Vortex Currents near the Boundary of a Two-Component Superconductor

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Master Thesis

Vortex Currents near the Boundary of a Two-Component Superconductor

Richard Edberg

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

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

Stockholm, Sweden 2017

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Typeset in L ^A TEX

TRITA-FYS 2017:82 ISSN 0280-316X

ISRN KTH/FYS/–17:82-SE Richard Edberg, June 2017 c

Printed in Sweden by Universitetsservice US AB, Stockholm June 2017

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Abstract

Superconductivity has been an active area of research since its discovery more than a century ago. Recently there has also been an increased interest in multicomponent superconductors and the fascinating properties of fractional vortices. This master thesis gives a theoretical investigation of such a system. We discuss the shape of a small cluster of fractional vortices near the boundary of a two-component superconductor. It is found that they can form stable configurations that rest at an equilibrium distance from the boundary. In particular, the vortices can form half-rings that collapse into rings when the external magnetic field is increased.

After this formation the system exhibits a hysteresis effect and does not return to its original half-ring state when the external field returns to its initial value.

Sammanfattning

Supraledning har varit ett aktivt forskningsomr˚ ade ¨ anda sedan dess uppt¨ ackt f¨ or mer ¨ an ett ˚ arhundrade sedan. P˚ a senare tid har det ocks˚ a funnits ett ¨ okat intresse f¨ or multikomponent-supraledare och de fascinerande egenskaperna hos fraktionella virvelstr¨ ommar. Detta masterexamensarbete behandlar p˚ a teoretisk v¨ ag ett s˚ adant system. Vi diskuterar formation av ett litet kluster av fraktionella virvelstr¨ ommar n¨ ara kanten av en tv˚ akomponents-supraledare. Det visar sig att dessa kan bil- da stabila konfigurationer p˚ a ett j¨ amviktsavst˚ and fr˚ an kanten. Specifikt kan vir- velstr¨ ommarna bilda halvringar som kollapsar till ringar d˚ a det externa f¨ altet ¨ okar.

Efter en s˚ adan formation uppvisar systemet en form av hysteresis och ˚ aterv¨ ander inte till halvrings-tillst˚ andet n¨ ar det externa f¨ altet ˚ aterg˚ ar till sitt ursprungsv¨ arde.

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Preface

This thesis is the result of five months of full-time studies from January 2017 to June 2017, for the degree of Master of Science in Engineering Physics, Teknisk Fysik, in the department of physics at KTH, Sweden.

I would like to thank my supervisor Prof. Egor Babaev for providing excellent guidance and support.

I would also like to thank my family for their support and my fellow thesis workers, in particular: Emil Mallmin, Simon Velander, Emil Blomquist, Alexander S¨ oderstrand and

^邓牧笛

for providing stimulating discussion and a good study environment.

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Introduction

“Superconductivity” refers to the property of a material to exhibit zero resistance to electrical currents. Superconductors were discovered by K.Onnes in 1911 who also used the term “supraconductivity”. He had successfully liquefied helium a few years earlier and discovered the phenomenon when it was used to cool mercury to temperatures near absolute zero. The resistance dropped to “practically zero”-as he wrote, when the mercury was cooled below some critical temperature T _c [1].

The zero resistance has gathered much interest due to its diverge applications, ranging in everything from electric power transmission to low signal dispersion which is a key aspect in microwave components, and communications technology [2]. Much research has been invested in the field since the original discovery, and the phenomenon has been found in diverse corners of nature.

An interesting feature of the zero resistance is the ability for a superconductor to sustain a persistent current [16]. In this thesis, we make a theoretical study of such currents in a Ginzburg-Landau model of a two-component type-II superconductor.

In particular we focus on the formation of vortices close to the boundary of a sample, figure 1.1.

Figure 1.1: Vortex current near the boundary of a sample.

1

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2 Chapter 1. Introduction

1.1 Different Superconductors

K.Onnes continued researching different materials and within a few years after his original discovery he had also found superconductivity in lead and tin [4]. By 1932, the phenomena was known to exist in a broad range of elements; Hg, Pb, Sn, Tl, In, Ga, Ta, Ti, Th [3]. – It seemed like as if superconductivity was a fairly common state of matter at low temperatures. Experimentalist had by then also started to look for superconductivity in alloys and found it in binary compounds such as SbSn, Sb ₂ Sn, Cu ₃ Sn and Bi ₅ Tl ₃ [4].

The early superconductors are now termed “type-I superconductors” and show perfect diamagnetic properties in an applied external magnetic field [6]. In 1950 breakthroughs were made with the introduction of the Ginzburg-Landau theory that modeled superconductors with a complex wave-function (discussed further in section 2.1). With it, A. A. Abrikosov was able to theoretically predict the existence of a “type-II superconductor”[7] (1952). The prediction turned out to be correct and a vast number of type-II superconductors have been discovered during the last 60 years [8]. The type-II superconductors differ from the type-I in how they behave in external fields, in particular vortex currents can form in these materials [18].

Conventional superconductivity is either of type-I or type-II. In addition to this there is also the aspect of multicomponent superconductivity, which can give rise to intermediate states such as type-1.5 superconductivity [9]. These systems are currently a very active area of research, displaying interesting phenomena such as phase frustration and skyrmion topological defects [47][10]. A whole range of new phenomena such as those described in [11]-[13] have also been investigated.

The two-component type-II superconductor, which is the central topic of this thesis, is explained to greater detail in section 2.5.

1.2 Chapters

The thesis is structured in the following way:

Chapter 2 gives a short introduction to the theory.

Chapter 3 gives an exact problem statement and derives the governing free energy.

Chapter 4 is devoted to minimizing this energy and understanding the structure of vortex currents.

Chapter 5 gives a short summary.

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Chapter 2

Mean-field theory

Physical systems involving many degrees of freedom are by nature hard to describe even when presented with the underlying deterministic laws. Indeed it is a highly non-trivial task to understand what causes phenomena such as superconductivity given what is known about single electron interaction. Phenomenological and mean- field theories present ways to study simplified models of a system. The essential mean-field approach is to reduce complex many-body problems to averaged systems with fewer degrees of freedom. Simplified models like these have been successful in describing phase transitions and make a reasonable first step in understanding the superconductor transition. [14]

2.1 Ginzburg-Landau theory

In 1950 V. L. Ginzburg and L. Landau introduced the Ginzburg-Landau theory for superconductivity [18]. It was originally a phenomenological model and it postu- lates that there exists a complex order-parameter ¹ ψ describing the superconductor transition. The transition of a material to a superconducting state is modeled by the order-parameter taking a non-zero value. The free energy is then assumed to have a continuous dependence on ψ. Close to the transition, one would write the free energy F of the system in form of a series expansion (2.1) in ψ, truncated at some finite order.

F (T ) = F _n (T ) + a(T )|ψ| ² + 1

2 b(T )|ψ| ⁴ + (. . . ) (2.1) F n is the free energy of the system in absence of superconductivity (ψ = 0). The coefficients a and b are temperature dependent phenomenological parameters, spe- cific to the material. A more complete version of the Ginzburg-Landau theory also allows for spatial variations in the order-parameter and electromagnetic interaction.

1

also called component, gap or condensate

3

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4 Chapter 2. Mean-field theory Writing ψ = ψ(r) and letting A(r) be the electromagnetic vector potential, it is postulated that

F (T ) = F n (T ) + Z

S

B ² 2µ d ³ r +

Z

S

~

2m ^∗ |(~∇ + iqA) ψ(r)| ² + a(T )|ψ| ² + 1

2 b(T )|ψ| ⁴

d ³ r

(2.2)

Here S is the superconducting regime, B = ∇ × A, µ is the magnetic perme- ability, q is the charge of the super-current carriers, m ^∗ is an effective mass and

~ is Planck’s constant divided by 2π. A central aspect of the spatially varying Ginzburg-Landau theory is the super-current flow v s . It is defined as

v s = ~

m ∇θ (2.3)

where m is the mass of the super-current carriers and θ is the phase of the order parameter ψ = |ψ|e ^iθ . Another important aspect is the coherence length ξ. The coherence length gives a length-scale for how quickly ψ can vary in space. Motivated by the solution of (2.2) near the boundary of a superconductor [18], it is defined as

ξ = s

~ ²

2m ^∗ |a(T )| (2.4)

The coherence length is important for all problems related to defects, interfaces and, as in the case of this thesis, vortices and boundaries.

2.2 External magnetic field

An external magnetic field B 0 can be applied to the superconductor. The total magnetic field is then the sum of the individual fields B total = B + B 0 . Since the energy is related to the square of the total field, it will be the sum of the individual energies of the two fields plus a cross term F Ext

F Ext = 1 µ

Z

B · B 0 d ³ r (2.5)

For a constant external field, the free energy F G which can be converted into work is then

F _G = F + F _Ext (2.6)

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2.3. Vortices and a two-dimensional system 5

2.3 Vortices and a two-dimensional system

When a homogeneous external field B 0 = −B 0 e z is applied to a type-II supercon- ductor, the magnetic field penetrates the sample through cylindrical channels [19]

[20]. These are known as vortex cores and super-current flow around these in a cir- cular motion. Since the channels are parallel to the external field, ψ is invariant of z and the system is two-dimensional by nature ² . It is therefore enough to consider the free energy per unit length in the z-direction, which is given by evaluation of the integral in F _G over only the xy-plane. In order to save notation space, we shall write r = (x, y) or r = (x, y, 0) depending on the situation.

v s

B 0

Figure 2.1: Super-current flowing around a flux channel (vortex core) parallel to the external magnetic field. The system is by nature two-dimensional as all physical quantities are invariant along the direction of B ₀ .

An interesting feature of vortices is that they carry quantized flux [21]. The phenomena is explained in Ginzburg-Landau theory by the property that the phase of the order parameter is determined only up to a multiple of 2π. Thus when integrating the super-current flow, which is proportional to the gradient of the phase, around any closed curve γ one obtains the following condition

I

γ

∇θ · dl = 2πN (2.7)

The integer N is determined by the number of vortex cores circulated ³ . It is convenient to introduce the rotated gradient ∇φ defined by the relation ∇θ = e _z × ∇φ. Property (2.7) is then equivalently written as

I

γ

∇φ · dn = 2πN (2.8)

Applying the two-dimensional divergence theorem, this is equivalent to (2.9), where Γ is the domain circulated by γ.

2

Assuming that we may neglect boundary effects in the z direction, this is a good approxima- tions for samples much thicker than ξ

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In principle a single channel could have a phase winding greater than 2π, however these excitations are unstable to splitting into multiple elementary 2π vortices [21]

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6 Chapter 2. Mean-field theory

Z Z

Γ

∆φ d ² r = 2πN (2.9)

By deformation of Γ into small regions around the vortex cores, this relation becomes

∆φ = 2π

N

X

i

δ(r − r _i ) (2.10)

where r i are the positions of the vortex cores. This famous Poisson equation is well known from electrostatics and the solution is a superposition of single vortex solutions φ _i = ln(r − r _i ) + (const).

2.4 The London limit

So far we have allowed the modulus of the order parameter to vary in space |ψ| =

|ψ|(r). An approximation often used when studying superconductors is the London model or hydromagnetostatic regime [21]. In this approximation, the modulus of

|ψ| is assumed to be constant in space. This means that the state of the matter field is described solely by the phase of the order-parameter. In regimes where superconductivity is destroyed, such as boundaries and vortex cores (channels) one then uses a cutoff at length ξ where ψ is set to zero. This allows for much simpler calculations since we do not have to consider in what way ψ decays. The London limit is of course not valid for all systems, but give good predictions when the coherence length is small.

2.5 Two-component superconductors

A multicomponent superconductor has multiple condensates ψ _α that can support super-current flow. They exhibit exotic phenomena which have been studied in recent works such as [12][24][27][47] among others. Ginzburg-Landau theory is naturally extended to these systems simply by summing the free energies of the individual components. In this thesis we focus in particular on two-component superconductors i.e. α = 1, 2. The most relevant example of this phenomenon is found in some crystals when super-current in different bands form independent condensates. These kind of systems have also gathered a lot of attention since the recent discovery of two-component superconductivity in MgB ₂ [22][23].

Just as for single-component systems, vortices appear in multicomponent su-

perconductors when exposed to an external magnetic field [24]. The theory is the

same as for one component, and the individual phases need to satisfy

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2.6. Inter-component interaction 7

I

γ

∇θ α · dl = 2πM α (2.11)

where γ is any curve circling M _α vortex cores in condensate α. Note that in general, the positions and number of vortices are not the same in the two condensates. This leads to the notion of fractional vortices, which refers to the case when there is a non-zero 2π phase winding in only one of the components ⁴ . These are the main topic of this thesis and their interaction is determined by the free energy per unit length in the z-direction:

F (T ) = F _n (T ) + Z

S

B ² 2µ d ² r +

Z

S 2

X

α=1

~

2m ^∗ |(~∇ + iqA) ψ α (r)| ² + a α (T )|ψ| ² + 1

2 b α (T )|ψ| ⁴

! d ² r (2.12) In the London limit at fixed temperature below the transition temperatures, F n , a α

and b α are constant and do not affect the physics of the system. Furthermore, a more complete model also features explicit inter-component interaction. That is;

addition of a phenomenological term coupling ψ 1 to ψ 2 . Up to a constant, the free energy of a two-component superconductor is then

F = 1 2 Z

S 2

X

α=1

|(∇ + iqA) ψ α (r)| ² + B ² + V (ψ 1 , ψ 2 )

!

d ² r (2.13) Here V (ψ ₁ , ψ ₂ ) is the explicit inter-component interaction. Note also that the expression has been normalized by setting µ = ~ = m ^∗ = 1. Moreover, as discussed in section 2.2, when the system is exposed to an external magnetic field, there is an additional contribution F _Ext to the free energy. The motion of the condensates is therefore governed by F G .

F G = 1 2

Z

S 2

X

α=1

|(∇ + iqA) ψ α (r)| ² + B ² + V (ψ 1 , ψ 2 )

! d ² r +

Z

B · B 0 d ² r (2.14)

2.6 Inter-component interaction

There are many different models for the coupling V (ψ 1 , ψ 2 ) between two conden- sates. Perhaps the most famous being the Josephson coupling proposed by B.D.

4

Fractional vortices with multiple phase windings in only one component are unstable to splitting into several vortices with single phase winding [21], just as for single-component superconductors.

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8 Chapter 2. Mean-field theory Josephson in 1962 [25]. Originally derived for quantum tunneling in weak links, it is also an important effect in multi-component systems. Its contribution V J to the free energy in a two-component superconductor was first considered in [26] and can be written as

V J = η

2 (ψ 1 ψ ^∗ ₂ + ψ ₁ ^∗ ψ 2 ) = η|ψ 1 ||ψ 2 | cos(θ 1 − θ 2 ) (2.15) Recent works such as [28] and [27] investigate some of the many fascinating implications this coupling has for two-component systems. In this thesis we also study the Andreev-Bashkin coupling related to dissipation-free drag between super- current components. It was first considered in a phenomenological study [30], but there are also microscopic theories to support this interaction [29]. Other backing for the existence of this drag can be found in [31]-[35].

In terms of free energy, it can be written as

V _D = ν |Im (ψ ^∗ ₁ (∇ + iqA)ψ ₁ ) + Im (ψ ₂ ^∗ (∇ + iqA)ψ ₂ )| ² (2.16) This interaction has been studied to great extent in [45], which also emphasizes the existence of stable skyrmions in such systems.

2.7 Skyrmions

“Skyrmions are topologically protected field configurations with particle-like prop- erties that play an important role in various fields of science.”-[37] In condensed matter field theory, one might think of skyrmions as a generalization vortices, with a topological invariant given by an integral over the xy-plane (rather than by a line integral as is the case of a vortex). They are defined for a vector field n(r) through a topological charge [21][37]

Q(n) = 1 4π

Z

n · ∂ _x n × ∂ _y n dxdy (2.17) The concept was originally introduced by T.Skyrme in 1962 in form of a hypo- thetical particle with topological charge related to the baryon number [38]. The interest in skyrmions has grown enormously since they have also been shown to exist as topological excitations in condensed matter systems early in [36] and in [39][40]. They are also important for the field of spintronics and data storage [41][42]. Skyrmions have as well been found in the context of superconductors and very recent investigations propose their existence in systems such as [10][43][44].

In the special case of a two-component superconductor, it can be shown via

mapping to an easy-plane non-linear σ-model that bound states of well separated

fractional vortices are skyrmions [21][45]. We shall therefore in the rest of this

thesis use the term “skyrmion” to refer to inter-component fractional vortex pairs.

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Chapter 3

Vortices near boundary

A recent work by M.A. Silaev [46] (2011) considers the equilibrium configuration of a single fractional vortex and of a skyrmion (inter-component vortex pair) near the boundary of a two-component superconductor. The model used was a Ginzburg- Landau theory (2.14) without inter-component coupling V = 0. In the study, it was found that fractional vortices will rest at an equilibrium distance from the boundary, while skyrmions are unstable and collapse. The aim of the present thesis is to look at the same problem, but with an added inter-component coupling V 6= 0.

The main consideration is the equilibrium configuration of close to the boundary skyrmions in the same model, but with an added Andreev-Bashkin coupling. We also give a few words about the phase configuration of a fractional vortex near the boundary with an added Josephson coupling.

Insulator

N

B ₀

Figure 3.1: Fractional vortices induced in a half-infinite superconductor under the influence of an external magnetic field.

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10 Chapter 3. Vortices near boundary The setup is as follows: Consider the region adjacent to a boundary of a two-component superconductor in an external homogeneous magnetic field B 0 =

−B 0 e z . The boundary is in the yz-plane and the region x < 0 is modeled as an insulator, figure 3.1. Vortex excitations in the superconductor occur when the mag- netic field increases. The placement of the vortices depend on the free energy F G , which is minimized in thermal equilibrium. This means that the fields A, θ 1 and θ 2

must be such that F reaches an extrema. From this we get two equations of motion δF

δA = 0 (3.1)

δF δθ α

= 0 (3.2)

Assuming that there are N α vortices in condensate α, we denote the positions of their cores by r αi = (x αi , y αi ), 0 < i < N α . In these, ψ α is set to zero as discussed in section 2.4. Therefore, the super-fluid regime ¹ S is given by (3.3), where Ξ is the set of points inside the vortex cores.

S = R ²⁺ \ Ξ, R ²⁺ ≡ {(x, y)|x > 0}

Ξ =

2,N

_α

[

α=1,i=1

{x, y|(x − x _αi ) ² + (y − y _αi ) ² < ξ ² _α } (3.3)

This is the region within which we wish to solve our equations of motion. Fur- thermore, since the electric current J cannot flow through the edge of the super- conductor and the super-current v _s together with B both go to zero at infinity, we get the following boundary conditions for the fields

∇ × A → 0 as |r| → ∞

∇θ α → 0 as |r| → ∞ e x · J = 0 at x = 0

(3.4)

We will in addition to this also have boundary conditions at the vortex cores, which will be discussed in more detail later.

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In principle, there is no super-fluidity within a distance x < ξ from the boundary. However this does not affect the physics since the superconductor is assumed to be half-infinite.

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3.1. Josephson interaction 11

3.1 Josephson interaction

In this section we give a few words on how to find the equilibrium phase config- uration for a single fractional vortex near the boundary, with an added bilinear Josephson coupling V _J .

V _J = η|ψ ₁ ||ψ 2 | cos(θ 1 − θ 2 ) (3.5) Adding this to the free energy (2.13) in the Ginzburg-Landau theory, and using the following variations



 

 

δ

δA R (∇ × A) ² d ² r = 2∇ × (∇ × A)

δ

δA R |(∇ + iqA)ψ α | ² d ² r = 2q|ψ α | ² (qA + ∇θ α )

δ

δA R η|ψ 1 ||ψ 2 | cos(θ 1 − θ 2 ) d ² r = 0

(3.6)

we, by (3.1), obtain the London-Maxwell relation (3.7).

∇ × B + q ² A |ψ 1 | ² + |ψ 2 | ² + q X

α

|ψ α | ² ∇θ α = 0 (3.7)

with aid of this relation, it is possible to separate F into two contributions F = F Θ + F A . This is done in appendix A.1, with the energies

F Θ = 1 2

Z |ψ 1 | ² |ψ 2 | ²

|ψ 1 | ² + |ψ ₂ | ² (∇Θ) ² + η|ψ 1 ||ψ 2 | cos Θ

d ² r

F _A = 1 2

Z "

(∇ × B) ²

q ² (|ψ 1 | ² + |ψ 2 | ² ) + B ²

# d ² r

(3.8)

with Θ = θ ₁ − θ 2 . Since only F _Θ depend on θ _α , the second equation of motion (3.2) becomes

δF _Θ

δΘ = 0 (3.9)

This leads to a Sine-Gordon equation for the phase difference, with boundary conditions (3.4).

∆Θ − m ² sin Θ = 0, m ² ≡ η |ψ 1 | ² + |ψ ₂ | ²

2|ψ 1 ||ψ 2 | (3.10) The boundary conditions for a single fractional vortex in (x 0 , 0) can be dealt with using a M¨ obius transformation M : R ² → R ² and map R ²⁺ to the inside of the unit circle. We may write it in complex form as a map from z = x + iy to ˆ z = ˆ x + iˆ y.

ˆ

z = M (z) = z − x 0

z + x 0

(3.11)

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12 Chapter 3. Vortices near boundary

r0 x-axis y-axis

M M

M (∞) M (0)

M (r0) x-axis y-axis

Figure 3.2: Illustration of the boundary value problem in the ˆ x, ˆ y coordinates. The right half-plane maps to the interior of the unit circle

The Laplace operator is invariant under such transformations, and the same Sine-Gordon equation is obtained in the transformed variables. Close to the core, it asymptotically becomes the Laplace equation since ∆Θ sin Θ and by rotational symmetry, we have the single variable boundary value problem (3.12).



 

 

∆Θ − m ² sin Θ = 0

∇Θ| r=ξ ˆ = ¹ _ξ e θ

qA

_r=1 _ˆ |ψ 1 | ² + |ψ 2 | ² + P _α |ψ α | ^∂θ _{∂ ˆ} _r

^α

_ˆ _r=1 = 0

(3.12)

Although the solution to this problem does minimize F Θ , it is hard to incorporate topological path invariance (2.11). The task of finding the equilibrium phase con- figuration of a Josephson vortex near the boundary of a superconductor is therefore rather complicated. This problem is not investigated further in this thesis as the behavior of skyrmions in the Andreev-Bashkin coupling gives a more interesting discussion.

3.2 Andreev-Bashkin interaction

Consider the Ginzburg-Landau energy (2.13), with an Andreev-Bashkin coupling

F = 1 2

Z

S

"

X

α

|(∇ + iqA)ψ α | ² + (∇ × A) ² + V _D

# d ² r V _D = ν |Im (ψ ^∗ ₁ (∇ + iqA)ψ ₁ ) + Im (ψ ₂ ^∗ (∇ + iqA)ψ ₂ )| ²

(3.13)

In the Lodon limit, the coupling can be written as

V _D = ν |ψ ₁ | ² (∇θ ₁ + qA) + |ψ ₂ | ² (∇θ ₂ + qA) ² (3.14) From the variations



 

 

δ

δA R (∇ × A) ² d ² r = 2∇ × (∇ × A)

δ δA

R P

α |(∇ + iqA)ψ _α | ² d ² r = 2q P

α |ψ _α | ² (qA + ∇θ _α )

δ

δA R V D d ² r = 2νq P

α |ψ α | ⁴ + |ψ 1 | ² |ψ 2 | ² (qA + ∇θ α )

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3.2. Andreev-Bashkin interaction 13 we get the London-Maxwell relation (3.15).

A + λ ² ∇ × B + qλ ² λ ⁻² ₁ ∇θ 1 + λ ⁻² ₂ ∇θ 2 = 0 (3.15)

where we have assumed that ν > 0 and introduced the notation ( λ ⁻² _α = |ψ α | ² 1 + ν |ψ 1 | ² + |ψ 2 | ²

λ ² = ¹

q

²

(λ

⁻²₁

+λ

⁻²₂

)

(3.16)

Using the London-Maxwell relation, the free energy is separated into two parts F Θ and F A , see derivation in appendix A.2.

F = F _Θ + F _A F Θ = 1

2 Z

S

|ψ 1 | ² |ψ 2 | ²

|ψ ₁ | ² + |ψ ₂ | ² (∇Θ) ²

d ² r F _A = 1

2 Z

R

²⁺

h λ ² (∇ × B) ² + B ² i d ² r

(3.17)

Here we have, just as in the previous section, introduced Θ = θ ₁ − θ 2 . We note that F _Θ can equivalently be written as

F _Θ = 1 2

Z

S

|ψ ₁ | ² |ψ ₂ | ²

|ψ 1 | ² + |ψ 2 | ² (∇Φ) ²

d ² r (3.18)

Where ∇Φ = ∇φ 1 − ∇φ 2 is gradient of the phase difference between the rotated fields φ α introduced in section 2.3. The energy F Θ is positively definite and is minimized when we constrain the fields with the equations of motion (3.2). For Φ, this implies that we have

δF Θ

δΦ = 0 =⇒ ∆Φ = 0 (3.19)

This Laplace equation also satisfies the topological path invariance (2.10). Fur- thermore, due to linearity in both the London-Maxwell relation (3.15) and the Laplace equation, the boundary condition (3.4) can be dealt with by adding image anti-vortices in the x < 0 half-plane (figure 3.3). When assuming that the vortex cores are small, the field φ α for the respective condensate will be a superposition of single vortex and anti-vortex solutions ² (3.20).

2

One could in principle add an arbitrary constant to (3.20). However, this constant can be set to zero without loss of generality.

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14 Chapter 3. Vortices near boundary

φ α =

N

_α

X

i=1

φ ⁺ _αi + φ ⁻ _αi φ ^± _αi = ± ln r − T ^± r αi

(3.20)

Here we have introduced the transformation

T ^± =

±1 0

0 1

(3.21)

The goal is then to find the total energy F G of the system for different vortex configurations. As a first step in doing so, we note that absolute values of both

∇Φ and B are invariant under the coordinate transformation T ⁻ . Hence we can get the same result by mathematically extending the domain of integration in F Θ

to V = S ∪ T ⁻ S and dividing by 2. In the same way, we also extend the domain of integration of F A to the full xy-plane. The following sections give the analytical result of carrying out these integrals.

Figure 3.3: Image anti-vortex fixes the boundary conditions.

3.2.1 Kinetic energy

The kinetic energy of relative motion of the two condensates F Θ is given by

F Θ = 1 4 Z

V

|ψ 1 | ² |ψ 2 | ²

|ψ 1 | ² + |ψ 2 | ² (∇Θ) ²

d ² r (3.22)

We shall mostly focus on the case when there is a non-zero number of vortices in each condensate, N 1 , N 2 > 0. Hence we need to evaluate

Z

V

(∇Φ) ² d ² r = Z

V





N

₁

X

i=1

∇φ ⁺ _1i + ∇φ ⁻ _1i −

N

₂

X

j=1

∇φ ⁺ _2j + ∇φ ⁻ _2j





2 d ² r (3.23)

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3.2. Andreev-Bashkin interaction 15 The evaluated expression for F Θ is given by (3.24), and is derived ³ in appendix A.3.1.

F Θ = π|ψ 1 | ² |ψ 2 | ²

|ψ ₁ | ² + |ψ ₂ | ²





N

₁

X

i

ln 2x 1i

λ

+

N

₂

X

j

ln 2x 1j

λ

+ N 1 ln λ ξ ₁

+ N 2 ln λ ξ ₂





+ π|ψ 1 | ² |ψ 2 | ²

|ψ 1 | ² + |ψ 2 | ²





N

₁

,N

₁

X

i6=k

ln |r _1i − T ⁻ r 1k |

|r 1i − r 1k |

+

N

₂

,N

₂

X

j6=l

ln |r _2j − T ⁻ r 2l |

|r 2j − r 2l |





− π|ψ 1 | ² |ψ 2 | ²

|ψ 1 | ² + |ψ 2 | ²

N

1

,N

2

X

i,j=1

2 ln |r 1i − T ⁻ r 2j |

|r 1i − r 2j |

(3.24) In appendix A.3.1 we also evaluate the energy for the special case N 1 = 1, N 2 = 0 with the result

F Θ = π|ψ 1 | ² |ψ 2 | ²

|ψ 1 | ² + |ψ 2 | ²

ln 2x ₁₁ λ

− ln λ ξ 1

(3.25)

3.2.2 Magnetic energy

The magnetic energy F A is given by F _A = 1

4 Z

R

²

h

λ ² (∇ × B) ² + B ² i

d ² r (3.26)

To evaluate this expression, we first use the London-Maxwell relation to obtain an expression for B that can then be integrated. Taking the curl of (3.15), gives a differential equation for B.

B + λ ² ∇ × (∇ × B) = −qλ ² λ ⁻² ₁ ∇ × ∇θ ₁ + λ ⁻² ₂ ∇ × ∇θ ₂

(3.27) This equation is solved in appendix A.3.2 with the boundary condition B(r) → 0 as r → ∞ (3.4). We there also assume that B(r) = B(r)e z due to the two- dimensional nature of the system. The result can be written as a superposition of single vortex solutions B α .

B(r) = X

α N

α

X

i

B α (r − r αi ) − B α (r − T ⁻ r αi )

(3.28) with

B _α (r) = −qλ ⁻² _α K ₀ (|r|/λ) (3.29) Here, K ₀ is the modified Bessel function of the second kind, which implies that the magnetic field decays rapidly when going away from a vortex core. In particular the

3

Assuming that the vortices are well separated, r

αi

− r

βj

> ξ

α

, ∀αβij

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16 Chapter 3. Vortices near boundary magnetic field decays much faster than |∇θ α | at large distances. One should note that (3.29) is not valid in the vortex core, and for |r| < ξ α we instead approximate with the field at the core boundary ⁴ .

B α (r) _|r|<ξ

α

≈ qλ ⁻² _α ln (ξ α /λ) (3.30) The physical reasoning for this is simply that there is no super-current inside the vortex cores i.e |ψ _α | = 0. Using the expressions (3.28-3.30) and inserting them in (3.26), the magnetic energy F _A can be calculated. This is done in appendix A.3.3 and the result is

F A =πq ² λ ² N 1 λ ⁻⁴ ₁ ln (λ/ξ 1 ) + N 2 λ ⁻⁴ ₂ ln (λ/ξ 2 ) +πq ² λ ² X

α,β

N

α

,N

β

X

(α,i)6=(β,j)

λ ⁻² _α λ ⁻² _β K 0 (|r αi − r βj |/λ)

−πq ² λ ² X

α,β N

_α

,N

_β

X

i,j

λ ⁻² _α λ ⁻² _β K 0 |r αi − T ⁻ r βj |/λ

(3.31)

3.2.3 External field energy

The energy F Ext due to the external field B 0 = −B 0 e z is given by F _Ext = −

Z

R

²⁺

B ₀ B(r) d ² r (3.32)

This integral is carried out in appendix A.3.4 by inserting the previously derived expression for B(r). The result is

F Ext = 2πqλ ² B 0

X

α N

_α

X

i

λ ⁻² _α

1 − e ^−x

^αi

^/λ

(3.33) It is worth noting that this energy is only relevant close to the boundary x ∼ λ due to the fast decaying exponential.

4

This follows from the behavior K

0

(r → 0) → − ln(r) of the modified Bessel function. Which in turn gives lim

|r|→0

B

α

(r) ≈ qλ

⁻² α

ln (|r|/λ). The expression then follows by setting |r| = ξ

α

also see [21]

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Chapter 4

Vortex configurations

We have now derived the full expression for F G in the Andreev-Bashkin coupling and are ready to analyze the equilibrium vortex placement. Our objective is to find minimum energy configurations {r _αi } to the function

F G ({r αi }) = F Θ ({r αi }) + F A ({r αi }) + F Ext ({r αi }) (4.1) We shall focus in particular on how the interaction strength ν and the external field B 0 affects the placement. Therefore we set |ψ 1 | = |ψ 2 | = |ψ|, ξ 1 = ξ 2 = ξ, i.e. assuming that both condensates are identical. From the results derived, the interaction energies for two vortices have the following qualitative shape

F _Θ ∼ ± ln(s/λ) F _A ∼ K ₀ (s/λ) (4.2) where s is the separation distance. The sign in F Θ is negative if the vortices are in the same condensate and otherwise positive. Since the modified Bessel function rapidly goes to zero at large s, F Θ will be dominant for big separations. This means that vortices in different condensates will attract each other at large distances. For smaller separations, however, the magnetic energy F A will not be negligible and give rise to a repulsive force between the vortices. As we shall see, there will in fact be an equilibrium separation distance depending on ν.

On the other hand, for a vortex and an image anti-vortex we have

F Θ ∼ ∓ ln(s/λ) F A ∼ −K 0 (s/λ) (4.3) where the sign in F Θ is positive if the vortices are in the same condensate and otherwise negative. That is, F Θ and F A both give rise to an attraction between a vortex and an anti-vortex in the same condensate. The only thing preventing a vortex from falling off the sample to annihilate with its anti-vortex is the external magnetic field energy F _Ext . We will see in the following that the combination of all these energies give rise to an equilibrium separation distance for vortices to the boundary.

17

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18 Chapter 4. Vortex configurations In the next few sections we analyze the minimum energy configuration for bound states of N = N 1 + N 2 vortices. Let us refer to such a bound state as a ‘vortex molecule’. As a first remark, we note that F A and F Ext do not depend on which condensate the vortices are in. The dependence is only via F Θ .

F Θ = |ψ| ² 8

Z

V

(∇θ 1 − ∇θ 2 ) ² d ² r (4.4) From the structure of this energy, we argue that N α − N β for a vortex molecule should be as small as possible. This lowers the energy since it gives the smallest

‘dipole’ moment at large distances, and hence faster decaying (∇θ 1 − ∇θ 2 ) ² . In fact molecules with an absolute winding of more than 2π in (θ α − θ β ) are unstable to splitting into smaller molecules due to the quadratic nature of F Θ , which is the only the attractive potential. It is thus enough to investigate the cases N α = N β

and N α = N β + 1 for even and odd N respectively.

We will look explicitly at F G to find the shape of molecules. To make things simpler, we note that removing a constant from F _G will not alter the equilibrium vortex configuration, and make the decomposition

F _G ({r _αi }) = F _C ({r _αi }) + F _ξ

F ξ = |ψ| ² (1 + ν|ψ| ² )πN ln (λ/ξ) (4.5) Here, F ξ are the terms in F G that depend on ξ. Since F ξ does not depend on {r αi }, we can equivalently minimize F C . That is; the equilibrium configuration does not depend on the size of the vortex cores. In addition, we shall work with normalized quantities and define

( ˜ F G = _|ψ| ^F

^G2

, ˜ F C = _|ψ| ^F

^C2

, ˜ F ξ = _|ψ| ^F

^ξ2

B ˜ 0 = − _q|ψ| ^B

⁰2

, ˜ r αi = ^r

^αi

_λ , ˜ ν = ν|ψ| ² (4.6)

4.1 Single fractional vortex

Consider the most simple case of having a single fractional vortex in one of the two identical condensates. Let the vortex core be positioned at r = λ(˜ x, ˜ y). Due to translation invariance along the y-axis, we obtain a single variable function for F ˜ C (˜ x). In thermal equilibrium, the position of the vortex must be such that it minimizes ˜ F C . Substituting to normalized quantities, we write ˜ F C (˜ x) as

F ˜ _C (˜ x) = π

2 ln 2x − ˜ π (1 + 2˜ ν)

2 K ₀ (2˜ x) − π ˜ B ₀ 1 − e ^−˜ ^x

(4.7) This energy in general has two minima, ˜ x = 0 and ˜ x = ˜ x ^∗ 6= 0. The trivial minimum

˜

x = 0 comes from the strong attraction of the vortex to its image anti-vortex at

short length scales. This region is not covered very well by (4.7), since it assumes

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4.1. Single fractional vortex 19 vortices to be separated from the boundary with distances x ξ. The physical situation ˜ x = 0, however, corresponds to the trivial state with zero super-current

∇θ 1 = ∇θ 2 = 0. Since such a configuration is not very interesting, we turn our focus to the non-trivial minimum ˜ x ^∗ . This solution is formed by equilibrium attraction of the vortex to the image anti-vortex and repulsion by the external magnetic field.

Figure 4.1a shows the form of ˜ F _C . With an increasing external field, we see how the non-trivial minimum appears. Figure 4.1b shows the region in parameter-space (˜ ν, ˜ B ₀ ) for which the non-trivial minimum exists. For each ˜ ν, there will be a critical lower external field ˜ B _lower below which the single fractional vortex is unstable to annihilation with its image anti-vortex. Such regions are likely to be present also in case with multiple fractional vortices, and non-trivial ¹ solutions cannot be assumed to exist for all choices of parameters.

1 ˜

x ^∗ + 2(1 + 2ν)K 1 (2˜ x ^∗ )

e ^x ^˜

^∗

− 2 ˜ B 0 = 0 (4.8)

Setting ^{d ˜} _d˜ ^F _x

^C

_x _˜

∗

= 0, we get condition (4.8) for the equilibrium distance to the boundary. The solution is plotted as a function of the external field in figure 4.2.

As can be seen, the equilibrium distance grows monotonically with ˜ B 0 which is to be expected by the form of F Ext . The interaction strength ˜ ν also affects the equilibrium position, and ˜ x ^∗ decreases with increasing ˜ ν. The reason for this is that a greater ˜ ν gives a stronger magnetic attraction to the image anti-vortex.

˜ x

0 2 4 6 8 10 12 14 16 18 20

˜ F(˜x)/(1+2˜ν)^C

-5 -4 -3 -2 -1 0 1 2

B˜0= 1, ˜ν = 1 B˜0= 1, ˜ν = 5 B˜0= 5, ˜ν = 1 B˜0= 5, ˜ν = 5

(a)

˜ν

0 1 2 3 4 5 6 7 8 9 10

˜ B⁰

1.5 2 2.5 3 3.5

Stable region

Blower˜

(b)

Figure 4.1: (a) Normalized energy plotted against the distance of the vortex core to the boundary. (b) Stable region in parameter space for which a non-trivial minimum exist.

1

With multiple fractional vortices, we extend the notion of non-trivial solution and define it as configurations satisfying x

αi

ξ ∀α, i.

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20 Chapter 4. Vortex configurations

B ˜

0

1.5 2 2.5 3 3.5 4 4.5 5

˜x∗

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

˜ν = 0

˜ν = 1

˜ν = 5

Figure 4.2: Plot of the solution ˜ x ^∗ to equation (4.8). Note how the equilibrium distance decreases with increasing coupling strength and that a stronger external field ˜ B 0 is required for solutions to exist for larger ˜ ν. The vertical lines show the points where the non-trivial minimum ceases to exist.

4.2 Single Skyrmion

Consider the case when there is one fractional vortex in each condensate (skyrmion).

With the positions denoted by r α = λ(˜ x α , ˜ y α ) for the respective condensate, the normalized free energy ˜ F C becomes

F ˜ C = π 2

ln (2˜ x 1 ) + ln (2˜ x 2 ) − ln (˜ x 1 + ˜ x 2 ) ² + (˜ y 2 − ˜ y 1 ) ² (˜ x ₂ − ˜ x ₁ ) ² + (˜ y ₂ − ˜ y ₁ ) ²

+ − π

2 (1 + 2˜ ν) (K 0 (2˜ x 1 ) + K 0 (2˜ x 2 )) + π(1 + 2˜ ν)

K ₀ p

(˜ x ₁ − ˜ x ₂ ) ² + (˜ y ₂ − ˜ y ₁ ) ²

− K ₀ p

(˜ x ₁ + ˜ x ₂ ) ² + (˜ y ₂ − ˜ y ₁ ) ²

− π ˜ B 0 2 − e ^−˜ ^x

¹

− e ^−˜ ^x

²

(4.9) As a first step in analyzing the structure of equilibrium configurations we consider a ‘free’ skyrmion far away from the boundary of the superconductor by setting

˜ r ₁ = (˜ x, 0), ˜ r ₂ = ˜ r ₁ + ˜ s and letting ˜ x → ∞. For |˜ s| = ˜ s ˜ x we then have

F ˜ C = π ln (˜ s) + π(1 + 2˜ ν)K 0 (˜ s) − 2π ˜ B 0 (4.10)

Note that the energy of a skyrmion in the interior of the superconductor is finite,

since (4.10) is finite for ˜ s > 0. This should be put in contrast with the energy of

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4.2. Single Skyrmion 21 a single fractional vortex which is infinite in the bulk by (4.7). Taking the deriva- tive d ˜ F C /d˜ s and setting it to zero gives an equation for the skyrmionic separation distance ˜ s ^∗

1/˜ s ^∗ = (1 + 2˜ ν)K 1 (˜ s ^∗ ) (4.11) The modified Bessel function K ₁ (˜ s) has the asymptotic behavior K ₁ (˜ s) → 1/˜ s as

˜

s → 0. Thus for ˜ ν → 0, we have the limiting solution ˜ s ^∗ → 0, since K ₁ (˜ s) < 1/˜ s for all ˜ s > 0. For ˜ ν > 0, ˜ s ^∗ monotonically increases with ˜ ν.

We turn to the stated problem of having the inter-component vortex pair lo- cated close to the boundary. Since the condensates are identical, there are only two possible configurations that can be extrema of ˜ F C . The skyrmion has to be either parallel or perpendicular to the boundary. Figure 4.3 illustrates the possible vortex configurations. We can conclude that the configuration in figure 4.3b has lower energy by the following argument: Consider the case when the equilibrium separation distance is large. i.e ˜ ν is large. Then, the free energy will be the sum of two individual fractional vortices. (since both ∇θ α and B α go to zero at large distances from the vortex core) By the results derived earlier, a single fractional vortex has infinite energy in the bulk (4.7). Hence we conclude that ˜ F _C in figure 4.3a will be greater than in figure 4.3b for larges separations. If then ˜ ν is decreased, the equilibrium has to change continually and the configuration in figure 4.3b will have a lower energy than that in figure 4.3a for all ˜ ν.

Insulator

^α ^β

(a)

Insulator ^α ^β

(b)

Figure 4.3: Demonstration of the two possible vortex configurations that are ex-

trema to ˜ F C by symmetry arguments. A fractional vortex in condensate α 6= β

is represented by an arrow circling the condensate number. α and β can be

exchanged.

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22 Chapter 4. Vortex configurations For the configuration in figure 4.3b, we denote the positions of the two fractional vortices with r 1 = λ(˜ x, 0), r 2 = λ(˜ x, ˜ y) in condensate 1 and 2 respectively. Note here that we have used that ˜ F C is invariant under translation of the whole configuration along the y-axis to set ˜ y 1 = 0. ˜ F C becomes

F ˜ _C (˜ x, ˜ y) = π ln 2˜ x˜ y p 4˜ x ² + ˜ y ²

!

− 2π ˜ B ₀ 1 − e ^−˜ ^x + π(1 + 2˜ ν)

K ₀ (|˜ y|) − K ₀ (2˜ x) − K ₀ p

4˜ x ² + ˜ y ²

(4.12)

Figure 4.4a shows a plot of ˜ F _C . From the figure one can see that the energy variation for changing the distance ˜ x to the boundary is many orders of magnitude smaller than that of changing the vortex separation ˜ y. The reason for this is that there in principle is a ‘dipole-dipole’ interaction between the pair and its image anti-vortex pair. The attraction to the boundary is thus much weaker than for the

‘monopole-monopole’ interaction of a single fractional vortex. Though the energy variations are small, there exists an equilibrium distance to the boundary, which can be seen by plotting the minimum value of ˜ F C as a function of ˜ x, figure 4.4b.

Note that this value reaches a constant as ˜ x → ∞, given by minimizing (4.10), which means that for a skyrmion far from the boundary, there is no (very weak) attraction to the boundary.

˜ x 10 20 3.5 30 3 y˜ 2 2.5 1 1.5

-28.5 -28 -27.5 -27 -26.5 -26

FC(˜x,˜y)

Minimum

(a) Energy surface F G (˜ x, ˜ y) plotted for ˜ y > 0.

There is a minimum at (˜ x ^∗ , ˜ y ^∗ ). Note that the variations along ˜ x are very small.

˜ x

5 10 15 20 25 30

miny(FC(˜x,˜y))

-28.21 -28.205 -28.2 -28.195 -28.19 -28.185 -28.18 -28.175 -28.17 -28.165

-28.16 ^min^y^(F^C^(˜^x,˜^y))

(b) Minimum value of F G (˜ x, ˜ y) plotted against ˜ x. Note that the variations along ˜ x are very small to the right of the minimum.

Figure 4.4: The figures illustrate the shape of the energy surface ˜ F C (˜ x, ˜ y) and

a zoom in on its minimum value plotted along ˜ x. ˜ ν = 1, ˜ B ₀ = 5. Note that the

equilibrium distance to the boundary ˜ x ^∗ is greater for the skyrmion than for a single

vortex (figure 4.2), which is explained by the weaker attraction to the boundary.

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4.3. Multiple fractional vortices 23 There will also be a region in parameter-space for which non-trivial solutions cease to exist, similar to that of one fractional vortex (figure 4.1b). The magnetic field in the stable regime can however take lower values due the weaker dipole character of the attraction to the boundary. In fact, as ˜ ν goes to zero, the lower critical field will also go to zero, since the attraction to the boundary vanishes.

4.3 Multiple fractional vortices

Consider a configuration with N > 2 vortices. Let the vortex cores be located at r _αi = λ(˜ x _αi , ˜ y _αi ). The free energy is a function of 2N − 1 variables since there are two coordinates for each vortex, and ˜ F _C is invariant under simultaneous translation of all cores along the y-axis. For many vortices ˜ F _C gets a rather complicated form, and it is not very illuminating to write it out in full. Instead, a symbolic program was used to obtain an expression for ˜ F C and its derivatives. The simplest approach of finding a minimum to this function is to employ a gradient descent algorithm.

˜ r ⁽ⁿ⁺¹⁾ _αi = ˜ r ⁽ⁿ⁾ _αi − γ∇ ˜ r

_αi

F ˜ C ({˜ r ⁽ⁿ⁾ _αi }) (4.13) Here γ is the step size and ˜ r ⁽ⁿ⁾ _αi is the vortex coordinates in the n:th iteration.

Using a random initial guess ˜ r ⁽⁰⁾ _αi over multiple descents, one then gets a Monte- Carlo style scheme for finding all local and the global minimum of ˜ F _C . If we suppose that vortices are initially formed by some random fluctuations or impurities, then there is also a physical argument for using randomized initial conditions. Since if vortices form randomly, they would drift towards the closest local minimum of ˜ F C . The algorithm (4.13) works in principle but has a slow convergence rate due to the shape of ˜ F C . Indeed, the problem with using a simple gradient descent method is that ˜ F C has two energy scales, which can differ by orders of magnitude:

1. The gain from forming a molecule.

2. The gain from rotation and translation of the molecule due to the attraction to the boundary.

In particular for the case when N 1 = N 2 , the first energy gain is much greater than the second. This poses a problem, since γ will be limited by the bigger energy scale, leading to that when a vortex molecule has formed, it will drift very slowly towards the boundary as the energy surface is essentially flat.

To circumvent this problem and get a faster convergence, we introduce a modi-

fied version of the gradient descent algorithm. We exploit the fact that rigid-body

translations of the entire configuration will be on the lower energy scale and in-

troduce what one might think of as a ‘rigid-body gradient descent’ method. It is

implemented according to the following scheme: Let I({˜ r _αi }, γ _rb ) be a rigid-body

transformation of the coordinates with some parameter γ _rb such that I({˜ r _αi }, 0) is

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24 Chapter 4. Vortex configurations the identity transformation. We then let the next iteration step be given by (4.14), where γ rb > 0 is small (step size).

˜ r ⁽ⁿ⁺¹⁾ _αi = I {˜ r αi }, −γ rb

d ˜ F C (I({˜ r αi }, γ _rb ⁰ )) dγ _rb ⁰

_γ

0

rb

=0

!

(4.14)

That is; we use a derivative to approximate the change of ˜ F C under a small rigid- body transformation, then perform it in the direction such that the change is neg- ative. The rigid-body transformations are rotations and translations of the vortex molecule. We implement this in the following way

˜ r ⁽ⁿ⁺¹⁾ _αi = R _h˜ _ri −γ _rot X

α N

_α

X

i

∇ ˜ r

αi

F ({˜ ˜ r ⁽ⁿ⁾ _αi }) · (e _z × ˜ r _αi )

!

˜ r ⁽ⁿ⁺¹⁾ _αi

˜ r ⁽ⁿ⁺¹⁾ _αi = ˜ r ⁽ⁿ⁾ _αi − γ trans

X

α N

_α

X

i

∇ ˜ r

_αi

F ˜ C ({˜ r ⁽ⁿ⁾ _αi })

(4.15)

Here, R r

0

(ϑ) is the 2D rotation matrix which rotates coordinates an angle ϑ around the point ˜ r ₀ . h˜ ri is the mean of all vortex positions ² , γ _rot and γ _trans are rigid-body gradient descent step sizes for rotation and translation respectively. The additional steps (4.15) are of course not necessary for finding a minimum of ˜ F _C . They are only used to give a faster convergence.

The initial points ˜ r ⁽⁰⁾ _αi were randomized in box B. To get a minimum proper length-scale ³ , B was chosen as

B = {(˜ x, ˜ y)|0 < ˜ x < ˜ x ^∗ + N ˜ s ^∗ , −N ˜ s ^∗ < ˜ s < N ˜ s ^∗ } (4.16) where ˜ x ^∗ is the normalized equilibrium distance to the boundary of a skyrmion, ˜ s ^∗ is the normalized equilibrium separation of a free skyrmion, see section 4.2.

The shape of vortex molecules depend greatly on the coupling ˜ ν. Indeed, for small ˜ ν, we will see that vortices form small skyrmions that then bind together in a larger cluster or a ‘skyrmion molecule’. In contrast, for large ˜ ν, the skyrmionic separation is comparable to the size of the molecule, and vortices form simpler equidistant configurations.

We start our discussion with investigating the large ˜ ν regime, as it has fewer local minima and is easier to understand.

2

In principle we could choose to rotate about any point.

3

We know that vortex molecules can at least not be smaller than the skyrmionic separation.

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4.3. Multiple fractional vortices 25 In this regime, the equilibrium configuration for a molecule of three vortices is a half-ring, figure ⁴ 4.5a. The inter-vortex distance varies with ˜ ν, while an in- creased magnetic field ˜ B 0 makes the configuration penetrate further into the bulk.

The vortices form a straight line when pushed far from the boundary, as there is no ‘bending’ force. For five vortices, the case is similar and the equilibrium con- figuration is again a half-ring, figure 4.5b. In figures 4.5c, 4.5d we see that the vortices in both cases are forming a ring-like configuration together with the image anti-vortices.

Insulator

˜

0 x 20

˜ y 8

−8

α

^β

(a)

Insulator

˜

0 x 20

˜ y 8

−8

α

β

(b)

α

^β

α

^β

(c)

α

β

α

β

(d)

Figure 4.5: Minimum energy configurations for three and five vortices at large ˜ ν.

˜

ν = 2, ˜ B = 2. The figures are in scale and with equal scaling on the ˜ x- and ˜ y-axis.

For higher odd number of vortices the equilibrium configuration is also a half- ring when ˜ ν is large. The first interesting behavior appears when the number of vortices is even. For small N , there are then two different local minimum energy configurations, the first one being a half-ring and the other being a ring. The phenomenon is most easily demonstrated with three vortices in each condensate, figure 4.6. Here there are two local minimum energy configurations (figure 4.6a- 4.6b), which of these is the global minimum depends on the parameters.

4

The figures in this section are the result of running the minimization algorithm until the absolute sum of all derivatives was zero to machine precision ∼ 10

⁻¹⁵

.

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26 Chapter 4. Vortex configurations

Insulator

˜

0 x 20

˜ y 8

−8

α

β β β

(a) half-ring, ˜ ν = 3, ˜ B 0 = 2

Insulator

˜

0 x 20

˜ y 8

−8

α

α α

β β

β

(b) ring, ˜ ν = 3, ˜ B 0 = 2

ν ˜

0.5 1 1.5 2 2.5 3 3.5 4

˜ B

⁰

0.5 1 1.5 2 2.5 3

Metastable region

Stable region

Bupper ˜ Blower ˜

(c) Stability of a half-ring of six vortices. The half-ring has lower energy than the ring in the ‘stable region’. The half-ring closes into a ring when ˜ B 0 > ˜ B upper and moves away from the boundary. For ˜ B 0 < ˜ B lower there exists no half-ring minimum.

Figure 4.6: For six vortices there are two minimum energy configurations. A half- ring and a ring. Note that the ring rests at a distance further away from the boundary, due to its lower dipole moment. Figures are to scale.

Running the minimization algorithm for a range of parameter-values, it was

found that there exists an upper critical magnetic field B upper above which the

half-ring closes into a ring and moves away from the boundary, an irreversible

transition. For fields lower that B upper the half-ring is either locally (metastable)

or globally stable. Figure 4.6c shows a plot of the stability in different regions of

parameter-space. The behavior at large ˜ ν is what we would intuitively expect from

our previous discussion: The lower critical field increases with ˜ ν as the magnetic

attraction to image anti-vortices get stronger. The upper critical field increases

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4.3. Multiple fractional vortices 27 with ˜ ν as a bigger half-ring has to penetrate further into the bulk before it can close into a ring.

However, the plot also shows a sudden increase of the upper field when ˜ ν becomes smaller. This increase can be explained from the fact that the skyrmionic separation is becoming smaller than the inter-skyrmion separation. (we get a ‘skyrmion ring’) For these values of ˜ ν, the attraction between skyrmions in the edges of the half-ring is not as strong and it is harder for it to close into a ring. See section 4.4 for illustrations. The effects are similar for both four and eight vortices, with only a half-ring and a ring minimum energy configuration for all values of ˜ ν.

Continuing to a larger even number of vortices in the large ˜ ν regime, the situa- tion becomes more complicated. We give an illustration for sixteen vortices (figure 4.7) where the half-ring can now collapse not only into a ring, but also into other configurations such as bound states of smaller rings, figures 4.7c-4.7d.

Insulator

˜

0 x 60

˜ y 30

−30

α α

α

β β

β β β β

(a)

Insulator

˜

0 x 60

˜ y 30

−30

α

α α

α

β

β β

β β β β

β

(b)

Insulator

˜

0 x 50

˜ y 20

−20

α α

α

α α α

α α

^β ^β ^β

β β β β

β

(c)

Insulator

˜

0 x 50

˜ y 20

−20

α

α α

α α α α

α

β β

(d)

Figure 4.7: Some of the minimum energy configurations for N = 16, ˜ ν = 10, ˜ B ₀ = 5.

Figures are to scale.

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28 Chapter 4. Vortex configurations We let this conclude our discussion of equilibrium configurations with an even number of vortices. The case of having small ˜ ν for this many vortices was not investigated, but it is speculated that we will have a the same kind of transition from vortex half-rings to skyrmion half-rings as was found for four, six, and eight vortices.

With an odd number of vortices the skyrmion transition is much more violent.

In particular the formation of skyrmions causes half-rings to ‘shatter’. Figure 4.8 illustrates the two different local minimum energy configurations for three vortices in the regime of small ˜ ν. Note that two of the vortices form a tightly bound skyrmion with weaker attraction to the boundary, leading to an asymmetric configuration.

See section 4.4 for illustrations of how larger odd N molecules shatter, in particular a lot of new local minima will spring into existence.

Insulator

˜

0 x 15

˜ y 6

−6

α

^β α

(a)

Insulator

˜

0 x 15

˜ y 6

−6

α α

^β

(b)

Figure 4.8: For smaller values of ˜ ν, the odd N half-rings shatter as the skyrmionic separation shrinks. This leads to asymmetry causing there to exist more local minima. Here (a) and (b) are the minimum energy configurations for three vortices.

(note that they are isomorphic) ˜ ν = 0.3, ˜ B ₀ = 2. Figures are to scale.

Vortex Currents near the Boundary of a Two-Component Superconductor

Master Thesis