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1

Ch. 7 SUPERCONDUCTIVITY

1911: Kammerling-Onnes discovered that below Tc = 4.2 K, the resistivity of Hg dropped suddenly to a value close to 0

r(T)

T

normal

supra

This behavior, called superconductivity,and which is found in many other metals,is still one of the challenging problemsin solid state physics (*)

(*) Nobel prize in physics 1972, 1987, 2003Tc

A microscopic understanding of the phenomenon was given only in 1957 by Bardeen, Cooper and Schrieffer: According to their so-called BCS theory, the e-ph coupling can mediate an effective attractive e-e interaction, which, below Tc, gives rise to a new correlated ground state with paired electrons

REMARKS

• The first hint that superconductivity could have been originated from lattice vibrations, is the so-called isotopeeffect discovered for Hg by Maxwell and Reynolds:

Tc is a function of the ion mass for different ions of the samemetal, yielding

DT/Tc ~ - d M/2M

• The second hint comes from the observation that, remarkably, the noble metals Cu, Ag and Au do NOT becomesuperconductors. This results also from their closed packed FCC crystalline structure which results is a small e-ph coupling,and hence a small phonon-induced resistivity compared to other elements.

Some elements as Pb, though with FCC structure becomesuperconducting. This is because of their very low modulusof elasticity.

2

Superconducting table of the elements

7.1 COOPER´S INSTABILTY

El. Hamiltonian with phonon mediated e-e interaction,

2

† †, 2 2 , , ' , , ' ', , ' , ,

, ', , , ' , ,

( ) ( )ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ( ))e n k p n k p n k n k n

k k p n k n k p

p pH H c c c c

p

g

• Attractive e-e interaction favours the formation of bound pairs• Would the bound pair formed at the Fermi surface be stable ?

1 2, ( )F Fk k

p

Filled Fermi sea(electronic ground state

of normal metal)' Indicates that sum is

restricted to

1 2 1 1 2 21 2

' † †12 1 2 ˆ ˆ( , )

k kk kF k k c c FS

i) Build a pair state:

Steps:

For electrons in band n interacting via a phonon in branch . From now on, n, omitted.

3

COOPER´S INSTABILITY II

1 2, ( )F Fk k

p

1 2 1 1 2 21 2

' † †12 1 2 ˆ ˆ( , )

k kk kF k k c c FS

1k

Most favourable situation corresponds to

021 kkK

2k

K

( )F p

F

• The interaction conserves wavevector, so that we are concerned with a situation in which all pairs have thesame total wavevector ;

further the two momenta must lie in a shell of thicknessabout F

21 kkK

2k

D

COOPER´S INSTABILITY II

1 2 1 2

' † †12 ˆ ˆ( )

k kkF k c c FS

021 kkK

1 2

'

, ( ) 0k

C F k

• The formation of a bound pair will require

This condition cannot be fulfilled if the spins are alligned since it would correspond to a spin part being symmetric, which would force the orbital part of the pair wave function to be antisymmetric

21

The bound pair is formed thus by time-reversal partners with opposite momentum and spin (Cooper pairs)

4

COOPER´S INSTABILITY III

Simplify the attractive interaction by writing

otherwise02))(()(

)()(','

22

'

2

Dkkkk

eff

kk

V

p

pp d

g

ii)

' † †

, , , , , ,, ,ˆ ˆ ˆ ˆ ˆ ˆ ˆ

2eff

k k k k p k p k kk k p

VH c c c c c c

Debyefrequency

Note: This is the potential form used by BCS (weak couplingtheory). In later approaches, once the undelying mechanismwas well understood, the full form of the interaction has been retained (strong coupling theory),

COOPER´S INSTABILTY IV

iii) Evaluate : 12 12 12ˆE H

Instability of Fermi surface if FEE 212

1 1 2 21 2

1 1 2 21

' '* † † †12 1 2, , , , , ,

' '* † † † †1 2, , , , , , , ,, ,

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )2

k k k k k k kk k k

eff

k k k p k p k k k kk k p k

E FS F k c c c c F k c c FS

VFS F k c c c c c c F k c c

2

'FS

2' ' *12 , ,

2 ( ) ( ) ( )effkk k qE F k V F k q F k

yielding

5

COOPER´S INSTABILTY V

iv) Minimize E12 with respect to F*, under the normalizationcondition

1)(2

k

kF

This can be done using the method of Lagrange multipliers:

or

01)()'(

' 2

12*

k

kFEkF

'

')()'()2(

keffkkFVkFE

'( ) , ( )

2eff

kk

V CF k C F k

or

'

2kk

eff

E

CVC

moreover,2' ' *

'' , '(2 ) ( ') ( ) ( ')effkk k k

F k V F k F k

COOPER´S INSTABILITY VI2 2' ' ' *

' '' ' ,(2 ) ( ') 2 ( ') ( ) ( )effk kk k k k q

F k F k V F k F k q

Recalling expression for E12 it follows: = E12'

12

12

eff

kk

V

E

v) Evaluate the restricted k-sum

12

12

1212 2

)(2ln

2

)(

2

1)(

2

)(1

EE

EEEDV

EEdEEDV

EE

EDVdE

F

DFFeffE

E

Feffeff

E

E

DF

F

DF

F

))(/2exp(22))(/2exp(1

))(/2exp(2212 FeffDF

Feff

FeffDF EDVE

EDV

EDVEE

1)( Feff EDV

density of states

• Instability occurs however weak Veff is !KEDVEE DFeffDFCP 4002))(/2exp(2212 D •

CBCP TkD• (see later)

6

REMARKS

• Cooper´s theory is not itself a theory of superconductivity,since it deals with pairs, and not with a many-body problem!

• Note that from the BCS theory it turns out that ~103 pairs are involved

7.2 MICROSCOPIC BCS THEORY

' , , ', ',, 'ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ2BCS k k k kk k k k kk k k

H c c V c c c c N

Starting point for BCS theory is the BCS Hamiltonian

with'

'

/2 ,,

0

eff Dk kk k kk

VV

otherwise

The key-idea is to treat the interaction within Hartee-Fockmeanfield theory

' , , ', ',, '

' , , ', ', ' , , ', ',, ' , '

*

,

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 2

ˆ ˆ ˆ ˆ

MFBCS k k k kk k k k kk k k

kk k k k k kk k k k kk k k k

k k k k kk

H N c c V c c c c

V c c c c V c c c c

c c c c

D

, , ,

ˆ ˆk k k kk k

c c

D

taken with respectto BCS Hamiltonian

7

BCS THEORY II

† * * † †

, , , ,ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆMF

BCS k k k k k k k k kk k kH N c c c c c c

D D

' ', ','ˆ ˆ2

k kk k kkV c c

D

Due to its quadratic form, can be diagonalized by a Bogoliubov transformation by theunitary transformation

MFBCSH

,† * * †

,

ˆˆ -v

ˆ +v ˆk

kk k k

k k k

cu

u c

with inverse transformation *,

†*†

,

ˆ ˆ+vˆ-vˆ

k kk k

kk kk

c u

uc

1v22

kk

u

BCS THEORY III

k

kk

k

kk

u

EΕ

1

2

1v,1

2

1 22where

Explicitly,

22

kkk D E

yielding † †ˆ ˆ ˆ ˆ ˆ ˆ( )MFBCS k k k k kk

H N const

E

kE

k k

kE

normalmetal

superconductor

No excitationpossible with energy < D

gap ctingsupercondukD

The quasiparticlesdescribed by the numberoperator are fermions

and are called bogoliubons

†ˆ ˆk k

kD

8

6.3 QUASIPARTICLES AND THE BCS GROUND STATE

The bogoliubons are operators associated to energy excitationsof the superconductor. Such excitations are generated by acting with a creation operator on the BCS ground state:

Similar to what one does for other excitations (cf. phonons), one associates a quasi-particle to such an excitation.

Due to , these are fermionic quasi-particles

In turn, the BCS ground state is defined as the vacuum of the quasiparticles:

†ˆBCSk

k

ˆ 0 ,BCSkk

†'' ' '

ˆ ˆ,k k kk d d

6.3 BCS GROUND STATE

The wave function which is the quasi-particle vacuum is then

† †ˆ ˆv 0BCS k k k kk

u c c

complex expansion coefficients defined by the Bogoliubov transformation

This is possible if we remember that is the vacuum for the quasi-particles has a definite phase. BCS

kku v,

BCS• is made up of zero momentum, spin singlet pairs(Cooper pairs) whose number is undetermined!

BCS

In situations in which not only the excitation spectrum but also other observables are required, one can formulate

9

PARTICLE CONSERVING BCS GROUND STATE

M

In situations in which not only the excitation spectrum but also other observables are required, one can formulate a theory of superconductivity with a particle-conserving ground state

See A.J. Leggett “Quantum liquids” (2006), Ch. 5

•

kk

PARTICLE CONSERVING BCS GROUND STATE

M

1 Cooper pair of momentum k NO Cooper pair of momentum k

can be rewritten as

comparing with BCS with

BCS

10

7.4 THE SUPERCONDUCTING GAP

The self-consistent solution for is found by explicit evaluation of the expectation value

kD

D

'''

*

''''

*

'''''

*

''

'

' '''*

''''*

''

''''

)(21vˆˆvˆˆv

ˆvˆˆvˆˆˆ

kkkkkkkkkkkkkk

kkk

k kkkkkkkkk

kkkkkkk

fuVuuV

uuVcV

E

c-

,','ˆˆ

kkcc

Bogoliubons are free fermions

k

k

k

k

kkk

kk

k

kk

uu

ΕΕEΕ

D

2

11

2

1v1

2

1v,1

2

12/1

2

2

*22

D

D

D

D

Dk

kk

kk

kFeff

k

k

keff dDV

fV

22

22

'

'

2

2tanh

)(2

)(211 E

E

E

THE SC GAP II

The superconducting gap can be evalauted numerically from

D

D

D

Dkk

kk

kFeff dDV

22

22

2

2tanh

)(1 E

• In the limit T 0 one finds the analytic result

D

D

k

DFeff

kk

kFeff DVdDVD

1

022

sinh)(1

)(1 EE

which, in the weak coupling limit VeffD(EF) << 1 yields

))(/1exp(2)0( FeffD DV ED

T

D0D(T)

Tc• note that by its definition is DDk

11

THE CRITICAL TEMPERATURE

An expression for the critical temperature is obtained from

D

D

D

Dkk

kk

kFeff dDV

22

22

2

2tanh

)(1 E

observing that as T rises from T = 0, the tanh decreases from 1.

CBDD Tk

FeffCB

Feff xx

dxDVTk

dDV2/

00

tanh1

)(2

tanh1

)(1

EE

Hence, the denominator must decrease as well, which can onlyhappen if D decreases.

TC is determined from below from (*) when D = 0+.

(*)

Weak coupling limit DCBTk )(/1exp2 Feff DV

DCB eTk E

= 0.577 Euler´s const

VALIDITY OF MEAN FIELD APPROACH?

Weak coupling limit 1)(, FeffDCB DVTk E

)(/1

)(/1)(/1

exp2)0(

exp13.1exp2

F

FeffFeff

fDVD

DVD

DVDCB eTk

Eef

EE

D

53.3)0(2

D

CBTk

Metal CBTk/)0(2D

Tin (Sn)Lead (Pb) Indium (In)

3.464.293.63

TETFCCTET

Structure

BCS theory explains superconductivity for elements and simple alloys with Tc close to 0K.However, it is inadequate to explain how superconductivity is occurring in so-called high Tc superconductors .

non perturbativein Veff !!!

12

High Tc superconductors

In 1986, a breakthrough discovery was made by Alex Müller and Georg Bednorz (IBM Research Laboratory in Rüschlikon, Switzerland). They created a brittle ceramic compound that superconducted at the highest temperature then known: 30 K.The Lanthanum, Barium, Copper and Oxygen compound that Müller and Bednorz synthesized, behaved in a not-as-yet-understood way. For their discovery of this first of the superconducting copper-oxides (cuprates) they won the Nobel Prize a year later.

Cuprates are characterized by a layered structure which is thought to be atthe origin of this high Tc behavior.

The world record Tc of 138 K is so far held by athallium-doped, mercuric-cupratecomprised of the elements Mercury, Thallium, Barium, Calcium, Copper and Oxygen.

… and novel superconducting materials

Nowadays (2019) a rush exists towards superconductivity in 2D materials and inmaterials made of few 2D-layers.

Astonishing example is the superconduciting behavior discovered in 2018 forbilayer graphene when the two layers are twisted ba the „magic angle“ of. 1,1°,see Nature volume 556, p. 43–50 (2018).

Other new research fields are related to topological superconductivity, which enables the existence of quasiparticles with non-abelian statistics, so calledMajorana quasiparticles. Such quasiparticles are considered to be ofoutermost relevance for fault-tolerant topological quantum computation.

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