Non-Pairwise Vortex Interactions in Ginzburg-Landau Theory...

Non-Pairwise Vortex Interactions in Ginzburg-Landau

Theory of Superconductivity

ALEXANDER EDSTRÖM

Master of Science Thesis

Stockholm, Sweden 2012

Thesis in theoretical physics for a Master of Science Degree in Engineering Physics.

Examensarbete inom ämnet teoretisk fysik för avläggande av civilingenjörsexameninom utbildningsprogrammet Teknisk fysik.

© Alexander Edström, April 2012Examiner: Mats WallinSupervisor: Egor Babaev

TRITA-FYS 2012:23ISSN 0280-316XISRN KTH/FYS/--12:23--SE

Department of Theoretical PhysicsSchool of Engineering Sciences

Royal Institute of Technology (KTH)AlbaNova University Center

SE-106 91 Stockholm, Sweden

Typeset in LATEX

Tryck: Universitetsservice US-AB

Abstract

Non-pairwise vortex interactions in Ginzburg-Landau

theory of superconductivity are studied by numerical

free energy minimization. In particular a three-body

interaction is defined as the difference between the total

interaction and sum of pairwise interactions in a system

of three vortices and such interactions are studied for

single and two-component type-1, critical κ, type-2 and

type-1.5 superconductors. The three-body interaction

is found to be short-range repulsive but long-range at-

tractive in the type-1 case, zero in the critical κ case,

attractive in the type-2 case and repulsive in the type-

1.5 case. Some systems of four and five vortices are also

studied and results indicate that the inclusion of three-

body interaction terms can improve the usual approxi-

mation of the total interaction by summation of pairwise

interactions.

Acknowledgements

I thank my supervisor Egor Babaev for introducing me to an interesting

research topic as well as for his advice and guidance. I thank Mats Wallin

for his feedback and for answering questions. I also want to thank Johan

Carlström for taking time to answer my many questions and give plenty of

helpful advice. Thanks also to Karl Sellin for discussions and company while

working on our projects.

Contents

1 Introduction 1

2 Theory 5

2.1 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Vortices - Type-1 and Type-2 Superconductivity . . . . . . . 72.1.2 Vortex Interactions . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Multicomponent Superconductivity . . . . . . . . . . . . . . . . . . . 92.2.1 Two-Component Ginzburg-Landau Theory . . . . . . . . . . 92.2.2 Vortices in Two-Component Systems . . . . . . . . . . . . . . 112.2.3 Type-1.5 Superconductivity . . . . . . . . . . . . . . . . . . . 11

3 Numerical Method 15

3.1 Energy Minimization Method . . . . . . . . . . . . . . . . . . . . . . 153.1.1 Discretization and Minimization . . . . . . . . . . . . . . . . 153.1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Vortex Configurations and Calculation of Interaction Energies . . . . 173.2.1 Two Vortex Configurations . . . . . . . . . . . . . . . . . . . 173.2.2 Three Vortex Configurations . . . . . . . . . . . . . . . . . . 173.2.3 Four Vortex Configurations . . . . . . . . . . . . . . . . . . . 183.2.4 Five Vortex Configurations . . . . . . . . . . . . . . . . . . . 19

4 Results 21

4.1 Three-Body Interaction Energy . . . . . . . . . . . . . . . . . . . . . 214.1.1 Type-1 Three-Body Interaction Energy . . . . . . . . . . . . 224.1.2 Critical Kappa Three-Body Interaction Energy . . . . . . . . 264.1.3 Type-2 Three-Body Interaction Energy . . . . . . . . . . . . 274.1.4 Type-1.5 Three-Body Interaction Energy . . . . . . . . . . . 30

4.2 Four-Body Interaction Energy . . . . . . . . . . . . . . . . . . . . . . 364.3 Five-Body Interaction Energy . . . . . . . . . . . . . . . . . . . . . . 394.4 Energy Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Conclusions 47

v

Appendices 49

A Units 51

B Convergence and Numerical Errors 53

B.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53B.2 Pairwise Interaction in a Critical Kappa System . . . . . . . . . . . 54

Bibliography 57

Chapter 1

Introduction

Since its discovery by Heike Kamerlingh Onnes in 1911, superconductivity has be-come one of the most widely studied topics in condensed matter physics. AfterAbrikosov’s 1957 discovery [1] that the group of superconductors known as type-2 allow for normal points known as vortices, these have been an important partin the theory of superconductivity. In addition to being perfect electrical conduc-tors, superconductors exhibit the Meissner effect meaning that they expel magneticfields. At the position of a vortex, magnetic fields can however penetrate into thesuperconducting material. Vortices are known to interact with each other and thetotal interaction between several vortices is often approximated by the sum of pair-wise interactions. This is however not exact due to nonlinearity in the equationsdetermining the interactions. The purpose of this thesis is to study to non-pairwisecontributions to the interaction between several vortices and in particular to studythe three-body interaction defined as the difference between total interaction andsum of pairwise interactions in a system of three vortices.

Abrikosov’s work was based on the Ginzburg-Landau (GL) theory of supercon-ductivity which is also the theory mainly used in this thesis and described in Sec.2.1. In the GL theory a superconductor can be parametrized by a single parameterκ known as the Ginzburg-Landau parameter. There is a critical value κc = 1√

2and

superconductors can be divided into two groups known as type-1 superconductorswith κ < 1√

2and type-2 superconductors with κ > 1√

2. As mentioned, Abrikosov

showed that type-2 superconductors allow for singular points known as vortices andthat repulsive interactions between these vortices result in the formation of a reg-ular lattice known as the Abrikosov lattice [1]. A type-1 system does not possess astable vortex state but it can be shown that vortex interactions in a type-1 systemwould be attractive [2]. Some description of vortices and vortex interactions canbe found in Sec. 2.1.1 and Sec. 2.1.2.

According to microscopic theory, superconductivity is due to coupling betweenelectron pairs forming a superconducting condensate which can flow without resis-tance. In certain materials it is possible for electrons from different energy bands to

1

CHAPTER 1. INTRODUCTION

form superconducting condensates resulting in multicomponent superconductivity.GL theory can be expanded to include also such systems as described in Sec. 2.2.In multicomponent superconductors it is no longer possible to parametrize the sys-tem by a single parameter κ and it has been suggested by Babaev and Speight [3]that it is not possible to divide superconductors only into type-1 and type-2. In-stead it is suggested that a multicomponent system can possess novel behaviorseparated from both type-1 and type-2 with non-monotonic, short-range repulsivebut long-range attractive interactions between vortex pairs. These non-monotonicinteractions can lead to formations of more complex equilibrium configurations suchas clusters or stripe patterns instead of the Abrikosov lattices observed in the type-2case. Experimental observations of such behavior has been done in clean samplesof MgB2 by Moshchalkov et. al. in [4] where the term type-1.5 superconductivitywas introduced.

It has been suggested in [5] that relatively strong non-pairwise interactions mightcause non-trivial changes to the to the equilibrium configuration of vortices incertain type-1.5 systems. It is shown that the existence of a repulsive three-bodyinteraction in addition to the non-monotonic pairwise interaction can make morestripe like patterns favored over cluster formations. This motivates a further studyof non-pairwise vortex interactions as will be done in this thesis.

Also in type-1 and type-2 superconductivity vortex interactions are nonlinearand non-pairwise contributions are expected to appear in systems of several vor-tices. A limited study of three-body interactions in single-component type-1 andtype-2 systems has been presented in [6], but only for two sets of parameters anda scaling of three vortices in an equilateral triangle. Even in single-componentsuperconductivity the complete form of the non-pairwise interactions is thereforestill unknown. In addition to studying more general configurations of vortices insingle-component superconductivity, the results here will be extended to cover non-pairwise interactions in two-component type-1 and type-2 systems previously notstudied. The question will be raised if non-pairwise interactions can affect theequilibrium vortex configurations also in type-1 or type-2 superconductivity in asimilar way as they have been suggested to do in type-1.5 superconductivity. Forexample the repulsive pairwise interaction in a type-2 system normally yields atriangular Abrikosov lattice but the question is whether non-pairwise contributionscould possibly stabilize other configurations.

Let Etot(R) be the total energy of a vortex pair with distance R and let E1

be the energy of a system with only a single vortex. The pairwise interactionenergy E2(R) of a vortex pair with distance R is then E2(R) = Etot(R) − 2E1.If Etot(R1, R2, R3) is the total energy of a system with three vortices separatedby distances R1, R2 and R3, then the total interaction energy of the system isEint(R1, R2, R3) = Etot(R1, R2, R3) − 3E1. The three-body interaction energyE3(R1, R2, R3) is defined as

E3(R1, R2, R3) = Eint(R1, R2, R3) − E2(R1) − E2(R2) − E2(R3). (1.1)

Similarly a four-body interaction is defined as the difference between the total

2

interaction in a system of four vortices and the sum of all pairwise and three-bodyinteractions. The highest order of many-body interaction studied is the five-bodyinteraction defined as the total interaction energy between five vortices minus thesum of all pairwise, three-body and four-body interactions. The approximation ofthe total interaction as the sum of pairwise interactions is

Uint ≈∑

i<j

E2(Rij), (1.2)

where Uint is the total interaction energy and Rij is the distance between twovortices labeled i and j.

The purpose of this thesis is to investigate non-pairwise vortex interaction inthe context of Ginzburg-Landau theory of superconductivity. This is done with anumerical energy minimization method described in Sec. 3. In particular three-body interactions are examined in single and two-component type-1 and type-2systems as well as two-component type-1.5 systems for which results are presentedin Sec. 4.1. Three-body interactions for these cases have already been studied tosome extent in [5, 6] but the aim of this thesis is to provide a more detailed studycovering a wider range of vortex configurations and parameter values. The caseof κ = κc is also briefly examined. In addition to this an investigation of fourand five-body interactions is done with results presented in Sec. 4.2 and Sec. 4.3.Energy densities of some vortex configurations are also studied in Sec. 4.4 in hopeof gaining a better understanding of the interactions.

3

Chapter 2

Theory

The theory in this thesis is mainly based on the Ginzburg-Landau theory of su-perconductivity described in section 2.1. This theory also needs to be expandedto describe two-component systems where electrons from different energy bandscontribute to the superconductivity as discussed in section 2.2. These multicompo-nent systems can allow for new phenomena such as type-1.5 superconductivity asdescribed in Sec. 2.2.3

2.1 Ginzburg-Landau Theory

The Ginzburg-Landau theory of superconductivity is a phenomenological theorywhich can be found in standard textbooks such as [7]. It is based on Landau’stheory of second order phase transitions and assumes that, close to the transitiontemperature, the free energy can be expanded in terms of an order parameter ψwhich is zero in the normal state and non-zero in the superconducting state. Thedensity of superconducting charge carriers, n, is related to the order parameter asn = |ψ|2. GL theory assumes that the order parameter is small and varies slowlyin space. The postulated expression for the free energy density is in SI-units

fs = fn + α|ψ|2 +β

2|ψ|4 +

1

2M|(~∇ + iqA)ψ|2 +

1

2µ0

|∇ × A|2, (2.1)

where q and M respectively denote effective charge and mass of the superconductingcharge carriers and A is the magnetic vector potential. α and β are temperaturedependent expansion coefficients of the theory and α changes sign at the criticaltemperature of the superconductor so that it is positive in the normal state andnegative in the superconducting state. It is preferable to work in reduced units

and rescaling quantities so that ψ =√

Mµ0

ψ, A = ~A, α = ~2

Mα, β = µ0~

2

M2 β and

f = ~2

µ0

f . By dropping all tilde and defining f = fs − fn, the free energy density is

f = α|ψ|2 +β

2|ψ|4 +

1

2|(∇ + iqA)ψ|2 +

1

2|∇ × A|2. (2.2)

5

CHAPTER 2. THEORY

Reduced units as those above are used in the rest of this thesis and further descrip-tion of the units can be found in Appendix A.

The different terms in Eq. 2.2 correspond to potential energy density

fpot = α|ψ|2 +β

2|ψ|4, (2.3)

kinetic energy density

fkin =1

2|(∇ + iqA)ψ|2 (2.4)

and magnetic energy density

fB =1

2|∇ × A|2. (2.5)

Furthermore, the kinetic energy density in Eq. 2.4 can, using ψ = |ψ|eiϕ, berewritten as two terms

fkin =1

2|(∇ + iqA)ψ|2 =

1

2

∣

∣(∇ + iqA) |ψ|eiϕ∣

∣

2=

=1

2

∣

∣eiϕ∇|ψ| + i|ψ|eiϕ∇ϕ+ iqA|ψ|eiϕ∣

∣

2=

=1

2

[

(∇|ψ|)2+ |ψ|2 (∇ϕ+ qA)

2]

. (2.6)

The second term, fJ = 12|ψ|2 (∇ϕ+ qA)

2, in Eq. 2.6 is the kinetic energy of

supercurrents while the first term, fgrad = 12

(∇|ψ|)2, is an increase in the kinetic

energy due to gradients in the magnitude of the order parameter.The total free energy of the system is the integral

F =

∫

V

(

α|ψ|2 +β

2|ψ|4 +

1

2|(∇ + iqA)ψ|2 +

1

2|∇ × A|2

)

dV (2.7)

over the volume V of the system. The problem to be solved in GL theory is toobtain the order parameter ψ and the vector potential A which minimize the energyfunctional in Eq. 2.7. Taking variational derivatives and minimizing the free energywith respect to ψ and A leads to the Ginzburg-Landau equations

1

2(∇ + iqA)

2ψ − αψ + βψ|ψ|2 = 0 (2.8)

and

J =iq

2(ψ∗∇ψ − ψ∇ψ∗) − q2|ψ|2A, (2.9)

where Ampere’s lawJ = ∇ × B = ∇ × ∇ × A (2.10)

has been used in Eq. 2.9 and B = ∇ × A is the magnetic flux density. Eq. 2.9describes the supercurrent density.

6

2.1. GINZBURG-LANDAU THEORY

2.1.1 Vortices - Type-1 and Type-2 Superconductivity

There are two naturally occurring length scales in the theory of superconductivity.These are the penetration depth

λ =

√

β

q2|α| , (2.11)

which describes the typical length into which a magnetic field can penetrate into asuperconductor and the coherence length

ξ =1

√

2|α|, (2.12)

which describes the length scale at which the order parameter varies in space. Theratio between these two lengths is the Ginzburg-Landau parameter

κ =λ

ξ=

√

2β

q2. (2.13)

The Ginzburg-Landau equations, Eq. 2.8 and Eq. 2.9, can be rewritten in a formwhere the only constant appearing in the equations is κ. Hence the behavior of thesystem is characterized by the value of the Ginzburg-Landau parameter and it turnsout that the equations show drastic differences depending on whether κ is greaterthen or smaller than a critical value κc = 1√

2. Hence superconductors are typically

divided into type-1 superconductors with κ < 1√2

and type-2 superconductors with

κ > 1√2. For more details see standard texts such as [7].

As first shown by Abrikosov in [1], a type-2 superconductor in a magnetic fieldallows for singular points where the order parameter goes to zero and the magneticfield can penetrate into the superconductor. This occurs if the magnetic field isgreater then a critical field Hc1, which separates the Meissner state from the socalled vortex state, but smaller than a critical field Hc2 where superconductivity isdestroyed and the material transitions to the normal state. These points are calledvortices and a repulsive interaction results in the formation of a so called Abrikosovlattice in the presence of many vortices. In type-1 superconductors a magnetic fieldwill destroy superconductivity without allowing for a stable vortex state. It canhowever be shown that there must be an attractive interaction between vortices ina type-1 superconductor [2].

The magnetic flux over a single vortex is

Φ =

∫

S

∇ × A · dS =

∮

γ

A · dl, (2.14)

where γ is the curve around the surface S which contains the vortex. RewritingEq. 2.9 yields

A = − J

q2|ψ|2 − ∇ϕq. (2.15)

7

CHAPTER 2. THEORY

Inserting this into Eq. 2.14 gives

Φ =

∮

γ

[

− J

q2|ψ|2 − ∇ϕq

]

· dl. (2.16)

However, the path γ can be chosen far from the vortex where the supercurrent J

can be neglected as it decays exponentially in the superconducting material. Thenthe magnetic flux depends only on the phase gradient and

Φ = −1

q

∮

γ

∇ϕ · dl = −2πN

q, (2.17)

where it is required that∮

γ

∇ϕ · dl = 2πN, (2.18)

with N being an integer for the order parameter ψ = |ψ|eiϕ to be a single-valuedcomplex scalar field. Hence the magnetic flux is quantized in units of Φ0 = 2π

q,

or Φ0 = 2π~

qin SI-units, and there is a 2πN phase winding around the vortex. In

a type-2 superconductor a vortex with N > 1 is not stable as it is energeticallyfavorable to decompose into several vortices with N = 1 so that the phase windingis 2π and the magnetic flux is Φ0.

2.1.2 Vortex Interactions

In a system with more than one vortex there is an interaction both due to repulsiveelectromagnetic interaction and an attractive interaction between the cores. Ithas been shown in [2] that for large distances the interaction energy between twovortices is

U(r) = 2πc2K0(r) − 2πd2

κ2K0(

√2κr), (2.19)

where c and d depend on the Ginzburg-Landau parameter κ, K0 is a modified Besselfunction and r is the vortex distance. The first term is the repulsive electromagneticinteraction and the second term is the core-core attraction. Furthermore it is shownthat for the critical value κ = 1

2, also known as the Bogomol’nyi point, d = c√

2

while

d <c√2

for κ >1√2

(2.20)

and

d >c√2

for κ <1√2. (2.21)

Hence U = 0 for κ = 1√2, U < 0 for κ < 1√

2and U > 0 for κ > 1√

2so there is an

attractive pairwise interaction in the type-1 case, a repulsive pairwise interaction inthe type-2 case and no pairwise interaction at the Bogomol’nyi point. Interactionsbetween many vortices are often treated by addition of pairwise interactions which

8

2.2. MULTICOMPONENT SUPERCONDUCTIVITY

might often be a good approximation but is not exact due to the non-pairwiseinteractions studied in this thesis. For the general case of interactions betweenseveral vortices, including short-range interactions, there are no analytical resultsand in general numerical treatments are required. The lack of pairwise interactionbetween vortices in a system with critical κ, also shown in [8], is mentioned againin Sec. 4.1.2 and also used in Appendix B to discuss the magnitude of numericalerrors.

2.2 Multicomponent Superconductivity

This section describes how the theory in the previous section changes when electronsfrom different energy bands can form superconducting condensates. In particularthe case with two superconducting bands is discussed. This requires a modificationof the Ginzburg-Landau free energy expression as described in Sec. 2.2.1. Multi-component superconductivity can allow for a new type of superconductivity knownas type-1.5 distinct from type-1 and type-2 as discussed in Sec. 2.2.3.

2.2.1 Two-Component Ginzburg-Landau Theory

In a multicomponent system where electrons from different energy bands contributeas superconducting charge carriers there is an order parameter ψi for each compo-nent. In the case of a two-component system there is a ψ1 and a ψ2 for the each ofthe two condensates. This leads to a new free energy expression replacing Eq. 2.7which also includes different αi and βi for the two condensates as well as a term dueto interband Josephson coupling between the condensates. The free energy densityin a two-component system is

f = 12

∑

i=1,2

[

|(∇ + iqA)ψi|2 +(

2αi + βi|ψi|2)

|ψi|2]

+

+ 12

(∇ × A)2 − η|ψ1||ψ2| cos(ϕ2 − ϕ1), (2.22)

where η is the strength of the interband coupling and ψi = |ψi|eiϕi . A derivationof such a model can be found for example in [9]. The free energy in Eq. 2.22 canalso include higher order coupling terms of the form η2|ψ1|2|ψ2|2 as well as mixedgradient terms which also cause a coupling between condensates. The studies inthis thesis are however limited to the terms in Eq. 2.22. Effects of other couplingterms are studied in [10].

When studying two-component superconductors the free energy integral to beminimized is

F =

∫

V

fdV, (2.23)

where V is the volume of the system and f is as in Eq. 2.22. The Minimizationshould be done with respect to both order parameters as well as the magneticvector potential A. This could be done by variational methods giving three coupled

9

CHAPTER 2. THEORY

differential equations but in this thesis numerical minimization is implemented asdescribed in Sec. 3.1.

Similarly to the one component case the different terms in Eq. 2.22 correspondto kinetic energy density

fkin =1

2

∑

i=1,2

|(∇ + iqA)ψi|2 , (2.24)

potential energy density

fpot =1

2

∑

i=1,2

(

2αi + βi|ψi|2)

|ψ|2 (2.25)

and magnetic energy density

fB =1

2(∇ × A)

2=

1

2B2. (2.26)

Now there is however also an interband Josephson coupling energy density

fcoup = −η|ψ1||ψ2| cos(ϕ2 − ϕ1). (2.27)

Again the kinetic energy density can be divided into two terms in a similar way asdone in Eq. 2.6 so

fkin =1

2

∑

i=1,2

|(∇ + iqA)ψi|2 =

=1

2

∑

i=1,2

[

(∇|ψi|)2+ |ψi|2 (∇ϕ+ qA)

2]

. (2.28)

The second term in Eq. 2.28

fJ =1

2

∑

i=1,2

|ψi|2 (∇ϕ+ qA)2

(2.29)

is due to the kinetic energy of supercurrents while

fgrad =1

2

∑

i=1,2

(∇|ψi|)2(2.30)

is an increase in the kinetic energy due to gradients in the order parameter magni-tude. In total there are five contributions to the free energy and the contributionsof these terms fJ, fgrad, fmag, fpot and fcoup are studied in Sec. 4.4.

In the single-component model α < 0 is required in the superconducting state.In the two band model it is possible to have an induced passive band with αi > 0but non-zero |ψi| > 0 contributing to the superconductivity due to coupling.

10


2.2.2 Vortices in Two-Component Systems

Similarly to the one-component case, the two-component model described aboveallows for vortex solutions where both order parameters ψi go to zero at somepoint and there is a phase winding

∮

γ

∇ϕi · dl = 2πNi (2.31)

for each condensate i = 1, 2. As described in [11, 12] a vortex in a two-componentsystem does not, in contrast to the one-component case, necessarily contain a mag-netic flux quantized in integer multiples of Φ0. Instead it is possible to have flux infractional values of Φ0 depending on the values N1 and N2 as well as the relativedensities of the two condensates.

A two-component superconductor can have strictly repulsive vortex interactionsif it is type-2 or attractive vortex interactions if it is type-1. As described in thenext section it is also possible to have one component with type-1 behavior and onecomponent with type-2 behavior so that the competition between these results ina new type of superconductivity with non-monotonic vortex interactions.

2.2.3 Type-1.5 Superconductivity

In a two-component model as the one described in the previous section it is impos-sible to parametrize the system by a single parameter κ = λ

ξsince there are now

three length scales. These are the penetration depth and the coherence lengths ofthe two superconducting condensates which are denoted λ, ξ1 and ξ2 respectively.It should be mentioned that ξi is strictly speaking only a coherence length in asystem with no coupling between the components. It is now possible for one con-densate to be in the type-1 regime while the other is type-2 allowing for new neithertype-1 nor type-2 behavior as suggested in [3].

In the one-component there is a repulsive attraction due to electromagneticinteraction and an attractive interaction due to interaction between vortex cores.This is true also in the two-component case but now there are two attractive con-tributions related to each of the condensates. In [10, 13] it is mentioned that theasymptotic long-range behavior of the interaction between two vortices with dis-tance r in a two-component model is

U(r) = 2π

[

q20K0(

r

λ) − q2

1K0(r

ξ1

) − q22K0(

r

ξ2

)

]

, (2.32)

where q0, q1 and q2 are some constants and K0 is a Bessel function. This interactionpotential is similar to that in Eq. 2.19 except now there are two attractive termsrelated to the two condensates while there is still one repulsive term related tothe magnetic vector potential. The range of these interactions will depend on thecorresponding length scales λ, ξ1 and ξ2. If one of the coherence lengths is thegreatest length scale so the corresponding interaction dominates at large distances

11

CHAPTER 2. THEORY

there will be a long-range attraction and the corresponding condensate shows atype-1 behavior. If the other condensate on the other hand exhibits type-2 behaviorand dominates at short distances there is a short-range repulsion resulting in a non-monotonic interaction between vortex pairs.

The penetration depth in a two-component system is λ =(

λ−21 + λ−2

2

)

−1

2 where

λi =√

βi

q2|αi| . In the special case of U(1)×U(1) symmetry where there is no coupling

between condensates, the coherence lengths are ξi = 1√2|αi|

. In the general case the

coherence lengths are more difficult to define. For the possibility of non-monotonicpairwise vortex interactions to exist, the requirement given in [3, 13] is

ξ1 <√

2λ < ξ2, (2.33)

so one condensate is type-1 and the other is type-2. In this regime it is possibleto have a long-range attractive but short-range repulsive interaction between vor-tex pairs with a minimum in the interaction energy at some equilibrium distance.This allows for a new semi-Meissner state additional to the Meissner and vortexstates [3]. In this semi-Meissner state, instead of the regular lattice observed intype-2 superconductors, more complicated vortex configurations are allowed andfor example stripes or cluster formations can appear [4, 5, 13–15]. This type ofbehavior with irregular distribution of vortices was observed experimentally in aclean sample of MgB2 with electrons from two energy bands forming superconduct-ing condensates with one being in the type-1 regime and the other in the type-2regime [4]. Fig. 2.1 is taken from [4] and shows the difference in a type-2 systemssuch as NbSe2 with vortices in an Abrikosov lattice and a type-1.5 system MgB2

with vortices tending to form more complicated patterns with clusters or stripes aswell as empty regions. Further experimental studies of type-1.5 superconductivityare presented in [14,16] and for a summary of the theory see [13].

It has been shown in [5] that the existence of a relatively strong, repulsive, non-pairwise three-body interaction in some type-1.5 systems can contribute to changingthe equilibrium vortex configurations from large clusters into more stripe-like pat-terns or division into smaller clusters. In particular if the three-body interactionis strong compared to the attractive binding energy in the pairwise interaction,then the non-pairwise interactions can be of importance. Fig. 2.2 shows how vor-tices in a system with weak non-pairwise interactions in 2.2a group together in acluster while vortices in a system with stronger non-pairwise interactions in 2.2bform stripes. This non-trivial effect to equilibrium vortex configurations motivatesa further study of non-pairwise vortex interactions which is the purpose of thisthesis.

12


Figure 2.1: Experimentally observed vortex configurations in MgB2 and NbSe2 aswell as numerical results for vortex configurations in type-1.5 and type-2 systemsin (a)-(d). (e) and (f) show experimental and numerical distributions of nearestneighbors P as function of vortex distance a. Figure taken from [4]. Resultsindicate type-1.5 behavior in MgB2.

13

CHAPTER 2. THEORY

(a) System with weakthree-body interaction.

(b) System with stronger three-body interaction.

Figure 2.2: Equilibrium vortex configurations in two different type-1.5 systems.The system in (a) has a weaker three-body interaction relative to the binding en-ergy of the pairwise interaction while the system in (b) has a stronger three-bodyinteraction relative to the pairwise binding energy. Figure taken from [5].

14

Chapter 3

Numerical Method

The numerical method used for minimizing the free energy in Eq. 2.22 is essentiallythat used in the first part of [5] where some description can also be found. Softwarefor the numerical energy minimization was provided by Johan Carlström. Themethod used is a finite difference minimization as described in Sec. 3.1 and itminimizes the Ginzburg-Landau free energy for a given vortex configuration. Thisis done for different vortex configurations as described in Sec. 3.2 in order tocalculate interaction energies.

3.1 Energy Minimization Method

3.1.1 Discretization and Minimization

The system is discretized on rectangular grid with lattice spacing h and grid sizeN = Nx ×Ny. The purpose is to minimize the free energy in Eq. 2.22 and Eq. 2.23with respect to the order parameters ψi and the magnetic vector potential A. Thevariables are discretized by a finite difference method with derivatives calculated as

f ′i =

f(i+ 1) − f(i)

h. (3.1)

A starting guess is made in which it is set that the order parameters should bezero at the vortex positions, magnetic field should be non-zero at vortex positionsand there should be a given phase winding around the vortices. The free energy isminimized by a Newton-Raphson method until convergence is reached as discussedin Sec. 3.1.2. Free boundary conditions are used as it minimizes boundary effects.It is important to use a large enough grid size so that all vortices are at sufficientdistance from the boundary as they can otherwise escape or be affected by theboundary. To check whether the grid size is large enough the energy of a systemcan be calculated for different grid sizes to see if the result is affected.

15

CHAPTER 3. NUMERICAL METHOD

When the vector potential A which minimizes the free energy has been found,magnetic flux is calculated as a line integral. As B = ∇ × A, by Stokes’ theorem

Φ =

∫

S

B · dS =

∫

S

∇ × A · dS =

∮

γ

A · dl, (3.2)

where γ is the curve enclosing the surface S. Hence the magnetic flux density Bij

at the square ω with corners in i, i+ 1, j and j + 1 can be calculated as

Bij =1

h2

∮

ω

A · dl. (3.3)

Vortex pinning is done by imposing the condition ψi = 0 at the given vortexpositions. For too short distances this method does however not work. At shortdistances vortices will move away from each other but keep small points with ψi = 0at the given vortex positions with the result shown in Fig. 3.1. Fig. 3.1a shows themagnetic flux of three vortices. Fig. 3.1b shows how these vortices have escapedand are not at their original positions but kept small points with ψi = 0 at theoriginal positions due to the pinning constraints. Due to this problem very short-range interactions can not be studied with the numerical method implemented here.By plotting the solutions as done in Fig. 3.1 it is however easy to determine whichresults are reliable.

0.5

1

1.5

2

(a) Magnetic flux density.

0

0.5

1

1.5

(b) Superconducting charge carrier density|ψ|.

Figure 3.1: Figure illustrating the problem with the vortex pinning method forshort distances. Densities of magnetic flux and superconducting charge carriers fora system with three vortices at short distances.

3.1.2 Convergence

Convergence can be determined by minimizing the energy for an initial system sizeN1 = Nx1 × Ny1 and lattice spacing h1 until the energy is seen to converge for

16

3.2. VORTEX CONFIGURATIONS AND CALCULATION OF INTERACTION

ENERGIES

this system size. More grid points are then interpolated, typically an increase by afactor two in both x and y-directions, to give a new system size N2 = Nx2 × Ny2

and lattice spacing h2 without changing the physical system size Lx = hi(Nxi − 1),Ly = hi(Nyi − 1). This gives a sequence of energy values E1 = E(h1), E2 = E(h2),etc. and convergence can be examined by the value

C =Ei − Ei+1

Ei

. (3.4)

Interpolation was done so that Nx and Ny were doubled until grid sizes in theorder of N ≥ 107 were reached. This typically resulted in convergence so thatC < 10−5. For more discussion on convergence and numerical errors with datafrom simulations see Appendix B.

3.2 Vortex Configurations and Calculation of Interaction

Energies

This section describes the choices of vortex configurations and how the variousinteraction energies are calculated for these configurations. The case with twovortices is simple and shortly mentioned in Sec. 3.2.1. Sec. 3.2.2 covers choicesof three vortex configurations and the calculation of three-body interactions. InSec. 3.2.3 the method for studying four-body interactions is described. Sec. 3.2.4describes the choice of five vortex configurations and how five-body interactionenergies are calculated.

3.2.1 Two Vortex Configurations

To calculate the energy of a system with two vortices simply choose a distance Rand place the two vortices for example at coordinates (R

2, 0) and (− R

2, 0). Let E(R)

denote the total energy of such a system and let E1 be the energy of a system withonly a single vortex. The pairwise interaction energy, E2(R), for a vortex pair withdistance R is

E2(R) = E(R) − 2E1. (3.5)

By first calculating the total energy of a single vortex and then the total energy ofa vortex pair, the pairwise interaction energy can be calculated by Eq. 3.5. Thisenergy was calculated for a number of different distances R to give the pairwiseinteraction energy as a function of vortex distance R.

3.2.2 Three Vortex Configurations

For a configuration of three vortices there are also three inter-vortex distances R1,R2 and R3. In order to calculate interaction energies for various configurationsa number of values were assigned to R1 and then the same set of values wereassigned to R2 and R3. The interaction energy was then calculated for all different

17


combinations of R1, R2 and R3 which fulfill the triangle inequality. To avoidredundant calculations it was further reduced such that R1 ≤ R2 ≤ R3. For eachfixed value of R1 the interaction energy of the three vortex configuration can beplotted as a function of the position of the third vortex.

Calculations are also done for a scaling of an equilateral triangle which leavesonly one degree of freedom and represents the case R1 = R2 = R3. The interactionenergy can then be studied as function of the triangle side length.

The vortices were placed so that for each value of R1 the first two vortices werefixed at (R

2, 0) and (− R

2, 0). For each pair of values of R2 and R3 the coordinates

(x3, y3) of the third vortex can then be calculated. If the first vortex is placed inthe first quadrant so x3, y3 ≥ 0, the coordinates are

(x3, y3) =

(

1

2R1

(R23 −R2

2),

√

R23 − 1

4R21

(R23 +R2

1 −R22)2

)

. (3.6)

After placing the three vortices in the grid, the total energy of the system iscalculated and denoted E(R1, R2, R3). If also the energy E1 of a single vortex isknown, the total interaction energy of the three vortex system with a given set ofvortex distances can be calculated as

Eint(R1, R2, R3) = E(R1, R2, R3) − 3E1. (3.7)

The three-body interaction energy of a given configuration is

E3(R1, R2, R3) = Eint(R1, R2, R3) − E2(R1) − E2(R2) − E2(R3). (3.8)

This can be calculated by first finding the pairwise interactions according to Eq.3.5 and the total interaction of three vortices according to Eq. 3.7.

3.2.3 Four Vortex Configurations

In this section the configurations of four vortices and calculations of four-body in-teraction are described. The four-body interaction is the difference between thetotal interaction energy of four vortices and the sum of pairwise and three-bodyinteractions. This is more complicated and computationally demanding than thecase of three vortices due to more degrees of freedom. Hence four-vortex config-urations are limited to a square configuration with side length R. The four-bodyinteraction energy is then studied as a function of R.

From a configuration of four vortices in a square, such as that in Fig. 3.2a, it ispossible to pick out three vortices in four different ways. These three vortices willbe in a triangle configuration with side lengths R, R and

√2R as shown in Fig.

3.2b. There are also four ways of picking a pair of vortices with distance R as wellas two different ways of picking a pair of vortices with distance

√2R.

Let Etot(R) denote the total energy of a system with four vortices in a squarewith side length R and let E1 denote the energy of a system containing only a single

18

3.2. VORTEX CONFIGURATIONS AND CALCULATION OF INTERACTION

ENERGIES

x

x

x

x

R

(a) Four vortices in a square with side R.

x x

x

��

��

R

R

√2R

(b) Three of the four vortices in (a).

Figure 3.2: There are four possible ways of picking three vortices in a triangle suchas the one in (b) out of the four vortices in (a).

vortex as in previous sections. The total interaction energy of the system is

Eint(R) = Etot(R) − 4E1. (3.9)

Let E3(R) = E3(R,R,√

2R) denote the three-body interaction energy of a vortextriangle as that in Fig. 3.2b. Let E2(R) be the interaction energy of a vortex pairwith distance R. E3(R) can be calculated using Eq. 3.8. Each of the four differentvortex triplets which can be picked out of the total four vortices contributes withE3(R) to the total interaction energy. Each of the four pairs with distance R

contributes with E2(R) and each of the two pairs with distance√

2R contributeswith E2(

√2R) to the total interaction energy. The four-body interaction energy is

E4(R) = Eint(R) − 4E3(R) − 4E2(R) − 2E2(√

2R). (3.10)

This is the method used to calculate the four-body interaction energies in Sec. 4.2.

3.2.4 Five Vortex Configurations

This section describes configurations of five vortices chosen to study five-body inter-actions and how these are calculated. Results from these calculations are presentedin Sec. 4.3.

The study of interactions between five vortices is limited to a configuration wherefour are in a square with side

√2R and one is in the middle so the nearest neighbor

distance is R as shown in Fig. 3.3a. From the five vortices it is possible to pick fourvortices in a square with side

√2R as in Fig. 3.3b in one way. This configuration

is labeled 4a. Four vortices in a configuration as in Fig. 3.3b can be picked infour ways and this configuration is labeled 4b. Configurations of three vortices asin Fig. 3.3d and Fig. 3.3e can be picked in four ways each and configurationsas in Fig. 3.3f can be picked in two ways. Let E4a(R) and E4b(R) be the four-body interaction of configuration 4a and 4b respectively. Let E3a(R), E3b(R) andE3c(R) be the three-body interaction of configurations 3a, 3b and 3c respectively.

19


Furthermore, let E2(R) be the pairwise interaction energy of a vortex pair withdistance R. Among the five vortices, there are four pairs with distance R, fourpairs with distance

√2R and two pairs with distance 2R. If the total interaction

energy of the five vortices in Fig. 3.3a is Etot(R), the five-body interaction energyE5(R) is

E5(R) = Etot(R)−E4a−4E4b−4E3a−4E3b−2E3c−4E2(R)−4E2(√

2R)−2E2(2R).(3.11)

This is how five-body vortex interactions are calculated in Sec. 4.3.

v

v

v

v

vR

(a) Five vortices with one in thecenter of a square.

v

v

v

v

��

@@

@@

��

√2R

(b) Vortex configuration 4a.

v

v

vv��

@@

R

√2R

(c) Vortex configuration 4b.

v

vv@@

R

√2R

(d) Vortex configuration 3a.

v

v

v��

@@

2R

√2R

(e) Vortex configuration 3b.

v vvR R

(f) Vortex configuration 3c.

Figure 3.3: Five vortices with one in the center of a square with side√

2R so thatnearest neighbor distance is R and the configurations of three and four vorticeswhich can be picked from the five.

Note that even when limiting the case of five vortices so that there is onlyone degree of freedom as done here, the energies of several configurations of two,three and four vortices must be determined before the five-body interaction canbe calculated. Hence it is computationally more demanding than the case of threeor four vortices also when removing degrees of freedom. Furthermore, since thecalculated energies of all configurations of two, three and four vortices are usedin calculating the five-body interaction, errors add up and the numerical error isgreater in the results of five-body interaction energies.

20

Chapter 4

Results

This chapter presents results of the numerical simulations. Sec. 4.1 contains plotsof three-body interaction energies, Sec. 4.2 contains results for four-body interac-tion energies and in Sec. 4.3 some results for five-body interactions are presented.Three-body interaction energies are studied for one and two-component type-1 andtype-2 superconductivity, one-component critical κ superconductors as well as two-component type-1.5 superconductors in Sec. 4.1.1 to Sec. 4.1.4. A significantthree-body interaction energy is found for all cases except that with critical κ.Four-body interaction results are presented for single-component type-1 and type-2systems as well as a two-component type-1.5 system while the five-body interactionis examined only for a two-component type-1.5 system. Sec. 4.4 contains plots ofenergy densities and describes how the various energy terms contribute to interac-tions. Results are presented in the units described in Appendix A, except energywhich is given in terms of the energy E1 of a single vortex in the system studied.

4.1 Three-Body Interaction Energy

This section contains results of three-body interaction energies as function of theposition of a third vortex when a first vortex pair is fixed at a distance R1 from eachother at coordinates (± R1

2, 0). Results are also presented for three-body interactions

as a function of side length R in an equilateral triangle. Results are presented forsingle and two-component type-1 superconductors in Sec. 4.1.1, single and two-component type-2 superconductors in Sec. 4.1.3 and for two-component type-1.5superconductors in Sec. 4.1.4. The three-body interaction energy is also examinedfor a system of critical κ but as explained in Sec. 4.1.2 the interaction is found tobe zero in this case.

21

CHAPTER 4. RESULTS

4.1.1 Type-1 Three-Body Interaction Energy

Fig. 4.1 shows the total interaction energy, the sum of pairwise interaction energiesand the three-body interaction energy of a single-component type-1 system. Thethird row is hence the difference between the first two rows. The interaction energyis shown as a function of the position of the third vortex when a first vortex pair isplaced with a distance R1 from each other at coordinates (± R1

2, 0). The parameters

used are α = −1, β = 1 and q = 2.5. This corresponds to a penetration depth

λ =

√

β

q2|α| =

√

1

2.52 · 1= 0.4 (4.1)

and a coherence length

ξ =1

√

2|α|=

1√2

≈ 0.71. (4.2)

The Ginzburg-Landau parameter is

κ =

√

2β

q2=

√

2

2.52≈ 0.6 < κc =

√

1

2. (4.3)

Hence these values represent type-1 superconductivity.The graphs in Fig. 4.1 and later also Fig. 4.5 and Fig. 4.9 have been made using

a triangle-based cubic interpolation from about one hundred data points. Data isabsent for vortex separations shorter than R = 0.6 so in a circle of radius 0.6 aroundthe positions of the first two vortices, the data is entirely due to interpolation andcan not be considered reliable. This is due to the problems with the pinning methodwhich does not allow studying short vortex distances as mentioned in Sec. 3.1.1.

Fig. 4.2 shows contour plots of the same data as in Fig. 4.1g - 4.1i. What canbe noted from the results in Fig. 4.1 and Fig. 4.2 is that the three-body interactionis repulsive when all vortices are close together but also has an attractive regionat longer distance. Both the pairwise interaction and the total interaction arehowever attractive at any distance. The plots for different values of R1 show howthe maximum magnitude of the three-body interaction decreases with increasingR1.

Fig. 4.3 shows interaction energies for two type-1 systems with three vortices inan equilateral triangle. The left column shows the three-body interaction energy ofsuch a triangle as function of side length R as well as pairwise interaction energy ofa single vortex pair with distance R. The right column shows the total interactionenergy of the system as well as the sum of pairwise interactions. The system in Fig.4.3a and Fig. 4.3b shows a single-component system with the same parametersas those used in 4.1. The system in Fig. 4.3c and Fig. 4.3d is a two-componentsystem with parameters α1 = −1.1, β1 = 1.0, α2 = −1.2, β2 = 1.0, η = 0 andq = 2.3. These plots show similar behavior as that in Fig. 4.1 with attractive total

22

4.1. THREE-BODY INTERACTION ENERGY

−20

2−2

02

−0.1

−0.05

0

xy

Ene

rgy

(a) Total interaction. R1 = 1.2.

−20

2−2

02

−0.1

−0.05

0

xy

Ene

rgy

(b) Total interaction. R1 = 1.8.

−20

2−2

02

−0.1

−0.05

0

xy

Ene

rgy

(c) Total interaction. R1 = 2.4.

−20

2−2

02

−0.1

−0.05

0

xy

Ene

rgy

(d) Sum of pairwise interactions.R1 = 1.2.

−20

2−2

02

−0.1

−0.05

0

xy

Ene

rgy

(e) Sum of pairwise interactions.R1 = 1.8.

−20

2−2

02

−0.1

−0.05

0

xy

Ene

rgy

(f) Sum of pairwise interactions.R1 = 2.4.

−20

2

−20

2

0

0.01

0.02

0.03

xy

Ene

rgy

(g) Three-body interaction.R1 = 1.2.

−20

2

−20

2

0

0.01

0.02

0.03

xy

Ene

rgy

(h) Three-body interaction.R1 = 1.8.

−20

2

−20

2

0

10

20

x 10−3

xy

Ene

rgy

(i) Three-body interaction.R1 = 2.4.

Figure 4.1: Total interaction energy, sum of pairwise interaction energies and three-body interaction energy for three vortices in a single-component type-1 supercon-ductor with a first vortex pair placed at a distance R1 from each other. Ginzburg-Landau parameters are α = −1, β = 1 and q = 2.5.

23

CHAPTER 4. RESULTS

x

y

−2 −1 0 1 2

−2

−1

0

1

2

0

5

10

15

20

25

30

(a) R1 = 1.2.

x

y

−2 −1 0 1 2

−2

−1

0

1

2

0

5

10

(b) R1 = 1.8.

x

y

−2 −1 0 1 2

−2

−1

0

1

2

−6

−4

−2

0

2

(c) R1 = 2.4.

Figure 4.2: Three-body interaction energy for three vortices in a single-componenttype-1 superconductor with a first vortex pair placed at a distance R1 from eachother. Ginzburg-Landau parameters are α = −1, β = 1 and q = 2.5. Energy scaleis 10−3 of a single vortex energy.

and pairwise interactions but short-range repulsive and long-range attractive three-body interaction. The two-component system shows a more stronger attractive partin the three-body interaction than the single-component system.

The results in Fig. 4.1 and Fig. 4.3 both show pairwise and total interactionsbeing attractive at all distances while the three-body interaction is shown to berepulsive for short distances but attractive for larger distances. That the pair-wise interaction is attractive is consistent with the analytical results discussed inSec. 2.1.1. The short-range behavior of the three-body interaction in the single-component system being opposite of the pairwise interaction so that it brings downthe magnitude of the total interaction compared to the sum of pairwise interac-tions is also consistent with studies of three-body interactions in [6]. Fig. 4.1 andFig. 4.2 however provide a more general picture of the three-body interaction insingle-component type-1 superconductivity than that given in [6] as more generalconfigurations are studied. The results in [6] also do not mention any attractiveregion in the three-body interaction. This attractive region in the three-body in-teraction is seen in all results presented for type-1 superconductivity here and it isespecially strongly present in the two-component system shown in Fig. 4.3c andFig. 4.3d.

Fig. 4.4 shows how the different terms in the free energy contribute to the totaland three-body interactions in the same single-component type-1 system as in Fig.4.1, Fig. 4.3a and Fig. 4.3b. Again it is shown as function of the side length R inan equilateral triangle. As described in Sec. 2.1, magnetic energy is

FB =

∫

V

1

2|∇ × A|2dV, (4.4)

kinetic energy is

Fkin =

∫

V

1

2|(∇ + iqA)ψ|2 dV (4.5)

24


0 1 2 3 4−0.2

−0.1

0

0.1

0.2

R

Ene

rgy

Pairwise InteractionThree−Body Interaction

(a) Pairwise and three-body interaction.

0 1 2 3 4−0.5

−0.4

−0.3

−0.2

−0.1

0

R

Ene

rgy

Total InteractionSum of Pair Interactions

(b) Total and sum of pairwise interactions.

0 1 2 3 4−1

−0.5

0

0.5

R

Ene

rgy


(c) Pairwise and three-body interaction.

0 1 2 3 4−2.5

−2

−1.5

−1

−0.5

0

R

Ene

rgy


(d) Total and sum of pairwise interactions.

Figure 4.3: Plots of interaction energies as function of side length R in an equilateraltriangle for various type-1 systems. The pairwise energy is for a single pair withdistance R and three-body interaction is for an equilateral triangle with side lengthR. (a) and (b) is a single-component system with parameters α = −1.0, β = 1.0and q = 2.5. (c) and (d) is a two-component system with parameters α1 = −1.1,β1 = 1.0, α2 = −1.2, β2 = 1.0, η = 0 and q = 2.3.

25

CHAPTER 4. RESULTS

and potential energy is

Fpot =

∫

V

(

α|ψ|2 +β

2|ψ|4

)

dV, (4.6)

while the total energy is the sum of the above contributions. Fig. 4.4a shows thethree-body interaction while Fig. 4.4b shows the total interaction.

0 1 2 3 4−0.1

0

0.1

0.2

0.3

R

Ene

rgy

TotalKineticMagneticPotential

(a) Three-body interaction.

0 1 2 3 4−1

−0.5

0

0.5

R

Ene

rgy


(b) Total interaction.

Figure 4.4: Plots of the different contributions to the interaction energy as functionof side length R in an equilateral triangle for a single-component type-1 systemwith parameters α = −1.0, β = 1.0 and q = 2.5. (a) shows three-body interactionwhile (b) shows the total interaction.

Fig. 4.4 shows that all the different energy terms contribute significantly to boththe total interaction as well as the three-body interaction. This indicates that allthe different terms in the free energy are affected by the insertion of another vortexdifferently from the addition of a single vortex energy. Hence the optimal solutionsfor both ψ and A must be affected by the insertion of another vortex in a waydifferent from superposition of single vortex solutions and pairwise interactions. Itcan also be noted that the kinetic energy contributes with attraction to the totalinteraction while potential and magnetic energy causes repulsion. This is true alsofor type-2 and type-1.5 systems as will be seen in Sec. 4.1.3 and Sec. 4.1.4. Hencewhat differs the different types of superconductivity is not how the terms contributeto interactions but which of the terms dominate. This was also suggested in Sec.2.1.2.

4.1.2 Critical κ Three-Body Interaction Energy

It can be shown analytically that the pairwise interaction energy is zero between avortex pair in a superconductor with κ at the critical value κc = 1√

2as mentioned in

26


Sec. 2.1.1. This is consistent with numerical results obtained here by calculationsof interaction energies in a critical κ superconductor and also used for error analysisin Appendix B. However, there could still exist a three-body interaction betweenvortices in a system with critical κ. Numerical results obtained here suggest thatno such three-body interaction exists in a critical κ system, at least within theorder of 10−5E1 where E1 is the energy of a single vortex. The case studied is asingle-component system with parameters α = −1, β = 1 and q = 2. This gives apenetration depth

λ =

√

β

q2|α| =

√

1

22 · 1= 0.25, (4.7)

a coherence length

ξ =1

√

2|α|=

1√2

≈ 0.71 (4.8)

and Ginzburg-Landau parameter

κ =

√

2β

q2=

√

2

22=

√

1

2= κc. (4.9)

4.1.3 Type-2 Three-Body Interaction Energy

Fig. 4.5 shows the total interaction, sum of pairwise interaction and three-body in-teraction energy of a single-component type-2 system with a first vortex pair placedat a distance R1 from each other. The interaction energy is shown as a function ofthe position of the third vortex. Fig. 4.6 shows the three-body interaction of thissystem as contour plots. The parameters used here are α = −1, β = 1 and q = 1.5.This gives a penetration depth

λ =

√

β

q2|α| =

√

1

1.52 · 1≈ 0.67 (4.10)

and a coherence length

ξ =1

√

2|α|=

1√2

≈ 0.71. (4.11)

The Ginzburg-Landau parameter is

κ =

√

2β

q2=

√

2

1.52≈ 0.9 > κc =

√

1

2. (4.12)

Hence these values represent type-2 superconductivity.Fig. 4.7 shows interaction energies for various type-2 systems with three vortices

in an equilateral triangle. The left column shows the three-body interaction energyof such a triangle as function of side length R as well as pairwise interaction energy

27

CHAPTER 4. RESULTS

−20

2−2

02

0

0.05

0.1

xy

Ene

rgy


−20

2−2

02

0

0.05

0.1

xy

Ene

rgy


−20

2−2

02

0

0.05

0.1

xy

Ene

rgy


−20

2−2

02

0

0.05

0.1

0.15

xy

Ene

rgy


−20

2−2

02

0

0.05

0.1

0.15

xy

Ene

rgy


−20

2−2

02

0

0.05

0.1

0.15

xy

Ene

rgy


−20

2

−20

2

−0.04

−0.02

0

xy

Ene

rgy


−20

2−2

02

−0.04

−0.02

0

xy

Ene

rgy


−20

2

−20

2

−0.02

−0.01

0

xy

Ene

rgy


Figure 4.5: Total interaction energy, sum of pairwise interaction energies and three-body interaction energy for three vortices in a type-2 superconductor with a firstvortex pair placed at a distance R1 from each other. Ginzburg-Landau parametersare α = −1, β = 1 and q = 1.5

28


of a single vortex pair with distance R. The right column shows the total interactionenergy of the system as well as the sum of pairwise interactions. The system in Fig.4.7a and Fig. 4.7b is a single-component system with the same parameters as thoseused in Fig. 4.5. The system in Fig. 4.7c and Fig. 4.7d is a two-component systemwith parameters α1 = −0.8, β1 = 1.5, α2 = −2.1, β2 = 2.0, q = 1.4 and η = 0 sotwo type-2 condensates without coupling. The system in Fig. 4.7e and Fig. 4.7f isa two-component system with parameters α1 = −0.8, β1 = 1.5, α2 = 2.1, β2 = 2.0,q = 1.4 and η = 7.0.

x

y

−2 0 2−3

−2

−1

0

1

2

3

−0.04

−0.03

−0.02

−0.01

0

(a) R1 = 1.2.

x

y

−2 0 2−3

−2

−1

0

1

2

3

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

(b) R1 = 1.8.

x

y

−2 −1 0 1 2

−2

−1

0

1

2

−10

−8

−6

−4

−2

0

2

(c) R1 = 2.4. Energy scale is10−3 of a single vortex energy.

Figure 4.6: Three-body interaction energy for three vortices in a single-componenttype-2 superconductor with a first vortex pair placed at a distance R1 from eachother. Ginzburg-Landau parameters are α = −1, β = 1 and q = 1.5.

In Fig. 4.5 - 4.7 it is seen that all the type-2 systems studied exhibit repulsivepairwise and total interactions consistent with known analytical and experimentalresults [7]. The three-body interaction on the other hand is at all distances at-tractive causing a reduction in the total interaction energy compared to the sumof pairwise interaction as seen in the right column of Fig. 4.7. This is consistentwith results presented for interactions between three vortices in a single-componenttype-2 system in [6] which however only presents results for a scaling of an equi-lateral triangle while more general configurations are studied here. Fig. 4.7 alsoindicates that similar results holds for two-component type-2 systems.

The system in Fig. 4.7a and Fig. 4.7b is the same single-component systempreviously studied in Fig. 4.5 and Fig. 4.2. Here the pairwise interaction is slightlystronger than the three-body interaction but the three-body interaction is stillstrong enough to show a significant difference between total and sum of pairwiseinteractions. The uncoupled two-component system in Fig. 4.7c and Fig. 4.7d has arelatively stronger three-body interaction compared to the pairwise interaction andhence possesses the greatest relative difference between total interaction and sumof pairwise interactions. The three-body interaction is expected to be of greatestimportance in such a system. The two-component system with one passive band,shown in Fig. 4.7e and Fig. 4.7f, has the strongest total interaction. This is howevermainly due to the strong pairwise interaction so the three-body interaction, which

29

CHAPTER 4. RESULTS

is relatively weak compared to the pairwise interaction, is expected to be of lessimportance.

The repulsive interaction between vortices in a type-2 superconductor is knownto result in the formation of an Abrikosov lattice in a system of many vortices [7].This is normally a triangular lattice which minimizes the free energy of the system.An interesting question to ask is whether a relatively strong attractive three-bodyinteraction, such as that seen in Fig. 4.7c and Fig. 4.7d, can cause a non-trivialstructural change in the equilibrium configuration of many vortices in a type-2system and stabilize another vortex configuration than the triangular lattice usuallyobserved.

Fig. 4.8 shows how the different terms in the free energy contribute to thetotal and three-body interactions in the same single-component type-2 system asin Fig. 4.5 and Fig. 4.7a - 4.7b. Energy is again shown as function of side lengthR in an equilateral triangle. As in the type-1 case all the different terms contributesignificantly to both three-body interaction and total interaction.

4.1.4 Type-1.5 Three-Body Interaction Energy

Fig. 4.9 shows the total interaction, sum of pairwise interactions and three-bodyinteraction energy of a two-component type-1.5 system with a first vortex pairplaced at a distance R1 from each other. The interaction energy is shown as afunction of the position of the third vortex. The parameters used are α1 = −1.0,β1 = 1.0, α2 = 3.0, β2 = 0.5, q = 1.5 and η = 7.0. As mentioned in Sec. 3.1.1, thenumerical method does not work when vortices are at too short distances from eachother and data is unavailable for vortex distances shorter than R = 0.7. Hence incircular regions of radius 0.7, centered at (± R1

2, 0), data is only due to interpolation.

Fig. 4.10 shows contour plots of the three-body interaction for the same data.Fig. 4.11 shows interaction energies for two type-1.5 systems with three vortices

in an equilateral triangle. The left column shows the three-body interaction energyof such a triangle as function of side length R as well as pairwise interaction energyof a single vortex pair with distance R. The right column shows the total interactionenergy of the system as well as the sum of pairwise interactions. The system inFig. 4.11a and Fig. 4.11b is the same as in Fig. 4.9. The system in Fig. 4.11cand Fig. 4.11d is a two-component system with parameters α1 = −1.0, β1 = 1.0,α2 = −0.0625, β2 = 0.25, q = 1.41 and η = 0.

All of the type-1.5 systems studied here show a pairwise and total interactionwhich is short-range repulsive and long-range attractive while the three-body inter-action is at all distances repulsive. These observations are consistent with resultsin [5] but a wider range of vortex configurations have been studied here, giving amore complete picture of the three-body interaction. In Fig. 4.9 and Fig. 4.10 itcan be seen how the maximum magnitude of the three-body interaction decreaseswith increasing R1.

Among the systems shown in 4.11 there is no significant change in the strengthof the three-body interaction energy. There is however a change in the strength

30


0 1 2 3 4−0.2

−0.1

0

0.1

0.2

0.3

R

Ene

rgy



0 1 2 3 40

0.2

0.4

0.6

0.8

R

Ene

rgy



0 1 2 3 4−0.4

−0.2

0

0.2

0.4

R

Ene

rgy



0 1 2 3 40

0.2

0.4

0.6

0.8

R

Ene

rgy



0 1 2 3 4−0.5

0

0.5

1

R

Ene

rgy


(e) Pairwise and three-body interaction.

0 1 2 3 40

0.5

1

1.5

2

2.5

R

Ene

rgy


(f) Total and sum of pairwise interactions.

Figure 4.7: Interaction energies as function of side lengthR in an equilateral trianglefor type-2 systems. Pairwise interaction is given for a single pair with distance Rand three-body interaction for an equilateral triangle with side R. Parameters areα = −1.0, β = 1.0, q = 1.5 in (a) - (b), α1 = −0.8, β1 = 1.5, α2 = −2.1, β2 = 2.0,η = 0, q = 1.4 in (c) - (d) and α1 = −0.8, β1 = 1.5, α2 = 2.1, β2 = 2.0, η = 7.0,q = 1.5 in (e) - (f).

31

CHAPTER 4. RESULTS

0 1 2 3 4−0.1

−0.05

0

0.05

0.1

0.15

R

Ene

rgy



0 1 2 3 4−0.6

−0.4

−0.2

0

0.2

0.4

R

Ene

rgy



Figure 4.8: Plots of the different contributions to the interaction energy as functionof side length R in an equilateral triangle for a single-component type-2 systemwith parameters α = −1.0, β = 1.0 and q = 1.5. (a) shows three-body interactionwhile (b) shows the total interaction.

of the attractive binding energy in the pairwise and total interactions which isparticularly weak in the system shown in the top row of Fig. 4.11. In a system witha three-body interaction which is relatively large compared to the binding energy ofthe pairwise interaction, the three-body interaction is expected to be particularlyimportant. It is suggested in [5] that in such systems non-pairwise interactionscan affect the equilibrium vortex configurations. This motivates including alsoa three-body interaction term when approximating the total interaction betweenmany vortices instead of using only pairwise interactions. For this approach to workwell it is however necessary that the situation does not change significantly whenadding more than three vortices to the system so that the inclusion of the three-body interaction is indeed an improvement over using only the pairwise interaction.Hence systems of four and five vortices are studied in Sec. 4.2 and Sec. 4.3.

Fig. 4.12 shows results of Monte Carlo simulations done by Karl Sellin with aninteraction potential calculated by curve fitting using the data from Fig. 4.9. Fig.4.12a shows the vortex configuration in a system without three-body interactionsso that the total interaction is just the sum of all pairwise interactions. Fig. 4.12bshows the result of including also a three-body interaction term with the samestrength as in Fig. 4.9. Fig. 4.12c illustrates what happens if the strength ofthis three-body interaction is significantly increased. What can be observed is thatthe three-body interaction favors more elongated vortex clusters and stripe-likepatterns compared to the configurations resulting from only pairwise interactions.The difference between Fig. 4.12a and Fig. 4.12b is however small while Fig. 4.12cexhibits very clear stripes. This indicates that three-body interactions can indeedhave an effect on the vortex configurations although the three-body interactions in

32


−20

2

−20

2

0

5

10

x 10−3

xy

Ene

rgy


−20

2

−20

2

0

5

10

x 10−3

xy

Ene

rgy


−20

2

−20

2

0

5

10

x 10−3

xy

Ene

rgy


−20

2

−20

2

−2

0

2

4

6

x 10−3

xy

Ene

rgy


−20

2

−20

2

−2

0

2

4

6

x 10−3

xy

Ene

rgy


−20

2

−20

2

−2

0

2

4

6

x 10−3

xy

Ene

rgy


−20

2

−20

2−1

0

1

2

3x 10

−3

xy

Ene

rgy


−20

2

−20

2−1

0

1

2

3x 10

−3

xy

Ene

rgy


−20

2

−20

2−1

0

1

2

3x 10

−3

xy

Ene

rgy


Figure 4.9: Total interaction energy, sum of pairwise interaction energies and three-body interaction energy for three vortices in a type-1.5 superconductor with a firstvortex pair placed at a distance R1 from each other. Ginzburg-Landau parametersare α1 = −1.0, β1 = 1.0, α2 = 3.0, β2 = 0.5, q = 1.5 and η = 7.0.

33

CHAPTER 4. RESULTS

−2 −1 0 1 2−2

−1

0

1

2

0.5

1

1.5

2

(a) R1 = 1.2.

−2 −1 0 1 2−2

−1

0

1

2

0

0.5

1

1.5

2

(b) R1 = 1.4.

−2 −1 0 1 2−2

−1

0

1

2

0.5

1

1.5

(c) R1 = 1.6.

−1 0 1

−1.5

−1

−0.5

0

0.5

1

1.5

0

0.2

0.4

0.6

0.8

1

1.2

(d) R1 = 1.8.

Figure 4.10: Three-body interaction energy for three vortices in a single-componenttype-1.5 superconductor with a first vortex pair placed at a distance R1 from eachother. Ginzburg-Landau parameters are α1 = −1.0, β1 = 1.0, α2 = 3.0, β2 = 0.5,q = 1.5 and η = 7.0. Energy scale is 10−3 of a single vortex energy.

systems studied here might be too small to make a clear difference.Fig. 4.13 shows how the different terms in the free energy contribute to the

total and three-body interactions in the same two-component type-1.5 system asin Fig. 4.9 and Fig. 4.11a - Fig. 4.11b. Again the energy is shown as a functionof side length R in an equilateral triangle. As described in Sec. 2.2.1 the energycontributions are kinetic energy

Fkin =1

2

∫

V

∑

i=1,2

|(∇ + iqA)ψi|2 dV, (4.13)

34


1 1.5 2 2.5 3 3.5−0.02

0

0.02

0.04

0.06

R

Ene

rgy



0 1 2 3 4−0.05

0

0.05

0.1

0.15

0.2

R

Ene

rgy



0 2 4 6 8−0.1

−0.05

0

0.05

0.1

R

Ene

rgy



0 2 4 6 8−0.2

−0.1

0

0.1

0.2

0.3

R

Ene

rgy



Figure 4.11: Plots of interaction energies as function of side length R in an equilat-eral triangle for two different two-component type-1.5 systems. The pairwise energyis for a single pair with distance R and three-body interaction is for an equilateraltriangle with side length R. (a) and (b) is the same system as that in Fig. 4.9.Parameters for (c) and (d) are α1 = −1.0, β1 = 1.0, α2 = −0.0625, β2 = 0.25,q = 1.41 and η = 0.

35

CHAPTER 4. RESULTS

(a) Vortex configuration with-out three-body interaction.

(b) Vortex configuration withweak three-body interaction.

(c) Vortex configuration withstrong three-body interaction.

Figure 4.12: Plots showing vortex configurations for a type-1.5 system with andwithout three-body interactions. Figures provided by Karl Sellin.

potential energy including Josephson coupling

Fpot =1

2

∫

V

∑

i=1,2

(

2αi + βi|ψi|2)

|ψ|2 − η|ψ1||ψ2| cos(ϕ2 − ϕ1)

dV, (4.14)

and magnetic energy

FB =1

2

∫

V

(∇ × A)2

dV. (4.15)

Fig. 4.13 exhibits similar traits as previously observed in the type-1 and type-2 systems. As expected, kinetic energy contributes with attraction to the totalattraction while magnetic and potential energy contributes with repulsion. In thethree-body interaction, the situation is opposite with attraction in the magnetic andpotential energy but repulsion in the kinetic energy. This was also observed in type-2 superconductivity in Fig. 4.8. Hence, in each of the terms, the sum of pairwiseinteractions overestimates the magnitude of the total interaction. The situation wasdifferent in the type-1 system in Fig. 4.4 where all three-body interaction energycontributions were non-monotonic. As in the type-1 and type-2 cases, all threeterms contribute significantly to the three-body interaction although the terms allgo to zero more rapidly here.

4.2 Four-Body Interaction Energy

This section contains the results calculated for the four-body interaction energy ofa square configuration of four vortices as described in Sec. 3.2.3. Results for single-component type-1 and type-2 and two-component type-1.5 superconductivity arepresented.

36

4.2. FOUR-BODY INTERACTION ENERGY

0 1 2 3 4−0.05

0

0.05

0.1

R

Ene

rgy



0 1 2 3 4−0.6

−0.4

−0.2

0

0.2

0.4

R

Ene

rgy



Figure 4.13: Plots of the different contributions to the interaction energy as functionof side length R in an equilateral triangle for a two-component type-1.5 system withparameters α1 = −1.0, β1 = 1.0, α2 = 3.0, β2 = 0.5, q = 1.5 and η = 7.0. (a)shows three-body interaction while (b) shows the total interaction.

Fig. 4.14 shows interaction energies of a configuration of four vortices in asquare with side length R. The left column shows interaction energy for a singlepair with distance R, a single triangle with sides R, R and

√2R as well as the

four-body interaction of the square. The right column shows the total interactionof the square, the sum of pairwise interactions as well as the sum of pairwise andthree-body interactions. The top row is the same single-component type-1 systempreviously studied with parameters α = −1, β = 1 and q = 2.5. The bottom row isa single-component type-2 system with parameters α = −1, β = 1 and q = 1.5. Fig.4.15 shows the same type of data as Fig. 4.14 but for a two-component type-1.5system with parameters α1 = −1.0, β1 = 1.0, α2 = 3.0, β2 = 0.5, q = 1.5 andη = 7.0.

The first thing that can be seen in Fig. 4.14 and Fig. 4.15 is that the four-body interaction is significantly smaller than pairwise and three-body interactions,except in the type-1 case where the four-body interaction is more prominent. It isin the type-1 case short-range attractive and long-range repulsive. In the type-2and type-1.5 cases it is difficult to say anything definite about the behavior of thefour-body interaction as it is not significantly larger than the order of numericalerrors. The most important conclusion to be drawn from Fig. 4.14 and Fig. 4.15 isthat the four-body interaction is smaller than pairwise and three-body interactionsexcept possibly in the type-1 case where it is of similar size as the three-bodyinteraction. In the right column of Fig. 4.14 and Fig. 4.15 it is seen however that,in all cases, the approximation of the total interaction by the sum of pairwise andthree-body interactions is significantly better compared to that of only the sum ofpairwise interactions, especially at short distances. It can be seen that adding also

37

CHAPTER 4. RESULTS

1 1.5 2 2.5 3 3.5−0.06

−0.04

−0.02

0

0.02

0.04

R

Ene

rgy

Pairwise InteractionThree−Body InteractionFour−Body Interaction

(a) Type-1.

1 1.5 2 2.5 3 3.5−0.4

−0.3

−0.2

−0.1

0

R

Ene

rgy

Sum of Pair InteractionsPairwise and Three−BodyTotal Interaction

(b) Type-1.

1 1.5 2 2.5 3 3.5−0.05

0

0.05

0.1

0.15

R

Ene

rgy


(c) Type-2.

1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

R

Ene

rgy


(d) Type-2.

Figure 4.14: Interaction energies for four vortices in a square with side R. Leftcolumn shows interaction for a single pair with distance R, three-body interactionfor a single triangle with sides (R,R,

√2R) and four-body interaction for the square.

Right column shows the total interaction, sum of pairwise interactions and sumof pairwise and three-body interactions. The top and bottom rows show single-component type-1 and type-2 systems with parameters α = −1, β = 1 and q = 2.5and α = −1, β = 1 and q = 1.5 respectively.

38

4.3. FIVE-BODY INTERACTION ENERGY

0.5 1 1.5 2 2.5 3−1

0

1

2

3x 10

−3

R

Ene

rgy


(a) Type-1.5.

0.5 1 1.5 2 2.5 3−0.01

0

0.01

0.02

0.03

R

Ene

rgy


(b) Type-1.5.

Figure 4.15: Interaction energies for four vortices in a square with side R. (a)shows interaction for a single pair with distance R, three-body interaction for asingle triangle with sides (R,R,

√2R) and four-body interaction for a square. (b)

shows the total interaction, sum of pairwise interactions and sum of pairwise andthree-body interactions. Parameters are α1 = −1.0, β1 = 1.0, α2 = 3.0, β2 = 0.5,q = 1.5 and η = 7.0.

the four-body interaction does not make as big difference. This implies that thethree-body interaction is of greater importance than four-body interactions.

4.3 Five-Body Interaction Energy

This section provides results for five-body interaction energies for a configurationof five vortices with four vortices in a square and one in the middle, as describedin Sec. 3.2.4. Results are presented for a two-component type-1.5 system and themain conclusion is that the five-body interaction in this case seems to have a smallsignificance compared to the pairwise and three-body interactions.

Fig. 4.16 shows the interaction energies of five vortices in the configurationdescribed in Sec. 3.2.4. The system is two-component type-1.5 with parametersα1 = −1.0, β1 = 1.0, α2 = 3.0, β2 = 0.5, q = 1.5 and η = 7.0, which is the samesystem as for which both three and four-body interactions have been studied inprevious sections. Fig. 4.16a shows the interaction energy for a single pair withdistance R, three-body interaction energy for a single triangle with distances R, Rand

√2R, four-body interaction for a square with side R and five-body interaction

for the previously mentioned configuration of five vortices. Fig. 4.16b shows totalinteraction for the configuration of five vortices, sum of pairwise interactions forthe same configuration as well as sum of pairwise and three-body interactions.

As the magnitudes of the four and five-body interaction energies are relatively

39

CHAPTER 4. RESULTS

small, while the errors are relatively large, it is difficult to say much about thebehavior of these interactions. What can be said about the data illustrated in Fig.4.16 is that both the four and five-body interactions are relatively small comparedto pairwise and three-body interactions. Furthermore, in Fig. 4.16b it is shown thatincluding three-body interactions is, at least for small distances, a significant im-provement in approximating the total interaction compared to using only pairwiseinteractions. Including four and five-body interactions, on the other hand, makesa small difference. In particular the region where the total interaction energy hasa minimum should be of particular interest. Here the three-body interaction doescause a significant shift in the energy while four and five-body interactions do not.

0.5 1 1.5 2 2.5 3−1

0

1

2

3x 10

−3

R

Ene

rgy

Pairwise InteractionThree−Body InteractionFour−Body InteractionFive−Body Interaction

(a) five-body, four-body, three-body and pairwiseinteractions.

0.5 1 1.5 2 2.5 3−0.01

0

0.01

0.02

0.03

R

Ene

rgy

Total InteractionSum of Pair InteractionsPair and Three−Body

(b) Total interaction, sum of pairwise interac-tions and sum of pairwise and three-body inter-actions.

Figure 4.16: Interactions in a system of five vortices in the configuration described inSec. 3.2.4 for a two-component type-1.5 system with Ginzburg-Landau parametersα1 = −1.0, β1 = 1.0, α2 = 3.0, β2 = 0.5, q = 1.5 and η = 7.0.

4.4 Energy Densities

To provide a better insight into the vortex interactions this section contains plotsof energy densities. As described in Sec. 2.2 the energy density consists of currentenergy density

fJ =1

2

∑

i=1,2

|ψi|2 (∇ϕ+ qA)2, (4.16)

kinetic energy density associated with gradients in the magnitude of the orderparameter

fgrad =1

2

∑

i=1,2

(∇|ψi|)2, (4.17)

40

4.4. ENERGY DENSITIES

magnetic energy density

fB =1

2(∇ × A)

2, (4.18)

potential energy density

fpot =1

2

∑

i=1,2

(

2αi + βi|ψi|2)

|ψ|2 (4.19)

and interband Josephson coupling energy density

fcoup = −η|ψ1||ψ2| cos(ϕ2 − ϕ1). (4.20)

All these terms can contribute to the interaction between vortices and interactionenergy densities can be calculated in a similar way as when calculating interactionenergies as described in Sec. 3.2. The only difference here is that the positions ofvortices need to be taken into account.

Figures 4.17-4.21 show the different energy densities and interaction energydensities for the type-1.5 system with parameters α1 = −1.0, β1 = 1.0, α2 = 3.0,β2 = 0.5, q = 1.5 and η = 7.0 previously studied. All vortex distances are R = 1.Such plots have also been produced for type-1 and type-2 systems but are notincluded for brevity as these plots show the same traits. The similarity of the plotsfor type-1, type-2 and type-1.5 systems indicate how interactions in these systemsare essentially the same with attractive and repulsive contributions from the samesources but with the difference in which contributions are the dominant ones.

Fig. 4.17 shows the magnetic energy density fB = 12

(∇ × A)2

= 12

|B|2 whichis positive and non-zero at the vortex positions where the magnetic flux density isnon-zero. In a system with several vortices the magnetic flux densities from thedifferent vortices add up while the magnetic energy density increases as the squareof the magnetic flux density. Hence addition of vortices makes the magnetic energydensity increase more than just the addition of a single vortex energy density andthis results in a positive, i.e. repulsive, contribution to the interaction as seen inFig. 4.17c and Fig. 4.17e. Fig. 4.17f illustrates how adding pairwise interactionsoverestimates the total interaction and hence leaves a three-body interaction energydensity which is mainly of opposite sign of the total interaction energy density.

The pairwise interaction energy density and total interaction energy densitybetween three vortices is repulsive also for the potential energy in Fig. 4.21 whileit is mainly attractive for the both of the kinetic energy densities as well as theJosephson coupling energy densities. This is consistent with earlier observations inFig. 4.4, Fig. 4.8 and Fig. 4.13. From (f) in each of Fig. 4.17-4.21 it seems thatthe three-body interaction energy density tends to be mainly of opposite sign ofthe pairwise interaction as the sum of pairwise interactions overestimates the totalinteraction between three vortices.

41

CHAPTER 4. RESULTS

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

5

10

15

20

25

30

35

(a) Energy density of a single vortex.

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

5

10

15

20

25

30

35

(b) Energy density of a vortex pair.

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

1

2

3

4

5

(c) Interaction energy density of a vortexpair.

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

5

10

15

20

25

30

35

(d) Energy density of three vortices.

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

2

4

6

8

10

(e) Total interaction energy density of threevortices.

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

−4

−3

−2

−1

0

(f) Three-body interaction energy density.

Figure 4.17: Magnetic energy densities.

42


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

2

4

6

8

10

12


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

2

4

6

8

10

12


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

2

4

6

8

10


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 −6

−5

−4

−3

−2

−1

0


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 −0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2


Figure 4.18: Kinetic energy densities of the form fgrad = 12(∇|ψ|)2.

43

CHAPTER 4. RESULTS

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

2

4

6

8

10

12


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

2

4

6

8

10


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

1

2

3

4

5

6

7

8

9


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

−6

−5

−4

−3

−2

−1

0


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4

−1

−0.5

0

0.5

1


Figure 4.19: Current energy densities.

44


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

5

10

15

20


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

5

10

15

20


x

y

−4 −2 0 2 4

−4

−2

0

2

4

−3

−2

−1

0

(c) Interaction energy density of a vortex pair.

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

5

10

15

20


x

y

−4 −2 0 2 4

−4

−2

0

2

4−6

−5

−4

−3

−2

−1

0

(e) Total interaction energy density of three vor-tices.

x

y

−4 −2 0 2 4

−4

−2

0

2

4

−0.4

−0.2

0

0.2


Figure 4.20: Josephson coupling energy densities.

45

CHAPTER 4. RESULTS

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4−12

−10

−8

−6

−4

−2

0


x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4−12

−10

−8

−6

−4

−2

0


x

y

−4 −2 0 2 4

−4

−2

0

2

4 0

1

2

3

4

(c) Interaction energy density of a vortex pair.

x

y

−4 −2 0 2 4

−4

−3

−2

−1

0

1

2

3

4−12

−10

−8

−6

−4

−2

0


x

y

−4 −2 0 2 4

−4

−2

0

2

40

2

4

6

8

(e) Total interaction energy density of three vor-tices.

x

y

−4 −2 0 2 4

−4

−2

0

2

4

−1

−0.8

−0.6

−0.4

−0.2

0

0.2


Figure 4.21: Potential energy densities.

46

Chapter 5

Conclusions

Non-pairwise interactions between vortices in single and two-component supercon-ductors of type-1, type-2 as well as type-1.5 have been been studied in the contextof Ginzburg-Landau theory. In particular three-body interactions have been thor-oughly studied for single and two-component type-1 and type-2 systems as well astwo-component type-1.5 systems. A summary of results are shown in table 5.1.

Type-1 systems show an attractive pairwise interaction and a three-body in-teraction which is short-range repulsive but long-range attractive. The pairwiseattraction is consistent with known analytical results [2,7]. The short-range repul-sion in the three-body interaction of single-component type-1 systems is consistentwith results in [6] where long-range attraction is however not mentioned and onlythe special case of three vortices in an equilateral triangle is studied. This long-range attractive part in the three-body interaction seems to be especially strong incertain two-component type-1 systems which are not discussed in [6].

Type-2 systems exhibit a pairwise interaction which is at all distances repulsivewhile the three-body interaction is attractive. This implies that using the sum ofpairwise interactions to approximate the total interaction in a system of severalvortices will overestimate the repulsion. Again the pairwise repulsion is consistentwith previously known results and the attractive three-body interaction is consis-tent with results in [5, 6]. In addition to covering both one and two-componentsystems, more general configurations of vortices than previously investigated havebeen studied to give a more complete picture of the three-body interactions in bothtype-1 and type-2 systems.

The results for type-1.5 systems show short-range repulsive but long-range at-tractive pairwise interaction while the three-body interaction is at all distancesrepulsive. This is consistent with previous theory and studies [3, 5, 13]. Here, how-ever, a more comprehensive study of the three-body interactions has been presentedcovering a wider range of vortex configurations.

Studying the different terms in the free energy consisting of kinetic, magneticand potential energy shows that all terms contribute significantly to both three-

47

CHAPTER 5. CONCLUSIONS

Pairwise Interaction Three-Body InteractionType-1 Attractive at all distances. Short-range repulsive and long-

range attractive.Type-2 Repulsive at all distances. Attractive at all distances.Type-1.5 Short-range repulsive and long-

range attractive.Repulsive at all distances.

Table 5.1: Summary of results regarding pairwise and three-body interactions indifferent superconducting systems.

body interactions as well as total interaction in all cases studied. The tendencyalso seems to be that the different terms contribute with opposite sign to the three-body interaction compared to the pairwise interaction. This implies that the sumof pairwise interactions tends to overestimate the magnitude of the total interactionin each of the terms.

In addition to the three-body interaction, systems of four and five vortices havebeen studied for certain cases to examine also four and five-body interactions. Theseresults indicate that three-body interactions are of significantly greater importancethan the four or five-body interactions. Adding three-body interactions gives aconsiderably better approximation of total interactions at short distances comparedto using only pairwise interactions while including also four or five-body interactionsdoes not greatly improve the approximation. It can not be said however, whetherthis changes for systems with more vortices.

The existence of non-pairwise interactions illuminates that the usual treatmentof interactions between several vortices by addition of pairwise interactions is notcorrect. The study was largely motivated by the finding in [5] that type-1.5 systemswith strong non-pairwise interaction can exhibit equilibrium vortex configurationsdifferent from those seen in systems with weaker non-pairwise interactions clearlyindicating that there are situations where the treatment of vortex interactions byaddition of pairwise interactions is not sufficient. The effects of non-pairwise inter-actions in the type-1.5 case is further supported by studies through Monte Carlosimulations done by Karl Sellin as mentioned in Sec. 4.1.4 and illustrated in Fig.4.12. It is however not clear from these results that the three-body interactions inthe systems studied here are strong enough to cause a significant effect on vortexconfigurations.

The use of only pairwise interactions in approximating the total interaction isinaccurate not only in the type-1.5 case but also in type-1 and type-2 systems.Hence an interesting question to ask is whether the non-pairwise interactions couldhave non-trivial effects to equilibrium vortex configurations also in these cases. Forexample it could be of interest to investigate whether attractive three-body inter-actions could have any effect on the triangular Abrikosov lattice usually observedin a type-2 system.

As mentioned in Sec. 3.1.1 the numerical method used here implements a vor-

48

tex pinning method which does not allow studying very short-range interactions.A way of improving and extending the studies done here could therefore includeimplementing a numerical method which allows for the study of interactions atshorter distances to see what effects non-pairwise interactions have in this region.

A difficulty which arises when studying interactions between several vortices isthe large number of degrees of freedom. The interaction between a vortex pairdepends only on one distance R being the separation between the two vortices. Ina system of N vortices the total interaction energy depends on

(

N2

)

different vortexdistances Rij , 1 ≤ i ≤ N , i < j ≤ N , i 6= j between vortex i and vortex j.This quickly increases to large numbers of configurations which need to be studiedin order to obtain the general picture without limitations to special cases.

The idea used here of discussing total interactions in terms of sums of two, threeand four-body interactions etc. is of course not entirely correct. Within Ginzburg-Landau theory the interaction between vortices appear because inserting a vortexwill change the optimal order parameter and vector potential which minimize theGinzburg-Landau free energy in a way so that the energy densities change differentlyfrom simply adding the free energy of a single vortex as discussed some in Sec. 4.4.The free energy of the system should in general depend on the total configurationof all vortices and cannot be described simply as a superposition neither of justpairwise nor pairwise and a limited order of many-body interactions. However, theresults in Sec. 4.2 and Sec. 4.3 indicate that using also a three-body interaction canbe a significant improvement over using only a superposition of pairwise interactionswhile adding four or five-body interactions should not be of as great importance.

49

Appendix A

Units

Ginzburg-Landau theory is often described in the standard SI or Gaussian units.The two-component Ginzburg-Landau free energy in Eq. 2.22 is

f = 12

∑

i=1,2

[

1Mi

|(~∇ + iqA)ψi|2 +(

2αi + βi|ψi|2)

|ψi|2]

+

+ 12µ0

(∇ × A)2 − η|ψ1||ψ2| cos(ϕ2 − ϕ1) (A.1)

in SI-units or

f = 12

∑

i=1,2

[

1Mi

∣

∣

(

∇ + i qcA)

ψi

∣

∣

2+(

2αi + βi|ψi|2)

|ψi|2]

+

+ 18π

(∇ × A)2 − η|ψ1||ψ2| cos(ϕ2 − ϕ1) (A.2)

in Gaussian units. Mi is the mass of the superconducting charge carriers in eachsuperconducting condensate, c is the speed of light and everything else is as definedin Sec. 2.1 and Sec. 2.2.1. A description of the Ginzburg-Landau theory in Gaussianunits as well as description of conversion to SI-units can be found in [7]. In order tosimplify equations and obtain more accurate numerical calculations, reduced unitsas those described below are used instead of SI or Gaussian units. These units aresame as those in for instance [5] and a description of the units and conversion fromGaussian units can also be found in [10].

To go from SI-units to the new reduced units, scaled quantities are introducedas

A = ~A ψi =

√

µ0

Mi

ψi

f =µ0

~2f αi =

Mi

~2αi

βi =M2

i

µ0~2βi η =

√M1M2

~2η. (A.3)

51

APPENDIX A. UNITS

Inserting these quantities into Eq. A.1 yields

f = 12

∑

i=1,2

[

∣

∣

(

∇ + iqA)

ψi

∣

∣

2+(

2αi + βi|ψi|2)

|ψi|2]

+

+ 12

(

∇ × A)2 − η|ψ1||ψ2| cos(ϕ2 − ϕ1), (A.4)

which upon dropping every tilde is identical to Eq. 2.22.In the single-component case ignore all quantities with an index i = 2 as well

as η and drop the index on quantities referring to the first condensate. The simplecase of α = −1 and β = 1 yields λ = 1

qand ξ = 1√

2. This case is studied in several

parts of this thesis and the Ginzburg-Landau parameter is

κ =λ

ξ=

√2

q. (A.5)

The Ginzburg-Landau parameter now depends only on the charge, to which it isinversely proportional. There is a critical charge qc = 2 for which κ = κc = 1√

2

and type-1 superconductivity corresponds to q > 2 while type-2 superconductivitycorresponds to q < 2.

52

Appendix B

Convergence and Numerical Errors

Here follows a discussion of errors in the numerical calculations which have beendone. First in terms of the convergence also mentioned in Sec. 3.1.2 and then interms of the pairwise interaction energy calculated for a critical κ-system.

B.1 Convergence

As also described in Sec. 3.1.2 energy minimization is done for a certain gridsize N1 = Nx1 × Ny1 until an energy value E1 is reached and does not change inseveral thousand iterations. More points are then interpolated giving a new gridsize N2 = Nx2 ×Ny2 and energy value E2. This is repeated to give a set of energyvalues E1, E2, E3, ..., Ei, Ei+1. Convergence is determined by

C =Ei − Ei+1

Ei

. (B.1)

This value should give an idea of the order of magnitude of the numerical errors.Alternatively this could be expressed in terms of the energy E0 of a single vortexas this is the unit of energy used in this thesis so

C =Ei − Ei+1

E0

. (B.2)

To give an idea of numbers used and actual values of errors some examples aregiven below.

To produce the data used in Fig. 4.14 and Fig. 4.15 the initial grid-size usedwas N1 = Nx1 ×Ny1 = 181 × 161 = 29141 with lattice spacing h1 = 0.05 giving aphysical size of the system Lx ×Ly = 9×8. The grid size was then doubled in bothx and y-direction five times while keeping the physical size of the system giving sixenergy values E1, E2, ..., E6. In the case of the data of the type-1.5 system in Fig.4.15, the largest value of C was obtained with E6 = 80.8911 and E5 = 80.8907 so

C =80.8911 − 80.8907

80.8907= 5 · 10−6, (B.3)

53

APPENDIX B. CONVERGENCE AND NUMERICAL ERRORS

or in terms of single vortex energy which in this case in reduced units is E0 =20.2360

C =80.8911 − 80.8907

20.2360= 2 · 10−5. (B.4)

For the data of the type-2 case in Fig. 4.14, E5 = E6 in all cases so all that can besaid is that C < 10−5 and C < 10−5. For the type-1 data in Fig. 4.14, the largestvalue of C was obtained with E6 = 2.86616 and E5 = 2.86617 so

C =2.86617 − 2.86616

2.86616= 3 · 10−6. (B.5)

This is the value for a single vortex so in this case C = C. All other results havesimilar accuracy and errors in the same order of magnitude.

B.2 Pairwise Interaction in a Critical κ System

As mentioned in Sec. 2.1.2 and in Sec. 4.1.2, the interaction energy between twovortices in a superconducting system at the Bogomol’nyi point (κ = κc = 1√

2)

should be zero according to analytical results [2, 8]. Numerically calculating theinteraction energy between a vortex pair for a superconducting system with κ = 1√

2

should therefore provide an estimate of the order of magnitude of numerical errors.Fig. B.1 shows numerically calculated values of the interaction energy of a vortexpair placed a distance R from each other in a system with parameters α = −1.0,β = 1.0 and q = 2.0 so that

κ =

√

2β

q2=

1√2. (B.6)

The by magnitude largest value of interaction energy seen in Fig. B.1 is|Eint(R = 3)| = 4 · 10−5 which should give an estimate of the numerical error. Thisis in the same order of magnitude as the values mentioned in Sec. B.1.

54

B.2. PAIRWISE INTERACTION IN A CRITICAL KAPPA SYSTEM

0.5 1 1.5 2 2.5 3−5

−4

−3

−2

−1

0

1x 10

−5

R

Ene

rgy

Figure B.1: Plot of the numerically calculated pairwise interaction energy for asingle-component system with critical κ. Parameters used are α = −1.0, β = 1.0and q = 2.0. Energy is given in terms of the energy E1 of a single vortex.

55

Bibliography

[1] A. A. Abrikosov, “On the magnetic properties of superconductors of the secondgroup,” Soviet Physics JETP, vol. 32, pp. 1442–1452, Dec 1957.

[2] L. Kramer, “Thermodynamic behavior of type-ii superconductors with smallκ near the lower critical field,” Phys. Rev. B, vol. 3, pp. 3821–3825, Jun 1971.

[3] E. Babaev and M. Speight, “Semi-meissner state and neither type-i nor type-iisuperconductivity in multicomponent superconductors,” Phys. Rev. B, vol. 72,p. 180502, Nov 2005.

[4] V. Moshchalkov, M. Menghini, T. Nishio, Q. H. Chen, A. V. Silhanek, V. H.Dao, L. F. Chibotaru, N. D. Zhigadlo, and J. Karpinski, “Type-1.5 supercon-ductivity,” Phys. Rev. Lett., vol. 102, p. 117001, Mar 2009.

[5] J. Carlström, J. Garaud, and E. Babaev, “Semi-meissner state and nonpairwiseintervortex interactions in type-1.5 superconductors,” Phys. Rev. B, vol. 84,p. 134515, Oct 2011.

[6] A. Chaves, F. M. Peeters, G. A. Farias, and M. V. Milošević, “Vortex-vortexinteraction in bulk superconductors: Ginzburg-landau theory,” Phys. Rev. B,vol. 83, p. 054516, Feb 2011.

[7] M. Tinkham, Introduction to superconductivity. McGraw-Hill, Inc., 2nd ed.,1996.

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Nuclear Physics, vol. 24, pp. 449–454, 1976.

[9] A. Gurevich, “Limits of the upper critical field in dirty two-gap superconduc-tors,” Physica C: Superconductivity, vol. 456, no. 1-2, pp. 160 – 169, 2007.

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[14] T. Nishio, V. H. Dao, Q. Chen, L. F. Chibotaru, K. Kadowaki, and V. V.Moshchalkov, “Scanning squid microscopy of vortex clusters in multiband su-perconductors,” Phys. Rev. B, vol. 81, p. 020506, Jan 2010.

[15] V. H. Dao, L. F. Chibotaru, T. Nishio, and V. V. Moshchalkov, “Giant vortices,rings of vortices, and reentrant behavior in type-1.5 superconductors,” Phys.

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