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.