Physica E 11 (2001) 323–331
www.elsevier.com/locate/physe
Near-eld imaging of pyramid-like nanoparticles at a surface
V. Lozovskia; ∗ , Yu. Nazaroka , S.I. Bozhevolnyib
a Institute
of Semiconductor Physics, National Academy of Sciences of Ukraine, Nauki pr. 45, 03650 Kyiv 03028, Ukraine
of Physics, University of Aalborg, Pontoppidanstraede 103, Aalborg Ost, DK9220, Denmark
b Institute
Received 8 March 2001; received in revised form 21 May 2001; accepted 23 May 2001
Abstract
An exact analytical solution of the self-consistent equation for the local eld is used to calculate the near-eld optical
images of pyramid-like nano-objects placed at a surface of a solid. The diagram method developed previously for near-eld
image formation is generalized in order to describe layered objects, which are treated as many-body systems. The near-eld
optical images of triangular and square pyramids are calculated for the illumination conguration as well as those of triangular
and square prisms. It is found that the near-eld images of nanoparticles having the dielectric constant close to that of the
c 2001 Elsevier Science B.V. All
substrate change rapidly and in a complicated manner with the probe–sample distance. rights reserved.
PACS: 07.79.Fc; 42.30.Va
Keywords: Near-eld imaging; Self-consistent eld; Pyramid-like object; Nanoparticles
1. Introduction
Quasi-zero-dimension objects (nanoparticles) at
a surface of a solid have been intensively studied during the last two decades [1]. Pyramid-like
nanoparticles obtained by di=erent techniques have
been also found promising for realization of photonic band gap systems [2]. There are a number of
techniques to fabricate the pyramid-like nanoparticles. One can employ etching of a surface of a solid.
Other well-known techniques are molecular beam
∗ Corresponding author. Tel.=fax: +380(44)265-5530.
E-mail address: [email protected] (V. Lozovski).
epitaxy [3] and metal-organic chemical vapor deposition [4]. Besides, the methods of direct fabrication
of nanoparticles by growth on microscopical ordered
facet surfaces [5,6] are widely used.
The electronic properties of pyramid-like quantum dots have been recently calculated [7]. Since
pyramid-like nano-objects can be used in photonic
band gap surface systems, it is of interest to study such
systems with scanning near-eld optical microscopy
[8]. This brings about the problem of identication
of (pyramid) objects placed at the surface from the
respective near-eld images (NFIs). There are a number of papers reporting the calculations of NFIs in
di=erent congurations of scanning near-eld optical
c 2001 Elsevier Science B.V. All rights reserved.
1386-9477/01/$ - see front matter PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 1 6 6 - 7
324
V. Lozovski et al. / Physica E 11 (2001) 323–331
microscopy (SNOM) [8–13]. We have recently developed a new approach that allows one to rigorously
calculate the NFIs in the illumination SNOM conguration [14 –16]. In particular, it was shown that
the polarization-resolved SNOM imaging provides an
additional information about the geometrical shape
of an object. However, in our previous work, only
the simplest shape of the objects, i.e., rectangular
parallelepiped, has been considered. In this work, the
developed approach is generalized to describe layered
objects and used to calculate the NFIs of pyramid-like
nano-objects situated at the surface of a solid.
2. The main equations
Pyramid-like nano-objects can be obtained by
multi-step etching of the surface of a solid. As a result, a step pyramid is produced at the surface. To
calculate the NFI of such an object, one has to rst
determine the self-consistent local eld inside the
object, i.e., inside the step pyramid. Let us consider
the pyramid as a set of rectangular parallelepipeds
(or triangular prisms) situated one on another
(Fig. 1). In this case, the Lippman–Schwinger equation for the self-consistent eld [14] can be written in
the following form:
N
(I )
Ei (R̃) = Ei (R̃) − i!0
d R̃ Rij (R̃; R̃ )
s=1
(s)
El (R̃ );
×jl
Vs
(1)
where
Rij (R̃; R̃ ) = Gij (R̃; R̃ ) + Gil (R̃; R̃p ) · Glj (R̃p ; R̃ ); (2)
is the generalized photon propagator which describes
both direct light scattering [the rst term on the
right-hand side of Eq. (2)] and indirect scattering of
light by a small (point-like) probe. In Eqs. (1) and
(2), the following notations are used:
Gij (R̃; R̃p ) = − i!0
d R̃ Gil (R̃; R̃ )li(p) ;
(3)
Vp
where the integration is performed over the probe
volume, R̃p is the coordinate of the probe center,
Gij (R̃; R̃ ; !) is the photon propagator describing the
electrodynamical properties of the medium in which
the object and probe are situated, ij(p) is the electrical
susceptibility of the probe, ij(s) is the susceptibility of
sth part of the object, N is the number of layered parts
composing the object. In Eq. (1), the illumination
eld is introduced according to
Ei(I ) (R̃) = Iij (R̃; R̃ )Ej(0) (R̃ );
(4)
where the illumination operator Iij (R̃; R̃ ) is used. It is
seen that this operator connects the long-range external
eld with the eld acting directly on the object. In
the general case, the illumination operator is of an
integral nature. Since, in this work, we consider the
source of illumination to be a suIciently small probe
(linear dimensions of the probe are much smaller than
the wavelength of the external long-range eld),
Fig. 1. Schematic representation of the SNOM illumination mode with (a) square pyramid and (b) triangular pyramid.
V. Lozovski et al. / Physica E 11 (2001) 323–331
the illumination operator can be approximated as
follows:
Iij (R̃; R̃p )Ej(0) (R̃) = Gij (R̃; R̃p ) · Ej(0) (R̃p ):
(5)
Similar to what has been done previously [14 –16], we
shall use the iteration procedure to obtain the exact
solution of the self-consistent equation [Eq. (1)] by
taking into account all terms of an innite iteration
set. Introducing the nonlocal scattering operator
N
(s) til (R̃; R̃ ) = − i!0
d R̃ Rij (R̃; R̃ )jl
(R̃ ) : : : ;
s=1
Vs
(6)
the iteration set can be written in the form
↔
Ẽ(R̃) = Ẽ (I ) (R̃) + [ t (R̃; R̃ )
↔
↔
+ t (R̃; R̃ ) t (R̃ ; R̃ ) + · · · ]Ẽ (I ) (R̃ ):
(7)
Furthermore, by using the “dressed” nonlocal scattering operator
N
(s) Til (R̃; R̃ ) = − i!0
d R̃ gij (R̃; R̃ )jl
(R̃ ) : : :
s=1
Vs
(8)
which is determined via an unknown function
gij (R̃; R̃ ), the iteration set [Eq. (7)] is reduced to the
simple equation
Ei (R̃) = Ei(I ) (R̃) + Tij (R̃; R̃ )Ej(I ) (R̃ ):
(9)
It is seen from Eq. (7) that the “dressed” operator
[Eq. (8)] must satisfy the following equation:
Tij (R̃; R̃ ) = tij (R̃; R̃ ) + Til (R̃; R̃ )tlj (R̃ ; R̃ ):
(10)
To obtain the unknown function gij (R̃; R̃ ) one
should act with the operators of Eq. (10) on an
arbitrary eld. Since one has to nd the e=ective response of an object on the non-long-range incoming
eld, one should consider the eld in the form of Eq.
(4) with the (arbitrary) external eld containing all
spatial harmonics characterized by wave vectors K̃a
Ei(I ) (R̃) = Iij (R̃; R̃p )Ej(A) eiK̃a R̃p ;
(11)
a
where Ej(A) is the amplitude of the spatial harmonic
with K̃a . Applying the operators in Eq. (10) to the
325
above eld results in the equation
N
−1
(s) d R̃ {gij (R̃; R̃ ) · jm
(R̃ ) · Q̃lm (R̃ )
s=1
Vs
a
(s)
−Rij (R̃; R̃ ) · jk
· Ikl (R̃ ; R̃p )}El(A) eiK̃a R̃p = 0; (12)
where
−1
Q̃lm (R̃ ) = Iml (R̃ ; R̃p ) + i!0
N
s=1
Vs
d R̃
(s)
×Rmn (R̃ ; R̃ )nk
Ikl (R̃ ; R̃p ):
(13)
Since the eld spatial harmonics are linearly independent, the expression in the brackets of Eq. (12) must
be zero to satisfy this equation. For each spatial harmonic characterized by wave vector K̃a ; one has then
the condition {: : :}K̃a = 0. In this way, the unknown
function gij (R̃; R̃ ) can be found, namely
(s) gij (R̃; R̃ )jm
(R̃ )
(s) = Rij (R̃; R̃ )jk
(R̃ )Ikl (R̃ ; R̃p )Q̃lm (R̃ ):
(14)
Since
Ikl (R̃ ; R̃p )Q̃lm (R̃ ) = Qkm (R̃ )
(15)
with
Qlm (R̃ )
(s)
= ml + i!0
d R̃ Rmn (R̃ ; R̃ )nk
s
Vs
−1
−1 ×Ikp (R̃ ; R̃p )Ipl
(R̃ ; R̃p )
;
(16)
tensor Qlm (R̃) does not depend on the wave vector K̃a .
Therefore, the e=ective susceptibility
(s)
Xjm (R̃) = jl
Qlm (R̃)
(17)
does not depend on the wave vector as well. This
means that the e=ective susceptibility expressed by
Eq. (17) can be used for any spatial harmonic of the
external eld and, thereby, for an arbitrary external
eld. One can thus assert that this susceptibility [Eq.
(17)] represents a linear response of the object on the
illumination eld [Eq. (4)]. Finally, using Eqs. (9)
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V. Lozovski et al. / Physica E 11 (2001) 323–331
and (14) one obtains
(K̃)
Ei (R̃) = Iim
(R̃; R̃p )
−i!0
N
s=1
Vs
s
d R̃ Rij (R̃; R̃ )jk
(R̃ )
(K̃) ×Qkl (R̃ )Ilm
(R̃ ; R̃p ) · Em(0) (R̃):
(18)
If the external eld contains only planar (x- and y-)
components, the nal equation [Eq. (18)] can be
rewritten in the form
Ei (R̃)=E (0) = Lix (R̃) cos ! + Liy (R̃) sin !;
(19)
with
(K̃)
Lim (R̃) = Iim
(R̃; R̃p ) − i!0
×
N
s=1
Vs
s
d R̃ Rij (R̃; R̃ )jk
(R̃ )
(K̃) ×Qkl (R̃ )Ilm
(R̃ ; R̃p )
(20)
being the local-led factor of the self-consistent problem and ! being the angle between the polarization of
the external eld and the x-axis of the coordinate system. Eq. (20) allows one to perform the calculations
in terms of the normalized arbitrary (Ei (R̃)=Ej(0) (R̃))
eld and to simulate the NFIs for di=erent polarization
angles !.
3. Modeling of illumination SNOM and numerical
calculations
In the illumination SNOM conguration considered
in our calculations, a sample is illuminated with radiation from an uncoated ber tip moved along the
scanning plane (Fig. 1). The optical signal is detected
in transmission by collecting the scattered light with
an objective. A small probe scatterer is considered as
a source of illumination of a pyramid-like mesoscopic
object. The detected optical signal is assumed to be
proportional to the intensity of the self-consistent eld
at the site of a remote detector located at the z-axis.
The detected intensity is calculated as a function of
the scanning probe coordinates, resulting thereby in
the NFI.
The developed model allows us to calculate the
NFIs of various nano-structures [15]. As an illustration, we consider a pyramid with a square basis of
120×120 nm2 ; and a triangular pyramid with the dimensions of 120 and 60 nm along the x- and y-axis,
respectively. Both pyramids are characterized by the
height h of 60 nm along the z-axis. For comparison,
the prisms with square and triangular basis are also
considered. The bases of the prisms are of the same
dimensions as those of the pyramids, but the height
of the prisms is 15 nm. The probe radius chosen
was much less than the wavelength of long-range
monochromatic external eld r0 %; namely r0 = 3 nm
at the wavelength % = 600 nm. The scanning planes
are considered to be parallel to the (x,y)-plane:
z1 = 102 nm (0:17%); z2 = 90 nm (0:15%); z3 = 78 nm
(0:13%); and z4 = 66 nm (0:11%) when calculating the NFIs of the pyramids. The following set
of zp -distances: z1 = 57 nm (0:095%); z2 = 45 nm
(0:075%); z3 = 33 nm (0:055%);
and
z4 = 21 nm
(0:035%) is used to calculate the NFIs of the prisms.
Taking into account the range of the distances involved, the photon propagator can be approximated in
numerical calculations by its near-eld contribution
[17,18], resulting in the following expression for the
direct part of the photon propagator:
2
↔(d)
c ↔
1
3c2
(21)
U − 2 3 ẽ R ẽ R ;
G (r̃;r̃ ; !) =
4& !2 R3
! R
↔
with U unit dyadic, R = |R̃ − R̃ | and ẽ R = R̃=R. The
indirect part of the photon propagator in this approximation is given by
↔(i)
↔(d)
G (R̃; R̃ ; !) = G
where
(R̃; R̃M ; !) · M̃ (!);

−1 0
+(!) − 1 
0 −1
(!)
=
M
+(!) + 1
0 0
↔

0
0;
1
(22)
(23)
R̃M = (x ; y ; −z ); and +(!) is the dielectric constant
of the substrate. It is clear from Eqs. (21) – (23) that,
in the near-eld approximation, the photon propagator consists only of a real part provided that the
substrate dielectric constant is real. This circumstance
signicantly simplied the numerical simulations
and allowed us to calculate some integrals analytically. The dielectric constants of the probe, objects
and substrate were chosen as follows (% = 600 nm) :
V. Lozovski et al. / Physica E 11 (2001) 323–331
327
+pr = 2:25 (glass); +obj = 13:2 (InAs), and +subs = 13:7
(GaAs).
The intensity of scattered light which is produced
by the currents generated inside the object by the local
(self-consistent) eld was calculated in the far zone
(as a function of the scanning coordinates) resulting
in the NFI of the object:
(s) 2
↔
J = −
(24)
G F-F (R̃; R̃ )Ẽ(R̃ ) ;
4& Vs
↔
where G F-F (R̃; R̃ ) is the far-eld part of Green dyadic,
and (s) =4& = +obj − 1. It should be noted that, since
the distance Dr between the object and a detector is
considerably larger than characteristic linear dimensions of the object a (Dr ∼ 107 nm; a ∼ 102 nm);
one can safely assume that only diagonal components
↔
of G F-F (R̃; R̃0 ) contribute to the intensity in Eq. (24).
This means that within the order of (a=Dr )2 ∼ 10−10
the detected intensity is given by
2 2
1 ↔
(s)
J = − G F-F (R̃; R̃0 ) d R̃ Ẽ(R̃ ) ;
4&
Vs
(25)
where R0 is the coordinate of the center of the object.
Taking into account that the distance Dr = |R̃ − R̃0 |
is constant during the scanning, the intensity of
i-polarized light, which forms the NFI for the jpolarized external eld, can be nally written down as
2
N(ij) (R̃p ) = d R̃ Lij (R̃ ) ;
(26)
Vs
where
N(ij) =
|(+obj −
J(i) (R̃p )
:
F
1)Gii F (R̃; R̃0 )|2 |Ej(0) (R̃p )|2
(27)
The calculations in the present work are based on
↔
Eq. (26), in which L (R̃) is the local-eld factor calculated in accordance with Eq. (20). In Eqs. (26) and
(27), the tensor indexes are introduced emphasizing
the fact that the scattered eld of di=erent polarizations can be recorded with a detector for a given
polarization of the external eld.
The results of numerical simulations of the NFIs for
triangular and square InAs pyramids and prisms situated at the plane surface of GaAs substrate are shown
in Figs. 2–7. The probe coordinates in the scanning
plane are given in arbitrary units—dimension=%. The
Fig. 2. Near-eld images of the triangular pyramid for the
y-polarized external eld. The scanning plane is located at
distances (a) zp = 0:17%; (b) zp = 0:15%; (c) zp = 0:13%; (d)
zp = 0:11%.
polarization of the external eld was oriented either along the x- or the y-axis. The cross-polarized
(yx- and xy-) components of the normalized far-eld
328
V. Lozovski et al. / Physica E 11 (2001) 323–331
Fig. 4. Near-eld images of the square pyramid for the x-polarized
external eld. The scanning plane is located at distances (a)
zp = 0:17%; (b) zp = 0:11%.
Fig. 3. Near-eld images of the triangular pyramid for the
x-polarized external eld. The scanning plane is located at
distances (a) zp = 0:17%; (b) zp = 0:15%; (c) zp = 0:13%; (d)
zp = 0:11%.
intensity N(ij) have been found to be considerably weaker (by two orders of magnitude) than the
co-polarized components N(ii) . For this reason, only
N(xx) values were calculated for the square pyramids
and prisms, and both N(xx) and N(yy) values were calculated for the triangular pyramids and prisms. The
bases of the latter were isosceles triangles with their
base side being parallel to the x-axis. The NFIs of
the triangular pyramids calculated for the di=erent
distances zp between the sample surface and the scanning plane are shown in Figs. 2 [N(yy) ] and 3 [N(xx) ].
The characteristic behavior of the NFI of the triangular pyramids as a function of the distance zp can be
explained by the interference of strong elds localized at vertexes and edges of elementary prisms. Note
that the NFIs for both polarizations are qualitatively
similar to each other at small distances zp (6 0:11%):
there are two bright spots at Pank facets of the pyramid (cf. Figs. 2d and 3d). On the other hand, the NFI
for the x-polarization of the external eld at a large
V. Lozovski et al. / Physica E 11 (2001) 323–331
Fig. 5. Near-eld images of the triangular prism for the y-polarized
external eld. The scanning plane is located at distances (a)
zp = 0:095%; (b) zp = 0:075%; (c) zp = 0:055%; (d) zp = 0:035%.
distance zp is analogous to that for the y-polarization
◦
with rotation of the latter by 90 (cf. Figs. 2a and
3a). But there appear signicant di=erences in the
NFIs for the x- and y-polarized illumination at intermediate values of distances zp (≈ 0:15–0:13%).
329
Fig. 6. Near-eld images of the triangular prism for the x-polarized
external eld. The scanning plane is located at distances (a)
zp = 0:095%; (b) zp = 0:075%; (c) zp = 0:055%; (d) zp = 0:035%.
Indeed, the simple “key hole” type of N(xx) changes
to the elaborate “bumblebee’s head” type of N(yy) .
First of all it is clear that, at large distances, the probe
interacts with the object as a whole and the resulted
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V. Lozovski et al. / Physica E 11 (2001) 323–331
The NFI image calculated at a small distance rePects
high symmetry of the object exhibiting a strong local
eld enhancement at the top of the pyramid (Fig. 4b).
The NFIs for thin triangular and square prisms
(Figs. 5 –7) furnish further insight into the underlying
physics of image formation, especially for the triangular pyramids, at intermediate distances zp . Note that
the edge enhancement is especially pronounced for
the edges that are perpendicular to the polarization of
the incoming eld (Figs. 5c and 7b) a circumstance
that has been established previously [16,19]. The edge
enhancement is though somewhat suppressed in the
NFI of the pyramid due to the interference of the elds
generated by di=erent layers (elementary prisms)
which form the pyramid, it should be emphasized that
this suppression is more distinct if the dielectric constants of the object and the substrate are close. It has
been found in the course of our simulations that, in
the case of notably di=erent dielectric constants, the
suppression e=ect is rather weak. Finally, we would
like to note that similar NFIs of square prisms have
been presented and discussed elsewhere [15,16,19].
4. Conclusions
Fig. 7. Near-eld images of the square prism for the x-polarized
external eld. The scanning plane is located at distances (a)
zp = 0:095%; (b) zp = 0:035%.
NFI should be similar to that of a point-like object,
implying that a similar NFI is expected for the square
pyramid. At intermediate distances, the interference
between local elds at di=erent vertexes and edges
becomes pronounced leading to an extremely complicated spatial structure of the total eld. At small
values of zp ; strong local elds at the vertexes of elementary prisms start to dominate over those at prism
edges (Figs. 5d and 6d) forming a very specic interference picture (which is determined by the object
geometry) and leading to a simple (and stable) form
of the NFI—two bright spots.
The NFI simulated for the square pyramid at a
large distance was indeed found to be rather similar
to that of the triangular pyramid (cf. Figs. 3a and 4a).
The near-eld imaging of variously shaped objects
situated on the surface of a solid has been considered. The exact analytical solution of the Lippmann–
Shwinger equation developed previously [15,16] has
been extended to include objects consisting of many
parts. This fact allowed us to consider the objects
having a more complicated form than in our previous
work. The theoretical results obtained have been used
for numerical calculations of the NFIs of di=erent
nano-objects placed at the surface of a solid, viz., triangular square pyramids and prisms. We have found
that the near-eld images of nanoparticles having a
dielectric constant close to that of the substrate change
quite rapidly and in a complicated manner with the
probe-sample distance. Finally, we would like to stress
that, in the framework of the proposed approach, the
numerical calculations have been signicantly simplied and reduced to a tabulation of the NFIs based
on the exact solution (for the self-consistent eld)
obtained analytically. Thus, the present approach has
certain advantages as compared with the widely used
method based on discretization of the Lippmann–
V. Lozovski et al. / Physica E 11 (2001) 323–331
Schwinger equation [9,10], whose usage requires
quite cumbersome numerical calculations at the stage
of determination of the self-consistent eld.
Acknowledgements
The authors (VL and SIB) gratefully acknowledge
nancial support from the Danish Natural Science
Research Council under Contract No. 9903131.
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