Impurity states and marginal stability of unconventional ...

Impurity states and marginal
stability of unconventional
superconductors
A.V.Balatsky(LANL)
• local =atomic or coherence length scale
• impurity or defect
•Impurity states inconventional supercondcutors
• Impurity states in unconventional superconductors
• Impurity states in pseudogap state
http://theory.lanl.gov
Rev Mod Phys, v 78, p 373 (2006)

See also Peter Hischfeld lectures

Motivation 1
Impurities as a local probe of the symmetry of the
superconducting order parameter:
30 Å
logarithmic intensity scale
0 Å
b
-1.2 mV
a
Zn atoms in
Bi 2 Sr 2 Ca(Cu 1-x Ni x ) 2 O 8+δ
Four-fold symmetry,
aligned with the gap nodes
S.H. Pan et al, Nature (2000)

Motivation 2
Impurities as a probe of the Pseudogap state of
high-Tc superconductors

Motivation 3
Most of the theory and experiment on
conventional superconductors was done *before*
Local probes, like Scanning Tunneling Microscopy were
Invented.
A lot of views have had “calcified” into standard
Textbook wisdom.
I will show some examples where local probes bring in
New and unexpected perspective, sometime new results.

1911 – H. Kamerlingh-Onnes:
superconductivity in
mercury, Tc = 4K
Brief history
1933 – Meissner effect:
expulsion of magnetic field
1957 – Bardeen-Cooper-Schrieffer
(BCS) theory
1962 – Josephson effect
1986 – first high-Tc superconductor,
LBCO Tc = 35K
1987 – YBCO, Tc = 93K – warmer than liquid N
1995 – Hg 0.8 Tl 0.2 Ba 2 Ca 2 Cu 3 O 8.33 Tc = 138K

Superconductivity primer
Metal
Superconductor
Cooper
pairs
(k & -k)
ε
n
ε
E n
E
EF
N(ε)
2∆
EF
N(E)
2
k
= 2 m
ε
k
− E 2 2
F
Ek
= ± ε
k
+ ∆
Electrons:
c , c
k↑ k↓
Bogoliubov q.particles:
γ , γ ( γ u +
↑ ↓ ↑ kc
v
↑
k
k
k
+
=
kc
k −k↓
)

Local Probes:Scanning tunneling microscopy/spectroscopy
ε
z
ε F + eV
I t∝∫LDOS sample (ε)dε
ε F
eV
I t
dI t
dV
∝ LDOS sample (ε)
Tip
DOS
sample
LDOS
Spectroscopic map (LDOS map) can
be obtained.

Bogoliubov-Nambu Hamiltonian
background
Bogoliubov angle measures the relative weight between particle
And hole component in Bogoliubov deGennes quasiparticle
† † †
( )
ks ks
(
k ks −k −s
)
H = ε k c c + ∆ c c + h c
γ
= uc + vc
*
∗
k↑ k k↑ k −k↓
2
| ( )| 1/2 (
uk = 1 −ε
( k)/ Ek ( ))
2
| ( )| 1/2 (
vk = 1 + ε ( k)/ Ek ( ))
True excitations in SC state γ
k
That are quantum mechanical †
γ
Mixture of particle and hole
k
Ψ =
0
Ψ
0;
= Ψ
0 1
Important! Dual
nature of qp inSC
† †
γ ↑
Ψ
0
= Ψ1 ∝c ( k)|
k ↑
Ψ1
↓
and at the same time ∝c ( −k) | Ψ
1

Bogoliubov Quasiparticles
1
2
2
=
+
+
=
−
=
↑
+
↓
−
+
↓
−
+
↓
−
↑
↑
i
i
i
i
i
i
i
i
i
i
i
i
v
u
v
u
v
u
ψ
ψ
γ
ψ
ψ
γ
mixing particles and holes
0
)
(
0
0 ∏
>
+
↓
−
+
↑
+
=
Ψ
i
i
i
i v i
u
ψ
ψ
GS:
Excited State:
↓
−
+
↓
−
↓
−
↑
>
+
↓
−
+
↑
+
↓
−
+
↓
−
↓
−
Ψ
=
Ψ
Ψ
=
Ψ
+
=
Ψ
=
Ψ
∏
1
1
0
1
1
1
0
1
1
1
0
1
1
,
0
)
(
u
v
v
u
i
i
i
i
i
ψ
ψ
ψ
ψ
ψ
γ
Positive bias
Negative bias

Tunneling into SC
• Particle and hole component in tunneling
sample the same Bogoliubov Nambu
quasiparticle. The only difference is relative
weight of these two processes.
Ψ
Ground state
0
Bogoliubov excitation
∏
γ ∗ ∗ ∗
↑
n↓
−
i
Ψ
0
= vnc ( ui+
vc
i ic
i)0
Electron spin down
Tunneling out

Tunneling into SC
• Particle and hole component in tunneling
sample the same Bogoliubov Nambu
quasiparticle. The only difference is relative
weight of these two processes.
Ψ
Ground state
0
Physically one is left with the same state (~1/N)
As a result of tunneling. Amplitudes (u vs v) are
different
Bogoliubov excitation
∏
γ ∗ ∗ ∗
↑
n↑
−
i≠n
Ψ
0
= unc∗ ( ui+
vc
i ic
i)0
Electron spin up
Tunneling in

Mind trick

Mind trick

Mind trick

Mind trick

Lets do it again

Lets do it again

Lets do it again

Lets do it again

You noticed!
•Few points on overall approach in this lecture:
•Impurities and defects are interesting.
•They are telling us a story about host
•They can be useful as a markers of new physics
•Controlled impurity physics is at the core of multibillion $
semiconducting industry
•Impurity states enable the new functionality
, that is used in devices. Case; my computer that allows me
to project this lecture.
•Impurities can be placed deliberately to destroy the state
we are trying to understand
•From response of unknown state to impurities we can
infer what “it is made of”
•Not average description but one at a time. Averaged quantities
can be useful. But they can be misleading.

Conventional, s-wave
superconductors (BCS)

T-matrix approximation
Single impurity of strength V at r = 0 (Yu Lu, Shiba, …):
G
r , r′ ; ω)
= G0 ( r − r′
; ω)
+ G0
( r;
ω)
T ( ω)
G ( r′
; ω)
(
0
G G 0 U
U U
r r’
=
r r’
+
r 0 r’
+
r 0 0 r’
+ …
T ( ω)
= U + U
2
G
0
(0, ω)
+ U
Impurity-induced resonance:
3
G
0
(0, ω)
2
+ ... =
1
1/ U − G
1
1/
U = G ( r = 0, ) = ∑
0
ω G0
( k,
ω)
N
k
0
(0, ω)

Bound state in 2D metal due to
weak attarctive potential
energy
size
Local DOS

Same prcedure is applied to SC
case, except Green’s functions
are now matrices

Theory of impurity states in s-
wave SC Yu Lu(65), Shiba(68)
H
= H BCS
+ JSimp
⋅σ
N
ω0
∆
1
= ±
1
−
+
( JN
( JN
0
0
S)
S)
2
2
1
Ψ ( r) = exp( −r/ ξω
)
0
r
ξ
ω
= ξ
N
ξ
( ω ∆ )
0 2
1 − /
0
∆
isotropic
gap
E
Magn
imp
∆ -ω ω
E
Isotropic
wave function

Imp S + Cooper pairs
continuum
|imp>
Ground state
|0>
Bound state formed due to magnetic gain
Bound state is FULLY spin polarized !

Intrinsic pi junction

Remark about Abrikosov-Gorkov theory
N
N
∆
Weak impurity
E
∆
Strong impurity
E

N
N
∆ E
Weak impurity
N
At finite imp concentration
∆
Strong impurity
N
E
∆
Imp band growth in Abrikosov-Gorkov
theory; one parameter: Nimp N(0) J S(S+1)
E
∆
General case:
Gapless SC much sooner
Lifshitz tails and gapless
Balatsky, Trugman, PRL, 97
E

Predicted behavior only occurs for weak impurities:
N
∆
E

Predicted behavior only occurs for weak impurities:
N
∆
E

Predicted behavior only occurs for weak impurities:
Gapless
After a large
concentration is added
N
∆
E

Gapless SC region

In Reality Imurity states can be at any energy
Energy of a single impurity state is a center of impurity band
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
Gd Cluster
Bare Nb
0
-0.01 -0.005 0 0.005 0.01
Votage[V]
A.Yazdani, Science,97

A.Yazdani et al, Science(1997)
Impurity intragap
States on Nb
Superconductor.
Gold has no effect on SC state
Wave function of imp state
Gd, Mn atoms on Nb surface
(Tc = 7K)

Lifshitz Tails and gapless supercoductivity in s-wave
Case.
Rare fluctuations are the ones that are nominally very unlikely.
However is the mean exectation of some observable is zero and
Fluctuaion will render this expectation nonzero then no matter
how low probability is this fluctuation is important
(hugely important in fact) = 0.0001/0-> infty
Rare fluctuations in the impurity distribution will make any
s-wave superconductor gapless, regardless how low impurity
Concentration is. It is gapless albeit with exponentially small
Density of States.
Rev Mod Phys, v 78, p 373 (2006)

Average theory of impurity scattering depend on
2 2
Only scattering rate Γ=nimpJ S
Band in case of strong
impurity scattering, J~1
Band in case of weak
impurity scattering J 1
( unitary)
Averaged theory
From textbooks does
not capture this
ditsinction
Γ= const
n large
imp
JS

Lifshits tails or rare fluctuations

Lifshitz tails in SC make it gapless s wave
superconductivity:
rare but important fluctuations are the ones where
local concentration of impurities exceed critical and
Locally we have a puddle of normal metal inside
SC. These regions always exist no matter how small
probability is.
Tail at E=0 grows with doping
Gapless SC
region

Unconventional Superconductors:
d-wave (High Tc supercondcutors)

Experimental Phase Diagram
of HTSC
T
AF
T * T cr
SC
AF – antiferromagnet
SC – d-wave superconductor
T cr – onset of spin fluctuations
T * – opening of pseudogap
n = 1
Doping, x

YBCO structure

99.659093
F i,
j
2.06115410 . 9
θ j
Earlier work: d-SC Impurity Resonances
On-site potential
U>0
On-site LDOS
LDOS Image at Ω
|Ψ| 2 for impuritystate
-
Rev. Mod Phys, 78, p 373, (2006)
Differential Conductance (nS)
2.5
2.0
1.5
1.0
0.5
typical region
on center of Zn atom
0.0
-100 -50 0 50 100
Sample Bias (mV)
E
E
∆
0
−1
=
2N
U
⎛
⎜
⎝ ln
F
8
Cross
shaped
state
1
⎞
( N U ) ⎟⎟ F ⎠
Impurities can be a useful tool
to investigate
the nature of an unknown state

Theory of impurity resonance in
d-wave SC
Balatsky, Salkola, Rosengren (95)
Hint = U n(0)
N
∆
E
Nonmagnetic
And magnetic imp

Theory of impurity resonance in
d-wave SC
Balatsky, Salkola, Rosengren (PRB,95)
Hint = U n(0)
N
ω 1
=
∆ πN U
0
ln
−1
( πN U )[1 + iπ
ln
0
−1
( πN U )]
0
N
∆
E
Nonmagnetic
And magnetic imp
ω
∆
E

Theory of impurity resonance in
d-wave SC
Balatsky, Salkola, Rosengren (PRB,95)
Hint = U n(0)
N
ω 1
= ln
∆ πN U
0
−1
( πN U )[1 + iπ
ln
0
−1
( πN U )]
0
N
∆
E
Nonmagnetic
And magnetic imp
Differential Conductance (nS)
2.5
2.0
1.5
1.0
0.5
ω
typical region
on center of Zn atom
∆
E
0.0
-100 -50 0 50 100
Sample Bias (mV)

Theory of impurity resonance in
d-wave SC
Balatsky, Salkola, Rosengren (95)
Hint = U n(0)
N
ω 1
= ln
∆ πN U
0
−1
( πN U )[1 + iπ
ln
0
−1
( πN U )]
0
N
Cross
shaped
state
∆
E
Nonmagnetic
And magnetic imp
Universal:Similar states will be created
in p,f,g…states with gap nodes.
ω
∆
Splitting due
to exchange J
E

Zn Impurity-State
~20
Zn atoms
0 560 Å
!
0 560 Å
SI-STM Image at E=–1.5mV
Bi 2 Sr 2 Ca(Cu 1-x Zn x ) 2 O 8+d : x ≅ 0.2%
Cross shaped
Impurity state
(due to 4 fold gap
Nature 403, 746 (2000).

Data by A.
Matsuda
NTT Research,
Japan

N(r) ~ 1/r
Remains to be seen experimentally if this is true in
2

Motivation 2: magnetic vs
nonmagnetic impurities
Impurities as a probe of the high-Tc mechanism
Differential Conductance (nS)
2.5
2.0
1.5
1.0
0.5
typical region
on center of Zn atom
ε = -1.5 meV
Differential Conductance (nS)
2.5
2.0
1.5
1.0
0.5
ε = 9 & 18 meV
typical region
on center of Ni atom
0.0
-100 -50 0 50 100
Sample Bias (mV)
Zn, non-magnetic, S = 0
suppresses superconductivity
0.0
-100 -50 0 50 100
Sample Bias (mV)
Ni, magnetic, S = 1, does not
suppress superconductivity

Impurity states in ANY Dirac point
materials

Impurity states in ANY Dirac point
materials

Impurity states in ANY Dirac point
materials

Impurity states in ANY Dirac point
materials

Local Density of states in Dirac Materials:
Nonmagnetic impurity in d-wave SC and in graphene will have
similar r dependence
Nr ( , ω)~Im Grr ( , , ω)~
Re[ G( rr , , ω) Gr ( , r, ω)]Im T( ω)
0
imp
imp
Grr ( , , ω) = G( rr , , ω) + G( rr , , ω) T( ω) G( r , r, ω)
T( ω) = U /(1 − G ( ω) U)
0 0 imp
0
2
= | Ψ( rr ,
imp
)| Zimpδω
( −Eimp
)
1 0
Eimp
=
U | LogU / W |
pole 1/U = G
0( ω)
sin( kFr
+ δ )
Ψ( r, ω) ~ Re G(,
0
r ω)~
r
0
2
LDOS
N(r) ~ 1/ r for all Dirac propagators
nature of local scattering center is not important:
G ( ω) = ω[ Log( ω/ W ) + i sign( ω)]
Kondo, potential scattering site, spin orbit impurity etc.
regardless being d-wave SC or graphene or anythng else
result of power counting
imp
T. Wehling, PRB, 75, 125425 (2007)
C. Bena, PRL 100, 076601 (2008)
T. Pereg-Barnea, A. H. MacDonald,
PRB 78, 014201 (2008)

H H +
H
H
Model
P.J. Stamp (1987)
Byers, Flatte, Scalapino (1993)
Balatsky, Salkola, Rosengren (1995)
Salkola, Balatsky, Schrieffer (1997)
t t’
= 0
H imp
0
= −∑
+
+ +
t + ∑
ijciσ
c
jσ
V c c c c + h.c.
i↓
j↑
j↑
i↓
i,
j,
σ
i,
j , σ
imp
= Vimp( n + n ) − Simp(
n − n )
0↑
0↓
0↑
0↓
V
Hopping:
Attraction:
nn: t = 400 meV
nnn: t’ = 0.3 t = 120 meV
nn: V = -0.525 t = - 210 meV

Ni,Theory vs. Expt – Images
Theory:
V imp = t ; S imp = 0.4 t;
doping 16%
Experiment:
E.W. Hudson , et.al.
9 mV - 9 mV

Zn: Strong potential imp
Theory
V imp = -10 t = -4 eV
doping 16%
Experiment
S.H. Pan et al
Impurity site
has to be bright!!!

Ni, weak potential + spin imp
Theory:
V imp = t ; S imp = 0.4 t;
doping 16%
Experiment:
Impurity level splitting
caused by Ni spin
Differential Conductance (nS)
2.5
2.0
1.5
1.0
0.5
typical region
on center of Ni atom
ε = 9 &
18 meV
ε 1 =0.052 t (21 mV), ε 2 =0.037 t (15 mV)
0.0
-100 -50 0 50 100
Sample Bias (mV)

Potential impurity
in BSCCO (0.16 doping)
Ni
Zn
Ni – weak repulsive
8 electrons in 3d
Zn – strong attractive
10 electrons in 3d
(filled shell)

Gap is suppressed on the atomic scale
Not only on scale of coherence length as GL would
Lead us to believe.

Impurity as a probe of
Pseudogap (PG) Regime
Pseudogap (Loram)
Vanishing of density of states at the fermi level is sufficient
for formation of impurity state. H. Kruis, I. Martin and AVB
PRB, 2002

T = U/(1-G0(Ω)U)
Hence the pole at G0(Ω) = 1/U
H. Kruis et al, PRB 64,

Role of the Interlayer
Tunneling
Ψ ij – value impurity state
wavefunction on site (i,j)
Intensity in CuO layer:
A ij = | Ψ ij | 2
Martin, Balatsky, Zaanen (2000)
Intensity in BiO layer:
A ij = | Ψ ij-1 + Ψ ij+1 - Ψ i-1j - Ψ i+1j | 2 ~ | cos kx – cos ky |
Direct tunneling (tip to CuO) is exponentially suppressed:
t ~ exp(-r/d), d~0.5A

Bi
Zn
tip
Cu
Hopping via virtual states on Bi and Zn(Ni)
eV
One uses the easiest available path. In this case
direct tunneling into Cu-O plane is exponentially
suppressed.
r, ω = dφ
t φ N r,
φ,
ω -
N eff
( ) ( ) ( )
surface seen Density of
∫
States

BiO vs. CuO plane image
Experiment!

Zn: Strong potential imp
Theory
V imp = -10 t = -4 eV
doping 16%
Experiment
S.H. Pan et al
Impurity site
has to be bright!!!

BiO vs. CuO plane image
Experiment!

Why local nanoscale probes are
useful

Examples ( incomplete list)
MIT in VO2 Basov
Science, v 318,
1759 (2007)
A. Yacobi
J. Stroscio
Gap inhomogeneity in
novel FeAs
superconductors,
J. Hoffman et al
Gap inhomogeneity
In high Tc oxides
J.C. Davie, S.H. Pan, A.
Yazdani
URu2Si2inhomo
geneity in dI/dV
at 35 mev
JC Davis et al

J.C. Davis Group
Gap Map, BSCCO (0.5%Ni)
55 meV
7000
6000
30 meV
512 Å
Frequency of Observation
Counts
5000
4000
3000
2000
1000
0
35 40 45 50 55 60 65
25 40 55
‘gap’
∆, Gap
magnitude
Magnitude
(mV)
(mV)

Cren, et al, PRL (2000)
Bi2212 film

YBCO - BaO plane
YBCO
BaO vs BiO
BSCCO - BiO plane
300 Å 280 Å
Do equivalent nanoscale electronic structure variations exist
in YBCO? Probably not. If they do they are probably
weaker and more ordered.

Impusity states
and STM on Graphene
Differential Conductance (nS)
2.5
2.0
1.5
1.0
0.5
typical region
on center of Zn atom
0.0
-100 -50 0 50 100
Sample Bias (mV)
HOPG
Y. Niimi et. al., PRL 97, 236804

Conclusion
Impurity and defects in correlated states
Are important in that enable new properties
Or make identification of an unknown state easier
One would need right set of tools: experiment…
Theory: investigate local features, no average lifetime
Average scattering potential etc.
A lot of correlated materials do show variety of
inhomogeneous competing states.

Thanks
M. Salkola
I. Martin
J.X. Zhu
I. Vekhter
J.R.Schrieffer
D. Scalapino
P. Kumar
J. Zaanen
J. Smakov
J.C. Davis and
Co
D. Morr
A. de Lozanne