50 years of bcs school lecture 2: overview of disorder · 50 years of bcs school overview of...
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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
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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
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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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July 2007 Cargese
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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July 2007 Cargese
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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July 2007 Cargese
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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July 2007 Cargese
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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July 2007 Cargese
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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July 2007 Cargese
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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July 2007 Cargese
Single static impurity
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July 2007 Cargese
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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July 2007 Cargese
T-matrix solution
kk ′U= + + …
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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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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 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)
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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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July 2007 Cargese
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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July 2007 Cargese
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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July 2007 Cargese
(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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July 2007 Cargese
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
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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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July 2007 Cargese
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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July 2007 Cargese
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July 2007 Cargese
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July 2007 Cargese
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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July 2007 Cargese
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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July 2007 Cargese
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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July 2007 Cargese
Graphical solution: potential scatterer
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July 2007 Cargese
Averaging over ensemble of impurities
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July 2007 Cargese
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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July 2007 Cargese
Self-consistent approximation
dilute impurities
1<<impn
sequential multiple scattering on impurities
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July 2007 Cargese
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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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+
+ r r’)(ˆ
1rU )(ˆ2rU )(ˆ
1rU
improbable: ignore
Then average over random positions of all impurities
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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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July 2007 Cargese
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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July 2007 Cargese
s-wave, magnetic scattering
Gap for excitations
transition temperature
order parameter
Skalski et al 1964
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July 2007 Cargese
Comparison with experiment
Skalski et al 1964
theory
M. Woolf and F. Reif, 1965
Pb-Gdexpt
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July 2007 Cargese
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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July 2007 Cargese
Experiment
L. Dumoulin et al., 1977
theory
expt expt
W. Bauriedl et al., 1981
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July 2007 Cargese
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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July 2007 Cargese
Common to unconventional SC
Finite DOS at ω=0 (gapless)
Impurity band ~γ
P. Hirschfeld et al., 1989,
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July 2007 Cargese
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.
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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)
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July 2007 Cargese