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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 interaction and 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

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Lecture 11: Anisotropic Superfluidity and Superconductivity

I.  Induced Interactions IA. Polarizer-Analyzer model IB. Phonon Mediated Electron-Electron Interaction

II. Generalization of BCS Theory

IIA. Singlet Energy Gap Equation IIB. Singlet Equation for Tc and Applications

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

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

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I. Induced Interactions

IA. Polarizer –Analyzer Model

Interactions between particles can usually be described as induced interactions in terms of the polarizer-analyzer model. 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 fermionic quasiparticles in a metallic vacuum in a similar way, but the induced interactions now include effects from all of the other particles in the system.

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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 −−−=−− χgis 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.

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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 test charge is moving (figure right).

Retarded positively charged screening cloud

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Examples of attraction by space avoidance include the van der Waals 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 superexchange interaction 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.

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We consider explicitly 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

211

0 ωω

ω

εωq

q

TFq

kqeV

IB. Phonon Mediated Electron-Electron 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.

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' where , 222

)'(22/1

)'('22/12

2

kkq

q

qqkkkqkkk

qindqV

q εεωωω

ω

ωεεεωεεεω

−=−

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

++−+

++−=

g

gg

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

g

g -q

k

-k

-k’ k’=k+q g

g q

-k

k

k’ -k’

time

qqqqkk gg =−=−=− & , ωωεε

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Because Vqω is in general a dynamical interaction, a full treatment requires the use of many-body Green functions or of Lagrangians and the path integral methods. We consider a simplified scheme where the interaction is represented in terms of weak coupling matrix elements that depend only on q and ω=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

/cε

€

− | g|

BCS Theory

Coulomb repulsion

Net attraction

Phonon frequency at wavevector q

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

where Vkk’ stands for the appropriate scattering amplitude* Note that Δk can depend on both the magnitude and direction of k near the Fermi surface. In conventional singlet superconductors Δk has the same sign over the Fermi surface. In unconventional superconductors Δk changes in sign over the Fermi surface.

∑−

⎟⎠

⎞⎜⎝

⎛−=' '2

))'(21(

k kEkEf

Vg

ΔΔ

∑−

−=' '

''' 2

))(21(

k kk

kkkk EEfV ΔΔ

II. Generalization of BCS Theory IIA. Singlet Gap Equation

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

*This applies also to the spin dependent interaction to be discussed in lecture 12 if Vkk’ stands for the spin-singlet amplitude (see e.g., Annett pp 154-156).

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The anisotropy of Δk 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-wave conventional (BCS) superconductors

+ +

–

–

d-wave (dx2-y

2) high-Tc

superconductors

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)/gap space- .,.( Fvkkei Δ=

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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 Tc & Applications

)0(1exp

13.1⎟⎟⎠

⎞⎜⎜⎝

⎛−≈

g NkcTB

cε

)62.01(*

)1( 04.1exp 45.1 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+−

+−≈

λµλ

λθDcT

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MgB2: Crystal Structure and Fermi Surface

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Graphite-like boron sheets

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A.

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Boron σ p-orbital

Boron σ p-orbital Fermi surface sheet

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Tem

pera

ture

Pressure

Tetragonal

“Collapsed” Tetragonal AFM

Orthorhombic

AFM = antiferromagnetic metal

CaFe2As2: Temperature-Pressure Phase Diagram and Fermi Surfaces

Dav

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Appendix I. Further Generalization of BCS Theory Appendix IA. Anisotropic BCS Hamiltonian

∑ ∑ −+−

++ +=σ

δγβααβγδ

αβγδσσε

kkkkk

kkkkkkk aaaaVaaH ''

''

ˆThe 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 Δk

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

x,dky,dk

z)

-kγ

kδ

-k’β

k’α

><= −∑ δγαβγδαβΔ ''' kkkkk aaV

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triplet |||| singlet, || *222222kkkkkkkk dddEE ×±+=+= ξΔξ

triplet singlet, 0

0⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

+

+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

↓↓↓↑

↑↓↑↑

yk

xk

zkk

zkk

yk

xk

k

k

kk

kkidd

d

d

id-d Δ

ΔΔ

Δ

ΔΔ

ΔΔ

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

↑↑, Δk↓↓ and Δk

↑↓ 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 |Δ| in the isotropic case. Note that we take the positive solution for Ek for the elementary excitations.

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Appendix IB. s, p and d Gap Functions The anisotropy of Δk 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-wave original BCS

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

d-wave (dx2-y

2) high-Tcs

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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.

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Appendix II. Spin-Triplet Pairing States Appendix IIA. Spin-Triplet Superfluidity in 3He

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.

Pict

ure

cred

its:

E. T

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berg

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(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

Pict

ure

cred

its:

J.F.

Ann

ett

Point nodes at north and south poles

Fully gapped, but direction of dk varies

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Appendix IIB. Spin-Triplet Superconductivity in Sr2RuO4 and UGe2

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

Pict

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Sr2RuO4

dxz,yz

dxy

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Superconductivity is normally destroyed by ferromagnetic order, but there are some exceptions. Perhaps the most surprising is UGe2 where the magnetic electrons are itinerant and 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.

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S. S

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Lab

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