Investigation of Order Parameter Structures of the Fulde-Ferrell …1241674/FULLTEXT01.pdf · 2018....

Master Thesis

Investigation ofOrder Parameter Structures of the

Fulde-Ferrell-Larkin-Ovchinnikov State inSuperconductors Using the Finite Element

Method

Mats Barkman

Condensed Matter Physics, Department of Theoretical Physics,School of Engineering Sciences

Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2018

Typeset in LATEX

Akademisk avhandling for avlaggande av teknologie masterexamen inomamnesomradet teoretisk fysik.

Scientific thesis for the degree of Master of Engineering in the subject areaof Theoretical physics.

TRITA-SCI-GRU 2018:325

c© Mats Barkman, August 2018Printed in Sweden by Universitetsservice US AB, Stockholm August 2018

Abstract

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is a superconductingphase that was theoretically predicted in 1964. In the presence of a strongmagnetic field, the paramagnetic interaction between the magnetic field andthe spin of the electrons leads to spin dependent Fermi surfaces. Conse-quently Cooper pairs with non-zero momentum are formed and the super-conducting order parameter becomes inhomogeneous. Similar effects havealso been discussed in the context of cold-atom gases and dense quark mat-ter. For most superconductors the orbital effect is much stronger than theparamagnetic and to this day there exists no unambiguous experimentalevidence of the FFLO state. However heavy fermion superconductors andlayered organic superconductors are candidate materials in which the statecould be observed.

In this thesis we investigate the FFLO state in the paramagnetic limitusing Ginzburg-Landau theory. Apart from the regular terms in the free en-ergy expansion, higher order terms in both momentum and order parametermodulus are included. We use coefficients that have been calculated from amicroscopic model. The finite element method is used to numerically solvethe nonlinear partial differential equations that arise in Ginzburg-Landautheory.

In agreement with previous studies of the FFLO state we found thatground state consists of one dimensional oscillations in the inhomogeneousregime. However, our findings also indicate that there exist additional typesof order parameter structures with similar free energies. These states areof equal physical interest, since it is possible to be trapped in local min-ima of the free energy. We also observed that as the applied magnetic fieldincreases, the amplitude of the oscillations become exponentially decayinginside the superconductor. Consequently the order parameter remains non-zero close to the boundary for larger magnetic fields strengths than previ-ously predicted.

Key words: Inhomogeneous superconducting states, FFLO, Ginzburg-Landau theory, finite element method.

iii

Sammanfattning

Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-tillstandet ar ett supraledande till-stand som forutspaddes 1964. I starka magnetfalt leder den paramagnetiskavaxelverkan mellan magnetfaltet och materialets elektroner till att Fermiy-tan beror av spin. Saledes bildas Cooperpar med nollskild rorelsemangd ochordningsparametern blir inhomogen. Liknande fenomen forvantas existera ikalla atomgaser och i material med hog kvarkdensitet. For de flesta suprale-dare ar den orbitala vaxelverkan med magnetfaltet mycket starkare an denparamagnetiska och det finns inga tydliga experiment som bevisat existen-sen av FFLO-tillstandet. Det finns dock material i vilka tillstandet skullekunna vara observerbar, sasom supraledare med tunga fermioner och skik-tade organiska supraledare.

I denna avhandling undersoks FFLO-tillstandet i den paramagnetiskagransen med hjalp av Ginzburg-Landau teori. Forutom de vanliga termer-na i fri energi-expansionen sa inkluderas det hogre ordningens termer irorelsemangd och belopp av ordningsparametern. Vi anvander koefficien-ter som beraknats utifran en mikroskopisk modell. Finita elementmetodenanvands for att numeriskt losa de icke-linjara partiella differential ekvationersom uppkommer fran Ginzburg-Landau teorin.

I enlighet med tidigare studier av FFLO-tillstandet bekraftar vara re-sultat att i regionen for inhomogena losningar beskrivs grundtillstandet aven endimensionellt oscillerande ordningsparameter. Vi har dessutom funnitytterligare typer av losningar med liknande fri energi. Dessa tillstand ar in-tressanta fran ett fysikaliskt perspektiv eftersom systemet kan fastna i lokalaminimipunkter till den fria energin. Vid hoga magnetfalt har vi observeratatt ordningsparameterns amplitud avtar exponentiellt inuti supraledaren.Ordningsparamatern kan saledes forbli nollskild nara randen for storre mag-netfaltsstyrkor an tidigare forutspatt.

Nyckelord: Inhomogena supraledande tillstand, FFLO, Ginzburg-Landauteori, finita elementmetoden.

iv

Preface

This thesis is the result of eight months of work form January 2018 to August2018 at the Department of Theoretical Physics at the Royal Institute ofTechnology. The work has been carried out under the supervision of EgorBabaev.

Acknowledgements

First of all, I would like to thank my supervisor Egor Babaev. Apart fromfruitful discussions on superconductivity, he has been very supportive andhelped me finding my future path in the scientific world. I would also liketo express my gratitude to Julien Garaud for introducing me to FreeFem++

and the finite element method and to Alexander Zyuzin for discussions onFFLO states. I am also very thankful for the time Anders Szepessy hastaken to discuss partial differential equations with me. I thank the groupof students I have had the pleasure to work with during the last two years.In particular Andrea who has studied topics similar to mine and has beenof great help. Lastly I would like to thank my family for comforting meduring many stressful periods. Especially my brother, who has supportedme countless times in the best way possible.

Outline

In Chapter 1 we present a brief summary of the discovery of superconductiv-ity and introduce the FFLO state. In Chapter 2 we firstly revise Ginzburg-Landau theory and BCS theory and secondly we will describe how thesetheories are modified in the presence of the paramagnetic effect. In Chap-ter 3 we introduce the finite element method and how it will be applied tothe modified Ginzburg-Landau theory. In Chapter 4 we present the result

v

vi Preface

from our numerical investigation of the FFLO state. Chapter 5 contains asummary of the thesis.

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiSammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Preface v

Contents vii

1 Introduction 11.1 Historical Introduction to Superconductivity . . . . . . . . . 11.2 The Fulde-Ferrell-Larkin-Ovchinnikov State . . . . . . . . . 31.3 Intention of Thesis . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory and Background 52.1 Landau and Ginzburg-Landau Theory . . . . . . . . . . . . 5

2.1.1 Landau Theory for The Ising Model . . . . . . . . . 52.1.2 Ginzburg-Landau Theory for Superconductivity . . . 7

2.2 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Paramagnetic Effect in Superconductors . . . . . . . . . . . 112.4 Modified Ginzburg-Landau Theory . . . . . . . . . . . . . . 13

2.4.1 Microscopically Derived Modified Ginzburg-Landau The-ory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Partial Differential Equations and Numerical Methods 213.1 Partial Differential Equations . . . . . . . . . . . . . . . . . 21

3.1.1 Weak Formulation of Poisson’s Equation . . . . . . . 213.1.2 Finite Element Method . . . . . . . . . . . . . . . . 233.1.3 The Biharmonic Equation . . . . . . . . . . . . . . . 25

3.2 Numerical Minimization Algorithms . . . . . . . . . . . . . 263.2.1 Nonlinear Conjugate Gradient Algorithm . . . . . . 273.2.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . 28

vii

viii Contents

3.3 Numerical Methods Applied to Ginzburg-Landau Theory . 29

4 Results 314.1 Order Parameter Structures . . . . . . . . . . . . . . . . . . 31

4.1.1 Initialization . . . . . . . . . . . . . . . . . . . . . . 314.2 Free Energy Comparison . . . . . . . . . . . . . . . . . . . . 364.3 Boundary Effects . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.1 Three Dimensional Coefficients . . . . . . . . . . . . 414.4 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . 43

4.4.1 Order Parameter Structures . . . . . . . . . . . . . . 434.4.2 Free Energy Comparison . . . . . . . . . . . . . . . . 464.4.3 Reversed Spanning Loop . . . . . . . . . . . . . . . . 50

4.5 Auxiliary Field Method . . . . . . . . . . . . . . . . . . . . 52

5 Conclusion 55

A Microscopically Derived Modified Ginzburg-Landau The-ory 57A.1 Modified Ginzburg-Landau Coefficients . . . . . . . . . . . . 57A.2 Rescaled Free Energy . . . . . . . . . . . . . . . . . . . . . . 59

B Boundary Effects 61B.1 System Size Effects . . . . . . . . . . . . . . . . . . . . . . . 61

Bibliography 63

Chapter 1

Introduction

1.1 Historical Introduction to Superconductivity

Superconductivity was first discovered by Kamerlingh Onnes in 1911 bymeasuring the electric resistivity of mercury at low temperatures. The mea-surements were made possible by his success in liquefying helium three yearsearlier, allowing him to experiment at temperatures in the range from 1 to14 K. At 4 K, the resistivity of the metal vanished, which differed from theexpected behaviour described by Drude-Lorentz theory (resistivity propor-tional to square-root of temperature) [1, 2].

In 1933, Meissner and Oschenfield discovered that superconductors expelmagnetic fields. However, increasing the field strength beyond some uppedbound leads to a phase transition from the superconducting state to thenormal metal state [3].

The expulsion of magnetic field was explained by the London brothersin 1935 [4]. They derived the following equation for the magnetic field Binside the superconductor

λ2∇×∇×B + B = 0. (1.1)

Solving this equations leads to exponentially decaying magnetic field insidethe superconductor where λ is the penetration length.

Although London’s equation described the magnetic properties of super-conductivity, it was not until 1950 that the two Russian physicists Ginzburgand Landau developed a phenomenological mean-field theory describing thesuperconducting phase transition which incorporated the electromagnetic

1

2 Chapter 1. Introduction

interaction [5]. This Ginzburg-Landau theory is a generalization of the Lan-dau theory from 1937 [6]. Ginzburg-Landau theory can be used to dividesuperconductor into two different classes. For the first class (type I) thereonly exists one superconducting state (the Meissner state) in which magneticfields are completely expelled. For the second class (type II) there exist anadditional state for larger applied magnetic fields in which the magneticfield partially penetrates the substrate. The experimental discovery of typeII superconductivity is credited to Shubnikov, Khotkevich, Shepelev andRyabinin [7].

In 1957, Abrikosov described the magnetic inhomogeneous structure inthe non-Meissner state of type II superconductors. He found that the mag-netic field would form a vortex lattice [8], where each vortex carries a specificmagnetic flux quantum. The vortex lattice was experimentally confirmedten years later in 1967 [9].

T

H

Meissner state

Normal state

Type I

T

H

Meissner state

Normal state

Type II

Figure 1.1: Phenomenological phase diagrams for type I and type II super-conductors. As the external field H is increased, the material transitionsfrom the Meissner state to the normal state. For type II superconductors,there exists an intermediate Abrikosov state in which vortices are formed.

A microscopic theory for superconductivity was developed in 1957 byBardeen, Cooper and Schrieffer [10, 11]. Named after its founders, BCStheory obtains an attractive interaction between electrons through the ex-change of a phonon. This leads to formation of Cooper pairs; two electronsbound to each other with opposite momentum and spin. Later Gor’kovshowed that near the phase transition, Ginzburg-Landau theory is recov-ered as a consequence of BCS theory [12].

1.2. The Fulde-Ferrell-Larkin-Ovchinnikov State 3

1.2 The Fulde-Ferrell-Larkin-Ovchinnikov State

In the aforementioned theories, the applied external magnetic field couplesto the superconductor as a consequence of the electric charge of the elec-trons. This interaction is referred to as the orbital interaction. Howeversince the electrons also carry an intrinsic spin, we would expect that a para-magnetic interaction should take place in addition to the orbital. In 1964,Fulde, Ferrell [13] and Larkin, Ovchinnikov [14] independently discussed theconsequences of the paramagnetic interaction, concluding the existence of anon-uniform state in very large applied magnetic fields. The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state has also been discussed in the context ofcold-atom gases [15, 16] and quantum chromodynamics [17].

T

H

BCS

Normal

Figure 1.2: Phenomenological phase diagram for a superconductor withdominant paramagnetic interaction. For low temperatures and high appliedfields the superconductor is in the inhomogeneous FFLO state.

The orbital effect usually stronger than the paramagnetic effect, whichrestrict the circumstances in which the inhomogeneous state can be ob-tained. The FFLO state is yet to be identified experimentally, althoughthere exists candidate material in which the state could be observed. Theseinclude heavy fermion superconductors such as CeCuIn5 [18] and variouslayered organic superconductors [19, 20, 21].

In 1996, Buzdin and Kachkachi derived the Ginzburg-Landau theory co-efficients from the microscopic theory in the paramagnetic limit (i.e. whenorbital effects are ignored completely) and analytically calculated the ex-act structure of the superconducting order parameter in a one dimensionalsystem [22].

4 Chapter 1. Introduction

1.3 Intention of Thesis

Even though the FFLO state was predicted more than fifty years ago, therestill remains many unanswered questions regarding the FFLO state. Cur-rent research within this field include the interplay between the orbital andparamagnetic effects [23, 24, 25, 26], anisotropy induced effects [27], multi-component FFLO states [28] and currents in FFLO superconductors [29].

In this thesis we will investigate possible order parameter structures inthe paramagnetic limit. Previous studies suggest that one dimensional pair-density-waves are ground states in the inhomogeneous regime [22, 30], butmore thorough numerical investigation is needed. We will study the inhomo-geneous superconducting phase using Ginzburg-Landau theory. In particu-lar, we will solve partial differential equations which arise from Ginzburg-Landau theory. The partial differential equations will be solved numericallyusing the finite element method, which has proven to be powerful and suc-cessful in the context of superconductivity [31, 32, 33, 34]. We will restrictour field of study to one and two dimensional systems. The free energies ofthe different structures will be compared in order to construct phase dia-grams.

Chapter 2

Theory and Background

In this chapter we will consider some parts of the underlying theory ofsuperconductors. There are many introductory books on superconductivitywhich will be the foundation for the majority of this chapter [2, 35, 36, 37].For a more detailed consideration we recommend [38]. In addition we willpresent the current research status of the FFLO state in superconductors inpreparation for the study in this thesis.

2.1 Landau and Ginzburg-Landau Theory

In 1937, Landau introduced a phenomenological theory to describe phasetransitions [6]. We will introduce the theory with a simple example. Thenwe will study Ginzburg-Landau theory in the context of superconductivity,which is a generalization of Landau theory introduced in 1950 [5].

2.1.1 Landau Theory for The Ising Model

Consider a one dimensional spin chain with simple nearest-neighbour-interaction.The Hamiltonian for the system is

H(σ) = −J∑<ij>

σiσj (2.1)

where J > 0 is the coupling constant, σi ∈ −1,+1 are the spins and< ij > restricts the summation to only nearest neighbours. This system isexactly solvable, but we will treat it in the mean field approximation. In

5

6 Chapter 2. Theory and Background

the mean field approximation, the spin σi interacts with the mean value ofthe spin. This corresponds to the following modification of Equation (2.1)

σiσj 7→ σi 〈σj〉 (2.2)

where 〈·〉 denotes the Boltzmann average

〈A〉 =

∑σA(σ)e−βH(σ)

∑σ

e−βH(σ)

where β = T−1 is the inverse temperature and∑σ

denotes summation over

all possible spin configurations. By translation symmetry 〈σj〉 is indepen-dent of spin site index. Simple calculations give the self-consistency equation

m = 〈σj〉 = tanh(2βJm) (2.3)

By analysing Equation (2.3) we see that m = 0 is always a solution, whichcorresponds to a completely disordered phase. However if T < 2J = Tc twonon-trivial solutions exist, corresponding to an ordered phase with net mag-netisation. Note that two non-trivial solutions exist due to the Z2 symmetryof the system1. For temperatures slightly below Tc the solution is given by

m2 = − TTc

T − Tc

Tc. (2.4)

We can see that the average spin m seems to describe the two phases ofthe magnetic material and can be thought of an order parameter for thesystem.

Landau theory can be used as a phenomenological description of thephase transition. Assuming that the order parameter vanishes as the phasetransition, Helmholtz free energy F can be expanded in powers of the orderparameter

F (m) =α(T )

2m2 +

β(T )

4m4 + . . . (2.5)

where we assume that the coefficients α and β are analytic functions ofthe temperature. The order parameter should be chosen such that the freeenergy is minimized. Note that the Z2 symmetry of the system is preserved

1The Hamiltonian in Equation (2.1) is invariant under the mapping σj 7→ −σj.

2.1. Landau and Ginzburg-Landau Theory 7

in the free energy expansion by only considering even powers in the orderparameter. By assuming

α(T ) = α0(T − Tc), β(T ) = β0

where α0 and β0 are positive constants, we can easily find the order param-eter

m2 = −α0

β0(T − Tc)

which has the same behaviour as the mean field solution in Equation (2.4).Note that the phase transitions is of second order, since the F and its

first derivative with respect to T are continuous in the phase transition (seeFigure 2.1), while the second derivative is not2.

−1.50 −0.75 0.00 0.75 1.50

m

0.0

0.5

1.0

1.5

2.0

F(m

)

T < Tc

T = Tc

T > Tc

−1.0 −0.5 0.0 0.5 1.0

T − Tc

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

m(T )

−1.0 −0.5 0.0 0.5 1.0

T − Tc

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

F (T )

Figure 2.1: Left figure: The Landau free energy in Equation (2.5) as afunction of the order parameter for various temperatures. We are usingα0 = β0 = 1. For temperatures smaller than the critical temperature, anon-zero order parameter minimizes the free energy.Center figure: The optimal order parameter as a function of the temperature.Right figure: The free energy as a function of the temperature.

2.1.2 Ginzburg-Landau Theory for Superconductivity

In the case of superconductivity the order parameter ψ is a complex fieldwhich is related to the density of Cooper pairs (for further details see Sec-tion 2.2). In Ginzburg-Landau theory the order parameter is allowed to

2The order parameter is continuous in the phase transition which also indicates asecond order transition.


vary in space and thus Helmholtz free energy F will be a generalized to afunctional as

F [ψ,A] =

∫ΩF(ψ(r),A(r)

)ddr

where Ω ⊂ Rd is the domain and the free energy density F is

F(ψ,A) =1

2|(∇+ ieA)ψ|2 + α|ψ|2 +

β

2|ψ|4 +

(∇×A)2

2(2.6)

where e is some charge and A is the vector potential to the magnetic field.In order to find ψ and A such that the free energy is minimized we expandthe free energy around the minimum

F [ψ+εvψ,A+εvA] = F [ψ,A]+ε·∫

Ω

(δF

δψ(r)vψ(r) +

δF

δA(r)· vA(r)

)ddr+O(ε2)

At a local minimum all terms proportional to ε should vanish for any choiceof the perturbations3. Thus we obtain a set of coupled nonlinear partialdifferential equations

0 = −1

2(∇+ ieA)2ψ + (α+ β|ψ|2)ψ (2.7)

0 = eIm [ψ∗((∇+ ieA)ψ)] +∇×∇×A (2.8)

Since the magnetic field B = ∇×A and the Maxwell equation ∇×B = Jholds, we can identify from Equation (2.8) the supercurrent

J = −eIm [ψ∗((∇+ ieA)ψ)] = −e|ψ|2∇θ − q2|ψ|2A

where θ is the phase of the order parameter ψ.Next let us comment briefly on the boundary conditions for solving these

partial differential equations. When deriving the equations of motion, oneneeds to carry out integration by parts, leading to surface integrals over the

3Carrying out this expansion is equivalent to setting the functional derivative to zero,where the functional derivative is defined as

δF [φ]

δφ(y)= limε→0

F [φ+ εδy]− F [φ]

ε

where δy is the Dirac delta distribution centred at y.

2.1. Landau and Ginzburg-Landau Theory 9

boundary ∂Ω. Since we need these boundary terms to vanish, we obtain aset of boundary conditions

(∇+ ieA(r))ψ(r) · n = 0

B(r)× n = 0if r ∈ ∂Ω

where n is the normal to the boundary4.If we assume that the superconducting order parameter were homoge-

neous we can find its magnitude in the same way as we did in the introduc-tory example of the Ising chain. We get that

n0 = |ψ0|2 =

−αβ if α < 0,

0 otherwise.

Assuming that the magnitude of the order parameter is equal to√n0 inside

the superconductor we get by taking the curl of Equation (2.8) that

∇×∇×B + q2n0B = 0

which we can identify as the London equation5 in Equation (1.1) with thepenetration length

λ =1

q√n0.

We can obtain an additional length scale from the derived partial differ-ential equation in Equation (2.7). By linearising the equation around thehomogeneous minimum ψ =

√n0 one obtains an easily solvable partial dif-

ferential equation for the perturbation from the homogeneous ground state.

4In the case of an applied external magnetic field H0, the boundary condition for themagnetic field would change into (B(r) −H0) × n = 0. This happens since under thiscircumstance we want to minimize the Gibbs free energy

G = F −∫

Ω

B ·H0ddr.

5We also assume that ∇×∇θ = 0, which is not always the case. If the order parameterwas in a vortex configuration, the phase would be singular in the center of the vortex,which would lead to a source term in the London equation.


The partial differential equation indicates that the perturbation decays ex-ponentially on a certain length scale ξ, referred to as the coherence length.It is given by

ξ =1

2√−α.

The ratio between the coherence length and the magnetic penetration lengthdetermines whether the superconductor is classified as type I or type II. TypeI superconductors have ξ > λ while type II superconductors have ξ < λ.

2.2 BCS Theory

In 1957, Bardeen, Cooper and Schrieffer introduced a microscopic theory ofsuperconductivity [10, 11]. They modelled the interaction between the elec-trons in the metal as a pairwise interaction with the exchange of a phonon.Since the typical energy scale of the phonon is the Debye energy ωD theinteraction is restricted to momenta k such that

|εk − εF| < ωD

where εk = k2/2m is the kinetic energy of an free electron and εF is theFermi energy. The interaction Hamiltonian is

Hint = −g2∑

ki,σi,q

c†k1+q,σ1c†k2−q,σ2

ck1,σ1 ck2,σ2

where g is the coupling constant and c†k,σ denotes the fermionic creationoperator for momentum k and spin σ. Remember that all involved electronenergies are restricted to lie within the energy range ±ωD of the Fermi sur-face.

By restricting the interaction to electrons with opposite spin and oppo-site momentum, Cooper showed that the creation of these pairs would beenergetically beneficial. The wave function of this two-particle state is

Ψ(r1, σ1, r2, σ2) = φ(r1 − r2) · 1√2

(|↑↓〉 − |↓↑〉

).

Due to the fermionic nature of electrons, the full wave function must beantisymmetric under exchange of the particles. Since the spin part is in the

2.3. Paramagnetic Effect in Superconductors 11

singlet configuration, this implies that the spatial part is symmetric. TheSchrodinger equation for this state reads(

p21

2m+

p22

2m

)φ+ V (|r1 − r2|)φ = (2εF + δε)φ

where δε is the shift in energy. Cooper showed that this shift is negative andconsequently showed that the formation of Cooper pairs is a stable process.

By restricting the interactions to those forming Cooper pairs, the fullHamiltonian can be written as

H =∑k,σ

εk,σ c†k,σ ck,σ − g2

∑k,k′

c†k,↑c†−k,↓c−k′,↓ck′,↑.

The BCS ground state |ΨBCS〉 is assumed to be a coherent state to the

creation operator of Cooper pairs P †k = c†k,↑c†−k,↓. The mean field approxi-

mation for the BCS Hamiltonian is

c†k,↑c†−k,↓c−k′,↓ck′,↑ 7→ 〈c

†k,↑c

†−k,↓〉 c−k′,↓ck′,↑ + c†k,↑c

†−k,↓ 〈c−k′,↓ck′,↑〉

similar to the one introduced in the case of the Ising chain in Equation (2.2).In the mean field description, one finds the existence of an order parameter

ψ = g2∑k

〈c−k,↓ck,↑〉 .

Note that the order parameter is related to the density of Cooper pairs. In1959, Gor’kov showed that the order parameter satisfies a partial differentialequation equivalent to those derived from the Ginzburg-Landau functionaland thus linking the microscopic and mean-field description of superconduc-tivity [12].

2.3 Paramagnetic Effect in Superconductors

In BCS and Ginzburg-Landau theory, the interaction with the magneticfield is a consequence of the charge of the electrons. However, since theelectrons also carry spin, we would also expect a paramagnetic interactionto occur. When applying an external magnetic field to the superconductor,a Zeeman splitting will occur resulting in spin dependent Fermi surfaces.Consequently the Cooper pairs will no longer have non-zero momentum asseen in Figure 2.2b.


↑

↓

(a) Cooper pair with zero momen-tum.

↑

↓

(b) Cooper pair with non-zero mo-mentum.

Figure 2.2: In regular BCS theory, the Cooper pair consists of two electronswith opposite momentum and spin. In the presence of an external magneticfield, Zeeman splitting takes place, resulting in the formation of Cooperpairs with non-zero total momentum.

The paramagnetic interaction with the Zeeman field has a pair-breakingeffect. For sufficiently high field strength, the Cooper pairs will be broken,resulting in a transition into the normal metal state [39, 40]. The non-zero momentum of the Cooper pairs will lead to an inhomogeneous orderparameter. It was shown that the inhomogeneous state allows for appliedfields stronger than the previously estimated transition field strength. Theinhomogeneous state is referred to as the Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) state, named after the two groups of physicists who originally pro-posed the existence of the state in 1964 [13, 14]. Similar inhomogeneousstates have been discussed in a more general context in which there isa coupling between two species of fermions with different Fermi surfaces[41, 42, 43]. Apart from superconductivity, this can be physically realizedin cold-atom gases [15, 16] and dense quark matter [15, 16, 44].

There is no clear experimental evidence proving the existence of theFFLO state. For most superconductors, the orbital effect is much strongerthan the paramagnetic effect, which makes the FFLO state difficult to ob-serve. Also, a very clean superconductor is necessary. The FFLO statecould exist in heavy fermion superconductors, where the paramagnetic ef-fect is non-negligible. Specifically CeCuIn5 has been investigated experi-

2.4. Modified Ginzburg-Landau Theory 13

mentally [18]. However the experiment only shows an indirect consequenceof the FFLO state. Other candidates are layered organic superconductors.By applying the Zeeman field parallel to the layers, the orbital effect isdecreased, making the paramagnetic effect more important. For these ex-perimental configurations a more direct observable consequence has beenproposed [45]. It has also been argued that the Josephon effect can be usedto detect the FFLO state [46]. Another proposed consequence of the FFLOstate is the formation of Andreev bound states which can be detected usingnuclear magnetic resonance [47]. Some organic superconductors that havebeen studied experimentally are λ-(BETS)2FeCl4 [19, 20] and β′′-salt [21].The FFLO state has also been predicted to host many rich physical phenom-ena including topological defects and various fluctuations [48, 49, 50]. For areview of the FFLO state, see references [51, 52, 53] and their bibliographies.

2.4 Modified Ginzburg-Landau Theory

This thesis is centred around a mean-field description of superconductivitybased on Ginzburg-Landau theory. The free energy density is

F(ψ) = α|ψ|2 + β|∇ψ|2 + γ|ψ|4 + δ|∇2ψ|2 (2.9)

where the coefficients α, β, γ and δ are functions of temperature and appliedmagnetic field. Note that we are considering the paramagnetic limit, inwhich we can ignore all coupling to the vector potential A and thus ignoringorbital effects. For a regular superconductor, the coefficient β is positive,resulting in homogeneous ground states. However when the paramagneticeffect is strong, β becomes negative, making inhomogeneous structures moreenergetically beneficial. The term δ|∇2ψ|2 is needed in order to stabilize thesolution, where δ is positive. The introduction of this additional term leadsto a so called modified Ginzburg-Landau theory. However in this thesis wewill use this interchangeably with Ginzburg-Landau theory.

Two possible inhomogeneous structures are the Fulde-Ferrell state

ψFF(r) = |ψ0|eiq·r (2.10)

and the Larkin-Ovchinnikov state

ψLO(r) = ∆ · cos(q · r). (2.11)

We can compute the average free energy density for these two states to seewhich is more energetically beneficial. For the Fulde-Ferrell state we obtain


that

FFF = FFF = (α+ βq2 + δq4)|ψ0|2 + γ|ψ0|4.where F denotes the spatial average of F . Minimizing FFF with respect toq2 and |ψ0|2 we obtain

q2 =−β2δ, |ψ0|2 = −α− α0

2γ, FFF = −(α− α0)2

4γ(2.12)

where

α0 =β2

4δ.

Similarly for the Larkin-Ovchinnikov state we obtain6

q2 =−β2δ, ∆2 = −α− α0

3γ/2, FLO = −(α− α0)2

6γ. (2.13)

Remember that we are assuming now that the quartic term γ is positive(otherwise we would have to include higher order terms in the free energyexpansion). Therefore these inhomogeneous solutions are only defined inthe region α < α0. In this region, we conclude that the Fulde-Ferrell stateis more energetically favourable. Note that we only obtain a criterion forthe modulus of the wavevector q. The Zeeman field changes the radii of thespin dependent Fermi surfaces while preserving the isotropy and thus thereis no preferred direction of propagation.

The energy for the uniform state is

|ψU|2 =

−α

2γif α < 0,

0 otherwise, FU =

−α2

4γif α < 0,

0 otherwise.

We will refer to the non-zero uniform state as the U state. The state wherethe magnitude of the order parameter is zero will be referred to as the normal(N) state. Since α0 > 0 we conclude that out of the three considered orderstructures, the Fulde-Ferrell state is the most energetically favourable for allvalues of α in this model.

6For this calculation we actually need to carry out the spatial average

FLO =q

2π

∫ 2π/q

0

F(ψLO(x))dx.


2.4.1 Microscopically Derived Modified Ginzburg-LandauTheory

As previously mentioned, in 1959 Gor’kov derived the Ginzburg-Landau co-efficients from the BCS theory [12]. Similarly, Buzdin and Kachkachi derivedthe modified Ginzburg-Landau coefficients when including paramagnetic in-teraction [22]. They found that the coefficient for the quartic term and thegradient term become negative simultaneously. Consequently, higher orderterms are needed in the free energy density expansion in Equation (2.9).The free energy density expansion then becomes

F(ψ) =α|ψ|2 + β|∇ψ|2 + γ|ψ|4 + δ|∇2ψ|2+

µ|ψ|2|∇ψ|2 + η((ψ∗)2(∇ψ)2 + c.c.

)+ ν|ψ|6

(2.14)

where c.c. denotes complex conjugation. Let H denote the strength of theZeeman field and T the temperature. The explicit form of the coefficients canbe found in Appendix A. The coefficients α, γ and ν are independent of thedimension d and are plotted in Figure 2.3. The coefficient α is dependenton both the Zeeman field strength and the temperature while the othersonly depend on the temperature. The free energy expansion in Equation(2.14) is only valid when ν > 0 . This inequality is fulfilled within a certaintemperature range as shown in Figure 2.3b.

The remaining coefficients β, δ, µ and ν are given as

β ∝ γ, δ, µ, η ∝ ν

where the proportionality constants are positive and depend on the dimen-sion of the system. The coefficients β and γ change sign simultaneouslyat some temperature Ttri, enabling non-uniform ground states for temper-atures less than Ttri. For temperatures larger than Ttri, the ground stateis uniform and the superconducting phase transition occurs at the Zeemanfield strength H0(T ) where the coefficient α changes sign. At the tricritical

point(T = Ttri,H = H0(Ttri)

)the non-uniform FFLO state, the uniform

state and the normal state meet.


0.0 0.2 0.4 0.6 0.8 1.0

T

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Ttri

H0(T )

(a)

0.0 0.2 0.4 0.6 0.8 1.0

T

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

Ttri

γ(T )

ν(T )

(b)

0.0 0.2 0.4 0.6 0.8 1.0

T

0.0

0.2

0.4

0.6

0.8

1.0

1.2

H

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

α(H,T

)

(c)

Figure 2.3: The coeffients α, γ and ν in the free energy expansion in Equation(2.14) are shown in (b) and (c). The explicit form for the coefficients can befound in Equation (A.3), Equation (A.4) and Equation (A.5) in Appendix A.The line H0(T ) where α changes sign is shown in (a) and is defined byEquation (A.2). The shaded regions in (a) and (b) indicate where the highestorder coefficient ν is positive and thus where the free energy expansion isvalid. The coefficient γ changes sign at some temperature Ttri, below whichinhomogeneous ground states are possible.

Using the same procedure as in the beginning of this section, we can findthe minimum free energies corresponding to various possible structures ofthe order parameter. However since the quartic term now is negative andhigher order terms need to be consider some changes are necessary in thecalculations. Assuming that the order parameter is uniform we find

|ψU|2 =

−γ +√γ2 − 3αν

3νif α <

γ2

4ν,

0 otherwise.(2.15)


The free energy density of the non-zero uniform order parameter is

FU =

(γ −

√γ2 − 3αν

)(γ2 − 6αν − γ

√γ2 − 3αν

)27ν2

. (2.16)

We can also compute the free energy densities for the FF state in Equation(2.10) and the LO state in Equation (2.11)

FFF = (α+ βq2 + δq4)|ψ0|2 + (γ + (µ− 2η)q2)|ψ0|4 + ν|ψ0|6.

In order to simplify calculations we will assume that the order parametervanishes close to the tricritical point and thus we can ignore the highestorder term ν|ψ0|6. However, this is only possible if we assume that thepre-factor for |ψ0|4 is positive. Under these restrictions, we obtain

q2 =−β2δ, |ψ0|2 = − α− α0

2γ(1− 3ζ/8), FFF = − (α− α0)2

4γ(1− 3ζ/8)

where α0 = β2/4δ and

ζ =β

γ

µ

δ. (2.17)

Note that ζ is a constant that depends on the dimension d. Since we areconsidering the region in which γ is negative we require that (1−3ζ/8) mustbe negative as well.

Similarly we find for the Larkin-Ovchinnikov state that

q2 =−β2δ, ∆2 = − α− α0

3γ/2(1− 5ζ/24), FLO = − (α− α0)2

6γ(1− 5ζ/24)(2.18)

Similarly as for the Fulde-Ferrell state, we get a requirement that (1 −5ζ/24) must be negative for the calculations to be valid. One can easilycheck that in the region in which both the calculations for the Fulde-Ferrellstate and the Larkin-Ovchinnikov state are valid (i.e. for ζ > 24/5) theLarkin-Ovchinnikov state has lower energy than the Fulde-Ferrell state. Wecould also consider a generalized Larkin-Ovchinnikov state which exhibitsoscillations in n dimensions7

ψLO,n(r) = ∆ ·n∏j=1

cos(qxj) (2.19)

7Note that the oscillation dimension n is less or equal than the total spatial dimension.


which gives

q2 =−β2nδ

, ∆2 = − α− α0

γ(1− 5ζ/24)

(2

3

)n2n−1, FLO,n =

(2

3

)n−1

FLO.

We can thus conclude that one dimensional oscillations (n = 1) are the mostenergetically favourable. The phase diagram can be computed by comparingthe free energy density of the uniform state in Equation (2.16) with the freeenergy density of the Larkin-Ovchinnikov state in Equation (2.18)8. Theobtained phase diagram is shown in Figure 2.4 for one and two dimensionalsystems.

0.3 0.4 0.5 0.6 0.7 0.8

T

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

H

Uniform

Normal

1D

(a)

0.3 0.4 0.5 0.6 0.7 0.8

T

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5H

LO

Uniform

Normal

2D

(b)

Figure 2.4: (a) is the phase diagram for a one dimensional system and (b)is the phase diagram for a two dimensional. The dashed line corresponds tothe FFLO→ N transition where α = α0. The dashdotted line indicates theU → FFLO transition and the solid line is the regular U → N transition.The dotted line is the line H0(T ) where α changes sign.

In one dimension, Buzdin and Kachkachi solved the nonlinear partialdifferential equation that arises from the free energy expansion in Equa-tion (2.14). The solutions belong to the family of elliptic sine functions,shown in Figure 2.5. At the FFLO→ N transition, the elliptic sine function

8To be precise, these two free energies will intersect at two different Zeeman fieldstrengths H1 and H2 (H1 < H2) for each temperature. For H > H2 the LO state ispreferred, for H1 < H < H2 the U state is preferred and for H < H1 the LO state ispreferred. However H1 is a unphysical intersection, arising from the fact that we ignoredthe higher order term ν|ψ|6 in our estimate for the inhomogeneous phases. Accounting forthis artefact, we conclude that the uniform state is energetically beneficial for all H < H2.


approaches a pure sine wave, in agreement with the estimates above. Simul-taneously the amplitude of the wave vanishes, indicating that the transitionis of second order. The phase diagram of a superconductor in the param-agnetic limit has been obtained previously under this assumption [2]. Thestructure of the order parameter has been computed by solving the quasi-classical Eilenberger equations numerically, under the assumption that themodulation is one dimensional [54].

0.0 0.2 0.4 0.6 0.8 1.0

x

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00sn(τ0x, 0)

sn(τ0.8x, 0.8)

sn(τ0.99x, 0.99)

Figure 2.5: In one dimension the order parameter is given by elliptic sinefunctions sn(x, k). In the figure above elliptic sine functions are plotted forvarious k and τk is the period of sn(x, k). In the limit k → 0 it approachesthe regular sine function, in agreement with the estimates presented.

The condition ζ > 24/5 is only fulfilled in one and two dimensions.Consequently, the sixth order term ν|ψ|6 cannot be neglected in the threedimensional case. A continued numerical study of the three dimensionalsuperconductor was conducted by Houzet et al. by expanding the orderparameter in harmonics [30]. They found that the one dimensional oscilla-tions still are the most energetically beneficial and that the order parameteris real9. Further analytical studies of the same system confirmed this [55].Houzet et al. also found that the transition FFLO→ U is of the first order.

By appropriately rescaling the free energy (see Appendix A for details)in the region T < Ttri, the system is described by a simplified free energydensity given by Equation (A.7) as

F =α|ψ|2 + β|∇ψ|2 + γ|ψ|4 + δ|∇2ψ|2 + µ|ψ|2|∇ψ|2

+ η((ψ∗)2(∇ψ)2 + c.c.

)+ ν|ψ|6

(2.20)

9A more accurate description would be to say that the oscillations resemble linearlypolarized waves. However the U(1) symmetry allows for a global phase shift such that theorder parameter always lies on the real axis.


where all coefficients except α are constants. The coefficient α can be inter-preted as a rescaled Zeeman field. In Figure 2.6 the free energy densities ofthe uniform state, the Fulde-Ferrell state and the Larkin-Ovchinnikov havebeen computed numerically when the sixth order term ν|ψ|6 has been in-cluded. In all dimensions, the transition U → LO seems to be first order.The transition LO → N is of the second order in one and two dimensions,while the numerical result suggest that in three dimensions the transition isof the first order.

0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

↵

0.4

0.3

0.2

0.1

0.0

0.1

U N 1D

FU

FFF

FLO

0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

↵

0.0

0.2

0.4

0.6

0.8

1.0

1.2

U N 1D

| U|| 0|

0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

↵

0.4

0.3

0.2

0.1

0.0

0.1

U N 2D

FU

FFF

FLO

0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

↵

0.0

0.2

0.4

0.6

0.8

1.0

1.2

U N 2D

| U|| 0|

0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

↵

0.4

0.3

0.2

0.1

0.0

0.1

U N 3D

FU

FFF

FLO

0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

↵

0.0

0.2

0.4

0.6

0.8

1.0

1.2

U N 3D

| U|| 0|

Figure 2.6: Comparison of the rescaled free energy densities of the uniform,Fulde-Ferrell and Larkin-Ovchinnikov state in one, two and three dimen-sions. The transition U → LO is of first order in all dimension while thetransition LO→ N is of second order in one and two dimensions and of firstorder in three dimensions.

Chapter 3

Partial Differential Equationsand Numerical Methods

3.1 Partial Differential Equations

A significant part of this thesis involves solving partial differential equations,abbreviated PDEs. In the context of superconductivity, partial differentialequations arise as a consequence of minimizing the Ginzburg-Landau freeenergy. In this section we will introduce weak (variational) formulationsand the finite element method for solving the variational form. Relevantreferences are [56, 57].

3.1.1 Weak Formulation of Poisson’s Equation

Let us start with an example; writing the weak formulation for Poisson’sequation with Dirichlet boundary conditions

−∇2u(r) = f(r) if r ∈ Ω, u(r) = 0 if r ∈ ∂Ω. (3.1)

Here Ω ⊂ Rd is our domain, ∂Ω is its boundary and f : Ω → R is a givenfunction. Suppose that the solution u : Ω → R (referred to as the trialfunction) lives in some space V and let us multiply Equation (3.1) by a test

21

22 Chapter 3. Partial Differential Equations and Numerical Methods

function v ∈ V . This gives us the weak formulation of Poisson’s equation;find u ∈ V such that for any v ∈ V the following equation holds

−∫

Ω(∇2u)vddr =

∫Ωfvddr. (3.2)

Note however that the boundary condition is not incorporated in Equation(3.2) but will be incorporated in the space V . For reasons that will becomeclearer later, one usually symmetrises the weak formulation by carrying outintegration by parts. Utilizing the boundary condition we find an equivalentintegral

a(u, v) =

∫Ω∇u · ∇vddr =

∫Ωfvddr = b(v). (3.3)

Now let us specify more precisely what the vector space V must be. Sincefirst order derivatives appear in the bilinear form a : V × V → R we mustrequire that the derivatives of the elements of V are square integrable. Notethat it is not necessary that the elements are differentiable, it is sufficientwith weak differentiability as in the theory of distributions. Thus we have

V = H10 (Ω) = u ∈ H1(Ω) | u(r) = 0 if r ∈ ∂Ω (3.4)

where H1(Ω) is the Sobolev space defined accordingly

Hk(Ω) =

u ∈ L2(Ω)∣∣∣D(κ)u ∈ L2(Ω),

d∑j=1

κj ≤ k

(3.5)

where L2(Ω) is the set of square integrable functions on Ω and the derivativeD(κ) is defined accordingly

D(κ) =∂κ1+...+κd

∂xκ11 · · · ∂xκdd

where each κj ∈ N. Note in fact that the Sobolev spaces are Hilbert spaceswith the inner product

〈u, v〉 =∑

κ|∑κj≤k

∫Ω

(D(κ)u

)∗D(κ)vddr. (3.6)

Let us comment on why this formulation of the PDE is called weak. Similarlyto the relation between differentiability and weak differentiability, if u is a

3.1. Partial Differential Equations 23

solution to the PDE, it will also be a solution to the weak formulation.However, the reverse is not true. Being a solution to the weak formulationdoes not imply satisfying the PDE. In general one cannot assume that thesolution to the weak formulation is unique. However in this specific exampleconcerning Poisson’s equation the solution is unique as a consequence of theLax-Milgram theorem [58].

3.1.2 Finite Element Method

In this section we present the finite element method (abbreviated FEM)which has been used in this thesis to solve PDEs numerically. FEM is basedon projecting the solution u onto a subspace Vh of V in order to find anapproximate solution to the weak formulation of the PDE.

Let us introduce FEM in the context of Poisson’s equation (that isEquation (3.1)) in one dimension. Adopting the notation in the previoussection we let our domain Ω = [0, 1] and we get the boundary conditionsu(0) = u(1) = 0. We discretize our domain by defining N + 1 vertices

qn =n

N, n = 0, 1, . . . , N.

These vertices form a mesh of our domain Ω. The subspace Vh is constructed

by defining its basisφ

(1)n

Nn=0

where the basis vector φ(1)n : Ω→ R is defined

as

φ(1)n (x) =

N(x− qn−1) if x ∈ [qn−1, qn],

N(qn+1 − x) if x ∈ [qn, qn+1],

0 otherwise.

(3.7)

The basis functions are plotted in Figure 3.1a. Note that the basis functionsare continuous and piecewise differentiable which is sufficient for the weakformulation of Poisson’s equation in Equation (3.3).


-0.5

0

0.5

1

1.5

2

(a) Piecewise linear basis functions.

-0.5

0

0.5

1

1.5

2

(b) Piecewise quadratic basis func-tions.

Figure 3.1: Basis functions for the subspace Vh.

By projecting u onto the subspace Vh

u(x) =

N∑n=0

cnφ(1)n (x) (3.8)

the weak formulation of Poisson’s equation reads; find coefficients cnNn=0

such that for any v ∈ VhN∑n=0

cn

∫ 1

0

∂φ(1)n

∂x

∂v

∂xdx =

∫ 1

0fvdx.

By choosing the test function v to one of the basis functions φ(1)m we obtain

a matrix equation for the coefficients

N∑n=0

Amncn = bm

where

Amn =

∫ 1

0

∂φ(1)n

∂x

∂φ(1)m

∂xdx, bm =

∫ 1

0fφ(1)

m dx.

Since the overlap between the basis functions is small the matrix equationbecomes very sparse. Of course other basis functions can be used, such aspiecewise quadratic functions as in Figure 3.1b.

The procedure in two dimensions in very similar. The vertices qn are

3.1. Partial Differential Equations 25

obtained from a triangulation of the domain as in Figure 3.2. Piecewiselinear and quadratic basis functions for each vertex are defined analogouslyas in one dimension.

Figure 3.2: Triangulation of a two dimensional domain Ω.

3.1.3 The Biharmonic Equation

When describing the paramagnetic effects in the superconductors, a higherorder derivative term was introduced in the free energy density expansion,which gives rise to a fourth order derivative term in the corresponding partialdifferential equation. The finite element method has proven to be a powerfultool in the context of superconductivity [31, 32, 33, 34]. However in thecited references, only partial differential equations involving second orderderivatives have been studied. In this section we present the numericaldifficulties that follow from the fourth order derivatives and methods onhow to resolve these issues.

For simplicity we consider the biharmonic equation

∇4u(r) = f(r) if r ∈ Ω, u(r) = ∇2u(r) = 0 if r ∈ ∂Ω. (3.9)

The weak formulation of Equation (3.9) is∫Ω

(∇2u)(∇2v)ddr =

∫Ωfvddr (3.10)


for each v ∈ Vbi. Note that the vector space Vbi now consists of functionsthat satisfy the boundary conditions and whose second order derivativesmust be square integrable. These requirements must be inherited by thefinite element subspace Vh ⊂ Vbi. Consequently we cannot use the basicfunctions in Figure 3.1. We will discuss two methods on how to resolve thisissue.

I. One possibility is to introduce an auxiliary unknown function u : Ω→R and writing Equation (3.9) as a set of coupled second order PDEs

∇2u(r) = u(r)

∇2u(r) = f(r)if r ∈ Ω, u(r) = u(r) = 0 if r ∈ ∂Ω.

(3.11)The weak formulation of Equation (3.11) is

−∫

Ω

(∇u · ∇v +∇u · ∇v + uv

)ddr =

∫Ωfvddr (3.12)

for each v, v ∈ V where V is the vector space defined in Equation (3.4)where square-integrable first order derivatives are sufficient. Thereforethe piecewise linear and piecewise quadratic basis functions can be usedto find an approximate solution to the coupled system of PDEs.

II. Another possibility is to define a set of basis functions that can handlesecond order derivatives. In the software FreeFem++ used throughoutthis thesis, Morley elements are implemented, which can be used tosolve the biharmonic equation [59].

3.2 Numerical Minimization Algorithms

In Landau and Ginzburg-Landau theory, the order parameter is chosen suchthat the free energy functional is minimized. Typically the minimizationcannot be carried out analytically and thus we are need of numerical mini-mization algorithms. In this section we will present Newton’s method andthe nonlinear conjugate gradient algorithm and how they are used combinedwith the finite element method.

3.2. Numerical Minimization Algorithms 27

3.2.1 Nonlinear Conjugate Gradient Algorithm

The nonlinear conjugate gradient algorithm is a generalization of the conju-gate gradient algorithm. The regular conjugate gradient algorithm can beused for quadratic minimization problems such as solving

x? = argminx

[F (x)] , F (x) =1

2xT(Ax− b)

where A ∈ Rm×m and b ∈ Rm are constant. By setting ∇F (x?) = 0 we findthe criterion Ax? = b. Consequently the optimization problem is equivalentto solving a linear system of equations. The conjugate gradient methoddoes not solve this linear equation exactly, but instead iteratively updatesan approximation for the optimum x?. The iterative procedure reads

1. Compute the step direction ∆xn = βn∆xn−1 −∇F (xn)

2. Compute step rescaling factor αn

3. Update the approximation xn+1 = xn + αn∆xn

The step size is thus determined by the scalar αn. For a quadratic mini-mization problem, there exists an analytical formula for αn such that thefunction F will decrease each iteration. The scalar βn is chosen such thatneighbouring step directions are conjugate to each other with respect to thematrix A, that is ∆xT

nA∆xn−1 = 0. This will improve the convergence ofthe algorithm, since it in some sense maximizes the information gained ineach step. In the quadratic minimization, an analytical formula for βn ex-ists.

The nonlinear conjugate gradient algorithm can be used in more com-plicated minimization problems. The iterative procedure described above isadopted, but choosing αn and βn is more complicated. The rescaling factoris found by performing a line search

αn = argminα

[F (xn + α∆xn)

].

There exist different methods for determining βn that are typically used.In this thesis, the default method in FreeFeem++’s implementation of thenonlinear conjugate gradient algorithm will be used.


3.2.2 Newton’s Method

Newton’s method is an iterative algorithm which is used to find the rootx? ∈ Rm to some function f : Rm → Rn. Suppose that we have an initialguess x0 for the root. The idea is to expand f in a Taylor series around theinitial guess

f(x) = f(x0) + J(x0) · (x− x0) +O(|x− x0|2) (3.13)

where J is the Jacobian matrix

Jij(x) =∂fi(x)

∂xj.

The approximation of the root is updated by solving finding the root tothe linear system in Equation (3.13), which leads to the iterative updateprocedure

xn+1 = xn + ∆xn, ∆xn = −J(xn)−1 · f(xn). (3.14)

Clearly x? is a fixed point to the iterative mapping and one can check thatit is locally attractive. However there exist various problems with Newton’smethod. For example, the algorithm can get stuck in loops and thereforenot converge to the root.

Newton’s method can be used in minimization problems, when we wantto find the minimum of some function F : Rm → R. We are thus searchingfor the roots to its gradient f = ∇F .

In order to avoid some of the convergence problems with Newton’smethod, a relaxed Newton’s method can be used. This means that theiterative mapping in Equation (3.14) can be modified to

xn+1 = xn + αn∆xn, ∆xn = −J(xn)−1 · f(xn) (3.15)

where αn should be chosen according to some scheme. For example, asproposed in [60]

αn = argminα

[F (xn + α∆xn)

].

However, in this thesis we have not succeeded in implementing this relax-ation.

3.3. Numerical Methods Applied to Ginzburg-Landau Theory 29

3.3 Numerical Methods Applied toGinzburg-Landau Theory

As previously mentioned, partial differential equations arise in Ginzburg-Landau theory by applying variational principles for the free energy func-tional. Starting from the free energy density in Equation (2.14), one obtainsthe partial differential equation

0 =αψ − β∇2ψ + 2|ψ|2ψ + δ∇4ψ − µ(ψ∗(∇ψ)2 + |ψ|2 +∇2ψ

)+

2η(ψ∗(∇ψ)2 − 2ψ|∇ψ|2 − ψ2∇2ψ∗

)+ 3ν|ψ|4ψ.

(3.16)

The field ψ will be approximated numerically by projecting it onto a set offinite element basis functions and thus the field ψ is parametrized by a setof degrees of freedom x ∈ Rm.

In general ψ is a complex field, which means that we need to considerit real and imaginary part as two separate fields. However in this thesis wewill assume that ψ is real, since real order parameter structures have beenshown to have the lowest free energies [22, 30]. Under this restriction, thevariational form for the partial differential equation in Equation (3.16) reads

0 = f [ψ(x), ϕ] =

∫Ω

(2αψϕ+ 2β∇ψ · ∇ϕ+ 4γψ3ϕ+

2δ((∂2xψ)(∂2

xϕ) + 2(∂x∂yψ)(∂x∂yϕ) + (∂2yψ)(∂2

yϕ))+

2(µ+ 2η)(ψϕ(∇ψ)2 + ψ2∇ψ · ∇ϕ

)+ 6νψ5ϕ

)ddr

where ϕ is any test function. We are using the boundary conditions thatnaturally arise when deriving the partial differential equation in Equation(3.16). Using the functional f we can construct the gradient of the freeenergy functional with respect to the degrees of freedom as

∂jF (x) = f [ψ(x), ϕj ], j ∈ 1, 2, . . . ,m .

Using the gradient ∇F we can apply the numerical minimization algorithmsdescribed in this chapter in order to find the structure of the order pa-rameter. Throughout the thesis we will be using the termination criterion|∇F | < ε = 10−3 in order to determine convergence1.

1We have also studied stricter termination criteria, but found that the structure of ψis not strongly affected by decreasing ε.


Even though the finite element method can be used to study very com-plicated geometries of our domain Ω, we will only study square systemsΩ = [−L,L]d where L ∈ R is some length scale. We have studied two differ-ent triangulations of this domain, illustrated in Figure 3.3. We will mainlybe using Morley elements and the nonlinear conjugate gradient algorithm,but some results obtained from the auxiliary field method and Newton’smethod can be found in Section 4.5.

(a) (b)

Figure 3.3: Two possible triangulations of a square.

Chapter 4

Results

In the previous chapters we have presented a modified Ginzburg-Landautheory which can exhibit inhomogeneous order parameter structures in theground state. We have also presented numerical methods which can be usedto analyse theses systems. In this chapter we will present our findings.

4.1 Order Parameter Structures

In Section 2.4 we discussed some inhomogeneous order parameter structuresthat can occur in systems described by a modified Ginzburg-Landau theorywhere the gradient coefficient β is negative and higher order terms are neededin the expansion. In this section we will further investigate the possibilitiesof inhomogeneous order parameters. We will restrict our analysis to therescaled free energy functional in Equation (2.20) where there is only onefree parameter α, which can be interpreted as the external Zeeman field.The remaining coefficients are constants and will be chosen to be those of atwo dimensional system.

4.1.1 Initialization

We will be using the nonlinear conjugate gradient algorithm described inSection 3.2.1 to find local minima to the free energy functional. In orderto use this method, an initial guess ψ0 for the order parameter needs tobe provided. Since we intend to explore many different order parameterstructures, a variety of initial guesses must be considered. We will fix theexternal field α = 1. At this particular value, the free energy density of

31

32 Chapter 4. Results

the uniform state vanishes and thus inhomogeneous phases may be morenumerically stable. We will be using a triangulation of our domain similarto Figure 3.3a.

Explicit Initialization

We have showed that one dimensional oscillations minimize the free energyfunctional. Therefore, one important initial guess to investigate is

ψ0(r) = ∆ · cos(q(x+ y)

). (4.1)

In Figure 4.1 such an initial guess and the local minimum it converged to isshown.

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

Figure 4.1: The left figure shows the initial guess which leads to the finalorder parameter structure in the figure to the right. The initial guess waschosen following Equation (4.1). The natural boundary conditions wereused.

Even though one dimensional oscillations seem to be the global minimumof the free energy, local minima are of equal interest from a physical point ofview. A system does not necessarily always lie in its global minimum, butcan be trapped in a local minimum. This motivates us to study differenttypes of initial guesses. One possibility is radial oscillations

ψ0(r) = ∆ · cos(q√x2 + y2

). (4.2)

4.1. Order Parameter Structures 33

We can compute the average free energy density of such a state by averagingover a circular disk of radius 2πn/q where n ∈ N. This gives us

F radial =1

2

(α+ βq2 + δ

(1 + g(n)

)q4)

∆2+

3

8

(γ +

µ+ 2η

3q2

)∆4 +O(∆6)

where g(n) is the integral

g(n) =1

π2n2

∫ 2πn

0

sin2(u)

udu.

Clearly g(n) goes to zero as n goes to infinity, which means that the averagefree energy density becomes the same as the one dimensional oscillations inthe thermodynamic limit. A radial initial guess and local minimum is shownin Figure 4.2.

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ


Another possible initial guess is an order parameter

ψ0(r) = ∆ · cos(qx) cos(qy) (4.3)

which has been used as an initial guess for the results in Figure 4.3.


−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ


Eigenfunction Initialization

Apart from the explicit initialization explained above, we have also studiedanother method, where the initial guess has been chosen as an eigenfunctionto the Laplace operator

−∇2ψ0 = λψ0 (4.4)

where λ is the corresponding eigenvalue. This particular choice is motivatedby the fact that near the FFLO→ N transition, the amplitude of the orderparameter is small, which means that we can linearise the partial differentialequation in Equation (3.16) to

(α− β∇2 + δ∇4)ψ = 0.

Combined with Equation (4.4), this gives us an equation for the appropriateeigenvalue

α+ βλ+ δλ2 = 0 =⇒ λ =−βδ±√α0 − αδ

.

In the rescaled units the transition field strength α0 = δ and −2δ = β whichgives

λ = 2±√

1− α

δ.

4.1. Order Parameter Structures 35

Therefore eigenfunctions with eigenvalues close to this particular λ mightbe suitable initial guesses for the numerical minimization1. Some orderparameter structure arising from this initialization is shown in Figure 4.4.

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−0.03

−0.02

−0.01

0.00

0.01

0.02

0.03

0.04

0.05

ψ−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

ψ

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

ψ

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−0.03

−0.02

−0.01

0.00

0.01

0.02

0.03

0.04

ψ

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75ψ

Figure 4.4: Pairs of initial guesses and final local minima where differenteigenfunctions to the Laplace operator has been used as initial guesses. Thenatural boundary conditions were used.

1We have explored eigenfunctions with different eigenvalues but they all seem to be ofsimilar structure, indicating that the particular choice of eigenvalue might not affect thefinal state significantly.


4.2 Free Energy Comparison

In the previous section we showed that there exists various local minima tothe free energy functional which correspond to non-uniform order parameterstructures. In this section we will compare the average free energy densitiesof these different structures in order to establish phase diagrams. Moreprecisely we will alter the applied Zeeman field α to see how the orderparameter structures and the corresponding free energy densities change.

We will start at α = 1, at which we already have found some local minimato the free energy density. Next we will increase and decrease α, using theprevious final state as the initial guess for the next. For each α we computethe average free energy density. A comparison between some different freeenergies is found in Figure 4.5, where we also have included the averagefree energy densities for the uniform state and the Larkin-Ovchinnikov statecomputed in Figure 2.6. The evolution of the corresponding order parameterstructures is shown in Figure 4.6.

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

−0.4

−0.3

−0.2

−0.1

0.0

FU

FLO

F 1D

FA

FB

FC

Figure 4.5: The average free energy densities for applied Zeeman field α ∈[0.6, 2.0]. The natural boundary conditions were used. The dotted line atα = 1 indicates the value at which the spans were initialized and the twoarrows denote the span directions. The evolution of the order parameter

structures corresponding to F1D, FA, FB and FC can be found in Figure4.6.

4.2. Free Energy Comparison 37

−1.0

−0.5

0.0

0.5

1.0

ψ

α = 0.6 α = 0.879 α = 1.162

−30 −20 −10 0 10 20 30

x

−1.0

−0.5

0.0

0.5

1.0

ψ

α = 1.444

−30 −20 −10 0 10 20 30

x

α = 1.717

−30 −20 −10 0 10 20 30

x

α = 2.0

(a) The order parameter for various applied Zeeman fields α. The evolution corre-

sponds to the free energy curve F1D in Figure 4.5. The natural boundary conditionswere used.

−30

−20

−10

0

10

20

30

y

α = 0.6 α = 0.804 α = 0.861

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

α = 1.0

−30 −20 −10 0 10 20 30

x

α = 1.283

−30 −20 −10 0 10 20 30

x

α = 2.0

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

(b) The order parameter for various applied Zeeman fields α. The evolution corre-

sponds to the free energy curve FA in Figure 4.5. The natural boundary conditionswere used.


−30

−20

−10

0

10

20

30

y

α = 0.6 α = 0.804 α = 0.861

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

α = 1.0

−30 −20 −10 0 10 20 30

x

α = 1.283

−30 −20 −10 0 10 20 30

x

α = 2.0

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

(c) The order parameter for various applied Zeeman fields α. The evolution corre-

sponds to the free energy curve FB in Figure 4.5. The natural boundary conditionswere used.

−30

−20

−10

0

10

20

30

y

α = 0.6 α = 0.804 α = 0.861

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

α = 1.0

−30 −20 −10 0 10 20 30

x

α = 1.283

−30 −20 −10 0 10 20 30

x

α = 2.0

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

(d) The order parameter for various applied Zeeman fields α. The evolution corre-

sponds to the free energy curve FC in Figure 4.5. The natural boundary conditionswere used.

Figure 4.6: Order parameter structures for various applied Zeeman fieldsα with associated free energy curves in Figure 4.5. The natural boundaryconditions were used.

4.3. Boundary Effects 39

Studying the order parameter structures, we see that two dimensionalstructures in Figures 4.6b, 4.6c and 4.6d transition into a nearly homoge-neous state as we decrease α. As α is increased these states become mainlyone dimensional oscillations. The one dimensional system in Figure 4.6a isoscillatory and as α increases the amplitude obtain some exponential decay.This decay can also be observed in the two dimensional structures for largeα.

The analysis from Section 2.4.1 predicts that the free energy density ofthe inhomogeneous states should vanish at α = 4/3, but in Figure 4.5 wesee that the free energy density remains negative for much larger α. Weconjecture that is a consequence of the exponential dampening of the orderparameter, which seems to give a negative contribution to the free energydensity. However in the thermodynamic limit this boundary effect shouldhave negligible impact on any thermodynamic quantities. The fact thatregular superconductors remain in the superconducting state at the bound-ary for applied fields beyond Hc2 was predicted in 1963 by Saint-James andGennes [61]. Our results indicate a similar effect for the FFLO state. Weexplore this boundary effect further in Section 4.3. The free energy densi-

ties FA, FB and FC are close to each other and merge completely as α isdecreased and transition into a nearly uniform state. Some boundary ef-fect seems to decrease the free energy densities, such that these free energy

curves do not merge with FU. The free energy F1D is larger than the free en-ergies of the two dimensional systems. Possibly the two dimensional systemcan obtain a smaller free energy density since we can have these boundaryeffects in two perpendicular directions. In addition, the one dimensionalsystem seem to get trapped in some sub-optimal state, since it intersects

FLO.

4.3 Boundary Effects

In the previous section we observed that an inhomogeneous state can existsbeyond the analytically predicted upper critical field α0 by obtaining anexponentially decaying amplitude as seen in Figure 4.6a. In this section wewill examine this effect further. Note that the order parameter in Figure4.6a is even and thus we will restrict ourselves to study only one boundary.Without loss of generality we choose the boundary at x = −L = −30. Wedefine a coordinate system x′ = x + L with origin at the left boundary.


In this coordinate system, we can parametrize the order parameter withexponential decaying amplitude as

ψp(x′) =(∆ + Γe−κx

′)cos(qx′ + φ) (4.5)

where we assume that the amplitudes ∆ and Γ are positive. Since we are onlyinterested in the left boundary we restrict this parametrization to x′ ∈ [0, L].The parameters ∆, Γ, q and κ are computed by carrying out a least squarefit to the order parameters in Figure 4.6a and are plotted in Figure 4.7. Thepurely oscillatory amplitude ∆ vanishes continuously at α ≈ 1.39, whichis close to the analytically predicted Zeeman field strength α0 = 4/3. Thedecaying amplitude Γ remains non-zero for higher field strengths.

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

0.0

0.2

0.4

0.6

0.8

2D

∆

Γ

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

0.2

0.4

0.6

0.8

1.02D

q

κ

Figure 4.7: Parameters for the order parameter structure given by Equa-tion (4.5), which have been computed by carrying out a least square fit tothe order parameters in Figure 4.6a. The purely sinusodial amplitude ∆vanishes close to the expected applied field, while the exponentially dampedwave has a non-zero amplitude Γ for larger field strengths. Two dimensionalcoefficients have been used as indicated by the box in the upper right corner.

We have investigated if this decaying boundary effect is a numericalartefact by performing analogous spans for various system sizes and carryingout a least square fit to Equation (4.5). We saw that for low α, the obtainedparameters ∆, Γ, κ and q differ slightly, while for larger α the parametersbecome the same, regardless of the system size L. Therefore we concludethat the decaying amplitude is not a numerical artefact. For further details,see Appendix B.

It is of interest to see whether the amplitude Γ vanishes for larger α ornot. A continued span was carried out, resulting in Figure 4.8. We see that

4.3. Boundary Effects 41

the amplitude Γ vanishes continuously at some large applied Zeeman fieldstrength α ≈ 6.34. As α is increased, the decay frequency κ increases, whichmeans that the penetration length of the oscillations becomes smaller.

1 2 3 4 5 6 7

α

0.0

0.2

0.4

0.6

0.8

2D

∆

Γ

1 2 3 4 5 6 7

α

0.2

0.4

0.6

0.8

1.0

1.22D

q

κ

Figure 4.8: Parameters for the order parameter structure given by Equa-tion (4.5), which have been computed by carrying out a least square fit toone dimensional order parameters in the range α ∈ [0.6, 7.0]. The purelysinusodial amplitude ∆ vanishes close to the expected applied field, whilethe exponentially damped wave has a non-zero amplitude Γ for larger fieldstrengths. Two dimensional coefficients have been used as indicated by thebox in the upper right corner.

4.3.1 Three Dimensional Coefficients

So far in this section, we have only been studying systems defined by thetwo dimensional Ginzburg-Landau coefficients. Now we will study a onedimensional system defined by the three dimensional Ginzburg-Landau co-efficients. The three dimensional coefficients could be of particular interest,since the predictions in Figure 2.6 indicate a discontinuity of the order pa-rameter amplitude at the phase transition. The decaying boundary effectmight have a non-trivial influence on this phase transition. Similarly asbefore, we start with an initial guess at α = 1 and carry out a span back-wards and forward originating from α = 1 such that we span the interval[0.6, 2.0]. Given the order parameters we carry out a least square fit toEquation (4.5). The obtained parameters ∆, Γ, κ and q are plotted in Fig-ure 4.9a. The amplitude ∆ vanishes discontinuously at α ≈ 1.18, which isclose to the transition field α0 ≈ 1.13 in Figure 2.6 for the three dimensional


coefficients. Simultaneously as ∆ vanishes, the amplitude Γ increases dis-continuously, such that the total amplitude at the boundary ∆ + Γ remainscontinuous. The average free energy density obtained from the span is givenin Figure 4.9b. We observe a kink at α ≈ 1.18, indicating a transition offirst order.

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

3D

∆

Γ

∆ + Γ

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

0.2

0.4

0.6

0.8

1.0 3D

q

κ

(a) Parameters for the order parameter structure given by Equation (4.5).

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

3D

F

(b) Average free energy density.

Figure 4.9: Free energy density and parameters which have been computedby carrying out a least square fit to one dimensional order parameters in therange α ∈ [0.6, 2.0]. The purely sinusodial amplitude ∆ vanishes close to theexpected applied field, while the exponentially damped wave has a non-zeroamplitude Γ for larger field strengths. Three dimensional coefficients havebeen used as indicated by the box in the upper right corner.

4.4. Periodic Boundary Conditions 43

Similarly as for two dimensional coefficients, we also continued the spanto larger Zeeman field strengths α. In Figure 4.10 the parameters definingthe order parameter given by Equation (4.5). The behaviour of the param-eters is very similar to the two dimensional coefficient case for large α andthe amplitude Γ vanishes at α ≈ 5.28.

1 2 3 4 5 6 7

α

0.0

0.2

0.4

0.6

0.8

1.0

1.2

3D

∆

Γ

1 2 3 4 5 6 7

α

0.2

0.4

0.6

0.8

1.0

1.23D

q

κ

Figure 4.10: Parameters for the order parameter structure given by Equa-tion (4.5), which have been computed by carrying out a least square fit toone dimensional order parameters in the range α ∈ [0.6, 7.0]. The purelysinusodial amplitude ∆ vanishes close to the expected applied field, whilethe exponentially damped wave has a non-zero amplitude Γ for larger fieldstrengths. Three dimensional coefficients have been used as indicated by thebox in the upper right corner.

4.4 Periodic Boundary Conditions

In this section we will carry out an analysis analogous to that of Section4.1 and Section 4.2, but using periodic boundary conditions. Using periodicboundary conditions may lower the influence of the boundary effects thatappeared before.

4.4.1 Order Parameter Structures

Firstly we will find a set of local minima at α = 1, using both explicit ini-tialization and eigenfunction initialization. A triangulation similar to Figure3.3b will be used.

In Figure 4.11 and Figure 4.12 we have used one dimensional and ra-dial oscillations as initial guesses. The corresponding final states seem to


have adopted these structures. In Figure 4.13 different eigenfunctions tothe Laplace operator have been used as initial guesses. We observe a largevariety of different structures, which might be a consequence of the newtriangulation or the periodic boundary conditions. Note that as the oscilla-tion frequency becomes smaller, the order parameters behaves more like theelliptic sine functions in Figure 2.5.

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

ψ

Figure 4.11: The left figure shows the initial guess which lead to the finalorder parameter structure in the figure to the right. The initial guess waschosen following Equation (4.1). Periodic boundary conditions were used.

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

ψ

Figure 4.12: The left figure shows the initial guess which lead to the finalorder parameter structure in the figure to the right. The initial guess waschosen following Equation (4.2). Periodic boundary conditions were used.


−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−3

−2

−1

0

1

2

3

ψ

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−4

−2

0

2

4ψ

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

ψ

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−2

−1

0

1

2

ψ

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

initial

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

final

−4

−3

−2

−1

0

1

2

3

4

ψ

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

Figure 4.13: Pairs of initial guesses and final local minima where differ-ent eigenfunctions to the Laplace operator has been used as initial guesses.Periodic boundary conditions were used.


4.4.2 Free Energy Comparison

Now we will carry out an free energy comparison of different order parameterstructures while spanning over the applied Zeeman field α. Similarly asbefore, we will use some of the local minima at α = 1 as our starting pointfor the span. The calculated average free energy densities are compared tothe energies of the uniform and Larkin-Ovchinnikov state in Figure 4.14.The evolution of the corresponding order parameter structures in shown inFigure 4.15.

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

α

−0.4

−0.3

−0.2

−0.1

0.0

FU

FLO

F 1D

FA

FB

FC

FD

Figure 4.14: The average free energy densities for applied Zeeman fieldsα ∈ [0.6, 1.5]. Periodic boundary conditions were used. The dotted line atα = 1 indicates the value at which the spans were initialized and the twoarrows denote the span directions. The evolution of the order parameter

structures corresponding to F1D, FA, FB, FC and FD can be found inFigure 4.15.


−0.5

0.0

0.5

ψ

α = 0.6 α = 0.782 α = 0.96

−30 −20 −10 0 10 20 30

x

−0.5

0.0

0.5

ψ

α = 1.141

−30 −20 −10 0 10 20 30

x

α = 1.318

−30 −20 −10 0 10 20 30

x

α = 1.5

(a) The order parameter for various applied Zeeman fields α. The evolution corre-

sponds to the free energy curve F1D in Figure 4.14. Periodic boundary conditionswere used.

−30

−20

−10

0

10

20

30

y

α = 0.6 α = 0.782 α = 0.851

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

α = 1.0

−30 −20 −10 0 10 20 30

x

α = 1.296

−30 −20 −10 0 10 20 30

x

α = 1.5

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

(b) The order parameter for various applied Zeeman fields α. The evolution corre-

sponds to the free energy curve FA in Figure 4.14. Periodic boundary conditionswere used.


−30

−20

−10

0

10

20

30

y

α = 0.6 α = 0.78 α = 0.853

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

α = 1.0

−30 −20 −10 0 10 20 30

x

α = 1.296

−30 −20 −10 0 10 20 30

x

α = 1.5

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

(c) The order parameter for various applied Zeeman fields α. The evolution corre-

sponds to the free energy curve FB in Figure 4.14. Periodic boundary conditionswere used.

−30

−20

−10

0

10

20

30

y

α = 0.6 α = 0.78 α = 0.853

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

α = 1.0

−30 −20 −10 0 10 20 30

x

α = 1.296

−30 −20 −10 0 10 20 30

x

α = 1.5

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

(d) The order parameter for various applied Zeeman fields α. The evolution corre-

sponds to the free energy curve FC in Figure 4.14. Periodic boundary conditionswere used.


−30

−20

−10

0

10

20

30

y

α = 0.6 α = 0.78 α = 0.853

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

α = 1.0

−30 −20 −10 0 10 20 30

x

α = 1.296

−30 −20 −10 0 10 20 30

x

α = 1.5

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

(e) The order parameter for various applied Zeeman fields α. The evolution corre-

sponds to the free energy curve FD in Figure 4.14. Periodic boundary conditionswere used.

Figure 4.15: Order parameter structures for various applied Zeeman fieldsα with associated free energy curves in Figure 4.14. Periodic boundaryconditions were used.

Studying the free energies in Figure 4.14, they all seem to vanish at theexpected value α = 4/3. As the external field is increased from α = 1,the two dimensional structure becomes oscillatory in only one direction, asseen in Figure 4.15b, Figure 4.15c, Figure 4.15d and Figure 4.15e. As theapplied field is decreased from α = 1, the free energies of the two dimensionalseparate. The order parameters in Figure 4.15b and Figure 4.15d becomeuniform while the order parameters in Figure 4.15c and 4.15e get trapped insome local minima. In the region in which non-uniform states are optimal,the free energy curves are very close to each other, but the free energy curvestogether with the order parameter structures suggest that a one dimensionaloscillation is optimal. The one dimensional system in Figure 4.15a is themost energetically beneficial only for high external fields. This could bea consequence of using periodic boundary conditions on a finite domainΩ = [−30, 30]. Since the order parameter must be periodic, it might beforced to choose a sub-optimal oscillation frequency. This might also explain


the free energy curves FB and FD, since they are of a similar oscillatorynature for small α.

4.4.3 Reversed Spanning Loop

Previously when spanning over α, we started at α = 1 and performed onespan backwards and one span forwards. We saw that as we increased α,the order parameter approached one dimensional oscillations and that as wedecreased α, more complicated structures could appear. In this section wewill first decrease α from 1 to 0.64 and then increase α from 0.64 to 1. Theinitial span from 1 to 0.64 will be taken from FA and Figure 4.15b. Thestate with α = 0.64 will be used as the starting point for the forward spanback to α = 1. The free energies for these two spans are seen in Figure 4.16and the related order parameter structures in Figure 4.17. We see that aswe increase the applied Zeeman field again, the order parameter forms somestripe pattern. Note that the free energy of the forward span is less thanthe backward loop.

0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

α

−0.35

−0.30

−0.25

−0.20

−0.15

−0.10

−0.05

FA←

FA→

Figure 4.16: Average free energy densities FA←

and FA→

obtained from the

backward span followed by the forward span. Periodic boundary conditionswere used.


−30

−20

−10

0

10

20

30

y

α = 0.64 α = 0.713 α = 0.786

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

α = 0.855

−30 −20 −10 0 10 20 30

x

α = 0.927

−30 −20 −10 0 10 20 30

x

α = 1.0

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

(a) Order parameter structures obtained by decreasing α from 1 to 0.64. The

associated free energy curve is FA←

in Figure 4.16.

−30

−20

−10

0

10

20

30

y

α = 0.64 α = 0.713 α = 0.786

−30 −20 −10 0 10 20 30

x

−30

−20

−10

0

10

20

30

y

α = 0.855

−30 −20 −10 0 10 20 30

x

α = 0.927

−30 −20 −10 0 10 20 30

x

α = 1.0

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ψ

(b) Order parameter structures obtained by increasing α from 0.64 to 1. The

associated free energy curve is FA→

in Figure 4.16.

Figure 4.17: Order parameter structures obtained by firstly decreasing theapplied Zeeman field and secondly increasing the applied Zeeman field toits original value. The corresponding average free energy densities can befound in Figure 4.16. Periodic boundary conditions were used.


4.5 Auxiliary Field Method

In this section we will present result obtained using method of introducing anauxiliary field as in Equation (3.11) with piecewise linear finite element basisfunctions. The studied free energy density is given in Equation (2.9) wherethe fourth order coefficient γ is positive. We have previously showed thatfor this free energy density, the Fulde-Ferrell state is energetically preferredover the Larkin-Ovchinnikov state. However, we will restrict our numericalanalysis to the subspace within which the order parameter is real. We willnow assume that the coefficients are

−α = −β = γ = δ = 1.

The average free energy densities for the Fulde-Ferrell state and Larkin-Ovchinnikov state given in Equation (2.12) and Equation (2.13) can becomputed

FFF =25

64≈ −0.3906, FLO =

25

96≈ −0.2604.

We can compare these energies to the average energies of the numerically ob-tained structures given in Figure 4.18. By comparing the average free energydensities we conclude that the Fulde-Ferrell state is the most energeticallyfavourable. The average free energy density for the order parameter struc-ture in the lower left corner in Figure 4.18 is very close to the energy of theLarkin-Ovchinnikov state. However there are more complicated structureswith energies comparable with the one dimensional oscillations.

4.5. Auxiliary Field Method 53

−40

−20

0

20

40

y

F = −0.2535 F = −0.2413 F = −0.2477

−40 −20 0 20 40

x

−40

−20

0

20

40

y

F = −0.2607

−40 −20 0 20 40

x

F = −0.2485

−40 −20 0 20 40

x

F = −0.255

−1.0

−0.5

0.0

0.5

1.0

ψ

Figure 4.18: Order parameter structures for the free energy density in Equa-tion (2.9), under the restriction that the order parameter is real. The domainΩ = [−40, 40]2 and the parameters are −α = −β = γ = δ = 1. The struc-tures have been obtained numerically by minimizing the free energy withNewton’s method and using the auxiliary field method and piecewise linearbasis functions.

Chapter 5

Conclusion

We have investigated the Fulde-Ferrell-Larkin-Ovchinnikov state in super-conductors. The paramagnetic interaction between the spins of the Cooperpairs and the applied Zeeman field leads to Cooper pairs with non-zero mo-mentum and consequently an inhomogeneous order parameter. The stateis long sought-after but so far there exists no unambiguous experimentalevidence for this inhomogeneous phase. Typically the orbital effect is muchstronger which may suppress the FFLO state. However there exist materialssuch as heavy fermion superconductors and layered organic superconductorin which the paramagnetic interaction is non-negligible and thus are possiblecandidates for the FFLO state.

We have studied the FFLO state using modified Ginzburg-Landau theorywith coefficients derived from the microscopic Hamiltonian. By appropri-ately rescaling the free energy and the order parameter, we found that thesystem has only one free parameter which can be interpreted as the effectiveZeeman field.

Using the finite element method, we numerically calculated the orderparameter and found various possible structures. We also varied the appliedZeeman field and calculated the average free energy densities of the differ-ent order parameter configurations. In agreement with previous research,we found that one dimensional oscillations seem to minimize the free energy.However, our result also indicate that there exists more complicated orderparameter structures with similar free energies. Even though these are lo-cal minima to the free energy, they are of physical relevance and should beunderstood and classified.

55

56 Chapter 5. Conclusion

We found a boundary effect that for large Zeeman fields, the order pa-rameter amplitude becomes exponentially decaying. Consequently, the or-der parameter will remain non-zero at the boundary for larger Zeeman fieldsthan analytically predicted. For an infinite system, this boundary effect willnot alter any thermodynamic quantities. However it might be observable infinite systems or by some other direct method.

Appendix A

Microscopically DerivedModified Ginzburg-LandauTheory

In section 2.4.1 we presented a modified Ginzburg-Landau theory, originallyderived from a microscopic Hamiltonian by Buzdin and Kachachi [22]. Inthis chapter we will present the coefficients in the free energy expansion inEquation (2.14) as a function of temperature T and external Zeeman fieldH. Furthermore, we will derive a rescaled free energy, analogously to theprocedure in [30].

A.1 Modified Ginzburg-Landau Coefficients

The microscopic mean-field Hamiltonian is given by

H =

∫ [(∑σ=±1

Φ†σ(r)−∇2

2mΦσ(r) + σHΦ†σ(r)Φσ(r)

)

+(ψ(r)Φ†1(r)Φ†−1(r) + h.c.

)]ddr

where Φσ(r) is the fermionic quantum field operator, ψ(r) is the order pa-rameter and m is the electron mass. From the microscopic Hamiltonian the

57

58 Appendix A. Microscopically Derived Modified Ginzburg-Landau Theory

Ginzburg-Landau coefficients were calculated using methods from quantumfield theory [62]. Let us introduce the functions

Kn(H, T ) =2T

(2πT )n(−1)n

(n− 1)!Re[Ψ(n−1)(z)

](A.1)

where Ψ(n) is the polygamma function of order n

Ψ(n)(z) =dn+1 ln Γ(z)

dzn+1

and

z =1

2− i H

2πT.

The Zeeman field strength H0(T ) where the coefficient α changes sign isdefined implicitly by the equation

ln

(Tc

T

)= Re

[Ψ(0)

(1

2− iH0(T )

2πT

)−Ψ(0)

(1

2

)](A.2)

where Tc is the critical temperature above which the metal always is in thenormal state.

The coefficients α, γ and ν are independent of the dimension and aregiven by in terms of the functions Kn defined in Equation (A.1) as

α(H, T ) = −πN(0)(K1(H, T )−K1(H0(T ), T )

)≈ N(0)

H−H0(T )

2πTIm

[Ψ(1)

(1

2− iH0(T )

2πT

)](A.3)

γ(T ) ≈ πN(0)K3(H0(T ), T )

4(A.4)

ν(T ) ≈ −πN(0)K5(H0(T ), T )

8(A.5)

where N(0) denotes the electron density of states at the Fermi surface. Inthe calculations it has been assumed that we are considering Zeeman fieldsstrengths close to H0(T ). Consequently α will depend on H and T , whilethe remaining coefficients depend only on T . The coefficients β, δ, µ and ηare dependent on the dimension d, but are proportional to the coefficientsγ and ν as

β = β(d) · v2F · γ, δ = δ(d) · v4

F · ν, µ = 8η = µ(d) · v2F · ν (A.6)

where vF is the Fermi velocity and β, δ and µ are positive coefficients, givenin Table A.1 in one, two and three dimensions. We will also define η(d) asη = η(d) · v2

F · ν.

A.2. Rescaled Free Energy 59

Dimension d β(d) δ(d) µ(d)

1 1 1/2 42 1/2 3/16 23 1/3 1/10 4/3

Table A.1: Proportionality coefficients defined in Equation (A.6).

A.2 Rescaled Free Energy

All the derived modified Ginzburg-Landau coefficients are proportional tothe electron density of states N(0). The coefficients related to gradients areproportional to the Fermi velocity vF to some power. By an appropriaterescaling of the spatial coordinates, the electron density and the Fermi ve-locity will only appear as an overall factor to the free energy. Using Equation(A.6) we find

F [ψ] = N(0)vdF

∫ [α′|ψ|2 + βγ′|∇′ψ|2 + γ′|ψ|4 + µν ′|ψ|2|∇′ψ|2

+ ην ′((ψ∗)2(∇′ψ)2 + c.c.

)+ ν ′|ψ|6

]ddr′

where

α′ =α

N(0), γ′ =

γ

N(0), ν ′ =

ν

N(0)

and

r = vFr′.

Next we will introduce a dimensionless order parameter ψ. Equation (2.15)gives that the uniform order parameter |ψU|2 = −γ/2ν at the Zeeman fieldstrength corresponding to α = γ2/4ν. We define the dimensionless orderparameter as

ψ =

√−γ2ν

ψ.

We also introduce the wavevector modulus

q20 =−β2δ

which is the optimal wavevector modulus for the Fulde-Ferrell state in Equa-tion (2.10) and the Larkin-Ovchinnikov state in Equation (2.11). Lastly we

60 Appendix A. Microscopically Derived Modified Ginzburg-Landau Theory

introduce the rescaled coefficient α as

α′ =γ2

4να.

By defining the spatial coordinate

r′ =r

q0

the free energy F can be written

F [ψ] =N(0)vdFqd0

· −γ3

8ν· F [ψ]

where F is the rescaled free energy

F [ψ] =

∫ [α|ψ|2 − 2β2

δ|∇ψ|2 − 2|ψ|4 +

β2

δ|∇2ψ|2 +

µβ

δ|ψ|2|∇ψ|2

+ηβ

δ

((ψ∗)2(∇ψ)2 + c.c.

)+ |ψ|6

]ddr.

(A.7)

For this rescaled free energy all but one coefficient are constants. The pro-portionality constant relating F and F is temperature dependent. The coef-ficient α can be thought of as an effective Zeeman field for each temperature.

Appendix B

Boundary Effects

In Section 4.3 we analysed the exponentially decaying order parameter thatoccur at the boundary of the superconductor. Here we will give some moredetails to the analysis.

B.1 System Size Effects

In order to ensure that the decaying order parameter is not a numericalartefact, we investigate if the parameters ∆, Γ, κ and q defined in Equa-tion (4.5) are dependent on the system size L, where we are using the domainΩ = [−L, L]. We span over α ∈ [0.6, 2.0] for L ∈ 30, 40, 50 and carry outa least square fit to calculate the parameters. In Figure B.1a we show theobtained parameters. The corresponding errors are shown in Figure B.1b,where we have defined

‖ψ − ψp‖ =1

N

N∑n=1

(ψ(x′n)− ψp(x′n)

)2where x′n = L(n− 1)/N and N = 500 is some large resolution number. Wesee that for small α, the different system sizes have different parameters,but as α increases, the parameters become independent of L, indicatingthat the decaying oscillations are not a numerical artefact. Studying theerrors in Figure B.1b, we conclude that the parametrization of Equation(4.5) becomes better as α increases. We also computed the average freeenergy densities for the different system sizes, which are shown in FigureB.1c. As expected, the boundary effects contribute less as the system sizeincreases.

61

62 Appendix B. Boundary Effects

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

0.0

0.2

0.4

0.6

0.8

2D

∆L=30

ΓL=30

∆L=40

ΓL=40

∆L=50

ΓL=50

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

0.2

0.4

0.6

0.8

1.02D

qL=30

κL=30

qL=40

κL=40

qL=50

κL=50

(a) Parameters defining the order parameter ψp in Equation (4.5).

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

0.000

0.002

0.004

0.006

0.008

0.010 2D

‖ψL=30 − ψp‖‖ψL=40 − ψp‖‖ψL=50 − ψp‖

(b) Distance between the optimal ψp in Equation (4.5) and the numerically calcu-

lated order parameter ψ.

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

α

−0.20

−0.15

−0.10

−0.05

0.00

2D

F L=30

F L=40

F L=50

(c) Average free energy densities.

Figure B.1: Results obtained by spanning the interval α ∈ [0.6, 2.0] forvarious system sizes L ∈ 30, 40, 50. A least square fit was carried out tocalculate the optimal parameters ∆, Γ, κ and q, defining the order parameterin Equation (4.5).

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