Non-Pairwise Vortex Interactions in Ginzburg-Landau Theory of Superconductivity

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Non-Pairwise Vortex Interactions in Ginzburg-Landau Theory of Superconductivity

ALEXANDER EDSTRÖM

Master of Science Thesis Stockholm, Sweden 2012

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

Supervisor: Egor Babaev

TRITA-FYS 2012:23 ISSN 0280-316X

ISRN KTH/FYS/--12:23--SE

Department of Theoretical Physics School of Engineering Sciences Royal Institute of Technology (KTH) AlbaNova University Center SE-106 91 Stockholm, Sweden Typeset in L^ATEX

Tryck: Universitetsservice US-AB

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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 approximation of the total interaction by summation of pairwise interactions.

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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.

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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. After Abrikosov’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 part in the theory of superconductivity. In addition to being perfect electrical conduc- tors, superconductors exhibit the Meissner effect meaning that they expel magnetic fields. At the position of a vortex, magnetic fields can however penetrate into the superconducting material. Vortices are known to interact with each other and the total interaction between several vortices is often approximated by the sum of pairwise interactions. This is however not exact due to nonlinearity in the equations determining the interactions. The purpose of this thesis is to study to non-pairwise contributions to the interaction between several vortices and in particular to study the three-body interaction defined as the difference between total interaction and sum of pairwise interactions in a system of three vortices.

Abrikosov’s work was based on the Ginzburg-Landau (GL) theory of superconductivity 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 parameter κknown as the Ginzburg-Landau parameter. There is a critical value κc=^√¹₂ and superconductors can be divided into two groups known as type-1 superconductors with κ < ^√¹₂ and type-2 superconductors with κ > ^√¹₂. As mentioned, Abrikosov showed that type-2 superconductors allow for singular points known as vortices and that repulsive interactions between these vortices result in the formation of a regular lattice known as the Abrikosov lattice [1]. A type-1 system does not possess a stable vortex state but it can be shown that vortex interactions in a type-1 system would be attractive [2]. Some description of vortices and vortex interactions can be found in Sec. 2.1.1 and Sec. 2.1.2.

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

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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 behavior separated from both type-1 and type-2 with non-monotonic, short-range repulsive but long-range attractive interactions between vortex pairs. These non-monotonic interactions can lead to formations of more complex equilibrium conﬁgurations such as clusters or stripe patterns instead of the Abrikosov lattices observed in the type-2 case. Experimental observations of such behavior has been done in clean samples of MgB2 by Moshchalkov et. al. in [4] where the term type-1.5 superconductivity was introduced.

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

Also in type-1 and type-2 superconductivity vortex interactions are nonlinear and non-pairwise contributions are expected to appear in systems of several vortices. A limited study of three-body interactions in single-component type-1 and type-2 systems has been presented in [6], but only for two sets of parameters and a scaling of three vortices in an equilateral triangle. Even in single-component superconductivity the complete form of the non-pairwise interactions is therefore still unknown. In addition to studying more general configurations of vortices in single-component superconductivity, the results here will be extended to cover non- pairwise interactions in two-component type-1 and type-2 systems previously not studied. The question will be raised if non-pairwise interactions can affect the equilibrium vortex configurations also in type-1 or type-2 superconductivity in a similar way as they have been suggested to do in type-1.5 superconductivity. For example the repulsive pairwise interaction in a type-2 system normally yields a triangular Abrikosov lattice but the question is whether non-pairwise contributions could 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 interaction energy 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 separated by distances R1, R2 and R3, then the total interaction energy of the system is E_int(R1, R₂, R₃) = Etot(R1, R₂, R₃) − 3E1. The three-body interaction energy E₃(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

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

Uint ≈X

i<j

E2(Rij), (1.2)

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

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

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Chapter 2

Theory

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

2.1 Ginzburg-Landau Theory

The Ginzburg-Landau theory of superconductivity is a phenomenological theory which can be found in standard textbooks such as [7]. It is based on Landau’s theory of second order phase transitions and assumes that, close to the transition temperature, the free energy can be expanded in terms of an order parameter ψ which is zero in the normal state and non-zero in the superconducting state. The density of superconducting charge carriers, n, is related to the order parameter as n= |ψ|². GL theory assumes that the order parameter is small and varies slowly in 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 superconducting charge carriers and A is the magnetic vector potential. α and β are temperature dependent expansion coeﬃcients of the theory and α changes sign at the critical temperature of the superconductor so that it is positive in the normal state and negative in the superconducting state. It is preferable to work in reduced units and rescaling quantities so that ψ =q

M µ0

ψ, A = ~ ˜˜ A, α = _M^~²α, β =˜ ^µ_M⁰^~2²β˜ and f =_µ^~²₀f˜. 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)

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Reduced units as those above are used in the rest of this thesis and further description 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

f_kin= 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|e^iϕ, be rewritten as two terms

fkin= 1

2|(∇ + iqA) ψ|²= 1 2

(∇ + iqA) |ψ|e^iϕ

2=

= 1 2

eîϕ∇|ψ| + i|ψ|eîϕ∇ϕ + iqA|ψ|eîϕ

2=

= 1 2

h(∇|ψ|)²+ |ψ|²(∇ϕ + qA)²i

. (2.6)

The second term, f_J = 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 =

Z

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 to obtain the order parameter ψ and the vector potential A which minimize the energy functional in Eq. 2.7. Taking variational derivatives and minimizing the free energy with 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 law

J= ∇ × B = ∇ × ∇ × A (2.10)

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

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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=

s β

q²|α|, (2.11)

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

ξ= 1

p2|α|, (2.12)

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

κ= λ ξ =

s2β

q². (2.13)

The Ginzburg-Landau equations, Eq. 2.8 and Eq. 2.9, can be rewritten in a form where the only constant appearing in the equations is κ. Hence the behavior of the system is characterized by the value of the Ginzburg-Landau parameter and it turns out that the equations show drastic diﬀerences depending on whether κ is greater then or smaller than a critical value κc= ^√¹₂. 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 field allows for singular points where the order parameter goes to zero and the magnetic field can penetrate into the superconductor. This occurs if the magnetic field is greater then a critical field Hc1, which separates the Meissner state from the so called vortex state, but smaller than a critical field Hc2 where superconductivity is destroyed and the material transitions to the normal state. These points are called vortices and a repulsive interaction results in the formation of a so called Abrikosov lattice in the presence of many vortices. In type-1 superconductors a magnetic field will destroy superconductivity without allowing for a stable vortex state. It can however be shown that there must be an attractive interaction between vortices in a type-1 superconductor [2].

The magnetic ﬂux over a single vortex is Φ =

Z

S∇ × A · dS = I

γ

A· dl, (2.14)

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

A= − J

q²|ψ|²−∇ϕ

q . (2.15)

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Inserting this into Eq. 2.14 gives Φ =

I

γ

− J

q²|ψ|² −∇ϕ q

· 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. Then the magnetic ﬂux depends only on the phase gradient and

Φ = −1 q I

γ∇ϕ · dl = −2πN

q , (2.17)

where it is required that

I

γ∇ϕ · dl = 2πN, (2.18)

with N being an integer for the order parameter ψ = |ψ|eîϕ to be a single-valued complex scalar field. Hence the magnetic flux is quantized in units of Φ0 = ^2π_q , or Φ0= ^2π~_q in 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 energetically favorable to decompose into several vortices with N = 1 so that the phase winding is 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 repulsive electromagnetic interaction and an attractive interaction between the cores. It has been shown in [2] that for large distances the interaction energy between two vortices is

U(r) = 2πc²K0(r) −2πd² κ² K0(√

2κr), (2.19)

where c and d depend on the Ginzburg-Landau parameter κ, K0is a modiﬁed Bessel function and r is the vortex distance. The ﬁrst term is the repulsive electromagnetic interaction and the second term is the core-core attraction. Furthermore it is shown that for the critical value κ = ¹₂, also known as the Bogomol’nyi point, d = √^c

while 2

d < c

√2 for κ > 1

√2 (2.20)

and

d > c

√2 for κ < 1

√2. (2.21)

Hence U = 0 for κ = ^√¹₂, U < 0 for κ < ^√¹₂ and U > 0 for κ > ^√¹₂ so there is an attractive pairwise interaction in the type-1 case, a repulsive pairwise interaction in the type-2 case and no pairwise interaction at the Bogomol’nyi point. Interactions between many vortices are often treated by addition of pairwise interactions which

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2.2. MULTICOMPONENT SUPERCONDUCTIVITY

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

2.2 Multicomponent Superconductivity

This section describes how the theory in the previous section changes when electrons from diﬀerent energy bands can form superconducting condensates. In particular the case with two superconducting bands is discussed. This requires a modiﬁcation of the Ginzburg-Landau free energy expression as described in Sec. 2.2.1. Multi- component superconductivity can allow for a new type of superconductivity known as 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 contribute as superconducting charge carriers there is an order parameter ψifor each compo- nent. In the case of a two-component system there is a ψ1and a ψ2for the each of the two condensates. This leads to a new free energy expression replacing Eq. 2.7 which also includes diﬀerent αiand βifor the two condensates as well as a term due to interband Josephson coupling between the condensates. The free energy density in a two-component system is

f = ¹₂P

i=1,2

h|(∇ + iqA) ψi|²+ 2α_i+ β_i|ψi|² |ψi|²i +

+¹₂(∇ × A)²− η|ψ1||ψ2| cos(ϕ2− ϕ1), (2.22) where η is the strength of the interband coupling and ψi = |ψi|e^iϕⁱ. A derivation of such a model can be found for example in [9]. The free energy in Eq. 2.22 can also include higher order coupling terms of the form η2|ψ1|²|ψ2|² as well as mixed gradient terms which also cause a coupling between condensates. The studies in this thesis are however limited to the terms in Eq. 2.22. Eﬀects of other coupling terms are studied in [10].

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

F = Z

V

fdV, (2.23)

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

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

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

fkin= 1 2

X

i=1,2

|(∇ + iqA) ψi|², (2.24)

potential energy density

fpot=1 2

X

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 as done in Eq. 2.6 so

fkin= 1 2

X

i=1,2

|(∇ + iqA) ψi|²=

= 1 2

X

i=1,2

h(∇|ψi|)²+ |ψi|²(∇ϕ + qA)²i

. (2.28)

The second term in Eq. 2.28 f_J =1

2 X

i=1,2

|ψi|²(∇ϕ + qA)² (2.29)

is due to the kinetic energy of supercurrents while f_grad= 1

2 X

i=1,2

(∇|ψi|)² (2.30)

is an increase in the kinetic energy due to gradients in the order parameter magnitude. In total there are ﬁve contributions to the free energy and the contributions of 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 α_i >0 but non-zero |ψi| > 0 contributing to the superconductivity due to coupling.

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2.2.2 Vortices in Two-Component Systems

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

I

γ∇ϕi· dl = 2πNi (2.31)

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

A two-component superconductor can have strictly repulsive vortex interactions if it is type-2 or attractive vortex interactions if it is type-1. As described in the next section it is also possible to have one component with type-1 behavior and one component with type-2 behavior so that the competition between these results in a 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_ξ since there are now three length scales. These are the penetration depth and the coherence lengths of the 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 a system with no coupling between the components. It is now possible for one condensate to be in the type-1 regime while the other is type-2 allowing for new neither type-1 nor type-2 behavior as suggested in [3].

In the one-component there is a repulsive attraction due to electromagnetic interaction 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 contributions related to each of the condensates. In [10, 13] it is mentioned that the asymptotic long-range behavior of the interaction between two vortices with dis- tance r in a two-component model is

U(r) = 2π

q₀²K0(r

λ) − q²1K0(r

ξ1) − q²2K0(r ξ2

)

, (2.32)

where q0, q1and q2are some constants and K0is a Bessel function. This interaction potential is similar to that in Eq. 2.19 except now there are two attractive terms related to the two condensates while there is still one repulsive term related to the magnetic vector potential. The range of these interactions will depend on the corresponding length scales λ, ξ1 and ξ2. If one of the coherence lengths is the greatest length scale so the corresponding interaction dominates at large distances

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there will be a long-range attraction and the corresponding condensate shows a type-1 behavior. If the other condensate on the other hand exhibits type-2 behavior and 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⁻²₂ ⁻

1 2 where λi=q

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-monotonic pairwise 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 possible to have a long-range attractive but short-range repulsive interaction between vortex 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 vortex states [3]. In this semi-Meissner state, instead of the regular lattice observed in type-2 superconductors, more complicated vortex conﬁgurations are allowed and for example stripes or cluster formations can appear [4, 5, 13–15]. This type of behavior with irregular distribution of vortices was observed experimentally in a clean sample of MgB2with electrons from two energy bands forming superconducting condensates with one being in the type-1 regime and the other in the type-2 regime [4]. Fig. 2.1 is taken from [4] and shows the diﬀerence in a type-2 systems such 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 as well as empty regions. Further experimental studies of type-1.5 superconductivity are 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 changing the equilibrium vortex configurations from large clusters into more stripe-like patterns or division into smaller clusters. In particular if the three-body interaction is 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 vortices in a system with weak non-pairwise interactions in 2.2a group together in a cluster while vortices in a system with stronger non-pairwise interactions in 2.2b form stripes. This non-trivial effect to equilibrium vortex configurations motivates a further study of non-pairwise vortex interactions which is the purpose of this thesis.

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Figure 2.1: Experimentally observed vortex conﬁgurations in MgB2 and NbSe2 as well as numerical results for vortex conﬁgurations in type-1.5 and type-2 systems in (a)-(d). (e) and (f) show experimental and numerical distributions of nearest neighbors P as function of vortex distance a. Figure taken from [4]. Results indicate type-1.5 behavior in MgB2.

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(a) System with weak three-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 energy of the pairwise interaction while the system in (b) has a stronger three-body interaction relative to the pairwise binding energy. Figure taken from [5].

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Chapter 3

Numerical Method

The numerical method used for minimizing the free energy in Eq. 2.22 is essentially that used in the first part of [5] where some description can also be found. Software for the numerical energy minimization was provided by Johan Carlström. The method used is a finite difference minimization as described in Sec. 3.1 and it minimizes the Ginzburg-Landau free energy for a given vortex configuration. This is done for different vortex configurations as described in Sec. 3.2 in order to calculate 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 size N= Nx× Ny. The purpose is to minimize the free energy in Eq. 2.22 and Eq. 2.23 with respect to the order parameters ψi and the magnetic vector potential A. The variables 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 be zero at the vortex positions, magnetic ﬁeld should be non-zero at vortex positions and there should be a given phase winding around the vortices. The free energy is minimized by a Newton-Raphson method until convergence is reached as discussed in 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 sufficient distance from the boundary as they can otherwise escape or be affected by the boundary. To check whether the grid size is large enough the energy of a system can be calculated for different grid sizes to see if the result is affected.

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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 = Z

S

B· dS = Z

S

∇ × A · dS = I

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 I

ω

A· dl. (3.3)

Vortex pinning is done by imposing the condition ψi = 0 at the given vortex positions. For too short distances this method does however not work. At short distances vortices will move away from each other but keep small points with ψi= 0 at the given vortex positions with the result shown in Fig. 3.1. Fig. 3.1a shows the magnetic ﬂux of three vortices. Fig. 3.1b shows how these vortices have escaped and are not at their original positions but kept small points with ψi = 0 at the original 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 which results 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 for short distances. Densities of magnetic ﬂux and superconducting charge carriers for a system with three vortices at short distances.

3.1.2 Convergence

Convergence can be determined by minimizing the energy for an initial system size N1 = Nx1× Ny1 and lattice spacing h1 until the energy is seen to converge for

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3.2. VORTEX CONFIGURATIONS AND CALCULATION OF INTERACTION ENERGIES

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

and lattice spacing h2without 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

E_i . (3.4)

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

3.2 Vortex Configurations and Calculation of Interaction Energies

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

3.2.1 Two Vortex Configurations

To calculate the energy of a system with two vortices simply choose a distance R and place the two vortices for example at coordinates (^R₂,0) and (−^R₂,0). Let E(R) denote the total energy of such a system and let E1be the energy of a system with only a single vortex. The pairwise interaction energy, E2(R), for a vortex pair with distance R is

E2(R) = E(R) − 2E1. (3.5)

By ﬁrst calculating the total energy of a single vortex and then the total energy of a vortex pair, the pairwise interaction energy can be calculated by Eq. 3.5. This energy was calculated for a number of diﬀerent distances R to give the pairwise interaction 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 configurations a number of values were assigned to R1 and then the same set of values were assigned to R2 and R3. The interaction energy was then calculated for all different

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combinations of R1, R2 and R3 which fulfill the triangle inequality. To avoid redundant calculations it was further reduced such that R1≤ R2 ≤ R3. For each fixed value of R1 the interaction energy of the three vortex configuration can be plotted as a function of the position of the third vortex.

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

The vortices were placed so that for each value of R1the first two vortices were fixed at (^R₂,0) and (−^R₂,0). 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 in the first quadrant so x3, y₃≥ 0, the coordinates are

(x3, y3) = 1 2R1

(R²₃− R²2), s

R²₃− 1

4R²₁(R²₃+ R²₁− R²2)²

!

. (3.6)

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

E_int(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 interaction are described. The four-body interaction is the difference between the total interaction energy of four vortices and the sum of pairwise and three-body interactions. This is more complicated and computationally demanding than the case 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-body interaction 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 is possible to pick out three vortices in four different ways. These three vortices will be 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 well as 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 square with side length R and let E1denote the energy of a system containing only a single

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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 such as 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 vortex triangle as that in Fig. 3.2b. Let E2(R) be the interaction energy of a vortex pair with distance R. E3(R) can be calculated using Eq. 3.8. Each of the four diﬀerent vortex triplets which can be picked out of the total four vortices contributes with E₃(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 contributes with 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 interactions and how these are calculated. Results from these calculations are presented in Sec. 4.3.

The study of interactions between ﬁve vortices is limited to a conﬁguration where four 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 four vortices 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 in four ways and this configuration is labeled 4b. Configurations of three vortices as in Fig. 3.3d and Fig. 3.3e can be picked in four ways each and configurations as 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) and E3c(R) be the three-body interaction of configurations 3a, 3b and 3c respectively.

Non-Pairwise Vortex Interactions in Ginzburg-Landau Theory of Superconductivity