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arXiv:cond-mat/0410717v1 [cond-mat.supr-con] 27 Oct 2004

Molecular Nanostructures on the Surface of a dx2−y2-Superconductor

Roy H. Nyberg and Dirk K. Morr

Department of Physics, University of Illinois at Chicago, Chicago, IL 60607

(Dated: June 17, 2011)

We study molecular nanostructures on the surface of a dx2−y2-wave superconductor. We show

that the interplay between the molecular nanostructure’s internal excitation spectrum and quantum

interference of the scattered host electrons leads to a series of novel effects in the local density of

states. We demonstrate that these effects give insight into both the nature of the superconducting

pairing correlations and of the intermolecular interactions.

PACS numbers: 73.22.-f, 73.22.Gk, 74.72.-h

Nanostructures provide the intriguing possibility of

manipulating in a controlled way the electronic struc-

ture of the host system they reside on [1]. This level of

control opens new venues for studying the complex elec-

tronic structure of many strongly correlated electron sys-

tems. For example, it was suggested that nanostructures

can elucidate the nature of superconducting correlations

in conventional [2] and unconventional superconductors

[3]. Nanostructures formed from more complex build-

ing blocks, such as molecules with internal degrees of

freedom, provide a novel way of manipulating and simul-

taneously gaining insight into the electronic structure of

complex systems. The internal vibrational and rotational

excitations of single molecules have been intensively stud-

ied by inelastic tunneling spectroscopy [4, 5]. Their in-

teraction with and effect on the electronic structure of

a metallic surface was recently investigated by Gross et

al. [6].

In this Letter we study nanostructures composed of

two molecules that reside on the surface of an unconven-

tional dx2−y2-superconductor that represents the family

of high-temperature superconductors (HTSC). We show

that the interplay between the molecular nanostructure’s

internal excitation spectrum and quantum interference

of the scattered host electrons leads to a series of novel

effects in the local density of states (LDOS) of the su-

perconductor. Our main results are threefold. First, we

identify three types of intermolecular interactions and

show that each of them leads to qualitatively distinguish-

able features in the LDOS. This, in turn, allows one

to identify the nature of intermolecular interaction from

experimental measurements of the LDOS. Second, we

demonstrate that the LDOS changes with the molecules’

distance and orientation relative to the underlying host

lattice. This effect permits us to probe the spatial depen-

dence of superconducting correlations. Third, we show

that by exciting specific energy levels of the molecular

structure, the LDOS can be manipulated in a controlled

manner. These results intricately relate the study of

strongly correlated electron systems with the further de-

velopment of molecular electronics.

A nanostructure consisting of N molecules, repre-

sented by local bosonic modes [7], on the surface of a two-

dimensional dx2−y2 superconductor possesses the Hamil-

tonian H = He+ Hb+ Hintwhere

He

=

?

k,σ=↑,↓

ǫkc†

k,σck,σ+

?

k

∆kc†

k,↑c†

−k,↓+ h.c. ,

Hb

= ω0

?

i

a†

riari+ J

?

i,j

Ψ?a†

ri,ari,a†

rj,arj

?,

Hint = g

?

i,σ

?a†

ri+ ari

?c†

ri,σcri,σ. (1)

He and Hb describe the unperturbed superconductor

and nanostructure, respectively, and Hint represents

the interaction between them. c†,a†are the fermionic

and bosonic creation operators, respectively.

∆0(coskx−cosky)/2 is the dx2−y2-wave superconducting

gap and ǫk= −2t(coskx+ cosky) − 4t′coskxcosky− µ

is the normal state tight-binding dispersion. ω0 is the

characteristic frequency of the N modes located at ri

(i = 1,..,N). Ψ is a quadratic functional in the bosonic

operators and describes the interaction between modes

at sites riand rjwith strength J (three specific forms of

Ψ are discussed below). g is the boson-fermion scatter-

ing vertex. The fermionic Greens function of the unper-

turbed (clean) system in Nambu notation is

∆k =

ˆG−1

0(k,iωn) = [iωnτ0− ǫkτ3]σ0+ ∆kτ2σ2, (2)

where σiand τiare the Pauli matrices in spin and Nambu

space, respectively. Diagonalizing Hb via a Bogoliubov

transformation to new operators b†

obtains Hb=?

l,bl(l = 1,...,N), one

lΩlb†

lbland

Hint=

?

i,l,σ

g(l)

ri

?

b†

l+ bl

?

c†

ri,σcri,σ, (3)

where g(l)

tex, and Ωl is the bosonic energy spectrum. The full

fermionic Green’s function is now given by

riis a site and mode dependent interaction ver-

ˆG(r,r′,ωn) = ˆG0(r,r′,ωn) +

?

i,j

ˆG0(r,ri,ωn)

×ˆΣ(ri,rj,ωn)ˆG0(rj,r′,ωn) ,(4)

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2

whereˆΣ is the full fermionic self-energy obtained from

the Bethe-Salpeter equation

ˆΣ(ri,rj,ωn) =ˆΣ0(ri,rj,ωn) +

?

p,q

ˆΣ0(ri,rp,ωn)

×ˆG0(rp,rq,ωn)ˆΣ(rq,rj,ωn) (5)

with p,q = 1,...,N and

ˆΣ0(ri,rj,ωn) = T

?

m,l

τ3g(l)

riˆG0(ri,rj,ωn+ νm)

×Dl(νm)τ3g(l)

rj. (6)

The retarded form of the bosonic propagator is given by

DR

l(ω) = [ω + iΓ(ω) − Ωl]−1−[ω + iΓ(ω) + Ωl]−1. (7)

The lowest order vertex corrections scale as δg/g ∼

(gkF/4vF)2F where F is a function of O(1) for bosonic

and fermionic frequencies smaller than ∆0[8]. For the pa-

rameter range considered below, gkF/4vF ≪ 1 and ver-

tex corrections can thus be neglected. The interaction,

Hint, not only leads to changes in the fermionic LDOS,

but also affects the bosonic excitation spectrum twofold.

First, Hintshifts the unperturbed bosonic frequencies, a

shift that we take to be included in the effective values for

Ωlconsidered below. Second, the bosonic modes acquire

a finite lifetime, Γ(ω) ?= 0. The lowest order bosonic self-

energy correction yields Γ(ω) = g2ω3/(6πv2

vF (v∆) is the Fermi (superconducting) velocity at the

nodal points perpendicular (parallel) to the Fermi sur-

face. Since for all cases considered below Γ(ω0) ≪ ω0,

which requires g ≪ 50∆0, Γ(ω) can be neglected and

we take for simplicity Γ(ω) = 0+. Finally, the LDOS,

N = A11+ A22 with Aii(r,ω) = −2ImˆGii(r,ω + iδ) is

obtained numerically from Eqs.(4) with δ = 0.2 meV.

We showed in Ref. [8] that a single local mode on the

surface of a dx2−y2-superconductor is pair-breaking and

induces a fermionic resonance state, whose spectroscopic

signature are two peaks in the LDOS. Introducing a sec-

ond mode leads to two new physical effects that introduce

qualitative changes in the LDOS. First, due to quantum

interference of the scattered electrons, the calculation of

the LDOS involves not only the local Greens function,

ˆG0(ri,rj), and self-energy,ˆΣ(ri,rj), with ri = rj, but

also the non-local ones with ri ?= rj. As a result, the

LDOS is determined by the distance and orientation of

the modes relative to the lattice, which is described by

∆r = ri− rj = (nx,ny) (we set the lattice constant

a0= 1). Second, the intermolecular interaction Ψ leads

to characteristic features in the LDOS that, as we show

below, depend on the specific form of Ψ.

In order to distinguish between the effects of quan-

tum interference, superconducting correlations and inter-

molecular interaction on the LDOS, we consider first a

simplified electronic bandstructure with t = 300 meV,

Fv2

∆), where

µ = t′= 0 and set J = 0. In this case, superconduct-

ing correlations, as represented by the non-vanishing of

the non-local anomalous Greens function, F∆r, and self-

energy, Φ∆r (the off-diagonal elements ofˆG0 andˆΣ0,

respectively), are present for (nx+ ny)mod2 = 1 (case

I) and vanish otherwise (case II). We find that the pres-

ence of superconducting correlations leads to qualitative

differences in the LDOS between case I and II that are

universal, i.e., independent of the specific realization of

case I or II considered. For definiteness, we consider be-

low two modes located at r1= (0,0) and r2= (1,0) (case

I) and r1= (0,0) and r2= (2,0) (case II) and present

the resulting LDOS at T = 0 in Fig. 1(a) and (b). The

FIG. 1: LDOS at R as a function of frequency for ω0 = 0.6∆0

and three intermolecular interactions: (a),(b) HO coupling,

(c),(d) FM coupling, and (e),(f) AFM coupling. Scaling of

the curves is indicated.

coupling of the resonance states due to quantum inter-

ference only (see curves for J = 0) leads to the formation

of bonding and antibonding states and an energy split-

ting between them. Only one of these states is located at

energies smaller than ∆0, resulting in a particle-like and

hole-like peak in the LDOS. Quantum interference leads

to oscillations in the frequency of the resonance peaks

when ∆r is changed (cf. Figs. 1(a) and (b)), similar to

the case of non-magnetic impurities [3]. Next, we discuss

the effects of three different intermolecular interactions

on the frequency and shape of the resonance peaks.

Harmonic Oscillator (HO) Coupling: A coupling of

the bosonic modes via a quadratic potential is the

quantum mechanical analog of two harmonic oscilla-

tors coupled by a spring.

?b†

tion, one has Ω1= ω0, Ω2= ω0

g(1)

In this case [9] Ψ=

r1+ br1− b†

r1,2= g/√2 and g(2)

r2− br2

?2with J ≥ 0. After diagonaliza-

√1 + 4λ with λ ≡ J/ω0,

r1,2= ±g/[√2(1 + 4λ)1/4]. The re-

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3

sulting LDOS for several values of J is shown in Fig. 1(a)

and (b) for cases I and II, respectively. With increasing

J, Ω2increases while |g(2)

tering hence becomes weaker and the resonance peaks are

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

to ±∆0as J increases from J = 0 to J = ∞, an effect

that is independent of g and directly related to a non-zero

F∆rand Φ∆r. In contrast, in case II, the frequency shift

is much smaller. For J = ∞, only the bosonic in-phase

mode couples to the fermionic system. This coupling is

much weaker in case I than in case II due to the super-

conducting correlations.

Ferromagnetic (FM) Coupling: A second type of in-

termolecular interaction arises if each molecule possesses

(2L+1) states that are represented by pseudo-spins L1,2,

whose interaction is given by the anisotropic Heisenberg

Hamiltonian

r1,2| decreases, the electronic scat-

Hspin= JzLz

1Lz

2+ J±(Lx

1Lx

2+ Ly

1Ly

2) . (8)

For Jz < 0 and |Jz| > |J±|, the pseudo-spins are ferro-

magnetically aligned (i.e., the molecules prefer to be in

the same state). Performing a Holstein-Primakoff trans-

formation followed by a large-L expansion up to order

O(L), we obtain Hb in Eq.(1) with ω0 = −JzS > 0,

J = J±S ≥ 0 and Ψ = b†

nalization, one has Ω1,2 = ω0∓ J, g(1)

g(2)

tations of the pseudo-spin dimer which scatter fermions

via Hint. For Jz < J±, i.e., J > ω0, the system be-

comes unstable since the ferromagnetic alignment of the

pseudo-spins is destroyed.

For small values of g, the resonance peaks move to-

wards lower energies in cases I and II with increasing

J (not shown). However, for intermediate g ≈ 30∆0,

this behavior changes qualitatively for case I; the res-

onance frequencies exhibit a non-monotonic dependence

on J [Fig. 1(c)], in contrast to case II [Fig. 1(d)]. For even

larger g, the resonance peaks shift to higher energies in

case I, and to lower energies in case II with increasing J.

A detailed analysis shows that this qualitative difference

arises from a non-zero F∆r and Φ∆r in case I. Note, in

case II (Fig. 1(d)), the width of the resonance peaks in-

creases when it shifts to lower energies, in contrast to the

effect expected from a decreasing residual DOS [3]. Here,

however, the width of the resonance peaks is also deter-

mined by the imaginary part of the self-energy which van-

ishes for |Ω| < Ω1. Since Ω1decreases with increasing J,

the resonance state becomes more strongly damped once

Ω1 < ωres. In Fig. 1(d), we have Ω1 = ω0 > ωres for

J = 0, but Ω1< ωresfor J = 2ω0/3 (Ω1= 0.2∆0) and

J = ω0(Ω1= 0), resulting in an increased peaks’ width

for the latter two values of J.

Antiferromagnetic (AFM) Coupling:

magnetic alignment of the pseudo-spins (Jz> 0, |Jz| >

J± > 0 in Eq.(8)), the molecules prefer to be in “op-

r1br2+ b†

r2br1. After diago-

r1,2= ∓g/√2 and

ri,brirepresent the exci-

r1,2= g/√2. The operators b†

For antiferro-

posite” states. Performing a Holstein-Primakoff trans-

formation followed by a large-L expansion, one obtains

Ψ = b†

r2+ br1br2resulting in two degenerate energy

dispersions Ω1,2 = ω0

±gu1,2, g(1,2)

For J > ω0, i.e., |Jz| < J±, the system becomes unstable

since the antiferromagnetic alignment of the molecular

pseudo-spins is destroyed.

With increasing J, Ω1,2decreases while g(1,2)

thus leading to an increase in the scattering strength.

As a result, the resonance peaks shift monotonically to

lower energies in case II [see Fig. 1(f)], while their width

decreases since ωres< Ω1,2for all J. In the limit J → ω0,

g(1,2)

r2

diverge, and the scattering becomes unitary. Since

the local and non-localˆG0andˆΣ vanish for ω = 0, the

resulting resonance peaks are located at zero energy and

the LDOS vanishes at the molecules’ sites. In contrast, in

case I [Fig. 1(e)], the resonance peaks first shift to lower

energies with increasing J, but eventually move back to

higher energies and broaden. This behavior arises from

the nonvanishing of the non-local Greens function and

self-energy for ω = 0.

At T = 0, the imaginary part of the normal self-

energy (the diagonal element ofˆΣ0) possesses logarith-

mic divergences that lead to dips in the LDOS at ±ω+

(ω±

1,2= ∆0± Ω1,2) [8, 10]. At T ?= 0, the positive en-

ergy levels of the bosonic modes, represented by the first

term in Eq.(7), become populated, opening a new chan-

nel for fermionic scattering.

logarithmic divergences at ±ω−

peaks or dips in the LDOS, depending on whether ω−

is smaller or larger than the frequency of the resonance

peaks, ±ωres. The LDOS for T ?= 0 is shown in Fig. 2.

In Figs. 2(a) and (b), we plot the LDOS for HO coupling

r1b†

√1 − λ2with λ ≡ J/ω0, g(1,2)

= ∓gu2,1 and u2

r1

=

r2

1,2= [1/√1 − λ2± 1]/2.

r2

increases,

1,2

As a result,ˆΣ0 acquires

1,2. This leads to either

i

FIG. 2:

0.6∆0and J = 2ω0/3: (a), (b) HO-coupling, (c) FM-coupling,

and (d) AFM-coupling.

Temperature dependence of the LDOS for ω0 =

in case II for g = 15∆0and g = 24∆0, respectively. For

g = 15∆0, ω−

1,2< ωres and the logarithmic divergences

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4

FIG. 3: LDOS at T = 0 for ω0 = 0.6∆0 and g = 15∆0.

(a),(b) The positive energy branches of the bosonic modes

are selectively populated for HO-coupling with n−

J = ω0/3. (c), (d) LDOS for a HTSC band structure and (c)

HO-coupling, and (d) AFM-coupling.

1,2= 1 and

lead to four peaks (indicated by arrows) in the LDOS.

Their intensity is governed by nB(Ωi) and thus larger at

±ω−

scattering channels leads to an increase in the imaginary

part of the self-energy, resulting in a shift of the reso-

nance peaks to higher energies and an increase in their

width. In contrast, for g = 24∆0, ω−

to a splitting of the resonance peaks and to a dip in the

LDOS. The second set of divergences at ω−

peaks in the LDOS, which due to the overall increase in

the LDOS at lower energies are barely perceptible. The

LDOS for FM coupling and AFM coupling in case II is

shown in Fig. 2(c) and (d), respectively. For FM cou-

pling, the LDOS exhibits two peaks and two dips since

ω−

2. In contrast, for AFM coupling, the

two modes are degenerate and since ω−

arithmic divergences result in a single set of dips. The

LDOS in case I (not shown) exhibits qualitatively sim-

ilar behavior to the ones discussed above for all three

couplings. Note that the frequencies of the features aris-

ing from the logarithmic divergences are independent of

g, which thus permits a direct measurement of ±ω±

The degeneracy of the modes for AFM coupling results

in a LDOS with only two features at ±ω−

spectively, in qualitative contrast to the LDOS for FM

and HO coupling. The latter two couplings, however,

can be distinguished by comparing ω0 obtained from

the LDOS near a single molecule [8] with ω±

the nature of the intermolecular interaction, similar to

a molecule’s internal structure [6], can be identified ex-

perimentally from the specific features it induces in the

LDOS.

Changing the population, n±

±Ω1,2, for example by optical means [11], opens intrigu-

ing venues to manipulate the local electronic structure of

the host material. In Fig. 3(a),(b), we present the LDOS

1than at ±ω−

2. Simultaneously, opening of the new

1≈ ωres, leading

2induces two

1< ωres < ω−

1,2> ωres, the log-

1,2.

1and ±ω+

1, re-

1,2. Thus

1,2, of the energy levels

for HO coupling with n−

n+

increase of n+

to the temperature induced changes discussed above.

1,2= 1 and n+

1,2?= 0. A non-zero

1, while an

2, similar

1[Fig. 3(a)] induces dips in the LDOS at ±ω−

2[Fig. 3(b)] leads to peaks at ±ω−

Finally, the use of a bandstructure representative of

the HTSC with t′/t = −0.4 and µ/t = −1.18 [12] mixes

the effects of intermolecular interaction, superconducting

correlations and quantum interference on the LDOS. For

example, the J-dependence of the LDOS for HO coupling

[Fig. 3(c)] is similar to the one shown in Fig. 1(a), while

the LDOS for AFM coupling [Fig. 3(d)] shows a depen-

dence on J that is qualitatively different from that in

Fig. 1(f).

In conclusion, we study the effects of molecular nanos-

tructures on the electronic structure of a dx2−y2-wave

superconductor. We show that different intermolecular

interactions lead to qualitatively distinguishable signa-

tures in the LDOS. By changing the population of the

bosonic excitations, one can manipulate the host’s elec-

tronic structure and gain further insight into the nature

of unconventional superconducting correlations.

We would like to thank J.C. Davis, T. Imbo, and K.-H.

Rieder for stimulating discussions. D.K.M. acknowledges

support from the Alexander von Humboldt foundation.

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