Page 1

arXiv:0808.0966v1 [cond-mat.supr-con] 7 Aug 2008

Extinction of impurity resonances in large-gap regions of inhomogeneous d-wave

superconductors

Brian M. Andersen1, S. Graser2, and P. J. Hirschfeld2

1Nano-Science Center, Niels Bohr Institute, University of Copenhagen,

Universitetsparken 5, DK-2100 Copenhagen, Denmark

2Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA

(Dated: August 7, 2008)

Impurity resonances observed by scanning tunneling spectroscopy in the superconducting state

have been used to deduce properties of the underlying pure state. Here we study a longstanding

puzzle associated with these measurements, the apparent extinction of these resonances for Ni and

Zn impurities in large-gap regions of the inhomogeneous Bi2Sr2CaCu2O8+x superconductor. We

calculate the effect of order parameter and hopping suppression near the impurity site, and find

that these two effects are sufficient to explain the missing resonances in the case of Ni. There

are several possible scenarios for the extinction of the Zn resonances, which we discuss in turn; in

addition, we propose measurements which could distinguish among them.

PACS numbers: 74.72.-h,74.81.-g,74.25.Jb,72.10.Fk

I.INTRODUCTION

An understanding of the low doped Mott region of

cuprate superconductors remains a key problem in con-

densed matter physics.1It has been proposed that elec-

tronic inhomogeneity is crucial for explaining experi-

ments performed in the underdoped regime,2and that

the inhomogeneity arises from strong Coulomb repulsion

which instigates the formation of spin- and charge den-

sity waves, enhanced and pinned by a random potential

generated by the doping process, or by external probes

which break the translational symmetry. Recent theo-

retical modelling has shown that several single-particle

and two-particle experimental probes can be explained

within a scenario where the underdoped superconduct-

ing region consists of a d-wave pair state coexisting with

a local cluster spin glass phase driven by disorder.3–7

Scanning tunneling

(STM/STS) measurements have provided a wealth

of detailed local density of states (LDOS) spectra on

the surface of Bi2Sr2CaCu2O8+x (BSCCO), revealing

an inhomogeneity in the spectral gap on the nanometer

scale.8–11Furthermore, the differential conductance has

been measured in the vicinity of Zn and Ni substituents

for planar Cu.11–13

Theoretically,

conductance pattern produced by the magnetic Ni ion

and its resonance energy of Ω0≃ ±10 meV can be well

described by a combination of potential and magnetic

scattering,14whereas no consensus has been reached

about the proper explanation of the pattern around the

Zn impurity, even though the local perturbation caused

by the ion should in principle be simpler.15–18

One way to gain further insight into the nature of the

physical state of small-gap versus large-gap regions in

cuprate superconductors, is to compare the LDOS near

Ni and Zn in these two different environments. Exper-

imentally, it is found that the Ni resonances, which are

clearly distinguishable in the small-gap regions, are com-

pletely absent in large-gap regions with ∆ ? 50meV for

microscopy/spectroscopy

the STS spatial

optimally doped BSCCO.11The strength of the reso-

nances which are observed does not depend strongly on

local gap size, but despite considerable noise appears to

anticorrelate slightly with the gap.19It was speculated11

that the absence of Ni resonances in large-gap regions

is due to the distinct nature of the (pseudogap) phase

characterizing these domains.

ence is not necessary for the creation of impurity reso-

nances; the existence of the pseudogap in the density of

states itself should result in well defined impurity reso-

nant states, as found in Ref. 20. Indeed, recent STS mea-

surements on native defects in underdoped single-layer

Bi2Sr2CuO6samples have shown that the resonances ex-

ist well above the superconducting critical temperature

Tcinto the pseudogap state.21

The LDOS near Zn is characterized by a sharp low-

energy resonance around −2meV in the small-gap regions

of optimally doped BSCCO.12The resonance weight and

width depend strongly on the local environment,19,23

and, as in the Ni case,11the resonances are never ob-

served in the large-gap regions.19,23The temperature de-

pendence of the Zn resonance has been measured within

the superconducting state. It was found that the evolu-

tion of the peak in the range 30mK < T < 52K < Tc

is consistent with thermal broadening of the peak.22

Within the nonlocal Kondo model for Zn impurities,16

it was recently argued that a gap-dependent exchange

coupling can lead to suppressed Zn resonances in large-

gap regions.24However, this model does not account for

the actual spatial variation of the superconducting gap,

and it is unclear how to explain the extinction of the Ni

resonances. In our view, the question of what causes the

extinction of the resonances and whether these effects

can indeed be used as probes of pseudogap physics13,20

is therefore still open.

For BSCCO materials, a significant part of the ob-

served gap inhomogeneity has been argued to originate

from interstitial oxygen dopant atoms;25detailed theo-

retical modelling of the LDOS spectra26,27has led to the

However, phase coher-

Page 2

2

remarkable conclusion that the pairing interaction itself

seems to be spatially varying in these materials on an

atomic scale. This scenario in which the pairing inter-

action is modulated locally by nearby dopant atoms also

explained the origin of the dominant so-called q1peak in

the Fourier transformed scanning tunneling spectroscopy

(FTSTS) data,28,29and the recently observed pair den-

sity wave driven by the structural supermodulation.30,31

In addition, it predicted the existence of islands with fi-

nite gap above Tc, as recently observed by Gomes et al.32

Such islands lead naturally to broadened thermodynamic

transitions although this point remains controversial at

present.33,34

Here, we show that within the modulated pairing sce-

nario, the absence of Ni resonances in large-gap regions

arises naturally when one properly includes the nano-

scale inhomogeneity of the superconducting order param-

eter. This occurs because in-gap impurity resonances in

a d-wave superconductor generically borrow their weight

from the coherence peaks. In the large-gap regions, these

peaks are broad and weak, so they overlap and swamp

the weaker resonances located away from the Fermi level.

In the small-gap regions, on the other hand, the struc-

tures normally referred to as coherence peaks are ac-

tually Andreev resonant states of the suppressed order

parameter,26,35and their height/width is significantly

larger/smaller than in the pure homogeneous case. Thus

in the small-gap regions the impurity resonances are well-

defined due to their smaller overlap with the contin-

uum. Within the modulated pairing scenario, which re-

produces the correlation of the coherence peaks’ weight

and position with the dopant atoms imaged by STS,25the

presence or absence of Ni resonances is therefore depen-

dent on whether or not interstitial out-of-plane oxygen

dopants exist nearby.

We further consider the effect of gap modulation on

the Zn resonance near the Fermi level. Here it appears

unlikely that the suppression and broadening of the co-

herence peaks in large-gap regions can alone suppress the

impurity resonance because of its weak coupling to the

continuum. There is very little experimental data on the

variation of the spectral form, weight, and position of

FIG. 1: (Color online) Schematic picture of a small-gap re-

gion. Circles are ionic sites, X represents an impurity, and the

bonds represent the absolute values of the bond order param-

eter ∆ on bonds in the bulk (top) and in a small gap patch

(bottom).

the resonance in correlation with the local gap size, but

it appears from what is available19,23that there is con-

siderable dispersion of all these quantities which makes

it difficult to extract systematics. Some of this disper-

sion may simply be due to the well-known very strong

interference of near-unitary impurity resonances in a d-

wave superconductor.36–38We therefore consider a series

of models for the microscopic nature of the Zn potential

which might possibly account for the experimental ob-

servations, and propose additional experimental tests to

distinguish between them.

II.FORMALISM

The starting point of our study is given by the d-wave

BCS Hamiltonian

H0=

?

k,σ

ξkˆ c†

kσˆ ckσ+

?

k

?

∆kˆ c†

k↑ˆ c†

−k↓+ H.c.

?

, (1)

where ξkdenotes the quasiparticle dispersion and ∆k=

∆0

2(coskx− cosky) is the d-wave pairing gap. In terms

of the Nambu spinorˆψ†

Matsubara Green’s function can be expressed as

k= (ˆ c†

k↑,ˆ c−k↓), the corresponding

G0(k,iωn) =iωnτ0+ ξkτ3+ ∆kτ1

(iωn)2− E2

k

,(2)

where E2

real-space, the perturbation due to δ-function potential

(magnetic) impurities of strength V3(V0) in the diagonal

τ3(τ0) channel is given by

k= ξ2

k+∆2

k, and τidenote the Pauli matrices. In

H′

imp(r,r′) =ˆψ†

r[(V3τ3+ V0τ0)δ(r)δ(r′)]ˆψr′. (3)

Likewise, local modulations in the hopping (δt) or super-

conducting gap (δ∆) enter as

H′

δ(r,r′) =ˆψ†

r[−δt(r,r′)τ3− δ∆(r,r′)τ1]ˆψr′. (4)

To obtain the resulting LDOS as a function of posi-

tion and energy, one needs to determine the full Green’s

function G(r,iωn) given by the Dyson equation

G(r,r′) = G0(r−r′)+G(r,r′′)H′(r′′,r′′′)G0(r′′′−r′), (5)

where H′= H′

indices is implied. Thus, by calculating the matrix ele-

ments of G0(r,iωn) =?

problem is that of a matrix inversion. The solution is

presented in terms of the T -matrix

imp+ H′

δ, and summation over repeated

kG0(k,iωn)eik·rthe remaining

G(r,r′) = G0(r − r′) + G0(r − r′′)T (r′′,r′′′)G0(r′′′− r′).

(6)

The poles of the T -matrix, or equivalently, the zeros in

the determinant of (1−H′G0), determine the bound state

energies. The total (spin summed) LDOS is given by

N(r,ω) = (−1/π)Im[G11(r,r,ω) + G22(r,r,−ω)]. (7)

Page 3

3

(e)

?0.4 ?0.200.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0t

∆t?0t

NN

?0.4 ?0.20 0.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t?0t

NN

?0.4 ?0.20 0.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t?0t

IS

?0.4 ?0.20 0.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0t

∆t?0t

IS

?0.4 ?0.20 0.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t?0t

IS

(b)(a)

(d)

(f)

(c)

?0.4 ?0.20 0.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t?0t

NN

FIG. 2: (Color online) LDOS in a d-wave superconductor with

a Ni impurity at the center of a 5×5 patch of (a-b) suppressed

(δ∆ = −0.05t), (c-d) constant (δ∆ = 0) and (e-f) enhanced

(δ∆ = +0.05t) superconducting gap. The Ni potentials used

for this study are V3 = −0.8t, V0 = 0.6t. The figures in the left

column (a,c,e) show spectra on the impurity site (IS) itself,

while right column (b,d,f) represents nearest neighbor (NN)

sites. In each case, the solid black curve shows the LDOS on

the site in question in the absence of the impurity, the solid

color curve indicates the total LDOS with the impurity, and

the dashed and dotted lines show the spin resolved LDOS for

up and down spins, respectively.

In explicit calculations we use the band ξk

−2t(coskx+cosky) −4t′coskxcosky−µ with a d-wave

order parameter ∆k =

2

the Fermi surface in the cuprates we take t′= −0.35t

and µ = −1.1t, with energy measured in units of near-

est neighbor hopping t. We also choose ∆0= 0.2t. The

modulations in the hopping are included on the bonds

near the impurity site assuming maximum modulations

δt on the nearest neighbor bonds, with a Gaussian decay

on other bonds with decay length of one lattice constant.

This is roughly consistent with the renormalization of the

=

∆0

(coskx− cosky). To model

(e)

?0.4 ?0.200.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0t

∆t??0.24t

NN

?0.4 ?0.200.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t??0.24t

NN

?0.4 ?0.200.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t??0.24t

IS

?0.4 ?0.200.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0t

∆t??0.24t

IS

?0.4 ?0.20 0.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t??0.24t

IS

(b)(a)

(d)

(f)

(c)

?0.4 ?0.200.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t??0.24t

NN

FIG. 3: (Color online) Same as Fig. 2 but including local

impurity-induced reduced hopping δt = −0.24t.

effective hopping found in recent studies of impurities in

a host described by the t − t′− J model.39Note this is

an atomic scale modulation caused by the in-plane impu-

rity due solely to magnetic correlations in the host. By

contrast, we also study a modulation of the order param-

eter, but assume that this is caused by an out-of-plane

influence such as a dopant atom, which gives rise to a

patch of modulated ∆kof roughly 20˚ A. Thus we study

the impurity added to a patch of reduced or enhanced

∆kon a 5×5 square as shown in Fig. 1 embedded in an

infinite system of fixed order parameter ∆0.

III.Ni IMPURITY

We first discuss the case of magnetic impurity ions such

as Ni. In Fig. 2 we show the LDOS at the magnetic im-

purity site using the parameters V3 = −0.8t,V0 = 0.6t,

and δ∆ = −0.05t,0,+0.05t corresponding to a local sup-

pressed, constant, and enhanced gap patch, respectively.

Page 4

4

The T -matrix corresponding to this potential has a pole

at an energy Ω0 ∼ 0.05t, but, as is evident from Fig.

2, resonance are visible neither on the impurity site (IS)

nor on the nearest neighbor (NN) site. Note that while

in Fig. 2 we have assumed a quasiparticle scattering rate

Γ(ω) = 0.1|ω| similar to that observed in experiment40,

the absence of resonance features is not caused by this

broadening. Instead, it is the fact that the coupling to

the continuum at Ω0is simply too strong. In Fig. 3, we

show the same sites with the same potentials in a sit-

uation where the hopping has been reduced around the

Ni impurity; it is clear that the resonance is observable,

and corresponds very closely in energy and weight to

experiment.13It appears, however, only in the case where

the order parameter has been suppressed [Fig. 3(a,b)]

over a patch, thus creating the sharp coherence peaks

which are then separated from the impurity feature. At

this point, the role of the suppressed hopping is purely

phenomenological, but we note that such a reduction was

also assumed in the more complicated set of potential pa-

rameters taken by Tang and Flatt´ e14to describe the Ni

resonance. The particular values of δt and δ∆ necessary

for the LDOS near the Ni impurity to resemble the ex-

perimental data are bandstructure-dependent.

From the point of view of experiment, scanning at

a bias voltage corresponding to the resonant frequency

ω = ±Ω0 of the Ni resonance should produce spatial

patterns as shown in Fig. 4. In the left column of pan-

els, we show the LDOS at resonance without suppressed

hopping; impurity states are hardly observable, even in

the reduced gap case. When the hopping around the

impurity site is reduced, the pattern is clearly seen in

the suppressed gap case but not in the large gap case.

Note that the images at positive and negative bias show

fourfold patterns which are rotated by 45◦with respect

to each other, as observed in experiment11and earlier

theories.14,15

IV.Zn IMPURITY

The LDOS pattern predicted for an unitary scatterer in

a d-wave superconductor with noninteracting quasiparti-

cles is well-known to have crucial qualitative differences

with the measured STS conductance maps G(eV,r), at

least if these are interpreted in the usual way as being

directly proportional to the LDOS. The primary discrep-

ancy is the existence of an observed intensity maximum

on the central site of the impurity pattern in the case

of Zn, which is impossible in the na¨ ıve theory for poten-

tials V3 larger than the bandwidth, since electrons are

effectively excluded from this site. There have been sev-

eral theoretical approaches to understand this apparent

paradox. The first is specific to the STM method and

relies on the fact that the impurity states are localized

in the CuO2 plane, two layers below the BiO surface

probed by the STM tip; the intervening layers are then

argued to provide a blocking layer-specific tunneling path

(l)

ω = −Ω

0

ω = +Ω

δ = −0.24

t

δ = 0

t

0

ω = −Ω

0

ω = +Ω

(a)

δ∆ = 0

(c)

δ∆ = 0.05

(d)

(b)

(e)(f)

(h)

(k)

0

(g)

(i)(j)

δ∆ = −0.05

FIG. 4: LDOS real-space pattern N(r,±Ω0) at the resonance

frequency for Ni impurities modelled as described in the text.

Left panel: no suppression of hopping by impurity assumed.

Subpanels (a-b), (c-d), and (e-f) show the resonance pattern

for gap sizes in the surrounding 5×5 patch corresponding to

δ∆ = +0.05t,0,and − 0.05t, respectively. Left column (a,c,e)

and right column (b,d,f) correspond to negative and positive

resonance frequency ±Ω0, respectively. Right panel: same

as left panel but for including additional modulations of the

hopping δt = −0.24t.

which samples not the Cu directly below the tip, but

preferentially the four nearest neighbors.41,42Some indi-

rect support for this point of view has been provided by

density functional theory,43which finds that the pattern

of LDOS near the impurity but close to the BiO sur-

face can be quite different than the LDOS in the CuO2

plane. On the other hand, this calculation, applicable

only to the normal state, suggested that the hybridiza-

tion of the wavefunctions involved is not only blocking

layer-specific, but also specific to the particular chemical

impurity in question. A second class of approaches ob-

tains LDOS patterns similar to experiment for both Zn

and Ni by simply assuming an ad hoc distribution of site

potentials and nearby hoppings to tune the weights of

on-site and nearest-neighbor LDOS.14,44While the bare

impurity potential has a much shorter range of order

∼ 1˚ A43compared to this ansatz, it is possible that the

phenomenological parameters used in these models repre-

sent dynamically generated quantities in a more complete

theory. Finally, the observed Zn conductance pattern has

also been obtained in theories which describe the Zn as

a Kondo impurity16and pairing impurity,18respectively.

We now discuss each of these scenarios in the context of

the resonance extinction problem.

Page 5

5

A.Zn as Kondo impurity

Motivated by NMR measurements45–49showing that

nonmagnetic Zn impurities induce a local spin 1/2 in

their vicinity, it was proposed that the LDOS data could

be understood within a nonlocal Kondo model.16,50With

a proper choice of a large magnetic potential coupling the

nonlocal spin associated with the impurity to the conduc-

tion electron bath, the observed spatial pattern could be

reproduced. This requires, however, the assumption of a

very weak potential scattering at the Zn site. An inter-

pretation of this work accounted also for the disappear-

ance of impurity resonances in large-gap patches16,19by

assuming that such regions were underdoped and there-

fore poorly screened. In such a case, the Kondo tempera-

ture TKwould fall below the measurement temperature,

leaving the impurity in the local moment (nonresonant)

regime. More recently, however, it has been observed by

STM that spatial charge variations are in fact quite small,

of order a few percent, and that the presence of dopants

correlates (rather than anticorrelates) with the large-

gap regions.25In addition, resonances that have been

observed in underdoped samples at temperatures well

above the Kondo temperature expected from NMR17are

similar to the low temperature impurity spectra.13,22,51

Nevertheless, recently Kir´ can proposed a mechanism by

which the size of the superconducting gap can modify

the impurity moment exchange coupling to the d-wave

quasiparticle bath.24In the large-gap regions, states are

pushed further away from the Fermi level, thus decreas-

ing the ability of the quasiparticle system to screen the

impurity, leading to a lower TK. This does not appear to

address the set of critiques above, but is consistent with

the STM results on BSCCO at low T. We return to this

scenario below, after discussing other possibilities.

B.Zn as screened Coulomb impurity

We now examine the conventional point of view that,

since Zn2+is a closed shell ion, it creates a strong local-

ized screened Coulomb potential43in a BSCCO host. A

strong impurity potential of this type is typically repre-

sented as a δ-function potential in the Hamiltonian with

V3 >> t and V0 = 0 in the notation of Eq.(3), and it

is well-known that such a perturbation generates an in-

gap resonant state in a d-wave superconductor.52In a

particle-hole symmetric normal state band, this LDOS

resonance can be tuned to Ω0= 0 with V3= ∞, but in a

more general band a specific fine-tuned value of the po-

tential is required to produce a resonance at zero energy

(unitarity) or any other particular subgap energy.53,54

For the band we have adopted here, which roughly re-

produces the correct Fermi surface of optimally doped

BSCCO, tuning the resonance to the nominal resonance

frequency observed by STS12of Ω0≃ −2meV≃ 0.013t re-

quires a potential of approximately V3= 2.5t. In Fig. 5,

we have plotted the resonance arising from such a poten-

(e)

?0.4 ?0.200.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0t

∆t?0t

NN

?0.4 ?0.20 0.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t?0t

NN

?0.4 ?0.20 0.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t?0t

IS

?0.4 ?0.200.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0t

∆t?0t

IS

?0.4 ?0.20 0.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t?0t

IS

(b)(a)

(d)

(f)

(c)

?0.4 ?0.20 0.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t?0t

NN

FIG. 5: (Color online) LDOS at impurity site (IS) and near-

est neighbor sites (NN) plotted in left and right panels re-

spectively for various gap modulations δ∆ = −0.05t,0,+0.05t

over a 5×5 patch around the impurity, described by an on-

site potential V3 = 2.5t. Black curves show the LDOS in the

absence of an impurity, while colored curves correspond to

the situation where an impurity is present. Hoppings are not

modulated, δt = 0

tial on both the impurity site (IS) and nearest neighbor

site (NN), for various values of the gap size in the local

patch around the impurity, as for the Ni case. There

are two obvious difficulties. The first is the well-known

problem discussed above, that the intensity on the IS

is substantially smaller than on the NN site, in contra-

diction to experiment. The second problem relates to

the behavior of the resonance in different types of local

patches. The resonance is suppressed somewhat in the

large-gap regions corresponding to Fig. 5(e-f), but is still

clearly visible. In fact, it is clear that the gap modulation

δ∆ has simply detuned the resonance and acts, via its

coupling through the T-matrix equations to the diagonal

channel, as a renormalization of the impurity potential.

Page 6

6

(e)

?0.4 ?0.20 0.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0t

∆t?0t

NN

?0.4 ?0.20 0.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t?0t

NN

?0.4 ?0.20 0.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t?0t

IS

?0.4 ?0.20 0.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0t

∆t?0t

IS

?0.4 ?0.200.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t?0t

IS

(b)(a)

(d)

(f)

(c)

?0.4 ?0.20 0.2 0.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t?0t

NN

FIG. 6: (Color online) Same as Fig.

mechanism applied as described in text.

5, but with a filter

The clear splitting of the resonance and shift of the peak

with increasing δ∆ is obvious in panels (b,d,f). The in-

ability of the conventional screened Coulomb model for

Zn to explain the real disappearance of the resonances in

the large-gap regions with this scenario was emphasized

recently by Kir´ can,24who studied a similar model.

We now recall that the correct spatial pattern can

be recovered by employing the assumption of a block-

ing layer “filter”,41,42which supposes that the tunneling

path to the CuO2 plane is not direct. We employ the

simplest of these approaches,41which assumes that the

LDOS measured on any given site is actually the average

of the LDOS on the four NN sites. In this case, as seen

from Fig. 6 the correct spatial pattern is recovered,15

and one sees that the resonance in the large-gap regions

is in fact hardly observable, particularly if one searches

for such objects by looking for values of the 0 meV con-

ductance which exceed the noise threshold, as in Ref. 19.

Of course, applying the same filter mechanism to the Ni

pattern will spoil the good agreement, so one is left with

(e)

?0.4 ?0.200.2 0.4

Ω?t

0

0.5

1

1.5

2

N?Ω?

∆??0t

∆t?0t,...,?1t

NN

?0.4 ?0.20 0.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t?0t

IS

?0.4 ?0.200.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆???0.05t

∆t?0t

NN

?0.4 ?0.2 00.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t??1t

IS

?0.4 ?0.200.20.4

Ω?t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N?Ω?

∆??0.05t

∆t??1t

NN

(b)(a)

(d)

(f)

(c)

?0.4 ?0.200.20.4

Ω?t

0

0.05

0.1

0.15

0.2

0.25

N?Ω?

∆??0t

∆t?0t,...,?1t

IS

FIG. 7: (Color online) LDOS for a Zn impurity represented

by a potential V3 = 5t and calculated with a constant scat-

tering rate η = 0.01t where the hopping is suppressed using

a Gaussian distribution with the maximum δt on the four

bonds around the impurity, and a decay length of one lattice

spacing. (a,c,e) show the LDOS at the IS site and (b,d,f) at

the NN site. (a) and (b) show the LDOS in a region of con-

stant gap (δ∆ = 0) for different hopping suppression where

the values of δt shown vary from 0 (top) to −t (bottom) in

steps of −0.1t. (c) and (d) show the LDOS in a small-gap

region without suppression of the hopping while (e) and (f)

show it in large-gap region with maximum suppression of the

hopping (δt = −t).

the unpleasant alternative of arguing that the effect of

the intervening layers may be different in the case of the

two impurities. This was in fact the conclusion reached

by Ref. 43, but it destroys the utility of comparing the

patterns for the two species as a way to extract informa-

tion.

The resonances observed in Fig. 6(e) still appear as

small peaks in the LDOS at a shifted energy, and so

might be observable; the case for such a scenario re-

Page 7

7

ally eliminating the resonances in the large-gap regions

is therefore not completely convincing. Before leaving

this scenario, therefore, we discuss briefly the case of re-

duced hopping. We remind the reader that due to the

antiferromagnetic correlations in the host material, the

hopping is expected to be reduced over the antiferromag-

netic correlation length, of order a few lattice spacings.39

This renormalization might be expected to be more im-

portant in the case of Zn than Ni, due to the stronger

bare potential.43We therefore plot in Fig. 7(a,b) the

resonance in a homogeneous background order parameter

(δ∆ = 0) for various values of the hopping near the im-

purity. It is seen that the reduced hopping dramatically

suppresses the weight in the impurity resonance on both

the impurity [Fig. 7(a)] and NN [Fig. 7(b)] sites, which

is lost to the antibound state outside the band. The res-

onance position will shift somewhat, however, since the

effective potential for the impurity which enters in the

determinant of the T-matrix is changing. As shown in

Fig. 7(c-f), assuming a sufficiently large renormalized

hopping in large-gap regions, it is possible to completely

wipe-out what would be a well-defined sharp resonance

in a small-gap region. Thus, provided the experimental

data are consistent with individual Zn’s displaying small

shifts in the resonance position in intermediate strength

gap patches, the conventional scenario reviewed in the

section, modified by kinetic energy renormalization from

electronic correlations, may well be consistent with the

data.

C.Zn as phase impurity

One of the more exotic proposals to describe the LDOS

measurements near Zn impurities is the so-called phase

−0.4−0.20

ω/t

0.20.4

0

0.5

1

1.5

N(ω)

Clean DOS

δ ∆=−0.05

δ ∆=0.0

δ ∆=0.05

δ ∆=0.1

δ ∆=0.2

−0.50 0.5

0

1

2

FIG. 8: (Color online) LDOS of a phase impurity with α = 1.5

as a function of the change in the local gap δ∆ within the 5×5

patch in which it lies. The red curve is similar to the results in

Ref. 18, which reported the LDOS near a phase impurity in

a homogeneous superconductor. The inset shows the LDOS

without the phase impurity (same color code).

impurity scenario.18As was shown in Ref.

sign changes of the pairing amplitude on the four near-

est bonds to the impurity site can generate low-energy

Andreev resonant states with the correct real-space dis-

tribution (without a filter mechanism). This is true only

for a sufficiently small conventional impurity potential

(see Ref. 18 for details).18Using the same notation as

in Ref.18, we introduce the parameter α which sets

the strength of the phase impurity by adding a potential

in the τ1 Nambu channel of V1 = −α∆0 which affects

only the four NN bonds to the impurity site in the center

of the 5 × 5 patch. As α increases, the Andreev reso-

nance moves to lower energy and sharpens up, and the

resonance energy Ω0can approach zero when α becomes

greater than 1. This means that the order parameter ac-

tually changes its phase by π on an atomic scale. The

question relevant to the present study is how these An-

dreev states depend on the local gap patch in which they

are positioned. In order to answer this, we have stud-

ied the LDOS near phase impurities with α = 1.5 in the

center of gap patches of size 5 × 5. Assuming that Zn

causes the same local changes of the gap irrespective of

the local gap environment (i.e. same α for all local gap

values of the 5×5 patch), then one would expect sharper

resonances in small-gap regions since there the phase im-

purity is effectively stronger. Indeed, the special case of

a phase impurity of strength α in a large-gap region with

increased local gap also of size α has effectively no phase

impurity at all, and hence no low-energy resonance.

In Fig. 8 we show the LDOS at the center of the phase

impurity as a function of the local gap change δ∆ in the

5×5 gap patch. As expected, small-gap regions support

well-defined Andreev resonant states, which are however

much weaker in large-gapregions. Thus at least at a qual-

itative level the phase impurity scenario seems to be fully

consistent with the present experimental STM data near

Zn impurities. Note that the results in Fig. 8 are shown

without the filter effect, and with δt = 0 and a constant

smearing factor η = 0.01t as opposed to an inelastic scat-

tering rate Γ(ω) = 0.1|ω| used in Figs. 2-6. We find that

local suppression of the hopping mainly renormalizes α

(δt < 0 leads to smaller effective α) whereas Γ(ω), which

dominates the large-gap regions,40obviously leads to fur-

ther smearing preferentially of the high-energy features

ω ? ∆.

18, local

V.DISCUSSION

Several rather general points and caveats need to be

mentioned in connection with the overall problem. First,

we have assumed that the impurity resonances are extin-

guished because of some effect of the electronic system,

i.e. we ignore the possibility that the impurities them-

selves are simply absent due to nonrandom processes dur-

ing the crystal growth (for a discussion, see Ref. 19).

Second, we have assumed that the spectral gap is a

quantity characterizing the strength of the same elec-

Page 8

8

tronic state in all gap patches.

gap patches even at optimal doping correspond to a

competing electronic state, a very different approach is

necessary. The na¨ ıve proposal that these gap patches

correspond to pseudogap regions does not, however,

suffice to explain the absence of impurity resonances,

since most models of the pseudogap state allow for such

resonances,20,55–57and in addition, by heating the sam-

ple through Tc, they have now been observed to survive

well in to the pseudogap state.21The Gaussian distri-

bution of gap sizes observed by STS also suggests that

there is no qualitative difference between, e.g. large-gap

regions and intermediate-gap regions.

Third, we note that all our calculations (and those

of others) have been performed for models of a single

impurity in some realization of a d-wave superconduc-

tor. In the real many-impurity problem, interference be-

tween different impurity resonances leads to distortions of

fourfold symmetry and shifts of resonance energies.36–38

These effects are particularly strong for near-unitary

scatterers like Zn.58Direct comparison of a single experi-

mental spectrum with a 1-impurity theoretical prediction

are therefore generally problematic. It is likely that these

effects are responsible for the fact that the existing exper-

imental data (for fields of view imaged by STS containing

tens of impurities) for resonance strength versus gap size

are extremely noisy19and show no clear evidence for a

continuous suppression of the resonance peak as the local

gap size is increased, a characteristic of all the theoretical

models discussed here. Improving these statistics, with

hundreds rather than tens of impurities, and binning over

a range of gap sizes, should enable more robust conclu-

sions to be drawn. According to Ref. 58, some of the

effects of interference may also be minimized by defining

the strength of the resonance via an integral over a small

energy window near the peak rather than simply by the

value of the conductance at a fixed bias.

Fourth, both the Kondo scenario and the phase impu-

rity scenario produce the same spatial pattern for the

Zn resonance as observed in experiment, whereas the

screened Coulomb impurity picture requires an assump-

tion of a filter mechanism. If the existence or nonexis-

tence of such a mechanism could be settled in the case

of the small-gap regions, it would enable one to rule out

at least one candidate. Refs. 18 and 37 proposed that

this could be settled if one can find isolated pairs of res-

onances in the same gap region.

Fifth, we discuss how the different scenarios for Zn

can be distinguished, independent of filter mechanism.

If truly isolated resonances were available, the shift of

resonance positions from patch to patch could suffice to

single out the model with Zn as a screened Coulomb scat-

terer. However, as discussed above, interference effects

will likely induce such shifts in either of the remaining

scenarios as well.

The Kondo scenario for the Zn moment can be tested

by a systematic measurement of the T-dependence of the

impurity resonances. According to Kirc` an,24the effective

Clearly, if the large-

Kondo scale depends on the gap size. Since the measured

distribution of gaps is continuous, it should be possible to

find resonances in intermediate-gap regions characterized

by a small enough Kondo scale, such that the resonance

will disappear with increasing temperature, still in the

superconducting state. In the conventional scenarios, or

in non-Kondo scenarios where the observed moment is

due to background correlations in the host system, in-

creasing temperature should broaden but not eliminate

the resonance until the gap disappears.

Another method to distinguish these scenarios is to

apply a magnetic field. In a magnetic field, resonances

due to an ordinary potential will be split by a Zeeman

coupling59linearly in the magnetic field strength. Split-

ting of the resonance peak also characterizes the phase

impurity and the Kondo resonance in a magnetic field.

However, as shown in Ref. 60, deviations from the linear

Zeeman splitting characterizes the Kondo resonance as

the field becomes of order the Kondo temperature and

the two split peaks develop a strong asymmetry in their

weight. For the intermediate gap patches with TK≪ Tc

discussed above, the corresponding field scale where the

crossover occurs should be small enough such that the

unusual field dependence predicted in Ref. 60 should be-

come observable.

Sixth, we remind the reader that none of the models

discussed here incorporate systematically the effects of

the approach to the Mott transition on the underdoped

side. We have attempted to include the correlation-

induced band narrowing near an impurity due to corre-

lations in a phenomenological way, but a more complete

treatment is clearly desirable.

Finally, we address the question of the static local mo-

ment induced by Zn in the underdoped regime.61In the

conventional model discussed above, such a moment is

not included in the theory. Recent studies however, have

extended this scenario (extended conventional scenario)

by including Hubbard correlations U in the host super-

conductor, and found that nonmagnetic scatterers can in-

deed generate–for sufficiently strong correlations–a local

S = 1/2 state tied to the impurity site,62,63in agreement

with results obtained from the t−J model treated in the

slave-boson mean-field64or Gutzwiller approximation.65

An important property of the LDOS resonance within

the latter scenario is that it is intrinsically split by the

local moment whereas no such effect exists in zero field

within the Kondo approach. Therefore, experimental ob-

servation of thermally broadened resonances which split

in zero field as the temperature is lowered would be

strong support for the extended conventional scenario.

We stress that local inhomogeneity may cause some res-

onances to exhibit this behavior while others will not.

VI. CONCLUSIONS

In this paper we have investigated a number of sce-

narios which might explain why STS experiments fail to

Page 9

9

observe Ni and Zn resonances in large-gap regions of the

BSCCO superconductors. In the case of Ni, it appears

that the physics of an inhomogeneous d-wave supercon-

ductor alone suffices to explain the phenomenon. That

is, only in the case of a coherence-length size small-gap

region embedded in a larger gap region can one expect to

have sufficiently sharp and strong features at the nomi-

nal gap positions for the impurity resonance to borrow

enough weight for it to be well-defined. This is a di-

rect consequence of Andreev states formed by partially

trapped quasiparticles in suppressed order parameter re-

gions, as pointed out by Nunner et al.26and Fang et al.35

In discussing the similar question for the Zn reso-

nances, we considered several scenarios. The first, due to

Kir´ can, assumes that the resonance is formed primarily

due to Kondo screening of the magnetic moment formed

in the correlated system around the Zn site, and pro-

poses that in the large-gap regions the moment is free

(unscreened). This seems unlikely to us, given that no

temperature dependence of these resonances has been ob-

served other than thermal broadening,21,22but we have

proposed further tests that should clarify this issue. The

second scenario is similar to the physics of the extinction

of the Ni resonances: if the mobility of quasiparticles

is reduced in the vicinity of the Zn due to correlations,

the resonance will be suppressed as the size of the order

parameter in the patch is increased, but shifted as well.

This scenario cannot presently be ruled out by existing

data. Finally, if the Zn ion primarily influences the pair

field locally, such as to cause a local π phase shift of

the order parameter, it will produce a resonance whose

weight depends directly on the size of the local gap, and

vanishes in sufficiently large gap regions.

Further theoretical and experimental work is clearly

necessary to answer the very fundamental challenge

posed here. From the theoretical side, we have included

strong electronic correlations only in a very phenomeno-

logical way, by introducing local suppressions of the hole

mobility around the impurity site, as found by more so-

phisticated treatments in the normal state.39As these

effects appear to be important, theoretical treatments of

impurities in the superconducting state capable of treat-

ing inhomogeneous order parameter situations together

with strong correlations are clearly desirable. On the

experimental side, we have discussed ways in which im-

proving statistics on impurity resonances and how they

are defined could provide important insights, and pro-

posed various tests of the scenarios treated here which

should enable one to distinguish them.

VII. ACKNOWLEDGEMENTS

The authorsare gratefulfordiscussions with

J.C. Davis, M. Vojta, and W. Chen. P.J.H. and S.G.

were funded by DOE Grant DE-FG02-05ER46236 and

S.G. acknowledges support by the Deutsche Forschungs-

gemeinschaft.B.M.A. acknowledges support from the

Villum Kann Rasmussen foundation.

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