17. Superconductivity / Linear Responselampx.tugraz.at/~hadley/ss2/lectures19/dec2.pdfImpulse response function (Green's functions) 142 exp sin 0 22 bt mk b gt t t mm m A Green's function

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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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.