Properties in n-component London Superconductors

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Master Thesis

Properties in n-component London Superconductors

Sergio Ampuero Felix

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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TRITA-FYS 2017:83 ISSN 0280-316X

ISRN KTH/FYS/–17:83—SE

Sergio Ampuero Felix, December 2017c

Printed in Sweden by Universitetsservice US AB, Stockholm December 2017

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Abstract

The purpose of this thesis is to analytically investigate n-component London superconductor properties that have previously been investigated analytically in cases involving a lower amount of components.

For vortex excitations in n-component London superconductor, we obtain expressions for magnetic flux, magnetic vector potential, microscopic magnetic field and circulation of velocity.

Furthermore, the critical applied field for vortex formation is found.

Finally, the response of a cylindrical specimen to rotation is derived both in terms of a n-component analogue of the London field but also by finding the critical angular frequency for vortex formation.

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Preface

This thesis assumes that the reader has some familiarity with quantum physics, functional derivation, the concept of minimizing appropriate free energy and the Ginzburg-Landau model for superconductivity.

The result of work done from roughly Mars 2017 to November 2017 is man- ifested through this thesis. I would like to take this opportunity to express my utmost gratitude to my thesis supervisor Egor Bavaev for his patience and excel- lent guidance.

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Introduction

Multicomponent London superconductivity has attracted interest for an extended period of time. See for example [4] and [5] for discussion of flux fractionalization and neutral sector.

The existence of a neutral sector motivated comparison to established results from analysis of neutral one-component superfluids, in particular Onsager’s [12, year 1949] and Feynman’s [14, year 1955] quantization of vortices and the associated circulation quanta _m^~. Notably, an investigation of the two-component equivalent was published in 2007 [9].

A distinct concern is how superconductors respond to rotation. In 1950, Lon- don provided pioneering analysis of rotating one-component superconductors [13].

Consequently, a dissection of the two-component correspondent was promulgated in 2007 [9]. In this thesis, the cynosure is an analytical examination of how n- component London superconductors respond to application of an external magnetic field and rotation respectively.

Among germane work, one finds an article from 2004 [11] in which expressions for the n-component magnetic flux Φ and current ~J were partial results.

Throughout this thesis, absence of interband couplings is assumed. An article [9] proposes that a physical example of a two-component version of a such system could be liquid metallic deuterium, liquid metallic hydrogen [15] and condensates in neutron stars [16] [23] [24]. Moreover, [11] advocates that higher component cognates should exist in ”metallic phases of light atoms under extreme pressure”.

Due to metallic hydrogen requiring extreme pressures in order to be produced, the task of creating and studying such matter through experiments is a difficult one. This is a subject of broad current research efforts. For example: a prominent article [21] published in the current year, 2017, provides information to ”stimulate theoretical predictions of how to retain metastably hydrogenous materials made at high pressure P on release to ambient”.

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A different article [22] aids by ”As such, the present study bridges the important but less well-explored intermediate regime between warm dense fluid and the solid phases at high P-T conditions.”.

A third article [20] analyzes the closely related topic of semimetallic hydrogen.

Most spectacular is an article describing a realized production of solid metallic hydrogen in a laboratory setting [17]. That claim was heavily criticized through arguments indicating that there existed severe experimental flaws [18]; as a reply the authors defended their claim [19].

These current experimental efforts motivate theoretical studies of multicomponent superconducting systems that are considered in this thesis.

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Chapter 2

General Considerations

2.1 The n-component free energy density F

_s

We will investigate the n-component superconductor using the Ginzburg-Landau and London models.

Following the well-known procedure by Ginzburg and Landau [25], by introduc- ing a macroscopic effective wave function ψ as an order parameter that through

|ψ|² corresponds to number density, the free energy density of a normal specimen in the absence of a magnetic field Fn, two temperature-dependent parameters α and β, charge q, mass m, magnetic vector potential ~A and magnetic field ~h one can express the free energy density of a 1-component superconductor as following:

Fs= Fn+ α|ψ|²+β

2|ψ|⁴+ 1 2m|(~

i

∇ − q ~~ A)ψ|²+ ~h²

2µ₀ (2.1)

An important remark is that ~h refers to the microscopic magnetic field. The magnetic flux density ~B is an average of ~h over a suitable volume.

Equation 2.1 is easily generalized to n components as following:

F_s= F_n+X

l

"

α_l|ψl|²+β_l

2|ψl|⁴+ 1 2ml

|(~ i

∇ − q~ lA)ψ~ _l|²

# + ~h²

2µ0

(2.2)

It is instructive to compare that equation to the one found at [11, equation 1]

and the more general one at [10, p. 167]. As an example one can mention the liquid metallic hydrogen case [15] where different components correspond to condensates of protonic and electronic cooper pairs, respectively. A third and fourth component can arise through cooper pairs of tritium and Bose-Einstein condensation of deuterium nuclei [11]. In a neutron star example [16] [23] [24], the different components correspond to condensates of neutronic and protonic cooper pairs, respectively.

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2.2 Deriving the Ginzburg-Landau equations

Recall following from your course in electrodynamics:

~h = ~∇ × ~A (2.3)

We will obtain the Ginzburg-Landau equations by varying the free energy functional with respect to the order parameter Ψ and the magnetic vector potential A.~

2.2.1 Variation with respect to ψ

_l

We will first minimize

R³d³rF_s with respect to a variation of ψ_l:

0 =

R³

d³r

"

αlψ_l^∗δψl+βl

2|ψl|²2ψ_l^∗δψl

+(_−i^~∇ − q~ lA)ψ~ _l^∗· (^~_i∇δψ~ l− qlAδψ~ l) 2m_l

# (2.4)

Next, recall that given any scalar φ and any vector ~T , we have following vector identity:

∇(φ ~~ T ) = ~∇φ · ~T + φ ~∇ · ~T (2.5)

Utilizing equation 2.5 in 2.4 for φ := ^~_iδψ_l and ~T := ⁽

~

−i∇−q~ _lA)~

2ml ψ_l^∗, we then obtain:

0 =

R³

d³rδψ_l

"

α_lψ^∗_l + β_l|ψ_l|²ψ_l^∗+(_−i^~∇ − q~ lA)ψ~ _l^∗· (−qlA)~ 2m_l

# +

R³

d³r ~∇ ·

"

~ iδψl

(_−i^~∇ − q~ _lA)~ 2ml

ψ_l^∗

#

−

R³

d³r~ iδψl∇ ·~

"

(_−i^~∇ − q~ _lA)~ 2ml

ψ^∗_l

# (2.6)

Keep now in mind that equation 2.6 is valid as well if we change integration region from R³ to the superconducting specimen (denoted by ”super”) since the order parameter ψ_l is 0 outside the specimen. We want to do this change because δψ_l is also 0 outside the specimen and we want to utilize the condition that we can choose arbitrary δψ_linside the specimen to find useful equations.

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2.2. Deriving the Ginzburg-Landau equations 5 Furthermore, recall the generalized divergence theorem [1, p. 93]:

d³r ~∇(...) =

d ~S(...) (2.7)

We can now rewrite equation 2.6 as:

0 =

super

d³rδψ_l

"

α_lψ^∗_l + β_l|ψ_l|²ψ_l^∗+(_−i^~∇ − q~ lA)~ ² 2ml

ψ_l^∗

# +

∂super

δψld ~S ·

"

~ i

(_−i^~∇ − q~ lA)~ 2ml

ψ^∗_l

# (2.8)

Which yields a Ginzburg-Landau equation and a boundary condition for each component of the superconductor:

(^~_i∇ − q~ lA)~ ² 2ml

ψl+ αlψl+ βl|ψl|²ψl= 0 (2.9)

d ~S · (~ i

∇ − q~ lA)ψ~ l= 0 (2.10) Recall the expression for the canonical momentum in the case of a particle in a magnetic field [2, p. 308]:

~

p = m~v + q ~A (2.11)

Also recall following expression for the momentum operator:

~ˆ p = ~

i

∇~ (2.12)

We can use equations 2.11 and 2.12 to express 2.10 as following:

d ~S · m_lv~_l= 0 (2.13)

Which means that the normal component of ~v_l through the surface of the superconducting specimen is 0.

2.2.2 Variation with respect to ~ A

Next we will utilize equation 2.3 to express ~h in terms of ~ A and then minimize

R³d³rF_swith respect to a variation of ~A:

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0 =X

l

"

R³

d³r

−q_lψ_lδ ~A · (_−i^~∇ − q~ _lA)~ 2ml

ψ_l^∗+−qlψ_l^∗δ ~A · (^~_i∇ − q~ lA)~ 2ml

ψ_l

#

+

R³

d³r

"

2( ~∇ × ~A) · ( ~∇ × δ ~A) 2µ₀

#

(2.14)

Recall now that for any ~C and ~D, the following vector identity holds:

∇ · ( ~~ C × ~D) = ~D · ( ~∇ × ~C) − ~C · (∇ × ~D) (2.15) By choosing ~C := _µ^~^h

0 = ^{∇× ~}^~_µ^A

0 and ~D := δ ~A and using equation 2.15 in 2.14, we obtain:

0 =X

l

"

R³

d³r−ql

2ml

δ ~A ·

ψl( ~

−i∇ − q~ lA)ψ~ ^∗_l + ψ_l^∗(~ i

∇ − q~ lA)ψ~ l

# +

R³

d³r

"

δ ~A · (∇ × ~h~ µ0

) − ~∇ ·(~h × δ ~A) µ0

#

(2.16)

Next, we again use the general divergence theorem shown at equation 2.7, which yields:

0 =

R³

d³rδ ~A ·

"

X

l

−ql

2m_l

ψl( ~

−i∇ − q~ lA)ψ~ ^∗_l + ψ_l^∗(~ i

∇ − q~ lA)ψ~ l

+

∇ × ~h~ µ₀

# +

∂R³

d ~S ·

"

−(~h × δ ~A) µ0

#

(2.17)

Assuming that ~h × δ ~A decays faster than _r¹₂ as r → ∞, the surface integral vanishes as r → ∞. The remaining integral yields a Ginzburg-Landau equation:

∇ × ~h~ µ0

=X

l

"

q_l~ 2iml

(ψ^∗_l∇ψ~ l− ψl∇ψ~ ^∗_l) − q_l² ml

|ψl|²A~

#

(2.18)

One should also keep in mind that when the displacement current is negligible and by defining ~J := ~Jf ree+ ~Jbound , a Maxwell equation yields [3, p. 330]:

J =~

∇ × ~h~

µ₀ (2.19)

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2.4. Supercurrents in the London limit 7

2.3 London limit

As commonly taught in literature regarding superconductivity, the conventional parametrization of the order parameter ψ is ψ = |ψ|e^iθ.

In the London limit, one models |ψ| to be constant with respect to space coordinates throughout the superconducting region and zero in normal regions [13].

The London limit is reasonable when the magnetic penetration length (length scale over which the magnetic field is reduced to a small fraction of its maximum value) is much larger than the coherence lengths (length scales over which the order parameters regain their corresponding bulk values) [27].

We can express the Ginzburg-Landau equations 2.9 and 2.18 as following in the London limit:

(~~∇θl− qlA)~ ²

2m_l ψ_l+ α_lψ_l+ β_l|ψ_l|²ψ_l= 0 (2.20) J =~ X

l

"

ql~

m_l|ψ_l|²∇θ~ _l− q_l² m_l|ψ_l|²A~

#

(2.21) Furthermore we can also simplify the expression for the free energy Fsat equation 2.2 as following in the London limit:

F_s= F_n+X

l

"

α_l|ψ_l|²+βl

2|ψ_l|⁴+ 1

2ml(~~∇θ_l− q_lA)~ ²|ψ_l|²

# + ~h²

2µ0

(2.22)

The part of F_sthat is relevant in the context of vortex formation in the London limit can be described as following:

F˜s:=X

l

"

1

2m_l(~~∇θl− qlA)~ ²|ψl|²

# + ~h²

2µ₀ (2.23)

2.4 Supercurrents in the London limit

A few words of caution: what one means by ”current” is a matter of convention.

Griffith [3, p. 212] refers to ~J as ”volume current density”, but we will refer to it as ”current” throughout this text.

Recall equation 2.21:

J =~ X

l

"

ql~

m_l|ψ_l|²∇θ~ _l− q_l² m_l|ψ_l|²A~

#

(2.24)

Following the procedure for two-component superconductors done at [4], we define supercurrents ~j_l as following:

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~jl:= ql~ ml

|ψl|²∇θ~ l− q²_l ml

|ψl|²A~ (2.25)

Given that definition it is easy to notice that following holds:

J =~ X

l

~j_l (2.26)

I.e. the current ~J that appears in Maxwell’s equations is in this case the sum of all supercurrents.

The definition makes sense from another perspective as well. Using equations 2.11, 2.12 and ψ_l= |ψ_l|e^iθ^l we obtain:

~ v_l= ~

m_l

∇θ~ _l− ql

m_l

A~ (2.27)

In general a charged current is obtained by multiplying a velocity with a charge and a density. Choosing |ψl|²as the density, ~vlas the velocity and qlas the charge one obtains:

~j_l= q_lv~_l|ψ_l|²= ql~

m_l|ψ_l|²∇θ~ _l− q²_l

m_l|ψ_l|²A~ (2.28) Which is the same expression as in equation 2.25.

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Chapter 3

Vortices

3.1 Definitions

[26, p. 202] essentially provides following definitions:

Vorticity is defined as the curl of the fluid velocity. A line drawn in R³ such that it is at each point parallel to the vorticity is called a ”vortex-line”. If we draw a closed curve in R³ and at each of its points draw a vortex line we have drawn a tube which is called a ”vortex-tube”. The fluid contained in a vortex-tube is called a ”vortex-filament” and is abbreviated as ”vortex”.

3.2 Vortex flux

Using the typical argument that ~J vanishes at large distances and using equation 2.24, we obtain following result for a contour sufficiently far away:

0 =

J · ~~ dl = X

l

"

q_l~ ml

|ψl|²

∇~~θl · ~dl

#

− X

l

"

q²_l ml

|ψl|²

A · ~~ dl

# (3.1)

Recall the definition of magnetic flux and Stokes’ theorem to obtain:

Φ :=

B · ~~ dS =

( ~∇ × ~A) · ~dS =

A · ~~ dl (3.2) Also recall following:

∇~~θ_l· ~dl = "

dx ∂

∂x + dy ∂

∂y + dz ∂

∂z

#

θ_l = ∆θ_l = 2πN_l, N_l ∈ Z (3.3) N_l is the so called ”winding number” which arises as a consequence of the single-valuedness of the order parameter ψ_l.

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Using equations 3.1, 3.2 and 3.3, we obtain a result which matches the one found at [11, equation 3]:

Φ = P

l ql~

m_l|ψ_l|²2πN_l P

j q_j² m_j|ψj|²

(3.4)

3.3 Calculating the magnetic vector potential in the London limit

We will proceed similar to how it is done in [5].

From [3, p. 44] we have for cylindrical parametrization (s, ϕ, z) following equation:

∇ × ~~ A = (1 s

∂Az

∂ϕ −∂Aϕ

∂z ) ˆes+ (∂As

∂z −∂Az

∂s ) ˆeϕ+1

s(∂(sAϕ)

∂s −∂As

∂ϕ) êz (3.5) We will be looking at two cases in particular: one where one applies an external field in the êzdirection and one where one maintains a rotation in the êzdirection.

Therefore we can set ~h = h_zeˆ_z. One way to achieve it is by following magnetic vector potential choice:

A = A~ ϕ(s, ϕ) ˆeϕ (3.6)

Equations 2.3, 3.5 and 3.6 together yield:

~h = 1 s

∂(sA_ϕ)

∂s eˆz (3.7)

From [3, p. 44], we have:

∇ · ~~ A = 1 s

∂(sAs)

∂s +1 s

∂Aϕ

∂ϕ +∂Az

∂z (3.8)

Using equation 3.6 in 3.8, we then obtain:

∇ · ~~ A = 1 s

∂Aϕ

∂ϕ (3.9)

Choosing the gauge ~∇ · ~A = 0 then leads to that Aϕ is independent of ϕ:

A = A~ _ϕ(s) ˆe_ϕ (3.10)

Through equation 3.7 we see that it means that ~h does not depend on ϕ. Using that together with 3.5 where we replace ~A with ~h, we obtain:

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3.3. Calculating the magnetic vector potential in the London limit 11

J =~

∇ × ~h~ µ0

=−1 µ0

∂hz

∂s eˆϕ=−1 µ0

∂

∂s(1 s

∂(sAϕ)

∂s ) ˆeϕ (3.11) One can use following ansatz as hinted by [5]:

∇~~θ_l=Nl

s eˆ_ϕ (3.12)

We verify that it yields correct winding number:

∇~~θl· ~dl = Nl

s eˆϕ· sdϕ ˆeϕ= 2πNl (3.13) Using equations 2.24, 3.11 and 3.12 we obtain:

−1 µ0

∂

∂s(1 s

∂(sA_ϕ)

∂s ) ˆe_ϕ=X

l

"

q_l~ ml

|ψl|²N_l s − q²_l

ml

|ψl|²A_ϕ

# ˆ

e_ϕ (3.14)

Recall the modified Bessel differential equation:

x²d²y dx² + xdy

dx− (x²+ n²)y = 0 (3.15) It is well-known that it has following general solution [7, equation 9.6.1]:

y = C1In(x) + C2Kn(x) (3.16) A relevant property is [7, equation 9.6.19] which shows that In(x) grows too quickly to be a valid solution for our purposes:

In(x) = 1 π

π 0

e^{x cos θ}cos(nθ)dθ (3.17)

A difficulty lies in connecting the flux Φ to the constant C2. It is done by Annett [6, p. 63] for following equation:

s²d²hz

ds² + sdhz

ds − (s²

λ² + 0²)h_z= 0 (3.18) That yielded following:

h_z(s) = Φ 2πλ²K₀(s

λ) (3.19)

Analogously, we can put equation 3.14 in a more useful form:

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s²d²A_ϕ

ds² + sdA_ϕ ds −

"

µ₀X

l

q_l² ml

|ψ_l|²s²

+ 1²

#

A_ϕ+ µ₀X

l

q_l~ ml

|ψ_l|²N_ls

= 0 (3.20)

That looks similar to the modified Bessel differential equation for n = 1. We will make following ansatz [5, equation 7] where D is a constant and following choice for λ:

Aϕ(s) = C2K1(s λ) +D

s (3.21)

λ := 1

q µ0P

l q²_l m_l|ψl|²

(3.22)

Inserting those in equation 3.20 yields:

s²d²(C₂K₁(^s_λ))

ds² + sd(C₂K₁(_λ^s))

ds −

"

s² λ² + 1²

#

C₂K₁(s λ)+

s²2D

s³ + s−D s² −

"

s² λ² + 1²

#D

s + µ₀X

l

q_l~ ml

|ψ_l|²N_ls

= 0 (3.23)

By realizing that the first row of equation 3.23 satisfies the modified Bessel differential equation and rearranging the 2nd row we obtain:

−s² λ²

D s + µ0

X

l

q_l~ ml

|ψl|²Nls

= 0 (3.24)

Which yields:

D = λ²µ₀X

l

q_l~ ml

|ψ_l|²N_l

(3.25) Inserting the result for D in the equation 3.21 for Aϕyields:

A_ϕ(s) = C₂K₁(s

λ) +λ²µ₀ s

X

l

q_l~ ml

|ψl|²N_l

(3.26)

3.4 Calculating the microscopic field in the London limit

The modified Bessel function of the second kind, K_n(x), has following two properties [7, equation 9.6.28]:

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3.5. Determining the constant C₂ 13

d

dx(xⁿKn(x)) = −xⁿKn−1(x) (3.27) d

dx(x⁻ⁿKn(x)) = −x⁻ⁿKn+1(x) (3.28) Recall equation 3.7 which is valid for our cases:

~h = 1 s

∂(sA_ϕ)

∂s eˆ_z (3.29)

Using equation 3.26 for A_ϕtogether with 3.29 in 3.27 for n = 1, yields:

~h = C2

s

d(sK1(_λ^s))

ds eˆz= −s λ

C2

s K0(s

λ) ˆez=−C2

λ K0(s

λ) ˆez (3.30)

3.5 Determining the constant C

₂

We will follow a similar approach to the one by de Gennes [8, p. 58-59]. We will first look at how ~∇ × (~∇ × ~h) looks like when we insert our expression 3.30 for ~h and use equation 3.5 to express derivatives in cylindrical coordinates:

∇ × ~h = −~ ∂hz

∂s eˆϕ=C2

λ

∂K0(_λ^s)

∂s eˆϕ (3.31)

Using equation 3.28 for the case n = 0 in 3.31 yields:

∇ × ~h = −~ C2

λ²K1(s

λ) ˆeϕ (3.32)

Using equation 3.5 again yields:

∇ × (~~ ∇ × ~h) = 1 s

∂(−s^C_λ₂²K₁(_λ^s))

∂s eˆ_z (3.33)

Using equations 3.27, 3.30 and 3.33 for the case n = 1 then yields:

∇ × (~~ ∇ × ~h) = 1 s s λ

C2

λ²K₀(s

λ) ˆe_z= C2

λ³K₀(s

λ) ˆe_z= −~h

λ² (3.34)

We recognize that as the well-known London equation. However, in the London model for vortices, there is a small cylindrical core of radius ξ << λ that is a normal region rather than a superconducting one. The symmetry axis of the core is parallel to ˆez. From Griffiths [3, p. 274] we have:

~h^||_above− ~h^||_below = µ0( ~K × ˆn) (3.35) It is an equation at the boundary between the core and the rest of the specimen where ~K is the surface current and ˆn is the normal direction. Furthermore, Griffiths [3, p. 267] provides following equation for the bound surface current ~K_b:

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K~b= ~M × ˆn (3.36) Assuming that the magnetization ~M just like ~h only has a z-component, we obtain:

K~_b = M_zeˆ_z× − ˆe_s= −M_zeˆ_ϕ (3.37) Inserting that in equation 3.35 yields:

~h^||_above− ~h^||_below= µ0( ~Kf ree× − ˆes) + µ0(−Mzeˆϕ× − ˆes) (3.38) Assuming that there is no free surface current and that the magnetization is zero at the boundary to the core, we thus obtain that the z-component of ~h and thus ~h itself is continuous at the boundary.

Furthermore, since the order parameter ψ_lis 0 in the normal region, then ~J is ~0 according to equation 2.21. Next to satisfy that result for ~J through ~∇ × ~h = µ₀J ,~ we will make the ansatz that ~h is constant in the normal region. Then due to the continuity of ~h and equation 3.30 following holds in the normal region of the vortex:

~h = −C2

λ K0(

λ) ˆez (3.39)

That expression can be simplified by using the low-argument limit of K₀ [7, equation 9.6.8]:

~h = −C₂ λ ln(λ

) ˆe_z (3.40)

Due to the discontinuity of ~∇ × ~h = µ₀J at the core boundary, the London~ Equation 3.34 only holds in the superconducting region. To generalize the London equation to also hold in the normal region one can approximate the core with a point and use:

λ²∇ × (~~ ∇ × ~h) + ~h = Φδ(s)

2πseˆz (3.41)

Outside of the core we can identify that as the regular London Equation. To make sure that it is a proper expression in the context of the normal region we integrate over a surface within the superconducting region but that is at a distance from the core much larger than λ:

dS · λ~ ²∇ × (µ~ 0J ) +~

dS · ~h =~

dS · Φ~ δ(s)

2πseˆz (3.42) Using Stokes’ theorem and using the expression for flux we obtain:

dl · λ~ ²µ0J + Φ =~

sdϕdsΦδ(s)

2πs (3.43)

Just like in 3.1, the circulation of the current disappears at large distances, which then as expected yields:

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3.6. Calculating the circulation of velocity in the London limit 15

0 + Φ = Φ (3.44)

Now that we have established that the suggested generalized London Equation is reasonable, we will look at 3.42 again but integrate this time in a circle just outside the core:

dl · λ~ ²µ0J +~

dS · ~h = Φ~ (3.45)

Since ξ << λ, the contribution from dS · ~h is negligible. Furthermore we can~ use 3.32 to express ~J explicitly. That yields:

ξdϕλ²−C2

λ² K1(ξ

λ) = Φ (3.46)

Using that K₁(_λ^s) = ^λ_s for small arguments [7, equation 9.6.9], that yields:

C₂= − Φ

2πλ (3.47)

Next using the equation 3.30 for ~h, equation 3.26 for Aϕ, equation 3.22 for λ, equation 3.4 for Φ, equation 3.32 for ~∇ × ~h = µ0J we obtain following useful~ expressions:

A =~

"

Φ 2πs− Φ

2πλK₁(s λ)

# ˆ

e_ϕ (3.48)

~h = Φ 2πλ²K0(s

λ) ˆez (3.49)

µ₀J = ~~ ∇ × ~h = Φ 2πλ³K₁(s

λ) ˆe_ϕ (3.50)

3.6 Calculating the circulation of velocity in the London limit

Recall equation 2.27:

~ v_l= ~

ml

∇θ~ l− q_l ml

A~ (3.51)

Using equation 3.48 for ~A and 3.12 for ~∇~θ_l, we can calculate the circulation of the velocity within the superconducting region:

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~

vl· ~dl = 1 m_l

sdϕ

"

~Nl

s −qlΦ 2πs+ qlΦ

2πλK1(s λ)

#

= 2π~

ml

"

Nl− qlΦ 2π~

# + 1

ml

sdϕ

"

qlΦ 2πλK1(s

λ)

#

(3.52)

For a large contour, the 2nd term vanishes due to the exponential decay of K1(^s_λ) [7, equation 9.7.2]. For that case, using equation 3.4 for Φ, we obtain:

~

v_l· ~dl = 2π~

ml

"

N_l−q_l P

n q_n

m_n|ψn|²Nn

P

j q²_j mj|ψj|²

#

=2π~

ml

P

n q_n

m_n|ψn|²(Nlqn− Nnql) P

j q²_j mj|ψj|²

(3.53)

3.6.1 2-component example

To test the derived equation 3.53, we will compare to the result for the 2-component case at [9, equation 2]. Therefore, we will choose N1 = 0, N2 = 1, q1 = e and q2= −e and a large contour, which yields:

~

v1· ~dl = 2π~

m1

h e

m₁|ψ1|²(0 − 0) +^−e_m

2|ψ2|²(0 − e)i

e²

m₁|ψ1|²+_m^e²

2|ψ2|² =2π~

m1

|ψ₂|² m₂

|ψ₁|² m₁ +^|ψ_m²^|²

2

(3.54)

~v2· ~dl = 2π~

m₂ h e

m1|ψ1|²(e − 0) +_m^−e

2|ψ2|²(−e + e)i

e²

m₁|ψ1|²+_m^e²

2|ψ2|² = 2π~

m₂

|ψ1|² m1

|ψ1|² m1 +^|ψ_m²^|²

2

(3.55)

That result agrees with [9, equation 2]!

3.6.2 The special case of exponentially decaying supercurrents

Looking at equation 2.28 we see that the supercurrent ~jl is proportional to the velocity ~v_l, hence such velocities being exponentially decaying within a superconducting region means that corresponding supercurrents within the same region are exponentially decaying as well.

Looking at equation 3.52 and 3.53, we notice that if N_lq_n − N_nq_l = 0 for all choices of n and l, then ~v_l is exponentially decaying and hence ~j_l is exponentially decaying as well. Evidently such exponential decay means that for a large contour the circulation of ~vl is zero.

The condition can be rephrased as: ^N_q^l

l =^N_qⁿ

n for all choices of n and l.

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3.7. Free energy of a vortex in the London limit 17

3.7 Free energy of a vortex in the London limit

Using the equation 2.23 for ˜Fs, 3.48 for ~A and 3.12 for ~∇θl, we obtain:

F˜_s:=X

l

"

1 2m_l ~

Nl

s − q_l

Φ 2πs− Φ

2πλK₁(s λ)

!2

|ψ_l|²

# + ~h²

2µ₀ = X

l

"

1 2ml

~Nl

s − ql

Φ 2πs

²

+ qlΦ 2πλK1(s

λ)

² +

2

~ N_l

s − ql

Φ 2πs

q_lΦ 2πλK₁(s

λ)

|ψl|²

# + ~h²

2µ0

(3.56)

Utilizing equation 3.4 for Φ then yields:

F˜s=X

l

"

|ψl|² 2m_l

~² s²

Pn qn

mn|ψn|²(q_nN_l− qlN_n) P

j q_j² m_j|ψ_j|²

²

+ qlΦ 2πλK1(s

λ)

² +

2~

s

P

n q_n

m_n|ψn|²(qnNl− qlNn) P

j q²_j m_j|ψj|²

q_lΦ 2πλK₁(s

λ)

# + ~h²

2µ0

(3.57)

Next, we want to integrate ˜Fs over all space. However, since the order parameters ψ_l is 0 outside of the superconducting region and ~h decays exponentially for large distances, it is enough that we integrate over the superconducting region and the vortex core, i.e. over the specimen:

specimen

d³r ˜Fs=

specimen

d³rX

l

"

|ψl|² 2ml

~² s²

Pn qn

j q²_j m_j|ψj|²

²

+ q_lΦ 2πλK₁(s

λ)

2

+2~

s

Pn q_n

j q²_j mj|ψj|²

q_lΦ 2πλK₁(s

λ)

# +

specimen

d³r~h² 2µ0

(3.58)

To simplify the expression, we want to use the generalized London Equation 3.41. To do so we got to simplify our expression. Something to also keep in mind is that ~J = ~0 and ψ_l = 0 holds true inside the core. We will start by using equation 3.50 for K₁(_λ^s) and equation 3.22 for λ:

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specimen

d³r ˜Fs=

specimen

d³rX

l

"

|ψl|² 2ml

~² s²

P

n qn

mn|ψ_n|²(q_nN_l− q_lN_n) P

j q²_j m_j|ψj|²

2

+

q_lλ²∇ × ~h~

² +2~

s

Pn q_n

j q²_j mj|ψj|²

q_lΦ 2πλK₁(s

λ)

# +

specimen

d³r~h² 2µ0

=

specimen

d³rX

l

"

|ψl|²~² 2mls²

P

n qn

mn|ψ_n|²(q_nN_l− q_lN_n) P

j q_j² m_j|ψj|²

2#

+

specimen

d³rX

l

"

|ψ_l|²~ mls

Pn q_n

j q_j² mj|ψj|²

q_lΦ 2πλK₁(s

λ)

# +

specimen

d³r λ² 2µ0

( ~∇ × ~h)²+

specimen

d³r~h² 2µ0

(3.59)

For the 3rd integral in equation 3.59, we can use the vector identity described by equation 2.15 for the case ~C := ~h and ~D := ~∇ × ~h as well as Gauss’ divergence theorem:

specimen

d³r λ² 2µ0

( ~∇ × ~h)²=

specimen

d³r λ² 2µ0

∇ · (~h × (~~ ∇ × ~h)) +

specimen

d³r λ² 2µ0

~h · (~∇ × (~∇ × ~h)) =

∂specimen

d ~S · λ²

2µ₀(~h × ( ~∇ × ~h)) +

specimen

d³r λ²

2µ₀~h · (~∇ × (~∇ × ~h)) (3.60) The surface integral in equation 3.60 disappears since both ~h and ~∇ × ~h decay exponentially at large distances. Combining the remaining integral in 3.60 with the 4th integral in equation 3.59 and using the generalized London Equation 3.41, we obtain:

specimen

d³r λ²

2µ₀( ~∇ × ~h)²+

specimen

d³r~h² 2µ₀ =

specimen

d³r 1 2µ₀~h ·h

λ²∇ × (~~ ∇ × ~h) + ~hi

=

specimen

d³r 1

2µ₀~h · Φδ(s)

2πseˆz (3.61) Assuming cylindrical specimen with radius R and height ∆z as well as a vortex core radius ξ and then using equation 3.49 for ~h and the low-argument limit for K₀ as shown by equation 3.40, then yields:

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specimen

d³r λ²

2µ₀( ~∇ × ~h)²+

specimen

d³r~h² 2µ₀ =

sdϕdsdz 1

2µ₀~h · Φδ(s)

2πseˆz= ∆zΦhz(ξ)

2µ₀ = ∆z Φ² 4πµ₀λ²ln(λ

ξ) (3.62)

We recognize that result as being in the form analogous to the vortex energy for the 1-component case as shown at [10, equation 5.85]. Next we will turn our attention to the 2nd integral in equation 3.59, use equation 3.28 for the case n = 0 to express K1as a derivative:

specimen

d³rX

l

"

|ψl|²~ m_ls

Pn qn

j q²_j m_j|ψ_j|²

qlΦ 2πλK1(s

λ)

#

=

sdϕdsdzX

l

"

|ψ_l|²~ mls

P

n q_n

j q²_j m_j|ψj|²

−q_lΦ 2π

d ds

K₀(s λ)#

= 2π∆zX

l

"

|ψl|²~ m_l

Pn qn

j q²_j m_j|ψ_j|²

−qlΦ 2π

K0(R

λ)−K0(ξ λ)#

(3.63)

We can simplify the result by again using the low-argument limit for K0as shown by equation 3.40 and utilizing that K0decays exponentially for large arguments [7, equation 9.7.2]:

2π∆zX

l

"

|ψl|²~ ml

P

n q_n

j q_j² m_j|ψj|²

−q_lΦ 2π(K₀(R

λ) − K₀(ξ λ)

#

=

∆zΦ ln(λ ξ)~X

l

"

|ψl|²ql

m_l

Pn q_n

mn|ψn|²(qnNl− qlNn) P

j q²_j mj|ψ_j|²

#

(3.64)

Finally, we will look at the first integral of equation 3.59:

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specimen

d³rX

l

"

|ψl|²~² 2m_ls²

Pn q_n

j q²_j mj|ψj|²

²#

=

2π∆z~² 2

ds s

X

l

"

|ψl|² ml

P

n qn

m_n|ψ_n|²(q_nN_l− q_lN_n) P

j q²_j m_j|ψj|²

2#

=

π∆z~²lnR ξ

X

l

"

|ψl|² m_l

Pn q_n

j q²_j mj|ψj|²

²#

(3.65)

Combining the results from equations 3.65, 3.64 and 3.62 we finally obtain:

specimen

d³r ˜F_s= π∆z~²lnR ξ

X

l

"

|ψl|² m_l

Pn q_n

j q_j² mj|ψj|²

²# +

∆zΦ ln(λ ξ)~X

l

"

|ψl|²ql

ml

P

n qn

m_n|ψ_n|²(q_nN_l− q_lN_n) P

j q_j² m_j|ψj|²

#

+ ∆z Φ² 4πµ0λ²ln(λ

ξ) (3.66)

Something to keep in mind is that the right-hand side of equation 3.66 is positive due to the squares in the expression 2.23 for ˜Fs.

3.7.1 2-component example

Like in our previous examples, we will choose N₁= 0, N₂= 1, q₁= e and q₂= −e and a large radius R. Since the first term in equation 3.66 grows the fastest with R, we will investigate it first since if it is non-zero the remaining terms would be small compared for it:

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π∆z~²lnR ξ

X

l

"

|ψl|² m_l

Pn q_n

j q²_j mj|ψj|²

²#

=

π∆z~²lnR ξ

"

|ψ1|² m1

P

n qn

m_n|ψ_n|²(0 − eN_n) P

j q²_j m_j|ψj|²

2

+|ψ2|² m2

P

n qn

m_n|ψ_n|²(q_n+ eN_n) P

j q_j² m_j|ψj|²

2#

= π∆z~²lnR ξ

"

|ψ1|² m₁

^−e

m₂|ψ2|²(−e) P

j q²_j mj|ψj|²

² +|ψ2|²

m₂

^e

m₁|ψ1|²(e) +_m^−e

2|ψ2|²(−e + e) P

j q_j² mj|ψj|²

²#

= π∆z~²lnR ξ

|ψ1|² m1

|ψ2|² m2

" |ψ₂|² m₂

(P

j

|ψj|² mj )² +

|ψ₁|² m₁

(P

j

|ψj|² mj )²

#

=

π∆z~²lnR ξ

|ψ1|² m1

|ψ2|² m2

1 (^|ψ_m¹^|²

1 +^|ψ_m²^|²

2 ) (3.67) That result is non-zero and using the argument above we do not need to calculate other contributions. This result is also the same as in [9]!

3.7.2 The special case of exponentially decaying supercurrents

Just like in chapter 3.6.2, we will look at the special case Nlqn− Nnql= 0 for all choices of l and n at large distances R. Looking at the equation 3.66 for the free energy of a vortex, we notice that the two first terms are zero for this case while the third remains:

specimen

d³r ˜F_s= ∆z Φ² 4πµ0λ²ln(λ

ξ) (3.68)

This scales as O(R⁰) rather than O(ln(R)) which would have been the case if the first term of equation 3.66 did not vanish.

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Properties in n-component London Superconductors

Master Thesis