manybody physics lab effective vortex mass from microscopic theory june seo kim
TRANSCRIPT
MANYBODY PHYSICS
Lab
Effective Vortex Mass from Microscopic Theory
June Seo Kim
MANYBODY PHYSICS
Lab
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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