van der Waals interaction between an atom and a metallic nanowire
M. Boustimi,
1
J. Baudon,
2,
* P. Candori,
1
and J. Robert
2
1
Dipartimento di Chimica dell’Universita `, 06123-Perugia, Italy
2
Laboratoire de Physique des Lasers (UMR-CNRS 7538), Universite ´ Paris 13, Avenue Jean-Baptiste Cle ´ment, 93430-Villetaneuse, France
Received 6 September 2001; published 22 March 2002
The nonretarded linear-response potential of a metallic cylindrical nanowire in the vicinity at which an atom,
assimilated to a fluctuating dipole, is placed is determined using i a complete orthogonal basis set over which
the potential inside the solid is expanded and ii the boundary conditions at the solid surface. Exact analytical
expressions of the reflection factors are obtained. The dipole-dipole van der Waals energy is then derived by
use of the propagator method. This energy is numerically calculated for an argon atom in the vicinity of an
aluminum wire. At short distance a repulsive potential, calculated by summing terms in r
-12
over the solid
lattice, is added to the previous one. The collision at thermal energies of Ar atoms with the Al wire is then
studied. The potential well makes a rainbow effect appear at angles easily accessible experimentally.
DOI: 10.1103/PhysRevB.65.155402 PACS numbers: 68.65.-k, 34.50.Dy
I. INTRODUCTION
For some years micro- and nanostructured solids of vari-
ous kinds microspheres,
1
tapered optical fibers,
2
metallic
nanowires,
3
carbon nanotubes,
4
and porous materials
5
have
become fabricated. A great deal of effort has been recently
devoted to the theoretical study of such solids interacting
with various external fields. While it is based upon the same
general principles as those used for macroscopic solids, the
interaction between a solid of a nanoscopic size and an atom,
a molecule, or an external electromagnetic field gives rise to
some difficulties precisely because of the small size along
one, two, or three dimensions. In particular the nonvalidity
of standard macroscopic Maxwell equations results in the
introduction of so-called additional boundary conditions.
6
As
a consequence the interaction has special characteristics,
among which are the enhancement of curvature and nonlo-
cality effects
7
and the manifestation in the optical properties
of surface and bulk resonant modes.
8
Important applications
can be found in near-field microscopy techniques, e.g., those
using metallized optical fibers.
9
Moreover, when deposited
on a substrate, these microstructures are the basic elements in
micro- nano- electronics and optronics. Two- or three- di-
mension periodically structured solids, scattering electro-
magnetic waves in the near-field regime, have been widely
studied.
10,11
They are known to behave as ‘‘photonic band-
gap’’ media, providing us with new types of optical guiding
devices. On another hand, nanosolids randomly distributed in
a homogeneous isotropic medium are able to produce a weak
localization of light, accompanied by a coherent backscatter-
ing effect,
12
or even a strong localization effect.
13
Most of
the experimental and theoretical studies in this domain deal
with optical waves, generally in the infrared range. Never-
theless, owing to both recent progress towards the reduction
of the solid sizes, and relatively large de Broglie wavelengths
made accessible by atom cooling techniques, most of these
applications should be readily transposed to atoms or mol-
ecules. Some of them have been already realized at a mi-
crometer scale.
14
Such atomic or molecular counterparts are
expected to give important advantages such as a high sensi-
tivity of interferometric devices, a sensitivity to gravity and
more generally to linear accelerations, and, above all, a wide
range of possibilities offered by the internal degrees of free-
dom interaction with external static fields and light, and
inelastic electronic or vibrational transition effects induced
by the solid
15,16
.
The first goal of this paper is to give a general and rigor-
ous treatment of the interaction of the van der Waals type
between an atom and a metallic or dielectric nanowire, at
intermediate distances a few atomic units a.u. up to a few
hundred a.u. where no retardation effect occurs. Actually, as
the atom is considered here as a fluctuating multipolar
source, the present treatment can be easily extended to polar
or nonpolar molecules as well as to electromagnetic waves,
provided that the wavelength is large compared to the solid
size. The first step in this treatment is the determination of
the linear- response potential of the solid, using expansions
over eigenmodes adapted to the geometry of the problem
Sec. II. Then the dispersion energy is determined using the
well-known and powerful ‘‘propagator’’ method
17,18
Sec.
III. Section IV is devoted to the second goal of the paper,
namely, the study of the collision, in the thermal energy
range, of an atom with a nanowire. For this application a
short-range repulsive potential is added to the previous one,
making a potential well appear in the vicinity of the solid. As
will be seen in the example of an argon atom impinging on
an aluminum wire, this well gives rise to a rainbow effect at
an angle easily accessible in an experiment. Conclusions and
perspectives are given in Sec. V.
II. RESPONSE POTENTIAL OF A METALLIC NANOWIRE
The metallic wire is assumed to be an infinitely long cyl-
inder of radius a, parallel to the z axis see Fig. 1. Owing to
the symmetry and for the sake of simplicity the atom placed
on the x axis at a distance R from the cylinder axis will be
considered as a fluctuating dipole moment. This is the main
contribution to the interaction, but higher moments could be
considered as well. The ‘‘source’’ potential
s
( r, ) pro-
duced by the atom obeys the Laplace equation. It can be
expanded as
19
PHYSICAL REVIEW B, VOLUME 65, 155402
0163-1829/2002/6515/1554026/$20.00 ©2002 The American Physical Society 65 155402-1