Molecular Nanostructures on the Surface of a d_(x2y2)Superconductor
ABSTRACT We study molecular nanostructures on the surface of a d_(x2y2)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.
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Article: Quantum Interference between Impurities: Creating Novel ManyBody States in swave Superconductors
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ABSTRACT: We demonstrate that quantum interference of electronic waves that are scattered by multiple magnetic impurities in an swave superconductor gives rise to novel bound states. We predict that by varying the interimpurity distance or the relative angle between the impurity spins, the states' quantum numbers, as well as their distinct frequency and spatial dependencies, can be altered. Finally, we show that the superconductor can be driven through multiple local crossovers in which its spin polarization, $$, changes between $=0, 1/2$ and 1. Comment: 4 pages, 4 figuresPhysical review. B, Condensed matter 05/2002;
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arXiv:condmat/0410717v1 [condmat.suprcon] 27 Oct 2004
Molecular Nanostructures on the Surface of a dx2−y2Superconductor
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−y2wave 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−y2superconductor that represents the family
of hightemperature 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−y2wave superconducting
gap and ǫk= −2t(coskx+ cosky) − 4t′coskxcosky− µ
is the normal state tightbinding 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 bosonfermion 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)
Page 2
2
whereˆΣ is the full fermionic selfenergy obtained from
the BetheSalpeter 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−y2superconductor is pairbreaking 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 selfenergy,ˆΣ(ri,rj), with ri = rj, but
also the nonlocal 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 nonvanishing of
the nonlocal anomalous Greens function, F∆r, and self
energy, Φ∆r (the offdiagonal 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 particlelike and
holelike 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 nonmagnetic 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
Page 3
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 nonzero
F∆rand Φ∆r. In contrast, in case II, the frequency shift
is much smaller. For J = ∞, only the bosonic inphase
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 pseudospins 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 pseudospins are ferro
magnetically aligned (i.e., the molecules prefer to be in
the same state). Performing a HolsteinPrimakoff trans
formation followed by a largeL 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 pseudospin dimer which scatter fermions
via Hint. For Jz < J±, i.e., J > ω0, the system be
comes unstable since the ferromagnetic alignment of the
pseudospins 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 nonmonotonic 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 nonzero 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 selfenergy 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 pseudospins (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 HolsteinPrimakoff trans
formation followed by a largeL 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
pseudospins 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 nonlocalˆG0andˆΣ 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 nonlocal Greens function and
selfenergy 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) HOcoupling, (c) FMcoupling,
and (d) AFMcoupling.
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
Page 4
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 HOcoupling with n−
J = ω0/3. (c), (d) LDOS for a HTSC band structure and (c)
HOcoupling, and (d) AFMcoupling.
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 selfenergy, 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 nonzero
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 Jdependence 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−y2wave
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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