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Effective Vortex Mass from Microscopic Theory

June Seo Kim

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Contents

1. Vortex Motion through a Type-II SuperconductorVortex Motion through a Type-II Superconductor

2. Vortex Dynamics

3. Self-consistent Field Method

4. Energy Spectra and Effective Mass

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Vortex Motion through a Type-II SuperconductorVortex Motion through a Type-II Superconductor

Imagine a small magnet with its north/south poles on either side of a thin slab of type-II superconductor. On dragging the magnet the vortex moves too. Force needed to execute the motion is (m+M) a

m = mass of magnetM = effective mass of vortex

M can be calculated within Caldeira-Leggett theory

2ee

S

N

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Vortex Dynamics

Hamiltonian including pairing of superconductivity is represented insecond quantization.

†k k k k k k k

k

H c c c c c c

k : Energy of the excitation

In real space,2

† † †( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 F

r

H r r r r r r r rm

†

†

( ) ( ) ( )

( ) ( ) ( )

m mm mm

m mm mm

r u r v r

r u r v r

†and are quasi-particle operators.

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2

2

( ) ( ) ( ) ( ) ( )2

( ) ( ) ( ) ( ) ( )2

F m m m m

m F m m m

u r r v r E u rm

r u r v r E v rm

The effect of magnetic field,

2 2 2 20 0

1 1 1 1( ) ( ) , ( ) ( )

2 2 2 2H P eA i eA H P eA i eA

m m m m

2

2

1( ) ( )

( ) ( )21 ( ) ( )

( ) ( )2

iF

iF

P eA r eu r u rm Ev r v r

r e P eAm

Bogoliubov-de Genne equation

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Put and , then 2

i

u e U

2

i

v e V

2 2 2 22 21 1

( ) ( ) , ( ) ( )2 2

i i

P eA u e P eA U P eA v e P eA V

We have to transform one more time. Ignoring and puttingeA

'

'

( , )

( , )zik z

U U re

V V r

, then

22

' '

' '22

1 1( ) ( )

2 2 2

1 1( ) ( )

2 2 2

zF

zF

kP r

U Um m EV Vk

r Pm m

d dP ir i

dr r d

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Put

22

2

22

2

1( )1 1 2( ( ) ) ( )

2 21

( )1 1 2( ) ( ( ) )2 2

kd dr r

u um r dr dr r m Ev v

kd dr r

m r dr dr r m

'

'

( )( , )

( )( , )i u rU re

v rV r

2 '' ' 2 2 2

2 '' ' 2 2 2

1[ ( 2 ) ( ) ] 2

21

[ ( 2 ) ( ) ] 22

r u ru r k mE u m v

r v rv r k mE v m u

2 2 2F zk k k

be half-odd integers by periodic boundary condition.

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Self-consistent Field Method

2 '' ' 2 2 2

2 '' ' 2 2 2

1[ ( 2 ) ( ) ] 2

21

[ ( 2 ) ( ) ] 22

r u ru r k mE u m v

r v rv r k mE v m u

How can we solve this coupled differential equation?

First of all, we have to treat the energy gap. If the energy gap isAbsent, then we can find solution of these equations.

, ,1 ,

2( ) ( )

( )j m m j mm j m

rr J

R J R

Where J is a Bessel function and R is a boundary.

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and we can find and as combination of Bessel functions. ( )nu r ( )nv r

, 1 , 1

2 2

( ) ( ) , ( ) ( )n n j j n n j jj j

r ru r c J v r d J

R R

Inserting into BdG equation and using orthogonality of Bessel functions,2

1 2,2

, , , ,21

21 2,2

, , , ,21

( )2 2

( )2 2

Ni

n n i n i i j n jj

Ni

n n i n i i j n jj

kE c c A d

mR m

kE d d B c

mR m

, 1 10 , ,2 2

, 1 1 ,0 , ,2 2

( )

( )

R

i ji j

R

i j j ii j

A r r dr

B r r dr A

Therefore we have to know and . ,i jA ,i jB

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Self-consistency requires that the r-dependent gap function obey the relation for a given choice of

the pairing interaction strength V. Put as initial condition.In iteration process, we can find exact gap function at zero temperatureand finite temperatures .

( ) ( ) ( )[1 2 ( )]r V u r v r f E

0( ) ( / )r tanh r

Moreover, and are change by the energy gap is changed. Therefore, we can calculate exact value of and . It is a self-consistent field method.

( )nu r ( )nv r( )nu r ( )nv r

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Energy Spectra and Effective Mass

Gap profile

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Energy Gap

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Energy Spectrum

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Mass Equation and Transition Matrix ElementsMass Equation and Transition Matrix Elements

Transition matrix element between localized and extended states arenon-zero due to vortex motion.

Second-order perturbation theory gives effective mass.2

03

( ) ( )

( )a b

vab a b v

f E f E HM a b

E E r

core-to-core

core-to-extended

extended-to-extended

Energ

y

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Effective Mass

At zero temperature,

20( )V e fM m k

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SummarySummary

We calculate the effective mass of a single quantized vortex in the BCS superconductor at finite temperature.

Based on self-consistent numerical diagonalization of the BDG equation we find the effective mass per unit length of vortex at finite temperatures.

The mass reaches a maximum value at and decreases continuously to zero at .

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