissipation-free curren t ro w. Of course there is a critical depinning curren t densit y j c whic h is alw a ys b ounded ab o v e b y the depairing curren t densit y j 0 The implications in a tec hnological con text are ob vious.

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7 Another kind of disorder is thermal ructuations. Quenc hed disorder as describ ed ab o v e is static while thermal ructuations are dynamic. Considering thermal ructuations op ens up the p ossibilit y for melting of the v ortex solid in to a v ortex liquid or the emergence of a glass phase. It is apparen t that quenc hed disorder and thermal ructuations are comp eting forms of disorder. Thermally activ ated jumps o v er the pinning barriers can cause the v ortices to mo v e, a phenomenon called rux creep, while thermal ructuations of a single v ortex line lead to a dynamical sampling and a v eraging of the conning p oten tial arising from disorder, whic h in turn reduce the critical curren t densit y j c All these eects mo dify the mean eld phase diagram considerably and at the same time enric h and complicate the ph ysics in v olv ed. In high temp erature sup erconductors the eects of the ructuations are more pronounced, due to w eak er pinning of the v ortices and enhanced thermal ructuations. This b ecomes immediately apparen t since the t ypical elemen tal sup erconductor has a critical temp erature T c 10 K compared to T c 100 K of a HTSC. As a consequence larger parts of the phase diagram con tain melted or depinned v ortex lattice, and also new exotic phases, suc h as en tangled or disen tangled v ortex liquids, along with a zo o of v ortex glasses, are p ossible. As the reader can infer from this limited in tro duction, the ph ysics asso ciated with the study of sup erconductivit y and the m ultitude of eects and phenomena it encompasses is dicult but the researc h is extremely fascinating and rew arding at the same time. 1.2 Ginzburg-Landau Theory The Ginzburg-Landau theory for sup erconductivit y [21] is an extension of Landau's theory of second order phase transitions to a spatially v arying complex order parameter ( r ). Originally prop osed w ell b efore the disco v ery of BCS theory it has b een pro v en an indisp ensable to ol for the phenomenological treatmen t of a v ariet y

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8 of problems in man y elds. Although conceptually simple, it is an elegan t theory of considerable predictiv e p o w er. The Ginzburg-Landau theory has initially a somewhat limited region of applicabilit y It is v alid near the transition temp erature T c Ho w ev er, this restriction can b e relaxed using v arious argumen ts or tec hniques, or ev en ignored for the sak e of making qualitativ e predictions. Tw o elds are necessary for the form ulation of the theory One is the complex order parameter ( r ) = f ( r ) e i ( r ) and the other is the v ector p oten tial A ( r ), related to the lo cal magnetic eld B ( r ) = r A ( r ). The order parameter is zero at the transition p oin t and has a nonzero v alue when the material is in the sup erconducting state. Based on the con tin uit y of the c hange of state in a phase transition of the second kind, an expansion of the free energy in terms of the order parameter is feasible. The Ginzburg-Landau free energy functional F [ ; A ] is built using symmetry argumen ts. Odd p o w ers of are immediately excluded as the free energy densit y (an observ able) should b e in v arian t under a transformation of the form e i ( x ) Another simplication one mak es is the assumption that the crystal has cubic symmetry The spatial v ariation of the order parameter then dictates the inclusion of gradien t terms in the free energy functional. First order deriv ativ es and second order deriv ativ es of the form @ 2 =@ x i @ x k are excluded b ecause these are essen tially surface terms. The lo w est order acceptable gradien t term is of the form j r j 2 whic h has to b e made gauge in v arian t b y com bining it with the v ector p oten tial resulting in j j 2 = j ( i r e A =c ) j 2 T aking in to accoun t the ab o v e considerations, the Ginzburg-Landau free energy functional can b e written no w in the follo wing form: F [ ; A ] = F n + Z d 3 x a j 2 j + b 2 j j 4 +2 2 m j j 2 + B 2 8 : (1.2)

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9 Here F n denotes the free energy of the normal state and b is a p ositiv e co ecien t without an y temp erature dep endence. On the other hand a is a function of the temp erature giv en b y a = a 0 ( T T c ) ; (1.3) with the constan t a 0 b eing p ositiv e. The constan ts e and m can b e iden tied with the c harge and the mass of a Co op er pair, namely e = 2 e and m = 2 m where e and m are resp ectiv ely the c harge and the mass of the bare electron. T o elucidate a few fundamen tal concepts I consider rst the case of a homogeneous sup erconductor with no external eld. The order parameter is then indep enden t of the co ordinates. The minim um of the free energy densit y o ccurs when j j 2 = a 0 b ( T c T ) : (1.4) One sees that the square of the order parameter, whic h is prop ortional to the sup erconducting electron densit y decreases linearly to zero at the transition p oin t. The dierence in the free energies of the sup erconductor and normal metal is found to b e F s F n = V a 20 2 b ( T c T ) 2 : (1.5) The dierence in the energies can b e equated to V H 2 c = 8 whic h is the energy densit y for the critical magnetic eld. This immediately yields the temp erature dep endence of the critical eld near the transition p oin t H c = 4 a 20 b 1 2 ( T c T ) : (1.6) Dieren tiating the dierence of the free energies with resp ect to the temp erature one obtains the dierence in the en tropies and nally the jump of the sp ecic heat at the

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10 transition p oin t C s C n = V a 20 T c b : (1.7) The reader should b e cautioned that suc h trivial manipulations are only p ossible in the simplest of cases. The solution of the Ginzburg-Landau equations, the equations of motion deriv ed from the Ginzburg-Landau free energy is a dicult task as will b e made apparen t shortly An in teresting observ ation is that only the mo dulus of the order parameter f ( r ) = j ( r ) j and the gauge in v arian t sup erv elo cit y Q = ( 0 = 2 m ) r ( r ) e A ( r ) =m c en ter the Ginzburg-Landau theory but not and A separately [25]. There is a completely equiv alen t form ulation of the Ginzburg-Landau in terms of Q and f implying that since only real and gauge in v arian t quan tities are in v olv ed, an y calculations and results (ev en appro ximations) should con tain gauge in v arian t equations. One can deriv e the equations of motion for the t w o elds and A minimizing the Ginzburg-Landau free energy functional with resp ect to and A One obtains2 2 m i r e c A 2 + a + b j j 2 = 0 (1.8) j = ie 2 m ( r r ) : (1.9) In the pro cess of deriving the equations of motion a b oundary condition has to b e imp osed to ensure that the surface in tegrals in the v ariation F are zero. With n b eing the normal v ector to the surface of the sup erconductor the b oundary condition has the form n i r e c A = 0 : (1.10)

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11 An immediate result is that the normal comp onen t of the sup ercurren t v anishes at the b oundary as exp ected: n j = 0, since one assumes that no curren t is fed in to the material. It can b e sho wn that the c haracteristic lengths and ha v e the follo wing forms within the Ginzburg-Landau theory: ( T ) = m a 0 ( T c T ) 1 2 ; (1.11) ( T ) = m c 2 b 4 e 2 a 0 ( T c T ) 1 2 : (1.12) In addition, quite generally one can include an electric eld and time dep endence through the assumption @ @ t + ie = r F ; (1.13) with b eing the scalar p oten tial and r b eing a constan t [26{28]. One then obtains the so called time dep enden t Ginzburg-Landau equations. A t this p oin t I ha v e to stress that the v alidit y criteria for the time dep enden t v ersion are dieren t from the simple Ginzburg-Landau equations. The in terested reader should consult the appropriate literature on the sub ject. The theory can also b e extended to sup erconductors with crystal symmetry other than cubic with the in tro duction of an anisotropic eectiv e mass tensor that m ultiplies the gradien t term in the free energy densit y [29, 30]. This sub ject will b e discussed extensiv ely along with other w a ys to accoun t for anisotrop y later in this w ork. The full solution of the Ginzburg-Landau equations, except in trivial cases, is p ossible only with the use of n umerical metho ds [31{34] since it is a nonlinear dieren tial equation. Ho w ev er, useful appro ximations can b e made whic h pro duce v aluable and accurate results. One frequen tly emplo y ed is the assumption that the nonlinear

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12 term j j 2 in the rst Ginzburg-Landau equation is small enough to b e neglected. This assumption w orks w ell in man y cases but should b e used with appropriate care. One case for whic h this term requires sp ecial treatmen t is in the calculation of the disp ersiv e elastic mo duli of the v ortex lattice, c 11 and c 44 where it in tro duces unph ysical div ergences that are remo v ed as so on as this term is included and prop erly treated [35, 36]. A w ell b eha v ed solution, in the symmetric gauge, that represen ts the v ortex lattice near H c 2 with one rux quan tum p er rux line w as found b y Brandt [35{40]: ( x; y ) = (constan t) exp B 2 o ( x 2 + y 2 ) Y ( x x ) + i ( y y ) : (1.14) Here the pro duct is o v er the (p erio dic) rux line p ositions ( x ; y ). Notice the similarit y with the Laughlin w a v efunction for a lled ( = 1) Landau lev el within the con text of the Quan tum Hall eect (see Chapter 6 later in this thesis). This particular solution can b e used to calculate the energy of structural defects in the v ortex lattice [39{41] and also the elastic mo duli [25, 35{38, 42] of the v ortex lattice. One needs to dene one v ery imp ortan t parameter here, the Abrik oso v parameter A It is the ratio A = hj j 4 i hj j 2 i 2 ; (1.15) where the brac k ets denote spatial a v eraging. By means of the Sc h w artz inequalit y the range of the Abrik oso v parameter is [1 ; 1 ). In the mixed phase the Abrik oso v parameter is a geometrical constan t, attaining dieren t v alues for dieren t t yp es of v ortex lattices. It is easily sho wn that for an isotropic sup erconductor the minimization of the Ginzburg-Landau free energy is equiv alen t to the minimization of the Abrik oso v parameter A F or example, for a triangular lattice A 1 : 1659, while for a square lattice A 1 : 1803, th us making the triangular lattice the lo w est energy conguration of the v ortices in an isotropic sup erconductor. One notable detail is the small

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13 dierence of the t w o v alues. Initially Abrik oso v obtained the incorrect result, namely the square lattice b eing the equilibrium conguration. This small dierence mak es the existence of structural transitions v ery plausible. F or an anisotropic sup erconductor the situation is more complicated and more in teresting b ecause the anisotrop y op ens up the p ossibilit y of phase transitions from the triangular phase to ev en more general rhom bic phases and ev en tually to the square phase under certain conditions. 1.3 London Mo del The Ginzburg-Landau theory b eing an expansion in p o w ers of the order parameter whic h is assumed to b e of small magnitude, is v alid near H c 2 Near H c 1 on the other hand, where the v ortex core o v erlap is not signican t one can put j ( r ) j = 1 outside the v ortex core. This is the starting p oin t for the form ulation of the London theory Either starting from the simplied Ginzburg-Landau free energy densit y or from rst principles one obtains the free energy functional of the London theory F [ B ] = 1 8 Z d 3 x B 2 + 2 ( r B ) 2 : (1.16) Here = [ m c 2 = 4 n S e 2 ] 1 2 is the London p enetration depth, with n S b eing the densit y of the sup erconducting electrons. In this section B B ( r ) is the lo cal magnetic eld. Minimizing (1.16) with resp ect to the lo cal magnetic eld B ( r ) using the fundamen tal relation r B = 0 and also adding the con tribution of the v ortex core in the form of function singularities one obtains the London equation for an isolated v ortex 2 r 2 B + B = 0 ( r ) : (1.17) Here 0 is a unit v ector along the rux line direction. The function appro ximates the core region whic h is of nite exten t (of the order of ). The solution of the equation

PAGE 19

14 is easy to obtain for a eld directed along the z axis, B = 0 2 2 K 0 r ; (1.18) where K 0 ( r = ) is the zeroth order mo died Bessel function of an imaginary argumen t. The asymptotic forms of the solution are more useful: B = 0 2 2 ln r + (constan t) for < r B = 0 2 2 r 2 r e r = for r The same metho d can b e used for t w o parallel v ortices directed along the z axis. One simply adds one more singularit y in the righ t hand side of equation (1.17) and the solution is obtained in a similar manner as b efore. One obtains for the magnetic eld of the i -th v ortex B i = 0 2 2 K 0 r r i ; (1.19) where r i is the p osition of the i -th v ortex. Substituting bac k in the London free energy densit y (1.16) after a short calculation one nds for the in teraction energy p er length of the t w o v ortices U int = 0 B 12 = 4 where B 12 is giv en b y B 12 = 0 2 2 K 0 r 1 r 2 : (1.20) Th us the in teraction b et w een t w o v ortices is repulsiv e. It decreases exp onen tially at large distances and div erges logarithmically at short distances. The London theory has b een successfully used in the in v estigation of the properties of the v ortex lattice. Actually it is relativ ely easier to obtain the elastic mo duli in the London limit than it is using the Ginzburg-Landau theory What is usually done is that some prop ert y is calculated in the high eld limit and in the lo w eld

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15 limit and nally the appropriate matc hing is done in the in termediate region in whic h b oth London and Ginzburg-Landau theory fail. Although in this w ork the Ginzburg-Landau theory is the main to ol used in the in v estigation, I use man y results obtained from the London theory 1.4 Anisotrop y and Structural Phase T ransitions There are man y sources of anisotrop y; all are found to mo dify fundamen tal prop erties suc h as the shap e of the v ortex cores, the in teractions b et w een the v ortices, the elastic resp onse and also the symmetry of the v ortex lattice. The mec hanisms are complicated and sometimes lead to signican t deviations from the b eha vior of the isotropic sup erconductor. The eect that this w ork is mainly fo cused on is the structural transitions of the v ortex lattice. The familiar arrangemen t of the v ortices in a hexagonal lattice is not unique, as exp erimen ts ha v e sho wn. Distorted rhom bic and square lattices are p ossible. The p ossibilit y for alternate arrangemen ts b ecomes apparen t once one realizes that the reason b ehind the formation of the hexagonal lattice 2 is the isotropic in teraction b et w een the v ortices. An additional indication w ould b e the small dierence in energy b et w een the square and hexagonal lattice. One source of anisotrop y is that whic h arises from the existence of an underlying crystal lattice of dieren t symmetry than the one usually assumed, namely cubic. This condition is met in the cuprate family whic h are la y ered, extreme t yp e-I I HTSCs. They exhibit uniaxial crystal symmetry A w a y to accoun t for that is to in tro duce a dieren t eectiv e mass along one direction (uniaxial symmetry). In that case the sup erconductor is c haracterized b y t w o p enetration lengths, ab for the a { b (or basal) plane and c along the c -axis, and also t w o coherence lengths ab and c 3 The anisotrop y ratio = c = ab = ab = c quan ties the strength of the anisotrop y 2 F or a xed area (densit y), v ortices on a hexagonal lattice are placed the farthest apart p ossible from eac h other. 3 In man y materials the situations is ev en more complicated, with a 6 = b

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16 vortex core PSfrag replacemen ts a b a b a b vortex core PSfrag replacemen ts c a c a c aFigure 1.3: V ortex proles for applied eld along the z and b axis resp ectiv ely T ypical v alues for a HTSC are 1 and ab 1000 A. The Ginzburg-Landau parameter is dened as = ab = ab An immediate consequence is that the proles of the v ortices are dieren t for elds along the z axis and along the a { b plane, see gure (1.3). Also, the critical elds H c 1 and H c 2 b ecome functions of the angle b et w een the applied eld and the c -axis of the material. The analysis in general is dicult but there exists a scaling sc heme that reduces the anisotropic problem to an isotropic one at the initial lev el of Ginzburg-Landau theory or London mo del [43]. Using that metho d, one essen tially generalizes isotropic results to the anisotropic case. The rux line lattice in anisotropic materials can b e distorted and dep ending on the presence of pinning, defects and other factors one ma y see the formation of domains of dieren t rux line conguration and also transitions from one lattice symmetry to another. Structural phase transitions can b e studied with the same metho d, b y promoting the mass m to a mass tensor, th us incorp orating the anisotrop y in the mo del. Ho w ev er, a signican t failure of this approac h is that cubic materials (e.g. V 3 Si) w ould not exhibit structural phase transitions, a fact whic h con tradicts exp eri-

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17 men ts. The results of the researc h presen ted in this w ork can easily b e generalized to an anisotropic mass tensor using the scaling sc heme [43]. Another source of anisotrop y is an anisotropic F ermi surface. This is t ypical in the b oro carbide family (RNi 2 B 2 C, with R=Y, Lu, Tm, Er, Ho, Dy), whic h are most lik ely con v en tional but highly anisotropic s -w a v e sup erconductors [44]. The anisotropic F ermi surface aects strongly the electromagnetic in teractions in the sup erconductor [45]. T o accoun t for an anisotropic F ermi surface, the London mo del has to b e amended in order to include nonlo cal corrections via the relation b et w een the curren t densit y j and the v ector p oten tial A F or tetragonal sup erconductors lik e the b oro carbides the corrections (in F ourier space) ha v e the follo wing form [46]: 4 c j i = 1 2 ( m 1 ij 2 n ij l m k l k m ) a j : (1.21) Here a j = A j + 0 @ j = 2 with the phase of the order parameter and A the v ector p oten tial. The in v erse mass tensor m 1 ij and the fourth rank tensor n ij l m whic h couples sup ercurren ts with the crystal are ev aluated as particular a v erages of the F ermi v elo cit y o v er the F ermi surface and in that w a y are strongly dep enden t on the shap e of the F ermi surface. More imp ortan tly the coupling p ersists ev en for cubic materials [47]. A qualitativ e explanation of the imp ortance of nonlo calit y is that the curren t resp onse of the sup erconductor m ust b e a v eraged o v er the nite size of the Co op er pair, whic h is of order in con trast to the usual lo cal electro dynamics. The London mo del is b oth successful and con v enien t, due to its simplicit y Among the goals of this w ork is to sho w that one obtains iden tical results emplo ying the Ginzburg-Landau theory with the inclusion of an anisotropic term. F or the description of con v en tional sup erconductors with anisotropic F ermi surface the additional term is purely phenomenological. F or the anisotrop y that arises from an un-

PAGE 23

18 con v en tional coupling the additional term will b e deriv ed from a more general mo del in subhead 2.1. The anisotrop y that is of particular in terest to me is the one arising from the existence of an uncon v en tional pairing or a m ulticomp onen t order parameter. It has b een established that the pairing mec hanism in the cuprates is of d w a v e nature [48{ 50], whic h results in a fourfold anisotrop y of the gap and anisotropic v ortex cores. The rst direct observ ation of a w ell ordered square v ortex lattice in a HTSC w as ac hiev ed v ery recen tly [51]. Ev en more exotic systems ha v e b een in v estigated suc h as Sr 2 RuO 4 with an uncon v en tional t w o comp onen t order parameter [52]. Co existence of phases of dieren t pairing symmetry and related phase transitions ha v e b een observ ed for the hea vy F ermion sup erconductors UPt 3 and U 1 x Th x Be 13 The in terested reader can consult the nice review b y Sigrist and Ueda [53] and also the review b y Jo yn t and T aillefer [54]. A v ery recen t w ork b y Agterb erg and Do dgson [55] generalizes the London mo del to uncon v en tional sup erconductors with m ultiple sup erconducting phase transitions. Although the main b o dy of this dissertation deals with the in v estigation of the eects of d x 2 y 2 pairing with an small admixture of s w a v e pairs in the condensate, it is reasonable to exp ect that similar results can b e obtained with the same metho dology for other pairing symmetries. In eac h case it w ould b e necessary to nd the form of the appropriate term that couples the co existing order parameters. 1.5 Elasticit y and Melting of the V ortex Lattice The structural phase transitions of the v ortex lattice are closely connected to its elastic prop erties. The elastic resp onse of the v ortex lattice, quan tied b y the appropriate elastic mo duli, is of great signicance for the phase diagram of the sup erconductor. The theory of elasticit y for the v ortex lattice in an isotropic sup erconductor w as deriv ed b y Brandt [35{38] using the Ginzburg-Landau theory for the high-induction

PAGE 24

19 regime and the London mo del for lo w inductions. The theory w as later conrmed b y the results of Larkin and Ov c hinnik o v [56], who used the microscopic theory as their starting p oin t. The theory has b een generalized to anisotropic sup erconductors with m uc h eort [42, 57{61]. In these w orks the anisotrop y w as incorp orated in the mass tensor. The elasticit y and the structural phase transitions are relev an t for the extremely in teresting sub ject of the melting of the v ortex lattice. Esp ecially in HTSCs, with the enhanced role of the thermal ructuations and the softness of the elastic mo duli, the region o v er whic h a melted v ortex lattice is exp ected is signican t. One can dene melting as the loss of long range correlations b et w een the v ortices. The melted v ortex lattice consists of highly mobile and rexible v ortex lines. A fundamen tal dierence in the b eha vior of HTSCs compared to con v en tional, lo wT c sup erconductors is that at high elds, the w ell dened and sharp sup erconductingnormal transition is replaced b y a smo oth crosso v er b et w een the v ortex liquid phase and the normal metallic phase. One criterion used to dene the onset of melting is the so-called Lindemann criterion, according to whic h melting of the v ortex lattice is exp ected once the mean squared amplitude of the thermal displacemen t of the v ortex lattice b ecomes comparable to the in ter-v ortex spacing (the lattice constan t). Although this criterion is not rigorous, it pro vides a simple and adequate to ol for the in v estigation of melting and the determination of the melting curv e B m ( T ). The elasticit y en ters in the calculation of the mean squared displacemen t h u 2 i F or an excellen t discussion of the issues p ertaining to the melting of the v ortex lattice and the relev an t theory the reader can consult the review b y Blatter and others [22].

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CHAPTER 2 GINZBUR G-LAND A U THEOR Y 2.1 Anisotropic Ginzburg-Landau Theory The isotropic Ginzburg-Landau theory has b een used extensiv ely in the study of isotropic, or s -w a v e sup erconductors. Ho w ev er, once it w as realized that alternate pairing mec hanisms could exist, new extended Ginzburg-Landau theories w ere prop osed and studied in the con text of highT c sup erconductivit y [62{69]. Although the latter are built using symmetry argumen ts, microscopic deriv ations do exist [70]. F or the highT c cuprates it is established that the ma jor pairing c hannel is the d x 2 y 2 pairing. The dominan t pairing mec hanism is of d -w a v e nature with a sub dominan t s -w a v e order parameter b eing induced b y gradien ts of the d -w a v e as will b e sho wn promptly The generalization of the Ginzburg-Landau theory to anisotropic sup erconductors in whic h the anisotrop y is due to s and d w a v e mixing con tains t w o elds, or order parameters. The corresp onding Ginzburg-Landau free energy can b e written in the follo wing form, whic h includes an anisotropic term with the appropriate symmetry [54, 66, 67]: f GL = s j s j 2 + d j d j 2 + 1 j s j 4 + 2 j d j 4 + 3 j s j 2 j d j 2 + 4 ( s 2 d 2 + s 2 d 2 ) + B 2 8 + r s ( s )( s ) + r d ( d )( d ) + r v [( y s ) ( y d ) ( x s ) ( x d ) + cc ] ; (2.1) where the brac k ets denote spatial a v eraging. Also I ha v e dened the parameters r i =2 = 2 m i i = s; d; v and = r =i e A =c is the co v arian t deriv ativ e as usual. F rom here on I set= c = 1. 20

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21 The critical parameter is d and it is generally assumed that T s T d meaning that d is the dominan t order parameter. The later treatmen t of s as a small correction arising from gradien ts of d is based on this assumption. The Euler-Lagrange equations of motion are obtained from equation (2.1) b y minimizing with resp ect to d and s One obtains ( r d 2 + d ) d + 2 2 j d j 2 d + 3 j s j 2 d + 2 4 s 2 d + r v ( 2y 2x ) s = 0 ; (2.2) ( r s 2 + s ) s + 2 1 j s j 2 s + 3 j d j 2 s + 2 4 d 2 s + r v ( 2y 2x ) d = 0 : (2.3) The b oundary conditions obtained during the deriv ation of the rst and second equation of motion are giv en b y the form ulas n f r d ( d ) + r v [ y ( y s ) x ( x s )] g = 0 ; (2.4) n f r s ( s ) + r v [ y ( y d ) x ( x d )] g = 0 ; (2.5) where n is the unit v ector normal to the surface of the sup erconductor. The second Ginzburg-Landau equation, whic h is the equation for the curren t, is obtained b y minimizing the Ginzburg-Landau free energy with resp ect to the v ector p oten tial A After a short calculation one obtains j = e r s [ s ( s ) + s ( s ) ] + r d [ d ( d ) + d ( d ) ] + y r v [ s ( y d ) + d ( y s ) ] x r v [ s ( x d ) + d ( x s ) ] + cc : (2.6) The solution of the Ginzburg-Landau equations is a v ery complicated problem ev en for the isotropic sup erconductor whic h can b e describ ed in terms of only one order parameter. The additional complications of ha ving t w o order parameters can b e a v oided with a drastic simplication: the reduction of the t w o-eld theory to an eectiv e one-eld theory In order to deriv e the single comp onen t free energy I will

PAGE 27

22 follo w Aec k and others [71] and in tegrate out the s eld. This task is ac hiev ed b y making the substitution s ( r v = s ) 2y 2x d: (2.7) This appro ximate equation is deriv ed from the equation of motion for the s eld up on neglecting higher order terms. It is a reasonable appro ximation, based on the smallness in magnitude of the s comp onen t when compared to the d comp onen t. This approac h is useful, for one is in terested primarily in the linearized equations whic h are easier to solv e. It will b e sho wn that the anisotropic term added to the rst GinzburgLandau equation b y this pro cedure is sucien t to capture the essen tial ph ysics of the problem. After all the substitutions, the linearized rst Ginzburg-Landau equation tak es the form ( r d 2 + d ) d r 2 v s ( 2x 2y ) 2 d = 0 ; (2.8) whic h is the cen tral equation and the starting p oin t in m y study F or the sak e of completeness the second Ginzburg-Landau equation is j e = r d [ d ( d ) + d ( d ) ] r 2 v s y ( y d )[( 2y 2x ) d ] + d [ y ( 2y 2x ) d ] x ( x d )[( 2y 2x ) d ] + d [ x ( 2y 2x ) d ] + cc : (2.9) The b oundary condition for the d -w a v e order parameter (2.4) can no w b e written as n r d ( d ) r 2 v s y y ( 2y 2x ) d x x ( 2y 2x ) d = 0 ; (2.10) whic h enforces the v anishing of the normal comp onen t of the sup ercurren t on the surface of the sup erconductor. This b oundary condition w ould b e correct for a sup erconductor-insulator in terface, but for an in terface b et w een a sup erconductor and

PAGE 28

23 a normal metal it has to b e mo died. The mo dication can b e found b y imp osing the condition n j = 0 on the curren t equation (2.9). It is con v enien t to switc h to dimensionless quan tities using the unit con v en tions that are frequen tly used in the literature r r ; (2.11) d d 0 d; j d 0 j 2 = j d j = 2 ; (2.12) A (c= e ) A ; (2.13) H (c= e ) H : (2.14) The ab o v e equations dene the natural lengthscales of the v arious quan tities in v olv ed. The Ginzburg-Landau parameter is dened = = with the p enetration depth and the coherence length giv en b y 2 = ( m c 2 = 4 e 2 )( 2 = j d j ) ; (2.15) 2 =2 = 2 m j d j : (2.16) The dimensionless Ginzburg Landau equations obtain a m uc h simpler form using these unit con v en tions. The rst equation is simply 2 d d ( 2x 2y ) 2 d = 0 ; (2.17) while the second Ginzburg-Landau equation, using the v ector p oten tial instead of the sup ercurren t, has the form r r A = A j d j 2 + i 2 ( d r d d r d ) 2 x ( x d )[( 2x 2y ) d ] + d [ x ( 2x 2y ) d ] y ( y d )[( 2x 2y ) d ] + d [ y ( 2x 2y ) d ] + cc ; (2.18)

PAGE 29

24 where is the dimensionless anisotrop y parameter, dened as = ( r 2 v j d j =r 2 d s ). Setting the anisotrop y parameter equal to zero one can immediately obtain the linearized Ginzburg-Landau equations for an isotropic sup erconductor. F rom no w on I switc h to the usual sym b ol for the order parameter instead of d since I eliminated s I also need to rotate the v ortex lattice b y an arbitrary angle ab out the direction of the applied eld. This is required b ecause the free energy is no longer in v arian t under rotations ab out the direction of the applied eld, due to the presence of the symmetry breaking anisotropic term. The preferred orien tation of the v ortex lattice will b e determined b y the minim um of the free energy|it will no longer b e arbitrary This crucial p oin t w as missed in some of the previous theoretical treatmen ts of this problem [72]. The anisotropic term pro vides the eectiv e coupling to the crystal lattice, thereb y making structural transitions p ossible. Without this coupling, the v ortex repulsion is isotropic for cubic or tetragonal materials resulting alw a ys in a hexagonal v ortex lattice. Although the anisotropic term w as deriv ed from the t w o comp onen t Ginzburg-Landau theory appropriate for highT c sup erconductors with d x 2 y 2 pairing it can serv e as a phenomenological term for the in v estigation of anisotrop y eects in tetragonal materials lik e the family of the b oro carbides. A useful transformation of the rst Ginzburg-Landau equation is eected b y in tro ducing creation and annihilation op erators through the follo wing relations = x i y with i [ x ; y ] = h= + ( ) is the creation (annihilation) op erator. The transformation of these op erators under rotations is sho wn b elo w. The primed quan tities corresp ond to quan tities in the rotated system. 0x = x cos + y sin ; (2.19) 0y = x sin + y cos ; (2.20) 0x 2 0y 2 = 1 2 e 2 i 2+ + e 2 i 2 : (2.21)

PAGE 30

25 Substituting in the rst Ginzburg-Landau equation and subsequen tly dropping the primes I obtain 2 4 e 2 i 2+ + e 2 i 2 2 = 0 ; (2.22) an equation whic h can b e reduced to the equation for an one dimensional harmonic oscillator with an appropriate p erturbation. The same transformation of co ordinates for the second Ginzburg-Landau equation yields a complicated result whic h after some manipulation can b e cast in the follo wing form r r A = A j j 2 + i 2 ( r r ) + b 3 3 4 j 0 j 2 1 4 ( z r ) + ( z r ) r 2 + 8 b 2 ; (2.23) where 0 is the zeroth order in appro ximation to the solution of equation (2.22), b = B =H c 2 is the reduced magnetic induction and the function is dened as ( x; y ; ) = cos (4 )( x 4 + y 4 6 x 2 y 2 ) + 4 sin (4 ) xy ( x 2 y 2 ) : (2.24) 2.2 Virial Theorem In this section I will presen t an alternativ e metho d to deriv e a compact and useful form of the free energy for the anisotropic Ginzburg-Landau mo del. The metho d is based on the virial theorem for sup erconductivit y disco v ered recen tly b y Doria and others [73]. In App endix A I deriv e in detail the generalization of the isotropic result. The virial theorem for an anisotropic sup erconductor has the follo wing simple form H B 4 = F k inetic + 2 F f iel d + 2 F anisotr opic ; (2.25)

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26 or the follo wing equiv alen t form, with the free energy comp onen ts expanded H B 4 = j ( x ) j 2 2 m + B 2 ( x ) 4 + 2 r 2 v s [ y ( x )][ y ( x 2 y 2 )( x )] [ x ( x )][ x ( x 2 y 2 )( x )] + cc ; (2.26) where r A ( x ) = B ( x ) is the microscopic magnetic eld. The magnetic induction is dened B = R d 3 x B ( x ) =V and the homogeneous applied eld is denoted b y H Using this extension of the virial theorem, as I sho w in detail in App endix A, one can cast the free energy in the follo wing form F = b 2 (1 b ) 2 (2 2 1) A + 1 1 (2 2 1) A + 1 ; (2.27) where b is the reduced magnetic induction, A is the Abrik oso v parameter and is the correction to the isotropic result, whic h has the follo wing form = 4 R e hj j 2 ( 2x 2y ) 2 i hj j 2 i 2 + A (2 2 1) R e h ( 2x 2y ) 2 i hj j 2 i : (2.28) It is straigh tforw ard to c hec k that if one sets the anisotrop y parameter equal to zero one obtains the familiar isotropic result. The ab o v e expressions seem to suggest that for hard, high, sup erconductors the eects of the anisotrop y are diminished. Ho w ev er, it will b e sho wn in the subsequen t analysis that ev en small deviations from isotrop y pro duce observ able and signican t eects on the structure and the prop erties of the v ortex lattice. This extension of the virial theorem allo ws one to easily generalize results obtained for the isotropic case. I sho w b elo w the form of the Gibbs free energy G = F H B = 4 the magnetization M = ( B H ) = 4 and the magnetic induction

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27 B for the anisotropic Ginzburg-Landau theory G = ( H ) 2 (2 2 1) A 1 (2 2 1) A ; (2.29) M = 1 4 H (2 2 1) A 1 (2 2 1) A ; (2.30) B = H H (2 2 1) A 1 (2 2 1) A : (2.31) The generalization of the second Abrik oso v iden tit y [4] is easily obtained in the pro cess of deriving the ab o v e relations. It reads H hj j 2 i 2 R e h ( 2x 2y ) 2 i = 1 2 2 1 hj j 4 i + 2 2 R e hj j 2 ( 2x 2y ) 2 i : (2.32) All the ab o v e results are in v aluable for the simplication of the calculations that are going to follo w. By itself, this generalization of the virial theorem can b e used in n umerical in v estigations of anisotropic sup erconductors presumably with the same success in reducing the o v erall computational complexit y of the problem as the virial theorem had for con v en tional sup erconductors. The original virial theorem has inspired a signican t amoun t of w ork. See for example [32, 73{77], and references therein.

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CHAPTER 3 MEAN FIELD THEOR Y In order to in v estigate the structure of the v ortex lattice one can start at the mean eld theory lev el. Although a simplied approac h, it pro vides useful insigh t ab out the problem and not surprisingly man y theoretical predictions whic h are in turn v alidated b y exp erimen t. 3.1 Solution of the First Ginzburg-Landau Equation In this section I will presen t the calculation of the Abrik oso v parameter A and the calculation of the correction to the free energy (2.27). T o that eect I will solv e the rst Ginzburg-Landau equation (2.22) p erturbativ ely in the anisotropic term. It has the form (in dimensionless units) 1 i r A 2 ( 2x 2y ) 2 = (1 j j 2 ) : (3.1) Rotating b y an angle ab out the z axis (the direction of the applied eld) and b y in tro ducing creation and annihilation op erators a y = i + = p 2 b a = i = p 2 b with = x i y one can write the rst linearized Ginzburg-Landau equation in the follo wing form a y a b 2 [ e 2 i a y 2 e 2 i a 2 ] 2 = 0 ; (3.2) without sp ecifying a particular gauge y et. The rotation ab out the z axis is necessary to accoun t for the orien tation of the v ortex lattice whic h is no longer arbitrary due to the four fold symmetric term that breaks rotational symmetry ab out the direction of the applied magnetic eld. Notice that the term w as group ed with the higher order term j j 2 in the righ t hand side 28

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29 of equation (3.1). This w a y the linearization of the rst Ginzburg-Landau equation is done in a consisten t manner, as p oin ted out b y Kogan [78]. T o pro ceed with the solution it is helpful to decide on a particular gauge. Tw o c hoices are useful in this problem, the Landau gauge A = (0 ; bx; 0) and the symmetric gauge A = ( by = 2 ; bx= 2 ; 0). I will sho w the results of the solution in t w o separate sections. The total w a v efunction will b e built b y sup erp osition, using appropriate linear com binations to ac hiev e the desired p erio dicit y of the observ able (from a quan tum mec hanical p oin t of view) j ( x ) j 2 The w a v efunction itself is neither p erio dic, nor gauge in v arian t. 3.1.1 Landau Gauge Throughout this section I will use the Landau gauge A = (0 ; bx; 0), whic h mak es the p erturbation theory calculation easy as it in v olv es only one dimension. In fact, all the gauge sensitiv e calculations that are presen ted in this w ork are done with this particular c hoice of gauge. I use standard p erturbation theory to obtain the w a v efunction up to second order in the anisotrop y parameter The relev an t matrix elemen ts are easy to calculate realizing that the underlying theory is that of the v ery familiar one dimensional harmonic oscillator. I obtain ( x ) = 0 ( x ) 1 + be 4 i 32 H 4 ( p b 2 x ) + ( b ) 2 32 e 8 i 64 H 8 ( p b 2 x ) (3.3) + 5 e 4 i H 4 ( p b 2 x ) + O ( 3 ) ; (3.4) where H n is the Hermite p olynomial of the n -th rank and 0 ( x ) = b 2 1 4 exp b 2 x 2 2 (3.5)

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30 -f PSfrag replacemen ts xVLyVLx CL y CL a b Figure 3.1: The geometry of the system: the v ortex lattice (VL) is rotated b y an angle ab out the c -axis with resp ect to the crystal lattice (CL). The ap ex angle is dened as the angle b et w een the t w o basis v ectors a and b is the unp erturb ed solution. Notice the explicit app earance of the orien tation angle in the expansion of the order parameter. The p erio dic solution is constructed follo wing Chang and others [79]. One can c ho ose one basis v ector, a to lie on the y axis and the second basis v ector b forming an angle with a The geometry is illustrated in gure (3.1). The a v erages hj j 2 i and hj j 4 i necessary for the calculation of the Abrik oso v parameter A and the correction to the free energy are calculated using the metho dology presen ted in the w ork of Chang and others [79]. Along with the kno wn rst order results, I presen t b elo w the results to second order in the anisotrop y parameter (the details of the calculations

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31 can b e found in App endix B): A = 0 + b 1 + ( b ) 2 r 1 + r 2 + 5 1 3 4 0 ; (3.6) = 8 b 2 A (2 2 1)[1 + 3 b ] + 0 1 + 3 b 2 + 1 [1 + 6 b ] + 2 b ( r 1 + r 2 ) : (3.7) The v arious quan tities that app ear in the ab o v e expressions are rapidly con v erging functions of three v ariables , whic h are in tro duced emplo ying the con v enien t parameterization + i = ( b = a ) exp ( i ) [1]. The functions 0 1 r 1 r 2 are p erio dic in with p erio d 1, and ha v e a ric h structure for 1. Their explicit forms are 0 = X nm 0 A nm ; (3.8) 1 = R e e 4 i X nm 0 A nm 8 2 2 n 4 6 n 2 + 3 8 ; (3.9) r 1 = R e e 8 i X nm 0 A nm 16 4 4 n 8 12 3 3 n 6 + 3 4 2 2 n 4 45 16 n 2 + 105 256 ; (3.10) r 2 = X nm 0 A nm 16 4 4 n 4 m 4 12 3 3 n 2 m 2 ( n 2 + m 2 ) + 3 4 2 2 ( n 4 + m 4 + 36 n 2 m 2 ) 45 16 ( n 2 + m 2 ) + 105 256 ; (3.11) where A nm = p e 2 i ( n 2 m 2 ) e 2 ( n 2 + m 2 ) The prime implies that there are t w o summations|the one sho wn, o v er n m and the other with n and m replaced b y ( n + 1 = 2) and ( m + 1 = 2) resp ectiv ely 3.1.2 Symmetric Gauge This is considered a more \natural" gauge, with no preferen tial treatmen t of one direction in space o v er another. It unfortunately comes with the disadv an tage that for this particular problem it is dicult to construct the p erio dic order parameter, in sharp con trast with the Landau gauge whic h readily allo ws the construction of the order parameter with its desired p erio dicit y Ho w ev er, this gauge c hoice b eing ph ys-

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32 ically more transparen t results in an order parameter whose form has an in teresting in terpretation. The solution of the unp erturb ed rst Ginzburg-Landau equation in the symmetric gauge reads 0 ( x; y ) = r b 2 2 exp b 2 4 ( x 2 + y 2 ) : (3.12) The correction to rst order in is easily obtained using the metho d of the previous section. I obtain ( x; y ) = 0 ( x; y ) 1 + b 3 4 32 e 4 i z 4 ; (3.13) where z = x + iy The second order correction is easy to calculate but it do es not add an ything to the ph ysical p oin t I am trying to mak e. Although the p erio dic order parameter is dicult to construct, one can p ostulate an amendmen t of the isotropic equation (1.14) to accoun t for the anisotrop y The form of the p erturbativ e result leads one to prop ose the follo wing form for the solution whic h accoun ts for fourfold anisotrop y ( x; y ) = exp b 2 4 ( x 2 + y 2 ) Y ( z z ) 1 + ( z z + r 1 )( z z r 1 ) ( z z + r 2 )( z z r 2 ) ; (3.14) with r 1 = r R + ir I = j r j e i! and r 2 = r I + ir R = j r j e i! + 2 and denoting the p osition of the v ortices|in other w ords z are the zero es of the unp erturb ed order parameter. The ph ysical in terpretation b ecomes apparen t once one is familiarized with the strange form of the solution. The term pro vides the four additional zeros of the induced s -w a v e order parameter, around the cen tral no de of the d w a v e order parameter. The p ositions of the no des are con trolled b y the r terms and they can accommo date v arious symmetries.

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33 I ha v e attempted to obtain the compression ( c 11 ) and tilt ( c 44 ) elastic mo duli using the solution (3.14) according to Brandt's prescription [35{38] without success. The complexit y unfortunately increases rapidly with eac h successiv e step. Ho w ev er, there are some in teresting results that w ere obtained that are w orth men tioning here. I rst need to dene t w o functions ( x; y ) and M ( x; y ) through ( x; y ) = X 4 + Y 4 6 X 2 Y 2 j r j 4 cos(4 ) ; (3.15) M ( x; y ) = 4 X Y ( X 2 Y 2 ) j r j 4 sin(4 ) ; (3.16) where X = x x and Y = y y with denoting the -th v ortex. The mo dulus of the order parameter can b e reduced to the follo wing form f 2 = f 2 isotr opic (1 + 2 g ) ; (3.17) where g is the correction due to the anisotrop y and it is dened as g = Q It is easy to sho w that the sup erv elo cit y and the quan tit y u = r f =f ha v e the form Q = Q isotr opic + r g z ; (3.18) u = u isotr opic + r g : (3.19) The equation Q = u z still holds. Other useful iden tities that ha v e a straigh tforw ard generalization in the anisotropic case are those rst disco v ered b y Brandt [35, 37] when deriving the elastic mo duli of the isotropic sup erconductors. I obtain for the sup erv elo cit y of the Abrik oso v solution (momen tarily I am follo wing Brandt's notation) Q B = A 1 z X r r j r r j 2 + 4 ( X 3 3 X Y 2 ) x + ( Y 3 3 X 2 Y ) y ) : (3.20)

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34 The follo wing cen tral iden tit y remains unc hanged: ( r ) 2 = 2 B 2 l + l r 2 l : (3.21) Note that l is the linearized order parameter, obtained from the Abrik oso v solution for the undisplaced v ortex lattice 0 up on substitution of the displaced co ordinates r and subsequen t expansion in the displacemen t s to linear order. The general v ortex latticeisdefnedbythelociofthezerosoftheorderparametergivenby: r ( z ) = R + s ( z ) : (3.22) Another v ery in teresting, from a ph ysical p oin t of view, result is that an arbitrary displacemen t s of a v ortex line and the anisotrop y app ear to mo dify the linearized order parameter in a similar manner. I obtain l ( x; y ) = 0 ( x; y ) 1 + 1 2 2 ; (3.23) with 0 ( x; y ) the solution for the undisplaced v ortex lattice. The generalized displacemen t term whic h in the isotropic theory accoun ts for the displacemen t only of the v ortex lattice, no w has the form ( r ) = )Tj/T1_16 11.9552 Tf9.2294 0 Td(2 X s ( r )Tj/T1_22 11.9552 Tf11.9894 0 Td(r ) + r W j r )Tj/T1_22 11.9552 Tf11.8694 0 Td(r j 2 ; (3.24) W = X 6 + Y 6 )Tj/T1_16 11.9552 Tf11.8694 0 Td(5 X 2 Y 2 ( X 2 + Y 2 ) )Tj11.9894 0 Td(j r j 4 [( X 2 + Y 2 ) cos(4 ) + X Y sin(4 )] : (3.25) The complexit y is signifcan t but the calculations are tractable so far. What hinders further progress is m y inabilit y to fnd the appropriate generalization of the free energy whic h serv es as the starting p oin t of the deriv ation of the disp ersiv e elastic mo duli c 11

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35 and c 44 In the isotropic theory it has the form [35, 37] f = h + 2 2 + ( r ) 2 4 2 + Q 2B h B h i + B 2 : (3.26) 3.2 Structure of the V ortex Lattice and Minimization of the F ree Energy In order to obtain the mean eld phase diagram and study the structural phase transitions formally one has to minimize the Gibbs free energy (2.29), whic h is the prop er free energy to minimize under the constrain t of a constan t external eld. I found that the minimization of the Gibbs free energy 1 G = F H B = 4 and the minimization of the Helmholtz free energy equation (2.27), giv e iden tical phase diagrams. I c ho ose to w ork with the Helmholtz free energy from whic h one obtains the elastic mo duli, as sho wn in Chapter 4. The particular form of the free energy (2.27) allo ws one to obtain the phase diagram in a direct w a y instead of minimizing only the Abrik oso v parameter A as w as done b y other w ork ers [72, 79{81]. Ho w ev er, for the sak e of comparison and as a consistency c hec k to m y algorithm, I minimized the Abrik oso v parameter A in order to obtain the phase diagram as a function of the anisotrop y The results are sho wn in gure (3.2). T o rst order I nd the transition p oin t at c 1 = 0 : 0240, in agreemen t with Chang et al. [79] who nd c 1 = 0 : 0235. The result of the n umerical solution for the phase b oundary of P ark and Huse [72] cannot b e compared with the previous results By solving for the phase b oundary n umerically they nd c 1 = 0 : 0367. 2 Ho w ev er, in their w ork the v ortex lattice w as not allo w ed to rotate, th us omitting from the calculation, an essen tial degree of freedom. The second order correction to the Abrik oso v parameter A mo v es the critical anisotrop y to c 2 = 0 : 0284 a relativ ely small correction whic h justies the use of p erturbation theory 1 After one mo dies G to b e a function of the reduced eld b 2 The rep orted result, c 1 = 0 : 0734, needs to b e divided b y 2 due to dieren t con v en tions for the anisotrop y parameter.

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36 00.02 0.04 0.06 0.080.10.12 0.14 0.16 0.180.2e 60 65 70 75 80 85 90q first order second order Apex angle q vs. anisotropy e 00.02 0.04 0.06 0.080.10.12 0.14 0.16 0.180.2e 25 30 35 40 45f first order second order Orientation angle f vs. anisotropy e ec1 ec2erFigure 3.2: Minimization of the Abrik oso v parameter A as a consistency c hec k for the computer co de. The ap ex angle of the unit cell and the orien tation angle of the v ortex lattice as a function of the anisotrop y parameter The critical p oin ts to rst and second order appro ximation in the anisotrop y parameter are c 1 = 0 : 0235 and c 1 = 0 : 0284. Notice the reorien tation transition in the inset panel whic h app ears only to second order in the anisotrop y parameter at r = 0 : 0492. The most in teresting and unan ticipated fact w as the observ ation of another phase transition at a still higher v alue of the anisotrop y r = 0 : 0492, whic h turns out to b e connected with the v anishing of the rotational mo dulus as will b e sho wn in Chapter 4. In particular, at this second transition p oin t, the orien tation of the square lattice c hanges con tin uously from the [110] direction, with = = 4 no w a lo cal maxim um and t w o degenerate minima on higher and lo w er angles. The v ortex lattice retains its square symmetry The minimization of the free energy shares the complications of the minimization pro cess for the Abrik oso v parameter A There exist man y lo cal minima that ha v e to

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37 b e a v oided in the searc h of the global minim um and also there are man y equiv alen t minima whic h corresp ond to the same lattice structure, with dieren t v alues for the three parameters , whic h determine the structure of the v ortex lattice. This problem can b e solv ed realizing that for the t w o subsets 2 [0 ; 1 = 2], 2 [1 = 2 ; 1], one obtains equiv alen t lattices. I c hose to w ork in the rst subset. The minimization w as carried out using the fmincon function for constrained minimization included in the optimization pac k age of Matlab A program w as written for the ev aluation of the free energy the calculation of the gradien t and the Hessian. A driv er w as then used to automate the minimization for dieren t v alues of the parameters of the problem. The in terested reader will nd the co de in App endix D. The results of the minimization of the free energy for = 0 : 11, = 5 are sho wn in gure (3.3). The dierence of m y results compared to previous theoretical predictions [79, 80] obtained from the minimization of the Abrik oso v parameter A is that the structure of the v ortex lattice is no w obtained as a function of the reduced eld b allo wing the comparison with exp erimen t. All the main features of the second order structural phase transition are there, along with the reorien tation transition in the region close to the upp er critical eld. In order to c hec k the results and the minimization algorithm I calculated the correction to second order in the anisotrop y parameter. The dierence b et w een the rst and second order in the anisotrop y parameter results is not signican t, as it can b e seen in gure (3.3). F or larger v alues of the anisotrop y parameter the transition o ccurs at a smaller critical eld, while for smaller anisotrop y the transitions o ccurs closer to b = 1. I c hose the particular v alue to sho w the reorien tation transition clearly The predictions of the anisotropic Ginzburg-Landau mo del are at v ariance with the predictions of the nonlo cal London mo del, with regard to the reorien tation of the v ortex lattice at lo w elds. The nonlo cal London mo del predicts [46, 82] that the reorien tation happ ens at a lo w eld H 1 with the orien tation angle b eing = 4

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38 0.2 0.4 0.6 0.81b 25 30 35 40 45f Orientation angle f 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91b 70 75 80 85 90q first order second order Apex angle q Figure 3.3: The ap ex angle of the unit cell, the orien tation angle the Abrik oso v parameter A and the dimensionless Helmholtz free energy F = 2 as a function of the reduced eld b for = 0 : 11 and = 5. In the rst panel on the top I include the second order result for the ap ex angle. The reorien tation transition is eviden t in the second panel on the top. throughout the rhom bic to square transition. This has b een recen tly v eried b y exp erimen t on annealed LuNi 2 B 2 C crystals [15]. In m y mo del the reorien tation of the v ortex lattice is con tin uous and is completed at the transition p oin t. This is not strange, as an y exp ectation of a quan titativ e agreemen t b et w een the t w o theories at lo w elds (the natural region of applicabilit y of the London theory) w ould b e unreasonable.

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CHAPTER 4 ELASTICITY THEOR Y F OR THE V OR TEX LA TTICE In this section I will deriv e the elastic mo duli of the v ortex lattice. This deriv ation is indep enden t of the particular mo del (Ginzburg-Landau theory or London mo del) that will b e used to ev aluate the elastic mo duli. As is done for the standard theory of elasticit y for crystals, one can in tro duce the displacemen t v ector u of a v ortex line from its equilibrium p osition and up on expanding the elastic energy in terms of deriv ativ es of the displacemen t u i;k @ k u i one obtains the v arious elastic mo duli. In principle the deriv ation is the same, ho w ev er one has to consider carefully the fundamen tal dierences b et w een the usual crystal and the v ortex lattice. Before pro ceeding with the deriv ation a few commen ts are in order. The full calculation of the elastic resp onse of the v ortex lattice is a complicated task. I will consider only the simplest part of that, namely straigh t and parallel v ortices, thereb y reducing the complexit y of the problem from three dimensional to the simpler t w o dimensional case. A further simplication is the so called incompressible limit, in whic h one considers deformations that conserv e the densit y of the v ortices. This puts the follo wing constrain t on the displacemen t eld: r u u i;i = 0. F or the shear mo duli calculated in this fashion it has b een pro v en that they are essen tially nondisp ersiv e [25]. One then has to consider only uniform deformations. The v ortices are inhabiting the sup erconductor and are th us inruenced b y the symmetry of the underlying crystal lattice. Moreo v er, rigid rotations of the v ortex lattice with resp ect to the crystal lattice ha v e an energy cost asso ciated with them, a strange prop ert y nev er encoun tered in ordinary solids whic h liv e in an isotropic space. The immediate consequence is that the usual decomp osition of the elastic energy in terms of strains and rotations is no longer con v enien t. A more useful form for the 39

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40 elastic energy has b een prop osed b y Mirano vi c and Kogan [83]. I will adopt that form and presen t here, in detail, the deriv ation of the elastic mo duli for the v ortex lattice. The form of the elastic energy densit y in terms of a general fourth rank strain tensor can b e written E = 1 2 r ik l m u i;k u l ;m : (4.1) The tensor r ik l m is not symmetric under the exc hange of i $ k or l $ m whic h w ould p ermit a decomp osition in to a symmetric and an an tisymmetric part. In fact the only symmetries that r ik l m p ossesses are those of the crystal lattice and of the equilibrium v ortex lattice along with the ob vious r ik l m = r l mik I will deal with the more symmetric case rst, the square v ortex lattice. The energy should b e in v arian t under an exc hange of x and y co ordinates. Th us, x and y app ear in ev en n um b ers in the elastic energy Under the constrain t of incompressibilit y the elastic energy reads E s = 1 2 [ r 1 s u 2x;x + r 2 s ( u 2x;y + u 2y ;x ) + 2 r 3 s u x;y u y ;x ] ; (4.2) with r 1 s = 2( r xxxx r xxy y ), r 2 s = r xy xy and r 3 s = r xy y x Similar considerations for the hexagonal v ortex lattice lead to an elastic energy of the form E h = 1 2 [ r 1 h u 2x;x + r 2 h u 2x;y + r 3 h u 2y ;x + 2 r 4 h u x;y u y ;x ] ; (4.3) with r 1 h = ( r xxxx + r y y y y 2 r xxy y ), r 2 h = r xy xy r 3 h = r y xy x and r 4 h = r xy y x In order to mak e the elastic mo duli more transparen t, it helps to consider v arious deformations and to study the form of the elastic energy for eac h c hoice. There are four imp ortan t deformations: Squash deformation: u sq = ( x x y y ), Simple shear along x : u x = y x

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41 Simple shear along y : u y = x y Rigid rotation ab out z : u r = ( x y y x ). Under eac h of these deformations, the elastic energy assumes a particularly simple form whic h allo ws one to c hose a con v enien t parameterization with elastic mo duli c sq c 66 x c 66 y and c r that are readily connected with a simple ph ysical situation. More sp ecically I ha v e for the square lattice rst E s [ u sq ] = 1 2 r 1 s 2 E s [ u x ] = 1 2 r 2 s 2 E s [ u y ] = 1 2 r 2 s 2 E s [ u r ] = 1 2 2( r 2 s r 3 s ) 2 I can then iden tify c sq = r 1 s c 66 x = c 66 y = r 2 s and c r = 2( r 2 s r 3 s ) as the relev an t elastic mo duli for the square v ortex lattice. In a similar fashion I obtain for the hexagonal lattice E h [ u sq ] = 1 2 r 1 h 2 E h [ u x ] = 1 2 r 2 h 2 E h [ u y ] = 1 2 r 3 h 2 E h [ u r ] = 1 2 ( r 2 h + r 3 h 2 r 4 s ) 2 Th us, the relev an t elastic mo duli for the hexagonal lattice are c sq = r 1 h c 66 x = r 2 h c 66 y = r 2 h and c r = ( r 2 h + r 3 h 2 r 4 h ). In gure (4.5) I presen t the action of the four particular displacemen t elds on the basis v ectors of an arbitrary lattice to clarify the ph ysical picture.

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42 1 0.5 0 0.5 1 y 1 0.5 0.5 1 x Squash deformation 1 0.5 0 0.5 1 y 1 0.5 0.5 1 x Rigid rotation 1 0.5 0.5 1 y 1 0.5 0.5 1 x Shear along x 1 0.5 0 0.5 1 y 1 0.5 0.5 1 x Shear along y Figure 4.5: The action of the four basic displacemen t elds, represen ted b y the small arro ws, on the basis v ectors of an arbitrary lattice, represen ted b y the large arro ws.

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43 There is another deformation, although not indep enden t from the ab o v e. It is the shear along the [110] direction for whic h one can easily pro v e that c 66 ( 4 ) = c sq + c r : (4.4) The ratio of c 66 x and c 66 ( 4 ) is a useful quan tit y that measures the anisotropic resp onse of the v ortex lattice to shear deformations of dieren t p olarizations and will b e of use later on. F or a general rhom bic lattice there are four indep enden t elastic mo duli c sq c 66 x c 66 y and c r rst prop osed b y Mirano vi c and Kogan [83]. The deformation resp onsible for the transformation of the general rhom bic lattice in to a square lattice is the so-called squash. The relev an t mo dulus v anishes at the p oin t where the structural transition tak es place, signaling the instabilit y with resp ect to the particular deformation. It will b e sho wn that another instabilit y o ccurs at higher elds. A t that p oin t the v ortex lattice b ecomes unstable with resp ect to rigid rotations ab out the c axis. All elastic mo duli will b e obtained from the second deriv ativ e of the GinzburgLandau free energy with resp ect to the amplitude of the deformation, in the limit that 0. I will mak e use of the Ginzburg-Landau free energy in the follo wing form F = b 2 (1 b ) 2 (2 2 1) A + 1 1 (2 2 1) A + 1 : (4.5) The elastic energy can b e written as E = 1 2 2 c ( ) j 0 = 1 2 2 @ 2 F j 0 ; (4.6) where c ( ) is the elastic mo dulus for eac h particular deformation. The amplitude of the deformation will en ter the free energy F through the three parameters and whic h in turn dep end on the lattice structure. The connection of a particular

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44 mo del of sup erconductivit y and the elastic mo duli is made at this stage. I c hose the Ginzburg-Landau theory ho w ev er the ev aluation of the elastic mo duli is along the same lines for the nonlo cal London mo del. It is straigh tforw ard to sho w that the second deriv ativ e of the Ginzburg-Landau free energy with resp ect to the amplitude of the deformation has the form @ 2 F = (2 2 1)( b 2 F ) (2 2 1) A + 1 @ 2 A + (1 b ) 2 [(2 2 1) A + 1] 2 @ 2 (2 2 1) (2 2 1) A + 1 [2 @ A @ + @ 2 A ] + 2(2 2 1) 2 ( @ A ) 2 [(2 2 1) A + 1] 2 : (4.7) Both A and dep end on the three parameters and The deriv ativ es of A and are th us giv en b y the c hain rule. One can dene = (2 2 1) = [(2 2 1) A + 1] and = (1 b ) 2 = [(2 2 1) A + 1] 2 and obtain the follo wing expression for eac h elastic mo dulus (denoted b y c for brevit y) using equations (4.6) and (4.7): c = C 2 + C 2 + C 2 + 2 C + 2 C + 2 C =0 ; (4.8) where C ij are the terms originating from the deriv ativ es of the Ginzburg-Landau free energy and are giv en b y C ij = (2 ij ) ( b 2 F ) A ;ij + ;ij ( A ;i ;j + A ;j ;i + A ;ij 2 A ;i A ;j ) : (4.9) Notice that the dieren tiations in the righ t hand side of equation (4.9) are with resp ect to the three lattice parameters , The calculation of , is purely geometrical in nature. One needs to express the parameters of the distorted v ortex lattice as a function of the magnitude of the deformation and ev aluate the deriv ativ e at the limit of v anishing deformation ( 0). The geometry of the system can b e seen in gure (3.1). The v ortex lattice is rotated b y an angle with resp ect to the crystal lattice. The direction of the

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45 rotation is completely arbitrary I c hose the particular one for con v enience. The angle is the ap ex angle of the (generally rhom bic) unit cell and a and b are the t w o basis v ectors. I will start with the geometrical denitions of and whic h ha v e no w b ecome functions of the amplitude of the deformation ( ) = a ( ) b ( ) j a ( ) j 2 ; (4.10) ( ) = j a ( ) b ( ) j j a ( ) j 2 ; (4.11) ( ) = cos 1 a ( ) y j a ( ) j ; (4.12) Under the inruence of eac h deformation the t w o basis v ectors are going to c hange direction and/or amplitude resulting in a deformed unit cell. The calculation of these eects on the unit cell as co ded in the parameters of the v ortex lattice for eac h individual deformation is straigh tforw ard, ho w ev er one should c ho ose carefully the branc h of the in v erse trigonometric functions that are in v olv ed in order to a v oid sign errors. The results are Squash: ( ; ; ) = ( 2 sin (2 ) ; 2 cos(2 ) ; sin(2 )) ; (4.13) Shear along x : ( ; ; ) = ( cos (2 ) ; sin(2 ) ; cos 2 ( )) ; (4.14) Shear along y : ( ; ; ) = ( cos (2 ) ; sin(2 ) ; sin 2 ( )) ; (4.15) Rigid rotation: ( ; ; ) = (0 ; 0 ; 1) : (4.16)

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46 A simple substitution of the lattice parameters in equation (4.8) yields for the nondisp ersiv e elastic mo duli of the v ortex lattice c sq = 4 C 2 sin 2 (2 ) + 4 C 2 cos 2 (2 ) + C sin 2 (2 ) 4 C 2 sin (4 ) 4 C sin 2 (2 ) + 2 C sin (4 ) ; (4.17) c 66 x = C 2 cos 2 (2 ) + C 2 sin 2 (2 ) + C cos 4 ( ) + C 2 sin (4 ) 2 C cos 2 ( ) cos(2 ) 2 C cos 2 ( ) sin(2 ) ; (4.18) c 66 y = C 2 cos 2 (2 ) + C 2 sin 2 (2 ) + C sin 4 ( ) + C 2 sin (4 ) + 2 C sin 2 ( ) cos(2 ) + 2 C sin 2 ( ) sin(2 ) ; (4.19) c r = C : (4.20) Up to no w the elastic mo duli ha v e b een indep enden t of the particular mo del that is used for the study of the v ortex lattice. The co ecien ts C ij whic h are mo del dep enden t will b e ev aluated in the next c hapter.

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CHAPTER 5 STR UCTURAL PHASE TRANSITIONS Using the results of mean eld theory from Chapter 3 and the elastic mo duli calculated in Chapter 4 I am in a p osition to calculate the elastic prop erties and study the phase transitions of the v ortex lattice within the anisotropic Ginzburg-Landau mo del. 5.1 Nondisp ersiv e Elastic Mo duli The pro cedure is v ery simple. First one has to minimize the free energy equation (2.27), with resp ect to the three parameters and This will pro duce the equilibrium energy and the equilibrium lattice for a giv en v alue of the reduced magnetic induction b The details of the minimization are presen ted in section 3.2. The second step is the calculation of the appropriate deriv ativ es of the free energy at the equilibrium v alues of and and the calculation of the four elastic mo duli, giv en b y equation (4.9), again for a giv en v alue of the reduced magnetic induction b One has to c ho ose v alues for the t w o parameters of the mo del, the anisotrop y parameter and The elastic mo duli w ere obtained as a function of the reduced eld for = 0 : 11 and = 5. They are ev aluated at eac h minim um of the free energy when obtaining the phase diagram in section 3. The calculated elastic mo duli for = 0 : 11 are sho wn in gure (5.1). The v anishing of the squash mo dulus signies the transition from rhom bic to square v ortex lattice. A t the same p oin t the t w o shear mo duli merge in to one b ecause of the higher symmetry of the square phase. A t a still higher eld I observ ed the reorien tation transition signied b y the v anishing of the rotation mo dulus c r This instabilit y of the v ortex lattice relativ e to rotations w as encoun tered in the nonlo cal London mo del also [83]. There are exp erimen tal 47

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48 indications for this instabilit y the in v estigation of whic h is hamp ered so far b y the high elds required for its observ ation [15]. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91b 00.002 0.004 0.006 0.0080.01 c sq c66,x c66,y c r 0.850.90.951b 01e-05 2e-05 c r Figure 5.1: The nondisp ersiv e elastic mo duli as a function of the reduced eld for = 0 : 11 and = 5. The rst transition p oin t at whic h the "squash" mo dulus c sq go es to zero is clearly seen in the left panel. The reorien tation transition starting at the p oin t where the rotational mo dulus c r v anishes is seen on the righ t panel. I can compare directly with the nonlo cal London mo del results of Mirano vi c and Kogan [83]. I can see the same qualitativ e b eha vior, the v anishing of the squash mo dulus c sq at the rst transition p oin t b, the merging of the t w o shear mo duli in to one after the rst structural transition and also the v anishing of the rotational mo dulus c r at the second transition p oin t b r whic h is ab out t w o times b. Ho w ev er, a quan titativ e dierence b et w een the t w o results is the c hange in the relativ e magnitude of the elastic constan ts after the phase transition. I observ e that the elastic mo duli

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49 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91b 00.511.522.533.5c66,x/c66, p/4 Figure 5.2: The ratio c 66 x =c 66 4 v ersus the reduced eld b in m y w ork constan tly diminish to w ards zero as one approac hes H c 2 ( T ), as exp ected. This b eha vior is attributed to the (1 b ) 2 factor in the free energy (2.27). I also note that the anisotropic elasticit y whic h manifests itself with a softer shear mo dulus for shearing along the sides of the square lattice than along the diagonals emerges naturally in this mo del. This resp onse of the v ortex lattice is measured b y the ratio c 66 x =c 66 4 whic h has a maxim um at the transition as can b e seen in gure (5.2). 1 It has b een suggested that this b eha vior explains the anisotropic orien tational long range order observ ed in decoration patterns in LuNi 2 B 2 C, whic h manifests itself as a signican t dierence in the correlation lengths along the [110] and [ 1 10] directions [15]. 1 Note that around b = 0 : 95 the theory breaks do wn b ecause the term can no longer b e considered a small correction. This is the origin of the discon tin uit y in the second p eak in gure (5.2).

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50 5.2 Fluctuations The thermal ructuations of the v ortex lattice are incorp orated in the anisotropic Ginzburg-Landau mo del through the three v ariables , c haracterizing the structure of the v ortex lattice. More sp ecically I will replace , in the expressions for the free energy and the squash mo dulus b y their thermally a v eraged v alues = [1 (2 ) u 2 ] ; (5.1) = [1 (2 ) u 2 ] ; (5.2) = + cot u 2 : (5.3) The deriv ation of the ab o v e equations go es along the lines of the deriv ation of the and presen ted in Chapter 4. An arbitrary displacemen t eld u corresp onding to the thermal ructuations is in tro duced with the follo wing constrain ts h u i ( x 1 ) u j ( x 1 ) i = 0 ; for i 6 = j; (5.4) h u i ( x 1 ) u j ( x 2 ) i = 0 ; (5.5) h u i ( x ) u i ( x ) i = 1 2 h u 2 ( x ) i : (5.6) Starting from the denitions of and (equations 4.10, 4.11, 4.12), under the ab o v e constrain ts it is straigh tforw ard to obtain after some simple algebra the thermally a v eraged and One then has to nd where the zero es of the squash mo dulus lie in the H { T plane. This pro cedure is analogous to the one carried out b y Gurevic h and Kogan [84] within the nonlo cal London mo del, taking the simplifying approac h of not calculating the phase transition line H( T ) selfconsisten tly That means that one do es not tak e in to accoun t the eects of the ructuations on the elastic mo duli themselv es. One is allo w ed to mak e this simplication based on the fact that the mean squared displace-

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51 men t of the v ortex u 2 is nite at H( T ) [84]. In order to sho w that explicitly I will presen t the deriv ation of the mean squared displacemen t. One starts from the elastic energy of the square v ortex lattice: 2 E = c sq u 2xx + c x u 2xy + c ! 2 xy + c 44 ( @ z u i ) 2 ; (5.7) written in the, con v enien t for this case, form in terms of strains u ij and rotations ab out the z axis xy u ij = 1 2 ( @ i u j + @ j u i ) = i 2 Z d 2 q d k (2 ) 3 e i q r + ik z ( q i ~ u j + q j ~ u i ) ; (5.8) xy = 1 2 ( @ x u y @ y u x ) = i 2 Z d 2 q d k (2 ) 3 e i q r + ik z ( q x ~ u y q y ~ u x ) ; (5.9) where the ~ u is the F ourier transformed displacemen t. F or a more general treatmen t of the problem, the tilt mo dulus c 44 related to b ending deformations of the v ortices, is included in the expression for the elastic energy The condition of incompressibilit y tak es the form @ x u y + @ y u x = 0 ) cos ~ u x = sin ~ u y ; (5.10) q = q (cos ; sin ) ; (5.11) from whic h one can obtain 2 E = Z d 2 q d k (2 ) 3 q 2 c ( ) 4 sin 2 + c 44 k 2 1 + cos 2 sin 2 ~ u x ( q ) ~ u x ( q ) ; (5.12) with c ( ) = c sq sin 2 (2 ) + c x cos 2 (2 ) + c In conclusion, the thermal a v erage u 2 will b e calculated from the general expression whic h relates the former with the comp onen ts

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52 xx and y y of the elastic tensor (see for example Blatter et al. [22]) u 2 = u 2x + u 2y = k B T Z d 2 q d k (2 ) 3 ( 1 xx + 1 y y ) : (5.13) In the case under in v estigation, the diagonal comp onen ts of the elastic tensor are equal to the co ecien t of ~ u x ( q ) ~ u x ( q ) in equation (5.12). I obtain u 2 = k B T Z d 2 q d k (2 ) 3 sin 2 q 2 c ( ) 4 + c 44 k 2 : (5.14) After p erforming the in tegral, the mean squared displacemen t has the form [84] u 2 K r c 66 x c sq c 66 x + c r ; (5.15) where K is the complete elliptic in tegral of the rst kind. It is eviden t that ev en at the instabilit y p oin t where c sq = 0, the mean squared displacemen t u 2 ( T ; B ) remains nite. There are t w o comp eting parameters, the strength of the thermal ructuations and the anisotrop y parameter whic h in this mo del is the equiv alen t of the nonlocalit y parameter nl of the nonlo cal London mo del. The comp etition arises b ecause thermal ructuations of increased strength tend to smear out the fourfold symmetric c haracter of the in terv ortex in teraction, making the in teractions isotropic and ev en tually restoring the rhom bic/hexagonal arrangemen t as the lo w est energy conguration of the v ortex lattice. F ollo wing Gurevic h and Kogan [84], I will tak e ( T ) = 0 = p 1 t 2 ( T ) = 0 = p 1 t 2 where t = T =T c I then ha v e = 0 t= p 1 t 2 2 The dimensionless mean 2 Other forms of and with dieren t temp erature dep endencies giv e similar results.

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53 squared displacemen t reads u 2 = 0 p (1 t 2 ) bt [(1 b ) 3 ln(1 + 1 p 2 b )] 1 2 ; (5.16) with 3 for LuNi 2 B 2 C [84]. The anisotrop y parameter w as connected to the nonlo calit y parameter nl via corresp ondence from the nonlo cal London mo del whic h is deriv ed from the extended Ginzburg-Landau theory [71]. One can obtain = 1 12 nl 0 2 (1 t 2 ) : (5.17) The ab o v e expression is dieren t from what one w ould obtain b y simply substituting the temp erature dep endence in = r 2 v j d j =r 2 d s Regardless of the fact that the results w ould b e similar ev en with a dieren t c hoice of temp erature dep endence, the motiv ation b ehind this particular c hoice w as to enable the comparison with the results of the nonlo cal London mo del, facilitated b y the in tro duction of the nonlo calit y parameter nl The results of the n umerical solution are sho wn in gure (5.3). The mean eld transition line without ructuations w as included in the graph also. Instead of c ho osing arbitrary v alues for all the parameters of the theory I used the same parameters used in the recen t w ork b y Gurevic h and Kogan [84] to facilitate the comparison of the t w o theories. The transition lines are obtained for nl = 2 : 5 o and o = 0 : 0064. One can see that the results of the t w o theories are in reasonable agreemen t th us v alidating the elegan t ph ysical picture concerning the comp etition b et w een the thermal ructuations and the nonlo calit y The transition line for the ructuating v ortex lattice ends abruptly at some p oin t. A t that p oin t the correction term equation (2.28), ceases to b e small enough to

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54 00.2 0.4 0.6 0.81T/Tc 00.2 0.4 0.6 0.81H/Hc2 (0) Hc2 (T) GL with fluctuations GL without fluctuations Non-local London model Transition lines for r=2.5x o c o =0.0064Figure 5.3: The phase b oundaries in the H T plane separating the square and rhom bic v ortex lattice obtained from Ginzburg-Landau theory and the nonlo cal London mo del. The line in tercepting H c 2 ( t ) is the mean eld GL result. b e considered a correction and at this p oin t the extended Ginzburg-Landau theory b ecomes unstable. This shortcoming of the mo del could b e corrected b y the inclusion of higher order terms or ev en b y the inclusion of an isotropic term of the same order in the anisotrop y parameter but that is not going to aect the qualitativ e b eha vior of the transition lines. Another imp ortan t observ ation is that for a Ginzburg-Landau parameter with v alue in the range appropriate for t ypical mem b ers of the cuprate family ( 100) the reorien tation transition do es no longer app ear. This seems to suggest that the reorien tation transition is irrelev an t for the cuprates. As a closing commen t I w ould lik e to address a recen t ob jection raised b y Nak ai and others [85] regarding the ph ysical mec hanism b ehind the reen tran t structural

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55 transition. The authors p erform a calculation based on the Eilen b erger theory and claim to ha v e found the true ph ysical origin of the phenomenon in another kind of comp etition. They claim that the reen tran t b eha vior is due to in trinsic comp etition b et w een sup erconducting gap and F ermi surface anisotrop y whic h pro vide t w o dieren t p ossible orien tations of the v ortex lattice, one with nearest neigh b ors along the gap minim um and another with nearest neigh b ors along the F ermi v elo cit y minim um. Their mo del, as they admit in the pap er, fails to predict the correct orien tation of the v ortex lattice in LuNi 2 B 2 C, at least for eld strengths used in exp erimen ts up to no w. Moreo v er, m y most serious ob jection is that their w ork is based on assumptions ab out the structure of the sup erconducting gap in the b oro carbides that remain to b e exp erimen tally v eried and whic h are not met in the cuprates. In m y opinion, the w ork of Gurevic h and Kogan [84] along with the presen t w ork ha v e unam biguously sho wn the imp ortance of thermal ructuations in the determination of the phase diagram, an issue that w ould still ha v e to b e addressed in the alternativ e scenario in order to ac hiev e a complete description.

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CHAPTER 6 FINGERING INST ABILITY OF ELECTR ON DR OPLETS F rom here on I dev ote m y atten tion to the ngering instabilit y of electron droplets in non uniform magnetic elds. This w ork w as inspired b y a recen t disco v ery b y Agam, Wiegmann and others [86, 87], whic h connects diusion limited gro wth in the limit of long diusion lengths (also called Laplacian gro wth) with the gro wth of a t w o-dimensional electron droplet in a high, non uniform magnetic eld. The calculation of the harmonic momen ts and the dev elopmen t of the Mon te Carlo co de w ere m y con tributions to a collab oration with Alan Dorsey and T a ylor Hughes [88]. Laplacian gro wth obtains its name from the description of a fron t propagating b et w een t w o incompressible liquids with high viscosit y con trast (e.g. w ater and oil). One considers a 2D ro w, where the less viscous ruid (w ater) is supplied b y a source at z = x + iy = 0 and the more viscous ruid (oil) is extracted at the same rate at innit y resulting in the motion of the b oundary b et w een the t w o phases. Assuming zero surface tension, the idealized surface mo v es with a normal v elo cit y v n whic h is prop ortional to the gradien t of the pressure p on the surface (Darcy's la w): v n = @ p @ n : (6.1) F or incompressible liquids the normal v elo cit y b ecomes prop ortional to the gradien t of a harmonic eld p giv en b y: r 2 p = 0 : (6.2) F or a nice review the reader can refer to [89]. Fingered patterns form at the in terface of t w o incompressible ruids with large viscosit y con trast only when the less viscous 56

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57 ruid is injected in to the more viscous ruid. It is essen tial that the system has a 2D geometry an exp erimen tal realization of whic h is the Hele-Sha w cell. It has b een found that an arbitrary deviation from a p erfectly circular shap e of the initial droplet will result in a pattern of gro wing ngers whic h gro w as more ruid is pump ed in to the droplet and these ngers ev en tually form cusp-lik e singularities at their tips. The same b eha vior has b een p ostulated for the Quan tum Hall droplet, the source of the instabilit y in this case b eing the c hanging gradien ts of the applied magnetic eld, pro ducing an inhomogeneous magnetic eld in the exterior of the droplet. The inruence of eld inhomogeneities on the shap e of the electron droplet is a manifestation of the Aharono v-Bohm eect. In this case also, singularities will dev elop in nite time. In the ruid system the singularities will b e con trolled b y the molecular scale of the particles and also b y the surface tension whic h is absen t in the idealized Laplacian gro wth. In the quan tum coun terpart of this phenomenon the singularities will b e cuto b y the only length scale in the problem, the magnetic length l B = p c=eB 0 whic h is the eectiv e size of the particles in v olv ed. The ngering instabilit y has b een studied for man y y ears since its disco v ery [90{ 94]. One can use conformal mapping metho ds to map the exterior of the unit circle ( w plane) on to the domain external to the droplet ( z -plane). One fundamen tal prop ert y for the later connection of the t w o phenomena is the conserv ation of the external harmonic momen ts dened for the ruid and electron droplet resp ectiv ely as t k = 1 k Z exter ior z k d 2 z ; (6.3) t k = 1 k Z exter ior B ( z ) z k d 2 z ; (6.4) with z = x + iy k = 1 ; 2 ; : : : and B ( z ) b eing the magnetic eld inhomogeneit y The momen t for k = 0 is prop ortional to the area of the droplet and the in tegrals

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58 for k = 1 ; 2 are assumed to b e prop erly regularized. These quan tities are of cen tral imp ortance for the theoretical treatmen t of the problem since they are used to relate the magnetic eld inhomogeneit y to the parameters of the conformal map. This part is divided in three sections. In the rst t w o sections I sho w the details of the deriv ation of the man y-b o dy w a v efunction of the 2D electrons in a p erp endicular, non uniform magnetic eld and the conformal mapping metho d. In the last section I discuss the Mon te Carlo sim ulation and presen t the comparison b et w een theory and exp erimen t. 6.1 Man y-b o dy W a v efunction F or a 2D system of electrons in a non uniform magnetic eld p erp endicular to the system, the N-particle Hamiltonian is H = N X j =1 ( i r j + A j ) 2 + g 2 B ( r j ) z ; (6.5) where the Coulom b in teractions of the electrons ha v e b een ignored and z is the P auli matrix. I ha v e also used appropriate units, namely the magnetic length l B = p c=eB 0 for lengths, the a v erage eld B 0 for magnetic elds,! c = 2 for energies with c = eB 0 =mc If in addition one tak es g = 2, equation (6.5) is the P auli Hamiltonian and the exact solution for the man y-b o dy groundstate is kno wn [95{99]. It is in teresting to p oin t out at this momen t that the single particle w a v efunctions, the w ell kno wn Landau lev els, are deriv ed in an almost iden tical metho d to the one presen ted in section 2.1. F or lling factor = 1, a lled Landau lev el, the ground state w a v e function can b e written as a Slater determinan t whic h reduces to [86, 97{99] ( z 1 ; : : : ; z N ) = 1 p N N Y i
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59 In the equation ab o v e, W ( z ) = j z j 2 = 4 + V ( z ), where the rst term comes from the homogeneous comp onen t of the magnetic eld and V ( z ) is a solution of r 2 V = B [ B ( z ) B 0 ] =B 0 The normalization factor 1 = p N N will not concern us in what follo ws. F or clarit y I will sho w explicitly the deriv ation. The homogeneous part of W ( z ) is giv en b y the solution to r 2 = B 0 whic h b ecomes (if one uses B 0 as the magnetic eld scale) r 2 = 1, whic h has the solution ( x; y ) = j z j 2 = 4. T o calculate the solution for the presence of a small inhomogeneit y so that B = ( B 0 + B ) z one has to solv e r 2 = 1 B from whic h b y setting = j z j 2 = 4 + V ( z ) one obtains r 2 V = B The formal solution is giv en b y V ( z ; z ) = 1 2 Z D ln j z z 0 j B ( z 0 ) d 2 z 0 : (6.7) It is essen tial to assume that the eld inhomogeneit y is far a w a y from the origin where the electron droplet will b e lo cated. Then, up on expanding the logarithm for j z j < j z 0 j ln j z 0 z j = ln z 0 + ln (1 z =z 0 ) = ln z 0 1 X k =1 1 k z z 0 k ; (6.8) I obtain the expansion of the p oten tial V ( z ) in terms of the harmonic momen ts of the domain D under the constrain t that the inhomogeneit y v anishes in the in terior of the domain D [100]: V ( z ; z ) = 1 2 1 X k =1 t k z k ; (6.9) t k = 1 k Z D B z k d 2 z : (6.10)

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60 The probabilit y densit y obtained from the w a v efunction (6.6) is j j 2 = exp 2 X i
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61 6.2 Conformal Mapping Metho d The idea is to use the p o w erful conformal mapping metho ds for 2D systems to transform the highly irregular domain of the exterior of the droplet in the complex plane z on to the exterior of the unit circle j w j = 1 in an auxiliary complex plane w The map is rev ersible b y construction. I will presen t v arious conformal maps that corresp ond to dieren t ngering patterns. F or a giv en conformal map z ( w ; t ) the ev olution of the b oundary in the Laplacian gro wth problem is go v erned b y the follo wing equation [91]: R e w @ z @ w @ z @ t = 1 2 : (6.15) F or a conformal map of the form z ( w ; t ) = M X n = 1 a n ( t ) w n ; (6.16) one obtains the follo wing dieren tial equations for the (real) co ecien ts a n ( t ) [91]: d d t M X n = 1 na 2n = 1 ; (6.17) M k X n = 1 na n d a k + n d t + ( k + n ) a k + n d a n d t ; for k = 1 ; : : : ; M + 1. (6.18) The solution of these will giv e the time ev olution of the b oundary As an example I will consider three simple conformal maps to illustrate the idea. F or the conformal map z = a + bw 1 one easily nds the solution z = e i q a 2 1 (0) + t + a 0 (0) ; (6.19) whic h corresp onds to a circular b oundary at all times. Th us the circular b oundary is stable.

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62 Considering the next order, namely M = 1, one can sho w that another stable b oundary has the form of an ellipse. The solution is z = [ a 0 (0) + ( B + 1) A ( t )] cos + i ( B 1) A ( t ) sin ; (6.20) with A ( t ) = (1 B 2 ) 1 2 p t; (6.21) B = a 1 (0) a 1 (0) ; (6.22) whic h corresp onds to an ellipse with a constan t eccen tricit y e = ( B 1) = ( B + 1). The mapping of order M = 2 pro duces the rst unstable b oundary one with three cusps. The solution for initial v alues a 2 (0) = 1 = 40, a 1 (0) = a 0 (0) = 0 and a 1 = 1 is z = [ a 2 cos (2 ) + a 1 cos ] + i [ a 2 sin (2 ) a 1 sin ] ; (6.23) with a 2 = a 2 1 40 ; (6.24) a 1 = q 400 p 159201 800 t: (6.25) The normal v elo cit y of the b oundary is giv en b y v n = j d w = d z j whic h div erges at the momen t the cusp is formed. This happ ens for an y p olynomial map suc h as the one I consider here. It can b e sho wn that the n um b er of cusps is equal to M + 1 whic h is the n um b er of ro ots of d z = d w The div ergence is caused b y precisely these ro ots as they approac h from the in terior of the unit circle and ev en tually fall on to the b oundary j w j = 1 at the critical time t cr itical

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63 The critical time for the M = 2 map and the ab o v e c hoice of parameters is t cr itical = 199. In gure (6.1) I sho w the ev olution of the b oundary for this particular map. F or higher orders in M the solution is p ossible only b y n umerical solution of the algebraic equations deriv ed from the initial dieren tial equations. 20 10 0 10 20 10 0 10 20 30Figure 6.1: The ev olution of the b oundary for the conformal map with M = 2 from time t = 0 to time t = t cr itical = 199. An alternativ e and more elegan t w a y to pro ceed is with use of the harmonic momen ts dened in the previous section, whic h facilitates the connection with the p oten tial V ( z ). The calculation of the harmonic momen ts for the v arious conformal maps that I used along with the connection to the p oten tial V ( z ) is presen ted separately in App endix C.

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64 6.3 Mon te Carlo Sim ulation The Mon te Carlo metho d is a v ery p o w erful to ol to sim ulate and in v estigate a system lik e the one I ha v e. See for example [101]. The ob jectiv e is to pro duce an arrangemen t of particles that follo ws the equilibrium distribution for the classical 2D plasma, equation (6.11). The standard sc heme is the Metrop olis algorithm whic h in a n utshell consists of the follo wing steps: Starting from an initial conguration of particles, a particle is c hosen at random and mo v ed to a new p osition at a random direction, for a distance equal to a fraction of the magnetic length l B If that mo v e results in a negativ e energy c hange E the mo v e is accepted. If not, it is accepted with a probabilit y exp ( 2 E ) since = 2. This sc heme ensures that in the limit where the n um b er of Mon te Carlo steps tends to innit y the particles will ha v e an equilibrium arrangemen t that corresp onds to the minim um of the energy In realit y one has to devise tests to ensure that the system has reac hed its equilibrium conguration, for the Mon te Carlo steps tak en. The co de that I dev elop ed for the Mon te Carlo sim ulation is presen ted in App endix E. Here I will fo cus on the details and the results of the sim ulation. The rst issue w as to c hec k the co de against kno wn analytical results for a uniform eld. It can b e sho wn that for the incompressible liquid state that the 2D plasma is in, the radius of the droplet whic h is formed scales as R = p 2 N where N is the n um b er of electrons in the droplet. 1 The densit y of the droplet is uniform and equal to = 1 = 2 In addition, the \energy" of the droplet 2 can also b e found. It has the form E = N 2 ln (2 N 1) = 4. Notice that in the ab o v e form ulas dimensionless 1 In realit y the particles in the sim ulated plasma gas are not electrons, as can b e inferred b y their in teraction energy and also the term \c harge" is more appropriate than solenoids. Ho w ev er, to a v oid confusion I retain the t w o terms, since the distinction is clear. 2 T o b e precise, it is the energy of the ctitious plasma.

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65 units ha v e b een used, for whic h l B = 1 and B = 1. All these theoretical predictions serv ed as consistency criteria for the sim ulation of the droplet in a uniform eld. There w as close agreemen t for v arious droplet sizes (n um b er of particles) and man y initial congurations of the droplet whic h indicated that the co de w as indeed w orking prop erly First, the system w as allo w ed to equilibrate and then measuremen ts w ere tak en. The time unit is one Mon te Carlo step, the mo v emen t of one particle in the sim ulation. After N mo v es, all the particles are mo v ed once on a v erage. This denes the Mon te Carlo sw eep. Equilibration for a t ypical droplet size of 500 particles to ok ab out 10 3 sw eeps, and measuremen ts w ere tak en t ypically for 4 10 4 sw eeps. This ensured go o d statistics. The quan tities measured w ere the total energy and the densit y In the end of the measuremen t cycle the a v erage densit y w as calculated from a previously calculated histogram of the particle p ositions. T o that end, a binning sc heme w as emplo y ed with bin sizes of 0 : 3 0 : 3. In gure (6.6) I sho w the results of a sim ulation for a uniform eld, starting with N = 100 electrons up to N = 400 electrons. The energy measured through the equilibration phase is sho wn for the last sim ulation N = 400, to illustrate the rapidit y of the equilibration pro cess. The measured energy is also sho wn. The densit y distribution is presen ted using pseudo-color pictures, in whic h blue is lo w densit y and red is high densit y I nd a droplet of uniform densit y and circular b oundary Before presen ting the sim ulations for the inhomogeneous magnetic eld a few commen ts are in order, on the t w o p ossible metho ds to implemen t the magnetic eld inhomogeneit y Initially I ran sim ulations with solenoids as the source of the inhomogeneit y The shortcomings of that metho d w ere that it w as hard to predict b eforehand what is the prop er v alue for the rux and at what distance the solenoids should b e placed to ha v e the desired eect, namely driv e the cusping of the droplet. These t w o v alues are in terrelated and the complication w as exacerbated b y the fact

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66 N = 100 N = 200 N = 300 0100 200 300 400 500 600 700 800 9001000sweeps -1.2e+05-1e+05-80000 -60000 -40000 -20000020000 40000 Equilibration energy for N=300 010000 20000 30000 40000sweeps -1.1018e+05 -1.1016e+05 Measured energy for N=300 Energy for N = 300 Figure 6.6: Gro wth of the electron droplet in a uniform magnetic eld. that for solenoids placed v ery close to the droplet and/or high ruxes, particles w ould break-o and attac h to the solenoids b efore the cusping o ccurred, an indication of an instabilit y Ho w ev er, this metho d pro v ed to b e in v aluable in assisting in c hoice of the parameters of the problem and the initial understanding of the system's complicated b eha vior. All the ab o v e problems w ere alleviated with the in tro duction of the inhomogeneit y in the magnetic eld through the p oten tial V ( z ) as describ ed in App endix C. As will b e shortly sho wn, the agreemen t b et w een the theory and the sim ulations is impressiv e. The gro wth of the droplet and the shap e of the b oundary w ere

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67 -50050100 -100-50050100 Figure 6.7: Illustration of the breakup of the droplet for a sim ulation with the n um b er of electrons N = 600 exceeding the critical n um b er N cr itical = 500. The droplet breaks up and the detac hed particles w ere accum ulated at the edge of the sim ulation b o x for coun ting purp oses. Eigh t y nine detac hed particles w ere coun ted, equally distributed among the three smaller droplets. found to b e consisten t with the predictions of the theory with uniform densit y in the in terior and a sharply dened b oundary Although the semiclassical theory breaks do wn at the cusping p oin t, the sim ulations w ere carried out past that p oin t with the same ndings in eac h case that I tried. The droplet w ould break up, with the excess n um b er of particles detac hing from the tips of the cusps and mo ving out w ards. In all cases the droplet w ould stabilize with a n um b er of particles equal to the critical n um b er N cr itical In gure (6.7) I sho w this b eha vior. In that particular sim ulation, an articial barrier w as placed at the edge of the sim ulation b o x in order to coun t the detac hed particles. The critical n um b er w as N cr itical = 500 and the sim ulation

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68 w as run for N = 600 particles. Eigh t y nine detac hed electrons w ere coun ted, equally distributed among the three smaller droplets at the b oundary I b egin with the droplet with three cusps. The appropriate conformal map is z ( w ) = aw + bw M with M = 2. The appropriate p oten tial for that situation is V ( z ) = t 3 z 3 = 2 and t 3 is related to the critical n um b er of electrons through N cr itical = 1 = 144 t 23 F or the details of the deriv ation the reader should consult App endix C. N = 100 N = 250 N = 350 N = 494 Figure 6.12: Gro wth of an electron droplet with three cusps. F or the sim ulations I c hose N cr itical = 500. In gure (6.12) I presen t the gro wth of the droplet for v arious n um b ers of electrons un til the cusping p oin t, sho wing a few represen tativ e densit y plots for N = 100, 250, 350 and 494. Notice that the cusping

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69 o ccurs for a n um b er of electrons N sligh tly less than the critical n um b er. In this case electrons started detac hing from the droplet for N = 494 instead of N = 500. In gure (6.13) I sho w the comparison b et w een the semiclassical curv es, colored red, with the results of the Mon te Carlo sim ulations, colored blue. 40 20 0 20 40 20 0 20 40 60Figure 6.13: Comparison of semiclassical (red) and MC (blue) results. Although in the ab o v e sim ulation the appropriate p oten tial w as used, it is also p ossible to obtain exactly the same results using six solenoids. The solenoids ha v e to ha v e alternate p ositiv e and negativ e rux and should b e placed at the v ertices of a regular hexagon, on a ring of of radius R s The rux and the distance R s are related to the relev an t harmonic momen t through t 3 = 4 =R 3 s Next, I turn to a sim ulation with t w o solenoids, placed on the x axis at opp osite p oin ts z 0 eac h carrying a rux = q ; this results in a droplet with t w o cusps. The appropriate conformal map is the mo died Jouk o wsky map z ( w ) = r w + 2 Qw = ( w 2

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70 N = 100 N = 250 N = 400 N = 465 Figure 6.18: Gro wth of an electron droplet with t w o cusps. w 2 0 ) and the t w o sets of parameters are related through z o = r w 0 + 2 Qw 0 1 w 4 0 ; (6.26) 2 q = r Q w 2 0 2 Q 2 (1 + w 4 0 ) (1 w 4 0 ) 2 ; (6.27) N = 1 2 r 2 4 Q (1 + w 4 0 ) (1 w 4 0 ) 2 : (6.28) F or this case also, the c hosen critical n um b er of electrons w as N cr itical = 500. In gure (6.18) I presen t the gro wth of the droplet for v arious n um b ers of electrons un til the cusping p oin t, sho wing a few represen tativ e densit y plots for N = 100, 250, 400 and

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71 465. In gure (6.19) I sho w the comparison b et w een theory and exp erimen t using the same color-co ding sc heme as b efore. Figure 6.19: Comparison of semiclassical (red) and MC (blue) results. I carried out another sim ulation whic h pro duced b eautiful pictures for whic h, unfortunately v ery little analysis can b e carried out. An ann ulus of 350 repulsiv e solenoids w as placed in the middle of the sim ulation b o x with man y solenoids distributed randomly in it. Fift y attractiv e solenoids w ere placed in the exterior of the sim ulation b o x, equiv alen t to a large distance, to enforce \c harge" neutralit y The sim ulation resulted in a v ery complicated densit y plot as can b e seen in gure (6.26). Finally a sim ulation for an also incompressible state of the F ractional Quan tum Hall Eect, with lling factor = 1 = 3 is presen ted. I used the same p oten tial V ( z ) = t 3 z 3 = 2 used in the sim ulation for the droplet with three cusps. The results w ere

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72 N = 100 N = 140 N = 180 N = 220 N = 260 N = 300 Figure 6.26: Gro wth of an electron droplet within a random arra y of solenoids.

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73 not what I initially exp ected. Based on theoretical grounds, the = 1 = 3 state w as exp ected to ha v e smaller densit y ructuations in the in terior of the droplet but the observ ed b eha vior in the sim ulations w as quite the con trary As seen in gure (6.33), there is an enhancemen t of the densit y near the b oundary and also, what is ev en more p erplexing, is that the breakup of the droplet o ccurs in a dieren t manner than for the = 1 case. It seems that a smaller droplet detac hes from the larger initial droplet. I do not ha v e an explanation for this b eha vior, although the enhancemen t of the densit y near the b oundary has b een encoun tered in previous n umerical sim ulations of the F ractional Quan tum Hall eect.

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74 N = 60 N = 120 N = 140 N = 160 N = 260 N = 167 Figure 6.33: Gro wth of an electron droplet for = 1 = 3.

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CHAPTER 7 CONCLUSION I ha v e in v estigated the structural phase transitions of the v ortex lattice within the anisotropic Ginzburg-Landau mo del, the anisotrop y arising from s and d mixing or from an anisotropic F ermi surface. I ha v e sho wn that the addition of the phenomenological higher order deriv ativ e term, is sucien t to accoun t for the observ ed b eha viors of the v ortex lattice. I deriv ed an extension of the virial theorem for sup erconductivit y to anisotropic sup erconductors, whic h enables one to write the free energy in a compact form from whic h the deriv ation of the nondisp ersiv e elastic mo duli is v ery easy The particular extension of the virial theorem is sho wn to b e a to ol whic h facilitates the generalization of the famous Abrik oso v iden tities along with other results obtained from the isotropic Ginzburg-Landau theory to the anisotropic theory The rhom bic to square structural transition w as studied and its eects on the elastic mo duli w ere analyzed. This mo del exhibits the same b eha vior at in termediate elds as the nonlo cal London mo del, v anishing of the squash mo dulus c sq at the rhom bic to square phase transition, v anishing of the rotational mo dulus c r at the p oin t where the v ortex lattice exhibits rotational instabilit y A t high elds, near H c 2 the mo duli v anish. I incorp orated the thermal ructuations in this mo del in a simplied manner, not taking in to accoun t the eect of the thermal ructuations on H( T ). Nonetheless, this approac h pro v es sucien t to sho w that the reen tran t transition from square to rhom bic v ortex lattice, rst encoun tered in the nonlo cal London mo del, is presen t in this mo del also. The mec hanism that causes the reen tran t b eha vior is the thermal smearing of the anisotrop y 75

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76 Although the anisotropic term is not deriv ed rigorously from a microscopic mo del, it succeeds in encompassing all the in teresting, and at times unexp ected, phenomena p ertaining to the structural transitions of the v ortex lattice. I also in v estigated the ngering of electron droplets in the Quan tum Hall regime driv en b y the inruence of an inhomogeneous magnetic eld via the Aharono v-Bohm eect. Carrying out detailed Mon te Carlo sim ulations of the gro wth of the electron droplets for v arious eld inhomogeneities, I w as able to compare the theoretical mappings I found for eac h inhomogeneit y to the exp erimen ts. I found that the dev elopmen t of the b oundary of the electron droplet up to the cusping p oin t w as describ ed with remark able accuracy b y the curv es obtained from the prop er conformal map for eac h inhomogeneit y In addition I w as able to predict the critical n um b er of electrons for eac h particular arrangemen t of solenoids outside the electron droplet. Tw o metho ds w ere dev elop ed for the sim ulation of the magnetic eld inhomogeneit y One in whic h the inhomogeneit y is pro vided b y thin solenoids distributed in the exterior of the droplet and another in whic h the eectiv e p oten tial that aects the electrons of the droplet is deriv ed in a systematic w a y Using the latter I w as able to study the gro wth of the electron droplet after the n um b er of electrons exceeded the critical n um b er at whic h cusping o ccurs. I ha v e found that the excess electrons w ere detac hed from the droplet and mo v ed to w ards the b oundary of the sim ulation b o x.

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APPENDIX A EXTENSION OF THE VIRIAL THEOREM In this app endix I sho w the deriv ation of the Virial Theorem for the anisotropic Ginzburg-Landau theory As will b ecome so on apparen t, the Virial Theorem facilitates greatly the simplication of the subsequen t analysis of the problem at hand. The deriv ation is essen tially the generalization to the anisotropic case of the deriv ation b y Doria et al. [73] for the isotropic theory The starting p oin t is the familiar Ginzburg-Landau free energy F = j jj ( x ) j 2 + 2 j ( x ) j 4 + j ( x ) j 2 2 m + [ r A ( x ) ] 2 8 + r 2 v s [ y ( x )][ y ( 2x 2y ) ( x )] [ x ( x )][ x ( 2x 2y ) ( x )] + cc ; (A.1) where r A ( x ) = B ( x ) is the microscopic magnetic eld. The magnetic induction is dened as B = R d 3 x B ( x ) =V and the homogeneous applied eld is denoted b y H ; the brac k ets denote spatial a v eraging. I can also dene the quan tities F k inetic = j ( x ) j 2 2 m ; (A.2) F f iel d = B 2 ( x ) 8 ; (A.3) F anisotr opic = r 2 v s [ y ( x )][ y ( 2x 2y ) ( x )] [ x ( x )][ x ( 2x 2y ) ( x )] + cc ; (A.4) b y iden tifying the ph ysical meaning of eac h comp onen t in the Ginzburg-Landau free energy 77

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78 Under a scaling transformation the v arious comp onen ts of the Ginzburg-Landau free energy transform in the follo wing manner x 0 = x ; (A.5) ( x 0 ) = ( x 0 ) ; (A.6) A ( x 0 ) = A ( x 0 ) ; (A.7) 0 ; (A.8) B 2 B : (A.9) The next step is to tak e the partial deriv ativ e with resp ect to and set it equal to zero. I neglect the -dep endence of the order parameter and the v ector eld b ecause the Ginzburg-Landau free energy is stationary under v ariations of these elds b y denition. Finally after setting = 1, I obtain 2 F k inetic 4 F f iel d 4 F anisotr opic + @ F @ B 2 B = 0 ; (A.10) whic h can b e recast in the follo wing form whic h con tains the applied eld, using the thermo dynamic relation H = 4 @ F =@ B : H B 4 = j ( x ) j 2 2 m + B 2 ( x ) 4 + 2 r 2 v s [ y ( x )][ y ( x 2 y 2 ) ( x )] [ x ( x )][ x ( x 2 y 2 ) ( x )] + cc : (A.11) The ab o v e expression can b e simplied ev en further with an in tegration b y parts. One has to use the appropriate b oundary condition (2.10) n r d ( d ) r 2 v s y y ( 2y 2x ) d x x ( 2y 2x ) d = 0 : (A.12)

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79 One is th us led to the follo wing simplied result, whic h will b e used in the deriv ation of the generalized Abrik oso v iden tities H B = 4 V Z d 3 x j ( x ) j 2 2 m + B 2 ( x ) 4 2 r 2 v s ( x )( 2x 2y ) 2 ( x ) + cc : (A.13) A.1 A Useful Iden tit y Starting from the reasonable assumption that the free energy is stationary under (1 + ) with b eing a constan t, v ariation with resp ect to leads to [3] F = 2 j j 2 2 m r 2 v s ( 2x 2y ) 2 + cc j jj j 2 + j j 4 = 0 ; (A.14) whic h m ust hold for ev ery In this w a y I obtain the generalization of a w ell kno wn and v ery useful iden tit y j j 2 2 m r 2 v s ( 2x 2y ) 2 + cc = j jj j 2 j j 4 : (A.15) The latter p ermits one to write the virial theorem in the follo wing compact form, from whic h the deriv ation of the Abrik oso v iden tities is trivial: H B = 4 j jj j 2 j j 4 + B 2 ( x ) 4 2 r 2 v s R e [ ( 2x 2y ) 2 ] : (A.16) A.2 Generalized Abrik oso v Iden tities The generalizations of the Abrik oso v iden tities are going to b e deriv ed as w as done for the isotropic sup erconductor b y Klein and P ottinger [74]. I b egin b y assuming that an expansion of the in ternal eld can b e written as B ( x ) = [ H ( x )] z and H = [ H c 2 A ] z with A and of order j j 2 and A constan t. Substituting in equation

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80 (A.16) and separating terms of dieren t orders in j j 2 I obtain Zeroth Order: H 2 c 2 = H 2 c 2 : (A.17) First Order: (2 A + ) H c 2 = 4 j jhj j 2 i + 2 H c 2 A + 2 H c 2 h i + 8 r 2 v s R e h ( 2x 2y ) 2 i : (A.18) Av eraging b oth sides and switc hing to the usual dimensionless units (see section 2.1) I obtain h i = 1 2 hj j 2 i R e h ( 2x 2y ) 2 i : (A.19) Here, is the dimensionless anisotrop y parameter dened as = r 2 v j j =r 2 d s F rom the relation B ( x ) = [ H ( x )] z it follo ws immediately that ( x ) = 1 2 j j 2 R e [ ( 2x 2y ) 2 ] ; (A.20) B = H 1 2 hj j 2 i + R e h ( 2x 2y ) 2 i : (A.21) Second Order: A 2 + A = 4 hj j 4 i + h A 2 i + h 2 i + 2 h A i Av eraging again and using A = H (switc hing once more to dimensionless units) I obtain after a short calculation H hj j 2 i 2 R e h ( 2x 2y ) 2 i = 2 2 1 2 2 hj j 4 i + 2 2 R e hj j 2 ( 2x 2y ) 2 i : (A.22) The familiar Abrik oso v iden tities for the isotropic sup erconductor are immediately obtained b y setting the anisotrop y equal to 0 in the generalized Abrik oso v iden tities (A.21,A.22).

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81 A v ery useful relation for the a v erage magnetic eld can b e obtained com bining the generalized Abrik oso v iden tities and it reads B = H H (2 2 1) A 1 4 R e h ( 2x 2y ) 2 i hj j 2 i + R e hj j 2 ( 2x 2y ) 2 i (2 2 1) A hj j 2 i 2 ; (A.23) where A is the Abrik oso v parameter dened as A = hj j 4 i = hj j 2 i 2 It is also con v enien t to dene whic h is of zeroth order in j j 2 as = 4 R e hj j 2 ( 2x 2y ) 2 i hj j 2 i 2 + A (2 2 1) R e h ( 2x 2y ) 2 i hj j 2 i ; (A.24) whic h simplies the follo wing expressions for the a v erage magnetic eld B the correction A and the applied eld whic h in turn will b e used in the deriv ation of the free energy: B = H H (2 2 1) A 1 (2 2 1) A ; (A.25) A = ( B ) (2 2 1) A + 1 (2 2 1) A + (2 2 1) A + 1 ; (A.26) H = B + B (2 2 1) A + 1 1 (2 2 1) A + 1 : (A.27) A.3 F ree Energy Magnetization, Gibbs Energy The free energy can b e written in a v ery manageable form if one uses iden tit y (A.15). I obtain F = 1 2 hj j 4 i + h B 2 i = 1 2 hj j 4 i + h ( A ) 2 i = ( A )( A 2 h i ) 1 2 hj j 4 i + hj j 2 i : (A.28)

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82 Using Abrik oso v's iden tities again, I obtain hj j 4 i 2 + hj j 2 i = A ( B A ) ; (A.29) and no w the free energy (A.28) can b e written F = ( B )( A ) + B : (A.30) Then, from (A.26) one obtains easily the generalization of the isotropic result for the free energy F = B 2 ( B ) 2 (2 2 1) A + 1 1 (2 2 1) A + 1 : (A.31) The magnetization M = ( B H ) = 4 and the Gibbs free energy G = F H B = 4 are calculated easily from the previous results. I nd M = 1 4 H (2 2 1) A 1 (2 2 1) A ; (A.32) G = ( H ) 2 (2 2 1) A 1 (2 2 1) A : (A.33) In short, I ha v e managed to signican tly reduce the complexit y of the original Ginzburg-Landau free energy using the generalized Virial Theorem and at the same time obtain simple expressions for imp ortan t relev an t quan tities.

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APPENDIX B CALCULA TION OF A h ( 2X 2Y ) 2 i hj j 2 ( 2X 2Y ) 2 i In this section I sho w the details of the calculation of the necessary spatial a v erages of the order parameter. The calculations are done follo wing the con v en tions and metho dology of Chang et al. [79]. The metho d is essen tially the same if one w an ts to include the second order correction. I will sho w the details for completeness. In order to construct the p erio dic solution one can form the follo wing linear com bination whic h can b e tuned so that j j 2 acquires the desired p erio dicit y that the simple solution of the linearized rst Ginzburg-Landau equation lac ks, as ( x; y ) = + 1 X n = 1 c n exp ( 2 in a y ) ( x 2 n b 2 a ) ; (B.1) where a and b are the t w o basis v ectors and b is the reduced eld. This function is p erio dic in the direction of a (the y direction). One can imp ose p erio dicit y in the b direction b y requiring that ( x b sin ; y + b cos ) = ( x; y ) holds. This is accomplished b y setting b sin = 2 =b 2 a, a condition whic h implies that one has one v ortex p er unit cell. I then c ho ose a co ordinate system ( X ; Y ) that coincides with the v ortex lattice directions so that ( x; y ) = ( Y sin ; X Y cos ). The spatial a v erages are going to b e calculated in tegrating o v er one lattice cell, namely 0 < Y < b and 0 < X < a, but the nal in tegrals will b e extended o v er all space taking adv an tage of the p erio dicit y of j j 2 The in tegrals in v olving j j 2 are the easiest to ev aluate. The in tegration o v er Y will result in a Kronec k er whic h will help in the ev aluation of the rst summation. 83

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84 The second summation will extend the in tegral o v er all space. I ha v e hj j 2 i = 1 b sin Z + 1 1 d x j ( x ) j 2 ; (B.2) h ( 2x 2y ) 2 i = 1 b sin Z + 1 1 d x ( 2x 2y ) 2 : (B.3) The calculation of hj j 4 i is sligh tly more complicated. No w a pro duct of four w a v efunctions is in v olv ed and I obtain hj j 4 i = 1 b sin X nml q Z 0 b sin d x n + l ;m + q exp 2 i b cos a + n 2 m 2 + l 2 q 2 2 n m l q ; (B.4) where j = ( x j b sin ). One can c ho ose new v ariables Z = n + l = m + q N = n l M = m q whic h facilitates the ev aluation of the sum under the constrain t of the particular Kronec k er The sum o v er n m l q breaks in to t w o sums, one o v er Z and one for ev en and o dd N and M resp ectiv ely In this case it is the in tegration o v er Z that will extend the in tegration o v er one cell to an in tegration o v er all space. Before calculating the in tegrals, one needs to rotate the deriv ativ e term b y an angle ab out the z axis to accoun t for the general orien tation of the v ortex lattice. The calculation of the in tegrals is tedious but straigh tforw ard. I sho w the results b elo w [the prime denotes a rotated term follo wing the notational con v en tion in tro duced in (2.1)]: hj j 2 i = 1 + 3 8 ( b ) ; (B.5) hj j 4 i = 0 + ( b ) 1 + ( b ) 2 [ r 1 + r 2 + 5 1 ] ; (B.6) h ( 2x 2y ) 0 2 i = 2 f 1 + 3( b ) g ; (B.7) hj j 2 ( 2x 2y ) 0 2 i = 2 0 + 1 + ( b ) 3 2 0 + 6 1 + 2( r 1 + r 2 ) ; (B.8)

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85 The functions 0 1 r 1 and r 2 ha v e the follo wing form 0 = X nm 0 A nm ; (B.9) 1 = R e e 4 i X nm 0 A nm 8 2 2 n 4 6 n 2 + 3 8 ; (B.10) r 1 = R e e 8 i X nm 0 A nm 16 4 4 n 8 12 3 3 n 6 + 3 4 2 2 n 4 45 16 n 2 + 105 256 ; (B.11) r 2 = X nm 0 A nm 16 4 4 n 4 m 4 12 3 3 n 2 m 2 ( n 2 + m 2 ) + 3 4 2 2 ( n 4 + m 4 + 36 n 2 m 2 ) 45 16 ( n 2 + m 2 ) + 105 256 ; (B.12) where A nm = p e 2 i ( n 2 m 2 ) e 2 ( n 2 + m 2 ) The prime has the meaning that there are t w o summations|the one sho wn, o v er n m and the other with n and m replaced b y ( n + 1 = 2) and ( m + 1 = 2) resp ectiv ely

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APPENDIX C MOMENTS In this app endix I presen t the calculation of the external and in ternal momen ts of v arious useful conformal maps. They are related to the harmonic momen ts through t k = c k k : (C.1) I mak e use of one little kno wn but imp ortan t theorem from the theory of complex v ariables. F or a function f whic h is analytic inside, outside and on a simple closed lo op C with the exception of a nite n um b er of n singular p oin ts z k inside C with k = 1 : : : n it holds I C d z f ( z ) = 2 i n X k =1 R es z = z k f ( z ) = 2 iR es z =0 1 z 2 f 1 z : (C.2) C.1 Conformal Map z = f ( w ) = aw + bw M The conformal map z = f ( w ) = aw + bw M is appropriate for the in v estigation of the cusping of the droplet with M + 1 cusps, with their tips residing on a ring at regular angles m = ( M + 1), with m = 0 ; 1 ; : : : ; M + 1. The deriv ativ e f 0 ( w ) = a M bw ( M +1) has zero es at w d k = M b a 1 M +1 e 2 ik M +1 ; with k = 0 : : : M (C.3) whic h are in the in terior of the unit circle j w j = 1 pro vided that M b=a < 1. When these ro ots fall on to the unit circle, the v elo cit y v = j d w = d z j div erges, signaling the formation of a singularit y 86

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87 C.1.1 Exterior Momen ts I need to ev aluate the external momen ts in order to nd the connection b et w een the conformal map and the harmonic momen ts of the solenoid distribution outside the droplet. I start with the denition c j = Z Z exter ior z j d x d y = 1 2 i I d w f f 0 f j : (C.4) The calculation for j = 0 is straigh tforw ard and yields c 0 = ( a 2 M b 2 ) : (C.5) F or the other momen ts I ha v e to ev aluate the in tegral c j = 1 2 i I d w w ( a 2 M b 2 ) | {z } I 1 M abw ( M +1) | {z } I 2 + abw ( M +1) | {z } I 3 w j M [ aw M +1 + b ] j ; (C.6) with f j ( w ) = w j M = [ aw M +1 + b ] j ha ving ( M + 1) p oles of order j in w f k = b a 1 M +1 e i (2 k +1) M +1 ; with k = 0 : : : M : (C.7) Using the result (C.2) I easily obtain for the three parts I 1 I 2 and I 3 : I 1 = R es w =0 w j ( M +1) j M 1 [ a + bw M +1 ] j = 0 ; (C.8) for j ( M + 1) j M 1 0 ) j 1 and also I 2 = R es w =0 w j ( M +1) j M + M [ a + bw M +1 ] j = 0 ; (C.9)

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88 for j ( M + 1) j M + M 0 ) j M along with I 3 = R es w =0 w j ( M +1) j M M 2 [ a + bw M +1 ] j = 0 ; (C.10) for j ( M + 1) j M + M 2 0 ) j M + 2 with the exception of j = M + 1 for whic h there is a residue at w = 0 with the v alue a j This is deriv ed from the expansion 1 w a j (1 + q M +1 ) j = 1 w a j 1 + ( 1) M +1 q M +1 + q 2( M +1) + : : : ; (C.11) with q = ( b=a ) 1 = ( M +1) w and as required j q j < 1. I nd th us that only t w o momen ts ha v e v alues other that zero c 0 = ( a 2 M b 2 ) ; (C.12) c M +1 = b a M : (C.13) F or the calculation of the critical n um b er of electrons N cr itical at whic h the cusps o ccur I notice that when the ro ots of the conformal map fall on to the unit circle I ha v e b = a= M Solving the t w o equations for the exterior momen ts for c 0 = 2 N I obtain N cr itical = M 1 2 M M c M +1 2 M 1 : (C.14) The time t in the Laplacian gro wth problem is equiv alen t to t w o times the n um b er of electrons 2 N in the Quan tum Hall droplet gro wth problem. C.1.2 In terior Momen ts The calculation of the in terior momen ts follo ws along the same line and is sho wn for completeness. Regardless of the closed form obtained, the results are not useful and it actually can b e sho wn that they formally div erge. Substituting j j in to

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89 the RHS of equation (C.6) I ha v e c j = 1 2 i I d w w ( a 2 M b 2 ) | {z } I 1 M abw ( M +1) | {z } I 2 + abw ( M +1) | {z } I 3 [ aw M +1 + b ] j w j M ; (C.15) Using again the theorem (C.2) I obtain for I 1 I 2 and I 3 I 1 = R es w =0 a j w j +1 1 + b a w M +1 j = R es w =0 a j w j +1 j X k =0 j k b a 1 M +1 w k ( M +1) : (C.16) The residue is dieren t from zero for j + 1 k ( M + 1) = 1 ) k = j = ( M + 1). So, for j = m ( M + 1) with m = 1 ; 2 ; : : : I ha v e I 1 = ( a 2 M b 2 )( a M b ) m m ( M + 1) m ; (C.17) and similarly for I 2 I 3 I obtain I 2 = M a 2 ( a M b ) m m mM + 1 m ( M + 1) m (C.18) I 3 = b 2 ( a M b ) m mM m + 1 m ( M + 1) m : (C.19) Com bining these three results I nd the nal expression for the in terior momen ts c m ( M +1) = ( a M b ) m m ( M + 1) m a 2 mM + 1 M b 2 m + 1 ; (C.20) with m = 1 ; 2 : : : One easily sees that these div erge for m 1

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90 C.2 Conformal Map z = f ( w ) = aw + P Mk =1 b k w k This is a generalization of the previous conformal map. I easily obtain follo wing the metho dology dev elop ed in the previous section the exterior momen ts c 0 = a 2 M X k =1 k b 2k (C.21) c j = a j ab j 1 M X k =1 k b k b k + j 1 ; (C.22) with j = 1 ; 2 ; : : : ; ( M + 1) and the implicit assumption that b 0 0 and b j >M 0. In this case only c 0 c 1 : : : c M +1 are non-zero. The calculation of the in terior momen ts is forbiddingly dicult. C.3 Conformal Map z = f ( w ) = r w + Q= ( w w 0 ) + Q= ( w + w 0 ) This is a mo died Jouk o wsky map whic h corresp onds to a droplet with t w o cusps on the p ositiv e and negativ e x axis resp ectiv ely It has the form z = f ( w ) = r w + Q w w 0 + Q w + w 0 = r w + 2 Qw w 2 w 2 0 : (C.23) The deriv ativ e of the conformal map has zero es at f 0 ( w ) = 0, whic h giv es w 2 = w 2 0 + Q r s Q r 2 + 4 Q r w 2 0 ; (C.24) under the constrain ts that 0 < w 0 < 1, Q=r 1 = 2 and w 0 p 2 Q=r The external momen ts can no w b e obtained as b efore c j = 1 2 i I d w w r w + 2 Qw 1 ( w w 2 0 ) r 2 Q ( w 2 + w 2 0 ( w 2 w 2 0 ) 2 ( w 2 w 2 0 ) j w j [ r ( w 2 w 2 0 ) + 2 Q ] j : (C.25) Although the general ev aluation of the preceding in tegral is out of the question in this case, I can ev aluate the rst few nonzero external momen ts. With the aid of Maple

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91 for the non trivial calculations, I obtained the follo wing expressions for the rst three nonzero external momen ts c 0 = r 2 4 Q (1 + w 4 0 ) (1 w 4 0 ) 2 ; (C.26) c 2 = 2 Q [ r (1 w 4 0 ) + 2 Qw 2 0 ] 2 r (1 w 4 0 ) 2 2 Qw 2 0 (1 + w 4 0 ) ; (C.27) c 4 = 2 Qw 2 0 (1 w 4 0 ) 2 [ r (1 w 4 0 ) + 2 Qw 2 0 ] 4 r (1 w 4 0 ) 2 2 Qw 2 0 (1 + w 4 0 ) : (C.28) C.4 Connection with Magnetic Field Inhomogeneit y In order to compare the theoretical results with the Mon te Carlo sim ulation I need to connect the harmonic momen ts calculated in the previous sections with the harmonic momen ts of the eld inhomogeneit y dened as t k = 1 k Z B z k d 2 z ; (C.29) where B is the magnetic eld inhomogeneit y This in turn will enable us to nd the expansion of the p oten tial V ( z ) whic h has the form V ( z ) = 1 2 1 X k =1 t k z k : (C.30) F or an arrangemen t of thin solenoids lik e the one I use in the sim ulations, the harmonic momen ts t k can b e ev aluated for the inhomogeneit y has the form B = X n n (2) ( z z n ) ; (C.31) where n is the rux that the n -th solenoid encloses. I nd that t k = 2 k X n n z k n : (C.32)

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92 The harmonic momen ts t k are connected to the exterior momen ts through the simple relation t k = c k k : (C.33) A t this stage there are t w o w a ys to pro ceed. The ob jectiv e is to compare the ev olution of the conformal map in \time" (equiv alen t to the n um b er N of electrons in this case) with the actual sim ulation. The \time" en ters through the zeroth-order exterior momen t c 0 = 2 N Therefore there is the ob vious option of solving the equations pro vided b y the calculation of the exterior momen ts whic h will b e pro v en dicult to carry out in general and the not so ob vious at rst glance option of calculating the p oten tial V ( z ) rst. Both metho ds will b e presen ted next. C.4.1 Exterior Momen ts I will fo cus on the solution for six solenoids on a ring of radius R s eac h of whic h carries a rux with alternating signs as one tra v els on the ring, distributed uniformly at angles whic h are m ultiples of = 3. The appropriate conformal map for this problem is z = f ( w ) = aw + bw M with M = 2. F rom equations (C.32,C.33) I ha v e c 0 = 2 N ; (C.34) c 3 = 12 R 3 s ; (C.35) whic h are the rst non-zero exterior momen ts for that particular arrangemen t of solenoids. The solution of equations (C.12,C.13) yields after c ho osing the appropriate ro ots a = 1 2 c 3 2 p 4 8 c 0 c 3 1 2 ; (C.36) b = 1 4 2 p 4 8 c 0 c 3 ; (C.37)

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93 with cusps forming for c 0 c 3 = 3 = 8 from whic h I can nd the critical n um b er of electrons N cr itical = R 3 s 48 2 : (C.38) I can obtain the same equation for the critical n um b er N cr itical b y direct substitution of the parameters in equation (C.14), whic h serv es as a consistency c hec k. The connection with the exp erimen t has b een established in this particular case as one can plot the curv es obtained from the conformal map z = f ( w ) = aw + bw 2 for a giv en n um b er of electrons and compare directly with the results of the Mon te Carlo sim ulation. Unfortunately an analytic approac h of this kind is p ossible for the simplest of cases and fails for higher M b ecause of the lac k of analytic solutions of high order p olynomials. In general, for giv en M one has to solv e a p olynomial of order 2 M There is a w a y around that dicult y and the metho d is presen ted in the next section. C.4.2 P oten tial V ( z ) First I need to in tro duce the Sc h w arz function for a curv e C A closed curv e C in the plane is describ ed b y an equation of the form f ( x; y ) = 0 whic h in terms of complex co ordinates can b e written as g ( z ; z ) = 0. F or a function g whic h is analytic, I can solv e for z and th us obtain the Sc h w arz function S ( z ), z = S ( z ) ; on C : (C.39) The Sc h w arz function is a v ery in teresting ob ject, with man y prop erties and applications. The in terested reader can refer to a nice b o ok b y Da vis [102]. F or m y purp oses I need the exterior expansion of the Sc h w arz function S (+) ( z ) whic h acquires meaning if I analytically con tin ue S ( z ) to a strip-lik e domain whic h con tains the curv e C It

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94 has the form [100] S (+) ( z ) = 1 1 X k =1 c k z k 1 ; (C.40) where the external momen ts can b e dened in terms of the Sc h w arz function as follo ws c k = 1 2 i I C z k S ( z ) d z ; with k = 1 ; 2 ; : : : : (C.41) Recalling that the p oten tial has the expansion [86] V ( z ) = 1 2 R e 1 X k =1 t k z k ; (C.42) I immediately nd that the p oten tial V ( z ) and the external expansion of S ( z ) are related b y V 0 ( z ) = 1 2 R eS (+) ( z ) : (C.43) The ob jectiv e no w is to nd the Sc h w arz function S (+) ( z ) for the conformal map that I c ho ose. I start with the conformal map z ( w ) = aw + bw M In v erting the equation and expressing w ( z ) as a Lauren t series I obtain w ( z ) = z a ba M 1 z M + O ( z (2 M +1) ) : (C.44) F or the Sc h w arz function I ha v e S ( z ) = a w ( z ) + bw M ( z ) b a M z M + a 2 M b 2 z + O ( z ( M +2) ) : (C.45) Comparing term b y term with the expansion S ( z ) = ( M + 1) t M +1 z M + t z + O ( z 2 ) ; (C.46)

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95 I obtain the expansion co ecien ts of the p oten tial V ( z ), namely t M +1 = ba M = ( M + 1) and t = a 2 M b 2 The connection with the n umerical exp erimen t has again b een established, as one can use the deriv ed p oten tial to sim ulate the droplet and subsequen tly compare with the theoretical results. What is most imp ortan t, all the diculties related to the analytical solution for this arrangemen t of solenoids ha v e v anished. With regard to the sim ulation, the route one follo ws is simple. First, b y the appropriate tuning of b I c ho ose a con v enien t v alue for the critical n um b er of electrons N cr itical equation (C.14). Then the sim ulation is carried out with the appropriate t M +1 In the end, the curv es obtained from the conformal map at dieren t \times" are compared to the b oundary of the droplet at the corresp onding n um b er of electrons. The mo died Jouk o wsky map is also simple to treat using this metho d. In v erting the conformal map I obtain w ( z ) = z 3 r + x 1 3 3 r + z 2 6 r Q + 3 r 2 w 2 0 3 r x 1 3 ; (C.47) where the parameter x has the form x = z 3 9 z r Q 9 z r 2 w 2 0 + 3 r q z 2 (30 r Qw 2 0 + 6 r 2 w 4 0 3 Q 2 ) 3 z 4 w 2 0 3 r ( r w 2 0 2 Q ) 3 : (C.48) The Sc h w arz function is S ( z ) = r =w ( z ) + 2 Qw ( z ) = [1 ( w ( z ) w 0 ) 2 ]. The rst term has no p oles and the p oles of the second term are found b y solving w 2 ( z ) = w 2 0 With the help of Maple I nd the simple result z 0 = r w 0 + 2 Qw 0 1 w 4 0 : (C.49)

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96 In order to isolate the p oles I write (neglecting the rst term) S ( z ) Qw 1 w w 0 + Qw 1 + w w 0 = Qw ( z 0 ) w 0 w 0 ( z 0 )( z z 0 ) + Qw ( z 0 ) w 0 w 0 ( z 0 )( z + z 0 ) ) (C.50) S ( z ) 2 q z z 0 2 q z + z 0 ; (C.51) where I ha v e dened 2 q = Qw ( z 0 ) w 0 w 0 ( z 0 ) = Q w 2 0 r 2 Qw 2 0 (1 + w 4 0 ) (1 w 4 0 ) 2 : (C.52) A simple in tegration yields the p oten tial V ( z ) V ( z ) = q ln( z z 0 ) q ln( z + z 0 ) ; (C.53) whic h is what one w ould exp ect for t w o solenoids lo cated at z 0 Summarizing, I sho w the relations b et w een the parameters of the p oten tial and the conformal map z 0 = r w 0 + 2 Qw 0 1 w 4 0 ; (C.54) 2 q = r Q w 2 0 2 Q 2 (1 + w 4 0 ) (1 w 4 0 ) 2 ; (C.55) N = 1 2 r 2 4 Q (1 + w 4 0 ) (1 w 4 0 ) 2 : (C.56) The last equation is deriv ed from c 0 = 2 N Once a suitable c hoice of q and z 0 has b een made, the results of the sim ulation can b e directly compared to the theoretical curv es obtained from the conformal map w ( z ) = r w + 2 Qw = ( w 2 w 2 0 ).

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APPENDIX D CODE F OR THE NUMERICAL CALCULA TIONS In this App endix I sho w the co de that w as dev elop ed and used for the n umerical part of the w ork presen ted in the rst part of this dissertation. Care w as tak en to ha v e a consisten t mapping b et w een the sym b ols used in the theoretical part and the v ariables used in the co de. If presen t, alternativ e sym b ol con v en tions are explained. The co de itself is Matlab 's programming language whic h is a form of pseudo-C. F or the most part the co de is self explanatory and requires minimal familiarit y with an y computer language for complete understanding. The co de ma y seem a wkw ard at some places but the h uman reader has to k eep in mind that the mec hanical reader needs sp ecial instruction to pro duce fast co de. The necessit y to do so b ecomes immediately apparen t if one considers that the minimization has to b e executed for as man y v alues of the magnetic eld necessary to pro duce the graphs in this dissertation. I dev elop ed t w o driv ers, driver1.m and driver2.m eac h assigned to a sp ecic task. The call tree is v ery simple: the rst driv er calls energy.m and moduli.m while the second driv er calls csquash.m The rst driv er driver1.m automates the minimization of the free energy and the calculation of the elastic mo duli. The second driv er driver2.m calculates the b oundary in the H { T plane of the phase transition b y nding the zero es of the squash mo dulus n umerically Eac h driv er mak es use of t w o in ternal Matlab functions resp ectiv ely The functions are fmincon for the minimization and fzero for ro ot nding. The reader requiring more information should consult the appropriate do cumen tation for details ab out the fmincon and fzero Matlab functions. 97

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98 D.1 driv er1.m clear all; format long; digits(64);global kappa; global aniso; global field; kappa=5.0;anisotropy=0.11;fd=fopen('helm.dat','w') ; gd=fopen('mod.dat','w'); % fmincon options options=optimset('LargeS cale ',' on', 'Dis play ',' iter ','G radO bj' ,'on '); options=optimset(options ,'To lX' ,1e11,' TolF un' ,1e11,' Hess ian ','o n'); options=optimset(options ,'Ma xIt er', 9000 ,'Ma xFu nEva ls', 9000 ); % Initial guess xo_r=0.43;xo_s=0.91;xo_p=35.12*pi/180.0;lb=[0.0,0.2,0.0*pi/180.0 ]; % Lower bound for the three variables ub=[0.5,1.0,45.0*pi/180. 0]; % Upper bound for the three variables step=0.001;for field=0.01:step:1.0, aniso=field*anisotropy;xo=[xo_r xo_s xo_p]; % The call to the minimizer [f,fval,xf,output,lambda ,gra d]= fmin con( 'ene rgy ',xo ,[], [],[ ],[ ], lb,ub,[],options); fprintf(fd,'%7.5f %6.2f %14.12f %7.5f %7.5f %6.2f\n',field, 180.0/pi*acot(f(1)/f(2) ),fv al,f (1), f(2 ),18 0.0/ pi*f (3) );

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99 % Here we set the found minimum as the % new trial point in the next iteration xo_r=f(1);xo_s=f(2);xo_p=f(3); % Calculate the elastic moduli and output to file [c_sq,c_x,c_y,c_r,b]=mod ([f( 1) f(2) f(3)]); fprintf(gd,'%7.5f %12.10f %12.10f %12.10f %12.10f %12.10f\n',field, c_sq,c_x,c_y,c_r,b); end;fclose(gd);fclose(fd); D.2 energy .m function [helm,G,H]=energy(x) N=5;global kappa; global aniso; global field; rho=x(1);sigma=x(2);phi=x(3);pi2=pi*pi;pi3=pi2*pi;pi4=pi3*pi;aniso2=aniso*aniso;sigma2=sigma*sigma;sigma3=sigma2*sigma;sigma4=sigma3*sigma;ss=sqrt(sigma);b_0_1=0.0; d_1_1=0.0; b_0_2=0.0; d_1_2=0.0; b_0_1_r=0.0; d_1_1_r=0.0; d_1_1_ss1=0.0;

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100 b_0_2_r=0.0; d_1_2_r=0.0; d_1_2_ss1=0.0; b_0_1_s=0.0; d_1_1_s1=0.0; d_1_1_ss2=0.0; b_0_2_s=0.0; d_1_2_s1=0.0; d_1_2_ss2=0.0; b_0_1_rr=0.0; d_1_1_s2=0.0; d_1_1_ss3=0.0; b_0_2_rr=0.0; d_1_2_s2=0.0; d_1_2_ss3=0.0; b_0_1_ss=0.0; d_1_1_rr=0.0; d_1_1_rs2=0.0; b_0_2_ss=0.0; d_1_2_rr=0.0; d_1_2_rs2=0.0; b_0_1_rs=0.0; d_1_1_rs1=0.0; b_0_2_rs=0.0; d_1_2_rs1=0.0; d_2_1=0.0;d_2_2=0.0;d_2_1_r=0.0; d_2_1_ss1=0.0; d_2_2_r=0.0; d_2_2_ss1=0.0; d_2_1_s1=0.0; d_2_1_ss2=0.0; d_2_2_s1=0.0; d_2_2_ss2=0.0; d_2_1_s2=0.0; d_2_1_ss3=0.0; d_2_2_s2=0.0; d_2_2_ss3=0.0; d_2_1_rr=0.0; d_2_1_rs2=0.0; d_2_2_rr=0.0; d_2_2_rs2=0.0; d_2_1_rs1=0.0;d_2_2_rs1=0.0;d_3_1=0.0;d_3_2=0.0;d_3_1_r=0.0; d_3_1_ss1=0.0; d_3_2_r=0.0; d_3_2_ss1=0.0; d_3_1_s1=0.0; d_3_1_ss2=0.0; d_3_2_s1=0.0; d_3_2_ss2=0.0; d_3_1_s2=0.0; d_3_1_ss3=0.0; d_3_2_s2=0.0; d_3_2_ss3=0.0; d_3_1_rr=0.0; d_3_1_rs2=0.0; d_3_2_rr=0.0; d_3_2_rs2=0.0; d_3_1_rs1=0.0;d_3_2_rs1=0.0;for n=-N:N, n2=n*n;n4=n2*n2;n6=n4*n2;n8=n6*n2;p2=(n+0.5)*(n+0.5);p4=p2*p2;p6=p4*p2;

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101 p8=p6*p2;f2=(8.0*pi2*sigma2*n4-6. 0*pi *si gma* n2+3 .0/8 .0) ; g2=(8.0*pi2*sigma2*p4-6. 0*pi *si gma* p2+3 .0/8 .0) ; f5=(16.0*pi4*sigma4*n8-5 6.0* pi3 *sig ma3* n6+1 05. 0/2. 0*pi 2*si gma 2*n4 105.0/8.0*pi*sigma*n2+10 5.0 /256 .0); g5=(16.0*pi4*sigma4*p8-5 6.0* pi3 *sig ma3* p6+1 05. 0/2. 0*pi 2*si gma 2*p4 105.0/8.0*pi*sigma*p2+10 5.0 /256 .0); f7=(16.0*pi2*sigma*n4-6. 0*pi *n2 ); g7=(16.0*pi2*sigma*p4-6. 0*pi *p2 ); f8=(64.0*pi4*sigma3*n8-1 68.0 *pi 3*si gma2 *n6+ 105 .0*p i2*s igma *n4 105.0/8.0*pi*n2); g8=(64.0*pi4*sigma3*p8-1 68.0 *pi 3*si gma2 *p6+ 105 .0*p i2*s igma *p4 105.0/8.0*pi*p2); f9=(192.0*pi4*sigma2*n8336. 0*p i3*s igma *n6+ 105 .0*p i2*n 4); g9=(192.0*pi4*sigma2*p8336. 0*p i3*s igma *p6+ 105 .0*p i2*p 4); for m=-N:N, m2=m*m;m4=m2*m2;m6=m4*m2;m8=m6*m2;q2=(m+0.5)*(m+0.5);q4=q2*q2;q6=q4*q2;q8=q6*q2;f3=(n2-m2);g3=(p2-q2);f32=f3*f3;g32=g3*g3;f4=(n2+m2);g4=(p2+q2);f42=f4*f4;g42=g4*g4;f1=exp(2.0*pi*i*f3*rho-2 .0*p i*f 4*si gma) ; g1=exp(2.0*pi*i*g3*rho-2 .0*p i*g 4*si gma) ; f6=(16.0*pi4*sigma4*n4*m 4-12 .0* pi3* sigm a3*n 2*m 2*(n 2+m2 )+ 3.0/4.0*pi2*sigma2*(n4+m 4+3 6.0* n2*m 2)-4 5.0 /16. 0*pi *sig ma* (n2+ m2)+

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102 105.0/256.0); g6=(16.0*pi4*sigma4*p4*q 4-12 .0* pi3* sigm a3*p 2*q 2*(p 2+q2 )+ 3.0/4.0*pi2*sigma2*(p4+q 4+3 6.0* p2*q 2)-4 5.0 /16. 0*pi *sig ma* (p2+ q2)+ 105.0/256.0); f10=(64.0*pi4*sigma3*n4* m4-3 6.0 *pi3 *sig ma2* n2* m2*( n2+m 2)+ 3.0/2.0*pi2*sigma*(n4+m4+3 6.0* n2*m 2)-4 5.0 /16. 0*pi *(n2 +m2 )); g10=(64.0*pi4*sigma3*p4* q4-3 6.0 *pi3 *sig ma2* p2* q2*( p2+q 2)+ 3.0/2.0*pi2*sigma*(p4+q4+3 6.0* p2*q 2)-4 5.0 /16. 0*pi *(p2 +q2 )); f11=(192.0*pi4*sigma2*n4 *m472. 0*pi 3*si gma* n2* m2*( n2+m 2)+ 3.0/2.0*pi2*(n4+m4+36.0*n2 *m2) ); g11=(192.0*pi4*sigma2*p4 *q472. 0*pi 3*si gma* p2* q2*( p2+q 2)+ 3.0/2.0*pi2*(p4+q4+36.0*p2 *q2) ); % beta_0 b_0_1=b_0_1+f1;b_0_2=b_0_2+g1; % beta_0_r b_0_1_r=b_0_1_r+f1*f3;b_0_2_r=b_0_2_r+g1*g3; % beta_0_s b_0_1_s=b_0_1_s+f1*f4;b_0_2_s=b_0_2_s+g1*g4; % beta_0_rr b_0_1_rr=b_0_1_rr+f1*f32 ; b_0_2_rr=b_0_2_rr+g1*g32 ; % beta_0_ss b_0_1_ss=b_0_1_ss+f1*f42 ; b_0_2_ss=b_0_2_ss+g1*g42 ; % beta_0_rs b_0_1_rs=b_0_1_rs+f1*f3* f4; b_0_2_rs=b_0_2_rs+g1*g3* g4; % delta_1 d_1_1=d_1_1+f1*f2;d_1_2=d_1_2+g1*g2; % delta_1_r d_1_1_r=d_1_1_r+f1*f2*f3 ; d_1_2_r=d_1_2_r+g1*g2*g3 ; % delta_1_s d_1_1_s1=d_1_1_s1+f1*f2* f4; d_1_2_s1=d_1_2_s1+g1*g2* g4; d_1_1_s2=d_1_1_s2+f1*f7;

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103 d_1_2_s2=d_1_2_s2+g1*g7; % delta_1_rr d_1_1_rr=d_1_1_rr+f1*f2* f32; d_1_2_rr=d_1_2_rr+g1*g2* g32; % delta_1_ss d_1_1_ss1=d_1_1_ss1+f1*f 2*f4 2; d_1_2_ss1=d_1_2_ss1+g1*g 2*g4 2; d_1_1_ss2=d_1_1_ss2+f1*f 4*f7 ; d_1_2_ss2=d_1_2_ss2+g1*g 4*g7 ; d_1_1_ss3=d_1_1_ss3+f1*n 4; d_1_2_ss3=d_1_2_ss3+g1*p 4; % delta_1_rs d_1_1_rs1=d_1_1_rs1+f1*f 2*f3 *f4 ; d_1_2_rs1=d_1_2_rs1+g1*g 2*g3 *g4 ; d_1_1_rs2=d_1_1_rs2+f1*f 3*f7 ; d_1_2_rs2=d_1_2_rs2+g1*g 3*g7 ; % delta_2 d_2_1=d_2_1+f1*f5;d_2_2=d_2_2+g1*g5; % delta_2_r d_2_1_r=d_2_1_r+f1*f3*f5 ; d_2_2_r=d_2_2_r+g1*g3*g5 ; % delta_2_s d_2_1_s1=d_2_1_s1+f1*f4* f5; d_2_2_s1=d_2_2_s1+g1*g4* g5; d_2_1_s2=d_2_1_s2+f1*f8;d_2_2_s2=d_2_2_s2+g1*g8; % delta_2_rr d_2_1_rr=d_2_1_rr+f1*f5* f32; d_2_2_rr=d_2_2_rr+g1*g5* g32; % delta_2_ss d_2_1_ss1=d_2_1_ss1+f1*f 5*f4 2; d_2_2_ss1=d_2_2_ss1+g1*g 5*g4 2; d_2_1_ss2=d_2_1_ss2+f1*f 4*f8 ; d_2_2_ss2=d_2_2_ss2+g1*g 4*g8 ; d_2_1_ss3=d_2_1_ss3+f1*f 9; d_2_2_ss3=d_2_2_ss3+g1*g 9; % delta_2_rs d_2_1_rs1=d_2_1_rs1+f1*f 5*f3 *f4 ; d_2_2_rs1=d_2_2_rs1+g1*g 5*g3 *g4 ; d_2_1_rs2=d_2_1_rs2+f1*f 3*f8 ; d_2_2_rs2=d_2_2_rs2+g1*g 3*g8 ;

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104 % delta_3 d_3_1=d_3_1+f1*f6;d_3_2=d_3_2+g1*g6; % delta_3_r d_3_1_r=d_3_1_r+f1*f3*f6 ; d_3_2_r=d_3_2_r+g1*g3*g6 ; % delta_3_s d_3_1_s1=d_3_1_s1+f1*f4* f6; d_3_2_s1=d_3_2_s1+g1*g4* g6; d_3_1_s2=d_3_1_s2+f1*f10 ; d_3_2_s2=d_3_2_s2+g1*g10 ; % delta_3_rr d_3_1_rr=d_3_1_rr+f1*f6* f32; d_3_2_rr=d_3_2_rr+g1*g6* g32; % delta_3_ss d_3_1_ss1=d_3_1_ss1+f1*f 6*f4 2; d_3_2_ss1=d_3_2_ss1+g1*g 6*g4 2; d_3_1_ss2=d_3_1_ss2+f1*f 4*f1 0; d_3_2_ss2=d_3_2_ss2+g1*g 4*g1 0; d_3_1_ss3=d_3_1_ss3+f1*f 11; d_3_2_ss3=d_3_2_ss3+g1*g 11; % delta_3_rs d_3_1_rs1=d_3_1_rs1+f1*f 6*f3 *f4 ; d_3_2_rs1=d_3_2_rs1+g1*g 6*g3 *g4 ; d_3_1_rs2=d_3_1_rs2+f1*f 3*f1 0; d_3_2_rs2=d_3_2_rs2+g1*g 3*g1 0; endendb_0=ss*real(b_0_1+b_0_2) ; b_0_r=2.0*pi*ss*real(i*( b_0_ 1_r +b_0 _2_r )); b_0_s=b_0*0.5/sigma-2.0* ss*p i*r eal( b_0_ 1_s+ b_0 _2_s ); b_0_rr=-4.0*ss*pi2*real( b_0_ 1_r r+b_ 0_2_ rr); b_0_ss=-0.25*b_0/sigma22.0/ ss* pi*r eal( b_0_ 1_s +b_0 _2_s )+ 4.0*ss*pi2*real(b_0_1_ss +b_0 _2_s s); b_0_rs=pi/ss*real(i*(b_0 _1_r +b_ 0_2_ r))4.0* ss* pi2* real(i*(b_0_1_rs+b_0_2_r s)); d_1=ss*(d_1_1+d_1_2);d_1_r=2.0*pi*ss*(i*(d_1_ 1_r+ d_1 _2_r )); d_1_s=0.5*d_1/sigma-2.0* ss*p i*( d_1_ 1_s1 +d_1 _2_ s1)+ ss* (d_1_1_s2+d_1_2_s2);

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105 d_1_rr=-4*ss*pi2*(d_1_1_ rr+d _1_ 2_rr ); d_1_ss=-0.25*d_1/sigma22.0/ ss* pi*( d_1_ 1_s1 +d_ 1_2_ s1)+ (d_1_1_s2+d_1_2_s2)/ss+4 .0*s s*pi 2*(d _1_ 1_ss 1+ d_1_2_ss1)-4.0*ss*pi*(d_ 1_1_ ss2+ d_1_ 2_s s2)+ 16.0*ss*pi2*(d_1_1_ss3+d _1_2 _ss3 ); d_1_rs=pi/ss*i*(d_1_1_r+ d_1_ 2_r )-4. 0*i* pi2* ss* (d_1 _1_r s1+ d_1_2_rs1)+2.0*i*pi*ss*( d_1_ 1_rs 2+d_ 1_2 _rs2 ); d_2=ss*(d_2_1+d_2_2);d_2_r=2.0*pi*ss*(i*(d_2_ 1_r+ d_2 _2_r )); d_2_s=0.5*d_2/sigma-2.0* ss*p i*( d_2_ 1_s1 +d_2 _2_ s1)+ ss*(d_2_1_s2+d_2_2_s2); d_2_rr=-4*ss*pi2*(d_2_1_ rr+d _2_ 2_rr ); d_2_ss=-0.25*d_2/sigma22.0/ ss* pi*( d_2_ 1_s1 +d_ 2_2_ s1)+ (d_2_1_s2+d_2_2_s2)/ss+4 .0*s s*pi 2*(d _2_ 1_ss 1+ d_2_2_ss1)-4.0*ss*pi*(d_ 2_1_ ss2+ d_2_ 2_s s2)+ ss* (d_2_1_ss3+d_2_2_ss3); d_2_rs=pi/ss*i*(d_2_1_r+ d_2_ 2_r )-4. 0*i* pi2* ss* (d_2 _1_r s1+ d_2_2_rs1)+2.0*i*pi*ss*( d_2_ 1_rs 2+d_ 2_2 _rs2 ); d_3=ss*real(d_3_1+d_3_2) ; d_3_r=2.0*pi*ss*real(i*( d_3_ 1_r +d_3 _2_r )); d_3_s=0.5*d_3/sigma-2.0* ss*p i*r eal( d_3_ 1_s1 +d_ 3_2_ s1)+ ss*real(d_3_1_s2+d_3_2_s2 ); d_3_rr=-4*ss*pi2*real(d_ 3_1_ rr+ d_3_ 2_rr ); d_3_ss=-0.25*d_3/sigma22.0/ ss* pi*r eal( d_3_ 1_s 1+d_ 3_2_ s1)+ real(d_3_1_s2+d_3_2_s2)/ ss+4 .0*s s*pi 2*r eal( d_3_ 1_ss 1+ d_3_2_ss1)-4.0*ss*pi*rea l(d_ 3_1_ ss2+ d_3 _2_s s2)+ ss* real(d_3_1_ss3+d_3_2_ss3 ); d_3_rs=pi/ss*real(i*(d_3 _1_r +d_ 3_2_ r))4.0* pi2 *ss* real (i* (d_3_1_rs1+d_3_2_rs1))+2 .0*p i*ss *rea l(i *(d_ 3_1_ rs2+ d_3_2_rs2)); c=cos(4.0*phi);s=sin(4.0*phi);d_1_p=-4.0*(s*real(d_1)+ c*im ag( d_1) ); d_1_pp=-16.0*(c*real(d_1 )-s* ima g(d_ 1)); d_1_rp=-4.0*(s*real(d_1_ r)+c *im ag(d _1_r )); d_1_sp=-4.0*(s*real(d_1_ s)+c *im ag(d _1_s )); ccc=cos(8.0*phi);sss=sin(8.0*phi);d_2_p=-8.0*(sss*real(d_2 )+cc c*i mag( d_2) ); d_2_pp=-64.0*(ccc*real(d _2)sss *ima g(d_ 2));

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106 d_2_rp=-8.0*(sss*real(d_ 2_r) +cc c*im ag(d _2_r )); d_2_sp=-8.0*(sss*real(d_ 2_s) +cc c*im ag(d _2_s )); % d_1 D_1=(c*real(d_1)-s*imag( d_1) ); d_1G1=(c*real(d_1_r)-s*i mag( d_1 _r)) ; d_1G2=(c*real(d_1_s)-s*i mag( d_1 _s)) ; d_1G3=d_1_p;d_1H11=(c*real(d_1_rr)-s *ima g(d _1_r r)); d_1H12=(c*real(d_1_rs)-s *ima g(d _1_r s)); d_1H13=d_1_rp;d_1H21=d_1H12;d_1H22=(c*real(d_1_ss)-s *ima g(d _1_s s)); d_1H23=d_1_sp;d_1H31=d_1H13;d_1H32=d_1H23;d_1H33=d_1_pp; % d_2 D_2=(ccc*real(d_2)-sss*i mag( d_2 )); d_2G1=(ccc*real(d_2_r)-s ss*i mag (d_2 _r)) ; d_2G2=(ccc*real(d_2_s)-s ss*i mag (d_2 _s)) ; d_2G3=d_2_p;d_2H11=(ccc*real(d_2_rr) -sss *im ag(d _2_r r)); d_2H12=(ccc*real(d_2_rs) -sss *im ag(d _2_r s)); d_2H13=d_2_rp;d_2H21=d_2H12;d_2H22=(ccc*real(d_2_ss) -sss *im ag(d _2_s s)); d_2H23=d_2_sp;d_2H31=d_2H13;d_2H32=d_2H23;d_2H33=d_2_pp; % d_3

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107 D_3=d_3;d_3G1=d_3_r;d_3G2=d_3_s;d_3G3=0.0;d_3H11=d_3_rr;d_3H12=d_3_rs;d_3H13=0.0;d_3H21=d_3H12;d_3H22=d_3_ss;d_3H23=0.0;d_3H31=d_3H13;d_3H32=d_3H23;d_3H33=0.0; % beta_A and its derivatives b=b_0+aniso*D_1+aniso2*( D_33.0 /4.0 *b_0 +D_2 +5. 0*D_ 1); bG1=b_0_r+aniso*d_1G1+an iso2 *(d _3G1 -3.0 /4.0 *b_ 0_r+ d_2G 1+ 5.0*d_1G1); bG2=b_0_s+aniso*d_1G2+an iso2 *(d _3G2 -3.0 /4.0 *b_ 0_s+ d_2G 2+ 5.0*d_1G2); bG3=aniso*d_1G3+aniso2*( d_2G 3+5 .0*d _1G3 ); bH11=b_0_rr+aniso*d_1H11 +ani so2 *(d_ 3H11 -3.0 /4. 0*b_ 0_rr + d_2H11+5.0*d_1H11); bH12=b_0_rs+aniso*d_1H12 +ani so2 *(d_ 3H12 -3.0 /4. 0*b_ 0_rs + d_2H12+5.0*d_1H12); bH13=aniso*d_1H13+aniso2 *(d_ 2H1 3+5. 0*d_ 1H13 ); bH21=bH12;bH22=b_0_ss+aniso*d_1H22 +ani so2 *(d_ 3H22 -3.0 /4. 0*b_ 0_ss + d_2H22+5.0*d_1H22); bH23=aniso*d_1H23+aniso2 *(d_ 2H2 3+5. 0*d_ 1H23 ); bH31=bH13;bH32=bH23;bH33=aniso*d_1H33+aniso2 *(d_ 2H3 3+5. 0*d_ 1H33 );

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108 fac_0=(2.0*kappa^2-1.0); % Gamma and its derivatives g=8.0*field*aniso*(fac_0 *b*( 1.0 +3.0 *ani so)+ b_0 *(1. 0+ 3.0/2.0*aniso)+D_1*(1.0+6. 0*a niso )+2. 0*an iso *(D_ 3+D_ 2)); gG1=8.0*field*aniso*(fac _0*b G1* (1.0 +3.0 *ani so) +b_0 _r*( 1.0+ 3.0/2.0*aniso)+d_1G1*(1. 0+6 .0*a niso )+2. 0*a niso *(d_ 3G1+ d_2 G1)) ; gG2=8.0*field*aniso*(fac _0*b G2* (1.0 +3.0 *ani so) +b_0 _s*( 1.0+ 3.0/2.0*aniso)+d_1G2*(1. 0+6 .0*a niso )+2. 0*a niso *(d_ 3G2+ d_2 G2)) ; gG3=8.0*field*aniso*(fac _0*b G3* (1.0 +3.0 *ani so) +d_1 G3*( 1.0+ 6.0 *ani so)+ 2.0*aniso*d_2G3); gH11=8.0*field*aniso*(fa c_0* bH1 1*(1 .0+3 .0*a nis o)+b _0_r r*(1 .0+ 3.0/2.0*aniso)+d_1H11*(1.0 +6.0 *ani so)+ 2.0 *ani so*( d_3H 11+ d_2H 11)) ; gH12=8.0*field*aniso*(fa c_0* bH1 2*(1 .0+3 .0*a nis o)+b _0_r s*(1 .0+ 3.0/2.0*aniso)+d_1H12*(1.0 +6.0 *ani so)+ 2.0 *ani so*( d_3H 12+ d_2H 12)) ; gH13=8.0*field*aniso*(fa c_0* bH1 3*(1 .0+3 .0*a nis o)+d _1H1 3*(1 .0+ 6.0*aniso)+2.0*aniso*d_2H1 3); gH21=gH12;gH22=8.0*field*aniso*(fa c_0* bH2 2*(1 .0+3 .0*a nis o)+b _0_s s*(1 .0+ 3.0/2.0*aniso)+d_1H22*(1.0 +6.0 *ani so)+ 2.0 *ani so*( d_3H 22+ d_2H 22)) ; gH23=8.0*field*aniso*(fa c_0* bH2 3*(1 .0+3 .0*a nis o)+d _1H2 3*(1 .0+ 6.0*aniso)+2.0*aniso*d_2H2 3); gH31=gH13;gH32=gH23;gH33=8.0*field*aniso*(fa c_0* bH3 3*(1 .0+3 .0*a nis o)+d _1H3 3*(1 .0+ 6.0*aniso)+2.0*aniso*d_2H3 3); % Free energy and its derivatives fac_1=(fac_0*b+1.0);fac_2=(1.0-field)^2/fac_ 1; fac_3=fac_0/fac_1;fl2=field^2;helm=fl2-fac_2*(1.0-g/fa c_1) ; fac_2=fac_2/fac_1;fac_4=fl2-helm;

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109 G1=fac_3*fac_4*bG1+fac_2 *(gG 1-f ac_3 *g*b G1); G2=fac_3*fac_4*bG2+fac_2 *(gG 2-f ac_3 *g*b G2); G3=fac_3*fac_4*bG3+fac_2 *(gG 3-f ac_3 *g*b G3); H11=-fac_3*(G1*bG1+G1*bG 1-fa c_4 *bH1 1)+f ac_2 *(g H11fac_ 3* (bG1*gG1+bG1*gG1+g*bH11) +2. 0*g* fac_ 3*fa c_3 *bG1 *bG1 ); H12=-fac_3*(G1*bG2+G2*bG 1-fa c_4 *bH1 2)+f ac_2 *(g H12fac_ 3* (bG1*gG2+bG2*gG1+g*bH12) +2. 0*g* fac_ 3*fa c_3 *bG1 *bG2 ); H13=-fac_3*(G1*bG3+G3*bG 1-fa c_4 *bH1 3)+f ac_2 *(g H13fac_ 3* (bG1*gG3+bG3*gG1+g*bH13) +2. 0*g* fac_ 3*fa c_3 *bG1 *bG3 ); H21=H12;H22=-fac_3*(G2*bG2+G2*bG 2-fa c_4 *bH2 2)+f ac_2 *(g H22fac_ 3* (bG2*gG2+bG2*gG2+g*bH22) +2. 0*g* fac_ 3*fa c_3 *bG2 *bG2 ); H23=-fac_3*(G2*bG3+G3*bG 2-fa c_4 *bH2 3)+f ac_2 *(g H23fac_ 3* (bG2*gG3+bG3*gG2+g*bH23) +2. 0*g* fac_ 3*fa c_3 *bG2 *bG3 ); H31=H13;H32=H23;H33=-fac_3*(G3*bG3+G3*bG 3-fa c_4 *bH3 3)+f ac_2 *(g H33fac_ 3* (bG3*gG3+bG3*gG3+g*bH33) +2. 0*g* fac_ 3*fa c_3 *bG3 *bG3 ); G=[G1,G2,G3];H=[H11,H12,H13;H21,H22,H 23;H 31, H32, H33] ; D.3 mo duli.m function [c_sq,c_x,c_y,c_r,b]=modul i(x) N=5;global kappa; global aniso; global field; rho=x(1);sigma=x(2);phi=x(3);pi2=pi*pi;pi3=pi2*pi;pi4=pi3*pi;

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110 aniso2=aniso*aniso;sigma2=sigma*sigma;sigma3=sigma2*sigma;sigma4=sigma3*sigma;ss=sqrt(sigma);b_0_1=0.0; d_1_1=0.0; b_0_2=0.0; d_1_2=0.0; b_0_1_r=0.0; d_1_1_r=0.0; d_1_1_ss1=0.0; b_0_2_r=0.0; d_1_2_r=0.0; d_1_2_ss1=0.0; b_0_1_s=0.0; d_1_1_s1=0.0; d_1_1_ss2=0.0; b_0_2_s=0.0; d_1_2_s1=0.0; d_1_2_ss2=0.0; b_0_1_rr=0.0; d_1_1_s2=0.0; d_1_1_ss3=0.0; b_0_2_rr=0.0; d_1_2_s2=0.0; d_1_2_ss3=0.0; b_0_1_ss=0.0; d_1_1_rr=0.0; d_1_1_rs2=0.0; b_0_2_ss=0.0; d_1_2_rr=0.0; d_1_2_rs2=0.0; b_0_1_rs=0.0; d_1_1_rs1=0.0; b_0_2_rs=0.0; d_1_2_rs1=0.0; d_2_1=0.0;d_2_2=0.0;d_2_1_r=0.0; d_2_1_ss1=0.0; d_2_2_r=0.0; d_2_2_ss1=0.0; d_2_1_s1=0.0; d_2_1_ss2=0.0; d_2_2_s1=0.0; d_2_2_ss2=0.0; d_2_1_s2=0.0; d_2_1_ss3=0.0; d_2_2_s2=0.0; d_2_2_ss3=0.0; d_2_1_rr=0.0; d_2_1_rs2=0.0; d_2_2_rr=0.0; d_2_2_rs2=0.0; d_2_1_rs1=0.0;d_2_2_rs1=0.0;d_3_1=0.0;d_3_2=0.0;d_3_1_r=0.0; d_3_1_ss1=0.0; d_3_2_r=0.0; d_3_2_ss1=0.0; d_3_1_s1=0.0; d_3_1_ss2=0.0; d_3_2_s1=0.0; d_3_2_ss2=0.0; d_3_1_s2=0.0; d_3_1_ss3=0.0; d_3_2_s2=0.0; d_3_2_ss3=0.0; d_3_1_rr=0.0; d_3_1_rs2=0.0; d_3_2_rr=0.0; d_3_2_rs2=0.0; d_3_1_rs1=0.0;d_3_2_rs1=0.0;

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111 for n=-N:N, n2=n*n;n4=n2*n2;n6=n4*n2;n8=n6*n2;p2=(n+0.5)*(n+0.5);p4=p2*p2;p6=p4*p2;p8=p6*p2;f2=(8.0*pi2*sigma2*n4-6. 0*pi *si gma* n2+3 .0/8 .0) ; g2=(8.0*pi2*sigma2*p4-6. 0*pi *si gma* p2+3 .0/8 .0) ; f5=(16.0*pi4*sigma4*n8-5 6.0* pi3 *sig ma3* n6+1 05. 0/2. 0*pi 2*si gma 2*n4 105.0/8.0*pi*sigma*n2+105 .0/ 256. 0); g5=(16.0*pi4*sigma4*p8-5 6.0* pi3 *sig ma3* p6+1 05. 0/2. 0*pi 2*si gma 2*p4 105.0/8.0*pi*sigma*p2+105 .0/ 256. 0); f7=(16.0*pi2*sigma*n4-6. 0*pi *n2 ); g7=(16.0*pi2*sigma*p4-6. 0*pi *p2 ); f8=(64.0*pi4*sigma3*n8-1 68.0 *pi 3*si gma2 *n6+ 105 .0*p i2*s igma *n4 105.0/8.0*pi*n2); g8=(64.0*pi4*sigma3*p8-1 68.0 *pi 3*si gma2 *p6+ 105 .0*p i2*s igma *p4 105.0/8.0*pi*p2); f9=(192.0*pi4*sigma2*n8336. 0*p i3*s igma *n6+ 105 .0*p i2*n 4); g9=(192.0*pi4*sigma2*p8336. 0*p i3*s igma *p6+ 105 .0*p i2*p 4); for m=-N:N, m2=m*m;m4=m2*m2;m6=m4*m2;m8=m6*m2;q2=(m+0.5)*(m+0.5);q4=q2*q2;q6=q4*q2;q8=q6*q2;f3=(n2-m2);g3=(p2-q2);f32=f3*f3;g32=g3*g3;f4=(n2+m2);

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112 g4=(p2+q2);f42=f4*f4;g42=g4*g4;f1=exp(2.0*pi*i*f3*rho-2 .0*p i*f 4*si gma) ; g1=exp(2.0*pi*i*g3*rho-2 .0*p i*g 4*si gma) ; f6=(16.0*pi4*sigma4*n4*m 4-12 .0* pi3* sigm a3*n 2*m 2*(n 2+m2 )+3. 0/4 .0* pi2*sigma2*(n4+m4+36.0*n2 *m2 )-45 .0/1 6.0* pi* sigm a*(n 2+m2 )+ 105.0/256.0); g6=(16.0*pi4*sigma4*p4*q 4-12 .0* pi3* sigm a3*p 2*q 2*(p 2+q2 )+3. 0/4 .0* pi2*sigma2*(p4+q4+36.0*p2 *q2 )-45 .0/1 6.0* pi* sigm a*(p 2+q2 )+ 105.0/256.0); f10=(64.0*pi4*sigma3*n4* m4-3 6.0 *pi3 *sig ma2* n2* m2*( n2+m 2)+3 .0/ 2.0* pi2*sigma*(n4+m4+36.0*n2 *m2 )-45 .0/1 6.0* pi* (n2+ m2)) ; g10=(64.0*pi4*sigma3*p4* q4-3 6.0 *pi3 *sig ma2* p2* q2*( p2+q 2)+3 .0/ 2.0* pi2*sigma*(p4+q4+36.0*p2 *q2 )-45 .0/1 6.0* pi* (p2+ q2)) ; f11=(192.0*pi4*sigma2*n4 *m472. 0*pi 3*si gma* n2* m2*( n2+m 2)+3 .0/ 2.0* pi2*(n4+m4+36.0*n2*m2)); g11=(192.0*pi4*sigma2*p4 *q472. 0*pi 3*si gma* p2* q2*( p2+q 2)+3 .0/ 2.0* pi2*(p4+q4+36.0*p2*q2)); % beta_0 b_0_1=b_0_1+f1;b_0_2=b_0_2+g1; % beta_0_r b_0_1_r=b_0_1_r+f1*f3;b_0_2_r=b_0_2_r+g1*g3; % beta_0_s b_0_1_s=b_0_1_s+f1*f4;b_0_2_s=b_0_2_s+g1*g4; % beta_0_rr b_0_1_rr=b_0_1_rr+f1*f32 ; b_0_2_rr=b_0_2_rr+g1*g32 ; % beta_0_ss b_0_1_ss=b_0_1_ss+f1*f42 ; b_0_2_ss=b_0_2_ss+g1*g42 ; % beta_0_rs b_0_1_rs=b_0_1_rs+f1*f3* f4; b_0_2_rs=b_0_2_rs+g1*g3* g4; % delta_1

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113 d_1_1=d_1_1+f1*f2;d_1_2=d_1_2+g1*g2; % delta_1_r d_1_1_r=d_1_1_r+f1*f2*f3 ; d_1_2_r=d_1_2_r+g1*g2*g3 ; % delta_1_s d_1_1_s1=d_1_1_s1+f1*f2* f4; d_1_2_s1=d_1_2_s1+g1*g2* g4; d_1_1_s2=d_1_1_s2+f1*f7;d_1_2_s2=d_1_2_s2+g1*g7; % delta_1_rr d_1_1_rr=d_1_1_rr+f1*f2* f32; d_1_2_rr=d_1_2_rr+g1*g2* g32; % delta_1_ss d_1_1_ss1=d_1_1_ss1+f1*f 2*f4 2; d_1_2_ss1=d_1_2_ss1+g1*g 2*g4 2; d_1_1_ss2=d_1_1_ss2+f1*f 4*f7 ; d_1_2_ss2=d_1_2_ss2+g1*g 4*g7 ; d_1_1_ss3=d_1_1_ss3+f1*n 4; d_1_2_ss3=d_1_2_ss3+g1*p 4; % delta_1_rs d_1_1_rs1=d_1_1_rs1+f1*f 2*f3 *f4 ; d_1_2_rs1=d_1_2_rs1+g1*g 2*g3 *g4 ; d_1_1_rs2=d_1_1_rs2+f1*f 3*f7 ; d_1_2_rs2=d_1_2_rs2+g1*g 3*g7 ; % delta_2 d_2_1=d_2_1+f1*f5;d_2_2=d_2_2+g1*g5; % delta_2_r d_2_1_r=d_2_1_r+f1*f3*f5 ; d_2_2_r=d_2_2_r+g1*g3*g5 ; % delta_2_s d_2_1_s1=d_2_1_s1+f1*f4* f5; d_2_2_s1=d_2_2_s1+g1*g4* g5; d_2_1_s2=d_2_1_s2+f1*f8;d_2_2_s2=d_2_2_s2+g1*g8; % delta_2_rr d_2_1_rr=d_2_1_rr+f1*f5* f32; d_2_2_rr=d_2_2_rr+g1*g5* g32; % delta_2_ss d_2_1_ss1=d_2_1_ss1+f1*f 5*f4 2; d_2_2_ss1=d_2_2_ss1+g1*g 5*g4 2;

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114 d_2_1_ss2=d_2_1_ss2+f1*f 4*f8 ; d_2_2_ss2=d_2_2_ss2+g1*g 4*g8 ; d_2_1_ss3=d_2_1_ss3+f1*f 9; d_2_2_ss3=d_2_2_ss3+g1*g 9; % delta_2_rs d_2_1_rs1=d_2_1_rs1+f1*f 5*f3 *f4 ; d_2_2_rs1=d_2_2_rs1+g1*g 5*g3 *g4 ; d_2_1_rs2=d_2_1_rs2+f1*f 3*f8 ; d_2_2_rs2=d_2_2_rs2+g1*g 3*g8 ; % delta_3 d_3_1=d_3_1+f1*f6;d_3_2=d_3_2+g1*g6; % delta_3_r d_3_1_r=d_3_1_r+f1*f3*f6 ; d_3_2_r=d_3_2_r+g1*g3*g6 ; % delta_3_s d_3_1_s1=d_3_1_s1+f1*f4* f6; d_3_2_s1=d_3_2_s1+g1*g4* g6; d_3_1_s2=d_3_1_s2+f1*f10 ; d_3_2_s2=d_3_2_s2+g1*g10 ; % delta_3_rr d_3_1_rr=d_3_1_rr+f1*f6* f32; d_3_2_rr=d_3_2_rr+g1*g6* g32; % delta_3_ss d_3_1_ss1=d_3_1_ss1+f1*f 6*f4 2; d_3_2_ss1=d_3_2_ss1+g1*g 6*g4 2; d_3_1_ss2=d_3_1_ss2+f1*f 4*f1 0; d_3_2_ss2=d_3_2_ss2+g1*g 4*g1 0; d_3_1_ss3=d_3_1_ss3+f1*f 11; d_3_2_ss3=d_3_2_ss3+g1*g 11; % delta_3_rs d_3_1_rs1=d_3_1_rs1+f1*f 6*f3 *f4 ; d_3_2_rs1=d_3_2_rs1+g1*g 6*g3 *g4 ; d_3_1_rs2=d_3_1_rs2+f1*f 3*f1 0; d_3_2_rs2=d_3_2_rs2+g1*g 3*g1 0; endendb_0=ss*real(b_0_1+b_0_2) ; b_0_r=2.0*pi*ss*real(i*( b_0_ 1_r +b_0 _2_r )); b_0_s=b_0*0.5/sigma-2.0* ss*p i*r eal( b_0_ 1_s+ b_0 _2_s ); b_0_rr=-4.0*ss*pi2*real( b_0_ 1_r r+b_ 0_2_ rr);

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115 b_0_ss=-0.25*b_0/sigma22.0/ ss* pi*r eal( b_0_ 1_s +b_0 _2_s )+4. 0*s s* pi2*real(b_0_1_ss+b_0_2_ ss); b_0_rs=pi/ss*real(i*(b_0 _1_r +b_ 0_2_ r))4.0* ss* pi2* real (i* (b_0_1_rs+b_0_2_rs)); d_1=ss*(d_1_1+d_1_2);d_1_r=2.0*pi*ss*(i*(d_1_ 1_r+ d_1 _2_r )); d_1_s=0.5*d_1/sigma-2.0* ss*p i*( d_1_ 1_s1 +d_1 _2_ s1)+ ss* (d_1_1_s2+d_1_2_s2); d_1_rr=-4*ss*pi2*(d_1_1_ rr+d _1_ 2_rr ); d_1_ss=-0.25*d_1/sigma22.0/ ss* pi*( d_1_ 1_s1 +d_ 1_2_ s1)+ (d_1_1_s2+d_1_2_s2)/ss+4 .0*s s*pi 2*(d _1_ 1_ss 1+ d_1_2_ss1)-4.0*ss*pi*(d_ 1_1_ ss2+ d_1_ 2_s s2)+ 16.0 ss*pi2*(d_1_1_ss3+d_1_2_ ss3) ; d_1_rs=pi/ss*i*(d_1_1_r+ d_1_ 2_r )-4. 0*i* pi2* ss* (d_1 _1_r s1+ d_1_2_rs1)+2.0*i*pi*ss*( d_1_ 1_rs 2+d_ 1_2 _rs2 ); d_2=ss*(d_2_1+d_2_2);d_2_r=2.0*pi*ss*(i*(d_2_ 1_r+ d_2 _2_r )); d_2_s=0.5*d_2/sigma-2.0* ss*p i*( d_2_ 1_s1 +d_2 _2_ s1)+ ss* (d_2_1_s2+d_2_2_s2); d_2_rr=-4*ss*pi2*(d_2_1_ rr+d _2_ 2_rr ); d_2_ss=-0.25*d_2/sigma22.0/ ss* pi*( d_2_ 1_s1 +d_ 2_2_ s1)+ (d_2 _1_ s2+ d_2_2_s2)/ss+4.0*ss*pi2* (d_2 _1_s s1+d _2_ 2_ss 1)-4 .0*s s*p i* (d_2_1_ss2+d_2_2_ss2)+ss *(d_ 2_1_ ss3+ d_2 _2_s s3); d_2_rs=pi/ss*i*(d_2_1_r+ d_2_ 2_r )-4. 0*i* pi2* ss* (d_2 _1_r s1+ d_2_2_rs1)+2.0*i*pi*ss*( d_2_ 1_rs 2+d_ 2_2 _rs2 ); d_3=ss*real(d_3_1+d_3_2) ; d_3_r=2.0*pi*ss*real(i*( d_3_ 1_r +d_3 _2_r )); d_3_s=0.5*d_3/sigma-2.0* ss*p i*r eal( d_3_ 1_s1 +d_ 3_2_ s1)+ ss* real(d_3_1_s2+d_3_2_s2); d_3_rr=-4*ss*pi2*real(d_ 3_1_ rr+ d_3_ 2_rr ); d_3_ss=-0.25*d_3/sigma22.0/ ss* pi*r eal( d_3_ 1_s 1+d_ 3_2_ s1)+ real(d_3_1_s2+d_3_2_s2)/ ss+4 .0*s s*pi 2*r eal( d_3_ 1_ss 1+ d_3_2_ss1)-4.0*ss*pi*rea l(d_ 3_1_ ss2+ d_3 _2_s s2)+ ss* real(d_3_1_ss3+d_3_2_ss3 ); d_3_rs=pi/ss*real(i*(d_3 _1_r +d_ 3_2_ r))4.0* pi2 *ss* real (i* (d_3_1_rs1+d_3_2_rs1))+2 .0*p i*ss *rea l(i *(d_ 3_1_ rs2+ d_3_2_rs2)); c=cos(4.0*phi);s=sin(4.0*phi);d_1_p=-4.0*(s*real(d_1)+ c*im ag( d_1) );

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116 d_1_pp=-16.0*(c*real(d_1 )-s* ima g(d_ 1)); d_1_rp=-4.0*(s*real(d_1_ r)+c *im ag(d _1_r )); d_1_sp=-4.0*(s*real(d_1_ s)+c *im ag(d _1_s )); ccc=cos(8.0*phi);sss=sin(8.0*phi);d_2_p=-8.0*(sss*real(d_2 )+cc c*i mag( d_2) ); d_2_pp=-64.0*(ccc*real(d _2)sss *ima g(d_ 2)); d_2_rp=-8.0*(sss*real(d_ 2_r) +cc c*im ag(d _2_r )); d_2_sp=-8.0*(sss*real(d_ 2_s) +cc c*im ag(d _2_s )); % d_1 D_1=(c*real(d_1)-s*imag( d_1) ); d_1G1=(c*real(d_1_r)-s*i mag( d_1 _r)) ; d_1G2=(c*real(d_1_s)-s*i mag( d_1 _s)) ; d_1G3=d_1_p;d_1H11=(c*real(d_1_rr)-s *ima g(d _1_r r)); d_1H12=(c*real(d_1_rs)-s *ima g(d _1_r s)); d_1H13=d_1_rp;d_1H21=d_1H12;d_1H22=(c*real(d_1_ss)-s *ima g(d _1_s s)); d_1H23=d_1_sp;d_1H31=d_1H13;d_1H32=d_1H23;d_1H33=d_1_pp; % d_2 D_2=(ccc*real(d_2)-sss*i mag( d_2 )); d_2G1=(ccc*real(d_2_r)-s ss*i mag (d_2 _r)) ; d_2G2=(ccc*real(d_2_s)-s ss*i mag (d_2 _s)) ; d_2G3=d_2_p;d_2H11=(ccc*real(d_2_rr) -sss *im ag(d _2_r r)); d_2H12=(ccc*real(d_2_rs) -sss *im ag(d _2_r s)); d_2H13=d_2_rp;d_2H21=d_2H12;

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117 d_2H22=(ccc*real(d_2_ss) -sss *im ag(d _2_s s)); d_2H23=d_2_sp;d_2H31=d_2H13;d_2H32=d_2H23;d_2H33=d_2_pp; % d_3 D_3=d_3;d_3G1=d_3_r;d_3G2=d_3_s;d_3G3=0.0;d_3H11=d_3_rr;d_3H12=d_3_rs;d_3H13=0.0;d_3H21=d_3H12;d_3H22=d_3_ss;d_3H23=0.0;d_3H31=d_3H13;d_3H32=d_3H23;d_3H33=0.0; % beta_A and its derivatives b=b_0+aniso*D_1+aniso2*( D_33.0 /4.0 *b_0 +D_2 +5. 0*D_ 1); bG1=b_0_r+aniso*d_1G1+an iso2 *(d _3G1 -3.0 /4.0 *b_ 0_r+ d_2G 1+5. 0*d _1G1 ); bG2=b_0_s+aniso*d_1G2+an iso2 *(d _3G2 -3.0 /4.0 *b_ 0_s+ d_2G 2+5. 0*d _1G2 ); bG3=aniso*d_1G3+aniso2*( d_2G 3+5 .0*d _1G3 ); bH11=b_0_rr+aniso*d_1H11 +ani so2 *(d_ 3H11 -3.0 /4. 0*b_ 0_rr +d_2 H11 + 5.0*d_1H11); bH12=b_0_rs+aniso*d_1H12 +ani so2 *(d_ 3H12 -3.0 /4. 0*b_ 0_rs +d_2 H12 + 5.0*d_1H12); bH13=aniso*d_1H13+aniso2 *(d_ 2H1 3+5. 0*d_ 1H13 ); bH21=bH12;bH22=b_0_ss+aniso*d_1H22 +ani so2 *(d_ 3H22 -3.0 /4. 0*b_ 0_ss +d_2 H22 + 5.0*d_1H22);

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118 bH23=aniso*d_1H23+aniso2 *(d_ 2H2 3+5. 0*d_ 1H23 ); bH31=bH13;bH32=bH23;bH33=aniso*d_1H33+aniso2 *(d_ 2H3 3+5. 0*d_ 1H33 ); fac_0=(2.0*kappa^2-1.0); % Gamma and its derivatives g=8.0*field*aniso*(fac_0 *b*( 1.0 +3.0 *ani so)+ b_0 *(1. 0+3. 0/2. 0*a niso )+ D_1*(1.0+6.0*aniso)+2.0*an iso *(D_ 3+D_ 2)); gG1=8.0*field*aniso*(fac _0*b G1* (1.0 +3.0 *ani so) +b_0 _r*( 1.0+ 3.0 /2.0 aniso)+d_1G1*(1.0+6.0*an iso )+2. 0*an iso* (d_ 3G1+ d_2G 1)); gG2=8.0*field*aniso*(fac _0*b G2* (1.0 +3.0 *ani so) +b_0 _s*( 1.0+ 3.0 /2.0 aniso)+d_1G2*(1.0+6.0*an iso )+2. 0*an iso* (d_ 3G2+ d_2G 2)); gG3=8.0*field*aniso*(fac _0*b G3* (1.0 +3.0 *ani so) +d_1 G3*( 1.0+ 6.0 aniso)+2.0*aniso*d_2G3); gH11=8.0*field*aniso*(fa c_0* bH1 1*(1 .0+3 .0*a nis o)+b _0_r r*(1 .0+ 3.0/2.0*aniso)+d_1H11*(1.0 +6.0 *ani so)+ 2.0 *ani so* (d_3H11+d_2H11)); gH12=8.0*field*aniso*(fa c_0* bH1 2*(1 .0+3 .0*a nis o)+b _0_r s*(1 .0+ 3.0/2.0*aniso)+d_1H12*(1.0 +6.0 *ani so)+ 2.0 *ani so* (d_3H12+d_2H12)); gH13=8.0*field*aniso*(fa c_0* bH1 3*(1 .0+3 .0*a nis o)+d _1H1 3*(1 .0+ 6.0*aniso)+2.0*aniso*d_2H1 3); gH21=gH12;gH22=8.0*field*aniso*(fa c_0* bH2 2*(1 .0+3 .0*a nis o)+b _0_s s*(1 .0+ 3.0/2.0*aniso)+d_1H22*(1.0 +6.0 *ani so)+ 2.0 *ani so*( d_3H 22+ d_2H22)); gH23=8.0*field*aniso*(fa c_0* bH2 3*(1 .0+3 .0*a nis o)+d _1H2 3*(1 .0+ 6.0*aniso)+2.0*aniso*d_2H2 3); gH31=gH13;gH32=gH23;gH33=8.0*field*aniso*(fa c_0* bH3 3*(1 .0+3 .0*a nis o)+d _1H3 3*(1 .0+ 6.0*aniso)+2.0*aniso*d_2H3 3); % Free energy fac_1=(fac_0*b+1.0);

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119 fac_2=(1.0-field)^2/fac_ 1; fac_3=fac_0/fac_1;fl2=field^2;helm=fl2-fac_2*(1.0-g/fa c_1) ; fac_2=fac_2/fac_1;fac_4=(fl2-helm)*fac_3; % Squash modulus c_sq=(fac_4*bH11+fac_2*( gH11 -fa c_3* (2.0 *bG1 *gG 1+g* bH11 -2.0 *fa c_3* g*bG1^2)))*4.0*sigma2*(sin (2.0 *phi ))^2 ; c_sq=c_sq+(fac_4*bH22+fa c_2* (gH 22-f ac_3 *(2. 0*b G2*g G2+g *bH2 2-2 .0* fac_3*g*bG2^2)))*4.0*sigma 2*(c os(2 .0*p hi) )^2; c_sq=c_sq+(fac_4*bH33+fa c_2* (gH 33-f ac_3 *(2. 0*b G3*g G3+g *bH3 3-2 .0* fac_3*g*bG3^2)))*(sin(2.0* phi) )^2; c_sq=c_sq-4.0*(fac_4*bH1 2+fa c_2 *(gH 12-f ac_3 *(b G1*g G2+b G2*g G1+ g* bH12-2.0*fac_3*g*bG1*bG2)) )*si gma2 *sin (4. 0*ph i); c_sq=c_sq-4.0*(fac_4*bH1 3+fa c_2 *(gH 13-f ac_3 *(b G1*g G3+b G3*g G1+ g* bH13-2.0*fac_3*g*bG1*bG3)) )*si gma* (sin (2. 0*ph i))^ 2; c_sq=c_sq+2.0*(fac_4*bH2 3+fa c_2 *(gH 23-f ac_3 *(b G2*g G3+b G3*g G2+ g* bH23-2.0*fac_3*g*bG2*bG3)) )*si gma* sin( 4.0 *phi ); % Shear along x c_x=(fac_4*bH11+fac_2*(g H11fac _3*( 2.0* bG1* gG1 +g*b H112.0* fac _3*g bG1^2)))*sigma2*(cos(2.0 *ph i))^ 2; c_x=c_x+(fac_4*bH22+fac_ 2*(g H22 -fac _3*( 2.0* bG2 *gG2 +g*b H222.0 fac_3*g*bG2^2)))*sigma2* (si n(2. 0*ph i))^ 2; c_x=c_x+(fac_4*bH33+fac_ 2*(g H33 -fac _3*( 2.0* bG3 *gG3 +g*b H332.0 fac_3*g*bG3^2)))*(cos(ph i)) ^4; c_x=c_x+(fac_4*bH12+fac_ 2*(g H12 -fac _3*( bG1* gG2 +bG2 *gG1 +g*b H12 2.0*fac_3*g*bG1*bG2)))*s igm a2*s in(4 .0*p hi) ; c_x=c_x-2.0*(fac_4*bH13+ fac_ 2*( gH13 -fac _3*( bG1 *gG3 +bG3 *gG1 +g* bH13 2.0*fac_3*g*bG1*bG3)))*s igm a*co s(2. 0*ph i)* (cos (phi ))^2 ; c_x=c_x-2.0*(fac_4*bH23+ fac_ 2*( gH23 -fac _3*( bG2 *gG3 +bG3 *gG2 +g* bH23 2.0*fac_3*g*bG2*bG3)))*s igm a*si n(2. 0*ph i)* (cos (phi ))^2 ; % Shear along y c_y=(fac_4*bH11+fac_2*(g H11fac _3*( 2.0* bG1* gG1 +g*b H112.0* fac _3*g bG1^2)))*sigma2*(cos(2.0 *ph i))^ 2; c_y=c_y+(fac_4*bH22+fac_ 2*(g H22 -fac _3*( 2.0* bG2 *gG2 +g*b H222.0

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120 fac_3*g*bG2^2)))*sigma2* (si n(2. 0*ph i))^ 2; c_y=c_y+(fac_4*bH33+fac_ 2*(g H33 -fac _3*( 2.0* bG3 *gG3 +g*b H332.0 fac_3*g*bG3^2)))*(sin(ph i)) ^4; c_y=c_y+(fac_4*bH12+fac_ 2*(g H12 -fac _3*( bG1* gG2 +bG2 *gG1 +g*b H12 2.0*fac_3*g*bG1*bG2)))*s igm a2*s in(4 .0*p hi) ; c_y=c_y+2.0*(fac_4*bH13+ fac_ 2*( gH13 -fac _3*( bG1 *gG3 +bG3 *gG1 +g* bH13 2.0*fac_3*g*bG1*bG3)))*s igm a*co s(2. 0*ph i)* (sin (phi ))^2 ; c_y=c_y+2.0*(fac_4*bH23+ fac_ 2*( gH23 -fac _3*( bG2 *gG3 +bG3 *gG2 +g* bH23 2.0*fac_3*g*bG2*bG3)))*s igm a*si n(2. 0*ph i)* (sin (phi ))^2 ; % Rotational modulus c_r=(fac_4*bH33+fac_2*(g H33fac _3*( 2.0* bG3* gG3 +g*b H332.0* fac _3*g bG3^2))); D.4 driv er2.m clear all; format long; digits(64);global kappa; global bfield; kappa=5.0;gd=fopen('phase.dat','w' ); options=optimset('Displa y',' ite r'); step=0.001; % Initial guess for b start_b=0.314256045; % Temperature range temp_1=0.0;temp_2=sqrt(0.9-start_b) ; for bfield=start_b:step:0.9, % The call to the root finder [zero_c_sq,fval,exitflag ,out put ]=fz ero( 'csq uas h',[ temp _1 temp_2],

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121 options) fprintf(gd,'%7.5f %12.10f\n',zero_c_sq,bfie ld) ; % This part of the code follows the root % by adjusting the temperature range temp_1=zero_c_sq-0.2;if(temp_1<0.0), temp_1=0.0; end; temp_2=zero_c_sq+0.1;if(temp_2>sqrt(0.9-start _b)) temp_2=sqrt(0.9-start_b) ; end; end;fclose(gd); D.5 csquash.m function [c_sq]=csquash(x) N=4;global kappa; global bfield; temp=x(1);field=bfield/(1.0-temp*t emp) ; % Consistency check if(field >= 1.0), error('Field out of bounds'), end % Fluctuation fluc=(0.0064*2.7)*sqrt(f ield )*t emp/ (sqr t(1. 0-t emp^ 2))/ sqrt((1-field)^3*log(2.0+1 .0/s qrt( 2.0* fie ld)) ); rho=0.0;sigma=1.0-2.0*fluc;phi=pi/4.0-fluc;pi2=pi*pi;pi3=pi2*pi;pi4=pi3*pi;aniso=(1.6/(1.0+temp^3)) ^2*b fie ld/1 2.0; aniso2=aniso*aniso;

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122 sigma2=sigma*sigma;sigma3=sigma2*sigma;sigma4=sigma3*sigma;ss=sqrt(sigma);b_0_1=0.0; d_1_1=0.0; b_0_2=0.0; d_1_2=0.0; b_0_1_r=0.0; d_1_1_r=0.0; d_1_1_ss1=0.0; b_0_2_r=0.0; d_1_2_r=0.0; d_1_2_ss1=0.0; b_0_1_s=0.0; d_1_1_s1=0.0; d_1_1_ss2=0.0; b_0_2_s=0.0; d_1_2_s1=0.0; d_1_2_ss2=0.0; b_0_1_rr=0.0; d_1_1_s2=0.0; d_1_1_ss3=0.0; b_0_2_rr=0.0; d_1_2_s2=0.0; d_1_2_ss3=0.0; b_0_1_ss=0.0; d_1_1_rr=0.0; d_1_1_rs2=0.0; b_0_2_ss=0.0; d_1_2_rr=0.0; d_1_2_rs2=0.0; b_0_1_rs=0.0; d_1_1_rs1=0.0; b_0_2_rs=0.0; d_1_2_rs1=0.0; d_2_1=0.0;d_2_2=0.0;d_2_1_r=0.0; d_2_1_ss1=0.0; d_2_2_r=0.0; d_2_2_ss1=0.0; d_2_1_s1=0.0; d_2_1_ss2=0.0; d_2_2_s1=0.0; d_2_2_ss2=0.0; d_2_1_s2=0.0; d_2_1_ss3=0.0; d_2_2_s2=0.0; d_2_2_ss3=0.0; d_2_1_rr=0.0; d_2_1_rs2=0.0; d_2_2_rr=0.0; d_2_2_rs2=0.0; d_2_1_rs1=0.0;d_2_2_rs1=0.0;d_3_1=0.0;d_3_2=0.0;d_3_1_r=0.0; d_3_1_ss1=0.0; d_3_2_r=0.0; d_3_2_ss1=0.0; d_3_1_s1=0.0; d_3_1_ss2=0.0; d_3_2_s1=0.0; d_3_2_ss2=0.0; d_3_1_s2=0.0; d_3_1_ss3=0.0; d_3_2_s2=0.0; d_3_2_ss3=0.0; d_3_1_rr=0.0; d_3_1_rs2=0.0; d_3_2_rr=0.0; d_3_2_rs2=0.0; d_3_1_rs1=0.0;

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123 d_3_2_rs1=0.0;for n=-N:N, n2=n*n;n4=n2*n2;n6=n4*n2;n8=n6*n2;p2=(n+0.5)*(n+0.5);p4=p2*p2;p6=p4*p2;p8=p6*p2;f2=(8.0*pi2*sigma2*n4-6. 0*pi *si gma* n2+3 .0/8 .0) ; g2=(8.0*pi2*sigma2*p4-6. 0*pi *si gma* p2+3 .0/8 .0) ; f5=(16.0*pi4*sigma4*n8-5 6.0* pi3 *sig ma3* n6+1 05. 0/2. 0*pi 2*si gma 2*n4 105.0/8.0*pi*sigma*n2+105 .0/ 256. 0); g5=(16.0*pi4*sigma4*p8-5 6.0* pi3 *sig ma3* p6+1 05. 0/2. 0*pi 2*si gma 2*p4 105.0/8.0*pi*sigma*p2+105 .0/ 256. 0); f7=(16.0*pi2*sigma*n4-6. 0*pi *n2 ); g7=(16.0*pi2*sigma*p4-6. 0*pi *p2 ); f8=(64.0*pi4*sigma3*n8-1 68.0 *pi 3*si gma2 *n6+ 105 .0*p i2*s igma *n4 105.0/8.0*pi*n2); g8=(64.0*pi4*sigma3*p8-1 68.0 *pi 3*si gma2 *p6+ 105 .0*p i2*s igma *p4 105.0/8.0*pi*p2); f9=(192.0*pi4*sigma2*n8336. 0*p i3*s igma *n6+ 105 .0*p i2*n 4); g9=(192.0*pi4*sigma2*p8336. 0*p i3*s igma *p6+ 105 .0*p i2*p 4); for m=-N:N, m2=m*m;m4=m2*m2;m6=m4*m2;m8=m6*m2;q2=(m+0.5)*(m+0.5);q4=q2*q2;q6=q4*q2;q8=q6*q2;f3=(n2-m2);g3=(p2-q2);f32=f3*f3;g32=g3*g3;

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124 f4=(n2+m2);g4=(p2+q2);f42=f4*f4;g42=g4*g4;f1=exp(2.0*pi*i*f3*rho)* exp( -2. 0*pi *f4* sigm a); g1=exp(2.0*pi*i*g3*rho)* exp( -2. 0*pi *g4* sigm a); f6=(16.0*pi4*sigma4*n4*m 4-12 .0* pi3* sigm a3*n 2*m 2*(n 2+m2 )+3. 0/4 .0* pi2*sigma2*(n4+m4+36.0*n2 *m2 )-45 .0/1 6.0* pi* sigm a*(n 2+m2 )+ 105.0/256.0); g6=(16.0*pi4*sigma4*p4*q 4-12 .0* pi3* sigm a3*p 2*q 2*(p 2+q2 )+3. 0/4 .0* pi2*sigma2*(p4+q4+36.0*p2 *q2 )-45 .0/1 6.0* pi* sigm a*(p 2+q2 )+ 105.0/256.0); f10=(64.0*pi4*sigma3*n4* m4-3 6.0 *pi3 *sig ma2* n2* m2*( n2+m 2)+ 3.0/2.0*pi2*sigma*(n4+m4 +36 .0*n 2*m2 )-45 .0/ 16.0 *pi* (n2+ m2) ); g10=(64.0*pi4*sigma3*p4* q4-3 6.0 *pi3 *sig ma2* p2* q2*( p2+q 2)+ 3.0/2.0*pi2*sigma*(p4+q4 +36 .0*p 2*q2 )-45 .0/ 16.0 *pi* (p2+ q2) ); f11=(192.0*pi4*sigma2*n4 *m472. 0*pi 3*si gma* n2* m2*( n2+m 2)+ 3.0/2.0*pi2*(n4+m4+36.0* n2* m2)) ; g11=(192.0*pi4*sigma2*p4 *q472. 0*pi 3*si gma* p2* q2*( p2+q 2)+ 3.0/2.0*pi2*(p4+q4+36.0* p2* q2)) ; % beta_0 b_0_1=b_0_1+f1;b_0_2=b_0_2+g1; % beta_0_r b_0_1_r=b_0_1_r+f1*f3;b_0_2_r=b_0_2_r+g1*g3; % beta_0_s b_0_1_s=b_0_1_s+f1*f4;b_0_2_s=b_0_2_s+g1*g4; % beta_0_rr b_0_1_rr=b_0_1_rr+f1*f32 ; b_0_2_rr=b_0_2_rr+g1*g32 ; % beta_0_ss b_0_1_ss=b_0_1_ss+f1*f42 ; b_0_2_ss=b_0_2_ss+g1*g42 ; % beta_0_rs b_0_1_rs=b_0_1_rs+f1*f3* f4; b_0_2_rs=b_0_2_rs+g1*g3* g4;

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125 % delta_1 d_1_1=d_1_1+f1*f2;d_1_2=d_1_2+g1*g2; % delta_1_r d_1_1_r=d_1_1_r+f1*f2*f3 ; d_1_2_r=d_1_2_r+g1*g2*g3 ; % delta_1_s d_1_1_s1=d_1_1_s1+f1*f2* f4; d_1_2_s1=d_1_2_s1+g1*g2* g4; d_1_1_s2=d_1_1_s2+f1*f7;d_1_2_s2=d_1_2_s2+g1*g7; % delta_1_rr d_1_1_rr=d_1_1_rr+f1*f2* f32; d_1_2_rr=d_1_2_rr+g1*g2* g32; % delta_1_ss d_1_1_ss1=d_1_1_ss1+f1*f 2*f4 2; d_1_2_ss1=d_1_2_ss1+g1*g 2*g4 2; d_1_1_ss2=d_1_1_ss2+f1*f 4*f7 ; d_1_2_ss2=d_1_2_ss2+g1*g 4*g7 ; d_1_1_ss3=d_1_1_ss3+f1*n 4; d_1_2_ss3=d_1_2_ss3+g1*p 4; % delta_1_rs d_1_1_rs1=d_1_1_rs1+f1*f 2*f3 *f4 ; d_1_2_rs1=d_1_2_rs1+g1*g 2*g3 *g4 ; d_1_1_rs2=d_1_1_rs2+f1*f 3*f7 ; d_1_2_rs2=d_1_2_rs2+g1*g 3*g7 ; % delta_2 d_2_1=d_2_1+f1*f5;d_2_2=d_2_2+g1*g5; % delta_2_r d_2_1_r=d_2_1_r+f1*f3*f5 ; d_2_2_r=d_2_2_r+g1*g3*g5 ; % delta_2_s d_2_1_s1=d_2_1_s1+f1*f4* f5; d_2_2_s1=d_2_2_s1+g1*g4* g5; d_2_1_s2=d_2_1_s2+f1*f8;d_2_2_s2=d_2_2_s2+g1*g8; % delta_2_rr d_2_1_rr=d_2_1_rr+f1*f5* f32; d_2_2_rr=d_2_2_rr+g1*g5* g32; % delta_2_ss d_2_1_ss1=d_2_1_ss1+f1*f 5*f4 2;

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126 d_2_2_ss1=d_2_2_ss1+g1*g 5*g4 2; d_2_1_ss2=d_2_1_ss2+f1*f 4*f8 ; d_2_2_ss2=d_2_2_ss2+g1*g 4*g8 ; d_2_1_ss3=d_2_1_ss3+f1*f 9; d_2_2_ss3=d_2_2_ss3+g1*g 9; % delta_2_rs d_2_1_rs1=d_2_1_rs1+f1*f 5*f3 *f4 ; d_2_2_rs1=d_2_2_rs1+g1*g 5*g3 *g4 ; d_2_1_rs2=d_2_1_rs2+f1*f 3*f8 ; d_2_2_rs2=d_2_2_rs2+g1*g 3*g8 ; % delta_3 d_3_1=d_3_1+f1*f6;d_3_2=d_3_2+g1*g6; % delta_3_r d_3_1_r=d_3_1_r+f1*f3*f6 ; d_3_2_r=d_3_2_r+g1*g3*g6 ; % delta_3_s d_3_1_s1=d_3_1_s1+f1*f4* f6; d_3_2_s1=d_3_2_s1+g1*g4* g6; d_3_1_s2=d_3_1_s2+f1*f10 ; d_3_2_s2=d_3_2_s2+g1*g10 ; % delta_3_rr d_3_1_rr=d_3_1_rr+f1*f6* f32; d_3_2_rr=d_3_2_rr+g1*g6* g32; % delta_3_ss d_3_1_ss1=d_3_1_ss1+f1*f 6*f4 2; d_3_2_ss1=d_3_2_ss1+g1*g 6*g4 2; d_3_1_ss2=d_3_1_ss2+f1*f 4*f1 0; d_3_2_ss2=d_3_2_ss2+g1*g 4*g1 0; d_3_1_ss3=d_3_1_ss3+f1*f 11; d_3_2_ss3=d_3_2_ss3+g1*g 11; % delta_3_rs d_3_1_rs1=d_3_1_rs1+f1*f 6*f3 *f4 ; d_3_2_rs1=d_3_2_rs1+g1*g 6*g3 *g4 ; d_3_1_rs2=d_3_1_rs2+f1*f 3*f1 0; d_3_2_rs2=d_3_2_rs2+g1*g 3*g1 0; endendb_0=ss*real(b_0_1+b_0_2) ; b_0_r=2.0*pi*ss*real(i*( b_0_ 1_r +b_0 _2_r )); b_0_s=b_0*0.5/sigma-2.0* ss*p i*r eal( b_0_ 1_s+ b_0 _2_s );

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127 b_0_rr=-4.0*ss*pi2*real( b_0_ 1_r r+b_ 0_2_ rr); b_0_ss=-0.25*b_0/sigma22.0/ ss* pi*r eal( b_0_ 1_s +b_0 _2_s )+ 4.0*ss*pi2*real(b_0_1_ss +b_0 _2_s s); b_0_rs=pi/ss*real(i*(b_0 _1_r +b_ 0_2_ r))4.0* ss* pi2* real(i*(b_0_1_rs+b_0_2_r s)); d_1=ss*(d_1_1+d_1_2);d_1_r=2.0*pi*ss*(i*(d_1_ 1_r+ d_1 _2_r )); d_1_s=0.5*d_1/sigma-2.0* ss*p i*( d_1_ 1_s1 +d_1 _2_ s1)+ ss* (d_1_1_s2+d_1_2_s2); d_1_rr=-4*ss*pi2*(d_1_1_ rr+d _1_ 2_rr ); d_1_ss=-0.25*d_1/sigma22.0/ ss* pi*( d_1_ 1_s1 +d_ 1_2_ s1)+ (d_1_1_s2+d_1_2_s2)/ss+4 .0*s s*pi 2*(d _1_ 1_ss 1+d_ 1_2_ ss1 )4.0*ss*pi*(d_1_1_ss2+d_1 _2_s s2)+ 16.0 *ss *pi2 (d_1_1_ss3+d_1_2_ss3); d_1_rs=pi/ss*i*(d_1_1_r+ d_1_ 2_r )-4. 0*i* pi2* ss* (d_1 _1_r s1+ d_1_2_rs1)+2.0*i*pi*ss*( d_1_ 1_rs 2+d_ 1_2 _rs2 ); d_2=ss*(d_2_1+d_2_2);d_2_r=2.0*pi*ss*(i*(d_2_ 1_r+ d_2 _2_r )); d_2_s=0.5*d_2/sigma-2.0* ss*p i*( d_2_ 1_s1 +d_2 _2_ s1)+ ss*( d_2_ 1_s 2+ d_2_2_s2); d_2_rr=-4*ss*pi2*(d_2_1_ rr+d _2_ 2_rr ); d_2_ss=-0.25*d_2/sigma22.0/ ss* pi*( d_2_ 1_s1 +d_ 2_2_ s1)+ (d_2 _1_ s2+ d_2_2_s2)/ss+4.0*ss*pi2* (d_2 _1_s s1+d _2_ 2_ss 1)-4 .0*s s*p i* (d_2_1_ss2+d_2_2_ss2)+ss *(d_ 2_1_ ss3+ d_2 _2_s s3); d_2_rs=pi/ss*i*(d_2_1_r+ d_2_ 2_r )-4. 0*i* pi2* ss* (d_2 _1_r s1+ d_2_2_rs1)+2.0*i*pi*ss*( d_2_ 1_rs 2+d_ 2_2 _rs2 ); d_3=ss*real(d_3_1+d_3_2) ; d_3_r=2.0*pi*ss*real(i*( d_3_ 1_r +d_3 _2_r )); d_3_s=0.5*d_3/sigma-2.0* ss*p i*r eal( d_3_ 1_s1 +d_ 3_2_ s1)+ ss* real(d_3_1_s2+d_3_2_s2); d_3_rr=-4*ss*pi2*real(d_ 3_1_ rr+ d_3_ 2_rr ); d_3_ss=-0.25*d_3/sigma22.0/ ss* pi*r eal( d_3_ 1_s 1+d_ 3_2_ s1)+ real(d_3_1_s2+d_3_2_s2)/ ss+4 .0*s s*pi 2*r eal( d_3_ 1_ss 1+ +d_3_2_s1)-4.0*ss*pi*rea l(d_ 3_1_ ss2+ d_3 _2_s s2)+ ss* real(d_3_1_ss3+d_3_2_ss3 ); d_3_rs=pi/ss*real(i*(d_3 _1_r +d_ 3_2_ r))4.0* pi2 *ss* real (i* (d_3_1_rs1+d_3_2_rs1))+2 .0*p i*ss *rea l(i *(d_ 3_1_ rs2+ d_3_2_rs2)); c=cos(4.0*phi);s=sin(4.0*phi);

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128 d_1_p=-4.0*(s*real(d_1)+ c*im ag( d_1) ); d_1_pp=-16.0*(c*real(d_1 )-s* ima g(d_ 1)); d_1_rp=-4.0*(s*real(d_1_ r)+c *im ag(d _1_r )); d_1_sp=-4.0*(s*real(d_1_ s)+c *im ag(d _1_s )); ccc=cos(8.0*phi);sss=sin(8.0*phi);d_2_p=-8.0*(sss*real(d_2 )+cc c*i mag( d_2) ); d_2_pp=-64.0*(ccc*real(d _2)sss *ima g(d_ 2)); d_2_rp=-8.0*(sss*real(d_ 2_r) +cc c*im ag(d _2_r )); d_2_sp=-8.0*(sss*real(d_ 2_s) +cc c*im ag(d _2_s )); % d_1 D_1=(c*real(d_1)-s*imag( d_1) ); d_1G1=(c*real(d_1_r)-s*i mag( d_1 _r)) ; d_1G2=(c*real(d_1_s)-s*i mag( d_1 _s)) ; d_1G3=d_1_p;d_1H11=(c*real(d_1_rr)-s *ima g(d _1_r r)); d_1H12=(c*real(d_1_rs)-s *ima g(d _1_r s)); d_1H13=d_1_rp;d_1H21=d_1H12;d_1H22=(c*real(d_1_ss)-s *ima g(d _1_s s)); d_1H23=d_1_sp;d_1H31=d_1H13;d_1H32=d_1H23;d_1H33=d_1_pp; % d_2 D_2=(ccc*real(d_2)-sss*i mag( d_2 )); d_2G1=(ccc*real(d_2_r)-s ss*i mag (d_2 _r)) ; d_2G2=(ccc*real(d_2_s)-s ss*i mag (d_2 _s)) ; d_2G3=d_2_p;d_2H11=(ccc*real(d_2_rr) -sss *im ag(d _2_r r)); d_2H12=(ccc*real(d_2_rs) -sss *im ag(d _2_r s)); d_2H13=d_2_rp;

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129 d_2H21=d_2H12;d_2H22=(ccc*real(d_2_ss) -sss *im ag(d _2_s s)); d_2H23=d_2_sp;d_2H31=d_2H13;d_2H32=d_2H23;d_2H33=d_2_pp; % d_3 D_3=d_3;d_3G1=d_3_r;d_3G2=d_3_s;d_3G3=0.0;d_3H11=d_3_rr;d_3H12=d_3_rs;d_3H13=0.0;d_3H21=d_3H12;d_3H22=d_3_ss;d_3H23=0.0;d_3H31=d_3H13;d_3H32=d_3H23;d_3H33=0.0; % beta_A and its derivatives b=b_0+aniso*D_1+aniso2*( D_33.0 /4.0 *b_0 +D_2 +5. 0*D_ 1); bG1=b_0_r+aniso*d_1G1+an iso2 *(d _3G1 -3.0 /4.0 *b_ 0_r+ d_2G 1+5. 0*d _1G1 ); bG2=b_0_s+aniso*d_1G2+an iso2 *(d _3G2 -3.0 /4.0 *b_ 0_s+ d_2G 2+5. 0*d _1G2 ); bG3=aniso*d_1G3+aniso2*( d_2G 3+5 .0*d _1G3 ); bH11=b_0_rr+aniso*d_1H11 +ani so2 *(d_ 3H11 -3.0 /4. 0*b_ 0_rr +d_2 H11 + 5.0*d_1H11); bH12=b_0_rs+aniso*d_1H12 +ani so2 *(d_ 3H12 -3.0 /4. 0*b_ 0_rs +d_2 H12 + 5.0*d_1H12); bH13=aniso*d_1H13+aniso2 *(d_ 2H1 3+5. 0*d_ 1H13 ); bH21=bH12;bH22=b_0_ss+aniso*d_1H22 +ani so2 *(d_ 3H22 -3.0 /4. 0*b_ 0_ss +d_2 H22 +

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130 5.0*d_1H22); bH23=aniso*d_1H23+aniso2 *(d_ 2H2 3+5. 0*d_ 1H23 ); bH31=bH13;bH32=bH23;bH33=aniso*d_1H33+aniso2 *(d_ 2H3 3+5. 0*d_ 1H33 ); fac_0=(2.0*kappa^2-1.0);fac_1=(fac_0*b+1.0); % Gamma and its derivatives g=8.0*field*aniso*(fac_0 *b*( 1.0 +3.0 *ani so)+ b_0 *(1. 0+3. 0/2. 0*a niso )+ D_1*(1.0+6.0*aniso)+2.0*an iso *(D_ 3+D_ 2)); if((g/fac_1)>=0.5), gg=g/fac_1,error('Gamma too large...'), end gG1=8.0*field*aniso*(fac _0*b G1* (1.0 +3.0 *ani so) +b_0 _r*( 1.0+ 3.0 /2.0 aniso)+d_1G1*(1.0+6.0*an iso )+2. 0*an iso* (d_ 3G1+ d_2G 1)); gG2=8.0*field*aniso*(fac _0*b G2* (1.0 +3.0 *ani so) +b_0 _s*( 1.0+ 3.0 /2.0 aniso)+d_1G2*(1.0+6.0*an iso )+2. 0*an iso* (d_ 3G2+ d_2G 2)); gG3=8.0*field*aniso*(fac _0*b G3* (1.0 +3.0 *ani so) +d_1 G3*( 1.0+ 6.0 aniso)+2.0*aniso*d_2G3); gH11=8.0*field*aniso*(fa c_0* bH1 1*(1 .0+3 .0*a nis o)+b _0_r r*(1 .0+ 3.0/2.0*aniso)+d_1H11*(1.0 +6.0 *ani so)+ 2.0 *ani so*( d_3H 11+ d_2H11)); gH12=8.0*field*aniso*(fa c_0* bH1 2*(1 .0+3 .0*a nis o)+b _0_r s*(1 .0+ 3.0/2.0*aniso)+d_1H12*(1.0 +6.0 *ani so)+ 2.0 *ani so*( d_3H 12+ d_2H12)); gH13=8.0*field*aniso*(fa c_0* bH1 3*(1 .0+3 .0*a nis o)+d _1H1 3*(1 .0+ 6.0*aniso)+2.0*aniso*d_2H1 3); gH21=gH12;gH22=8.0*field*aniso*(fa c_0* bH2 2*(1 .0+3 .0*a nis o)+b _0_s s*(1 .0+ 3.0/2.0*aniso)+d_1H22*(1.0 +6.0 *ani so)+ 2.0 *ani so*( d_3H 22+ d_2H22)); gH23=8.0*field*aniso*(fa c_0* bH2 3*(1 .0+3 .0*a nis o)+d _1H2 3*(1 .0+ 6.0*aniso)+2.0*aniso*d_2H2 3); gH31=gH13;gH32=gH23;gH33=8.0*field*aniso*(fa c_0* bH3 3*(1 .0+3 .0*a nis o)+d _1H3 3*(1 .0+ 6.0*aniso)+2.0*aniso*d_2H3 3);

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131 field=bfield; % Free energy and its derivatives fac_2=(1.0-field)^2/fac_ 1; fac_3=fac_0/fac_1;fl2=field^2;helm=fl2-fac_2*(1.0-g/fa c_1) ; fac_2=fac_2/fac_1;fac_4=(fl2-helm)*fac_3;% Squash modulus c_sq=(fac_4*bH11+fac_2*( gH11 -fa c_3* (2.0 *bG1 *gG 1+g* bH11 -2.0 *fa c_3* g*bG1^2)))*4.0*sigma2*(sin (2.0 *phi ))^2 ; c_sq=c_sq+(fac_4*bH22+fa c_2* (gH 22-f ac_3 *(2. 0*b G2*g G2+g *bH2 2-2 .0* fac_3*g*bG2^2)))*4.0*sigma 2*(c os(2 .0*p hi) )^2; c_sq=c_sq+(fac_4*bH33+fa c_2* (gH 33-f ac_3 *(2. 0*b G3*g G3+g *bH3 3-2 .0* fac_3*g*bG3^2)))*(sin(2.0* phi) )^2; c_sq=c_sq-4.0*(fac_4*bH1 2+fa c_2 *(gH 12-f ac_3 *(b G1*g G2+b G2*g G1+ g* bH12-2.0*fac_3*g*bG1*bG2)) )*si gma2 *sin (4. 0*ph i); c_sq=c_sq-4.0*(fac_4*bH1 3+fa c_2 *(gH 13-f ac_3 *(b G1*g G3+b G3*g G1+ g* bH13-2.0*fac_3*g*bG1*bG3)) )*si gma* (sin (2. 0*ph i))^ 2; c_sq=c_sq+2.0*(fac_4*bH2 3+fa c_2 *(gH 23-f ac_3 *(b G2*g G3+b G3*g G2+ g* bH23-2.0*fac_3*g*bG2*bG3)) )*si gma* sin( 4.0 *phi );

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APPENDIX E MONTE CARLO CODE In this App endix I presen t the Mon te Carlo co de that w as dev elop ed for the study of the gro wth pro cess of the QH electron droplet. It is an implemen tation of the standard Metrop olis algorithm. There are man y switc hes that mak e the program quite v ersatile, all are explained in commen ts in the program. The language of c hoice w as C and use w as made of the optimization rags of the compiler for optim um p erformance. Other optimizations implemen ted at the co de lev el w ere textb o ok minimizations of the n um b er of op erations (m ultiplications and esp ecially divisions) p erformed. This sligh tly obscures the co de at places but the corresp ondence b et w een the equations from the theoretical treatmen t and the implemen tation is clear. The compiled co de is v ery fast, allo wing the sim ulation of large droplets within reasonable time, of the order of one da y for 1000 electrons and 10 8 Mon te Carlo steps. I initially carried out exp erimen ts to c hec k the predictions of the sim ulation of a system without solenoids against theoretical results and to determine ho w the equilibration time scales with the n um b er of electrons. The quan tit y measured w as the energy of the droplet. Then the data taking pro cess w as automated to obtain the shap e of the droplet for a series of particle n um b ers. T ypical n um b ers used w ere 3 10 4 equilibration sw eeps and 4 10 4 measuremen t sw eeps at whic h the densit y distribution of the droplet w as measured b y binning the particles. F or clarication, b y sw eep I imply N Mon te Carlo steps, with N the n um b er of particles. In other w ords in one sw eep all the particles (on a v erage) ha v e mo v ed once. All the statistical analysis w as p erformed at the end of eac h run. I emplo y ed t w o tests to mak e sure that the system had reac hed an equilibrium conguration. I did a few v ery long runs and also started the sim ulation from dieren t 132

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133 initial distributions for the droplet. Both tests conrmed in eac h case that equilibrium had b een reac hed. When at the critical n um b er of electrons, the breakup of the droplet happ ened at nite time, with a distinct signature in the measured energy as one particle w as detac hed from the droplet. The only routine the user m ust supply in the follo wing co de is a random n um b er generator: ran1 /* Magnetic field B=1.0 and also magnetic length l_B=1.0 */ #include#include#include /* Switches */ //#define BARE /* No solenoids (for testing purposes) */ //#define COLLISION_CHECK /* Check for collisions with solenoids */ //#define CHECK_BIN //#define PRINT_ENERGY_WHILE_EQUILI BRAT ING #define PRINT_ENERGY_WHILE_MEAS URIN G //#define INIT_FROM_SAVE /* Assumes that data for N_START -> N_MAX */ /* exist in current directory. Uses last */ /* state to continue taking measurements */ //#define INIT_SQUARE /* Disk is default */ /* Parameters */ #define N_START 100 /* Starting number of electrons */ #define N_MAX 500 /* Maximum number of electrons */ #define N_STEP 50 /* Increment step of electrons */ //#define N_RESTART 600/* Restart simulation from different N */ /* without changing the grid. N_START */ /* and N_MAX should be the same. */ #define N_AVG 30000L /* Measurement sweeps */ #define N_S 6 /* Number of solenoids */ #define MAX_STEP 0.5 /* Maximum step for movement (measured in */ /* units of magnetic length) */ #define PREFACTOR 0.25 /* Potential energy prefactor: 1/4l_B */ #define PI 3.1415926536 #define MAX_THETA 2.0*PI

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134 #define EPS 1.0e-8 /* Epsilon for collision check */ #define EPS2 EPS*EPS #define BIN_OFFSET 20.0 /* Extra space after theoretical R_MAX */ #define BIN_STEP 0.5 /* Size of bins (units of magnetic lengths) */ int N; /* Number of particles */ long SWEEPS; /* Equilibration sweeps */ int SWEEP_FACTOR; /* N*SWEEP_FACTOR=total number of sweeps */ double R_T; /* Theoretical expectation for maximum radius */ /* after equilibration */ double R_INIT; /* Maximum radius used for the initialization */ /* of the particles */ double R_MAX; /* Theoretical radius for N_MAX */ unsigned int N_BINS; /* Number of bins in a single direction */ typedef struct e { double x; double y; } electron; #ifndef BARE typedef struct s { double x; double y; double flux; } solenoid; void Print_Solenoids(solenoi d *); void Init_Energy(electron *,solenoid *); #ifdef INIT_FROM_SAVE void Init_Ensemble_From_Save( elec tron *); void Init_Solenoids_From_Save (sol enoi d *); #else void Init_Solenoids_Custom(sol enoi d *); #endif #else void Init_Energy(electron *); #endif #ifdef INIT_SQUARE

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135 void Init_Square(electron *); #else void Init_Disk(electron *); #endif void Monte_Carlo(void); void Print_Ensemble(electron *); void Bin_Electrons(double **, electron *); int Check_Coincidence_Init(e lec tron *, int); #ifdef COLLISION_CHECK int Collision(solenoid *, double, double, unsigned int); #endif double Sum(electron *, unsigned int, double, double); double ran1(long *); double potential_energy,e_energ y; #ifndef BARE double s_energy; #endif long collisions; /* Seed */ long dum=-2; int main(void) { FILE *otf; unsigned int i; double x,r_max_temp; /* Construct grid */ #ifndef BARE otf=fopen("grid","w");#else otf=fopen("bare_grid","w" );

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136 #endif/* Choose convenient R_MAX */ R_MAX=sqrt(2.0*((double) N_MAX))+BIN_OFFSET; r_max_temp=0.0;for(i=0;i<1000000;i++) { r_max_temp+=BIN_STEP;if(r_max_temp > R_MAX) break; } if(i==1000000) { printf("Error: too small BIN_STEP..."); exit(-1);} N_BINS=2*(i+1);R_MAX=r_max_temp;x=-R_MAX+BIN_STEP/2;for(i=0;i
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137 printf("N=%d\n",N);SWEEP_FACTOR=N/10;SWEEPS=N*SWEEP_FACTOR;R_T=sqrt(2.0*((double) N)); R_INIT=0.01*R_T;Monte_Carlo();} #endifreturn 0; }void Monte_Carlo(void) { FILE *otf,*inf; #ifdef PRINT_ENERGY_WHILE_MEASU RING FILE *ef; #endif char string[80]; unsigned int i,iter,counter; double x_try,y_try,x_temp,y_tem p,te mp_p oten tia l_en ergy ; double denergy,temp_energy,**bi ns,s um; electron *ensemble; #ifndef BARE int j; double solenoid_contribution,s_x ,s_y ; double dx_try,dy_try,dx_temp,dy_ temp ; solenoid *solenoids; #endif/* Memory Allocation */ ensemble=(electron *) malloc((size_t) (N*sizeof(electron))); if(!ensemble) { printf("Allocation error 1...\n");

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138 exit(-1);} #ifndef BARE solenoids=(solenoid *) malloc((size_t) (N_S*sizeof(solenoid))) ; if(!solenoids) { printf("Allocation error 2...\n"); exit(-1);} #endifbins=(double **) malloc((size_t) (N_BINS*sizeof(double *))); if(!bins) { printf("Allocations error 3...\n"); exit(-1);} for(i=0;i
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139 collisions=0;#ifdef INIT_FROM_SAVE Init_Ensemble_From_Save(e nse mble ); fprintf(inf,"Initializing from saved data...\n"); #else #ifdef INIT_SQUARE Init_Square(ensemble);fprintf(inf,"Initial distribution: square...\n"); #else Init_Disk(ensemble);fprintf(inf,"Initial distribution: disk...\n"); #endif #endif fprintf(inf,"# of collisions while initializing: %ld\n",collisions); collisions=0; /* Initialize bins */ for(i=0;i
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140 ef=fopen(string,"w"); #endif #endif #ifndef BARE Init_Energy(ensemble,sole noi ds); #else Init_Energy(ensemble); #endif/* Main */ fprintf(inf,"\n# of electrons: %d\n",N); #ifndef BARE fprintf(inf,"# of solenoids: %d\n\n",N_S); #endif fprintf(inf,"Equilibrati on Sweeps: %ld\n",SWEEPS); fprintf(inf,"Equilibrati on Steps: %ld\n",SWEEPS*N); fprintf(inf,"Samples: %ld\n",N_AVG); fprintf(inf,"Sampling Steps: %ld\n",N_AVG*N); fprintf(inf,"\nExpected radius (theoretical): %3.1f\n",R_T); fprintf(inf,"Droplet energy (theoretical): %g\n",-0.25*((double) (N*N))*(log(2.0*((double) N))-1)); fprintf(inf,"\nNumber of bins: %d\n",N_BINS*N_BINS); printf("Equilibrating system...\n"); /* Equilibration */ for(iter=0;iter
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141 x_temp=ensemble[i].x;y_temp=ensemble[i].y;x_try=MAX_STEP*ran1(&dum) ; y_try=MAX_STEP*ran1(&dum) ; if(ran1(&dum) >= 0.5) x_try*=-1.0; if(ran1(&dum) >= 0.5) y_try*=-1.0; x_try+=x_temp;y_try+=y_temp; #ifdef COLLISION_CHECK /* Check for collision to avoid infinities in the energy */ /* If there is a collision try another move since the */ /* proposed one is not in the phase space of the problem */ if(Collision(solenoids,x_ try, y_tr y,i) ) { ++collisions;--counter;continue;} #endif temp_potential_energy=PRE FACT OR*( x_tr y*x _try +y_t ry*y _tr yx_temp*x_temp-y_temp*y_tem p); temp_energy=Sum(ensemble, i,x_ try, y_tr y); #ifndef BARE solenoid_contribution=0.0 ; for(j=0;j
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142 }/* Divide solenoid contribution by 2 since we */ /* use |z_i-z_j|^2 instead of |z_i-z_j| */ solenoid_contribution*=0. 5; denergy=temp_energy+temp_ pote ntia l_en erg y+ solenoid_contribution; #else denergy=temp_energy+temp_ pote ntia l_en erg y; #endif /* Metropolis */ if(denergy <= 0.0) { #ifndef BARE s_energy+=solenoid_contrib utio n; #endif potential_energy+=temp_pot enti al_e ner gy; e_energy+=temp_energy;ensemble[i].x=x_try;ensemble[i].y=y_try;continue;} if(ran1(&dum) < exp(-2.0*denergy)) { #ifndef BARE s_energy+=solenoid_contrib utio n; #endif potential_energy+=temp_pot enti al_e ner gy; e_energy+=temp_energy;ensemble[i].x=x_try;ensemble[i].y=y_try;} } } printf("Equilibration complete...\nTaking measurements...\n"); /* Measurements */

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143 for(iter=0;iter= 0.5) x_try*=-1.0; if(ran1(&dum) >= 0.5) y_try*=-1.0; x_try+=x_temp;y_try+=y_temp; #ifdef COLLISION_CHECK /* Check for collision to avoid infinities in the energy */ /* If there is a collision try another move since the */ /* proposed one is not in the phase space of the problem */ if(Collision(solenoids,x_ try, y_tr y,i) ) { ++collisions;--counter;continue;} #endif temp_potential_energy=PRE FACT OR*( x_tr y*x _try +y_t ry*y _tr yx_temp*x_temp-y_temp*y_tem p); temp_energy=Sum(ensemble, i,x_ try, y_tr y);

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144 #ifndef BARE solenoid_contribution=0.0 ; for(j=0;j
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145 if(ran1(&dum) < exp(-2.0*denergy)) { #ifndef BARE s_energy+=solenoid_contrib utio n; #endif potential_energy+=temp_pot enti al_e ner gy; e_energy+=temp_energy;ensemble[i].x=x_try;ensemble[i].y=y_try;} } } #ifdef PRINT_ENERGY_WHILE_EQUIL IBRA TING fclose(otf);#endif #ifdef PRINT_ENERGY_WHILE_MEASU RING fclose(ef);#endif #ifdef COLLISION_CHECK fprintf(inf,"# of collisions: %ld\n\n",collisions); #endif Print_Ensemble(ensemble) ; /* Normalize distribution */ for(i=0;i
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146 fprintf(otf,"%f\t",bins[N _BIN S-1i][i ter ]); fprintf(otf,"\n");} fclose(otf); /* Construct and print contour */ sum=0.0;counter=0;for(i=0;i
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147 int sgn_x,sgn_y; for(i=0;i=0.0) ? 1 : -1; sgn_y=(y>=0.0) ? 1 : -1; x=fabs(x);y=fabs(y);for(x_try=R_MAX-BIN_STEP, x_i =N_B INS1;x_ try >=0. 0;x_ try=BI N_ST EP ,x_i--) if(x >= x_try) break; for(y_try=R_MAX-BIN_STEP, y_i =0;y _try >=0. 0;y _try -=BI N_ST EP, y_i+ +) if(y >= y_try) break; #ifdef CHECK_BIN if(x_i < N_BINS/2 || y_i < N_BINS/2) { printf("Error in Bin_Electrons... %f %d %f %d\n",x,x_i,y,y_i); exit(-1);} #endif if(sgn_x==1 && sgn_y==1) { bins[y_i][x_i]+=1.0;continue;} if(sgn_x==1 && sgn_y==-1) { bins[N_BINS-1-y_i][x_i]+= 1.0; continue;} if(sgn_x==-1 && sgn_y==-1) { bins[N_BINS-1-y_i][N_BINS -1-x _i]+ =1.0 ; continue;} if(sgn_x==-1 && sgn_y==1) { bins[y_i][N_BINS-1-x_i]+= 1.0; continue;}

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148 printf("Bin_Electrons: Should not have reached this point...\n"); exit(-1);} return;}#ifndef BARE void Init_Solenoids_Custom(sole noid *s) { double radius=sqrt(2.0*((double ) N_MAX)); int i; for(i=0;i<3;i++) { s[i].x=(radius)*cos(2.0*P I/3 .0*i ); s[i].y=(radius)*sin(2.0*P I/3 .0*i ); s[i].flux=-50.0;s[3+i].x=(radius)*cos(PI/ 3.0 +2.0 *PI/ 3.0* i); s[3+i].y=(radius)*sin(PI/ 3.0 +2.0 *PI/ 3.0* i); s[3+i].flux=50.0;} return;} #endif #ifdef INIT_FROM_SAVE void Init_Ensemble_From_Save(el ectr on *e) { FILE *inf; char string[80]; unsigned int i;

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149 double x,y; #ifndef BARE sprintf(string,"final_sta te_ %d", N); #else sprintf(string,"bare_fina l_st ate_ %d", N); #endif inf=fopen(string,"r");if(!inf) { printf("Error: File %s not found...\n",string); exit(-1);} i=0;while(!feof(inf)) { if(i==N) { printf("Error: number of particles read exceeds N...\n"); exit(-1);} fscanf(inf,"%lf %lf\n",&x,&y); e[i].x=x;e[i].y=y;++i;} if(i!=N) { printf("Error: number of particles read exceeds N...\n"); exit(-1);} fclose(inf);return;}void Init_Solenoids_From_Save(s olen oid *s) { FILE *inf; double x,y,flux;

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150 int i; inf=fopen("solenoids_pos ","r "); if(!inf) { printf("Error: could not find solenoids_pos...\n"); exit(-1);} i=0;while(!feof(inf)) { if(i==N_S) { printf("Error: number of solenoids found exceeds N_S...\n"); exit(-1);} fscanf(inf,"%lf %lf %lf\n",&x,&y,&flux); s[i].x=x;s[i].y=y;s[i].flux=flux;++i;} fclose(inf);inf=fopen("solenoids_neg ","r "); if(!inf) { printf("Error: could not find solenoids_neg...\n"); exit(-1);} while(!feof(inf)) { if(i==N_S) { printf("Error: number of solenoids found exceeds N_S...\n"); exit(-1);} fscanf(inf,"%lf %lf %lf\n",&x,&y,&flux); s[i].x=x;s[i].y=y;s[i].flux=flux;++i;}

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151 fclose(inf);if(i!=N_S) { printf("Error: less solenoids found than expected in files...\n"); exit(-1);} printf("Solenoids read:\n"); for(i=0;i= 0.5) ensemble[i].x*=-1.0;

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152 if(ran1(&dum) >= 0.5) ensemble[i].y*=-1.0; } while(Check_Coincidence _Ini t(en semb le, i)); return;} #endif #endif double Sum(electron *ensemble, unsigned int k, double x_try double y_try) { unsigned int i; double sum,x_k,y_k,x_i,y_i,dx_k ,dy_ k,dx _try ,dy _try ; sum=0.0;x_k=ensemble[k].x;y_k=ensemble[k].y;for(i=0;i
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153 }/* Divide by 2 since we use |z_i-z_j|^2 instead of |z_i-z_j| */ return 0.5*sum; }#ifdef COLLISION_CHECK int Collision(solenoid *s, double x_try, double y_try, unsigned int i) { unsigned int j; double x_j,y_j,dx,dy; for(j=0;j
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154 potential_energy=0.0;e_energy=0.0;s_energy=0.0;for(i=0;i
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155 for(i=0;i
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156 for(i=0;i 0.0) { fprintf(otf,"%f %f %f\n",s[i].x,s[i].y,s[i]. flux ); fprintf(ftf,"%f\t",s[i].x ); fprintf(gtf,"%f\t",s[i].y ); } fclose(otf);fclose(ftf);fclose(gtf);sprintf(string,"solenoid s_ne g") ;

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157 otf=fopen(string,"w");sprintf(string,"s_n_x");ftf=fopen(string,"w");sprintf(string,"s_n_y");gtf=fopen(string,"w");for(i=0;i
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158 } } return 0; }#undef N_START #undef N_MAX #undef N_STEP #undef PI #undef MAX_THETA #undef EPS #undef EPS2 #undef N_AVG #undef MAX_STEP #undef N_S #undef BIN_OFFSET #undef BIN_STEP #undef PREFACTOR #undef SWEEP_FACTOR #ifdef COLLISION_CHECK #undef COLLISION_CHECK #endif#ifdef BARE #undef BARE #endif#ifdef PRINT_ENERGY_WHILE_EQUIL IBRA TING #undef PRINT_ENERGY_WHILE_EQUIL IBRA TING #endif#ifdef PRINT_ENERGY_WHILE_MEASU RING #undef PRINT_ENERGY_WHILE_MEASU RING #endif#ifdef CHECK_BIN #undef CHECK_BIN #endif#ifdef INIT_SQUARE #undef INIT_SQUARE

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159 #endif#ifdef INIT_FROM_SAVE #undef INIT_FROM_SAVE #endif#ifdef N_RESTART #undef N_RESTART #endif

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PAGE 172

BIOGRAPHICAL SKETCH I w as b orn in A thens, Greece in 1974. Immediately after high sc ho ol I enrolled in the Ph ysics Departmen t of the Univ ersit y of Crete in 1992, where I obtained a B.S. in ph ysics in 1997. As an undergraduate m y in terest w as fo cused in the elds of computational ph ysics, neural net w orks and condensed matter. In 1997 I joined the Ph ysics Departmen t of the Univ ersit y of Florida. After one y ear of core courses I to ok and successfully completed the qualifying exams for the Ph.D. degree. After a short p erio d of indecision, I made up m y mind to study condensed matter, a decision I ha v e not regretted. 167


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Title: Structural transitions of the vortex lattice in anisotropic superconductors and fingering instability of electron droplets in an inhomogeneous magnetic field
Physical Description: Mixed Material
Creator: Klironomos, Alexios ( Author, Primary )
Publication Date: 2003
Copyright Date: 2003

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Material Information

Title: Structural transitions of the vortex lattice in anisotropic superconductors and fingering instability of electron droplets in an inhomogeneous magnetic field
Physical Description: Mixed Material
Creator: Klironomos, Alexios ( Author, Primary )
Publication Date: 2003
Copyright Date: 2003

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Full Text









STRUCTURAL TRANSITIONS OF THE VORTEX LATTICE IN ANISOTROPIC
SUPERCONDUCTORS AND FINGERING INSTABILITY OF ELECTRON
DROPLETS IN AN INHOMOGENEOUS MAGNETIC FIELD












By

ALEXIOS KLIRONOMOS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2003













ACKNOWLEDGMENTS

First and foremost I would like to thank Professor Alan T. Dorsey for his kind

and motivating mentorship. His dedication, ability and encouragement have been

inspiring in difficult times.

I would also like to extend my gratitude to my immediate family and to my

close and dear friends for their support and tolerance.
















TABLE OF CONTENTS

ACKNOWLEDGMENTS .. ............. . ii

ABSTRACT . . . . . . . .. v

CHAPTERS


1 SUPERCONDUCTIVITY . ..
1.1 Conventional Superconductivity ..........
1.2 Ginzburg-Landau Theory ..............
1.3 London M odel ....................
1.4 Anisotropy and Structural Phase Transitions .
1.5 Elasticity and Melting of the Vortex Lattice .

2 GINZBURG-LANDAU THEORY . .....
2.1 Anisotropic Ginzburg-Landau Theory . .
2.2 Virial Theorem . ..............

3 MEAN FIELD THEORY . .
3.1 Solution of the First Ginzburg-Landau Equation
3.1.1 Landau Gauge . ..........
3.1.2 Symmetric Gauge . . .


3.2 Structure of the Vortex Lattice and Minimization c


. . . I
2
. . . 7
.. . 13
. . 15
. . 18

. 20
.. . 20
.. . 25

.. . 28
. . . 28
.. . 29
. . . 3 1
f the Free Energy 35


4 ELASTICITY THEORY FOR THE VORTEX LATTICE .. .....


5 STRUCTURAL PHASE TRANSITIONS . .....
5.1 Nondispersive Elastic Moduli . ...........
5.2 Fluctuations ............... .......

6 FINGERING INSTABILITY OF ELECTRON DROPLETS .
6.1 Many-body Wavefunction . .............
6.2 Conformal Mapping Method ................
6.3 Monte Carlo Simulation . ..............

7 CONCLUSION . . . . . .

A EXTENSION OF THE VIRIAL THEOREM . ....
A.1 A Useful Identity . ................
A.2 Generalized Abrikosov Identities ..............
A.3 Free Energy, Magnetization, Gibbs Energy . ...


. 47
. . 47
. . 50

. . 56
. . 58
. . 61
. . 64


. 77
. . 79
. . 79
. . 81










B CALCULATION OF /A, ( X *(l- n )2'), \ 2*(II 2 I)22). .

C M OM ENTS . . . . . . .
C.1 Conformal Map z = f(w) = aw bw-M ...............
C.I.1 Exterior M om ents ........................
C.1.2 Interior M om ents ........................
C.2 Conformal Map z = f(w) .= aw M .-..
C.3 Conformal Map z = f(w) = rw Q/(w wo) + Q/(w wo) .
C.4 Connection with Magnetic Field Inhomogeneity .. .........
C.4.1 Exterior M moments .. ....................
C.4.2 Potential V (z) . ....................


D CODE FOR THE NUMERICAL CALCULATIONS .
D.1 driverl.m. .................. ..
D.2 energy.m ... .. ................
D .3 m oduli.m . . . . . .
D.4 driver2.m ....... ........... ...
D.5 csquash.m .....................

E MONTE CARLO CODE .. ........

REFERENCES . . . . .


. . 97
. . . 98
. . . 99
. . . .. 109
. . . 120
. . . 12 1


. 160


BIOGRAPHICAL SKETCH . . . . . 167


S 83

















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

STRUCTURAL TRANSITIONS OF THE VORTEX LATTICE IN ANISOTROPIC
SUPERCONDUCTORS AND FINGERING INSTABILITY OF ELECTRON
DROPLETS IN AN INHOMOGENEOUS MAGNETIC FIELD

By

Alexios Klironomos

Mi.,- 2003

Chairman: Alan T. Dorsey
lM.. ir Department: Physics

I present a derivation of the nondispersive elastic moduli for the vortex lattice

within the anisotropic Ginzburg-Landau model. I derive an extension of the virial

theorem for superconductivity for anisotropic superconductors, with the anisotropy

arising from s-d mixing or an anisotropic Fermi surface. The structural transition

from rhombic to square vortex lattice is studied within this model along with the

effects of thermal fluctuations on the structural transition. The reentrant transition

from square to rhombic vortex lattice for high fields and the instability with respect

to rigid rotations of the vortex lattice, predicted by calculations within the nonlocal

London model, are also present in the anisotropic Ginzburg-Landau model.

I also study the fingering of an electron droplet in a special Quantum Hall

regime, where electrostatic forces are weak. Performing Monte Carlo simulations I

study the growth and fingering of the electron droplet in an inhomogeneous magnetic

field as the number of electrons is increased. I expand on recent theoretical results

and find excellent agreement between my simulations and the theoretical predictions.














CHAPTER 1
SUPERCONDUCTIVITY

The fascinating phenomenon of superconductivity, although discovered in 1911

by H. Kamerlingh Onnes, has in the last two decades begun to play a very impor-

tant role in everyday life through technology. This technological revolution was even

more apparent to scientists and laymen alike after the discovery of high temperature

superconductors (HTSCs) by Bednorz and Miller in 1986.

The theory of conventional superconductivity is well understood today. There

exists an accepted microscopic theory, the so called BCS theory, which explains ad-

equately the behavior of conventional or isotropic, low temperature superconductors

like Pb, Al, Nb, Nb3Sn. However, HTSCs have been proven difficult beasts to tame.

No microscopic theory with the degree of success of BCS in describing the relevant

phenomena exists as of today. There are many phenomenological models that describe

the phenomena in both kinds of superconductors with various degrees of success.

One very complicated and extremely important issue for technological applica-

tions is the morphology and properties of the vortex lattice in type-II superconductors.

The vortex lattice is an arrangement of magnetic filaments inside the bulk of the su-

perconductor which enter under certain conditions when the superconductor is placed

inside a magnetic field. The distinction between type-I and type-II superconductors

will be clarified later in this introductory survey.

In this work I will use one particular phenomenological model, the Ginzburg-

Landau theory, to study the elastic properties of the vortex lattice in anisotropic

superconductors-in other words, the response of the vortex lattice to deformations.

I also address the issue of structural phase transitions of the vortex lattice and the

effects of thermal fluctuations.







2

The survey of superconductivity that follows will attempt to define and ex-

plain the basic concepts required and thus provide the theoretical background for the

understanding of the main body of the research that is reported here. For a more

detailed introduction to superconductivity, the reader can refer to various sources on

the subject [1-3].

1.1 Conventional Superconductivity

The two distinctive phenomena associated with superconductivity are the dis-

appearance of resistance at some critical temperature T, (perfect conductivity) and

the expulsion of magnetic fields from the bulk of a metal once it enters the supercon-

ducting phase (perfect diamagnetism or Meissner effect). Conversely, the application

of a sufficiently strong magnetic field is found to destroy superconductivity.

There exists a classification scheme for superconductors according to their be-

havior when an external magnetic field is applied. For type-I superconductors, the

magnetic field cannot enter the bulk of the material until the so-called thermody-

namic critical field H,(T) is applied. The thermodynamic critical magnetic field

is related to the free energy difference between the normal and the superconduct-

ing phases. For type-II superconductors the magnetic field penetration begins at a

stronger field, called H-2(T), and persists down to a smaller critical field, H, i(T), with

H,2 > H, > H,1. The characteristic behavior of each type of superconductor when a

magnetic field is applied is illustrated in figure (1.1).

Type-II superconductors and especially the intermediate phase, also called the

mixed phase or Shubnikov phase, are of interest to me. In the mixed phase the

magnetic field enters the bulk of the material in the form of flux tubes, each carrying

a quantum of magnetic flux


he
o= = 2.07 x 10-7 gauss-cm2. (1.1)
2e























Type II





Type I


0 /HI H, Hc2 H

Figure 1.1: Flux penetration behavior for type-I and type-II superconductors with
the same thermodynamic critical field H,.

For most of the region between the two critical fields H~i(T) and Hc2(T), the flux

tubes form a regular array that in a sense resembles the ( '--I.1l lattice of ordinary

( v -I .,l- The major difference is that the vortex lattice is essentially two dimensional

as the flux tubes can be considered as straight lines which, however, can bend and

deform.

The existence of magnetic fields inside the superconductor and superconductiv-

ity may at first seem to be two mutually exclusive events. However, the magnetic

tubes have a vortex of supercurrent swirling around them which screens the magnetic

field from the rest of the bulk. Superconductivity is completely destroyed in the core

of the magnetic flux tube where the magnetic field attains its largest value.

The interactions between the vortex lines are repulsive. This interaction, bal-

anced by the external magnetic field pressure, stabilizes the vortex lattice. Alexei









Abrikosov, in a seminal work [4], first studied and predicted the existence of the

vortex lattice in conventional, type-II superconductors.

There are two very important length scales associated with superconductivity.

The penetration length A determines the electromagnetic response of the supercon-

ductor providing a measure of the spatial extent of the magnetic field in the bulk of

the material. The coherence length C provides a measure of the size of the normal

core. The coherence length, C is also the characteristic lengthscale associated with

variations of the superconducting phase in more general situations. It can be shown

that H,1 (o/47iA2 and Hc2 I- 0/27r2. In addition, K = A/C is a dimensionless

q(111.nl ilt approximately independent of temperature, called the Ginzburg-Landau pa-

rameter. For a typical elemental superconductor A 500 A and C 3000 A which

gives a Ginzburg-Landau parameter K < 1.

The value of the Ginzburg-Landau parameter facilitates the characterization of

a material as type-I or type-II. Actually, c = 1//2 serves as the boundary value

between the two regimes. For smaller values one has a type-I superconductor while

for larger values a type-II. Typical HTSCs are extreme type-II superconductors with

K of the order of 102.

There are several methods that one can use to observe and study the static

properties and the morphology of the vortex lattice. Small angle neutron scattering

(SANS) [5-14], Bitter decoration [15, 16], scanning tunneling microscopy (STM) [17,

18], magneto-optical methods [19] and muon spin rotation methods (p/SR) [20] are

the ones widely used.

Of all these methods, small angle neutron scattering stands out as the most

useful to make bulk measurements as the majority of the mentioned methods are

limited to imaging the flux lines at the surface of the superconductor.1 In short,

the SANS method takes advantage of the weak interaction of neutrons with mat-
1One can probe the bulk of the superconductor using pSR. However, sophisticated modeling is
necessary to interpret the data.









ter through short range nuclear forces and dipole interactions with spatially varying

magnetic fields. The neutrons are scattered from the bulk of the material revealing

information about ( ',-I.,l structure, the vortex lattice and magnetic ordering. More

specifically, from the diffraction pattern one can deduce the symmetry of the vortex

lattice. Measuring the scattered intensity enables one to determine the characteristic

length scales A and and the field profile around the flux lines, which in turn allows

the determination of the critical fields. Also, from the width of the diffraction peaks

one can estimate the vortex lattice correlation lengths.

The existence of this new superconducting phase and the phase transition which

accompanies it leads naturally to the idea of the order parameter. Although proposed

by Ginzburg and Landau [21] before the discovery of the BCS theory, the order pa-

rameter can be thought of as the wavefunction for the center of mass motion of the

Cooper pairs. I will devote a separate section to the study of the Ginzburg-Landau

theory.

For type-II superconductors the mean field H-T phase diagram is rather simple,

as illustrated in figure (1.2). The characterization min .111 field" implies that fluctua-

tion effects are not considered, which is a drastic simplification.

Although this dissertation deals with mainly static properties of the vortex

lattice, I will address here some of the dynamic properties of the vortex system to

unveil the complexity that emerges once one chooses to depart from the simple (but

nonetheless important) approach of mean field theory. For an excellent extensive

review the interested reader might refer to the classic work of Blatter and others [22].

For a more pedagogical introduction two other nice reviews are available [23, 24]

When an external current density is applied to the vortex system, the flux

lines will start to move under the influence of the Lorentz force F = j x B/c. In

a homogeneous system the only resistance to the motion is provided by the friction

force, which is proportional to the steady state velocity v of the vortex system. The









H


H,2(T)


Normal Metal



Mixed Phase






H, (T)

Meissner Phase

0 T T

Figure 1.2: The H-T phase diagram of a type-II superconductor. At high fields and
temperatures the material is in the normal metallic phase. At low temperatures and
fields the material is in the Meissner phase while for intermediate temperatures and
fields the material is in the mixed phase.

steady state velocity will then be: v = j x B/plC, where pt is a generalized friction

coefficient. However, as the flux tubes move, an electric field E = B x v/c appears

parallel to the external current density. With the current density j and the electric E

in parallel there is power dissipated in the system which implies that the fundamental

property of dissipation-free current flow in a superconductor is lost. Disorder comes

to the rescue, pinning the flux lines and restoring dissipation-free current flow. Of

course there is a critical depinning current density je which is always bounded above

by the depairing current density jo. The implications in a technological context are

obvious.









Another kind of disorder is thermal fluctuations. Quenched disorder as de-

scribed above is static while thermal fluctuations are dynamic. Considering thermal

fluctuations opens up the possibility for melting of the vortex solid into a vortex liq-

uid or the emergence of a glass phase. It is apparent that quenched disorder and

thermal fluctuations are competing forms of disorder. Thermally activated jumps

over the pinning barriers can cause the vortices to move, a phenomenon called flux

creep, while thermal fluctuations of a single vortex line lead to a dynamical sampling

and averaging of the confining potential arising from disorder, which in turn reduce

the critical current density j,. All these effects modify the mean field phase diagram

considerably and at the same time enrich and complicate the 1ir, -i. involved.

In high temperature superconductors the effects of the fluctuations are more pro-

nounced, due to weaker pinning of the vortices and enhanced thermal fluctuations.

This becomes immediately apparent since the typical elemental superconductor has

a critical temperature T,c 10 K compared to T,c 100 K of a HTSC. As a conse-

quence larger parts of the phase diagram contain melted or depinned vortex lattice,

and also new exotic phases, such as entangled or disentangled vortex liquids, along

with a zoo of vortex glasses, are possible.

As the reader can infer from this limited introduction, the 1.li-,-i. associated

with the study of superconductivity and the multitude of effects and phenomena it

encompasses is difficult but the research is extremely fascinating and rewarding at the

same time.

1.2 Ginzburg-Landau Theory

The Ginzburg-Landau theory for superconductivity [21] is an extension of Lan-

dau's theory of second order phase transitions to a spatially varying complex order

parameter 1(r). Originally proposed well before the discovery of BCS theory it has

been proven an indispensable tool for the phenomenological treatment of a variety









of problems in many fields. Although conceptually simple, it is an elegant theory of

considerable predictive power.

The Ginzburg-Landau theory has initially a somewhat limited region of applica-

bility. It is valid near the transition temperature T,. However, this restriction can be

relaxed using various arguments or techniques, or even ignored for the sake of making

qualitative predictions.

Two fields are necessary for the formulation of the theory. One is the complex

order parameter T(r) = f(r)e4(r) and the other is the vector potential A(r), related

to the local magnetic field B(r) = V x A(r). The order parameter is zero at the

transition point and has a nonzero value when the material is in the superconducting

state. Based on the continuity of the change of state in a phase transition of the second

kind, an expansion of the free energy in terms of the order parameter is feasible.

The Ginzburg-Landau free energy functional F[I, A] is built using symmetry

arguments. Odd powers of T are immediately excluded as the free energy density

(an observable) should be invariant under a transformation of the form ei "(x)4.
Another simplification one makes is the assumption that the crystal has cubic sym-

metry. The spatial variation of the order parameter then dictates the inclusion of

gradient terms in the free energy functional. First order derivatives and second order

derivatives of the form 02/XzizXk are excluded because these are essentially surface

terms. The lowest order acceptable gradient term is of the form IV I'2, which has

to be made gauge invariant by combining it with the vector potential resulting in

Infl 2 = (-iV e*A/hc)l|2. Taking into account the above considerations, the
Ginzburg-Landau free energy functional can be written now in the following form:


F[I, A] = F, + dxL a12 1ip4 2 h+ B12 (1.2)
2 2m* 87 J









Here F, denotes the free energy of the normal state and b is a positive coefficient

without any temperature dependence. On the other hand a is a function of the

temperature given by

a = ao(T T,), (1.3)

with the constant ao being positive. The constants e* and m* can be identified with

the charge and the mass of a Cooper pair, namely e* = 2e and m* = 2m, where e and

m are respectively the charge and the mass of the bare electron.

To elucidate a few fundamental concepts I consider first the case of a homoge-

neous superconductor with no external field. The order parameter is then independent

of the coordinates. The minimum of the free energy density occurs when


-l (T,~ T). (1.4)
b


One sees that the square of the order parameter, which is proportional to the super-

conducting electron density, decreases linearly to zero at the transition point. The

difference in the free energies of the superconductor and normal metal is found to be

2
F F =-V (T- T)2. (1.5)


The difference in the energies can be equated to -VH,2/87, which is the energy density

for the critical magnetic field. This immediately yields the temperature dependence

of the critical field near the transition point


H, (T T). (1.6)


Differentiating the difference of the free energies with respect to the temperature one

obtains the difference in the entropies and finally the jump of the specific heat at the









transition point

Cs c, = v (1.7)
b

The reader should be cautioned that such trivial manipulations are only possible

in the simplest of cases. The solution of the Ginzburg-Landau equations, the equations

of motion derived from the Ginzburg-Landau free energy, is a difficult task as will be

made apparent shortly.

An interesting observation is that only the modulus of the order parameter

f(r) = I(r)l and the gauge invariant supervelocity Q = (hMo/27m*)Vq(r)-

e*A(r)/m*c enter the Ginzburg-Landau theory, but not T and A separately [25].

There is a completely equivalent formulation of the Ginzburg-Landau in terms of

Q and f, implying that since only real and gauge invariant quantities are involved,

any calculations and results (even approximations) should contain gauge invariant

equations.

One can derive the equations of motion for the two fields T and A minimizing

the Ginzburg-Landau free energy functional with respect to I* and A. One obtains


h i2 2- 20
-iV ?A + a+ b112 = 0 (1.8)
2m* ch )
-ieh
=h(f*Vf IV\*). (1.9)
2m*


In the process of deriving the equations of motion a boundary condition has to be

imposed to ensure that the surface integrals in the variation 6F are zero. With n

being the normal vector to the surface of the superconductor the boundary condition

has the form

n iV -A 0. (1.10)
\hC )









An immediate result is that the normal component of the supercurrent vanishes at

the boundary as expected: n j = 0, since one assumes that no current is fed into the

material.

It can be shown that the characteristic lengths and A have the following forms

within the Ginzburg-Landau theory:


(T) (T- T), (1.11)

F m*c2b 1
A(T) M b (T- T) (1.12)
[47c*2e* OI

In addition, quite generally, one can include an electric field and time dependence

through the assumption
( ie* DF


with D being the scalar potential and 7 being a constant [26-28]. One then obtains the

so called time dependent Ginzburg-Landau equations. At this point I have to stress

that the validity criteria for the time dependent version are different from the simple

Ginzburg-Landau equations. The interested reader should consult the appropriate

literature on the subject.

The theory can also be extended to superconductors with crystal symmetry

other than cubic with the introduction of an anisotropic effective mass tensor that

multiplies the gradient term in the free energy density [29, 30]. This subject will be

discussed extensively along with other ways to account for anisotropy later in this

work.

The full solution of the Ginzburg-Landau equations, except in trivial cases, is

possible only with the use of numerical methods [31-34] since it is a nonlinear differ-

ential equation. However, useful approximations can be made which produce valuable

and accurate results. One frequently "'apl-vid is the assumption that the nonlinear









term T1|12 in the first Ginzburg-Landau equation is small enough to be neglected.

This assumption works well in many cases but should be used with appropriate care.

One case for which this term requires special treatment is in the calculation of the

dispersive elastic moduli of the vortex lattice, cll and c44, where it introduces un-

pll, .-i .,l divergences that are removed as soon as this term is included and properly

treated [35,36].

A well behaved solution, in the symmetric gauge, that represents the vortex

lattice near Hc2 with one flux quantum per flux line was found by Brandt [35-40]:


'(x, y) (constant) exp [- 2 (x2 2+ )] 2 [( x) + iy )]. (1.14)


Here the product is over the (periodic) flux line positions (x,, y,). Notice the similarity

with the Laughlin wavefunction for a filled (v = 1) Landau level within the context of

the Quantum Hall effect (see Chapter 6 later in this thesis). This particular solution

can be used to calculate the energy of structural defects in the vortex lattice ;', 11]

and also the elastic moduli [25, 35-38, 42] of the vortex lattice.

One needs to define one very important parameter here, the Abrikosov param-

eter PA. It is the ratio

PA = (1.15)
(12) 2'

where the brackets denote spatial averaging. By means of the Schwartz inequality the

range of the Abrikosov parameter is [1, oo). In the mixed phase the Abrikosov param-

eter is a geometrical constant, attaining different values for different types of vortex

lattices. It is easily shown that for an isotropic superconductor the minimization of

the Ginzburg-Landau free energy is equivalent to the minimization of the Abrikosov

parameter AA. For example, for a triangular lattice PA 1.1659, while for a square

lattice PA 1.1803, thus making the triangular lattice the lowest energy configura-

tion of the vortices in an isotropic superconductor. One notable detail is the small









difference of the two values. Initially Abrikosov obtained the incorrect result, namely

the square lattice being the equilibrium configuration. This small difference makes

the existence of structural transitions very plausible.

For an anisotropic superconductor the situation is more complicated and more

interesting because the anisotropy opens up the possibility of phase transitions from

the triangular phase to even more general rhombic phases and eventually to the square

phase under certain conditions.

1.3 London Model

The Ginzburg-Landau theory, being an expansion in powers of the order param-

eter which is assumed to be of small magnitude, is valid near H,2. Near Hj1 on the

other hand, where the vortex core overlap is not significant one can put I1(r)l = 1

outside the vortex core. This is the starting point for the formulation of the London

theory. Either starting from the simplified Ginzburg-Landau free energy density, or

from first principles one obtains the free energy functional of the London theory


F[B] = -I dx B2 +2(V x B)2 (1.16)
87i J


Here A = [m*c2/4 nse*2] is the London penetration depth, with ns being the density

of the superconducting electrons. In this section B = B(r) is the local magnetic field.

Minimizing (1.16) with respect to the local magnetic field B(r) using the fun-

damental relation V B = 0 and also adding the contribution of the vortex core in

the form of 6 function singularities one obtains the London equation for an isolated

vortex

-A2V2B + B = o6(r). (1.17)

Here 4o is a unit vector along the flux line direction. The 6 function approximates the

core region which is of finite extent (of the order of ). The solution of the equation









is easy to obtain for a field directed along the z axis,


(Do
B -( K (1.18)


where Ko(r/A) is the zeroth order modified Bessel function of an imaginary argument.

The ..i- mptotic forms of the solution are more useful:

B =2( In +(constant) ,for < r < A,
I ro irA -_ A

SB- C/A for r >>A.
2 A2 2r

The same method can be used for two parallel vortices directed along the z axis.

One simply adds one more singularity in the right hand side of equation (1.17) and

the solution is obtained in a similar manner as before. One obtains for the magnetic

field of the i-th vortex
Go r T
Bi 272 Ko (1.19)


where ri is the position of the i-th vortex. Substituting back in the London free energy

density (1.16) after a short calculation one finds for the interaction energy per length

of the two vortices Ui, = 4oB12/4r, where B12 is given by


#o ri r2
B12= 2 Ko (l (1.20)


Thus the interaction between two vortices is repulsive. It decreases exponentially at

large distances and diverges logarithmically at short distances.

The London theory has been successfully used in the investigation of the prop-

erties of the vortex lattice. Actually it is relatively easier to obtain the elastic moduli

in the London limit than it is using the Ginzburg-Landau theory. What is usually

done is that some property is calculated in the high field limit and in the low field









limit and finally the appropriate matching is done in the intermediate region in which

both London and Ginzburg-Landau theory fail.

Although in this work the Ginzburg-Landau theory is the main tool used in the

investigation, I use many results obtained from the London theory.

1.4 Anisotropy and Structural Phase Transitions

There are many sources of anisotropy; all are found to modify fundamental

properties such as the shape of the vortex cores, the interactions between the vortices,

the elastic response and also the symmetry of the vortex lattice. The mechanisms are

complicated and sometimes lead to significant deviations from the behavior of the

isotropic superconductor.

The effect that this work is mainly focused on is the structural transitions of the

vortex lattice. The familiar arrangement of the vortices in a hexagonal lattice is not

unique, as experiments have shown. Distorted rhombic and square lattices are pos-

sible. The possibility for alternate arrangements becomes apparent once one realizes

that the reason behind the formation of the hexagonal lattice 2 is the isotropic inter-

action between the vortices. An additional indication would be the small difference

in energy between the square and hexagonal lattice.

One source of anisotropy is that which arises from the existence of an under-

lying crystal lattice of different symmetry than the one usually assumed, namely

cubic. This condition is met in the cuprate family, which are layered, extreme type-II

HTSCs. They exhibit uniaxial ( ,i -I.,1 symmetry. A way to account for that is to

introduce a different effective mass along one direction (uniaxial symmetry). In that

case the superconductor is characterized by two penetration lengths, Aab for the a-b

(or basal) plane and A, along the c-axis, and also two coherence lengths ,ab and &,.3

The anisotropy ratio F = Ac/Xab = Cab/Cc quantifies the strength of the anisotropy.
2For a fixed area (density), vortices on a hexagonal lattice are placed the farthest apart possible
from each other.
3In many materials the situations is even more complicated, with Ana A'b.


























Figure 1.3: Vortex profiles for applied field along the z and b axis respectively.

Typical values for a HTSC are F > 1 and Aab 1000A. The Ginzburg-Landau pa-

rameter is defined as K = Xb/ab. An immediate consequence is that the profiles of

the vortices are different for fields along the z axis and along the a-b plane, see figure

(1.3). Also, the critical fields Hc1 and Hc2 become functions of the angle between

the applied field and the c-axis of the material. The analysis in general is difficult

but there exists a scaling scheme that reduces the anisotropic problem to an isotropic

one at the initial level of Ginzburg-Landau theory or London model [43]. Using that

method, one essentially generalizes isotropic results to the anisotropic case.

The flux line lattice in anisotropic materials can be distorted and depending

on the presence of pinning, defects and other factors one may see the formation of

domains of different flux line configuration and also transitions from one lattice sym-

metry to another. Structural phase transitions can be studied with the same method,

by promoting the mass m* to a mass tensor, thus incorporating the anisotropy in the

model. However, a significant failure of this approach is that cubic materials (e.g.

VaSi) would not exhibit structural phase transitions, a fact which contradicts experi-









ments. The results of the research presented in this work can easily be generalized to

an anisotropic mass tensor using the scaling scheme [43].

Another source of anisotropy is an anisotropic Fermi surface. This is typical

in the borocarbide family (RNi2B2C, with R Y, Lu, Tm, Er, Ho, Dy), which are

most likely conventional but highly anisotropic s-wave superconductors [44]. The

anisotropic Fermi surface affects strongly the electromagnetic interactions in the su-

perconductor [45].

To account for an anisotropic Fermi surface, the London model has to be

amended in order to include nonlocal corrections via the relation between the cur-

rent density j and the vector potential A. For tetragonal superconductors like the

borocarbides the corrections (in Fourier space) have the following form [46]:


47 1
-3 ,2 (m %- A2rijlmklkm)aj. (1.21)


Here aj = Aj + (Io0O/27, with 0 the phase of the order parameter and A the vector

potential. The inverse mass tensor m?1 and the fourth rank tensor nijim which cou-

ples supercurrents with the crystal are evaluated as particular averages of the Fermi

velocity over the Fermi surface and in that way are strongly dependent on the shape

of the Fermi surface. More importantly, the coupling persists even for cubic materi-

als [47]. A qualitative explanation of the importance of nonlocality is that the current

response of the superconductor must be averaged over the finite size of the Cooper

pair, which is of order in contrast to the usual local electrodynamics.

The London model is both successful and convenient, due to its simplicity.

Among the goals of this work is to show that one obtains identical results employ-

ing the Ginzburg-Landau theory with the inclusion of an anisotropic term. For the

description of conventional superconductors with anisotropic Fermi surface the addi-

tional term is purely phenomenological. For the anisotropy that arises from an un-









conventional coupling the additional term will be derived from a more general model

in subhead 2.1.

The anisotropy that is of particular interest to me is the one arising from the

existence of an unconventional pairing or a multicomponent order parameter. It has

been established that the pairing mechanism in the cuprates is of d wave nature [48

50], which results in a fourfold anisotropy of the gap and anisotropic vortex cores.

The first direct observation of a well ordered square vortex lattice in a HTSC was

achieved very recently [51].

Even more exotic systems have been investigated such as Sr2RuO4 with an un-

conventional two component order parameter [52]. Coexistence of phases of different

pairing symmetry and related phase transitions have been observed for the heavy

Fermion superconductors UPt3 and U1_xThxBei3. The interested reader can consult

the nice review by Sigrist and Ueda [53] and also the review by Joynt and Taillefer [54].

A very recent work by Agterberg and Dodgson [55] generalizes the London model

to unconventional superconductors with multiple superconducting phase transitions.

Although the main body of this dissertation deals with the investigation of the effects

of d,2_y2 pairing with an small admixture of s wave pairs in the condensate, it is

reasonable to expect that similar results can be obtained with the same methodology

for other pairing symmetries. In each case it would be necessary to find the form of

the appropriate term that couples the coexisting order parameters.

1.5 Elasticity and Melting of the Vortex Lattice

The structural phase transitions of the vortex lattice are closely connected to its

elastic properties. The elastic response of the vortex lattice, quantified by the appro-

priate elastic moduli, is of great significance for the phase diagram of the superconduc-

tor. The theory of elasticity for the vortex lattice in an isotropic superconductor was

derived by Brandt [35-38] using the Ginzburg-Landau theory for the high-induction









regime and the London model for low inductions. The theory was later confirmed by

the results of Larkin and Ovchinnikov [56], who used the microscopic theory as their

starting point. The theory has been generalized to anisotropic superconductors with

much effort [42, 57-61]. In these works the anisotropy was incorporated in the mass

tensor.

The elasticity and the structural phase transitions are relevant for the extremely

interesting subject of the melting of the vortex lattice. Especially in HTSCs, with the

enhanced role of the thermal fluctuations and the softness of the elastic moduli, the

region over which a melted vortex lattice is expected is significant.

One can define melting as the loss of long range correlations between the vor-

tices. The melted vortex lattice consists of highly mobile and flexible vortex lines. A

fundamental difference in the behavior of HTSCs compared to conventional, low-T,

superconductors is that at high fields, the well defined and sharp superconducting-

normal transition is replaced by a smooth crossover between the vortex liquid phase

and the normal metallic phase. One criterion used to define the onset of melting

is the so-called Lindemann criterion, according to which melting of the vortex lat-

tice is expected once the mean squared amplitude of the thermal displacement of the

vortex lattice becomes comparable to the inter-vortex spacing (the lattice constant).

Although this criterion is not rigorous, it provides a simple and adequate tool for

the investigation of melting and the determination of the melting curve Bm(T). The

elasticity enters in the calculation of the mean squared displacement (u2). For an

excellent discussion of the issues pertaining to the melting of the vortex lattice and

the relevant theory, the reader can consult the review by Blatter and others [22].














CHAPTER 2
GINZBURG-LANDAU THEORY

2.1 Anisotropic Ginzburg-Landau Theory

The isotropic Ginzburg-Landau theory has been used extensively in the study

of isotropic, or s-wave superconductors. However, once it was realized that alternate

pairing mechanisms could exist, new extended Ginzburg-Landau theories were pro-

posed and studied in the context of high-Tc superconductivity [62-69]. Although the

latter are built using symmetry arguments, microscopic derivations do exist [70].

For the high-Tc cuprates it is established that the major pairing channel is

the d,2_y2 pairing. The dominant pairing mechanism is of d-wave nature with a

subdominant s-wave order parameter being induced by gradients of the d-wave as will

be shown promptly. The generalization of the Ginzburg-Landau theory to anisotropic

superconductors in which the anisotropy is due to s and d wave mixing contains two

fields, or order parameters. The corresponding Ginzburg-Landau free energy can be

written in the following form, which includes an anisotropic term with the appropriate

symmetry [54, 66, 67]:


fGL = S 2 ? +1 + S 4 + d2 4 3 S 212 4 Sd2+ S2d*2)
8 (2.1)
+ 7(ns)(ns)* + d(nd)(nd)* + 7,[(ns)*(nd) (n,s)*(n,d) + cc] ,


where the brackets denote spatial averaging. Also I have defined the parameters

7, = h2/2mn, i = s, d, v and I = V/i e*A/hc is the covariant derivative as usual.

From here on I set h = c = 1.









The critical parameter is ad and it is generally assumed that Ts < Td, meaning

that d is the dominant order parameter. The later treatment of s as a small correction

arising from gradients of d is based on this assumption.

The Euler-Lagrange equations of motion are obtained from equation (2.1) by

minimizing with respect to d* and s*. One obtains


(7di2 + ad)d + 2/d2 d + 31 2d + 2 2d* + 7, (f I s 0, (2.2)

(7yI2 t+ ,)s + 2 11s 2S + 3d2 21 4 + 7(II -_ In)d 0. (2.3)


The boundary conditions obtained during the derivation of the first and second equa-

tion of motion are given by the formulas


n {yd(Hd) + 7,[y(Iys) x(IIs)]} = 0, (2.4)

n {%((Hs) + -. [v(nIId) x(IId)]} 0, (2.5)


where n is the unit vector normal to the surface of the superconductor.

The second Ginzburg-Landau equation, which is the equation for the current, is

obtained by minimizing the Ginzburg-Landau free energy with respect to the vector

potential A. After a short calculation one obtains


j e [s*(ns) + s(Hs)*] + 7d[d*(Hd) + d(Hd)*] + yY[s*(fd) + d(Iys)*]
1 (2.6)
x-Y[s*(n,d) + d(n,s)*] + cc .


The solution of the Ginzburg-Landau equations is a very complicated problem

even for the isotropic superconductor which can be described in terms of only one

order parameter. The additional complications of having two order parameters can

be avoided with a drastic simplification: the reduction of the two-field theory to an

effective one-field theory. In order to derive the single component free energy I will









follow Affleck and others [71] and integrate out the s field. This task is achieved by

making the substitution

s > (',/as) (UI I ) d. (2.7)

This approximate equation is derived from the equation of motion for the s field

upon neglecting higher order terms. It is a reasonable approximation, based on the

smallness in magnitude of the s component when compared to the d component. This

approach is useful, for one is interested primarily in the linearized equations which are

easier to solve. It will be shown that the anisotropic term added to the first Ginzburg-

Landau equation by this procedure is sufficient to capture the essential 'lli, -i. of the

problem. After all the substitutions, the linearized first Ginzburg-Landau equation

takes the form


(I2 + d (I ly)2d = 0, (2.8)
as

which is the central equation and the starting point in my study. For the sake of

completeness the second Ginzburg-Landau equation is


= 7 [d*(nd) + d(nd)*] 1 y [(nd)[(n )d] + d[,(n nl)d]
a, (2.9)
-x [(Id)[( nI)d]* + d[In,(nI n )d]* + cc.


The boundary condition for the d-wave order parameter (2.4) can now be written as


n < [d7(Hd) l [y-1 (II lf2)d xii,(I2 f12)d = 0, (2.10)
.. .. na)d -n (2.10)


which enforces the vanishing of the normal component of the supercurrent on the

surface of the superconductor. This boundary condition would be correct for a

superconductor-insulator interface, but for an interface between a superconductor and









a normal metal it has to be modified. The modification can be found by imposing the

condition n j = 0 on the current equation (2.9).

It is convenient to switch to dimensionless quantities using the unit conventions

that are frequently used in the literature


r Ar, (2.11)

d d dod, do 2 Id /\2, (2.12)

A (hc/e*)A, (2.13)

H (hc/Ae*)H. (2.14)


The above equations define the natural lengthscales of the various quantities involved.

The Ginzburg-Landau parameter c is defined c = A// with the penetration depth A

and the coherence length C given by


A2- (m*c2*47r*2) 2/ d), (2.15)

C2 h2/2m* ad|. (2.16)


The dimensionless Ginzburg Landau equations obtain a much simpler form using these

unit conventions. The first equation is simply


i2d d f2 )2d 0, (2.17)


while the second Ginzburg-Landau equation, using the vector potential instead of the

supercurrent, has the form


V x V x A -Ad|2 + | (dVd* d*Vd) dx1 [ )(12 12)d*
2K 2 I x (2.18)
+d[II,(I2- I 2)(] -y (IIyd)[(II- )d]* + d[IIy(II II)d]* + cc









where e is the dimensionless anisotropy parameter, defined as e = ( ?l.'y|/as).

Setting the anisotropy parameter c equal to zero one can immediately obtain the

linearized Ginzburg-Landau equations for an isotropic superconductor.

From now on I switch to the usual symbol T for the order parameter instead of

d since I eliminated s. I also need to rotate the vortex lattice by an arbitrary angle

0 about the direction of the applied field. This is required because the free energy

is no longer invariant under rotations about the direction of the applied field, due to

the presence of the symmetry breaking anisotropic term. The preferred orientation of

the vortex lattice will be determined by the minimum of the free energy-it will no

longer be arbitrary. This crucial point was missed in some of the previous theoretical

treatments of this problem [72]. The anisotropic term provides the effective coupling

to the crystal lattice, thereby making structural transitions possible. Without this

c"..ilii-. the vortex repulsion is isotropic for cubic or tetragonal materials resulting

always in a hexagonal vortex lattice. Although the anisotropic term was derived from

the two component Ginzburg-Landau theory appropriate for high-Tc superconductors

with d12_y2 pairing it can serve as a phenomenological term for the investigation of

anisotropy effects in tetragonal materials like the family of the borocarbides.

A useful transformation of the first Ginzburg-Landau equation is effected by

introducing creation and annihilation operators through the following relations II =

fI, F illy, with i[II,, II] = -h/v. niI(n_) is the creation (annihilation) operator.

The transformation of these operators under rotations is shown below. The primed

quantities correspond to quantities in the rotated system.


I, = I, cos + Iy sin (2.19)

II' = -fI sin + Ily cos 9, (2.20)

I 2 n2 i + -2i (2.21)









Substituting in the first Ginzburg-Landau equation and subsequently dropping the

primes I obtain

II2p p ie + 2 -2i n2 ] i 0, (2.22)

an equation which can be reduced to the equation for an one dimensional harmonic

oscillator with an appropriate perturbation.

The same transformation of coordinates for the second Ginzburg-Landau equa-

tion yields a complicated result which after some manipulation can be cast in the

following form

i3 1 + ( 1
V x V x A = -A|4|2 *V ) + o 2 -(zx VA)
2K (2.23)
+(z x r) r2 b2


where To is the zeroth order in c approximation to the solution of equation (2.22),

b = B/Hc2 is the reduced magnetic induction and the function A is defined as


A(x, y, q) = cos (4))(x4 + y4 62y2) + 4 sin (4 ,,(x2 y2). (2.24)


2.2 Virial Theorem

In this section I will present an alternative method to derive a compact and use-

ful form of the free energy for the anisotropic Ginzburg-Landau model. The method

is based on the virial theorem for superconductivity discovered recently by Doria and

others [73]. In Appendix A I derive in detail the generalization of the isotropic result.

The virial theorem for an anisotropic superconductor has the following simple form

H B
= Fkinetic t 2Ffield t 2Fanisotropic, (2.25)
4x7









or the following equivalent form, with the free energy components expanded

H.B In p(x)l12, + B2(X) + 2-y2 ([n (x)][II(II2 f n2)T(X)]*
4x 2m* 4x as
(2.26)
[n UI(x)][n,(n,2 _- )(x)]* + CC )


where V x A(x) = B(x) is the microscopic magnetic field. The magnetic induction
is defined B = f dx B(x)/V and the homogeneous applied field is denoted by H.
Using this extension of the virial theorem, as I show in detail in Appendix A,
one can cast the free energy in the following form


F 2-2 K(I 2 _r I b (2.27)
(22 1)A + 1 (2K2 1)A + 1

where b is the reduced magnetic induction, PA is the Abrikosov parameter and F is
the correction to the isotropic result, which has the following form
F R4 ,(|2*(H _2H_)22 R( *(2 )2t2
F Re i f + A(2 1) P2) (2.28)
I|W|PA2 12 2- 2 2

It is straightforward to check that if one sets the anisotropy parameter c equal to zero
one obtains the familiar isotropic result.
The above expressions seem to -II.w-. -1 that for hard, high-K, superconductors
the effects of the anisotropy c are diminished. However, it will be shown in the
subsequent analysis that even small deviations from isotropy produce observable and
significant effects on the structure and the properties of the vortex lattice.
This extension of the virial theorem allows one to easily generalize results
obtained for the isotropic case. I show below the form of the Gibbs free energy
G = F HB/47, the magnetization M = (B H)/4r and the magnetic induction








B for the anisotropic Ginzburg-Landau theory


(G H) (2- 2 IA (2.29)
(2K2_-1) A 2
i >- { (- )} (2.30)

B =H (2 A} (2.31)
B -(2K2 1)A l-(2 2 1) A "

The generalization of the second Abrikosov identity [4] is easily obtained in the process
of deriving the above relations. It reads

(Kc H) I{ 2) -2Re{(4U U (1 )2 )} (1 2T 1)) {(.32
\2K Y68 ( 2 2 (2.32)
+ 2 f1P 12,(fl f)2,p).

All the above results are invaluable for the simplification of the calculations
that are going to follow. By itself, this generalization of the virial theorem can be
used in numerical investigations of anisotropic superconductors presumably with the
same success in reducing the overall computational complexity of the problem as the
virial theorem had for conventional superconductors. The original virial theorem has
inspired a significant amount of work. See for example [32, 73-77], and references
therein.














CHAPTER 3
MEAN FIELD THEORY

In order to investigate the structure of the vortex lattice one can start at the

mean field theory level. Although a simplified approach, it provides useful insight

about the problem and not surprisingly, many theoretical predictions which are in

turn validated by experiment.

3.1 Solution of the First Ginzburg-Landau Equation

In this section I will present the calculation of the Abrikosov parameter ?A and

the calculation of the correction F to the free energy (2.27). To that effect I will solve

the first Ginzburg-Landau equation (2.22) perturbatively in the anisotropic term. It

has the form (in dimensionless units)

1
-V A e(I 2 I)2 (1- I_ 2),p. (3.1)
KV ) y

Rotating by an angle 9 about the z axis (the direction of the applied field) and by

introducing creation and annihilation operators at = -ill/ 2b, a = il_/v2, with

nII = IIx iFl,, one can write the first linearized Ginzburg-Landau equation in the

following form

ata_ a 2 -_ e-2iOa212: 0, (3.2)

without specifying a particular gauge yet.

The rotation about the z axis is necessary to account for the orientation of the

vortex lattice which is no longer arbitrary due to the four fold symmetric term that

breaks rotational symmetry about the direction of the applied magnetic field. Notice

that the term 1 was grouped with the higher order term ILT121 in the right hand side









of equation (3.1). This way, the linearization of the first Ginzburg-Landau equation

is done in a consistent manner, as pointed out by Kogan [78].

To proceed with the solution it is helpful to decide on a particular gauge. Two

choices are useful in this problem, the Landau gauge A = (0, bcx, 0) and the sym-

metric gauge A = (-b y/2, brcx/2, 0). I will show the results of the solution in two

separate sections. The total wavefunction will be built by superposition, using appro-

priate linear combinations to achieve the desired periodicity of the observable (from

a quantum mechanical point of view) I~(x) 2. The wavefunction itself is neither

periodic, nor gauge invariant.

3.1.1 Landau Gauge

Throughout this section I will use the Landau gauge A = (0, bKx, 0), which

makes the perturbation theory calculation easy, as it involves only one dimension. In

fact, all the gauge sensitive calculations that are presented in this work are done with

this particular choice of gauge.

I use standard perturbation theory to obtain the wavefunction up to second

order in the anisotropy parameter e. The relevant matrix elements are easy to calcu-

late realizing that the underlying theory is that of the very familiar one dimensional

harmonic oscillator. I obtain

_e4 2 (cb)2 F e8
I(x) = o(x) + 2H4 bx) 2 64 H8 H(b2x) (3.3)
32 32 64

+ 5e4H4(b2x) + O(e3)}, (3.4)


where H, is the Hermite polynomial of the n-th rank and


O(X) ( b exp bK } (3.5)
x 2









YCL


&
^t


XCL


Figure 3.1: The geometry of the system: the vortex lattice (VL) is rotated by an
angle -0 about the c-axis with respect to the crystal lattice (CL). The apex angle is
defined as the angle 0 between the two basis vectors a and b.

is the unperturbed solution. Notice the explicit appearance of the orientation angle

c in the expansion of the order parameter.

The periodic solution is constructed following Chang and others [79]. One can

choose one basis vector, a, to lie on the y axis and the second basis vector b forming an

angle 0 with a. The geometry is illustrated in figure (3.1). The averages (l1I2) and

(1 '4), necessary for the calculation of the Abrikosov parameter IA and the correction

to the free energy F, are calculated using the methodology presented in the work of

Chang and others [79]. Along with the known first order results, I present below the

results to second order in the anisotropy parameter (the details of the calculations








can be found in Appendix B):


3A = o + ebi + (eb)2 71 72 + 51 3o (3.6)

r= 8cb2 {3A( 22 1)[1+ 3cb] +3o[1 [ + 6b] i c 2b(i + 2)}. (3.7)


The various quantities that appear in the above expressions are rapidly converging
functions of three variables p, a, 9, which are introduced impl-,ving the convenient
parameterization p + i = (b/a) exp (iO) [1]. The functions 3o, 31, i7, 2 are periodic
in p with period 1, and have a rich structure for a ~ 1. Their explicit forms are


Ao > Anm, (3.8)

nrn
1 R]L C A^4io LYA.Tn 8 2(-2 n 4 67rTn2 + 3]I, (3.9)






36n2r 2) --a(n2 2)
71 C eL8io A.Tn[167r 4(-4 n8 1273a-3n6 + 37r 2(-24 -- 4_ 45 7an2 + 105]| (3.10)

72 'A. 1674 (iT4n4 127 3(3n2M2(n2 + M2) + 3 ( T2n4 + m4


16 256

where Anm = /ae2i7P(n2-2) 'e-27a(n2 2). The prime implies that there are two
summations-the one shown, over n, m and the other with n and m replaced by
(n + 1/2) and (m + 1/2) respectively.

3.1.2 Symmetric Gauge

This is considered a more "natural" gauge, with no preferential treatment of one
direction in space over another. It unfortunately comes with the disadvantage that
for this particular problem it is difficult to construct the periodic order parameter, in
sharp contrast with the Landau gauge which readily allows the construction of the
order parameter with its desired periodicity. However, this gauge choice being phys-









ically more transparent results in an order parameter whose form has an interesting
interpretation.
The solution of the unperturbed first Ginzburg-Landau equation in the sym-
metric gauge reads


po(x, y) =V exp (x 2 +) (3.12)
lb}C|-V 2 bp--^(+y (3.12)


The correction to first order in c is easily obtained using the method of the previous
section. I obtain

) (x,y) = o(x, y) 1+ e4 -4 (3.13)

where z = x iy. The second order correction is easy to calculate but it does not add
anything to the 1ir, -i. .,1 point I am trying to make.
Although the periodic order parameter is difficult to construct, one can postulate
an amendment of the isotropic equation (1.14) to account for the anisotropy. The form
of the perturbative result leads one to propose the following form for the solution which
accounts for fourfold anisotropy


I(x, y) = exp [- b (x2 2) (Z z ) {1 C (z 7 Z 1)( 7 1)
J (3.14)
X (z + 72)(+ -v 72) ,


with i7 = 7R+i7I = I and 72 -tI+i R = I I '- and v denoting the position
of the vortices-in other words z, are the zeroes of the unperturbed order parameter.
The 1.li,--i .,1 interpretation becomes apparent once one is familiarized with the
strange form of the solution. The c term provides the four additional zeros of the in-
duced s-wave order parameter, around the central node of the d wave order parameter.
The positions of the nodes are controlled by the 7 terms and they can accommodate
various symmetries.








I have attempted to obtain the compression (cil) and tilt (C44) elastic moduli
using the solution (3.14) according to Brandt's prescription [35-38] without success.
The complexity unfortunately increases rapidly with each successive step. However,
there are some interesting results that were obtained that are worth mentioning here.
I first need to define two functions A,(x, y) and M (x, y) through


A,(x, y) = X4 + Y4 -6X2Y2 14 cos(4w), (3.15)

M,(x, y) = 4XY(X2_ Y2)_ 4 sin(4aw), (3.16)


where X = x and Y = y y, with v denoting the v-th vortex. The modulus of
the order parameter can be reduced to the following form


f2 ftropic( + 2g), (3.17)


where g is the correction due to the anisotropy and it is defined as g = [F A,. It is
easy to show that the supervelocity and the quantity u = Vf/ f have the form


Q = Qisotropic + -Vg x z, (3.18)

U = Uisotropic + -Vg. (3.19)


The equation Q = u x z still holds. Other useful identities that have a straightfor-
ward generalization in the anisotropic case are those first discovered by Brandt [35, 37]
when deriving the elastic moduli of the isotropic superconductors. I obtain for the
supervelocity of the Abrikosov solution (momentarily I am following Brandt's nota-
tion)

1 r-rr
QB = A -z 2x i {- 2 + 4e [(X3 3XY2)x + (y 3X2Y)y)] (3.20)









The following central identity remains unchanged:


(Vw)2 12Bw + wV2w. (3.21)


Note that ci is the linearized order parameter, obtained from the Abrikosov solution

for the undisplaced vortex lattice cLo, upon substitution of the displaced coordinates r,

and subsequent expansion in the displacement s, to linear order. The general vortex

lattice is defined by the loci of the zeros of the order parameter given by:


r,(z) R, + s,(z). (3.22)


Another very int rI -lin_. from a ]llr,-i. .,1 point of view, result is that an ar-

bitrary displacement s, of a vortex line and the anisotropy appear to modify the

linearized order parameter in a similar manner. I obtain


1 2
wi(x,y) o(x,y) 1 + 5 (3.23)


with wjo(x, y) the solution for the undisplaced vortex lattice. The generalized displace-

ment term q, which in the isotropic theory accounts for the displacement only of the

vortex lattice, now has the form


(r) 2- -2 (r r- ) +--VW (3.24)
r r,2

W = X6 + Y6 5X2Y2(X2 + y2) i4[(X2 + y2) cos(4w)) + XY sin(4w:)].

(3.25)


The complexity is significant but the calculations are tractable so far. What hinders

further progress is my inability to find the appropriate generalization of the free energy

which serves as the starting point of the derivation of the dispersive elastic moduli cl1









and C44. In the isotropic theory it has the form [35, 37]


f = + + ( )2 Q hBh) B2 (3.26)
2 4K2L,


3.2 Structure of the Vortex Lattice and Minimization of the Free Energy

In order to obtain the mean field phase diagram and study the structural phase

transitions formally, one has to minimize the Gibbs free energy (2.29), which is the

proper free energy to minimize under the constraint of a constant external field.

I found that the minimization of the Gibbs free energy 1 G = F HB/47, and

the minimization of the Helmholtz free energy, equation (2.27), give identical phase

diagrams. I choose to work with the Helmholtz free energy, from which one obtains

the elastic moduli, as shown in Chapter 4.

The particular form of the free energy (2.27) allows one to obtain the phase

diagram in a direct way instead of minimizing only the Abrikosov parameter 3A as

was done by other workers [72, 79-81]. However, for the sake of comparison and as a

consistency check to my algorithm, I minimized the Abrikosov parameter PA in order

to obtain the phase diagram as a function of the anisotropy. The results are shown in

figure (3.2). To first order I find the transition point at ecl = 0.0240, in agreement

with Chang et al. [79] who find qci = 0.0235. The result of the numerical solution

for the phase boundary of Park and Huse [72] cannot be compared with the previous

results By solving for the phase boundary numerically they find qci = 0.0367.2

However, in their work the vortex lattice was not allowed to rotate, thus omitting 9

from the calculation, an essential degree of freedom. The second order correction to

the Abrikosov parameter 3A moves the critical anisotropy to c2 = 0.0284 a relatively

small correction which justifies the use of perturbation theory.
1After one modifies G to be a function of the reduced field b.
2The reported result, cl = 0.0734, needs to be divided by 2 due to different conventions for the
anisotropy parameter.






















0 75


65 F


Apex angle 0 vs. anisotropy E


-- first order
second order Orientation angle vs. anisotropy e

45 r

I --- first order
SI-second order
'45" r ---------------
40 -


1 35


IEr
30



S50 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

/ e


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Figure 3.2: Minimization of the Abrikosov parameter IA as a consistency check for
the computer code. The apex angle 0 of the unit cell and the orientation angle 9 of
the vortex lattice as a function of the anisotropy parameter c. The critical points to
first and second order approximation in the anisotropy parameter c are Ec1 = 0.0235
and c1i = 0.' 1'- Notice the reorientation transition in the inset panel which appears
only to second order in the anisotropy parameter at c, = 0.0492.


The most interesting and unanticipated fact was the observation of another

phase transition at a still higher value of the anisotropy c, = 0.0492, which turns

out to be connected with the vanishing of the rotational modulus as will be shown

in Chapter 4. In particular, at this second transition point, the orientation of the

square lattice changes continuously from the [110] direction, with 9 = r/4 now a

local maximum and two degenerate minima on higher and lower angles. The vortex

lattice retains its square symmetry.

The minimization of the free energy shares the complications of the minimization

process for the Abrikosov parameter 3A. There exist many local minima that have to









be avoided in the search of the global minimum and also there are many equivalent

minima which correspond to the same lattice structure, with different values for the

three parameters p, a, o which determine the structure of the vortex lattice. This

problem can be solved realizing that for the two subsets p E [0, 1/2], p E [1/2, 1], one

obtains equivalent lattices. I chose to work in the first subset. The minimization was

carried out using the fmincon function for constrained minimization included in the

optimization package of Matlab. A program was written for the evaluation of the free

energy, the calculation of the gradient and the Hessian. A driver was then used to

automate the minimization for different values of the parameters of the problem. The

interested reader will find the code in Appendix D. The results of the minimization

of the free energy for e 0.11, K = 5 are shown in figure (3.3).

The difference of my results compared to previous theoretical predictions [79, 80]

obtained from the minimization of the Abrikosov parameter 3A is that the structure

of the vortex lattice is now obtained as a function of the reduced field b, allowing the

comparison with experiment. All the main features of the second order structural

phase transition are there, along with the reorientation transition in the region close

to the upper critical field. In order to check the results and the minimization algo-

rithm I calculated the correction F to second order in the anisotropy parameter. The

difference between the first and second order in the anisotropy parameter results is

not significant, as it can be seen in figure (3.3).

For larger values of the anisotropy parameter the transition occurs at a smaller

critical field, while for smaller anisotropy the transitions occurs closer to b = 1. I

chose the particular value to show the reorientation transition clearly.

The predictions of the anisotropic Ginzburg-Landau model are at variance with

the predictions of the nonlocal London model, with regard to the reorientation of

the vortex lattice at low fields. The nonlocal London model predicts [46, 82] that

the reorientation happens at a low field H1, with the orientation angle being 4/4











Apex angle 0


- I
first order
S-second order


45 -


40


35


30


25

0.2


Orientation angle )


0.4 0.6 0.8
b


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

b

Figure 3.3: The apex angle 0 of the unit cell, the orientation angle 9, the Abrikosov
parameter IA and the dimensionless Helmholtz free energy F/i2 as a function of the
reduced field b for e 0.11 and K = 5. In the first panel on the top I include the
second order result for the apex angle. The reorientation transition is evident in the
second panel on the top.


throughout the rhombic to square transition. This has been recently verified by

experiment on annealed LuNi2B2C crystals [15]. In my model the reorientation of

the vortex lattice is continuous and is completed at the transition point. This is

not strange, as any expectation of a quantitative agreement between the two theories

at low fields (the natural region of applicability of the London theory) would be

unreasonable.


' '














CHAPTER 4
ELASTICITY THEORY FOR THE VORTEX LATTICE

In this section I will derive the elastic moduli of the vortex lattice. This deriva-

tion is independent of the particular model (Ginzburg-Landau theory or London

model) that will be used to evaluate the elastic moduli. As is done for the stan-

dard theory of elasticity for crystals, one can introduce the displacement vector u of

a vortex line from its equilibrium position and upon expanding the elastic energy in

terms of derivatives of the displacement ui,k akUi, one obtains the various elastic

moduli. In principle the derivation is the same, however one has to consider carefully

the fundamental differences between the usual ( I--l.1l and the vortex lattice. Before

proceeding with the derivation a few comments are in order. The full calculation of

the elastic response of the vortex lattice is a complicated task. I will consider only

the simplest part of that, namely straight and parallel vortices, thereby reducing the

complexity of the problem from three dimensional to the simpler two dimensional

case. A further simplification is the so called incompressible limit, in which one con-

siders deformations that conserve the density of the vortices. This puts the following

constraint on the displacement field: V u ui, = 0. For the shear moduli calculated

in this fashion it has been proven that they are essentially nondispersive [25]. One

then has to consider only uniform deformations.

The vortices are inhabiting the superconductor and are thus influenced by the

symmetry of the underlying crystal lattice. Moreover, rigid rotations of the vortex

lattice with respect to the (I, -I .,l lattice have an energy cost associated with them, a

strange property never encountered in ordinary solids which live in an isotropic space.

The immediate consequence is that the usual decomposition of the elastic energy in

terms of strains and rotations is no longer convenient. A more useful form for the









elastic energy has been proposed by Miranovi6 and Kogan [83]. I will adopt that form

and present here, in detail, the derivation of the elastic moduli for the vortex lattice.

The form of the elastic energy density in terms of a general fourth rank strain

tensor can be written
1
E iklmUi,kUl,m (4.1)
2

The tensor Tiklm is not symmetric under the exchange of i +- k or 1 +- m, which

would permit a decomposition into a symmetric and an antisymmetric part. In fact

the only symmetries that Tikim possesses are those of the ( 1--I..1 lattice and of the

equilibrium vortex lattice along with the obvious Yiklm lmik-

I will deal with the more symmetric case first, the square vortex lattice. The

energy should be invariant under an exchange of x and y coordinates. Thus, x and y

appear in even numbers in the elastic energy. Under the constraint of incompressibility

the elastic energy reads


1
Es [21s,2 + t 72s(Uy + u2t ) + 2m3sUx,yUy1], (4.2)


with s = 2(,, ,, ), 72s = -, and ,

Similar considerations for the hexagonal vortex lattice lead to an elastic energy

of the form
1 2 2 U2
Eh = [/lh U, + 72h ,y + 73h y, 274hUx,yUy,], (4.3)

with lh = (Y + r 2.,, ), 72h = 73h = r and 74h r,

In order to make the elastic moduli more transparent, it helps to consider various

deformations and to study the form of the elastic energy for each choice. There are

four important deformations:

Squash deformation: u, = p(xx yy),


* Simple shear along x: u


- pyx,









Simple shear along y: uy = pxy,

Rigid rotation about z: u, = p(xy yx).

Under each of these deformations, the elastic energy assumes a particularly simple

form which allows one to chose a convenient parameterization with elastic moduli Csq,

C66x, C66y and cr, that are readily connected with a simple 1ir, -i, .1l situation. More
specifically I have for the square lattice first

Es [uq] = 752,

E [u,] = "72s2,

E~[uy] = 172s2

E [ur] = 2(y72s 7s)2.

I can then identify Csq = 'y7, C661 = C66y 72s and c, = 2(72s 73s) as the relevant

elastic moduli for the square vortex lattice.

In a similar fashion I obtain for the hexagonal lattice

Eh[Usq] 1= /al[2

Eh [ux] = 72h 2

Eh[Uy] = T73h 2

Eh[Ur,] 1(2h + 73h 274 )2.

Thus, the relevant elastic moduli for the hexagonal lattice are Csq = l1h, C66x 72h,

C66y 72h and cr = (72h + 73h 24h).
In figure (4.5) I present the action of the four particular displacement fields on

the basis vectors of an arbitrary lattice to clarify the 1.li, -i, .l1 picture.















































Squash deformation


Shear along x


/ ///l/a a -


/////( *'''
///////r -

// / // 0





'-0
II I I I l i -


\^ \\ ,-


\1\\\\s _o

\\\\ss.>.--


.' tttt\\\\
... 1s \\\\\


1111rr /


Rigid rotation


Shear along y


Figure 4.5: The action of the four basic displacement fields, represented by the small

arrows, on the basis vectors of an arbitrary lattice, represented by the large arrows.


////////I I f tPII \\\\\
//'/// / t I t t 11 \ \
I--.''' I I # t f t I \ N N \ N \\
m f / I I t I I I k N N N N N





S- 5 ^ r fc4


O.N 1 /I I I % I I I N/ /NN
1\\\\ 1 111 % /




~~~~~~~\\\\\\\\//1 J
I If 0 V le e e e/J
L~~~~\WIII/A/


-1-----0---------5-----
- --- -- - -- -- ---- -




. . .. . . .
--1 85 4. I
-

. . .

-- .7- - -


I .1 t t t

o* t tt t


I'n 1 t s t t t
I, 1 1 t ,t tt
J, 1 i t I t t
4 I U t t t t




', ', t t t
, ~ t t


JSia ... u*lttt
1 t t
I r I t ttf

11 3tt. .t t

4, 1 1 t1 I t t tt









There is another deformation, although not independent from the above. It is

the shear along the [110] direction for which one can easily prove that


C66() = Csq + Cr. (4.4)


The ratio of c66; and C66(-) is a useful quantity that measures the anisotropic response

of the vortex lattice to shear deformations of different polarizations and will be of use

later on.

For a general rhombic lattice there are four independent elastic moduli Csq, c66X,

c66y and cr, first proposed by Miranovib and Kogan [83]. The deformation responsi-

ble for the transformation of the general rhombic lattice into a square lattice is the

so-called squash. The relevant modulus vanishes at the point where the structural

transition takes place, signaling the instability with respect to the particular deforma-

tion. It will be shown that another instability occurs at higher fields. At that point

the vortex lattice becomes unstable with respect to rigid rotations about the c axis.

All elastic moduli will be obtained from the second derivative of the Ginzburg-

Landau free energy with respect to the amplitude pt of the deformation, in the limit

that ft -- 0. I will make use of the Ginzburg-Landau free energy in the following form


F b2- (2b2 i { F(2 i (4.5)
_1)A 2 A 1 I


The elastic energy can be written as


E =1 2 p2C(k ) 0 202 2a-F (4.6)


where c(p) is the elastic modulus for each particular deformation. The amplitude ft

of the deformation will enter the free energy F through the three parameters p, a

and 0, which in turn depend on the lattice structure. The connection of a particular









model of superconductivity and the elastic moduli is made at this stage. I chose the

Ginzburg-Landau theory, however the evaluation of the elastic moduli is along the

same lines for the nonlocal London model. It is straightforward to show that the

second derivative of the Ginzburg-Landau free energy with respect to the amplitude
of the deformation has the form

(22 -1)(b2-F) 2 ( b)2 r
S (22 1)A 1 [(22 1)A + 112 F
(4.7)
(2K2 1) 1 2 P 2(2K2 1)2 F(0/ A)2
(2K2 1) 1 A + [(2K2 1)/A + 1]2

Both PA and F depend on the three parameters p, a and 0. The derivatives of /A and
F are thus given by the chain rule. One can define ( = (22 1)/[(22 1) A+1] and

S= (1 b)2/[(22 1)A + 1]2 and obtain the following expression for each elastic

modulus (denoted by c for brevity) using equations (4.6) and (4.7):


c pp + + + 2Cp + 2C+2p ,0 + 2C0,7 09 (4.8)


where Cij are the terms originating from the derivatives of the Ginzburg-Landau free

energy and are given by


Ci, (2 6ij) (b2 -F)/A,j + ,ij (I)(A,,J +/3A,j,i +/3A, 2(/A,lA,j) }.
(4.9)

Notice that the differentiations in the right hand side of equation (4.9) are with respect

to the three lattice parameters p, a, 9.

The calculation of pp, a/, 0, is purely geometrical in nature. One needs to

express the parameters of the distorted vortex lattice as a function of the magnitude

of the deformation and evaluate the derivative at the limit of vanishing deformation

(t -- 0). The geometry of the system can be seen in figure (3.1). The vortex lattice

is rotated by an angle -9 with respect to the ( v--I.1l lattice. The direction of the








rotation is completely arbitrary, I chose the particular one for convenience. The angle
0 is the apex angle of the (generally rhombic) unit cell and a and b are the two basis
vectors.
I will start with the geometrical definitions of p, a and 0 which have now become
functions of the amplitude of the deformation


p(1 ( 2 (4.10)

2 ,iit) (4.11)

cos- a(f)y }, (4.12)
( )I a^ ft) I )1 I

Under the influence of each deformation the two basis vectors are going to change
direction and/or amplitude resulting in a deformed unit cell. The calculation of these
effects on the unit cell as coded in the parameters of the vortex lattice for each
individual deformation is straightforward, however one should choose carefully the
branch of the inverse trigonometric functions that are involved in order to avoid sign
errors. The results are


Squash: (p,, (a,, ,) (-2a sin(20), 2a cos(20), sin(20)), (4.13)

Shear along x: (p1, (,, ,) = (a cos(20), a sin(20), cos2()), (4.14)

Shear along y: (p,, o,, 0/) = (a cos(20), a sin(20), sin2 ()), (4.15)

Rigid rotation: (p,, (a, 0) = (0, 0,1). (4.16)









A simple substitution of the lattice parameters in equation (4.8) yields for the nondis-

persive elastic moduli of the vortex lattice


Csq = 4Cppa2 sin2(2) + 4C,,(2 cos2 (29) + Co sin2 (29) 4CoU2 sin (49)
(4.17)
4CpO, sin2(20) + 2Ca sin(40),

C66, Cpp(2 Cos2(20) + C(U2 sin2 (20) + C cos4 () + C,(T2 sin (4)
(4.18)
2Cpa cos2(C) cos(29) 2CO- COS2 (S) sin(29),

c66y = Cp2 cos2 (2)) + Cga2 sin2 (29) + Co sin4 (0) + Ca2 sin(49)
(4.19)
+ 2Cpa sin2(C) cos(20) + 2C0,O sin2(0) sin(2 ),

C, = Coo. (4.20)


Up to now the elastic moduli have been independent of the particular model

that is used for the study of the vortex lattice. The coefficients C, which are model

dependent will be evaluated in the next chapter.















CHAPTER 5
STRUCTURAL PHASE TRANSITIONS

Using the results of mean field theory from Chapter 3 and the elastic moduli

calculated in Chapter 4 I am in a position to calculate the elastic properties and study

the phase transitions of the vortex lattice within the anisotropic Ginzburg-Landau

model.

5.1 Nondispersive Elastic Moduli

The procedure is very simple. First one has to minimize the free energy, equa-

tion (2.27), with respect to the three parameters p, a and 0. This will produce the

equilibrium energy and the equilibrium lattice for a given value of the reduced mag-

netic induction b. The details of the minimization are presented in section 3.2. The

second step is the calculation of the appropriate derivatives of the free energy at the

equilibrium values of p, a and ( and the calculation of the four elastic moduli, given

by equation (4.9), again for a given value of the reduced magnetic induction b.

One has to choose values for the two parameters of the model, the anisotropy

parameter e and K. The elastic moduli were obtained as a function of the reduced

field for e 0.11 and c = 5. They are evaluated at each minimum of the free energy

when obtaining the phase diagram in section 3. The calculated elastic moduli for

c = 0.11 are shown in figure (5.1). The vanishing of the squash modulus signifies the

transition from rhombic to square vortex lattice. At the same point the two shear

moduli merge into one because of the higher symmetry of the square phase. At a

still higher field I observed the reorientation transition signified by the vanishing of

the rotation modulus c,. This instability of the vortex lattice relative to rotations

was encountered in the nonlocal London model also [83]. There are experimental










indications for this instability, the investigation of which is hampered so far by the

high fields required for its observation [15].

0.01 I I I I

sq
66,x
0.008 C 6
66,y
S ... C 2e-05 -




I \
0.006 le-05




0.004 -\
S0.85 0.9 0.95 1
b

0.002 ..




0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

b

Figure 5.1: The nondispersive elastic moduli as a function of the reduced field for
c = 0.11 and K = 5. The first transition point at which the "-ijl.i-h" modulus Csq
goes to zero is clearly seen in the left panel. The reorientation transition starting at
the point where the rotational modulus c, vanishes is seen on the right panel.


I can compare directly with the nonlocal London model results of Miranovi6

and Kogan [83]. I can see the same qualitative behavior, the vanishing of the squash

modulus Csq at the first transition point b:, the merging of the two shear moduli

into one after the first structural transition and also the vanishing of the rotational

modulus c, at the second transition point br, which is about two times b:. However, a

quantitative difference between the two results is the change in the relative magnitude

of the elastic constants after the phase transition. I observe that the elastic moduli
































0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
b


Figure 5.2: The ratio cs6,/c,.,. versus the reduced field b.


in my work constantly diminish towards zero as one approaches H, (T), as expected.

This behavior is attributed to the (1 b)2 factor in the free energy (2.27).

I also note that the anisotropic elasticity which manifests itself with a softer

shear modulus for shearing along the sides of the square lattice than along the diago-

nals emerges naturally in this model. This response of the vortex lattice is measured

by the ratio c66r/c,.,.:, which has a maximum at the transition as can be seen in figure

(5.2). 1 It has been -i.'-. -led that this behavior explains the anisotropic orienta-

tional long range order observed in decoration patterns in LuNi2B2C, which manifests

itself as a significant difference in the correlation lengths along the [110] and [110]

directions [15].

SNote that around b = 0.95 the theory breaks down because the F term can no longer be considered
a small correction. This is the origin of the discontinuity in the second peak in figure (5.2).









5.2 Fluctuations

The thermal fluctuations of the vortex lattice are incorporated in the anisotropic

Ginzburg-Landau model through the three variables p, a, 9, characterizing the struc-

ture of the vortex lattice. More specifically, I will replace p, a, 9, in the expressions

for the free energy and the squash modulus by their thermally averaged values


S= p[l (2 p)u2], (5.1)

I = a[1 a(2 p)U], (5.2)

S= a cot u2. (5.3)


The derivation of the above equations goes along the lines of the derivation of the p,,

ac and presented in Chapter 4. An arbitrary displacement field u corresponding

to the thermal fluctuations is introduced with the following constraints


(u(xi)U(xi)) = 0, for i / j, (5.4)

(u4(Xi)Uj(X2)) = 0, (5.5)
12
(ui(x)u(x)) (u2(x)). (5.6)
2

Starting from the definitions of p, a and 9 (equations 4.10, 4.11, 4.12), under the above

constraints it is straightforward to obtain after some simple algebra the thermally

averaged p, a and q.

One then has to find where the zeroes of the squash modulus lie in the H-T

plane. This procedure is analogous to the one carried out by Gurevich and Kogan [84]

within the nonlocal London model, taking the simplifying approach of not calculating

the phase transition line Ho(T) selfconsistently. That means that one does not take

into account the effects of the fluctuations on the elastic moduli themselves. One is

allowed to make this simplification based on the fact that the mean squared displace-









ment of the vortex u2 is finite at Ha(T) [84]. In order to show that explicitly, I will

present the derivation of the mean squared displacement. One starts from the elastic

energy of the square vortex lattice:


2E = CqU + c + c + c44( 2, (5.7)


written in the, convenient for this case, form in terms of strains ui and rotations

about the z axis Lcwy,


Vuj 2- 1(i + = 2 (q2uj qeqr+i + g jiij), (5.8)
1 i d2qdkqr
Lxmy = (0uy 0yuX) = 2 dk eiq.r+ikz(qxuiy qyix), (5.9)
2 2f(27)3

where the u is the Fourier transformed displacement. For a more general treatment

of the problem, the tilt modulus c44, related to bending deformations of the vortices,

is included in the expression for the elastic energy.

The condition of incompressibility takes the form


Oxuy + Oyu 0 => cos f u, = sin ity, (5.10)

q = q(cos 9, sin ), (5.11)


from which one can obtain


2E (2) + c44k2 1+ Cs2 j ix (q)i x,(-q), (5.12)
(27)3 4 sin2 sin 2

with c(o) = Csq sin2(29)+cx Cos2(2)+c,. In conclusion, the thermal average u2 will be

calculated from the general expression which relates the former with the components









x,, and yy, of the elastic tensor (see for example Blatter et al. [22])

7 7 7 Tf, d2qdk -(
2 8U + u =qT (2 ) ; .D(5.13)
(2i7)3 D)


In the case under investigation, the diagonal components of the elastic tensor are

equal to the coefficient of ui(q)ui(-q) in equation (5.12). I obtain


/ d2 qdk sin2 (5.1
)u2 = kB T 2(5.14)
1(27?)3 q2c()


After performing the integral, the mean squared displacement has the form [84]


U2 K Cx C,q (5.15)
66 + Cr

where K is the complete elliptic integral of the first kind. It is evident that even at

the instability point where Csq = 0, the mean squared displacement u2(T, B) remains

finite.

There are two competing parameters, the strength of the thermal fluctuations

X and the anisotropy parameter c which in this model is the equivalent of the nonlo-

cality parameter pn of the nonlocal London model. The competition arises because

thermal fluctuations of increased strength tend to smear out the fourfold symmetric

character of the intervortex interaction, making the interactions isotropic and eventu-

ally restoring the rhombic/hexagonal arrangement as the lowest energy configuration

of the vortex lattice.

Following Gurevich and Kogan [84], I will take A(T) = Ao/ v -t, C(T)

Co/l/- t2, where t = T/TC. I then have X = Xot/ 2.2 The dimensionless mean
2Other forms of A and E with different temperature dependencies give similar results.









squared displacement reads


Xorl bt
uq2 =- (5.16)
(1 -t2) [(1 b)3ln(1+ 2)] '

with r ~ 3 for LuNi2B2C [84].

The anisotropy parameter c was connected to the nonlocality parameter pn via

correspondence from the nonlocal London model which is derived from the extended

Ginzburg-Landau theory [71]. One can obtain


= 21 t2). (5.17)


The above expression is different from what one would obtain by simply substitut-

ing the temperature dependence in c = ? /. I ., 'yas. Regardless of the fact that the

results would be similar even with a different choice of temperature dependence, the

motivation behind this particular choice was to enable the comparison with the re-

sults of the nonlocal London model, facilitated by the introduction of the nonlocality

parameter pl.

The results of the numerical solution are shown in figure (5.3). The mean field

transition line without fluctuations was included in the graph also. Instead of choosing

arbitrary values for all the parameters of the theory I used the same parameters used

in the recent work by Gurevich and Kogan [84] to facilitate the comparison of the two

theories. The transition lines are obtained for pnl = 2.5C0 and Xo = 0.0064. One can

see that the results of the two theories are in reasonable agreement thus validating the

elegant 1.r, -i, .,1 picture concerning the competition between the thermal fluctuations

and the nonlocality.

The transition line for the fluctuating vortex lattice ends abruptly at some point.

At that point the correction term F, equation (2.28), ceases to be small enough to









Transition lines for p=2.50 xo=0.0064


0.2 0.4 0.6 0.8


Figure 5.3: The phase boundaries in the H-T plane separating the square and rhombic
vortex lattice obtained from Ginzburg-Landau theory and the nonlocal London model.
The line intercepting H, (t) is the mean field GL result.


be considered a correction and at this point the extended Ginzburg-Landau theory

becomes unstable. This shortcoming of the model could be corrected by the inclusion

of higher order terms or even by the inclusion of an isotropic term of the same order

in the anisotropy parameter c, but that is not going to affect the qualitative behavior

of the transition lines.

Another important observation is that for a Ginzburg-Landau parameter t with

value in the range appropriate for typical members of the cuprate family ( c 100)

the reorientation transition does no longer appear. This seems to -ii,.'-. -1 that the

reorientation transition is irrelevant for the cuprates.

As a closing comment I would like to address a recent objection raised by Nakai

and others [85] regarding the ]lli-i .,1 mechanism behind the reentrant structural









transition. The authors perform a calculation based on the Eilenberger theory and

claim to have found the true pli, -i. .il origin of the phenomenon in another kind of

competition. They claim that the reentrant behavior is due to intrinsic competition

between superconducting gap and Fermi surface anisotropy, which provide two differ-

ent possible orientations of the vortex lattice, one with nearest neighbors along the

gap minimum and another with nearest neighbors along the Fermi velocity minimum.

Their model, as they admit in the paper, fails to predict the correct orientation of

the vortex lattice in LuNi2B2C, at least for field strengths used in experiments up to

now. Moreover, my most serious objection is that their work is based on assumptions

about the structure of the superconducting gap in the borocarbides that remain to

be experimentally verified and which are not met in the cuprates. In my opinion, the

work of Gurevich and Kogan [84] along with the present work have unambiguously

shown the importance of thermal fluctuations in the determination of the phase di-

agram, an issue that would still have to be addressed in the alternative scenario in

order to achieve a complete description.














CHAPTER 6
FINGERING INSTABILITY OF ELECTRON DROPLETS

From here on I devote my attention to the fingering instability of electron

droplets in nonuniform magnetic fields. This work was inspired by a recent discovery

by Agam, Wiegmann and others [86, 87], which connects diffusion limited growth in

the limit of long diffusion lengths (also called Laplacian growth) with the growth of

a two-dimensional electron droplet in a high, nonuniform magnetic field. The calcu-

lation of the harmonic moments and the development of the Monte Carlo code were

my contributions to a collaboration with Alan Dorsey and Taylor Hughes [88].

Laplacian growth obtains its name from the description of a front propagating

between two incompressible liquids with high viscosity contrast (e.g. water and oil).

One considers a 2D flow, where the less viscous fluid (water) is supplied by a source

at z = x + iy = 0 and the more viscous fluid (oil) is extracted at the same rate at

infinity, resulting in the motion of the boundary between the two phases. Assuming

zero surface tension, the idealized surface moves with a normal velocity vI which is

proportional to the gradient of the pressure p on the surface (Darcy's law):


n (6.1)
On


For incompressible liquids the normal velocity becomes proportional to the gradient

of a harmonic field p given by:

V2p = 0. (6.2)

For a nice review the reader can refer to [89]. Fingered patterns form at the interface

of two incompressible fluids with large viscosity contrast only when the less viscous









fluid is injected into the more viscous fluid. It is essential that the system has a 2D

geometry, an experimental realization of which is the Hele-Shaw cell. It has been

found that an arbitrary deviation from a perfectly circular shape of the initial droplet

will result in a pattern of growing fingers which grow as more fluid is pumped into

the droplet and these fingers eventually form cusp-like singularities at their tips.

The same behavior has been postulated for the Quantum Hall droplet, the source

of the instability in this case being the changing gradients of the applied magnetic

field, producing an inhomogeneous magnetic field in the exterior of the droplet. The

influence of field inhomogeneities on the shape of the electron droplet is a manifesta-

tion of the Aharonov-Bohm effect. In this case also, singularities will develop in finite

time.

In the fluid system the singularities will be controlled by the molecular scale of

the particles and also by the surface tension which is absent in the idealized Laplacian

growth. In the quantum counterpart of this phenomenon the singularities will be cutoff

by the only length scale in the problem, the magnetic length 1B = hc/eBo, which

is the effective size of the particles involved.

The fingering instability has been studied for many years since its discovery [90

94]. One can use conformal mapping methods to map the exterior of the unit circle (w-

plane) onto the domain external to the droplet (z-plane). One fundamental property

for the later connection of the two phenomena is the conservation of the external

harmonic moments defined for the fluid and electron droplet respectively as


tk -1 zk d2z, (6.3)
7t k J exterior

tk 1 B(z)z-k d2z, (6.4)
7k "7Jeterior


with z x + iy, k = 1,2,... and 6B(z) being the magnetic field inhomogeneity.

The moment for k = 0 is proportional to the area of the droplet and the integrals









for k = 1, 2 are assumed to be properly regularized. These quantities are of central

importance for the theoretical treatment of the problem since they are used to relate

the magnetic field inhomogeneity to the parameters of the conformal map.

This part is divided in three sections. In the first two sections I show the

details of the derivation of the m.iii. -body wavefunction of the 2D electrons in a

perpendicular, nonuniform magnetic field and the conformal mapping method. In the

last section I discuss the Monte Carlo simulation and present the comparison between

theory and experiment.

6.1 Many-body Wavefunction

For a 2D system of electrons in a nonuniform magnetic field perpendicular to

the system, the N-particle Hamiltonian is

N
H = (- V + A)2 -B(rj)Uj, (6.5)
j= 1

where the Coulomb interactions of the electrons have been ignored and a( is the Pauli

matrix. I have also used appropriate units, namely the magnetic length 1B = hc/eBo

for lengths, the average field Bo for magnetic fields, hca/2 for energies with wcL

eBo/mc. If in addition one takes g = 2, equation (6.5) is the Pauli Hamiltonian and

the exact solution for the many-body groundstate is known [95-99]. It is interesting

to point out at this moment that the single particle wavefunctions, the well known

Landau levels, are derived in an almost identical method to the one presented in

section 2.1. For filling factor v = 1, a filled Landau level, the ground state wave

function can be written as a Slater determinant which reduces to [86, 97-99]


(i..., ZN) = (z z e W(z (6.6)
i








In the equation above, W(z) = -1z|2/4 + V(z), where the first term comes from

the homogeneous component of the magnetic field and V(z) is a solution of V2V

-6B -[B(z) Bo]/Bo. The normalization factor 1/IN!TN will not concern us in

what follows.

For clarity I will show explicitly the derivation. The homogeneous part of W(z)

is given by the solution to V2 = -Bo which becomes (if one uses Bo as the magnetic

field scale) V2 = -1, which has the solution O(x, y) = -Izl2/4. To calculate the

solution for the presence of a small inhomogeneity, so that B = -(Bo + 6B)z, one

has to solve V2 = -1 6B, from which by setting = -z 24 + V(z) one obtains

V2V = -6B. The formal solution is given by


V(z, z = f In z z'| 6B(z') d2. (6.7)
27 D


It is essential to assume that the field inhomogeneity is far away from the origin where

the electron droplet will be located. Then, upon expanding the logarithm for z| < z'l


In z' z| = In z' + ln(1 z/z') In z' (6.8)
k=

I obtain the expansion of the potential V(z) in terms of the harmonic moments of

the domain D, under the constraint that the inhomogeneity vanishes in the interior

of the domain D [100]:


V(z, z) 2Z tk k (6.9)
k=1

tk If 6Bz-k d2z. (6.10)
7k D









The probability density obtained from the wavefunction (6.6) is

N N
P2 -exp 2 In Izi zj- i zi2 + IV ) (6.11)
i
which can be interpreted as the probability density for a classical 2D plasma in a

background potential of the form

N N
UB = z, I2 ReV(z), (6.12)
j=1 j=1

with a constant Boltzmann factor / = 2. This is where the road for Monte Carlo

simulations opens. The only remaining issue is the choice of V(z). One possibility is

the use of thin solenoids in the form

N,
6B = 6(2) (z -Zn), (6.13)
n=l

where Ns is the number of solenoids and DT is the number of flux quanta carried by

the solenoid. The potential V(z) after an integration is found to be

N,
V(z) = In Iz Z. (6.14)
n=l

This form has both advantages and disadvantages with regard to the Monte Carlo sim-

ulation. A discussion of these will follow in the section that deals with the simulation

details.

Another, by far more elegant, way to find a suitable V(z) to carry out the sim-

ulation is to start from a suitable chosen conformal map that describes the evolution

of the boundary. First, a discussion of the conformal mapping within the context of

this problem is in order.









6.2 Conformal Mapping Method

The idea is to use the powerful conformal mapping methods for 2D systems to

transform the highly irregular domain of the exterior of the droplet in the complex

plane z onto the exterior of the unit circle |wl 1 in an auxiliary complex plane

w. The map is reversible by construction. I will present various conformal maps

that correspond to different fingering patterns. For a given conformal map z(w,t)

the evolution of the boundary in the Laplacian growth problem is governed by the

following equation [91]:

Re w-Oz (6.15)
( Ow 0t 2

For a conformal map of the form

M
z(w, t) = an(t)w", (6.16)
n=-l

one obtains the following differential equations for the (real) coefficients a,(t) [91]:


d, --1, (6.17)




dt da
n=-1
M-k /




The solution of these will give the time evolution of the boundary. As an example I

will consider three simple conformal maps to illustrate the idea.

For the conformal map z = a + bw-1 one easily finds the solution


z = e-0 al (0) + t + ao(0), (6.19)


which corresponds to a circular boundary at all times. Thus the circular boundary is

stable.









Considering the next order, namely M = 1, one can show that another stable

boundary has the form of an ellipse. The solution is


z = [ao(0) + (B + 1)A(t)] cos + i(B 1)A(t) sin 0, (6.20)


with


A(t) (1- B2) (6.21)

B (0) (6.22)
al(0)

which corresponds to an ellipse with a constant eccentricity e = (B 1)/(B + 1).

The mapping of order M = 2 produces the first unstable boundary, one with

three cusps. The solution for initial values a2(0) = 1/40, al(0) = ao(0) = 0 and

a_ = 1 is

z [a2 cos(20) + a cos 0 + i[a2 sin(20) a_ sin 0], (6.23)

with

2
a2 1 (6.24)
40
a_1 400 159201- 800t. (6.25)


The normal velocity of the boundary is given by v, = Idw/dz|, which diverges at the

moment the cusp is formed. This happens for any polynomial map such as the one I

consider here. It can be shown that the number of cusps is equal to M+ 1 which is the

number of roots of dz/dw. The divergence is caused by precisely these roots as they

approach from the interior of the unit circle and eventually fall onto the boundary

|w| = 1 at the critical time critical.










The critical time for the M = 2 map and the above choice of parameters is

tcritical = 199. In figure (6.1) I show the evolution of the boundary for this particular

map. For higher orders in M the solution is possible only by numerical solution of

the algebraic equations derived from the initial differential equations.




20




10




0- O




-10




-20



-10 0 10 20 30
Figure 6.1: The evolution of the boundary for the conformal map with M = 2 from
time t = 0 to time t = teritical = 199.


An alternative and more elegant way to proceed is with use of the harmonic

moments defined in the previous section, which facilitates the connection with the

potential V(z). The calculation of the harmonic moments for the various confor-

mal maps that I used along with the connection to the potential V(z) is presented

separately in Appendix C.









6.3 Monte Carlo Simulation

The Monte Carlo method is a very powerful tool to simulate and investigate a

system like the one I have. See for example [101]. The objective is to produce an

arrangement of particles that follows the equilibrium distribution for the classical 2D

plasma, equation (6.11). The standard scheme is the Metropolis algorithm which in

a nutshell consists of the following steps:


Starting from an initial configuration of particles, a particle is chosen at random

and moved to a new position at a random direction, for a distance equal to a

fraction of the magnetic length Il.

If that move results in a negative energy change AE, the move is accepted. If

not, it is accepted with a probability exp (-2AE), since / = 2.

This scheme ensures that in the limit where the number of Monte Carlo steps tends

to infinity, the particles will have an equilibrium arrangement that corresponds to the

minimum of the energy. In reality, one has to devise tests to ensure that the system

has reached its equilibrium configuration, for the Monte Carlo steps taken.

The code that I developed for the Monte Carlo simulation is presented in Ap-

pendix E. Here I will focus on the details and the results of the simulation.

The first issue was to check the code against known analytical results for a

uniform field. It can be shown that for the incompressible liquid state that the 2D

plasma is in, the radius of the droplet which is formed scales as R = 2N, where N

is the number of electrons in the droplet.1 The density of the droplet is uniform and

equal to p = 1/27. In addition, the ;-" of the droplet 2 can also be found. It has

the form E -N2 In (2N 1)/4. Notice that in the above formulas dimensionless

'In reality, the particles in the simulated plasma gas are not electrons, as can be inferred by their
interaction energy and also the term "charge" is more appropriate than solenoids. However, to avoid
confusion I retain the two terms, since the distinction is clear.
2To be precise, it is the energy of the fictitious plasma.









units have been used, for which 1B 1 and B = 1. All these theoretical predictions

served as consistency criteria for the simulation of the droplet in a uniform field.

There was close agreement for various droplet sizes (number of particles) and many

initial configurations of the droplet which indicated that the code was indeed working

properly.

First, the system was allowed to equilibrate and then measurements were taken.

The time unit is one Monte Carlo step, the movement of one particle in the simulation.

After N moves, all the particles are moved once on average. This defines the Monte

Carlo sweep. Equilibration for a typical droplet size of 500 particles took about 103

sweeps, and measurements were taken typically for 4 x 104 sweeps. This ensured good

statistics. The quantities measured were the total energy and the density. In the

end of the measurement cycle the average density was calculated from a previously

calculated histogram of the particle positions. To that end, a inning scheme was

eimpl'l-id with bin sizes of 0.3 x 0.3.

In figure (6.6) I show the results of a simulation for a uniform field, starting

with N = 100 electrons up to N = 400 electrons. The energy measured through

the equilibration phase is shown for the last simulation N = 400, to illustrate the

rapidity of the equilibration process. The measured energy is also shown. The density

distribution is presented using pseudo-color pictures, in which blue is low density and

red is high density. I find a droplet of uniform density and circular boundary.

Before presenting the simulations for the inhomogeneous magnetic field a few

comments are in order, on the two possible methods to implement the magnetic

field inhomogeneity. Initially, I ran simulations with solenoids as the source of the

inhomogeneity. The shortcomings of that method were that it was hard to predict

beforehand what is the proper value for the flux and at what distance the solenoids

should be placed to have the desired effect, namely drive the cusping of the droplet.

These two values are interrelated and the complication was exacerbated by the fact
































N 100 N = 200
Equilibration energy for N=300
40000 Measuredenergy forN=300

20000

400

o 1 1016e 05
20000



-60000

0000 0 10000 20000 30000 40000

-40 l+weep

-
........ ........ .... 00
sweeps
N 300 Energy for N =300

Figure 6.6: Growth of the electron droplet in a uniform magnetic field.



that for solenoids placed very close to the droplet and/or high fluxes, particles would

break-off and attach to the solenoids before the cusping occurred, an indication of an


instability. However, this method proved to be invaluable in assisting in choice of the


parameters of the problem and the initial understanding of the system's complicated

behavior.


All the above problems were alleviated with the introduction of the inhomo-

geneity in the magnetic field through the potential V(z) as described in Appendix


C. As will be shortly shown, the agreement between the theory and the simula-


tions is impressive. The growth of the droplet and the shape of the boundary were









100 I





50
50 -







6*



-50





-100 ,4 I
-50 0 50 100

Figure 6.7: Illustration of the breakup of the droplet for a simulation with the number
of electrons N = 600 exceeding the critical number Nritical = 500. The droplet breaks
up and the detached particles were accumulated at the edge of the simulation box for
counting purposes. Eighty nine detached particles were counted, equally distributed
among the three smaller droplets.


found to be consistent with the predictions of the theory, with uniform density in the

interior and a sharply defined boundary. Although the semiclassical theory breaks

down at the cusping point, the simulations were carried out past that point with the

same findings in each case that I tried. The droplet would break up, with the excess

number of particles detaching from the tips of the cusps and moving outwards. In

all cases the droplet would stabilize with a number of particles equal to the critical

number Neritical. In figure (6.7) I show this behavior. In that particular simulation,

an artificial barrier was placed at the edge of the simulation box in order to count

the detached particles. The critical number was Neritical = 500 and the simulation







68

was run for N = 600 particles. Eighty nine detached electrons were counted, equally

distributed among the three smaller droplets at the boundary.

I begin with the droplet with three cusps. The appropriate conformal map is

z(w) = aw + bw-M, with M = 2. The appropriate potential for that situation is

V(z) = t3z3/2 and t3 is related to the critical number of electrons through Ncritical

1/144t2. For the details of the derivation the reader should consult Appendix C.


N =100


N = 250


N = 350
Figure 6.12: Growth


of an electron droplet


N 494
with three cusps.


For the simulations I chose Ncitical = 500. In figure (6.12) I present the growth of

the droplet for various numbers of electrons until the cusping point, showing a few

representative density plots for N = 100, 250, 350 and 494. Notice that the cusping


la ~s










occurs for a number of electrons N slightly less than the critical number. In this

case electrons started detaching from the droplet for N = 494 instead of N = 500.

In figure (6.13) I show the comparison between the semiclassical curves, colored red,

with the results of the Monte Carlo simulations, colored blue.




40



20



0-



-20



-40



-20 0 20 40 60
Figure 6.13: Comparison of semiclassical (red) and MC (blue) results.


Although in the above simulation the appropriate potential was used, it is also

possible to obtain exactly the same results using six solenoids. The solenoids have to

have alternate positive and negative flux ( and should be placed at the vertices of a

regular hexagon, on a ring of of radius R,. The flux ( and the distance R, are related

to the relevant harmonic moment through ts = 4(./R>.

Next, I turn to a simulation with two solenoids, placed on the x axis at opposite

points zo, each carrying a flux ( = q; this results in a droplet with two cusps. The

appropriate conformal map is the modified Joukowsky map z(w) = rw + 2Qw/(w2

























N =100


N 400
Figure 6.18: Growth


N 465
of an electron droplet with two cusps.


w5) and the two sets of parameters are related through


r
zo +
wo

2q r -
rQ
wN
1 f
N Ir2
2


2Qwo
1 wo
2Q2(1 w4)
(1- wo)2
4Q(1+ )
(1 w4)2


(6.26)


(6.27)


(6.28)


For this case also, the chosen critical number of electrons was Ncritical = 500. In figure

(6.18) I present the growth of the droplet for various numbers of electrons until the

cusping point, showing a few representative density plots for N = 100, 250, 400 and


BO


80


N = 250









465. In figure (6.19) I show the comparison between theory and experiment using the

same color-coding scheme as before.




6':4 40


-4: -11 II 1 1

Figure 6.19: Comparison of semiclassical (red) and MC (blue) results.


I carried out another simulation which produced beautiful pictures for which,

unfortunately, very little analysis can be carried out. An annulus of 350 repulsive

solenoids was placed in the middle of the simulation box with many solenoids dis-

tributed randomly in it. Fifty attractive solenoids were placed in the exterior of the

simulation box, equivalent to a large distance, to enforce "charge" neutrality. The

simulation resulted in a very complicated density plot as can be seen in figure (6.26).


Finally, a simulation for an also incompressible state of the Fractional Quantum

Hall Effect, with filling factor v = 1/3 is presented. I used the same potential V(z) =

t3z3/2 used in the simulation for the droplet with three cusps. The results were




























N =100


N =180


N 220


N 260

Growth of an


N = 300

a random .i i., .- of solenoids.


40

30-


~o~ro


Figure 6.26:


electron droplet within


N = 140







73

not what I initially expected. Based on theoretical grounds, the v = 1/3 state was

expected to have smaller density fluctuations in the interior of the droplet but the

observed behavior in the simulations was quite the contrary. As seen in figure (6.33),

there is an enhancement of the density near the boundary and also, what is even

more perl : ::ii _. is that the breakup of the droplet occurs in a different manner than

for the = 1 case. It seems that a smaller droplet detaches from the larger initial

droplet. I do not have an explanation for this behavior, although the enhancement of

the density near the boundary has been encountered in previous numerical simulations

of the Fractional Quantum Hall effect.








74


















-W -W a0 -a C 20 4 60 W -40 -6 40 -W 1 20 l N K


N 60 N 120
















-a -W 40 =2 C 20 4 60 N -40 -W 40 -M f 20 4 N K


N 140 N 160
















-60 40 2O 60 a -40 -M 40 -2 20 4O 6


N 260 N 167

Figure 6.33: Growth of an electron droplet for v = 1/3.














CHAPTER 7
CONCLUSION

I have investigated the structural phase transitions of the vortex lattice within

the anisotropic Ginzburg-Landau model, the anisotropy arising from s and d mixing or

from an anisotropic Fermi surface. I have shown that the addition of the phenomeno-

logical higher order derivative term, is sufficient to account for the observed behaviors

of the vortex lattice. I derived an extension of the virial theorem for superconduc-

tivity to anisotropic superconductors, which enables one to write the free energy in

a compact form from which the derivation of the nondispersive elastic moduli is very

easy. The particular extension of the virial theorem is shown to be a tool which facil-

itates the generalization of the famous Abrikosov identities along with other results

obtained from the isotropic Ginzburg-Landau theory to the anisotropic theory.

The rhombic to square structural transition was studied and its effects on the

elastic moduli were analyzed. This model exhibits the same behavior at intermediate

fields as the nonlocal London model, vanishing of the squash modulus cq at the

rhombic to square phase transition, vanishing of the rotational modulus c, at the

point where the vortex lattice exhibits rotational instability. At high fields, near H,2

the moduli vanish.

I incorporated the thermal fluctuations in this model in a simplified manner,

not taking into account the effect of the thermal fluctuations on H:(T). Nonetheless,

this approach proves sufficient to show that the reentrant transition from square to

rhombic vortex lattice, first encountered in the nonlocal London model, is present in

this model also. The mechanism that causes the reentrant behavior is the thermal

smearing of the anisotropy.









Although the anisotropic term is not derived rigorously from a microscopic

model, it succeeds in encompassing all the interesting, and at times unexpected, phe-

nomena pertaining to the structural transitions of the vortex lattice.

I also investigated the fingering of electron droplets in the Quantum Hall regime

driven by the influence of an inhomogeneous magnetic field via the Aharonov-Bohm

effect. Carrying out detailed Monte Carlo simulations of the growth of the electron

droplets for various field inhomogeneities, I was able to compare the theoretical map-

pings I found for each inhomogeneity to the experiments. I found that the development

of the boundary of the electron droplet up to the cusping point was described with

remarkable accuracy by the curves obtained from the proper conformal map for each

inhomogeneity. In addition I was able to predict the critical number of electrons for

each particular arrangement of solenoids outside the electron droplet.

Two methods were developed for the simulation of the magnetic field inhomo-

geneity. One in which the inhomogeneity is provided by thin solenoids distributed in

the exterior of the droplet and another in which the effective potential that affects the

electrons of the droplet is derived in a systematic way. Using the latter I was able to

study the growth of the electron droplet after the number of electrons exceeded the

critical number at which cusping occurs. I have found that the excess electrons were

detached from the droplet and moved towards the boundary of the simulation box.














APPENDIX A
EXTENSION OF THE VIRIAL THEOREM

In this appendix I show the derivation of the Virial Theorem for the anisotropic

Ginzburg-Landau theory. As will become soon apparent, the Virial Theorem facili-

tates greatly the simplification of the subsequent analysis of the problem at hand. The

derivation is essentially the generalization to the anisotropic case of the derivation by

Doria et al. [73] for the isotropic theory.

The starting point is the familiar Ginzburg-Landau free energy

-/ ll 12 l) 4 I Vf (X) 12 [V x A(X)12 _,2,\ 2
F a-I (x) H(x)2 (x)[Vx+A(x) +
2 2m* 8r (A.1)

InT I.(x) -[,(I [ '(x)][II(I ) (x)] + cc


where V x A(x) = B(x) is the microscopic magnetic field. The magnetic induction

is defined as B =f d3x B(x)/V and the homogeneous applied field is denoted by H;

the brackets denote spatial averaging. I can also define the quantities


Fkinetic W (X) 12 (A.2)

/B2(X) (A.3)
87

Fanisotropic __ a, [n ( y X.
(A.4)
[nI'. .(x)][I,( I -H (x)]*+cc
lx y )] + CC ,


by identifying the ]lr, -i, .,1 meaning of each component in the Ginzburg-Landau free

energy.









Under a scaling transformation the various components of the Ginzburg-Landau

free energy transform in the following manner

X
x' =. -(A.5)

Y(x') (Ax'), (A.6)

A,(x') AA(Ax'), (A.7)

nH (A.8)

B A A2B. (A.9)


The next step is to take the partial derivative with respect to A and set it equal to

zero. I neglect the A-dependence of the order parameter and the vector field because

the Ginzburg-Landau free energy is stationary under variations of these fields by
definition. Finally, after setting A = 1, I obtain

OF
-2Fketic 4Ffield 4Fanisotropic + O2B = 0, (A.10)
aB

which can be recast in the following form which contains the applied field, using the

thermodynamic relation H = 47rF/OB:


H-B H_ (x) 2 B2 X) 2 H
+ + (27, fY(x) [II,(I,2 2) )X)
S2m* 47 a

[T, ,./(x)][I,(I,2 _- 2)(x)]* + cc (A.11)


The above expression can be simplified even further with an integration by parts. One
has to use the appropriate boundary condition (2.10)


~n {q[(Yd) [yly(II 2 ]
n. ..(n C n nl)d xn,(n nl ) o. (A.12)
a, s L X








One is thus led to the following simplified result, which will be used in the derivation
of the generalized Abrikosov identities


B 47 d3 H I (x) 12 B2(X)
H.B 2* 4
V 2m* 47


) [I(xk Y- y +cc1 (A.13)
\ Qs JJ


A.1 A Useful Identity

Starting from the reasonable assumption that the free energy is stationary under
y -+ (1 + 6)y, with 6 being a constant, variation with respect to 6 leads to [3]


F 2 I 2 In1"212
\2m*


) [*(I 2 -n )2+C a 2 + i4
(-,,2)


0, (A.14)


which must hold for every 6. In this way I obtain the generalization of a well known
and very useful identity


\ 2m*
Knfl 2
2 rn*


-( ) [*( l )2 + cc]
(-,,2) lz


(A.15)


a\ -2 1 4 .


The latter permits one to write the virial theorem in the following compact form, from
which the derivation of the Abrikosov identities is trivial:


H B a 4 |a| 2 J1 4 + B ( [ 2 f(n 2- .
4 7 a ,


(A.16)


A.2 Generalized Abrikosov Identities

The generalizations of the Abrikosov identities are going to be derived as was
done for the isotropic superconductor by Klein and P6ttinger [74]. I begin by assuming
that an expansion of the internal field can be written as B(x) = [H x(x)]z and
H = [H2 A]z with A and 0 of order 11 2 and A constant. Substituting in equation









(A.16) and separating terms of different orders in |1|2 I obtain


Zeroth Order: H2 = HH. (A.17)

First Order: (2A + 0)H2 -47aa 1(1 12) + 2HC2A + 2HI2

+8 Rte(Y*(I2 I)2y). (A.18)
Oas

Averaging both sides and switching to the usual dimensionless units (see section 2.1)

I obtain

() (= II2 Re(*(Un n2)2y). (A.19)

Here, c is the dimensionless anisotropy parameter defined as c = 21 ?| I/'yas. From

the relation B(x) =[H q(x)]z it follows immediately that


(x) 1= 12- e n H2)2], (A.20)
1K 2
B= H- y 2)+ Rey*(H F)2y). (A.21)


Second Order: A2 + A0 -43p( f 4) + (A2) + (2) + 2(Aq). Averaging again and

using A = K H (switching once more to dimensionless units) I obtain after a short

calculation

H H_ 2K2 1
12 2x*(n Y) 22 k 2 U

(A.22)

The familiar Abrikosov identities for the isotropic superconductor are immedi-

ately obtained by setting the anisotropy e equal to 0 in the generalized Abrikosov

identities (A.21,A.22).








A very useful relation for the average magnetic field can be obtained combining
the generalized Abrikosov identities and it reads

K H Re(*( n2IJ R- 2)2 e I 2 2 2)2
(2 2 1)A f2) (22 1)/A(1122 2
(A.23)
where 3A is the Abrikosov parameter defined as 3A = (114)/(112)2. It is also conve-
nient to define F which is of zeroth order in | |2, as


S f 4 e( n2)2Y) + ReiA(2( )) 1) (A.24)
(Ifp 2)2 ( I2)

which simplifies the following expressions for the average magnetic field B, the cor-
rection A, and the applied field which in turn will be used in the derivation of the
free energy:


B =H -(2 (A.25)
S- (2K2 1) A (22 1) A
A (22 B) 1(2K2 1)3A + F (A.26)
S(212 1) A+

H B (2 1 -(22 (A.27)
(22 1)A + I T2 K_ 1)A + I


A.3 Free Energy, Magnetization, Gibbs Energy

The free energy can be written in a very manageable form if one uses identity
(A.15). I obtain

F 1
F = -(14) + (B2) 14 + ((- A- 0)2)
2 2
S(K A)(K A 2(9)) -( 1 +- | ) 2). (A.28)
2









Using Abrikosov's identities again, I obtain


I + (1 2) = A(B A), (A.29)
2

and now the free energy (A.28) can be written


F = (B K)(r A) + BK. (A.30)


Then, from (A.26) one obtains easily the generalization of the isotropic result for the

free energy
F(B2 -B)2 F
F K B2_- (1 (- 1 } (A.31)
(2 1) + 1 (2 K2_ -)A + 1

The magnetization M = (B H)/47 and the Gibbs free energy G = F-HB/47

are calculated easily from the previous results. I find


M--H F (A.32)
47(2K2 -1)A (2K22- 1)/A (A.3
(G 1 } (A. 33)
(2K2 1)A (2 2 1)3A

In short, I have managed to significantly reduce the complexity of the original

Ginzburg-Landau free energy, using the generalized Virial Theorem and at the same

time obtain simple expressions for important relevant quantities.














APPENDIX B
CALCULATION OF /A, ( *(n2 II2)2), ( 2ip*(n I2 )21).

In this section I show the details of the calculation of the necessary spatial

averages of the order parameter. The calculations are done following the conventions

and methodology of Chang et al. [79]. The method is essentially the same if one wants

to include the second order correction. I will show the details for completeness.

In order to construct the periodic solution one can form the following linear

combination which can be tuned so that |Il2 acquires the desired periodicity that the

simple solution of the linearized first Ginzburg-Landau equation lacks, as

S27min 27n
P(x, y) c, exp ( y)(x ), (B.1)
n=-oo

where a and b are the two basis vectors and b is the reduced field. This function

is periodic in the direction of a (the y direction). One can impose periodicity in

the b direction by requiring that T(x b sin 0, y + b cos 0) = (x, y) holds. This is

accomplished by setting b sin 0 = 2r/bcK2a, a condition which implies that one has one

vortex per unit cell.

I then choose a coordinate system (X, Y) that coincides with the vortex lattice

directions so that (x, y) = (-Y sin 0, X Y cos 0). The spatial averages are going to

be calculated integrating over one lattice cell, namely 0 < Y < b and 0 < X < a, but

the final integrals will be extended over all space taking advantage of the periodicity

of ||I2.

The integrals involving I 12 are the easiest to evaluate. The integration over Y

will result in a Kronecker 6 which will help in the evaluation of the first summation.









The second summation will extend the integral over all space. I have







The calculation of (I1 T4) is slightly more complicated. Now a product of four

wavefunctions is involved and I obtain


('Fl4) bsin 0 +i 22 ib cos0 n2 m 2 + 12 q2
2 d| (B.4)

where j (x -jb sin 0).
One can choose new variables Z n m + dxq, N 1, M m- q, which
facilitates the evaluation of the sum under the constraint of the particular Kronecker
w. The sum over i i 1, q, breaks into two sums one over Z and one for even and

odd N and M, respectively. In this case it is the integration over Z that will extend
the integration over one cell to an integration over all space. Before calculating the

integrals, one needs to rotate the derivative term by an angle 9 about the z axis
to account for the general orientation of the vortex lattice. The calculation of the
integrals is tedious but straightforward. I show the results below [the prime denotes
a rotated term following the notational convention introduced in (2.1)]:


(2) + (cb), (B.5)
8
(| |4) =fo + (e6b)31 + (eb)2[ 1 7+/2 + 521], (B.6)

(p*(n~ n_ )'f ) = 2{1 + 3(cb)}, (B.7)

(2* 2 y'2 I + o 1 + (eb) 23o + 631 + 2(71 + 72) (B.8)










The functions 3o, 31, 71 and 72 have the following form


A = Anm,
nm

S3i Re e4i Anm [8722 4

'y e~e8CPAnm -161Fc7
71 Re e 8 A.n 16n4(4 n8
nffn


(B.9)


- 6ran2 + ,


-12_r33 n6 2+ 3 _2(
4


(B.10)

45 2 1051
16 256J


72 Anm 4 [16rU4 nm4 12n 1 3n2M2(n2 + )+ 72( + m4
nm
45 o 2 105]
+ 36n2m2) (n2 + m2) + 05
16 256


(B.11)



(B.12)


where Anm = e2i"p(n2-' 'e-27(n2+2). The prime has the meaning that there are

two summations-the one shown, over n, m and the other with n and m replaced by

(n + 1/2) and (m + 1/2) respectively.














APPENDIX C
MOMENTS

In this appendix I present the calculation of the external and internal moments

of various useful conformal maps. They are related to the harmonic moments through


t k (C.1)
7ik

I make use of one little known but important theorem from the theory of complex

variables. For a function f which is analytic inside, outside and on a simple closed

loop C, with the exception of a finite number of n singular points Zk inside C, with

k = 1...n, it holds


dz f(z) = 2 Res f(z) = 2Res f(o (C.2)



C.1 Conformal Map z = f(w) = aw + bw-M

The conformal map z = f(w) = aw+bw-M is appropriate for the investigation of

the cusping of the droplet with M+ cusps, with their tips residing on a ring at regular

angles m7/(M+ 1), with m = 0, 1,..., M+1. The derivative f'(w) a- Mbw-(M+l)

has zeroes at
(Mb M+I 2'ik
.,= e+ with k 0 ... M, (C.3)

which are in the interior of the unit circle wl 1 provided that Mb/a < 1. When

these roots fall onto the unit circle, the velocity v = |dw/dz| diverges, signaling the

formation of a singularity.









C.1.1 Exterior Moments

I need to evaluate the external moments in order to find the connection between
the conformal map and the harmonic moments of the solenoid distribution outside
the droplet. I start with the definition


Cj -- exterior


z-3 dx dy= Idwff'f- .


(C.4)


The calculation for j = 0 is straightforward and yields


co r(a2 -Mb2).


For the other moments I have to evaluate the integral


c f dw (a2 Mb2) Mabw-(M+l) + abw(M+l) jM
S 2i w V)- [awM+1 + b]l'
II I2 13

with f- (w) = wM/[awM+1 + b]j having (M + 1) poles of order j in

f M+ i,(2k+1)
,c- e M+I with k = 0... M.


Using the result (C.2) I easily obtain for the three parts 1i, I2 and I3:

1{ j(M+I)-jM-1
Il = Resw=o, 00,


forj(M + ) jM- 1 > 0 = j > 1 and also


SR o j(M+ )-jM+M 0
I2 =Res w=o [=1.I 0,


(C.5)


(C.6)


(C.7)


(C.8)


(C.9)









for j(M + 1) -M + M > 0 = j > -M, along with

1( o j(M+1)-jM-M-2
Is = Resw=o[a + =1 0, (C.10)


for j(M +) -jM + M- 2>0 j > M + 2 with the exception of j =M + 1

for which there is a residue at w = 0 with the value a-~. This is derived from the

expansion


w 1 qI +(-i)M+lqM+l q2(M+l) (C.11)
waJ(1 + qM+1 )j ""'

with q = (b/a)1(M+l)w and as required |q| < 1.

I find thus that only two moments have values other that zero


co 7(a2 Mb2), (C.12)

CM+1 M. (C.13)


For the calculation of the critical number of electrons Ncritical at which the cusps

occur I notice that when the roots of the conformal map fall onto the unit circle I

have b = a/M. Solving the two equations for the exterior moments for co = 2N I

obtain
M 1 7 7 -1
Neritical l (C.14)
2M MC t

The time t in the Laplacian growth problem is equivalent to two times the

number of electrons 2N in the Quantum Hall droplet growth problem.

C.1.2 Interior Moments

The calculation of the interior moments follows along the same line and is shown

for completeness. Regardless of the closed form obtained, the results are not useful

and it actually can be shown that they formally diverge. Substituting j -j into









the RHS of equation (C.6) I have


c I dw (a2 Mb2)-
1


Mabw-(M+1) + bw(M+l)[aw M+ b (C.15)
12 13


Using again the theorem (C.2) I obtain for 1i, I2 and 13


1 = Reswo
j+I


b M+1 '
a j J


The residue is different from zero for j + 1


j = m(M + 1) with m = 1,2,... I have


li -r (a2 Mb2)(aMb) (MM +
m


RS ( M k(M+)1
s,,- (.16)
(C.16)


k(M + 1) -1 = k = j/(M + 1). So, for


(C.17)


1)),


and similarly for I2, 13 I obtain


I2 = -Ma2(aMb)" M(M + 1)
mM + 1) m)

m+1 m


(C.18)

(C.19)


Combining these three results I find the final expression for the interior moments


(M) (a m(M ) a2 Mb2 (20)
C-(M+) = (aMb) M( + l b2 (C.20)
m ) +I MM+ I1


with m 1, 2.... One easily sees that these diverge for m -- oo.









C.2 Conformal Map z = f(w) =aw + M 1 -k

This is a generalization of the previous conformal map. I easily obtain following

the methodology developed in the previous section the exterior moments


SM
co = ( a2 kb
k 1

c3 = abj_1 --
a e-


(C.21)


(C.22)


M
Skbkbk+j-
7,^ 1 )


with j = 1, 2,..., (M + 1) and the implicit assumption that bo 0 and bj>M 0. In

this case only Co, Cl, ..., CM+1 are non-zero.

The calculation of the interior moments is forbiddingly difficult.

C.3 Conformal Map z = f(w) = rw + Q/(w wo) + Q/(w + Wo)

This is a modified Joukowsky map which corresponds to a droplet with two

cusps on the positive and negative x axis respectively. It has the form


Q Q 2Qw
z = f(w) = rw + + w +
w Wo W + Wo W W


(C.23)


The derivative of the conformal map has zeroes at f'(w) = 0, which gives


w w + + 2 4( w2,
0 ?g


(C.24)


under the constraints that 0 < wo < 1, Q/r < 1/2 and wo > 2Qr.

The external moments can now be obtained as before


L dw r 2QW
cyj- ( W-W r2
2iJw w 1 -(ww) JL


2Q(w22 (2 w 2)
(w2 w)2 J[( (C.25)
(~w2 W2\2 2 + 2Q


Although the general evaluation of the preceding integral is out of the question in this

case, I can evaluate the first few nonzero external moments. With the aid of Maple









for the nontrivial calculations, I obtained the following expressions for the first three

nonzero external moments


Co = rrr -4Q(1 w (C.26)

c2 r W4)2 2QW(1 + ) (C. 27)
t z- z, (C.29)
2~Qw 2(1 w )2
C4 2IQwg( 0)2 I_4 w 2Qw2(+ W4( (C.28)
[r(1 w4) 2Qw1]4 00


C.4 Connection with Magnetic Field Inhomogeneity

In order to compare the theoretical results with the Monte Carlo simulation I

need to connect the harmonic moments calculated in the previous sections with the

harmonic moments of the field inhomogeneity defined as


tk f 6 -k d2z, (C.29)


where 6B is the magnetic field inhomogeneity. This in turn will enable us to find the

expansion of the potential V(z) which has the form

1 0O
V(z) = tkZk. (C.30)
k=1

For an arrangement of thin solenoids like the one I use in the simulations, the

harmonic moments tk can be evaluated for the inhomogeneity has the form


6B = j (2) (-Z n), (C.31)


where (, is the flux that the n-th solenoid encloses. I find that


tk k (C.32)









The harmonic moments tk are connected to the exterior moments through the simple

relation

tk (C.33)
7rk

At this stage there are two ways to proceed. The objective is to compare the evolution

of the conformal map in "time" (equivalent to the number N of electrons in this case)

with the actual simulation. The "time" enters through the zeroth-order exterior

moment co = 2N. Therefore there is the obvious option of solving the equations

provided by the calculation of the exterior moments which will be proven difficult to

carry out in general and the not so obvious at first glance option of calculating the

potential V(z) first. Both methods will be presented next.

C.4.1 Exterior Moments

I will focus on the solution for six solenoids on a ring of radius Rs each of which

carries a flux ( with alternating signs as one travels on the ring, distributed uniformly

at angles which are multiples of 7/3. The appropriate conformal map for this problem

is z = f(w) = aw + bw-M, with M = 2. From equations (C.32,C.33) I have


co = 27N, (C.34)
127r(I
ca 2 (C.35)
RS

which are the first non-zero exterior moments for that particular arrangement of

solenoids. The solution of equations (C.12,C.13) yields after choosing the appropriate

roots

2
a 2 i V/ 8coC3 (C.36)

b +1 2 / SCC3 (C.37)
47T









with cusps forming for coc3 = r3/8 from which I can find the critical number of

electrons

Ncritica ( ) (C.38)

I can obtain the same equation for the critical number Nritical by direct substitution

of the parameters in equation (C.14), which serves as a consistency check.

The connection with the experiment has been established in this particular case

as one can plot the curves obtained from the conformal map z = f(w) = aw + bw-2

for a given number of electrons and compare directly with the results of the Monte

Carlo simulation. Unfortunately an analytic approach of this kind is possible for the

simplest of cases and fails for higher M because of the lack of analytic solutions of

high order polynomials. In general, for given M one has to solve a polynomial of

order 2M. There is a way around that difficulty and the method is presented in the

next section.

C.4.2 Potential V(z).

First I need to introduce the Schwarz function for a curve C. A closed curve

C in the plane is described by an equation of the form f(x, y) = 0 which in terms of

complex coordinates can be written as g(z, z) = 0. For a function g which is analytic,

I can solve for z and thus obtain the Schwarz function S(z),


= S(z), on C. (C.39)


The Schwarz function is a very interesting object, with many properties and applica-

tions. The interested reader can refer to a nice book by Davis [102]. For my purposes

I need the exterior expansion of the Schwarz function S() (z) which acquires meaning

if I analytically continue S(z) to a strip-like domain which contains the curve C. It









has the form [100]

S(+z) -= S ck-1, (C.40)
k-1
where the external moments can be defined in terms of the Schwarz function as follows


Ck- I I -kS(z)dz, with k = 1,2,.... (C.41)


Recalling that the potential has the expansion [86]

1
V(z) -Re t, (C.42)
k=1

I immediately find that the potential V(z) and the external expansion of S(z) are

related by

V'(z) = ReS(+)(z). (C.43)
2

The objective now is to find the Schwarz function S) (z) for the conformal map that

I choose.

I start with the conformal map z(w) = aw + bw-M. Inverting the equation and

expressing w(z) as a Laurent series I obtain

Z 1,if-1
w(z) + O(z-(2M1)) (C.44)
a zM


For the Schwarz function I have


a b M a2 M!b2
S(z) = a + '(z) -zM + 2 + O(z-(M+2)). (C.45)
w(z) aM z

Comparing term by term with the expansion


S(z) = (M + 1)tM+ilM + t + O(Z-2), (C.46)
z









I obtain the expansion coefficients of the potential V(z), namely tM+l = ba-M/(M +

1) and t = a2 Mb2. The connection with the numerical experiment has again

been established, as one can use the derived potential to simulate the droplet and

subsequently compare with the theoretical results. What is most important, all the

difficulties related to the analytical solution for this arrangement of solenoids have

vanished.

With regard to the simulation, the route one follows is simple. First, by the

appropriate tuning of b I choose a convenient value for the critical number of electrons

Ncritical, equation (C.14). Then the simulation is carried out with the appropriate

tM+1. In the end, the curves obtained from the conformal map at different lim. ,"

are compared to the boundary of the droplet at the corresponding number of electrons.

The modified Joukowsky map is also simple to treat using this method. Inverting

the conformal map I obtain


z x3 z2 6rQ+ 3r22
W(z) = + + -(C.47)
3r 3r 3rxz

where the parameter x has the form


x z3- 9z9zrQ zr2w + 3r z2(3OrQw2 + 6r2w 3Q2) 3z4w 3r(rw 2Q)3.

(C.48)

The Schwarz function is S(z) = r/w(z) -2Qw(z)/[l (w(z)wo)2]. The first term has

no poles and the poles of the second term are found by solving w2 () = w2. With

the help of Maple I find the simple result


o + ~ (C.49)
w( 1 0)