Last lecture (#10) · in a simple model for jellium, for which the real part of the Fourier transform of the impulse interaction is given by where kTF is the Thomas-Fermi wavevector

11

Last lecture (#10):Last lecture (#10):

We presented the BCS theory of superconductivity for a weak attractive interaction that is isotropic, spin independent and finite only in a thin shell around the Fermi surface. The ground state is a coherent state made up of spin-singlet Cooper pairs and the excited states have an isotropic energy gap function. The order parameter can be taken to be

or equivalently <a-k ↓ ak ↑ > , which defines the gap function.

We now consider the possible origins of the attractive interactionand generalize the BCS theory to describe complex materials such as the high-temperature superconductors MgB2 and the copper-oxide and iron-pnictide compounds.

)(ˆ)(ˆ)( >=<↑↓

rrrs ψψΨ

22

Lecture 11: Anisotropic Lecture 11: Anisotropic SuperfluiditySuperfluidity and and SuperconductivitySuperconductivity

I. Induced InteractionsIA. Polarizer-Analyzer modelIB. Phonon Mediated Electron-Electron Interaction

II. Generalization of BCS TheoryIIA. Singlet Energy Gap EquationIIB. Singlet Equation for Tc and Applications

Appendix I. Further Generalization of BCS TheoryAppendix II. Spin-Triplet Pairing States

Literature: Annett ch 7; Waldram selections from chs 7, 11-17

33

I. Induced InteractionsI. Induced Interactions

IA. Polarizer IA. Polarizer ––Analyzer ModelAnalyzer Model

Interactions between particles can usually be described as induced interactions in terms of the polarizer-analyzermodel. For example, the electron-electron interaction in the electromagnetic vacuum may be described as follows. One electron, the polarizer induces a change in the electromagnetic field and the second electron, the analyzer, samples this charge.

One can think of the interactions between fermionicquasiparticles in a metallic vacuum in a similar way, but the induced interactions now include effects from all of the other particles in the system.

44

Basic analyser-polarizer model: let g be the interaction between

a particle and a field and χ(r,t) the impulse response for the field, then the interaction potential between two particles induced by a test particle moving at velocity u is

'')''()','(),( dtdrutrttrrVtr −−−= ∫ δV

g

χ)','(2)','( ttrrttrrV −−−=−− χg

is the impulse interaction

g

In general, we include the dependence of the interaction on spin, the initial states and the momentum and energy transfers. Nearby in space and time we expect the dominant interaction to be repulsive. To get a bound state the quasi-particles must take advantage of attractions that appear at a finite separations in time or space, i.e., by avoidance in time or space.

55

An example of attraction by time avoidance is the retarded interaction between charges moving in a deformable lattice, i.e., the induced interaction arising from the virtual emission and absorption of phonons. The figure below shows the form of the interaction potential V(r,t) versus r at a moment in time due to the screened Coulomb interaction in the Thomas-Fermi model plus the retarded induced interaction due to a deformable positively charged medium (jellium). An attractive tail appears if the testcharge is moving (figure right).

Retarded positively chargedscreening cloud

66

Examples of attraction by space avoidance include the van derWaals attraction in liquid 3He and the induced spin-spin interactions in liquid 3He and in nearly magnetic metals. On the border of a Mott transition in the cuprates, for example, the relevant spin-spin interaction is known as the superexchangeinteraction acting between nearest neighbours in a square lattice.The figures below show the forms of the triplet and singlet interaction potentials for carriers on the border of ferromagnetism and anti-ferromagnetism, respectively, for homogeneous media. Note that due to the Pauli principle there is effectively a hard core repulsion in the triplet as well as in the singlet case.

77

We consider explicitly only the form of the overall interaction in a simple model for jellium, for which the real part of the Fourier transform of the impulse interaction is given by

where kTF is the Thomas-Fermi wavevector and ωq is the phonon frequency. The first term represents the instantaneous screened Coulomb repulsion while the second term gives the electron-phonon mediated attraction.* The overall interaction is attractive in the frequency range 0 < ω < ωq. This is the origin of the attractive term near the Fermi level in the BCS model.

−−

+=

22

2

22

21

1

0 ωω

ω

εωq

q

TF

qkq

eV

IB. Phonon Mediated ElectronIB. Phonon Mediated Electron--Electron InteractionElectron Interaction

*We replace ω in Vqω by the energy transfer in a scattering process so that the scattering matrix elements depend on the initial momentum as well as the momentum transfer.

88

qqqqkkkkq

q

qqkkkqkkk

qindqV

q ggg

gg

=−=−=−−=−

−=

++−+

++−=

& , ; ' , 22

2

)'(2

1

)'('2

1

2

ωωεεεεωωω

ω

ωεεεωεεεω

hh

hh

The form of the phonon mediated interaction potential can be obtained from second order perturbation theory in which the initial and final states are represented by (k,-k) and (k’,-k’) , respectively, and the intermediate states involve a virtual excitation (a phonon) as in the following diagrams

gg

-q

k

-k

-k’

k’=k+q gg

q

-k

k

k’

-k’

tim

e

99

Because Vqω is in general a dynamical interaction, a full treatment requires the use of many-body Green functions or of Lagrangiansand the path integral methods. We consider a simplified scheme where the interaction is represented in terms of weak couplingmatrix elements that depend only on q and hω=εk+q-εk. The strong coupling extension of the BCS theory will not be developed but will be reviewed below (see, e.g., Waldram ch 11).

ωqV

0 ω

qω

ωqV

0 ω

h/cε

− | g|

BCSTheory

Coulomb repulsion

Net attraction

Phonon frequencyat wavevector q

1010

The self-consistent gap equation can be generalized in a similar way. For example, for spin-singlet pairing the expression for ∆ given in lecture 10

generalizes to

where Vkk’ stands for the appropriate singlet scattering amplitude

∆k can depend on both the magnitude and direction of k near the Fermi surface. In conventional spin-singlet superconductors ∆k

has the same sign over the Fermi surface. In unconventional superconductors ∆k changes in sign over the Fermi surface.

∑−

−=' '2

))'(21(

k kEkEf

V

g∆∆

∑−

−=' '

'''

2

))(21(

k k

kkkkk

E

EfV ∆∆

IIII. Generalization of BCS TheoryGeneralization of BCS TheoryIIA. Singlet Gap EquationIIA. Singlet Gap Equation

h/)'( & ' where , ' .,. kkkkqqVkkVge εεωω −=−==

1111

The anisotropy of ∆k near the Fermi surface is usually illustrated as in the following figure in which the k-space gap at the Fermi surface is a relative measure of the energy gap function .

If the starting Hamiltonian has rotational symmetry then the labels s and d stand for angular momentum.

+

s-waveconventional (BCS)

superconductors

+ +

–

–

d-wave (dx2

-y2)

high-Tc

superconductors

Pic

ture

cre

dits:

C. Berg

em

ann, Cavendis

h L

abora

tory

)/gap space- .,.( Fvkkei h∆=

1212

The equation for Tc can also be generalized. For example, for electron-phonon mediated superconductivity the expression for Tc

given in lecture 10

generalizes, in the Eliashberg-McMillan strong-coupling theory, to

where θD is a Debye temperature, λ is the electron-phonon mass renormalization factor and µ* arises from the effect of the direct Coulomb repulsion, corrected for screening and recoil.

The McMillan formula is found to be in reasonable accord with experiment in a number of s-wave superconductors from Hg to the high-temperature superconductor MgB2, Tc=39 K.

IIB. Singlet Equation for IIB. Singlet Equation for TTcc & Applications& Applications

g Nk

cT

B

c )0( , 1

exp 13.1

=

−≈ λλ

ε

)62.01(*

)1( 04.1exp

45.1

+−

+−≈

λµλ

λθD

cT

13131313

MgBMgB22: Crystal Structure and Fermi Surface: Crystal Structure and Fermi Surface

Top:

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mgb2/c

choose.s

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: w

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Graphite-likeboron sheets

Ferm

i Surf

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xperim

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: A. Carr

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Boron σ p-orbital

Boron σ p-orbitalFermi surface sheet

14141414

Pressure

Tetragonal

“Collapsed”TetragonalAFM

Orthorhombic

AFM = antiferromagnetic metalDavid Tompsett, Quantum Matter Group, Cavendish Laboratory

CaFeCaFe22AsAs22: Temperature: Temperature--PressurePressurePhase Diagram and Fermi SurfacesPhase Diagram and Fermi Surfaces

1515

Appendix IAppendix I. Further Generalization of BCS TheoryFurther Generalization of BCS TheoryAppendix IA. Anisotropic BCS HamiltonianAppendix IA. Anisotropic BCS Hamiltonian

∑ ∑ −+−

++ +=σ

δγβααβγδ

αβγδσσε

kkkkk

kkkkkkk aaaaVaaH

'''

'ˆ

The generalization of the BCS Hamiltonian of lecture 10 is thus

where α, β, γ, δ are spin indices.

The generalization of the gap equation of lecture 10 is then

For systems with separate singlet and triplet pairing states, it is helpful to write the four gap components ∆k

αβ in terms of a scalar ∆k and a 3D vector dk = (dk

x,dky,dk

z)

-kγ

kδ

-k’β

k’α

><= −∑ δγαβγδαβ∆ ''' kkkkk

aaV

1616

triplet |||| singlet, || *222222kkkkkkkk

dddEE ×±+=+= ξ∆ξ

triplet singlet, 0

0

+

+

+−

+

−=

↓↓↓↑

↑↓↑↑

yk

xk

zkk

zkk

yk

xk

k

k

kk

kk

idd

d

d

id-d ∆

∆∆

∆

∆∆

∆∆

The scalar ∆k and the components of the vector dk can be found in terms of the starting gap functions ∆k

↑↑, ∆k↓↓ and ∆k

↑↓ by inversion.

In this scheme the excitation energies are found to satisfy

In superfluid 3He the last term vanishes so that |dk| plays the same role as |∆| in the isotropic case. Note that we take the positive solution for Ek for the elementary excitations.

1717

Appendix Appendix IBIB. s. s, , pp and and dd Gap FunctionsGap Functions

The anisotropy of ∆k or dk near the Fermi surface is usually illustrated as follows

If the starting Hamiltonian has rotational symmetry then the labels s, p and d stand for angular momentum. Pairing can arise in the presence of short range repulsion either by time avoidance in an s-state or by space avoidance in a non-s or finite angular momentum state in which the pair wavefunction vanishes when the interacting particles get too close.

+ + +

–

–

+

–

+–

i

–i

s-waveoriginal BCS

p-wave (py) (px+ipy)(dk shown)

d-wave (dx2

-y2)

high-Tcs

Pic

ture

cre

dits:

C. Berg

em

ann, Cavendis

h L

abora

tory

1818

Non-s-wave states tend to be important in strongly correlated electron systems in which low Fermi velocities reduce the relative effectiveness of the time avoidance mechanism for pairing.

In the absence of rotational invariance, we label gap functions according to the irreducible representations of the symmetry group of the lattice. Thus, s-wave corresponds to the simple representation invariant under all the relevant symmetry operations, while dx

2-y

2 corresponds to the B1 representation of the tetragonal group D4 of the lattice relevant to the cuprates. Other representations are of course possible in general and the lowest energy representation is to be determined by experiment and interpreted by theory.

1919

Appendix II. SpinAppendix II. Spin--Triplet Pairing StatesTriplet Pairing StatesAppendix IIA. SpinAppendix IIA. Spin--Triplet Triplet SuperfluiditySuperfluidity in in 33HeHe

The temperature-pressure-magnetic field phase diagram of superfluid 3He shows that there are two basic superfluid phases, A and B, with different gap functions dk.

The A phase has point nodes (at opposite poles) while the B phase is fully gapped, but the direction of dk rotates around the Fermi surface.

Pic

ture

cre

dits:

E. Thuneberg

2020

(the arrows in the figures give the directions of dk)

Some aspects of superfluid hydrodynamics are similar to those of He-II, however, the complex nature of the order parameter leads to non-trivial topological defects.

3He-A 3He-B

Pic

ture

cre

dits:

J.F. Annett

Point nodesat north and south poles

Fully gapped, butdirection of dk varies

2121

Appendix Appendix IIB. SpinIIB. Spin--Triplet SuperconductivityTriplet Superconductivityin Srin Sr22RuORuO44 and UGeand UGe22

Sr2RuO4 becomes superconducting below 1.5 K and is believed to order in a state similar to the A phase of 3He in 2D. The crystal structure is similar to that of the La2CuO4 family of the cuprates and has a quasi-2D Fermi surface with three sheets corresponding to the one-electron orbitals dxy, dxz and dyz.

Sr

RuO6

octahedra

Crystal structure Fermi surface

Pic

ture

cre

dits:

C. Berg

em

ann, Cavendis

h L

abora

tory

Sr2RuO4

dxz,yz

dxy

2222

Superconductivity is normally destroyed by ferromagnetic order, but there are some exceptions. Perhaps the most surprising is UGe2 where the magnetic electrons are itinerantand form distinct majority and minority spin Fermi surfaces. Since the states (k,↑) and (-k,↓) are not degenerate spin-singlet pairing is energetically unfavourable. Pairing is expected to be in a spin-triplet state, but the detailed nature of the order parameter is still unknown.

Pic

ture

cre

dits:

S. S. Saxena

et

al., Cavendis

h L

abora

tory