Abstract-In this paper we compare the optical absorption for nanospheres made from a range of transition and alkali metals from Li (A=3) to Au (A=79). Numerical solutions to Mie theory were used to calculate the absorption efficiency, Q abs , for nanospheres varying in radii between 5 nm and 100 nm in vacuum. We show that, although gold is the most commonly used nanoparticle material, its absorption efficiency at the plasmon resonance is not as strong as materials such as the alkali metals. Of all the materials tried, potassium spheres with a radius of 21 nm have an optimum absorption efficiency of 14.7. In addition we also show that, unlike gold, the wavelength of the plasmon peak in other materials is sensitive to the sphere radius. In potassium the peak position shifts by 100 nm for spheres ranging from 5 nm to 65 nm, the shift is less than 10 nm for gold spheres

Martin G Blaber

Michael B Cortie

Michael J Ford

Nadine Harris

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Optimisation of absorption efficiency for varying
dielectric spherical nanoparticles
Martin G Blaber, Nadine Harris, Michael J Ford*and Michael B Cortie
Institute for Nanoscale Technology, University of Technology, Sydney
PO Box 123, Broadway, NSW, 2007, Australia
*Email: Mike.Ford@uts.edu.au
Abstract—In this paper we compare the optical absorption for
nanospheres made from a range of transition and alkali metals
from Li (A=3) to Au (A=79). Numerical solutions to Mie theory
were used to calculate the absorption efficiency, Q abs,  for
nanospheres varying in radii between 5 nm and 100 nm in
vacuum. We show that, although gold is the most commonly used
nanoparticle material, its absorption efficiency at the plasmon
resonance is not as strong as materials such as the alkali metals.
Of all the materials tried, potassium spheres with a radius of 21
nm have an optimum absorption efficiency of 14.7. In addition
we also show that, unlike gold, the wavelength of the plasmon
peak in other materials is sensitive to the sphere radius. In
potassium the peak position shifts by 100 nm for spheres ranging
from 5 nm to 65 nm, the shift is less than 10 nm for gold spheres.
Keywords- nanoparticle; plasmon; absorption efficiency; Au; K
 I. INTRODUCTION
Metallic nanoparticles have potential applications in
thermal therapeutic techniques [1-4], nanolithography [5] and
energy efficient window coatings [6]. Considerable progress
has been made in recent years not only in synthesizing
nanoparticles [7-9] with a variety of geometries, but also in
modeling their optical properties [10-16]. The utility of
nanoparticles with dimensions in the range of 5 nm to 50 nm
lies in their ability to absorb incident light very efficiently with
minimal scattering. Excitation of a surface mode, or plasmon
resonance, is the underlying mechanism for such large
absorption cross-sections [17]. Moreover, the wavelength of
the plasmon resonance is sensitive to particle geometry and can
therefore be tuned.
Predominately Au has been used to make spherical
nanoparticles because the synthesis process is relatively simple
[18], Au is inert, and absorption occurs within the visible
spectrum. The peak absorption efficiency, Qabs, for Au
nanospheres with radius of 10 nm is 0.66 and occurs at a
wavelength of 506 nm in vacuum. For gold the resonance
position is relatively insensitive to sphere size over the size
range of 5 nm to about 50 nm radius. For this reason Au
nanoshells and nanorods have received considerable attention
as they have much larger absorption efficiencies and the peak
position can be tuned by varying the shell or rod aspect ratio
[19]. For example a 25 nm radius, 0.9 aspect ratio nanoshell is
capable of producing a Qabs of 19 within the near-infrared
spectrum [20]. However, synthesis of monodisperse nanorods,
and nanoshells with uniform shell thickness is difficult.
In this paper Mie’s analytical solution to Maxwell’s
equations is used to compare Qabs for nanospheres composed of
different metallic elements. For each element an optimum
nanosphere diameter producing a maximum Qabs is established
and the shift in the plasmon resonance with increased sphere
size is discussed. The results show that, in terms of absorption
efficiencies alone, gold is far from the ideal material for
nanoparticle plasmonic applications.
 II. METHOD
The numerical implementation of Mie theory [21] in the
program BHCOAT [22] was used to calculate the absorption
and scattering efficiencies, Qabs and Qsca respectively. These
quantities are the absorption and scattering per unit cross-
sectional area of the particle. Hence a value greater than 1
implies absorption (or scattering) that is larger than the
geometric area the particle presents to the incident radiation.
Particles were modeled with radii between 5 nm and 100
nm in 1 nm steps. A strong plasmon absorption occurs for
particle sizes below about 50 nm. At larger radii scattering
begins to dominate and higher order terms (beyond dipole)
contribute. Tabulated values for the dielectric function in
vacuum were taken from Weaver and Frederikse [23].
Calculations were performed between wavelengths of 100 nm
and 1000 nm in 1 nm steps. This range encompasses the
plasmon resonance and consequently the peak Qabs for the
various dielectric nanospheres.
The main numerical consideration is that calculated cross-
section is sufficiently converged, that is, sufficient terms have
been retained in the multipole expansion of Mie theory. For the
particle size range and wavelengths considered here only the
first few terms of the expansion are required to achieve this.
Since Mie theory is the exact solution to the problem for any
particle size and wavelength, we can have confidence in our
numerical results.
 III. RESULTS AND DISCUSSION
Table 1 gives the calculated Qabs for each metal at the
maximum of the plasmon resonance, together with sphere
radius and wavelength corresponding to this maximum.
TABLE I.  COMPARISON OF CALCUALTED OPTIMUM ABSORPTION
EFFICENCES FOR VARIOUS METALLIC SPHERES
Atomic
Number Element Qabs
Sphere
Radius
(nm)
Wavelength
(nm)
3 Li 4.8 30 401
11 Na 12.4 16 387
13 Al 13.0 6 143
19 K 14.7 21 555
22 Ti 2.4 36 315
23 V 2.8 32 309
24 Cr 2.7 18 183
25 Mn 1.9 28 203
26 Fe 2.3 22 194
27 Co 2.4 22 200
28 Ni 2.4 23 208
29 Cu 2.3 23 204
41 Nb 2.4 17 153
42 Mo 3.0 14 150
44 Ru 2.9 16 169
45 Rh 3.0 18 187
46 Pd 2.7 22 217
47 Ag 5.8 23 361
72 Hf 2.1 32 256
73 Ta 2.3 19 162
74 W 2.5 17 151
75 Re 2.4 17 152
76 Os 2.6 18 164
77 Ir 2.5 22 202
78 Pt 2.1 31 247
79 Au 3.3 49 516
This data has been extracted from complete calculations of
the optical spectrum for each metallic nanoparticle over the
range of 100 nm to 1000 nm wavelengths. A representative
subset of the optical spectra is shown in Fig. 2. In the visible
spectrum K and Na are excellent absorbers, while Al has the
best absorption in the UV. Cu is a poor absorber while Ag, Li
and Au are average. Fig. 2 shows that the plasmon resonance in
K, Na and Al is very sharp, while in Cu it is broad and ill-
defined. Au falls somewhere between these two extremes.
Comparing K and Au nanospheres, the optimum particle sizes
from table 1 are 21 nm and 49 nm, respectively. The difference
in optimum size influences the proportion of absorption to
scattering. Larger particles as a general rule tend to scatter
more. This can be seen in Fig. 2 with the K particle absorbing
around 5 times the amount it scatters, whereas the Au particle
absorbs around 1.7 times the amount it scatters.
Figure 1.  Complete absorption spectra for a small number of nanoparticles in
table 1.
The wavelength position, size and width of the plasmon
absorption peak is often described by (1) [17],
€ 
σ ext (λ) = 12
πr 3εm
3/ 2
λ
ε2 (λ)
[ε1 (λ) + 2εm ]
2 +ε2 (λ)
2 (1)
where σext is the extinction cross section resulting for dipole
absorption only, εm is the dielectric function of the surrounding
medium (assumed constant and non-absorbing), r is the
nanosphere radius, and ε(λ) is the dielectric function of the
nanosphere. In vacuum, according to (1) the peak occurs
approximately where the real part of the dielectric function of
the sphere is –2 and is independent of size. Strictly, (1) is true
only in the quasistatic limit, i.e. as the size factor (particle
radius divided by wavelength) approaches zero. As we shall
show this is only a reasonable approximation in some cases.
Materials with a small imaginary component at the plasmon
resonance will have a large, narrow absorption peak, for
example K (ε = -2.14 + 0.147i), Al (ε = -2.20 + 0.179i) and Na
(ε = -2.15 + 0.191i). Conversely materials with large imaginary
components at the plasmon resonance will have small, broad
absorption peaks, for example Cu (ε = -1.37 + 3.17i).
The plasmon peak position as a function of particle size for
K and Au spheres from 5 nm to 65 nm is shown in Fig. 3. For
Au particles the plasmon position, as described by (1) is quite
size independent and occurs where ε’ = –2. However, for K the
peak shifts about 100 nm over the same range of particle sizes.
Closer inspection of our calculations shows that contributions
from higher order terms in the multipole expansion are not the
cause of this shift, the dipole term dominates across this size
range. Furthermore, this phenomenon is not unique to K with
the position of the plasmon resonances of most other metallic
nanospheres experiencing a significantly greater dependence
on sphere radius than Au.
Figure 2.  Qabs and Qsca for a 21 nm K and 49 nm Au nanoparticle.
The result that the plasmon resonance occurs at ε = -2
(relative to vacuum) clearly does apply to K, and indeed to
many of the other metallic nanoparticles investigated here. This
is not surprising given that almost all metallic nanoparticles
have a plasmon resonance at a wavelength of order 100 nm
with particles sizes of order 10 nm. Hence, it is questionable,
even for gold, whether the quasistatic limit, and (1), can be
applied to real metallic nanoparticles.
From Mie theory it is possible to derive a correction term
that gives the shift of the plasmon resonance with particle size.
The value of the real part of the dielectric function, ε’, at which
the plasmon peak now occurs is given by (2).
2
2224
5
122
λ
επ
εε mm
r
−−=′ (2)
where λ is the wavelength, r  the sphere radius, and εm is
dielectric of the surrounding medium (assumed constant and
non-absorbing). The results of (2) for Au and K are also plotted
in figure 3. They agree well with the full Mie calculation.
Taking the derivative of (2) and removing constants gives the
following proportionality between the shift in the plasmon
resonance position, λp, with particle size, r, and gradient of the
dielectric function,
€ 
δλ p
δr
∝−
δλ p
δ ′ ε 
= −1/ δ ′ ε 
δλ p
(3) 
In other words (3) demonstrates that the amount the
resonance wavelength shifts for a given change in particle size
is inversely proportional to the gradient of the real part of
dielectric function around the value –2. Moreover, if the
gradient is negative, as it invariably is for Drude-type metals,
the resonance shifts to longer wavelengths as the particle size
increases.
Figure 3.  Plasmon peak shift for Au and K with increasing radius
Note, that gold is not very Drude-like below about 550 nm
due to interband transitions and the plasmon resonance actually
shifts, albeit slightly, in both directions as particle size
increases.
The real part of the dielectric function for gold is quite
steep around the value –2, and has a gradient of about –65 µm-1
in this region, whereas potassium is relatively flat with a
gradient of only –10 µm-1. It is for this reason that the plasmon
peak is relatively size insensitive in gold spheres of about 5 nm
to 50 nm radius, and not because the quasistatic approximation
is valid in this size region. This is quite fortuitous, and for most
other metals the gradient is sufficiently small that the peak
shifts considerably over the same size range, deviations from
the quasistatic limit are then more apparent.
 IV. CONCLUSION
We have used Mie theory to calculate the optical response
of nanosphere of varying radii and composed of a range of
metallic elements. The results show that for the different metals
there is an optimum radius that maximizes the absorption
efficiency, Qabs. Potassium and sodium nanospheres produce
the greatest Qabs within the visible spectrum, achieving
magnitudes of 14.7 and 12.4, respectively. These results far
exceed those for Au particles, where the maximum value of
Qabs is only 3.3.
In addition, if a large Qabs and minimal Qsca within the
visible region are desired then we have shown that K and other
metallic nanospheres offer better optical responses than Au
nanospheres. For example, potassium particles absorb five
times the amount they scatter at their optimum size of 21 nm.
Because the optimum Au particle size is larger, 49 nm, they
scatter relatively more, the ratio of Qabs to Qsca in this case is
1.7.
The calculations also demonstrate that gold nanospheres are
rather unique in having a plasmon resonance that is relatively
size independent in the size range of 5 nm to about 50 nm. The
reason for this is not so much that the quasistatic
approximation applies across this range of particle sizes, but
that the shift in peak position is small. For K nanospheres, by
contrast the peak shift is about 50 nm and the quasistatic
approximation is clearly not valid. We have shown that the
underlying reason for this behaviour is the gradient of the
dielectric function, with larger gradients as for example in gold
giving smaller peak shifts.
While the synthesis of Au nanoparticles is relatively
simple, the synthesis of K nanoparticles is more problematic
due to their extreme instability. However, work has already
begun in this direction with a report by Beitia et al [24]. These
authors have investigated the effect of the substrate on the
polarisability of 18 nm radius K particles. The particles were,
however, produced by evaporation in ultra-high vacuum onto a
silicon substrate at -80oC. Whether it is feasible to synthesize K
particles by capping in some protective barrier remains to be
seen. However, the computational work reported here provides
motivation for pursuing such a course.
 V. ACKNOWLEDGEMENT
This work was supported by the Australian Research
Council, and the University of Technology, Sydney.
Computing resources were provided by the Australian Centre
for Advanced Computing and Communication (ac3) in New
South Wales and the national facility at the Australian
Partnership for Advanced Computing (APAC).
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Optimisation of absorption efficiency for varying dielectric spherical nanoparticles

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Optimisation of absorption efficiency for varying dielectric spherical nanoparticles

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