Institute of Solid State Physics Technische Universität Graz 17. Superconductivity / Linear Response Dec. 2, 2019
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Institute of Solid State PhysicsTechnische Universität Graz
17. Superconductivity/ Linear Response
Dec. 2, 2019
Vortices in Superconductors
F q E v B
1F j Bn
j nqv
dVdt
Lorentz force
Faraday's law
j
Defects are used to pin the vortices
Superconducting Magnets
Whole body MRI
Magnets and cables
Maglev trains
ITER
Magnet wire
Nb3Sn Magnet
Superconducting magnets
Largest superconducting magnet, CERN21000 Amps
ac - Josephson effect
10 V standard
Brian Josephson
http://www.nist.gov/pml/history-volt/superconductivity_2000s.cfm
DOI: 10.1140/epjst/e2009-01050-6
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0 2d nhfV n fdt e
SQUID
Superconducting quantum interference device
10-6 0 / (Hz)1/2
10-20 m/ (Hz)1/2
Gravity wave detectorSensitive detectors
Institute of Solid State PhysicsTechnische Universität Graz
Linear Response Theory
Fourier transformsImpulse response functions (Green's functions)Generalized susceptibilityCausalityKramers-Kronig relationsFluctuation - dissipation theoremDielectric functionOptical properties of solids
Institute of Solid State Physics
Classical linear response theory
Technische Universität Graz
http://lampx.tugraz.at/~hadley/num/ch3/3.3a.php
Notations for Fourier Transforms
f(r) is built of plane waves
http://lamp.tu-graz.ac.at/~hadley/ss1/crystaldiffraction/ft/ft.php
Properties of Fourier transforms
Convolution (Faltung)
( )* ( ) ( ) ( )f r g r f r g r r dr
Impulse response function (Green's functions)
21 4( ) exp sin 02 2
bt mk bg t t tm m m
A Green's function is the solution to a linear differential equation for a -function driving force
has the solution
For instance,2
2 ( )d g dgm b kg tdt dt
Green's functions
( )f t t t f t dt
A driving force f can be thought of a being built up of many delta functions after each other.
By superposition, the response to this driving function is superposition,
( ) ( ) ( )u t g t t f t dt
21 4( ) exp sin ( )
2 2b t t mk bu t t t f t dt
m m m
has the solution
For instance,2
2 ( )d u dum b ku f tdt dt
Green's function converts a differential equation into an integral equation
Generalized susceptibility
A driving function f causes a response u
If the driving force is sinusoidal,
0( ) ( ) ( ) ( ) i tu t g t t f t dt g t t F e dt
( )( )
i t
i t
g t t e dtuf e
0( ) i tf t F e
The response will also be sinusoidal.
The generalized susceptibility at frequency is
Generalized susceptibility
( ) uf
http://lampx.tugraz.at/~hadley/physikm
/apps/resonance.en.php
Generalized susceptibility
Since the integral is over t', the factor with t can be put in the integral.
( )( )
i t
i t
g t t e dtuf e
Change variables to = t - t', d = -dt', reverse the limits of integration
The susceptibility is the Fourier transform of the Green's function.
( )( ) ( ) i t tg t t e dt
( ) ( ) ig e d
1( ) ( )2
i tg t e d
F1,-1
First order differential equation
1( ) ( ) exp 0bt bg t H tm m m
The Fourier transform of a decaying exponential is a Lorentzian
( ) ( ) i tg t e dt
22
1( )
b im
m bm
( )dgm bg tdt
Susceptibility
00 2 2 2
F b i mA Fb i m b m
0i mA bA F
( )dum bu F tdt
22
1b iu m
F m bm
Assume that u and F are sinusoidal 0 i t i tu Ae F F e
The sign of the imaginary part depends on whether you use eit or e-it.
Susceptibility
1i m b
( )dgm bg tdt
22
1b im
m bm
Fourier transform the differential equation
1b i m
2
2 ( )d g dgm b kg tdt dt
2 42
b b mkm
Damped mass-spring system
21 4( ) ( ) exp sin 2 2
bt mk bg t H t tm m m
2 1m i b k
2
2 22
1k bim m
m k bm m
Fourier transform pair
tg e
dx Mxdt
More complex linear systems
Any coupled system of linear differential equations can be written as a set of first order equations
The solutions have the form itix e
where are the eigenvectors and i are the eigenvalues of matrix M.
ix
Re(i) < 0 for stable systems
i is either real and negative (overdamped) or comes in complex conjugate pairs with a negative real part (underdamped).
More complex linear systems
Low frequency "1/f noise"resonances
frequency
ampl
itude
Any function f(t) can be written in terms of its odd and even components
Odd and even components
The Fourier transform of E(t) is real and evenThe Fourier transform of O(t) is imaginary and odd
E(t) = ½[f(t) + f(-t)]
f(t) = E(t) + O(t)
f(t) = ½[f(t) + f(-t)] + ½[f(t) - f(-t)]
O(t) =½[f(t) - f(-t)]
( ) cos ( ) sinE t tdt i O t tdt
( ) ( ) ( ) cos sini tf t e dt E t O t t i t dt
( ) sgn( ) ( )( ) sgn( ) ( )
O t t E tE t t O t
odd component
even component exp( )t
sgn( )exp( )t t
22
1( )
b im
m bm
Causality and the Kramers-Kronig relations (I)
The real and imaginary parts of the susceptibility are related.
If you know ', inverse Fourier transform to find E(t). Knowing E(t) you can determine O(t) = sgn(t)E(t). Fourier transform O(t) to find ".
( ) ( ) ( ) cos( ) ( ) sin( )ig e d E d i O d i
Kramers-Kronig relations
https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations
If you know any of these for just positive frequencies, you can calculate all the others.
Causality and the Kramers-Kronig relation (II)
( ) sgn( ) ( )O t t E t
( ) sgn( ) ( )E t t O t
* ,i i *ii
Take the Fourier transform, use the convolution theorem.
1 ( )( ) P d
1 ( )( ) P d
Singularity makes a numerical evaluation more difficult.
P: Cauchy principle value (go around the singularity and take the limit as you pass by arbitrarily close)
Real space Reciprocal space
http://lamp.tu-graz.ac.at/~hadley/ss2/linearresponse/causality.php
Kramers-Kronig relations (III)
1 ( )( ) P d
1 ( )( ) P d
Kramers-Kronig relations II
( ) ( )( ) ( )
Real part is evenImaginary part is odd
0
0
1 ( ) 1 ( )( ) P d P d
change variables '-'(4 minus signs)
Kramers-Kronig relations (III)
0 0
1 ( ) 1 ( )( ) P d P d
2 20
2 ( )( ) P d
2 2
1 1 2
2 20
2 ( )( ) P d
Singularity is stronger in this form.
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