Scattering of plasmons at the intersection of two metallic nanotubes: implications for tunneling.

Abstract

We study theoretically the plasmon scattering at the intersection of two metallic carbon nanotubes. We demonstrate that, for a small angle of crossing theta<1, the transmission coefficient is an oscillatory function of lambda/theta, where lambda is the interaction parameter of the Luttinger liquid in an individual nanotube. We calculate the tunnel density of states nu(omega,x) as a function of energy omega and distance x from the intersection. In contrast with a single nanotube, we find that, in the geometry of crossed nanotubes, conventional "rapid" oscillations in nu(omega,x) due to the plasmon scattering acquire an aperiodic "slow-breathing" envelope which has lambda/theta nodes.

Full text

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arXiv:0806.2152v2 [cond-mat.mes-hall] 8 Jan 2009
Scattering of plasmons at the intersection of two nanotubes: Implications for
tunnelling
V. V. Mkhitaryan, Y. Fang, J. M. Gerton, E. G. Mishchenko, and M. E. Raikh
Department of Physics, University of Utah, Salt Lake City, UT 84112
We study theoretically the plasmon scattering at the intersection of two metallic carbon nan-
otubes. We demonstrate that for a small angle of crossing, θ ≪ 1, the transmission coefficient is
an oscillatory function of λ/θ, where λ is the interaction parameter of the Luttinger liquid in an
individual nanotube. We calculate the tunnel density of states, ν(ω,x), as a function of energy, ω,
and distance, x, from the intersection. In contrast to a single nanotube, we find that, in the geome-
try of crossed nanotubes, conventional “rapid” oscillations in ν(ω,x) due to the plasmon scattering
acquire an aperiodic “slow-breathing” envelope which has λ/θ nodes.
PACS numbers: 71.10.Pm, 73.40.Gk,72.15.Nj
Introduction. By now, observation of Luttinger liquid
in 1D systems has been reported for single-walled car-
bon nanotubes [1, 2, 3, 4, 5, 6] and GaAs-based semi-
conductor wires [7]. Conclusions about Luttinger liq-
uid behavior have been drawn from analysis of the data,
which can be divided into two groups: (i) power-law,
∝ (max{V,T})α, behavior of tunnel or source-drain con-
ductance [1, 2, 3, 4, 5, 6], where parameter α is the mea-
sure of deviation from the Fermi liquid behavior, and
(ii) momentum-resolved tunnelling in a parallel magnetic
field [7].
On the conceptual level, the difference between the
techniques (i) and (ii) is that (i) probes a single-
point Green function, G(x,x,ω), while (ii), by mapping
?dx?dx′G(x,x′,ω)exp[−iqB(x − x′)], with qB propor-
tional to applied field, yields information about two-point
Green function, and thus is more informative.
With regard to quantitative determination of the Lut-
tinger liquid parameter, g, which is related to α as [8]
α = (g−1+ g − 2)/8, it is desirable to identify an effect,
which would depend on g stronger than a power law.
An example of such an effect was given by Ussishkin and
Glazman in Ref. 9, where, due to electron backscattering,
g appears in the argument of sine; this sine describes the
amplitude modulation of the probe-induced Friedel oscil-
lations [10], ∝ cos(2kFx) in the local density of states; kF
is the Fermi momentum.
In the present paper we demonstrate that the geom-
etry of the crossed nanotubes (see Fig. 1) offers a qual-
itatively new manifestation of the Luttinger liquid be-
havior. In particular, the oscillatory dependence on g,
similar to that in Ref. 9, emerges in the geometry of
crossed nanotubes even without electron backscattering
[5, 11, 12, 13]. More precisely, we show that, in this
geometry, the envelope, “breathing” with g, modulates
not cos(2kFx) oscillations, but much slower oscillations
resulting from the plasmon backscattering.
There is an important difference between scattering
of plasmons and electrons: for an obstacle bigger than
k−1
F
electron scattering is exponentially suppressed, while
y
0
0
dR
t
r
1
2
1
d
x
dL
θ
FIG. 1: (Color online) Intersecting nanotubes, separated by
a distance, d; the angle of intersection is θ. Directions of
incident, reflected, r, transmitted, t, and deflected, dL, dR,
plasmon waves are shown with solid red arrows. Dashed ar-
rows illustrate two contributions to the reflected wave.
plasmon scattering is efficient as long as the size of the
obstacle does not exceed the plasmon wavelength. This
scattering gives rise to the oscillations of the local density
of states δν(ω,x) ∝ cos(2ωx/vF) where vF is the Fermi
velocity. It is these oscillations that acquire a breathing
envelop in the geometry of crossed nanotubes, Fig. 2.
Our main finding is that, with regard to this modula-
tion, making the crossing angle θ small, effectively en-
hances the Luttinger liquid parameter. To describe this
enhancement quantitatively, we first consider an auxil-
iary problem of plasmon scattering at the intersection
and later utilize it for the calculation of δν(ω,x).
Plasmon scattering at the intersection. Assume that d
is the minimal distance between the nanotubes. Even in
the absence of electron tunnelling, a plasmon propagat-
ing towards x = 0 in the nanotube 1 can: (i) pass x = 0
(transmission); (ii) excite a plasmon in the nanotube 2,
which propagates away from the intersection x = 0 either
to the left or to the right (deflection); (iii) get reflected.
Incorporating the plasmon scattering into the formalism
of the Luttinger liquid gives rise to the breathing enve-
lope in Fig. 2. The underlying reason is that the interac-
tion between the tubes strengthens towards intersection
[14]. This leads to the x-dependent splitting of velocities
in each tube. The resulting x-dependent phase accumu-
lation near the intersection translates into nontrivial de-
pendence of δν(ω,x). Moreover, the phase accumulation
increases rapidly with decreasing angle θ, thus simulating

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2
??
?
??
?
?
??
?
?
??
?
FIG. 2: (Color online) Oscillating corrections to the tun-
nelling conductance is plotted from Eq. (15) versus dimen-
sionless bias w = V x/s, where x is the distance from the
intersection. Periodic oscillations (black) are modulated by
”breathing” envelope (red), with ”period” determined by λ/θ,
where λ is the interaction parameter; (d) illustrates suppres-
sion of oscillations at finite separation, d = 0.01x, between
the nanotubes.
the enhancement of the Luttinger parameter.
Collective modes of intersecting nanotubes. As a result of
long-range interaction, e2?
is the displacement of the electron position from the equi-
librium, the plasmon spectrum of an individual tube is
ω(q) = qs(q) with velocity [8] s = vF(1 + λln(qr))1/2.
Here r is the nanotube radius and λ = 8e2/(π?vF) is
the interaction constant [8]. Following Ref. 8, we ne-
glect the relative change of ln(qr). At a given frequency,
ω, displacement u(x) is the eigenmode,
?ω2/v2
ˆD{f} = −∂2
∂x2
dxdx′
|x−x′|
∂u(x)
∂x
∂u(x′)
∂x′ , where u(x)
ˆD{u(x)} =
F
?u(x), of the operator
∂x2f(x) − λ∂2
?∞
−∞
dy
|x − y|f(y).(1)
For two crossed nanotubes, the eigenmodes are described
by the system of two coupled equations,
?ω2/v2
F
?u1,2(x) =ˆD{u1,2(x)} +ˆF{u2,1(x)},
ˆF{f} = −λ∂
∂y
−∞
(2)
?∞
dx
?d2+ x2+ y2− 2xycosθ
∂f(x)
∂x
. (3)
The operatorˆF has a meaning of longitudinal force cre-
ated by the density fluctuation, ∂u1(x)/∂x, in the nan-
otube 1, at point y of the nanotube 2. The scattering
problem corresponds to the solution of Eqs. (2) which
has the following asymptotes at large x and y:
u1(x)??
u2(y)??
x→−∞= eikx+ re−ikx,
y→−∞= dLe−iky,
u1(x)??
x→∞= teikx,
y→∞= dReiky.u2(y)??
(4)
Born approximation. For small λ, the elements of scat-
tering matrix can be found in the Born approximation in
momentum space. To the first order in λ, only dR and
dLare non-zero. They are given by matrix elements of
the operatorˆF, Eq. (3), namely, dR(k) = (i/2k)ˆFk,kand
dL(k) = −(i/2k)ˆF−k,k. Analytical expression forˆFp,qis
ˆFp,q= 2πλpq e−
d
sin θ(p2+q2−2pq cosθ)
(p2+ q2− 2pqcosθ)1/2
1/2
.(5)
This leads to the final result for deflection coefficients
dR= iπλe−
kd
cos(θ/2)
2sin?θ
2
? ,dL= −iπλe−
kd
sin(θ/2)
2cos?θ
2
? .(6)
An apparent consequence of Eq. (6) is that deflection is
exponentially small when the plasmon wavelength is ≪ d.
Less obvious is that, for kd < 1 and small θ, coefficients
dRand dLcan differ exponentially. This is because the
exponent, exp(−2kd/θ), in dLcan be small if kd is small.
Noteworthy, in the long-wavelength limit, kd ≪ θ, we
still have dR/dL ≈ 1/θ ≫ 1. The underlying reason is
that dL corresponds to the wave which travels almost
in the opposite direction to the incident wave, while dR
travels almost along the incident wave. From Eq. (6) we
conclude that the Born approximation applies at λ ≪ θ.
The reflection coefficient, r(k), in the second Born ap-
proximation, is expressed via the matrix elements Eq. (5)
r(k) =
1
4iπk
?∞
−∞
dp
p2− k2− iǫ
ˆF−k,pˆFp,k.(7)
The integral in Eq. (7) is the sum, (2πλ)2k(I1+ iI2), of
contributions from the poles p = ±k and the principal
value, which can be cast in the form
I1= P
?
dpp2
p2− k2
sin θ(p2+k2+2pk cosθ)1/2,
π
2sinθe−kd[
e−
d
sin θ(p2+k2−2pk cosθ)1/2
?(p2+ k2)2− 4p2k2cos2θ?1/2
×e−
d
I2=
1
cos(θ/2)+
1
sin(θ/2)]. (8)
For
dence
−(λ2/2)(kdsinθ/π)−3/2e−2kd/sin θ.
wavelength limit, kd ≪ 1, one can replace the exponent
e−kd/sinθby 1. In what follows, we will focus on small θ,
where dLand r diverge. Note that the pole contribution
in Eq. (8) diverges for θ → 0 much stronger that the
principal value contribution, which is ∝ ln(1/θ), so
that r ≈ π2λ2/2θ. We also notice that in the small-θ
domain, the relation r ≈ dLdRholds. This relation can
be understood from the following reasoning.
There are two contributions to the reflected wave in the
second Born approximation. (i) The wave deflected into
the second tube to the right with the amplitude (solid ar-
row in Fig. 1), undergoes a secondary deflection back into
the first tube (dashed arrow in Fig. 1) with amplitude dL.
(ii) The wave deflected into the second tube to the left,
dL, is subsequently deflected back into the first tube with
short
r(k)
wavelengths,
isdominated
kd≫
the
1,
integral I1
In
thedepen-
by≈
thelong-

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3
the amplitude dR, Fig. 1. The sum of the two contribu-
tions amounts to r = (c1+ c2)dLdR. Remarkably, both
numerical factors c1 and c2 are equal to 1/2. This is a
consequence of a strong difference in distances at which
formation of the primary left- and right-deflected waves
takes place. The wave dLis formed within ∼ 1/k from
the intersection, while the wave, dR, is formed within a
much broader interval ∼ 1/(kθ). Therefore, in second
tube, at some distance y from the intersection, such that
1/k ≪ y ≪ 1/(kθ), the amplitude of the left-deflected
wave is already dL, while the amplitude of the right-
deflected wave is only1
2dR. Subsequent formation of the
contribution (ii) occurs at y ∼ 1/(kθ), so that the corre-
sponding amplitude is?1
other hand, formation of the contribution (i) takes place
only over negative −1/(kθ) < y < 0, and thus results in
dL
?1
Semiclassical description. From Eq. (6) one can see that
for θ < πλ the Born approximation renders an unphysi-
cal result, namely, dR> 1, suggesting that this approxi-
mation is not applicable for small θ. This manifests the
change in the mechanism of the plasmon scattering which
takes place for θ ? λ. Indeed, at small θ, incident wave
travels closely to the wave dR over a long distance, so
that their amplitudes get mutually redistributed. Im-
portantly, in describing this redistribution one can: (i)
neglect both left-deflected, dL, and reflected, r, waves
and (ii) employ semiclassical approach, which yields
2dR
?dL, i.e., c2= 1/2. On the
2dR
?, i.e., c1= 1/2.
t(k) = cos
?2λ
θ
?2λ
?∞
0
dzK0
??
k2d2+ z2??
??
, (9)
dR(k) = isin
θ
?∞
0
dzK0
k2d2+ z2??
,
where K0is the MacDonald function. A remarkable fea-
ture of this result is that t and r oscillate strongly with θ,
and that the oscillations scale with the interaction con-
stant. Note, that in the short-wavelength limit kd ≫ 1,
the perturbative result Eq. (6) is valid even for λ > θ.
Using the large-argument asymptote of K0(z), it is easy
to see that Eq. (9) reproduces Eq. (6) in this limit. For
long-wavelengths, kd ≪ 1, Eq. (9) yields dR= sin(πλ/θ),
t = cos(πλ/θ), so that the perturbative and semi-classical
results match at λ/θ ? 1.
To outline the derivation of Eq. (9), we note that the
system of equations, Eqs. (2), can be rewritten as two
independent closed equations,
?ω2/v2
F
?u±(x) =ˆD{u±(x)} ±ˆF{u±(x)},
where combinations u±(x) = u1(x) ± u2(x) are intro-
duced. Searching for the semiclassical solution of Eq. (10)
in the form u±(x) = eikx+iϕ±(kx), with slowly varying
phase, ϕ′
(10)
±≪ 1, we find
2ϕ′
±(kx) = ∓λK0
??1 + ϕ′
±(kx)?k
?
d2+ x2θ2?
. (11)
In evaluating the r.h.s. we assumed that θ is small. We
see that when λ is small, the assumption, ϕ′
justified. Then the smallness of ϕ′
it in the argument of K0. Upon integrating Eq. (11),
we find ϕ±. Then transforming back to u1 and u2, we
recover Eq. (9). The expression for r(k) generalized to
the domain θ < λ < 1 follows from Eqs. (6) and (9)
±≪ 1, is
±allows one to neglect
r(k)??
θ<λ= dRdL
=πλ
(12)
2e−2kd
θ sin
?2λ
θ
?∞
0
dzK0
??
k2d2+ z2??
,
and in the long-wavelength limit simplifies to r|θ<λ =
(πλ/2)sin(πλ/θ).
Tunnel density of states. Most importantly, the non-
trivial dependence of the plasmon scattering on λ and
θ manifests itself in the observables, e.g., in the depen-
dence of tunnel density of states, ν(ω,x), on the distance,
x, from the intersection. To illustrate this, consider first
a single nanotube with inhomogeneity at x = 0 which
scatters plasmons with reflection coefficient ˜ r(k). Then
the correction to the tunnel density of states reads
δν(ω,x)
ν0(ω)
= Γ(α + 1)
?
α2+ α/2 (13)
×
|˜ r(ω/s)|
(2ωx/s)α+1sin
?2ωx
s
− ϕ(ω/s) −πα
2
?
,
where ϕ(k) = arg(˜ r). Eq. (13) follows from the expres-
sion for interaction contribution to the local Green func-
tion which takes into account the plasmon scattering,
G(x,t) =
?
(14)
exp−π
?
dk
?|?xuk|2sk
8vF
+|uk(x)|2vF
8sk
??
1 − e−is|k|t??
.
Here uk(x) are the plasmon eigenmodes: uk(x) = (eikx+
˜ r(k)e−ikx)/√2π, and uk(x) =
kx < 0 and kx > 0, respectively. Expanding the ex-
ponent in Eq. (14) with respect to ˜ r, and evaluating
ν(ω,x) = π−1Re?∞
Eq. (14) emerges upon representing electrons via the
dual bosonic fields θiαand φiα, ψiα∼ ei(φiα±θiα); i = 1,2
labels the two bands, α =↑,↓ are the spins [8].
teraction is completely described by the charged field
θc=1
2
?
φc(x) =
?√n0
modes, uk(x), reduces the interacting Hamiltonian to a
system of harmonic oscillators{ˆQk,ˆPk} yielding Eq. (14).
A simple reasoning allows to generalize Eq. (13) to the
case of two intersecting nanotubes. Indeed, with inter-
section playing the role of inhomogeneity, instead of one
reflected wave with reflection coefficient ˜ r we have two
independent modes, u±(x), solutions of Eq. (10), with
reflection coefficients r±. It is important that while the
?(1 − |˜ r|2)/2π eikx, for
0dteiωtG(x,t), we arrive at Eq. (13).
In-
iαθiα, while the three neutral sectors are non-
interacting. Expansion [15] θc(x) = π√n0
1
?dk[?xdyuk(y)]ˆPkover the plasmon eigen-
?dkuk(x)ˆQk,

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absolute values r+and r−are the same and equal to |dL|,
given by Eq. (6), their phases are different and are equal
to ϕ+(kx) − ϕ+(−kx) and ϕ−(kx) − ϕ−(−kx) + π, re-
spectively, where ϕ±are determined by Eq. (11). Due to
this difference in phases, the oscillations ∝ sin[2ωx/s+ϕ]
in Eq. (13) transform into a beating pattern
δν(ω,x)
ν0(ω)
= −Γ(α + 1)
?
α2+ α/2πλ
2
e−2ωd/sθ
(2ωx/s)α+1
(15)
×sin


2λ
θ
ωxθ/s
?
0
dzK0
??
(ωd/s)2+ z2
?


cos
?2ωx
s
−πα
2
?
.
Eq. (15) is our main result. Remarkably, the shape of the
envelope of cos(2ωx/s) oscillations depends strongly on
the interaction parameter, λ, offering a unique signature
of Luttinger liquid behavior. In particular, the number
of nodes in the envelope is equal to λ/θ. Examples of
oscillations Eq. (15) are plotted in Fig. 2 in terms of tun-
nelling conductance, G(V,x) ∝ ν(ω = V,x), for different
interaction parameters. Note, that the language of re-
flected plasmons, r+, r−, applies at distances x ≫ s/V ,
over which the reflection coefficient is formed. Since the
characteristic scale of the envelope is s/V θ, Eq. (15) is
valid as long as θ ≪ 1. For large x ≫ s/V θ, the ar-
gument of the sine in Eq. (15) saturates at πλ/θ. On
the physical grounds, the magnitude of the cos(2V x/s)
oscillations at large x should be given by Eq. (13), with
element of scattering matrix r instead of ˜ r. From Eq. (12)
we realize that this is indeed the case.
Implications. Our main prediction is that, for purely
capacitive coupling between nanotubes, the conductance
G(V ) into one or both ends of crossed nanotubes must
exhibit a structure, like shown in Fig. 2, with a large
characteristic ”period” V ∼ s/(Lθ), where L is the dis-
tance from the end to the intersection. Smallness of θ
insures that this structure (envelope in Fig. 2) is distin-
guishable from size-quantization-like ”filling” of the en-
velope [8, 16], which changes with the period V = πs/L.
Also, an important prediction is that the envelope beat-
ing structure in Fig. 2 vanishes with temperature much
slower than the filling, which vanishes at T ∼ s/L.
The loop geometry of Ref. 17 offers another possible
experimental implication. For this geometry, the easi-
est way to compare the Sagnac oscillations in Ref. 17
and our finding Eq. (15) is to assume that interaction
is weak. Then the contribution to the differential con-
ductance from the Sagnac effect is roughly the product
of ”size-quantization” oscillations, ∝ cos(2V L/vF), and
the envelope ∝ cos(2V Lug/v2
drain bias; L is the loop perimeter and ug is the gate-
induced detuning of the ”left” and ”right” velocities. Our
Eq. (15) for this geometry contains the same first cosine
cos(2V L/vF), while the envelope is entirely due to inter-
actions. Thus, common feature of the two effects is that
F), where V is the source-
envelopes survive at “high” temperatures when Fabry-
Perot oscillations vanish.
Concluding remarks. Adding a second parallel nanotube
to a given one leads [18] to a reduction of α in G(V ) by
a factor of 2. One could expect that for a finite crossing
angle, the effect of the second nanotube is weaker. We
found, however, that G(V ) depends on θ in a nonanalyt-
ical fashion when θ → 0. This nonanalyticity translates
into a peculiar bias dependence of G, as shown in Fig.
2. Thus, for crossed nanotubes, G(V ) is extremely sensi-
tive to the value of intratube Luttinger liquid parameter,
g. In armchair nanotubes, the currently accepted value
[1, 2, 3] is in the range 0.19 ÷ 0.26. We emphasize that
changing g from 0.19 to 0.26 leads to the increase of the
interaction parameter, λ, by a factor of 2, which would
have a drastic effect on the shape of envelope in δG(V ),
Fig. 2.
Concerning relation between our study and earlier
studies [5, 13] of crossed nanotube junctions, this rela-
tion is exactly the relation between plasmon and elec-
tron scattering. In the above papers anomalies were due
to either direct passage of electrons through the crossing
point [13] or due to crossing-induced electron backscatter-
ing [5]. Scattering of plasmons was disregarded in Ref. 5.
This is justified for perpendicular nanotubes of Ref. 5.
As shown in our manuscript, scattering of plasmons be-
comes important at small angles.
The work was supported by the Petroleum Re-
search Fund (grant 43966-AC10), DOE (grant DE-FG02-
06ER46313) and by the Research Corporation (JMG).
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