50 years of BCS SchoolOverview of disorder

Ilya Vekhter

Louisiana State University, USA

A. Balatsky, I. Vekhter, J.-X. Zhu, Rev. Mod. Phys. 78, 373 (2006)

July 2007 Cargese

July 2007 Cargese

Disorder is part of life

Undergraduate:periodic lattice,

translational invariance,

traveling phonons,

Bloch’s theorem

)(),( rrrr ′−=′ AA

)(kνωh

)()( rr kkr

k uei=ψ

Graduate: Postdoc:

Impurities, dislocations, disclinations, twin boundaries, vacant sites, chemical inhomogeneities, interstitials

July 2007 Cargese

Disorder is part of life

Undergraduate:periodic lattice,

translational invariance,

traveling phonons,

Bloch’s theorem

)(),( rrrr ′−=′ AA

)(kνωh

)()( rr kkr

k uei=ψ

Graduate: Postdoc:

Impurities, dislocations, disclinations, twin boundaries, vacant sites, chemical inhomogeneities, interstitials

Too hard

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Graduate level: impurity atoms

∑ ++

++=+

↓↑↓↑

kk

iiiiimp

chcVd

nUnnnEH

..

)(0

σσ

Different mass: local phonon modes

Different electronic configuration:

a) fixed electronic configuration of impurity

Potential scattering

Magnetic scattering

b) Possibility of hybridization

rrrr dUHimp ∫ −= )()( 0 ρ

)( 0rSσJHimp =

Uµ

Anderson, Kondo etc

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Main questions

• Do dirty superconductors superconduct?• Observable changes due to impurities.

This talk: density of states• Impurity-induced bands• Superconducting transition temperature.• Mostly not in this talk: see the list at the end

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Plan

• Single impurity problem: T-matrix solution

• Overview of results on superconductors

• Ensemble of impurities and averaging

• Abrikosov-Gor’kov theory and its extensions

• List of some issues that did not fit

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Theorist’s pride

Important disorder effects in superconductors: bound states around impurities, gapless

superconductivity, universal conductivity, have been predicted by theorists before they were

observed experimentally!

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Scattering

We are solving a scattering problem

∑∑∫∫∫

′′

+′

′−

′′

+

+

==

ΨΨ==

kkkkkk

rkk

kkkkrr

rrrrrrr

σσσσ

σσρ

ccUeccUd

UddUHi

imp

)()(

)()()()()(

kk ′U

k′ k

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Nambu formalism and matrices

•BCS hamiltonian

•Matrices ii τσ , in spin and particle-hole space respectively

•Matrix structure of the impurity scattering:

Potential:

Magnetic:

3)()(ˆ τrr UU ⇒ e.g attracts electrons/repels holes

( ) ( )[ ] 2/11 3333 σσττ σσα −++=αSσS ⋅⇒⋅

•Pure BCS

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Single static impurity

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Multiple scattering..chccUH kk

kkkkimp += ′

+

′′∑ σσ• Key: multiple scattering

change of momentum/spin at each scattering event

can include all the scattering events … in principle

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T-matrix solution

kk ′U= + + …

July 2007 Cargese

T-matrix solution

kk ′U= + + …

)(ˆ),(ˆˆˆ)(ˆ0 ωωω TGUUT k

k∑+= UGUT

1

0 ),(ˆˆ1)(ˆ−

⎥⎦⎤

⎢⎣⎡ −= ∑ ωω k

k

UU =′kk ,

structure is especially simple for

isotropic scatterers, T-matrix

depends on ω only. )(, ωTT =′kk

T-matrix includes all the effects of multiple scattering on a single impurity

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DOS and T-matrix

);,(ˆIm),( 111 ωπω rrr GN −−=• Density of states

• T-matrix in real );,()();,();,();,( 00000 ωωωωω rrrrrrrr GTGGG +=space

[ ]);,()();,(Im)(),( 00001

0 ωωωπωω rrrrr GTGNN −−=• With impurity

0);,(Im)( 01

0 =−= − ωπω rrGN• If for some ω2

0 ),()(Im),( rr ωωωδ GTN ∝• New states

Impurity-induced new states appear at energies

where T-matrix has imaginary part: poles of T(ω)

July 2007 Cargese

DOS and T-matrix

);,(ˆIm),( 111 ωπω rrr GN −−=• Density of states

• T-matrix in real );,()();,();,();,( 00000 ωωωωω rrrrrrrr GTGGG +=space

[ ]);,()();,(Im)(),( 00001

0 ωωωπωω rrrrr GTGNN −−=• With impurity

0);,(Im)( 01

0 =−= − ωπω rrGN• If for some ω

• New states 20 ),()(Im),( rr ωωωδ GTN ∝ Friedel

oscillations

Impurity-induced new states appear at energies

where T-matrix has imaginary part: poles of T(ω)

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2D Metal: experiment

P. Sprunger et al. 1997

Be

Fourier transformreal space

Spatial oscillations with kFr: Fourier transform gives image of the Fermi surface

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Scattering strength

3

1

031

0

1

0 ),ˆ(ˆˆ)(ˆ),(ˆˆ1)(ˆ τωτωω−

−−

⎥⎦

⎤⎢⎣

⎡−=⎥⎦

⎤⎢⎣⎡ −= ∫∑

FSgdUNUGUT kkk

k

integral over Fermi surface

∫=FS

Gdg ),,ˆ(ˆ),ˆ(ˆ 00 ωξξω kkphase shift of scattering

011

0 cot)( δ−− =UN generally depends on band structure

1)( 10 <<−UN 2/0 πδ ≈ unitaritystrong scatterers

1)( 10 >>−UN 00 ≈δ Bornweak scatterers

expand

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(Un)conventional pure SC• Conventional / isotropic /

s-wave superconductors

0)ˆ( =∆∑ Fkk

constF =∆ )ˆ(k

• Unconventional / anisotropic / nodal superconductors

0/)( ∆∝ωωN0)ˆ( =∆ nodek

line nodes

direction at the Fermi surface

φ2cos0∆=∆

∫∆−FS

d22

Reωωϕ

∫∆−FS

dφω

ωφ2cos

Re22

02

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Simple example: potential scattering4x4 → 2x2

21

20

20

31

01100

)()()(ˆggUN

UNggT−−

−−= −

− τττωspin is not “active”

221

)ˆ(

)ˆ(ˆ)(k

kk∆−

∆−= ∫

ωω

FSdg

220

)ˆ(ˆ)(

kk

∆−−= ∫

ω

ωω idgFS

vanishes0)( 1g⇒=∆∑ kk

Different structure of T-matrix for conventional and nodal superconductors:

check for new poles

July 2007 Cargese

Potential scatterer: s-wave

∑′

′+=

kkkk σσccUHimp const=∆ )(k

1)()(ˆ

20 +

= −

∑UN

aT i

iiτω no new poles

↓−↑ kk ,Physics: we are pairing time-reversed states: potential impurity makes states not simply|k>, but does not violate time-reversal. ↑↑ nTn ,

The only situation where impurities are not harmful to

superconductivity: Anderson’s theorem

P. W. Anderson, 1957

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Classical magnetic scattering: s-wave

∑′

′+=

kkkk σS βαβα ccJHimp const=∆ )(kclassical spin S

( )[ ]20

20

2

2/)(ˆ1

)(ˆ2/)(ˆω

ωωgJS

gJST−

∝ new poles

Time-reversal violated: new states below the gap edge

10 ≈JSN state in midgap

10 <<JSN state near gap edge

00 ≈E 00 ∆≈E

A. Rusinov, 1968; H. Shiba, 1968, L. Yu 1965

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Experiment: s-wave

A. Yazdani et al, 1997

Mn & Gd magnetic, Ag non-magnetic

Asymmetric spectra: extract/inject e

2kv

2ku

Decay of the state on the scale:

)/exp(ψ 02 rr−∝

20000 )/(1/ ∆−≈ Er ξ

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Potential scatterer: d-wave0)ˆ(ˆ =∆∫

FSFd kk∑

′′

+=kk

kk σσccUHimp φφ 2cos)( 0∆=∆

Wait, no gap! Poles of T-matrix ? Yes

⎥⎦⎤

⎢⎣⎡ +∆−=Ω+Ω=Ω

ci

cci

ππ

ππ

/8ln2/1

/8ln2/

0210

energy

10)( −= UNc

resonance lifetime

12 Ω<<Ωstrong scatterers sharp

12 Ω≥Ω smearedweak scatterers

P. Stamp 1987, J. Byers and D. Scalapino 1993, A. V. Balatsky et al. 1995

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Experiment: d-wave

S. Pan, E. Hudson et al., 2000

far from Zn

on Zn

Zn impurity in BSCCO Spatial dependenceis poorly understood:see P. Hirschfeld and A. Pasupathy lectures

tails

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Graphical solution: potential scatterer

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Averaging over ensemble of impurities

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Impurities and superconductivity-

++

-

Scattering mixes gaps at

different points at the FS

d-waveanisotropic s-wave

Anisotropy smeared out, Tc slightly suppressed

Gap and Tc suppressed

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Self-consistent approximation

dilute impurities

1<<impn

sequential multiple scattering on impurities

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Self-consistent approximation

dilute impurities

1<<impnprobability for electron to return to a previously encountered impurity is at least of order impimp nn <<2

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Self-consistent approximation

dilute impurities 1<<impn

= +r r’ rr r’ r’)(ˆ

1rU+ r r’

)(ˆ1rU )(ˆ

1rU

r r’+)(ˆ

1rU )(ˆ1rU )(ˆ

2rU

r r’)(ˆ

1rU )(ˆ2rU )(ˆ

2rU+

+ r r’)(ˆ

1rU )(ˆ2rU )(ˆ

1rU

improbable: ignore

Then average over random positions of all impurities

July 2007 Cargese

Self-consistent approximation

dilute impurities 1<<impn

= +r r’ rr r’ r’)(ˆ

1rU+ r r’

)(ˆ1rU )(ˆ

1rU

r r’+)(ˆ

1rU )(ˆ1rU )(ˆ

2rU

r r’)(ˆ

1rU )(ˆ2rU )(ˆ

2rU+ +…

Self-consistent T-matrix

Full Green’s function with scattering on all other impurities: need self-consistency

Gap and order parameter are not the same.

P. Hirschfeld et al., S. Schmitt-Rink et al. 1986

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Abrikosov-Gorkov theoryS-wave. Weak scatterers: Born approximation

Potential scattering does not affect Tc or gap: Anderson’s theorem

Destroys superconductivity and the gap)1(20 += SSJNnimpsα

Order parameter (solution of self-consistency equation) exists up to

Gap for excitations (pole of Green’s function) vanishes at

)4/exp(0 παα −∆== gs gcs ααα 1.12/0 ≈∆==<

There exists a regime of gapless superconductivity!

Abrikosov, Gorkov, 1960

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s-wave, magnetic scattering

Gap for excitations

transition temperature

order parameter

Skalski et al 1964

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Comparison with experiment

Skalski et al 1964

theory

M. Woolf and F. Reif, 1965

Pb-Gdexpt

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Shiba bands)1(2

0 += SSJNnimpsαAbrikosov-Gorkov: smearing out of the gap edge

Growth of impurity band from the position of the bound state: hopping

Weak scattering: bound state near the gap edge, smearing of the gap

Strong scattering: growth of impurity band from the position of the bound state in the gap

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Experiment

L. Dumoulin et al., 1977

theory

expt expt

W. Bauriedl et al., 1981

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Unconventional superconductorsAlways gapless (in SCTM)All impurities are pairbreaking

Born limitL. Gorkov and P. Kalugin, 1985, T. Rice and K. Ueda, 1985

Unitarity limit

P. Hirschfeld et al., 1986, S. Schmitt-Rink et al, 1986

Experimental manifestations: penetration depth, universal transport, …

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Common to unconventional SC

Finite DOS at ω=0 (gapless)

Impurity band ~γ

P. Hirschfeld et al., 1989,

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Summary I• One protected phenomenon: non-magnetic impurities in

isotropic superconductors do not influence mean field transition temperature (Anderson’s theorem).

• Bound states around impurities: test by STM

• In all other cases impurities are pairbreaking: reduce gap, reduce Tc, not necessarily at the same rate

• Gapless superconductivity: a common phenomenon. Finite density of states at the Fermi level, still superconducting.

July 2007 Cargese

Summary II: have not considered• Impurities with internal dynamics: Kondo impurity,

Anderson impurity, precessing spin…

• Beyond self-consistent T-matrix: localization etc.

• Beyond simple averaging: optimal fluctuation. All superconductors are gapless

• Effect of impurities on transport etc.

• Impurities in correlated systems (P. Hirschfeld)

July 2007 Cargese