Seite 1 von 1

Technische Universität München

Physics Department

Walther-Meiÿner-Institute for Low Temperature Research

Bachelor thesis

Gap suppression in superconducting

cylinders: analytic results from the

Ginzburg-Landau theory

Luzia Höhlein

Garching, August 7, 2013

Advisor: PD Dr. Dietrich Einzel

First supervisor: PD Dr. Dietrich Einzel

Second supervisor: Prof. Dr. Christine Papadakis

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Ginzburg-Landau equations for ψ(r) and ∆(r) in the presence of B0 . . 3

3 Boundary condition for ψ(r) and ∆(r) for a superconduction-vacuum

interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Ginzburg-Landau equations for superconducting cylinders . . . . . . . . 9

5 Solution of the linearized Ginzburg-Landau equations for g(r) and Aφ(r) 11

6 Discussion of the analytical results for f(r) and Bz(r) . . . . . . . . . . 21

7 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

A Ginzburg-Landau equations for superconducting full cylinders . . . . . . 29

B Solution of the linearized Ginzburg-Landau equations for g(r) and Aφ(r) 31B.1 derivation of the Ginzburg-Landau equations in cylindrical coordinates 31B.2 particular solution gp(r) . . . . . . . . . . . . . . . . . . . . . . . . . . 33B.3 relative error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

C Discussion of the analytical results for f(r) and Bz(r) . . . . . . . . . . 35

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

iii

Chapter 1

Introduction

The general aim of this work is to derive and solve the Ginzburg-Landau (GL)di�erential equation (DEQ) for a full superconducting cylinder of radius R(0 ≤ r ≤R) bounded by a vacuum in the presence of an externally applied magnetic �eld.If an external magnetic �eld B0 = B0z is applied from the vacuum side, it willgenerally decay away from the surface inside the superconductor, which is nothingbut the screening side of the Meiÿner-Ochsenfeld e�ect [1].The early phenomenological theories of superconductivity by F. and H. London [2]

and V.L. Ginzburg and L.D. Landau [3] are based on the description of the supercon-ducting Cooper-pair condensate by a macroscopic quasi-boson or pair �eld ψ(r, T ),which may at the same time be viewed as the superconducting order parameter, since2 |ψ|2 = ns is related to the condensate particle density ns. The order parameterrises continuously from zero at the critical temperature Tc to ψ(r, 0) at T = 0. TheLondon's approach to the description of magnetic �eld screening is to assume thatthe order parameter ψ(r) = ψ0 remains una�ected by the presence of B0 and staysat its equilibrium value ψ0 =

√ns/2.

On the contrary, the GL approach allows for a spatial dependence of the normal-ized order parameter f(r) = ψ(r)/ψ0, induced by the external �eld. Later in 1957J. Bardeen, L.N. Cooper and J.R. Schrie�er (BCS) proposed the �rst microscopictheory of superconductivity [4]. L.P. Gor'kov showed that the phenomenological GLequations follow from the microscopic theory of superconductivity in the neighbor-hood of the critical temperature Tc [5]. He related the macroscopic wave functionand hence the order parameter of the GL theory to the energy gap ∆(r) of thesuperconductor and hence the order parameter of the microscopic BCS theory.The GL equations, expressed through the macroscopic order parameter ψ(r), cantherefore be rewritten in terms of the normalized microscopic BCS order parameterf(r) = ∆(r)/∆0.

Previous attempts to tackle the general problem analytically include the applica-tion of a Taylor expansion method [6], which was, however, restricted to thin slabs(d < ξGL) and wires (R < ξGL) and does not in particular allow conclusions aboutthe half-space limit. A numerical solution of the cylinder problem was provided by

1

Chapter 1 Introduction

R. Doll and P. Graf [7] as well as by G.F. Zharkov [8] and is currently dealt withby R. Doll [9]. Analytical results for full superconducting slabs were provided by D.Einzel [10].

The goal of this work is a rigorous analytic calculation of the linear response ofthe order parameter f(r) to externally applied magnetic �elds in the GL temper-ature regime (temperatures near the critical temperatures Tc) for superconductingfull cylinders. Linear order parameter response implies that f(r) does not deviatesigni�cantly from 1. This is the low-�eld or London-limit. As a consequence, themagnetic induction decay pro�le inside the superconductor is known analytically forvarious sample geometries because it is equivalent to the solution of London's the-ory. It can be inserted into the linearized second-order di�erential equation for theorder parameter f(r), where it enters in the form of an inhomogeneity. Its solutionprovides us with the information on the spatial dependence of the superconductingorder parameter in the presence of external magnetic �elds for various material para-meters, such as transition temperature Tc, GL parameter κ = λL/ξGL, the samplesize R, and for various experimental parameters such as the external �eld B0 andthe temperature T . The analytic form of these results has not been treated in textbooks before. However, the calculations follow in part the lecture notes "Problemsin the theory of superconductivity" by D. Einzel [10].My contributions to this work are essentially the investigation of the half space

limit for the full cylinder (the case of large radii R→∞ turns out to be topologicallyequivalent to a superconducting half space) and the numerical results of the spatiallydependent order parameter.

2

Chapter 2

Ginzburg-Landau equations for ψ(r)and ∆(r) in the presence of B0

Let us start from the well-known form of the two coupled Ginzburg-Landau equations[5, 10]

αψ + β |ψ|2 ψ +1

4m

(h

i∇− 2e

cA

)2

ψ = 0

∇× (∇×A) =4π

c

[eh

2mi{ψ∗∇ψ − ψ∇ψ∗} − e2

mc2 |ψ|2 A

]=

4π

cjse,

in which ψ(r) represents the superconducting order parameter. The �rst GL equationdescribes the spatial variation of ψ caused by the presence of an externally appliedmagnetic �eld. The second GL equation describes the shielding of magnetic �eldsfrom the interior of the superconductor by the presence of a macroscopic supercurrentjse. The macroscopic wave function ψ can be decomposed into amplitude a(r) andphase φ(r) in a way introduced for the �rst time in quantum mechanics by ErwinMadelung [11].

ψ(r) = a(r)eiφ(r)

By inserting the Madelung ansatz we get after some manipulations

ξ2GL

[∇2 −

(2π

Φ0

)2(A− Φ0

2π∇φ

)2]a(r) + a(r)− a3(r)

a20

= 0

∇× (∇×A) +a2

a20

1

λ2L

(A− Φ0

2π∇φ

)= 0,

where a0 is the equilibrium value of the order parameter in the absence of a mag-netic �eld. Φ0 = hc/2e denotes the �uxoid quantum generated by the pair charge.

3

Chapter 2 Ginzburg-Landau equations for ψ(r) and ∆(r) in the presence of B0

ξGL is the so-called Ginzburg-Landau coherence length and λL London's magneticpenetration depth. A discussion of these lengths will follow later.The super�uid density ns(r, T ) is given by

ns(r, T ) = 2 |ψ(r)|2 .

In contrast to ordinary quantum mechanics, |ψ(r)|2 does not represent the probabilitybut the actual local density of superconducting electron pairs. Using the BCS-resultfor the super�uid density in equilibrium ns(T ) involves the so-called Yosida functionY (T ) [12]

ns(T ) = n [1− Y (T )]

Y (T ) =

∞∫0

dt

cosh2

√t2 +

(∆2

0(T )2kBT

)2,

which is expressed in terms of the equilibrium energy gap ∆0(T ). The GL expansion(with ∆0 → 0) of the Yosida function with respect to the normalized equilibriumenergy gap ∆0(T )/πkBT allows to relate the macroscopic wave function ψ(r) =a0(T ), and hence the order parameter of the GL theory ns = 2a2

0, to the equilibriumenergy gap ∆0(T ) of the superconductor, and hence the order parameter of themicroscopic BCS theory, as follows [10]:

a20(T ) =

ns(T )

2= n

7ζ(3)

8

[∆0(T )

πkBT

]2

.

This relation may be generalized in a straightforward manner to apply to the non-equilibrium situation [5, 10]

a2(r) = n7ζ(3)

8

[∆(r)

πkBT

]2

,

where we have introduced a spatially dependent gap function ∆(r) related to theMadelung amplitude a(r) in the presence of an externally applied �eld. The �rstand second of these GL equation can be rewritten for the normalized BCS orderparameter

f(r, T ) =∆(r, T )

∆0(T )

and are expected to yield results for the spatial dependence of the magnetic �eldinside the superconductor and the spatial dependence of the dimensionless BCS

4

order parameter induced by the magnetic �eld

0 = ξ2GL

{∇2 −

(2π

Φ0

)2(A− Φ0

2π∇φ

)2}f(r) + f(r)− f3(r)

0 = ∇× (∇×A) +f2(r)

λ2L

(A− Φ0

2π∇φ

).

It turns out that the phase gradient term (∝∇φ) gives rise to the quantization ofthe so-called �uxoid Φ = nΦ0. For a hollow cylinder one has [13, 14]

hc

2e

∮c

∇φ · dr

︸ ︷︷ ︸n·2π

= λ2L

∮c

jse · dr +

∮c

A · dr

︸ ︷︷ ︸fluxoid

= nΦ0.

In the Doll-Näbauer experiment [15] the quantization of the �uxoid was meas-ured, using a superconducting hollow cylinder. Doll and Näbauer [16] and Deaverand Fairbank [17] showed independently and simultaneously, that the magnetic �uxthrough a hollow cylinder only appears in multiples of the �uxoid quantum Φ0. Ana-lytic results for hollow cylinders were provided by R. Doll and D. Einzel in a paperthat was dedicated to V. L. Ginzburg on the occasion of his 90th birthday in 2006 [13].

In what follows, we wish to restrict our considerations to the case that no �uxoidquantum is present inside the superconducting full cylinder and we therefore considerhere the case n = 0, in which the phase gradient term is absent. The GL equationcan then be written in the compact form

ξ2GL∇2f + f − f3 =

(2πξGL

Φ0

)2

A2f

∇2A− f2

λ2L

A = 0,

which describes the screening of the magnetic �eld B(r) = ∇ × A(r) inside thesuperconductor and the accompanied reduction of the dimensionless order parameterf(r). The treatment by Fritz and Heinz London was restricted to the special casef(r) = 1 and hence to the solution of the second GL screening equation only [2]. Newin our approach is the fact that f(r) may vary spatially. Note that the normalizedorder parameter f(r) does not only depend on the spatial variable r, but also on theexternal �eld B0, the sample size R, the temperature T , the transition temperature

5

Chapter 2 Ginzburg-Landau equations for ψ(r) and ∆(r) in the presence of B0

Tc and the GL parameter κ, which allows the distinction between type-I (κ < 1/√

2)and type-II superconductors (κ > 1/

√2):

f(r) = f(r;B0, R, Tc, T, κ).

In order to proceed with an analytical treatment of the coupled GL equations, it isconvenient to investigate linearized versions of these. There exist two possibilities fora linearization of the coupled GL equations. First, one may treat the limit in whichthe order parameter f(r) � 1 is strongly reduced, which happens in the case ofstrong externally applied magnetic �elds. Second, one may linearize with respect toa nearly fully developed superconducting ground state, represented by the conditionf(r) = 1−g(r) with g(r)� 1. This happens in the case of a weak externally appliedmagnetic �eld (low-�eld or London-limit).In the low-�eld-limit the magnetic �eld almost can not penetrate the superconductor,if the magnetic penetration depth λL is small compared to the typical sample size.The order parameter is supressed only near the boundary, whereas for su�centlyhigh �elds, superconductivity is destroyed and the �eld penetrates completely intothe sample. At a certain critical �eld for H = Hc2, superconducting regions beginto nucleate spontaneously [9, 18]. The order parameter f(r) is �nite only near thecylinder axis for radii r within a few coherence lengths, which can be referred to as"local nucleation of superconductivity" [9, 18]. This behaviour of the order parameteris in sharp contrast to the solution in the low �eld limit.A detailed analysis of the two limits applied for full superconducting slabs and fullcylinders is provided by D.Einzel [10] and P.G. de Gennes [18].

6

Chapter 3

Boundary condition for ψ(r) and ∆(r)for a superconduction-vacuum interface

In deriving the �rst and second GL equations from the GL functional of the freeenergy by calculus of variations, there arises a surface integral (after using Green'stheorem) [19]. The boundary condition arises from the condition, that this surfaceintegral has to vanish [3, 18]

ih

2m∗

∮∂V

dS ·(−ih∇ψ − e∗A

cψ

)δψ∗.

The surface contribution vanishes only if one imposes the boundary condition{n ·(−ih∇− e∗A

c

)ψ(r)

}r∈∂V

= 0.

To see the physical content of this boundary condition, we make use of Madelung'sdecomposition ψ = aeiφ and obtain the alternative condition{

n ·(h∇φ− e∗A

c− ih∇a

a

)ψ(r)

}r∈∂V

= 0.

The supercurrent density reads

jse = e∗|ψ|2vs =e∗

m∗a2

(h∇φ− e∗A

c

)⇔ h∇φ− e∗A

c=m∗

e∗1

a2jse,

so that the boundary condition can be reexpressed in terms of jse as follows{[m∗

e∗1

a2n · jse −

ih

an · (∇a)

]ψ(r)

}r∈∂V

= 0.

Both, the real part and the imaginary part have to vanish. From the real part acondition is found that no supercurrent passes normal to the surface at the boundary

{n · jse}r∈∂V = 0

7

Chapter 3 Boundary condition for ψ(r) and ∆(r) for a superconduction-vacuum

interface

and from the imaginary part a condition of vanishing directional derivative of theamplitude a(r) at the surface

{n · (∇a)}r∈∂V = 0.

In the special case of a full cylinder of radius R with a wave function f(r) = 1− g(r)this boundary condition leads to g′(R) = dg/dr(R) = 0. We will make use of thiscondition later when determining unknown integration constants.The condition used above{

n ·(−ih∇− e∗A

c

)ψ(r)

}r∈∂V

= 0,

which assures that no current passes through the surface, is appropriate for asuperconductor-vacuum interface. For other possible interfaces, for example asuperconductor-normal metal interface, P.G. de Gennes has shown that the con-dition must be generalized to read [18]{

n ·(−ih∇− e∗A

c

)ψ

}r∈∂V

= − ihbψ,

with b the so called extrapolation length. It is the length which states how far theorder parameter extends into the normal metal (via the proximity e�ect [20]). It canbe interpreted as the magnitude of superconducting electron pairs penetrating thenormal metal. This condition implies a dramatic reduction of the wave function andhence the order parameter near the surface, independent of the magnetic �eld. Therealso exists a boundary induced reduction of the order parameter, connected with theboundary condition f(R) = 0 at the boundary. In such a case the external magnetic�eld is of minor importance for the description of order parameter reduction nearthe surface.The nature of our approach for a superconducting-vaccum interface is, however,that the magnetic �eld is the driving force for the reduction of the order parameternear the boundary. We will therefore not pursue the de Gennes boundary conditionany further, instead we shall concentrate on the case of a superconductor-vacuuminterface.

8

Chapter 4

Ginzburg-Landau equations for

superconducting cylinders

After having clari�ed the boundary condition, let us now return to the known com-pact form of the two GL equations

ξ2GL∇2f + f − f3 =

(2πξGL

Φ0

)2

A2f

∇2A− f2

λ2L

A = 0.

The coupled GL equations can be rewritten in terms of cylindrical coordinates asfollows (for intermediate calculations see appendix A)

ξ2GL

(f ′′ +

f ′

r

)+ f − f3 =

(2πξGL

Φ0

)2

A2φf

A′′φ +A′φr−Aφr2− f2

λ2L

Aφ = 0,

where f ′ = ∂f(r)/∂r and f ′′ = ∂2f(r)/∂r2. The spatial dependence of Aφ(r) inthe second GL equation is characterized by London's magnetic penetration depthλL, which provides a measure of how far the �eld penetrates into the sample. Thecharacteristic length scale for the variation of the order parameter is described bythe GL coherence length ξGL in the �rst GL equation.This pair of DEQ's can not be solved analytically in this general form. We alreadydiscussed the two possibilites for a linearization of the GL equation. In the fol-lowing, we will only consider the low-�eld or London-limit in which one linearizeswith respect to the low-�eld-limit, according to f(r) = 1 − g(r), which means thatsuperconductivity is nearly everywhere fully developed.

9

Chapter 5

Solution of the linearized

Ginzburg-Landau equations for g(r)and Aφ(r)

With the new function g(r)� 1 from the linearization procedure

f(r) = 1− g(r),

the new length

ξ =ξGL√

2

and the critical �eld

Bc2(T ) =Φ0

2πξ2GL(T )

the two GL equations can be written in the following form (for intermediate calcu-lations see appendix B.1)

g′′ +g′

r− g

ξ2= − 1

2ξ2

A2φ

B2c2ξ

2GL

A′′φ +A′φr−Aφr2−Aφλ2

L

= 0 ⇔ B′′z +B′zr− Bzλ2L

= 0.

Note that in deriving the pair of linearized GL equations, we have omitted a termof the order O(gAφ). This can be justi�ed by observing that the solution for g isexpected to be of the order O(|Aφ|2) and we have therefore gAφ = O(|Aφ|2Aφ),which can be neglegted in the leading order.We will solve the second GL equation for Aφ(r) and use the result as an inhomo-

geneity in the �rst GL equation.

11

Chapter 5 Solution of the linearized Ginzburg-Landau equations for g(r) and Aφ(r)

In order to solve the DEQs for Aφ(r) and Bz(r), it is convenient to rewrite themin terms of Bessel's DEQ by introducing the dimensionless variable z = r/λL. Theresults read

z2A′′φ + zA′φ −(ν2 + z2

)Aφ = 0 ; ν = 1

z2B′′z + zB′z −(ν2 + z2

)Bz = 0 ; ν = 0.

The solution for Aφ is a linear combination of the modi�ed or hyperbolic Besselfunctions I1(z) and K1(z)

Aφ(r) = a1I1

(r

λL

)+ a2K1

(r

λL

).

The solution for Bz(r) is a linear combination of the hyperbolic Bessel functionsI0(z) and K0(z)

Bz(r) = b1I0

(r

λL

)+ b2K0

(r

λL

)Making use of the following identities [21], which hold for the hyperbolic Bessel

functions

dI0(z)

dz= I1(z)

dK0(z)

dz= −K1(z)

dI1(z)

dz= I0(z)− I1(z)

zdK1(z)

dz= −K0(z)− K1(z)

z

we may state that since

Bz(r) = A′φ(r) +Aφ(r)

r

=1

λL

[dAφ(z)

dz+Aφ(z)

z

]=

1

λL

[a1

(I0(z)− I1(z)

z

)+ a2

(−K0(z)− K1(z)

z

)+a1I1(z) + a2K1(z)

z

]=

1

λL[a1I0(z)− a2K0(z)]

12

we may identify the coe�cients

b1 =a1

λL; b2 = − a2

λL.

The regularity requirement for the solutions on the cylinder axis at r = 0 impliesthat the coe�cient of K0 vanishes and we are left with

Aφ(r) = a1I1

(r

λL

)Bz(r) =

a1

λLI0

(r

λL

)Bz(R) = B0 =

a1

λLI0

(R

λL

)a1 =

λLB0

I0

(RλL

) .The �nal result for Aφ(r) and Bz(r) is:

Aφ(r) =λLB0

I0

(RλL

)I1

(r

λL

)

Bz(r) = B0

I0

(rλL

)I0

(RλL

) .This result is reminiscent of the slab case [10]

Ay(x) =λLB0

cosh(

d2λL

) sinh

(x

λL

)

Bz(x) = B0

cosh(xλL

)cosh

(d

2λL

) .The vector potential Aφ(r) may now be inserted into the inhomogeneous second

order DEQ for g(r)

g′′ +g′

r+

g

ξ2= − b20

2ξ2I2

1

(r

λL

), b0 =

κ

I0

(RλL

) B0

Bc2.

13

Chapter 5 Solution of the linearized Ginzburg-Landau equations for g(r) and Aφ(r)

The general solution of an inhomogeneous DEQ is the sum of the general solutiongh(r) of the related homogeneous DEQ and a particular solution gp(r) [22]:

g(r) = gh(r) + gp(r).

First, one has to determine the fundamental system of linearly independent func-tions gh1(r) and gh2(r) of the homogeneous solution

gh(r) = c1gh1(r) + c2gh2(r).

The solutions are linearly independent, if the associated Wronski determinant (or"Wronskian") does not vanish

W (r) = detW(r) = gh1(r)g′h2(r)− gh2(r)g′h1(r),

where

W(r) =

(gh1(r) gh2(r)g′h1(r) g′h2(r)

)denotes the Wronski matrix. The homogeneous solution reads

gh(r) = c1I0

(r

ξ

)+ c2K0

(r

ξ

)with the Wronskian

W (r) = I0

(r

ξ

)K ′0

(r

ξ

)−K0

(r

ξ

)I ′0

(r

ξ

)= −1

r6= 0.

Second, one has to construct the particular solution gp(r), using the Lagrangeprocedure of "variation of constants"

gp(r) = c1(r)gh1(r) + c2(r)gh2(r),

where the constants c1 and c2 in the homogeneous solution are replaced by the twounknown functions c1(r) and c2(r). The result for these functions reads [19]

c1(r) = −r∫d

dtgh2(t)h(t)

W (t)= − b20

2ξ2

r∫d

dttK0

(t

ξ

)I2

1

(t

λL

)

c2(r) = −r∫d

dtgh1(t)h(t)

W (t)=

b202ξ2

r∫d

dttI0

(t

ξ

)I2

1

(t

λL

)

14

such that we �nally arrive at the following solution of the inhomogeneous DEQ

gp(r) =b202ξ2

−I0

(r

ξ

) r∫d

dttK0

(t

ξ

)I2

1

(t

λL

)+K0

(r

ξ

) r∫d

dttI0

(t

ξ

)I2

1

(t

λL

) .

Since the integrals do not seem to be manageable analytically for arbitraty r/λL,we decide now to proceed �rst with the asymptotic limit of large radii r/λL, in whichthe hyperbolic Bessel functions Iν(z) and Kν(z) simplify considerably [21]:

Iν(z) =ez√2πz

{1− 4ν2 − 1

8z+

(4ν2 − 1)(4ν2 − 9)

2!(8z)2− ...

}Kν(z) =

√π

2ze−z

{1 +

4ν2 − 1

8z+

(4ν2 − 1)(4ν2 − 9)

2!(8z)2+ ...

}.

In the half space limit R/λL � 1 and R/ξ � 1 of a superconducting cylinder onemay write

limRξ→∞

I0

(R

ξ

)= lim

Rξ→∞

I1

(R

ξ

)=

√ξ

2πReRξ

limRξ→∞

K0

(R

ξ

)= lim

Rξ→∞

K1

(R

ξ

)=

√ξπ

2Re−Rξ

limRλL→∞

I21

(R

λL

)=

λL

2πRe

2RλL .

After inserting the asymptotic forms into the expression for gp(r), we obtain thecompact result

gp(r) =b20λL

8ξπ

1√r

−e rξr∫d

dt1√te− tξ e

2tλL + e

− rξ

r∫d

dt1√tetξ e

2tλL

=b20λL

8ξπ

1√r

∑s=±1

se−s r

ξ

r∫d

dt√teαst

︸ ︷︷ ︸Is(r)

; αs =2

λL+s

ξ.

We have to deal with integrals of the error function type

Is(r) =

r∫d

dt√teαst =

2√αs

√αsr∫

√αsd

dxex2

︸ ︷︷ ︸=√π

2ierf(ix)

=

√π

αs

(1

ierf(ix)

)√αsr√αsd

; x =√αsr.

15

Chapter 5 Solution of the linearized Ginzburg-Landau equations for g(r) and Aφ(r)

We will see that the arbitrary integration limit d will contribute to the generalsolution of the homogeneous DEQ only and can therefore be ignored in the particularsolution.

Making use of the asymptotic behavior of the hyperbolic error function erf(ix)/i

1

ierf(ix)

x�1=

1√π

ex2

x=

1√π

eαsr√αsr

,

we may write the result for the integral Is(r) in the simple �nal form

Is(r) =eαsr

αs√r− eαsd

αs√d.

Inserting Is(r) into the expression for gp(r) leaves us with (for intermediate calcula-tions see appendix B.2)

gp(r) =b20λL

8ξπ

1√r

∑s=±1

se−s r

ξ Is(r) = ab202I2

1

(r

λL

)+ c3(d)I0

(r

ξ

)+ c4(d)K0

(r

ξ

).

We have introduced two constants which depend on the lower integration limit dthrough

c3(d) =b202ξ2

I21

(d

λL

)K0

(d

ξ

)d

α−

c4(d) = − b202ξ2

I21

(d

λL

)I0

(d

ξ

)d

α+.

This linear combination contributes to the general solution of the homogeneous DEQonly and can therefore be ignored. To summarize, we have obtained an approxim-

ate form for gp(r) ≈ g(0)p (r), which becomes exact in the asymptotic limit r/λL →∞.

The inhomogeneous second order DEQ for g(r) is, as before

h(r) = g′′ +g′

r− g

ξ2= − b20

2ξ2I2

1

(r

λL

)⇔ g − ξ2 1

rg′ − ξ2g′′ =

b202I2

1

(r

λL

)= d[g] = dg.

In the last step we de�ned the linear di�erential operator

d = 1− ξ2 1

r

d

dr− ξ2 d

2

dr2

dg := d[g] = g − ξ2 1

r

dg

dr− ξ2d

2g

dr2= g − ξ2 g

′

r− ξ2g′′.

16

We have already obtained an approximate form gp(r) ≈ g0p(r), which becomes exact

in the asymptotic limit r/λL →∞

g0p =

b202aI2

1

(r

λL

).

In order to extend this result systematically to lower values r/λL, we start from thefollowing ansatz for an iteration procedure [10]:

g1p(r) = g0

p(r) + δg0p(r)

Then the inhomogeneous DEQ can be rewritten in the form

d[g1p(r)] = d[g0

p(r)] + d[δg0p(r)] =

b202I2

1

(r

λL

)h0(r) :=

b202I2

1

(r

λL

)− d[g0

p(r)] = d[δg0p(r)].

h0(r) serves as an inhomogeneity in the second order DEQ for δg0p(r)

d[δg0p(r)] = δg0

p(r)− ξ2δg0′p (r)

r− ξ2δg0′′

p (r).

The function h0(r) is given by a Taylor expansion of the equation

h0(r) :=b202I2

1

(r

λL

)− d[g0

p(r)]

and yields

h0(r) =b202

[a− 1

4+

(1− a16λ4

L

+15aξ2

64λ6L

)r4 +

(5(1− a)

768λ6L

+7aξ2

288λ8L

)r6 +O

(r8)].

Taking only the leading order term into account, we obtain

h0(r) =b202

a− 1

4= const,

which immediately implies the following result for δg0p(r)

δg0p(r) =

b202

a− 1

4

and we may write

g1p(r) = g0

p(r) + δg0p(r) =

b202

[aI1

(r

λL

)2

+a− 1

4

].

17

Chapter 5 Solution of the linearized Ginzburg-Landau equations for g(r) and Aφ(r)

From the second iteration procedure one gets

h1(r) =b202

[(1− a16λ4

L

+15aξ2

64λ6L

)r4 +

(5(1− a)

768λ6L

+7aξ2

288λ8L

)r6 +O

(r8)].

In summary, we have been able to prove, that gp(r) assumes the following approx-imate form, valid for all r/λL (Figure 5.1)

gp(r) ≈ g1p(r) =

b202

[aI2

1

(r

λL

)+a− 1

4

].

This form for gp(r), though approximate, represents a very accurate interpolationbetween the two limits r/λL →∞ (asymptotic limit) and r/λL → 0 (behavior nearthe cylinder axis).

0 20 40 60 80 1000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

r

ΛL

rela

tive

erro

r

Figure 5.1: relative error plotted versus the dimensionless variable r/λL

The relative error err(r) never exeeds the value of 7 percent and vanishes near thecylinder axis and in the asymptotic limit (for detailed calculations see appendix B.3).

With the knowledge of this particular solution, we may now proceed and con-struct the general solution of the inhomogeneous DEQ in the usual way:

18

g(r) = cgh(r) + gp(r) = cI0

(r

ξ

)+b202

[aI2

1

(r

λL

)+a− 1

4

]

g′(r) =c

ξI1

(r

ξ

)+

2a

λL

b202I1

(r

λL

)I0

(r

λL

)−I1

(rλL

)rλL

.The GL boundary condition g′(R) = 0 �xes the unknown integration constant c

to be

c = −ab20

2

2ξ

λL

I1

(RλL

)I1

(Rξ

)I0

(R

λL

)−I1

(RλL

)Rλ L

and we have the following result for the general solution g(r)

g(r) =b202

aI21

(r

λL

)+a− 1

4− a 2ξ

λL

I1

(RλL

)I1

(Rξ

)I0

(R

λL

)−I1

(RλL

)RλL

I0

(r

ξ

) .

19

Chapter 6

Discussion of the analytical results for

f (r) and Bz(r)

In summary, we obtained the following results for the behaviour of the order para-meter

f(r) = 1− b202

aI21

(r

λL

)+a− 1

4− a 2ξ

λL

I1

(RλL

)I1

(Rξ

)I0

(R

λL

)−I1

(RλL

)RλL

I0

(r

ξ

)and the magnetic �eld penetration in a superconducting full cylinder

Bz(r) = B0

I0

(rλL

)I0

(RλL

) .In order to proceed with a numerical analysis of those results, we have to introduce

dimensionless quantities. We start with the speci�cation of the ratio

ζ =R

ξGL(0).

Then we know the dimensionless quantities

R

ξGL(T )=

R

ξGL(0)

√1− T

Tc= ζ

√1− T

Tc

R

λL(T )=ζ

κ

√1− T

Tc

r

ξGL(T )=

r

R

R

ξGL(T )=

r

Rζ

√1− T

Tc

r

λL(T )=

r

R

ζ

κ

√1− T

Tc

with the aid of which the spatial dependence of the normalized order parameter f(r)may be analysed (for detailed calculations see appendix C).

21

Chapter 6 Discussion of the analytical results for f(r) and Bz(r)

R�ΞGLH0L = 1

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

r

R

BzHrLB0

R�ΞGLH0L = 2

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

r

R

BzHrLB0

R�ΞGLH0L = 4

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

r

R

BzHrLB0

R�ΞGLH0L = 8

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

r

R

BzHrLB0

R�ΞGLH0L = 15

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

r

R

BzHrLB0

R�ΞGLH0L = 30

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

r

R

BzHrLB0

Figure 6.1: spatial dependence of the normalized magnetic induction Bz(r)/B0 insidea superconducting full cylinder with radius R for κ = 0.4, B0/Bc2(0) = 0.04, for sixdi�erent geometry parameters ζ, in each case for three di�erent reduced temperaturesT/Tc = 0.95 (blue), T/Tc = 0.9 (red), T/Tc = 0.85 (green).

22

R�ΞGLH0L = 1

-1.0 -0.5 0.0 0.5 1.00.988

0.990

0.992

0.994

0.996

0.998

1.000

r

R

f HrLR�ΞGLH0L = 2

-1.0 -0.5 0.0 0.5 1.00.988

0.990

0.992

0.994

0.996

0.998

1.000

r

R

f HrL

R�ΞGLH0L = 4

-1.0 -0.5 0.0 0.5 1.00.988

0.990

0.992

0.994

0.996

0.998

1.000

r

R

f HrLR�ΞGLH0L = 8

-1.0 -0.5 0.0 0.5 1.00.988

0.990

0.992

0.994

0.996

0.998

1.000

r

R

f HrL

R�ΞGLH0L = 15

-1.0 -0.5 0.0 0.5 1.00.988

0.990

0.992

0.994

0.996

0.998

1.000

r

R

f HrL R�ΞGLH0L = 30

-1.0 -0.5 0.0 0.5 1.00.988

0.990

0.992

0.994

0.996

0.998

1.000

r

R

f HrL

Figure 6.2: spatial dependence of the normalized order parameter f(r) = ∆(r)/∆0

inside a superconducting full cylinder with radius R for κ = 0.4, B0/Bc2(0) = 0.04,for six di�erent geometry parameters ζ, in each case for three di�erent reducedtemperatures T/Tc = 0.95 (blue), T/Tc = 0.9 (red), T/Tc = 0.85 (green).

23

Chapter 6 Discussion of the analytical results for f(r) and Bz(r)

In �gure 6.1 we have plotted the normalized magnetic induction Bz(r)/B0 and in�gure 6.2 the normalized order parameter f(r) = ∆(r)/∆0 in a superconducting fullcylinder of radius R against r/R. The parameters used in these plots are κ = 0.4,B0/Bc2 = 0.04 and six di�erent values for ζ. The results are shown for threedi�erent temperatures T/Tc = 0.95 (blue), T/Tc = 0.9 (red), T/Tc = 0.85 (green).For small values of ζ (thin wires), the magnetic �eld can penetrate into the fullcylinder. The order parameter pro�le is nearly constant for ζ ≤ 2 and then becomesbell-sharped for larger ζ. This is the �eld penetration regime (emerging for up toζ ≈ 8). For large values of ζ (thick wires), the �eld can penetrate only near thesuperconductor-vaccum interface. The order parameter becomes �at and equal to 1near the symmetry center. It is only suppressed near the boundaries. This is the�eld screening regime. The expulsion of the magnetic �eld from the superconductingfull cylinder is part of the Meiÿner-Ochsenfeld e�ect.

There emerge two important limiting values for this order parameter pro�le,namely the order parameter on the cylinder axis f(0) = 1− g(0)

f(0) = 1− b202

a− 1

4− a 2ξ

λL

I1

(RλL

)I1

(Rξ

)I0

(R

λL

)−I1

(RλL

)RλL

; b0 =κ

I0

(RλL

) B0

Bc2

and the order parameter at the boundary f(R) = 1− g(R)

f(R) = 1− b202

aI21

(R

λL

)+a− 1

4− a 2ξ

λL

I1

(RλL

)I1

(Rξ

)I0

(R

λL

)−I1

(RλL

)RλL

I0

(R

ξ

) .

In the limit of an in�nite radius R (thick wires) the result for f(r) should beconnected to that of the half space geometry[10]. Using the asymptotic form for thehyperbolic Bessel functions indeed yields the half-space limit

limRλL→∞

f(0) = 1

limRλL→∞

f(R) = 1− κ2

2

B20

B2c2

1ξ

1ξ + 2

λL

.

This can be interpreted such that the only e�ect of an externally magnetic �eld B0

on a superconducting full cylinder in the limit of a large radius R is the suppressionof the order parameter at the boundary by an amount

g(R) =κ2

2

B20

B2c2

1ξ

1ξ + 2

λL

,

24

which depends on the ratios κ and B0/Bc2 .

Finally we consider the temperature dependence of the order parameter f(r).The sources of temperature dependence are the lengths λL(T ) and ξGL(T ), theexplicit temperature dependence of f(R→∞) in the GL regime is

limRλL→∞

f(R) = 1− κ2

2

κ

κ+√

2

B20

Bc2(0)2

1(1− T

Tc

)2 .

This temperature dependence is quite strong: if B0/Bc2(0) = 0.04 thenB0/Bc2(0.95Tc) = 0.8. For a certain temperature, the critical magnetic �eld Bc2

must always be lower than B0, otherwise no superconducting phase can appear

B0

Bc2(T )=

B0

Bc2(0)

1

1− TTc

!< 1.

25

Chapter 7

Summary and conclusion

This work provides an analytic calculation of the linear response of the order para-meter f(r) = ψ(r)/ψ0, with ψ(r) the pair �eld, to externally applied low magnetic�elds in the GL temperature regime for superconducting full cylinders. Linear re-sponse means that one linearizes with respect to a nearly fully developed supercon-ducting ground state. As a consequence the analytical result for Aφ(r) is equal tothe solution of London's theory. Calculations at least for superconducting slabs cantherefore be found in many textbooks on superconductivity [20].The novelty in this work is the result for the spatially dependent order parameterf(r), which was assumed to be constant in London's approach. The order para-meter f(r) is not only dependent of the variable r but also of physical quantities likethe experimental parameters T and B0 as well as the sample parameters R, Tc andκ = λL/ξGL. We restricted our consideration to type-I-superconductors (κ < 1/

√2),

so no �uxoid quantum is present inside the superconductor. The GL theory is validonly near the transition temperature. For the parameter T/Tc we chose therefore therange 0.85 < T/Tc < 1. For the superconducting phase to occur, the ratio B0/Bc2(T )must be smaller than 1, therefore B0/Bc2(0) must be set su�ciently small because ofthe strong temperature dependence. R/ξGL and R/λL respectively, varies accordingto whether one considers thick or thin superconducting full cylinders.The pro�les of f(r) generally describe the order parameter reduction near the sur-face. For small values of ζ = R/ξGL(0), the magnetic �eld penetrates into the fullcylinder (�eld penetration regime) and the order parameter f(r) is bell-shaped. Forlarge values of ζ, the Meiÿner-Ochsenfeld e�ect emerges. The �eld can penetrateonly near the boundary (�eld screening regime). The order parameter becomes �atand equal to 1 and is only suppressed near the boundary. In the limit of large ζ, theorder parameter pro�le f(r) can be well characterized by the half space limit.

27

Appendix A

Ginzburg-Landau equations for

superconducting full cylinders

For solving the coupled GL equation for a full cylinder in a magnetic �eld, the useof cylindrical coordinates is recommended.For an arbitrary function a(r) the Laplace operator ∇2 in cylindrical coordinates

reads

∇2a(r) =1

r

∂

∂r

(r∂

∂r

)a(r) = a′′ +

a′

r.

We assume that the magnetic �eld Bz(r) is oriented parallel to the cylinder axisalong the z-direction and is independent of the azimuthal angular coordinate φ. Theassociated vector potential is A(r) = Aφ(r)φ

B = ∇×A =

(1

r

∂Az∂φ−∂Aφ∂z

)r︸ ︷︷ ︸

=0

+

(∂Ar∂z− ∂Az

∂r

)φ︸ ︷︷ ︸

=0

+1

r

(∂

∂r(rAφ)− ∂Ar

∂φ

)z︸ ︷︷ ︸

=Bz z

B = Bz(r)z =

(Aφr

+∂Aφ∂r

)z

∇×B = ∇× (∇×A) = ∇ (∇ ·A)−∇2A = −∂Bz∂r

φ = −

(A′′φ +

A′φr−Aφr2

)φ.

With those tools the GL equations can be rewritten in the form

ξ2GL

(f ′′ +

f ′

r

)+ f − f3 =

(2πξGL

Φ0

)2

A2φf

A′′φ +A′φr−Aφr2− f2

λ2L

Aφ = 0.

29

Appendix B

Solution of the linearized

Ginzburg-Landau equations for g(r)and Aφ(r)

B.1 derivation of the Ginzburg-Landau equations in

cylindrical coordinates

After the linearization procedure the GL equations can be written in the followingform:

g′′ +g′

r− g

ξ2= − 1

2ξ2

A2φ

B2c2ξ

2GL

A′′φ +A′φr−Aφr2−Aφλ2

L

= 0 ⇔ B′′z +B′zr− Bzλ2

L

= 0.

Let us brie�y prove the last relationship:

Aφ = λ2L

(A′′φ +

A′φr−Aφr2

)

A′φ = λ2L

(A′′′φ +

A′′φr−

2A′φr2

+ 2Aφr3

).

As we calculated before

Bz(r) =

(Aφr

+∂Aφ∂r

)= A′φ +

Aφr

31

Appendix B Solution of the linearized Ginzburg-Landau equations for g(r) and

Aφ(r)

0.0 0.5 1.0 1.5 2.0 2.5 3.00

5

10

15

r

Kn@rD

0 1 2 3 4 50

5

10

15

r

In@rD

Figure B.1: left: hyperbolic Bessel function K0(r) (red), K1(r) (green), right: hy-perbolic Bessel function I0(r) (red), I1(r) (green)

and we get the di�erential equation in B

Bz = A′φ +Aφr

= λ2L

(A′′′φ + 2

A′′φr−A′φr2

+Aφr3

)= λL

(B′′z +

B′zr

).

We introduce the dimensionless variable z = r/λL

d2

dr2Aφ =

d

dr

(d

drAφ (z(r))

)=

d

dr

(dAφdz

dz

dr

)=

d

dr

(A′φ(z)

1

λL

)=dA′φdz

dz

dr

1

λL⇒ d2

dr2=

1

λ2L

d2

dz2

1

λLA′′φ +

1

λLz

1

λLA′φ −

(1

zλL

)2

Aφ −Aφλ2

L

= 0.

We rewrite the DEQ for Aφ(r) and Bz(r) in terms of Bessel's DEQ

z2A′′φ + zA′φ −(ν2 + z2

)Aφ = 0 ; ν = 1

z2B′′z + zB′z −(ν2 + z2

)Bz = 0 ; ν = 0.

The solution for Aφ and Bz(r) is a linear combination of hyperbolic Bessel functions(�gure B.1).

32

B.2 particular solution gp(r)

B.2 particular solution gp(r)

After inserting the asymptotic forms into the expression for gp(r) we get the compactresult

gp(r) =b20λL8ξπ

1√r

∑s=±1

se−s r

ξ

r∫d

dt√teαst

︸ ︷︷ ︸Is(r)

; αs =2

λL+s

ξ.

We may write the result for the integral Is(r) in the simple �nal form

Is(r) =eαsr

αs√r− eαsd

αs√d.

Inserting Is(r) into the expression for gp(r) yields

gp(r) =b20λL

8ξπ

1√r

−e rξ e

( 2λL− 1ξ

)r(2λL− 1

ξ

)√r− e

( 2λL− 1ξ

)d(2λL− 1

ξ

)√d

+ e− rξ

e( 2λL

+ 1ξ

)r(2λL

+ 1ξ

)√r− e

( 2λL

+ 1ξ

)d(2λL

+ 1ξ

)√d

=b204ξ

λL

2πre

2rλL︸ ︷︷ ︸

=I21

(rλL

)

− 1(2λL− 1

ξ

) +1(

2λL

+ 1ξ

) +

√r

d

e2d−2rλL

+ r−dξ

2λL− 1

ξ

−√r

d

e2d−2rλL− r−d

ξ

2λL

+ 1ξ

=b202I2

1

(r

λL

)[ 1ξ2

1ξ2− 4

λL

]︸ ︷︷ ︸

=a

+b204ξ

λL

2πde

2dλL︸ ︷︷ ︸

I21

(dλL

)√d

r

[1

2λL− 1

ξ

− 12λL

+ 1ξ

]

=b202aI2

1

(r

λL

)+

b202ξ2

I21

(d

λL

)[√πξ

2de− dξ

d2λL− 1

ξ

√ξ

2πrerξ −

√πξ

2re− rξ

d2λL

+ 1ξ

√ξ

2πdedξ

]

= ab202I2

1

(r

λL

)+ c3(d)I0

(r

ξ

)+ c4(d)K0

(r

ξ

).

33

Appendix B Solution of the linearized Ginzburg-Landau equations for g(r) and

Aφ(r)

B.3 relative error

corr(r) := h(r) = g − ξ2 1

rg′ − ξ2g′′ =

b202I2

1

(r

λL

)approx(r) := g1

p − ξ2 1

rg1′p − ξ2g1′′

p

g1p(r) =

b202

[aI2

1

(r

λL

)+a− 1

4

], with a =

1ξ2

1ξ2− 4

λL

.

We considered type-I superconductors. In the plots for the order parameter f(r)and Bz(r)/B0(r) we chose κ to be 0.4.

κ =λL

ξGL=

λL√2ξ

= 0.4 → ξ = 10λ = 4√

2

err(r) =corr(r)− approx(r)

approx(r)

34

Appendix C

Discussion of the analytical results for

f (r) and Bz(r)

We get for the order parameter the following �nal result:

g(r) =b202

aI21

(r

λL

)+a− 1

4− a 2ξ

λL

I1

(RλL

)I1

(RλL

)I0

(R

λL

)−I1

(RλL

)RλL

I0

(r

ξ

) .

To plot g(r) we introduce some dimensionless variables

κ =λL

ξGLζ =

R

ξGL(0)h =

B0

Bc2(0)t =

T

Tcκ =√

2κ y =r

R

Bc2(T ) =Φ0

2πξ2GL(T )

ξGL(T ) =ξGL(0)√(

1− TTc

) Bc2(T ) = Bc2(0) ·(

1− T

Tc

)

a =

1ξ2

1ξ2− 4

λL

=κ2

κ2 − 4

2ξ

λL=

2

κ

α :=r

λL=

1

κyξ√

1− t β :=r

ξ=√

2yζ√

1− t

γ :=R

λL=

1

κζ√

1− t δ :=R

ξ=√

2ζ√

1− t

µ :=κ2h2

2

1

I20 (γ)

1

2κ2 − 4

g(y, t) =µ

(1− t)2

{2κ2I2

1 (α) + 1− 2√

2κI1(γ)

I1(δ)

[I0(γ)− I1(γ)

γ

]I0(β)

}.

35

Appendix C Discussion of the analytical results for f(r) and Bz(r)

For the behaviour of the magnetic �eld Bz(r) in cylinders one gets

Bz(r) = B0

I0

(rλL

)I0

(RλL

) = B0I0(α)

I0(γ).

36

Bibliography

[1] W. Meiÿner and R. Ochsenfeld. `Ein neuer E�ekt bei Eintritt der Supraleit-fähigkeit'. In: Naturwissenschaften 21 (1933), pp. 787�788.

[2] F. London and H. London. `The Electromagnetic Equations of the Supracon-ductor'. In: Royal Society of London Proceedings Series A 149 (1935), pp. 71�88.

[3] V. L. Ginzburg and L. D. Landau. `On the theory of superconductivity'. In:Zh. Eksp. Teor. Fiz. 20 (1950), p. 1064.

[4] J. Bardeen, L. N. Cooper and J. R. Schrie�er. `Theory of Superconductivity'.In: Physical Review 108 (1957), pp. 1175�1204.

[5] L. P. Gor'kov. `Microscopic derivation of the Ginzburg Landau equations in thetheory of Superconductivity'. In: Sov.Phys.JETP 36 (1959), pp. 1364�1367.

[6] J. Woste. Analytic Ginzburg-Landau description of superconducting slabs and

cylinders. Bachelor Thesis, Technische Universität München, 2012.

[7] R. Doll and P. Graf. `Eigenschaften der Lösungen der nichtlinearisier-ten Ginsburg-Landau-Gleichungen in Zylindersymmetrie'. In: Zeitschrift fürPhysik 197 (1966), pp. 172�191.

[8] G. F. Zharkov, V. G. Zharkov and A. Yu. Zvetkov. `Ginzburg-Landau calcula-tions for a superconducting cylinder in a magnetic �eld'. In: Phys. Rev. B 61(2000), pp. 12293�12301.

[9] R. Doll. private communication. (2012).

[10] D. Einzel. Lecture Notes: "Problems in the theory of superconductors". unpub-lished, 2013.

[11] E. Madelung. `Eine anschauliche Deutung der Gleichung von Schrödinger'. In:Naturwissenschaften 14 (1926), pp. 1004�1004.

[12] K. Yosida. `Paramagnetic Susceptibility in Superconductors'. In: Phys.Rev. 110(1958), pp. 769�770.

[13] R. Doll and D. Einzel. `Ginzburg�Landau Analysis of the Doll�Näbauer Ex-periment'. In: Journal of Superconductivity and Novel Magnetism 19 (2006),pp. 173�179.

37

Bibliography

[14] F. London. Super�uids: Macroscopic theory of superconductivity. Structure ofmatter series. John Wiley & Sons, 1950.

[15] R. Doll and M. Näbauer. `Experimenteller Nachweis der Quantisierung einge-frorener Magnet�üsse im supraleitenden Hohlzylinder'. In: Zeitschrift für

Physik 169 (1962), pp. 526�563.

[16] R. Doll and M. Näbauer. `Experimental Proof of Magnetic Flux Quantizationin a Superconducting Ring'. In: Physical Review Letters 7 (1961), pp. 51�52.

[17] B. S. Deaver and W. M. Fairbank. `Experimental Evidence for Quantized Fluxin Superconducting Cyclinders'. In: Physical Review Letters 7 (1961), pp. 43�46.

[18] P. G. de Gennes. Superconductivity of Metals and Alloys. New York: W. A.Benjamin, 1966.

[19] C. W. Wong. Introduction to Mathematical Physics - Methods and Concepts.Oxford University Press, 2013.

[20] M. Tinkham. Introduction to Superconductivity. New York: McGraw-Hill Inc.,1996.

[21] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. NewYork: Dover, 1964.

[22] H. Heuser. Gewöhnliche Di�erentialgleichungen: Einführung in Lehre und Geb-

rauch. B.G. Teubner, 1989.

38

Acknowledgments

I would like to thank my supervisor PD Dr. Dietrich Einzel for the very interestingtopic I was permitted to discuss in my Bachelor thesis. I would like to express mysincere gratitude to him for supervising me extraordinarily well. I am grateful for hispatience, motivation and his great support. Advice and comments given by him havebeen a great help in understanding the subject. Without his guidance and persistenthelp this thesis would not have been possible on this scale. Furthermore I would liketo thank him for taking time to proofread my thesis.Lastly I owe special thanks to my boyfriend Pascal, who supported and encouraged

me not only for this thesis, but for the last three years throughout my whole studies.

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