Chapter 3 - Walther Meißner Institut - Homepage I/SL_Chapte… · TT1-Chap2 - 1 Chapter 3 3....

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TT1-Chap2 - 1

Chapter 3

3. Phenomenological Models of Superconductivity

3.1 London Theory

3.1.1 The London Equations

3.2 Macroscopic Quantum Model of Superconductivity

3.2.1 Derivation of the London Equations

3.2.2 Fluxoid Quantization

3.2.3 Josephson Effect

3.3 Ginzburg-Landau Theory

3.3.1 Type-I and Type-II Superconductors

3.3.2 Type-II Superconductors: Upper and Lower Critical Field

3.3.3 Type-II Superconductors: Shubnikov Phase and Flux Line Lattice

3.3.4 Type-II Superconductors: Flux Lines

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TT1-Chap2 - 2Fritz Wolfgang London (1900 – 1954)

3.1 London Theory

* 7 March 1900 in Breslau † 30 March 1954 in Durham,

North Carolina, USA

study: Bonn, Frankfurt, Göttingen, Munichand Paris.

Ph.D.: 1921 in Munich1922-25: Göttingen and Munich1926/27: Assistent of Paul Peter Ewald at

Stuttgart, studies at Zurich and Berlin withErwin Schrödinger.

1928: Habilitation at Berlin 1933-36: London1936-39: Paris1939: Emigration to USA,

Duke Universität at Durham

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TT1-Chap2 - 3

Heinz and Fritz London

3.1 London Theory

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TT1-Chap2 - 4

3.1 London Theory

1935 Fritz and Heinz London describe the Meißner-Ochsenfeld effect

and perfect conductivity within phenomenological model

homogeneous pair condensate

• starting point is equation of motion of charged particles with mass m and charge q

(t = momentum relaxation time)

• two-fluid model: - normal conducting electrons with charge qn and density nn

- superconducting electrons with charge qs density ns

• normal state: nn = n, ns = 0

• superconducting state nn = 0, ns = max (for T → 0), t → ∞ , Js = ns qs vs

1st London equation

London coefficient

BCS theory:ms = 2me, qs = -2ens = n/2

3.1.1 London Equations

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TT1-Chap2 - 5


• take the curl of 1st London equation and use

flux through an arbitrary area inside a sample with infinite conductivitystays constant

e.g. flux trapping when switching off the external magnetic field

• Meißner-Ochsenfeld effect tells us: not only but itself must be zero expression in brackets must be zero

2nd London equation

• use Maxwell‘s equation

with

London penetration depth

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TT1-Chap2 - 6


• example: B = Bz

y

Bz,0

Bz

z

xlL

• solution:

• Bz decays exponentially withcharacteristic decay length lL

• T dependence of lL

empirical relation:

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TT1-Chap2 - 7


• with 2nd London equation

we obtain for Js:

y

z

Bz

xlL

Jy,0

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TT1-Chap2 - 8


• example: thin superconducting sheet of thickness d with B ǁ sheet

• Ansatz:

y

Bext,z

Bext,z

x

z

lL

d

• boundary conditions:

• solution:

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TT1-Chap2 - 9


• remarks to the London model:

1. normal component is completely neglected not allowed at finite frequencies !

2. we have assumed a local relation between Js , E and B

- Js is determined by the local fields for every position r

- this is problematic since ℓ → ∞ for t → ∞

nonlocal extension of London theory by A.B. Pippard (1953)

1st London equation London coefficient

2nd London equation London penetration depth

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TT1-Chap2 - 10

3.2 SC as Macroscopic Quantum Phenomenon

• more solid derivation of London equations by assuming that superconductor can bedesrcribed by a macroscopic wave function

- Fritz London (> 1948)- derived London equations from basic quantum mechanical concepts

• basic assumption of macroscopic quantum model of superconductivity

complete entity of all superconducting electrons can be described by macroscopicwave function

amplitude phase

• hypothesis can be proven by BCS theory (discussed later)

• normalization condition: volume integral over |y|² is equal to the number Ns of superconducting electrons

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TT1-Chap2 - 11

• general relations in electrodynamics:

electric field:

flux density:

A = vector potentialf = scalar potential

• canonical momentum:


• electrical current is driven by gradient of electrochemical potential:

• kinematic momentum:

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TT1-Chap2 - 12

• Schrödinger equation for charged particle:

electro-chemical potential

• insert macroscopic wave-function (Madelung transformation)

replacements: Y y, q qs, mms


• calculation yields after splitting up into real and imaginary part two fundamental equations

- current-phase relation: connects supercurrent density with gauge invariant phase gradient

- energy-phase gradient: connects energy with time derivative of the phase

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TT1-Chap2 - 13

3.2 Special Topic: Madelung Transformation

• we start from Schrödinger equation:

electro-chemical potential

• we use the definition and obtain with

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TT1-Chap2 - 14


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TT1-Chap2 - 15


• equation for real part:

energy-phase relation (term of order ²ns is usually neglected)

∆

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TT1-Chap2 - 16

• interpretation of energy-phase relation


corresponds to action

in the quasi-classical limes the energy-phase-relation becomesthe Hamilton-Jacobi equation

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TT1-Chap2 - 17

• equation for imaginary part:


continuity equation for probabilitydensityandprobability current density

conservation law for probability density

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TT1-Chap2 - 18


• we define supercurrent density Js by multiplying Jr with charge qs of superconductingelectrons :

vs Js = ns qs vs

gauge invariant phase gradient

current-phase relation

• canonical momentum: 𝐩 = 𝑚𝑠𝐯𝑠 + 𝑞𝑠𝐀

𝐩 = 𝑚𝑠

ℏ

𝑚𝑠𝛁𝜃 𝐫, 𝑡 −

𝑞𝑠

𝑚𝑠𝐀 𝐫, 𝑡 + 𝑞𝑠𝐀

𝐩 = ℏ 𝛁𝜃 𝐫, 𝑡

𝐯𝑠

zero total momentum state for vanishing phase gradient: Cooper pairs 𝐤 ↑, −𝐤 ↓

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TT1-Chap2 - 19

• energy-phase relation

• supercurrent density-phase relation

(i) superconductor with Cooper pairs of charge qs = -2e

(ii) neutral Bose superfluid, e.g. 4He

(iii) neutral Fermi superfluid, e.g. 3He

equations (1) and (2) have general validity for charged and uncharged superfluids

(London parameter)

1

2

we use equations (1) and (2) to derive London equations


• note: k drops out !!

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TT1-Chap2 - 20

• take the curl second London equation:

which leads to:

• describes Meißner-Ochsenfeld effect: applied field decays exponentially inside superconductor,

decay length:

with Maxwell´s equations:

(London penetration depth)

2nd London equation and the Meißner-Ochsenfeld effect:

3.2.1 Derivation of London Equations

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TT1-Chap2 - 21

1st London equation and perfect conductivity:

• take the time derivative

first London equation:

• neglecting 2nd term:

• interpretation: for a time-independent supercurrent the electric field inside the superconductor vanishes dissipationless dc current

(linearized 1st London equation)

• inserting

and substituting yields


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TT1-Chap2 - 22

3.2.1 London Equations - Summary

• energy-phase relation

• supercurrent density-phase relation (London parameter)

1

2

• take the curl

which leads to:

2nd London equation and the Meißner-Ochsenfeld effect:

2nd Londonequation

1st London equation and perfect conductivity:

• take the time derivative

which leads to: 1st Londonequation

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TT1-Chap2 - 23


• the assumption that the superconducting state can be described by a macroscopicwave function leads to a general expression for the supercurrent density Js

• London equations can be directly derived from the general expression for thesupercurrent density Js

• London equations together with Maxwell‘s equations describe the behavior ofsuperconductors in electric and magnetic fields

• London equations cannot be used for the description of spatially inhomogeneoussituations Ginzburg-Landau theory

• London equations can be used for the description of time-dependent situations Josephson equations

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TT1-Chap2 - 24

• Processes that could cause a decay of Js (plausibility consideration):

example: Fermi sphere in two dimensions in the kxky – plane

• T = 0: all states inside the Fermi circle are occupied• current in x-direction shift of Fermi circle along kx by ±kx

• normal state: relaxation into states with lower energy (obeying Pauli principle) centered Fermi sphere, current relaxes

• superconducting state: Cooper pairs with the same center of mass moment (discussion later)

only scattering around the sphere no decay of supercurrent

normal state superconducting state


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TT1-Chap2 - 25

kinetic energy of superelectrons

• usually, 1. London equation is linearized:

allowed if |E| >> |vs| |B|condition is satisfied in most cases

3.2.1 Additional Topic: Linearized 1. London Equation

• in order to discuss the origin of the extra term (nonlinearity) we use the vector identityto write

• then, by using the second London equation, we can rewrite the 1. London equation as

• with and we obtain

(Lorentz law)

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TT1-Chap2 - 26

we can conclude:

the nonlinear first London equation results from the Lorentz's law and the second London equation exact form of the expression describing the phenomenon of zero dc resistance in

superconductors the first London equation is derived using the second London equationMeißner-Ochsenfeld effect is the more fundamental property of

superconductors than the vanishing dc resistance

3.2.1 Additional Topic: Linearized 1. London Equation

• we can neglect the nonlinear term if

• as variations of Js occur on lL, we have and obtain the condition

with 2. London equation:

typically, vs < 1 m/s even at very high current densities of the order of 106 A/cm² due to the large values of ns

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TT1-Chap2 - 27

3.2.1 Additional Topic: The London Gauge

• 1. London equation:

• frequently, we have no conversion of Js in Jn or no supercurrent flow throuh sample surface

• in some cases it is convenient to choose a special gauge often used: The London Gauge

• if the macroscopic wavefunction is single valued (this is the case for a simply connected superconductor containing no flux) we can choose such that

a vector potential that satisfies is said to be in the London gauge

everywhere

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TT1-Chap2 - 28

• Stoke‘s theorem (path C in simply or multiply connected region):

• if 𝑟1 → 𝑟2 (closed path) integral to zerobut: phase is only specified within modulo 2𝜋 of its principle value −𝜋, 𝜋 : 𝜃𝑛 = 𝜃0 + 2𝜋𝑛

• integration of expression for supercurrent density arounda closed contour


• integral of gradient:

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TT1-Chap2 - 29

• then:

• flux quantum:

fluxoid is quantized in units of F0

• quantization condition holds for all contour linesincluding contour that can be shrunk to single point

𝑟1 = 𝑟2: 𝑟1𝑟2 𝛁𝜃 ⋅ 𝑑ℓ = 0

• contour line can no longer be shrunk to single point inclusion of nonsuperconducting region in contour 𝑟1 = 𝑟2: we have built in „memory“ in integration path

𝑛 ≠ 0 possible, 𝑟1𝑟2 𝛁𝜃 ⋅ 𝑑ℓ =2𝜋𝑛


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TT1-Chap2 - 30

rr

Js

screeningcurrent oninner wall

Hextmagneticflux

Js


screeningcurrent onouter wall

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TT1-Chap2 - 31

• Fluxoid Quantization:total flux = external applied flux + flux generated by induced supercurrentmust have discrete values

• Flux Quantization:- superconducting cylinder, wall much thicker than 𝜆𝐿

- application of small magnetic field at 𝑇 < 𝑇𝑐 screening currents, no flux inside

application of 𝐵cool during cool down: screening current on outer and inner wall amount of flux trapped in cylinder: satisfies fluxoid quantization condition wall thickness ≫ 𝜆L: closed contour deep inside with 𝐽𝑠 = 0 then:

remove field after cooling down trapped flux = integer multiple of flux quantum

flux quantization


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TT1-Chap2 - 32

• Flux Trapping: why is flux not expelled after switching off external field

𝜕𝐽𝑠

𝜕𝑡= 0 according to 1st London equation: 𝑬 = 0 deep inside SC

(supercurrent flow only on surface within 𝜆L)

• with and we get:

Φ: magnetic flux enclosed in loopcontour deep inside the superconductor: 𝑬 = 0 and therefore

flux enclosed in cylinder stays constant


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TT1-Chap2 - 33

• 1961 Doll/Näbauer at Munich, Deaver/Fairbanks at Stanford quantization of magnetic flux in a hollow cylinder Cooper pairs with 𝒒𝒔 = −𝟐𝒆

cylinder with wall thickness ≫ 𝜆𝐿

different amounts of flux are frozen in during coolingdown in 𝑩cool

measure amount of trapped flux

required: large relative changes of magnetic flux small fields and small diameter d

for 𝑑 = 10 𝜇𝑚, we need 𝑩cool = 2 × 10−5 T for oneflux quantum

measurement of (very small) torque D = ¹ x Bp due toprobe field Bp

resonance method

amplitude of rotary oscillation ≈ exciting torque

d ≈ 10 µm

Pb

quartzcylinder

Bp

quartzthread


Paarweise im Fluss D. Einzel, R. Gross, Physik Journal 10, No. 6, 45-48 (2011)

http://www.pro-physik.de/Phy/pjtoc/84681/3

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TT1-Chap2 - 34

Paarweise im Fluss D. Einzel, R. Gross, Physik Journal 10, No. 6, 45-48 (2011)


(a) (b)

R. Doll, M. NäbauerPhys. Rev. Lett. 7, 51 (1961)

B.S. Deaver, W.M. FairbankPhys. Rev. Lett. 7, 43 (1961)

F0

prediction by F. London: h/e experimental proof for existence of Cooper pairs

qs = 2e

-0.1 0.0 0.1 0.2 0.3 0.4-1

0

1

2

3

4

reso

na

nce

am

plitu

de

(m

m/G

au

ss)

Bcool

(Gauss)

0.0 0.1 0.2 0.3 0.40

1

2

3

4

tra

pp

ed

ma

gn

etic f

lux (

h/2

e)

Bcool

(Gauss)

http://www.pro-physik.de/Phy/pjtoc/84681/3

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TT1-Chap2 - 35

Brian David Josephson (born 1940)

3.2.3 Josephson Effect (1962)

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TT1-Chap2 - 36

• what happens if we weakly couple two superconductors?- coupling by tunneling barriers, point contacts, normal conducting layers, etc.- do they form a bound state such as a molecule?- if yes, what is the binding energy?

• B.D. Josephson in 1962 (nobel prize with Esaki and Giaever in 1973)

finite supercurrent at zero applied voltage

oscillation of supercurrent at constant applied voltage

finite binding energy of coupled SCs = Josephson coupling energy

Josephson effects


Cooper pairs can tunnel through thin insulating barrierexpectation:- tunneling probability for pairs ∝ 𝑇 2 2

extremely small ∼ 10−4 2

Josephson:- tunneling probability for pairs ∝ 𝑇 2

- coherent tunneling of pairs („tunneling of macroscopic wave function“)

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TT1-Chap2 - 37

• coupling is weak supercurrent density is small 𝜓 2 = 𝑛𝑠 is not changed in 𝑆1 and 𝑆2

• supercurrent density depends on gauge invariant phase gradient:

• simplifying assumptions:- current density is homogeneous- 𝛾 varies negligibly in electrodes- Js same in electrodes and junction area

𝛾 in superconducting electrodesmuch smaller than in insulator I

• then:- replace gauge invariant phase gradient 𝛾

by gauge invariant phase difference:


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TT1-Chap2 - 38

• we expect:

• for Js = 0: phase difference must be zero:

therefore:

Jc: critical/maximumJosephson current density

• in most cases: keep only 1st term (especially for weak coupling):

1. Josephson equation:

• generalization to spatially inhomogeneous supercurrent density:

derived byJosephson forSIS junctions

supercurrent density varies

sinusoidally with𝜑 = 𝜃2 − 𝜃1

w/o externalpotentials

general formulation of 1st Josephson equation: current-phase relation

first Josephson equation:


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TT1-Chap2 - 39


• other argument why there are only sin contributions to Josephson current

time reversal symmetry

• if we reverse time, the Josephson current should flow in opposite direction- 𝑡 → −𝑡, 𝐽𝑠 → −𝐽𝑠

• the time evolution of the macroscopic wave functions is ∝ exp 𝑖𝜃 𝑡 = exp 𝑖𝜔𝑡- if we reverse time, we have

𝑡 → −𝑡

• if the Josephson effect stays unchanged under time reversal, we have to demand

satisfied only by sin-terms

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TT1-Chap2 - 40

• time derivative of the gauge invariant phase difference:

• substitution of the energy-phase relationgives:

• supercurrent density across the junction is continuous (Js(1) = Js(2)):

2. Josephson equation:

(term in parentheses = electric field)

second Josephson equation:

voltage drop

voltage – phaserelation


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TT1-Chap2 - 41

• for a constant voltage across the junction:

• Is is oscillating at the Josephson frequency n = V/F0:

voltage controlled oscillator

• applications: - Josephson voltage standard- microwave sources


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TT1-Chap2 - 42

Macroscopic wave function 𝝍: describes ensemble of macroscopic number of superconducting electrons𝜓 2 = 𝑛𝑠 is given by density of superconducting electrons

Current density in a superconductor:

Gauge invariant phase gradient:

Phenomenological London equations:

flux/fluxoid quantization:

3.2 Summary

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TT1-Chap2 - 43

maximum Josephson current density Jc:wave matching method

Josephson equations:

3.2 Summary

more detail in chapter 5

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TT1-Chap2 - 44


• London theory: suitable for situations where ns = const. how to treat spatially inhomogeneous systems?

• example: step-like change of wave function at surfaces and interfaces associated with large energy gradual change on characteristic length scale expected

• Vitaly Lasarevich Ginzburg and Lew Davidovich Landau (1950)

description of superconductor by- complex, spatially varying order parameter 𝝍 𝒓

with 𝝍 𝒓 𝟐 = 𝒏𝒔 𝒓 (density of superconducting electrons)

- no time dependence (cannot describe Josephson effect) based on Landau theory of 2nd order phase transitions

• Alexei Alexeyevich Abrikosov (1957) prediction of flux line lattice for type-II superconductors

• Lev Petrovich Gor'kov (1959) GL-theory can be inferred from BCS theory for T ≈ Tc

Ginzburg-Landau- Abrikosov-Gor'kov (GLAG) theory

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TT1-Chap2 - 45


A: spatially homogeneous superconductor in zero magnetic field

• develop free enthalpy density 𝑔𝑠 of superconductor in power series of Ψ 2

free entalpy densityof normal state

higher order terms can beneglected for 𝑇 ≃ 𝑇𝑐 as 𝜓 small

• discussion of coefficients 𝛼 and 𝛽:- 𝛽 > 0, otherwise large |Ψ| always results in 𝑔𝑠 < 𝑔𝑛

minimum of 𝑔𝑠 always for Ψ → ∞- 𝛼 changes sign at phase transition

𝑇 > 𝑇𝑐: 𝛼 > 0, since 𝑔𝑠 > 𝑔𝑛

𝑇 < 𝑇𝑐: 𝛼 < 0, since 𝑔𝑠 < 𝑔𝑛

Ansatz:

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TT1-Chap2 - 46


-0.8 -0.4 0.0 0.4 0.8-1

0

1

2

3

4

g s - g

n (

arb

. un

its)

Y (arb. units)

-0.8 -0.4 0.0 0.4 0.8-1

0

1

2

3

4

g s - g

n (

arb

. un

its)

Y (arb. units)

Bcth2 / 2µ0

𝝍𝟎

a < 0T < Tc

a > 0T > Tc

(a) (b)

𝒈𝒔 − 𝒈𝒏 for 𝑻 < 𝑻𝒄 and 𝑻 > 𝑻𝒄

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TT1-Chap2 - 47


• equilibrium condition yields

describes homogeneous equilibrium state at T

• physical meaning of coefficients:

gs – gn corresponds to condensation energy which, in turn, is given by

• solve for 𝛼 and 𝛽:

= (gs-gn)/(ns/2)

condensation energyper Cooper pair

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TT1-Chap2 - 48


Temperature dependence of Bcth:

• GLAG theory prediction:

• experimental observation:

with

reasonable agreement for T close to Tc

GLAG expressions only valid for T close to Tc

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TT1-Chap2 - 49


B: spatially inhomogeneous superconductor in external magnetic field 𝜇0𝐇ext

• additional terms in free enthalpy density for 𝑩ext = 𝜇0𝐇ext, 𝐛 = 𝜇0 𝐇ext + 𝐌

takes into accountspatial variations of local

flux density b

are associated with Js

according to 2nd London eq. additional kinetic energy prevents spatial variations of 𝜓 stiffness of order parameter

with

gradient of amplitude

gradient of phase

vs²

work required forfield expulsion (Bext – b)²

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TT1-Chap2 - 51


• minimization of free enthalpy 𝐺𝑠: integration of enthalpy density 𝑔𝑠 over whole volume 𝑉 of superconductor

• boundary condition during minimization of free enthalpy no current density through surface of superconductor

replace by for superconductor/normal conductor interfacevs

= 0

divergence theorem

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TT1-Chap2 - 54


• minimization of free enthalpy:

• minimization of Gs with respect to variations

= 0, since equation must be satisfied for all

1. Ginzburg-Landau equation

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TT1-Chap2 - 55


• minimization of Gs with respect to variation

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TT1-Chap2 - 56


• integral of last term:

• with London gauge we obtain from Maxwell equation :

= 0, since equation must be satisfied for all

2. Ginzburg-Landau equation

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TT1-Chap2 - 57


Ginzburg-Landau equations:

1. GL-equation

2. GL-equation

• Summary:minimization of free enthalpy 𝐺𝑠 = 𝑉 𝑔𝑠 𝑑𝑉 with

yields

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TT1-Chap2 - 59

energy-phase relation

supercurrent density-phase relation

(London parameter)

• Note:- GL theory can well describe spatially inhomogeneous situations- but cannot account for time-dependent phenomena (there is no time derivative)

- macroscopic quantum model cannot account for spatially inhomogeneous situations- but can describe time-dependent phenomena (Josephson effect)

Ginzburg-Landau theory:

1. GL-equation

2. GL-equation

Macroscopic quantum model:

3.3 GL-Theory vs. Macroscopic Quantum Model

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TT1-Chap2 - 60


Characteristic length scales:

• 2nd GL equation:

for

expression for supercurrent density 1st and 2nd London equation characteristic screening length for 𝐵ext:

London penetration depth• 1st GL equation:

normalization and use of

2nd characteristic length scale

GL coherence length 𝝃𝑮𝑳

𝑛𝑠 = −𝛼/𝛽

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TT1-Chap2 - 61


Temperature dependence of characteristic length scales:

• Ansatz for 𝛼 and 𝛽:

• with and

both length scales diverge for 𝑻 → 𝑻𝒄

experimental T-dependence:

reasonable agreement for T close to Tc

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TT1-Chap2 - 62


Ginzburg-Landau parameter:

solve for 𝐵cth

relation between GL and BCS coherence length:

- 𝛼 = condensation energy / (ns /2)

correct BCS result:

- BCS: average condensation energy per electron pair

𝛼/2 corresponds to ≈ −3Δ2(0)/4𝐸𝐹

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TT1-Chap2 - 63


Superconductor-normal metal interface:

• solve for x > 0

• assumptions: no magnetic field (𝑨 = 0), 𝜓 𝑥 = 0 = 0

superconductor extends at 𝑥 > 0

• boundary conditions:

and

solution:

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TT1-Chap2 - 64


• type-I superconductors:

expel magnetic field until Bcth

only Meißner phase single critical field Bcth

• type-II superconductors:

partial field penetration above Bc1

Bi > 0 for Bext > Bc1

Shubnikov phase between Bc1 ≤ Bext ≤ Bc2

upper and lower critical fields Bc1 and Bc2

experimental facts:

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TT1-Chap2 - 65


• thermodynamic critical field defined as:(for type-I and type-II superconductors)

• area under 𝑀 𝐻ext curve is the same for type-I and type-II superconductor with the same condensation energy:

condensation energy

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TT1-Chap2 - 66


difference between type-I and type-II superconductors: NS boundary energy

• savings in field expulsion energy (per area)

• loss in condensation energy (per area)

• resulting boundary energy

always positive for 𝜆L < 𝜉GL up to Bext = Bcth

can be negative for 𝜆L > 𝜉GL for Bext < Bcth formation of NS boundary lowers energy already for Bext < Bcth

type-II SC: 𝜿 = 𝝀𝐋/𝝃𝐆𝐋 > 𝟏/ 𝟐 (numerical result)- boundary energy negative for Bext < Bcth

- formation of boundary favorable

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TT1-Chap2 - 67


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TT1-Chap2 - 68

3.3.1 Type-I and Type-II Superconductors: Demagnetization Effects and Intermediate State

• ideal Bi (Hext) dependence valid only forvanishing demagnetization effects- e.g. long cylinder of slab

• for finite demagnetization

with (perfect diamagnetism)

long cylinder: N ≈ 0 Hlok ≈ Hext

flat disk: N ≈ 1 Hlok ∞sphere: N = 1/3 Hlok = 1.5 Hext

sphere:Hlok = 1.5 Hext @ equatorµ0Hlok = Bcth @ µ0Hext = 2/3 Bcth

intermediate state can have complex structure

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TT1-Chap2 - 69

3.3.1 Type-I and Type-II Superconductors: Demagnetization Effects and Intermediate State

magneto-optical image

of the intermediate state of In

bright: normal regions

(from Buckel)

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TT1-Chap2 - 70


Task: derive expressions for Bc1 and Bc2 from GLAG-equations

• (a) high magnetic fields Bc2

approximation: y is small for Bext close to Bc2 neglect 𝛽 𝜓 2𝜓 term in 1st GL-eq.

linearized 1st GL-equation:

further approximations: B = µ0Hext, since M 0 for µ0Hext Bc2

Hext = Hz A = (0,Ay,0) with Ay = µ0 Hz x = Bz x

corresponds to Schrödinger equation of charged particle in magnetic field

Ansatz:

1st GL-equation:

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TT1-Chap2 - 71


• differential equation for u(x):

with:

spring constant

• energy eigenvalues of harmonic oscillator:

• and solve for Bz:

• maximum Bz for n = 0 and kz = 0:

Bc2 > Bcth

for 𝜅 > 1/ 2

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TT1-Chap2 - 72


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TT1-Chap2 - 73


𝑩𝒄𝒕𝒉 and 𝝀𝑳 of type-I superconductors

𝑩𝒄𝟐 and 𝝀𝑳 of type-II superconductors

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TT1-Chap2 - 74


𝑩𝒄𝟐 of type-II superconductors

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TT1-Chap2 - 75


• (b) low magnetic fields Bc1

derivation of lower critical field Bc1 is more difficult (no linearization of GL equations)

• more precise result based on GL theory:

we use simple argument, that flux generated by Bc1 in area 𝜋𝜆𝐿2 must be equal to Φ0

here, we have assumed (London approximation)

Bc1 given by the fact that theminimum amount of flux throughnormal area is F0

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TT1-Chap2 - 76

3.3.3 Type-II Superconductors: Shubnikov Phase and

flux Line Lattice

• solution of the GL-equations in intermediate field regime complicated linearization of GL-equations is no longer

a good approximation qualitative discussion

• How is the magnetic flux arranged in Shubnikov phase above Bc1 ?

- maximize NS boundary, since boundary energy is negative many small normal conducting “N” regions

- upper limit is set by flux quantization: flux content in a “N” region cannot be smaller than F0

- flux lines can be viewed as small bar magnets which repel each otherarrange flux lines with maximum distance

regular lattice of flux lines with flux content F0

Abrikosov flux line lattice hexagonal lattice

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TT1-Chap2 - 77


flux Line Lattice

k = 2, 5, 20 B(r) / Bc1

ns(r)

-4 -2 0 2 4

0.0

0.4

0.8

1.2

1.6

2.0

ns ,

B /

Bc1

r / lL

µ0Hext

200 nm

(a)

(c)

(b)

(d)

H.F. Hess et al., Phys. Rev. Lett. 62, 214 (1989) E.H. Brandt, Phys. Rev. Lett. 78, 2208 (1997)

flux line, vortex

NbSe2

contour lines of ns

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TT1-Chap2 - 78


flux Line Lattice

• distance between flux lines is maximum in hexagonal lattice energetically most favorable state square lattice also often observed, since other effects (e.g. Fermi surface

topology) play a significant role

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TT1-Chap2 - 79

Right: STM-images showing the flux line lattice of ion irradiated NbSe2 (T=3 K, I=40 pA, U=0.5 mV) taken during increasingthe applied magnetic filed to 70, 100, 200, 300 mT. The images always show the same sample area of 2 x 2 µm.(source: University of Basel)


flux Line Lattice

200 nm

NbSe2: flux-line lattice of non-irradiated single crystal at 1 T

distortion of ideal flux-linelattice by defects flux-line pinning

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TT1-Chap2 - 80

Bitter technique:decoration of flux-linelattice by „Fe smoke“ imaging by SEM

U.Essmann, H.Träuble (1968) MPI MetallforschungNb, T = 4 Kdisk: 1mm thick, 4 mm ø Bext = 985 G, a =170 nm

D. Bishop, P. Gammel (1987 )AT&T Bell Labs YBCO, T = 77 K Bext = 20 G, a = 1200 nm

similar:- L. Ya. Vinnikov, ISSP Moscow- G. J. Dolan, IBM NY


flux Line Lattice

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TT1-Chap2 - 82


• in flux line lattice:how does 𝑛𝑠 𝑟 and 𝐛 𝑟 look like across a flux line ?

• radial dependence of 𝜓:we consider isolated flux line and use the Ansatz

• insertion into nonlinear GL equations yields expression for 𝑓 𝑟 :

with 𝑐 ≈ 1 𝜓 𝑟 2 = 𝑓2(𝑟)

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TT1-Chap2 - 83


• radial dependence of b (London vortex, we use approximation 𝑓 𝑟 = 𝛿2 𝑟 ):

we use 2nd London equation

accounts for the presence of vortex core• interpretation

with Maxwell eqn. we obtain

𝜆𝐿2 𝛁 × 𝛁 × 𝐛 + 𝐛 = 𝐳 Φ0𝛿2 𝑟

integration over circular area S with 𝑟 ≫ 𝜆𝐿 perpendicular to 𝐳 yields

𝑆

𝐛 ⋅ 𝑑𝑆 + 𝜆𝐿2

𝜕𝑆

(𝛁 × 𝐛) ⋅ 𝑑ℓ = 𝐳 Φ0 ⇒ Φ = Φ0

Φ 0 since 𝛁 × 𝐛 = 𝜇0𝐉𝒔 and 𝐽𝑠 ≃ 0 for 𝑟 ≫ 𝜆𝐿

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TT1-Chap2 - 84


• solution of

0th order modified Bessel function of 2nd kind

is exact result only if we assume 𝜉𝐺𝐿 → 0

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TT1-Chap2 - 85


• solution of

exact result if we assume finite 𝜉𝐺𝐿

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TT1-Chap2 - 86

3.3 Summary

Literature:• P.G. De Gennes, Superconductivity of Metals and Alloys• M. Tinkham, Introduction to Superconductivity• N.R. Werthamer in Superconductivity, edited by R.D. Parks

The Ginzburg-Landau Theory explains:

• all London results• type-II superconductivity (Shubnikov or vortex state): 𝜉𝐺𝐿 < 𝜆𝐿

• behavior at surface of superconductors and interfaces tonon-superconducting materials

The Ginzburg-Landau Theory does not explain:

• 𝑞𝑠 = − 2𝑒• microscopic origin of superconductivity• not applicable for T << Tc

• non-local effects

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