Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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6 Impurity Nanostructures and Quantum Interference 247

Fig. 6.15. Spatial plot of the LDOS for

(a) N = 10 impurities aligned along the

(110)-direction; (b), (c)twolineswith

N = 10 impurities each separated by

d =4

√

2; (d), (e)and(f)twolineswith

N = 34 impurities each separated by

d =6

√

2. Non-magnetic impurities are

represented by filled yellow circles.Re-

sults are obtained within the

T-matrix

formalism

trast,theimpurity statesshown in Fig.6.16(c) and (d)

are “star-like” and extend almost radially outwards.

Note that the“global”(i.e.,larger length scale) spatial

structure ofthe LDOS at§

res

= ±5meV[Figs.6.16(c)

and (d)] is identical, and thus particle-hole symmet-

ric. This symmetry also exists for other impurity res-

onances, such as the ones at §

res

= ±11 meV. Only

in the local structure of the impurity resonances can

we identify differences in the spatial LDOS pattern

between particle-like and hole-like energies, such as

a45

◦

rotation of the LDOS intensity, similar to the

one observed near single impurities [28]. This di-

chotomy of global particle-hole symmetry and local

particle-hole asymmetry is also exhibited by other

nanostructure geometries.

When the square is rotated by 45

◦

such that

its sides are parallel to the (100) direction [see

Fig. 6.16(e)], specularly reﬂected electrons do not

248 D.K.Morr

Fig. 6.16. (a)–(d) Spatial plot of the

LDOS in the superconducting state for

asquarewithN = 32 impurities and

sides of length a =8

√

2 parallel to the

(110) and symmetry related directions.

(e), (f) Spatial plot of the LDOS for a

squarewithN = 40 impurities in (e)the

normal state, and (f) the superconduct-

ing state. Results are obtained within

the

T-matrix formalism

experience a sign change in the superconducting

pairing potential and hence no impurity resonances

are formed. However, the square can still excite an

eigenmode in the superconducting state that is also

present in the normal state at energies |!| < 

Consider for example the normal state eigenmode

at ! = −3 meV shown in Fig. 6.16(e). In the su-

perconducting state, this eigenmode is located at

! = −1 meV [see Fig. 6.16(f)]. Note that this

eigenmode in the superconducting state does not

arise from pairbreaking effects, such as for exam-

ple, the impurity pairbreaking resonances shown in

Figs. 6.15(a) and (b). It is therefore important to

distinguish between impurity states arising from

eigenmodes of the nanostructure and those that

are pairbreaking resonances. In general, the spatial

structure of impurity states arises from an inter-

play of eigenmodes and pairbreaking resonances,

6 Impurity Nanostructures and Quantum Interference 249

Fig. 6.17. Spatial plot of the LDOS at

res

= 0 meV for two right isosceles tri-

angles whose hypotenuse is aligned (a)

along the (100) direction, and (b)along

the(110)direction.Results are obtained

within the

T-matrix formalism

with eigenmodes (pairbreaking resonances) deter-

mining the spatial structure of the LDOS inside (out-

side) the nanostructure. The above results demon-

strate that the resonances (and in particular the

ZBCP-like state) induced by impurity nanostruc-

tures exhibit a strong dependence on the orienta-

tion of the nanostructure, which directly reﬂects

the momentum dependence of the superconduct-

ing order parameter. Hence we conclude that nanos-

tructures provide a valuable new tool for identify-

ing the symmetry of unconventional superconduc-

tors.

The above results suggest that it is already suffi-

cient to construct a single nanostructure in order to

identify the symmetry of the superconducting gap.

Consider, for example, the right isosceles triangle

shown in Fig.6.17.When the hypotenuse of the trian-

gle is aligned along the (100) direction, and the two

shorter legs are aligned along the (110) direction, a

ZBCP-like resonance emerges only from the shorter

sides, as shown in Fig. 6.17(a). After rotating the tri-

angle by 45

◦

,such that the hypotenuse is now aligned

along the(110)direction,westill observea ZBCP-like

resonance, now emerging from the hypotenuse only,

as shown in Fig. 6.17(b). Thus, it is possible to use a

single nanostructure with appropriate geometry in

order to identify the symmetry of the superconduct-

ing order parameter, and in particular, its nodal and

antinodal directions.

Another important quantum phenomenon aris-

ing from the interplay between the nanostructure’s

geometry and the momentum dependence of the su-

perconducting gap is the spatial screening of impu-

rity resonances in unconventional superconductors.

This screening effect immediately follows from the

observation that certain nanostructures, such as a

square whose sides are parallel to the (100)-direction

shown in Fig. 6.16(e), do not induce resonant impu-

rity states. Consider, for example, the resonance state

at §

res

= 0 meV shown in Fig. 6.18(a) that is induced

by a single magnetic impurity. When the impurity

is enclosed by a square whose sides are parallel to

the (100)-direction,the single impurity resonance is

completely confined to the interior of the square, as

shown Fig. 6.18(b). Hence, the impurity resonance is

screened by the nanostructure square. We find that

even spatially more complex impurity resonances,

such as ZBCP-like state shown in Fig. 6.18(c) can

be screened when enclosed by a second square, as

shown in Fig. 6.18(d). This new quantum phenom-

ena of screening impurity states permits us to em-

ploy impurities or nanostructures to locally probe

the electronic structure of an unconventional super-

conductor without inducing long-range changes in

the superconductor due to its nodal structure.

More complex nanostructures provide the possi-

bility to createan abundance of novel spatial patterns

of impurity resonances and, in particular, to form

resonances that are confined to distinctively differ-

ent parts of a nanostructure. For example, the two

congruent squares shown in Fig. 6.19 induce impu-

rity states that are either confined to the interior of

the inner square [Fig. 6.19(a)], to the area between

the two squares [Fig. 6.19(b)], or to the exterior of

the outer square [Fig.6.19(c)],wherethe latter corre-

sponds to the ZBCP-like resonance ofthis nanostruc-

250 D.K.Morr

Fig. 6.18. Spatial plot of the LDOS for

(a)and(b) an unscreened and screened

impurity resonance of a single mag-

netic impurity,respectively,and (c)and

(d) an unscreened and screened im-

purity resonance of a nanostructure

square, respectively. Results are ob-

tained within the

T-matrix formalism

ture.Moreover,we also observe more complex spatial

patterns of impuritystates,such as the ones shown in

Figs. 6.19(d), (e) and (f). These results exemplify the

possibility of custom-designing the spatial structure

of impurity resonances and thereby gaining further

insight into the complex electronic structure of the

cuprate superconductors.

6.6 Molecules on the Surface of d

−y

-wave

Superconductors

Before investigating quantum interference phenom-

ena in a molecular (dimer) nanostructure, we study

the effects of a single molecule located on the sur-

face of a d

−y

-wave superconductor, on the super-

conductor’s local electronic structure. As discussed

earlier, the predominant effect of molecules that are

located on the surface of a host systems is to provide a

localized dynamic defect in the form of a vibrational

bosonic mode. Hence, the possibility that supercon-

ductingelectronscanhoponto theelectronicorbitals

of the molecule, and thus “proximity-induce”super-

conducting correlations in the molecule, will not be

considered here; an investigation of this effect, how-

ever, is the topic of currently ongoing work.

To simplify the discussion we consider a particle-

hole symmetric dispersion

with t



=  =0.In

Fig. 6.20 we present the LDOS at the site of the

molecule, r =(0, 0). Similar to a non-magnetic im-

purity, we find that the molecule induces a fermionic

resonance state in its vicinity. In order to derive an

equation for the resonance frequency, §

res

,wefirst

calculate £

loc

(!) ≡ £(r, r, !) whose imaginary part

for |§|≥!

=0isgivenby

Im£

loc

(§)=−



|§| − !





|§| − !





, (6.23)

A band dispersion with strong particle-hole asymmetry, characteristic of the cuprate superconductors, yields qualita-

tively similar results.

6 Impurity Nanostructures and Quantum Interference 251

Fig. 6.19. Spatial plot of the LDOS for

two congruent squares whose sides are

aligned along the (110) direction (U =

2 eV). Results are obtained within the

T-matrix formalism

where K is the complete elliptic integral of the

first kind, while Im£

loc

≡ 0 for |§| < !

Performing a Kramers–Kronig transform, we find

for §/!

 1, Re£

loc

≈ −A§/!

where A =

−





∞

dx Im£

loc

(x)/x

.Inthelimit§

res

 

,the

resulting equation for the resonance frequencies is

res





A log



4

|§

res



−1

. (6.24)

We thus find that §

res

shifts to higher energies with

increasing !

, in agreement with our numerical re-

sults shown in Fig. 6.20(a). Moreover, for !

→∞

the LDOS approaches that of the unperturbed clean

superconductor,as can be seen,for example,from the

results for !

= 200 meV.A qualitative difference to

the effects of static impurities is that the logarithmic

divergence in £

loc

at |§| = 

+ !

leads to a dip in

the LDOS, which is indicated by arrows in the inset

252 D.K.Morr

Fig. 6.20. LDOS at r = 0 as a function of frequency for

 =0

, g =1.2eV,and(a)atT = 0 for several values

of !

(inset: arrows indicate the“dip” in the DOS), and

(b)for!

= 10 meV and several T

of Fig. 6.20(a) (this dip was also found by Balatsky,

Abanov and Zhu [84]).

At non-zero temperature,a new scattering channel

is opened since the ! =+!

branch of the bosonic

mode becomes populated. This yields significant

changes in the LDOS. Specifically, if the resonance

frequency at T =0,satisfies|§

res

(T =0)| < §

−

with §

−

= 

− !

we find that the resonance peak

is shifted to lower energies with increasing tempera-

ture,asshowninFig.6.20(b).Inaddition,theopening

of a second scattering channel for T =0leadstoa

logarithmic divergence in £

loc

, and consequently a

dip in the LDOS at |§| = §

−

= 15 meV, as indicated

by the arrows in Fig. 6.20(b).

Consider next a molecular nanostructure con-

sisting of two molecules located at r

=(0, 0) and

=(1, 0) (case I) and r

=(0, 0) and r

=(2, 0)

(case II). Note that for two molecules,the calculation

of the LDOS involves not only the local Greens func-

tion,

, r

), and self-energy,

£(r

, r

), with r

= r

but also the non-local ones with r

= r

.Asaresult,

the LDOS is determined by the distance and orien-

tation of the modes relative to the lattice, which is

described by r = r

− r

=(n

, n

). For the band-

structure considered here, superconducting corre-

lations, as represented by the non-vanishing of the

non-local anomalous Greens function, F

r

,andself-

energy, ¥

r

(the off-diagonal elements of

and

, respectively), are important for the form of the

LDOS if (n

+ n

)mod2 = 1 which is satisfied for the

molecules in case I, but not in case II.

The LDOS for several values of the inter-molecular

coupling J is shown in Fig. 6.21(a) and (b) for cases

I and II, respectively. Since with increasing J, §

in-

creases while |g

(2)

1,2

| decreases, the electronic scatter-

ing becomes weaker and the resonance peaks are

shifted to higher energies. In case I, the peaks move

close to ±

as J increases from J =0toJ = ∞,an

effect that is independent of g and directly related

to a non-zero F

r

and ¥

r

. In contrast, in case II, the

frequency shift is much smaller. For J = ∞,only

the bosonic in-phase mode couples to the fermionic

6 Impurity Nanostructures and Quantum Interference 253

Fig. 6.21. LDOS at R as a function of frequency for !

0.6

and g =15

at T = 0 and in equilibrium [(a)and

(b)] and (c)forJ = !

/3 whenthe molecules areexternally

excited, leading to changes in the populations n

1,2

system. This coupling is much weaker in case I than

in case II due to the non-vanishing superconducting

correlations.

The fact that the coupled energy levels of the two

molecules can be externally excited, for example, by

optical means [95], opens an intriguing possibility

for manipulating the local electronic structureof the

host material. In order to demonstrate this effect we

assume that through excitations of the energy lev-

els ±§

1,2

we can arbitrarily change their population,

1,2

.InFig.6.21(c)wepresenttheLDOSwithn

−

1,2

=1,

= 0 and several values of n

.Forn

=0,which

corresponds to the equilibrium case at T =0,weob-

serve two peaks in the LDOS. With increasing n

,we

find thateach of the peaks issplit,and a dip is induced

in the LDOS at ±!

−

with !

−

= 

−§

.Thisnewlo-

cal method of manipulating the electronic structure

of the host system can provide important insight into

the complex behavior not only of the cuprate super-

conductors, but of strongly correlated electron sys-

tems in general.

6.7 Conclusions

Over the last 50 years, the investigation of impu-

rity effects in superconductors has provided im-

portant insight into the nature of superconductiv-

ity. In the past, the main emphasis has been on the

study of either disorder-averaged physical quanti-

ties, or of single impurity effects. However, recent

experimental as well as theoretical studies suggest

that novel quantum interference phenomena emerge

when nanoscale impurity structures are placed on

the surface of conventional or unconventional su-

perconductors.

While superconductorswith s-wave symmetry are

in general well understood, quantum interference in

nanostructures leads quite unexpectedly to a series

of novel and fundamental quantum effects. In par-

ticular, the interplay between particle-hole mixing

in the superconducting state, the presence of a full

gap in the LDOS and quantum interference leads to

new types of quantum imaging and optical selection

rules.Of particular importance for these phenomena

is the fact that magnetic and non-magnetic impu-

rities exert qualitatively different effects on the su-

perconductor’s electronic structure,thus permitting

the study of quantum interference separately from

the creation of fermionic impurity states. One of our

most important results pertains to the pairbreaking

nature of a single magnetic impurity in a supercon-

ductor which was believed to be one of its most fun-

damental (and unalterable) properties. We showed,

however, that when a magnetic impurity is located

inside a (non-magnetic) nanostructure, destructive

quantum interference can lead to a complete sup-

pression of its pairbreaking ability. In other words,

254 D.K.Morr

non-magnetic nanostructures can render magnetic

impurities “harmless”for the superconductor.More-

over, we demonstrated that nanostructures can be

used for the creation of quantum interference in-

duced first order phase transitions. Since in these

transitions, the spin polarization of the supercon-

ductor’s ground state changes, spin resolved scan-

ning tunneling spectroscopy should be the primary

tool for their investigation.

Unconventional superconductors, on the other

hand, give rise to novel types of quantum interfer-

ence phenomena due to the momentum dependence

of their order parameter. In particular, we showed

that in superconducting host systems with d

−y

wave symmetry,the nature and existence of impurity

resonances is determined by the interplay between

the momentum dependence of the superconducting

gap and the geometry and orientation of a nanos-

tructure. We identified two important consequences

of this interplay. First, by changing the geometry and

orientation of the nanostructure, it is possible to

identify the symmetry of the superconducting order

parameter through the creation of a zero bias con-

ductance peak. In particular, using a nanostructure

in the form of a square as an example,we showed that

the presence or absence of a ZBCP-like state depends

on the relative orientation of the square with respect

to the order parameter’s nodal directions. In other

words, the nanostructure’s ZBCP is a phase sensitive

probe of the order parameter’s symmetry. Second,

while constructive quantum interference is respon-

sible for the emergence of a ZBCP in nanostructures,

destructive quantum interference leads to a com-

plete suppression of impurity resonances around the

nanostructure.To demonstratethis effect,we consid-

ered a square of impurities whose sides are aligned

along the (100) direction. An important corollary of

this effect is the possibility to screen impurity reso-

nances in the superconducting state.In particular,we

found that when a single magnetic impurityis placed

inside the square, the impurity resonance induced

by the single impurity is screened by the square, i.e.,

completely confined to its interior.

Finally, we considered the effects of a new class

of defects, namely molecules that possess an inter-

nal vibrational mode, on the local electronic struc-

ture of d

−y

-wave superconductors. Dueto the scat-

tering of the host system’s electrons off the vibra-

tional mode, a fermionic resonance state is created

in the vicinity of the molecule. While this effect is

similar to that of conventional impurities, we identi-

fied a number of important qualitative differences,

such as the emergence of dips in the supercon-

ductor’s LDOS with increasing temperature. These

dips at frequencies ! < 

arise from the open-

ing of a second scattering channel due to thermal

population. We also considered two molecules that

are coupled by a “quantum mechanical spring” and

demonstrated that their effect on the superconduc-

tor’s LDOS strongly depends on distance and ori-

entation of the molecules; a dependence that arises

from the spatial form of the superconducting corre-

lations, as reﬂected in the anomalous Greens func-

tion, F(r). In a related study [85] (not discussed

above), we recently showed that the LDOS can also

provide insight into the nature of intermolecular

interactions, about which very little is known at

present. Moreover, we demonstrated that by exter-

nally exciting the energy levels of the molecular

dimer, for example by optical means, it is possible

to manipulate the superconductor’s local electronic

structure. Hence, molecular nanostructures provide

us for the first time with the ability to probe and po-

tentially manipulate superconducting correlations in

real space.

How can one experimentally realize those quan-

tum phenomena presented above that require a su-

perconducting host with d

−y

-wave symmetry, i.e.,

a cuprate superconductor? One of the main exper-

imental obstacles is the creation of nanostructures

in close vicinity to the CuO

planes. However, one of

the candidate materials for the experimental realiza-

tion of nanostructures in d

−y

-wave superconduc-

tors might be given by Ca

2−x

CuO

in which

(after cleaving) the topmost CuO

plane is sepa-

rated from the vacuum only by a single (Ca,Na)Cl

plane [96]. Placing nanostructures on the surface of

or inside the (Ca,Na)Cl layer might be sufficient to

project localized scattering potentials into the CuO

plane [97]. In contrast, for the extensively studied

compound Bi

CaCu

8+ı

placing nanostructures

close to the CuO

layer might be somewhat more dif-

6 Impurity Nanostructures and Quantum Interference 255

ficult since a BiO plane and a SrO plane separate the

topmost CuO

layer from the vacuum.

The results presented above demonstrate that the

interactionof nanoscale impurity structures withsu-

perconducting host systems leads to a series of novel

quantum phenomena. We expect these phenomena,

though derived for the specific case of a d

−y

-wave

superconductor, to exist in many other unconven-

tional superconductors, since their existence only

requires the vanishing of the local superconduct-

ing gap and the presence of line nodes. Moreover,

the fact that these quantum effects as well as the

superconductor’s local electronic structure can be

manipulated by changing the geometry or orienta-

tion of a nanostructure or by exciting its internal

degrees of freedom, provides us with fundamentally

new possibilities to explore the nature and origin of

superconductivity, and in particular, to identify the

symmetry of the superconducting order parameter.

These possibilities are of great importance for many

unconventional superconductors in which the mi-

croscopic mechanism responsible for superconduc-

tivity is still unclear, such as the high-temperature,

ferromagnetic, heavy-fermion or organic supercon-

ductors.

The studies presented here suggest a number of

different directions for future work. For example, up

to now, nanostructures, and in particular quantum

corrals, were employed for quantum imaging char-

acteristic features (quantum candles)intheLDOS.

However, some recent studies [98,99] have extended

this idea and addressed the question whether su-

perconducting correlations and order parameters in

general can be quantum imaged. While quantum

imagingof aspatiallyvarying orderparameter might

be difficult to achieve [98], it was found that order

parameter modes can be induced through their cou-

pling to the LDOS [99]. Testing of these predictions

is likely possible in the near future due to recent ad-

vances in the development of Josephson scanning

tunneling spectroscopy [100]. Moreover, the above

studies suggest that the emergence of novel quantum

phenomena can also be expected for other host ma-

terials possessing broken-symmetry states with non-

trivial electronic correlations, such as charge- and

spin-density-wave systems,or phases with coexisting

orders. Indeed, first studies of two-impurity quan-

tum interference effects in the d-density wave phase

[101] and triplet p-wave superconductors [102] have

recently emerged. Another very interesting class of

host materials include quantum critical systems, in

particular those where competing phases near quan-

tum critical points are present. Here, the ability of

nanostructures to manipulate the local electronic

structure can potentially identify the nature and mi-

croscopic origin of the competition. Moreover, in

many strongly correlated materials, the question of

which collective modes are responsible for the emer-

gence of complex phases is at the center of intense

scientific debates. For example, in the cuprate su-

perconductors, it was argued that the unconven-

tional behavior observed in the pseudogap region

can be attributed to the presence of collective spin

or charge modes [103]. The question thus naturally

arises whether nanostructures can be used to iden-

tify the relevant collective modes, for example, by

choosing a geometry which is commensurate with

the modes’characteristic wave-vectors.Furthermore,

we assumed above that the predominant effect of

nanostructures is to provide localized scattering po-

tentials.Whether superconducting electrons can hop

onto the electronic orbitals of a nanostructure and

thus “proximity-induce” superconducting correla-

tions, is an open question and the topic of currently

ongoing work. This new effect could potentially lead

tointeresting new pairingsymmetriesdueto the pro-

jection of the host system’s pairing symmetry on the

geometry of the nanostructure, and possible feed-

back effects. Finally, we recently showed [104] that

quantum corrals provide a new venue for manipulat-

ing (on the atomic length scale) one of the most fun-

damental quantum phenomena in condensed matter

physics,theKondo effect.Specifically,it ispossible,by

using the spatial form of low energy eigenmodes of a

quantum corral, to custom-design Kondo-screening

by changing the location of a single magnetic impu-

rity inside the corral. This idea can easily be gener-

alized, for example, in order to study the possibility

of manipulating the two-impurity Kondo problem

inside a nanoscale impurity structure.

In conclusion, research at the intersection of

strongly correlated electron systems and nanoscale

256 D.K.Morr

structures provides many possibilities for the emer-

gence of novel quantum phenomena that will have

important impact on a large number of fields, rang-

ing from the study of quantum critical systems

and unconventional phases to quantum information

technology and spin electronics.

Acknowledgements

We would like to thank A. de Lozanne, A.V. Bal-

atsky,K.Bennemann,J.C.Davis,L.Glazman,A.Hein-

rich,A.Kapitulnik,R.H.Nyberg,M.Randeria,K.-H.

Rieder,E.Rossi,J.Schmalian,Q.Si,N.A.Stavropou-

los,R.Wiesendanger,A.Yazdani,J.Yoon,and J.-X.Zhu

for stimulating discussions. D.K. M. acknowledges

financial support by the Alexander von Humboldt

Foundation, the National Science Foundation under

Grant No. DMR-0513415 and the U.S. Department of

Energy under Award No. DE-FG02-05ER46225.

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