\hfill{}\\ \hfill{}\vspace*{0.5cm}\\ \sc Photonic crystals of coated metallic spheres \vspace*{0.3cm}
{\sc Alexander Moroz}\thanks{www.amolf.nl/research/photonic\_materials\_theory} \thanks{ Address after Sept. 1
1999: I. Institut f\"{u}r Theoretische Physik
Jungiusstrasse 9
Universit\"{a}t Hamburg, D-20355 Hamburg, Germany} \vspace*{0.3cm}
coateh.tex
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\begin{document}
\title{
\hfill{}\\
\hfill{}\vspace*{0.5cm}\\
\sc
Photonic crystals of coated metallic spheres
\vspace*{0.3cm}}
\author{ {\sc Alexander 
Moroz}\thanks{www.amolf.nl/research/photonic\_materials\_theory}
\thanks{
Address 
after Sept. 1, 1999:
I. Institut f\"{u}r Theoretische Physik,  Jungiusstrasse 9, 
Universit\"{a}t Hamburg, D-20355 Hamburg, Germany}
\vspace*{0.3cm}}
\date{
\protect\normalsize
%\it FOM-Instituut voor  Atoom- en Molecuulfysica, 
\it FOM Institute AMOLF, 
Kruislaan 407,  1098 SJ Amsterdam,  The Netherlands}

\maketitle

\begin{center}
{\large\sc abstract}
\end{center}
It is shown that simple face-centered-cubic (fcc) structures of 
both metallic and coated metallic spheres are ideal candidates to achieve
a tunable complete photonic bandgap (CPBG) for optical wavelengths using
currently available experimental techniques. 
For coated microspheres with the coating width to plasma 
wavelength ratio $l_c/\lambda_p \leq  10\%$
and the coating and host refractive indices $n_c$ and $n_h$, respectively,  
between $1$ and $1.47$, one can always find a
sphere radius $r_s$ such that the relative gap width $g_w$ 
(gap width to the midgap frequency ratio) is larger
than $5\%$ and, in some cases, $g_w$ can exceed $9\%$.
Using different coatings and supporting liquids, the width and 
midgap frequency of a CPBG can be tuned considerably.



\vspace*{0.6cm}

\noindent PACS numbers:  42.70.Qs - Photonic bandgap materials \hfill\\
\noindent PACS numbers:  82.70.Dd - Colloids\hfill  


\vspace*{1.9cm}




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\input colbody

I should like to thank  Dr. H. Bakker, Prof. A. van Blaaderen,  
Dr. A. Tip, and Dr. K. P. Velikov for careful reading of the manuscript 
and useful comments and M.J.A. de Dood for help with figures.
SARA computer facilities are also gratefully acknowledged.
This work is part of the research program by  the Stichting voor 
Fundamenteel Onderzoek der Materie  (Foundation for Fundamental 
Research on Matter) which  was made possible by financial support from the 
Nederlandse Organisatie voor Wetenschappelijk Onderzoek 
(Netherlands Organization for Scientific Research).


\newpage

%%%%%%%%%%%  %%%%%%%%%%%  %%%%%%%%%%%  %%%%%%%%%%%  %%%%%%%%%%% 
\begin{thebibliography}{99}

\bibitem{OG}Graydon O., OLE June 1998, p. 31.

\bibitem{Y}Yablonovitch E.,
Phys. Rev. Lett. {\bf 58} (1987) 2059.

\bibitem{HCS}Ho K. M., Chan C. T. and Soukoulis C. M.,
Phys. Rev. Lett. {\bf 65} (1990) 3152.

\bibitem{YGL}Yablonovitch E., Gmitter T. J. and Leung K. M.,
Phys. Rev. Lett. {\bf 67} (1991) 2295.

\bibitem{AMe}Moroz A., Europhys. Lett. {\bf 46} (1999) 419.

\bibitem{Tip}Glauber R. J. and Lewenstein M.,  
Phys. Rev. A {\bf 43} (1991) 467,
Eq. (6.12); Tip A., Phys. Rev. A {\bf 56} (1997) 5022;
ibid. {\bf 57} (1998) 4818.

\bibitem{WW}Weisskopf V. and Wigner E., Z. Phys. {\bf 63} (1930) 54.

\bibitem{By}Bykov V. P., Sov. J. Quant. Electron. {\bf 4} (1975) 861.

\bibitem{LZM}Lewenstein M., Zakrzewski J., Mossberg T. W. and Mostowski J.,
J. Phys. B{\bf 21} (1988) L9.

\bibitem{RSS}Russel W. B., Saville D. A. and Schowalter W. R.,
{\em Colloidal Dispersions} (Cambridge University Press, Cambridge, 1995).

\bibitem{WV}Wijnhoven J. E. J. G. and Vos W. L.,
Science {\bf 281} (1998) 802.

\bibitem{BSS}Biswas R., Sigalas M. M., Subramania G. and Ho K.-M., 
Phys. Rev. B {\bf 57} (1998) 3701.

\bibitem{MS}Moroz A. and Sommers C., 
J. Phys.: Condens. Matter {\bf 11} (1999) 997. 

\bibitem{Lev}Levi B. G., Phys. Today, January 1999, p. 17.

\bibitem{Lin}Lin S. Y. et al., Nature {\bf 394} (1998)  251.

\bibitem{AlM}Moroz A., Phys. Rev. Lett. {\bf 83} (1999) 5274.

\bibitem{BH}Bohren C. F. and Huffman D. R., {\em Absorption
and Scattering of Light by Small Particles}
(Wiley, New York, 1984), Chap. 9, 12.

\bibitem{AsM}Ashcroft N. W. and Mermin N. D., {\em Solid State Physics}
(Saunders College, 1976) pp. 73-75.

\bibitem{PHS}Pendry J. B., Holden A. J., Stewart W. J. and Youngs I.,
Phys. Rev. Lett. {\bf 76} (1996) 4773.

\bibitem{JC}Johnson P. B. and Christy R. W.,
Phys. Rev. B {\bf 6} (1972) 4370; ibid. {\bf 9} (1974) 5056.

\bibitem{PKA}Pan G., Kesavamoorthy R. and Asher S. A.,
Phys. Rev. Lett. {\bf 78} (1997) 3860;
Grier D., Phys. World, July 1997, p. 24.

\bibitem{KML}Leung K. M., Phys. Rev. A {\bf 33} (1986) 2461;
Chemla D. S. and Miller D. A., Opt. Lett. {\bf 11} (1986) 522.

\bibitem{ULM}Ung T., Liz-Marzan L. M. and Mulvaney P., 
Langmuir {\bf 14} (1998) 3740.

\bibitem{Mo}Moroz A., 
Phys. Rev. B {\bf 51} (1995) 2068.

\bibitem{KKR}Korringa J.,
Physica {\bf 13} (1947) 392; Kohn W. and Rostoker N.,
Phys. Rev. {\bf 94} (1954) 1111. 

\bibitem{KMP1}Kuzmiak V. and Maradudin  A. A.,
Phys. Rev. B {\bf 55} (1997) 7427.

\bibitem{AvBV}van Blaaderen A. and Vrij A., 
Langmuir {\bf 8} (1992)  2921.

\bibitem{AvB}van Blaaderen A., MRS Bulletin {\bf 23}, October (1998) 39;
Dassanayake U., Fraden S., and  van Blaaderen A.,
J. Chem. Phys. {\bf 112} (2000) 3851.

\bibitem{LM}van der Lem H. and Moroz A., to appear in J. Opt. A: Pure Appl.

\bibitem{GMa}Goia D. V. and Matijecic E., 
New J. Chem. {\bf 146} (1999) 139.

\bibitem{KPV}Velikov K. P., private communication.


\end{thebibliography}

\newpage

\begin{center}
{\large\bf Figure captions}
\end{center}

\vspace*{2cm}

\noindent {\bf Fig. 1.-} Calculated photonic-band-structure of a close-packed 
fcc lattice of silica coated ($n_c=1.47$) 
metallic spheres embedded in a host 
dielectric with refractive index $n_h=1.4$
for $\omega_p=11$ ($r_s/\lambda_p \approx 1.238$). 
For convenience, on the right y-axis the angular frequency is shown 
in units $\omega_p$.
The coating width $l_c$ to the sphere radius ratio is 
$l_c/r_s\approx 7.39\%$ ($l_c/\lambda_p\approx 9.15\%$).
This corresponds to a metallic (core) filling fraction 
of $f_m\approx 0.588$ (to be compared with the fcc close-packed
filling fraction $f\approx 0.7405$). 
Note that there are three CPBG's, one with $g_w=3.2\%$ at midgap 
frequency $7.01$, the second with $g_w=8.4\%$ at midgap frequency $6.47$,
and the third with $g_w=2.59\%$ at midgap frequency $6.1$.

\vspace{0.6cm}


\noindent {\bf Fig. 2.-} The relative gap width $g_w$ 
of the CPBG's of a close-packed fcc lattice of silica 
coated metallic spheres as a function of the host refractive index $n_h$
for $l_c/\lambda_p \approx 9.15\%$ (i.e., $l_c=30$ nm for silver). 
For a close-packed fcc lattice, the value of $r_s/\lambda_p$ 
can be recovered by multiplying the numerical values of $\omega_p$
by $1/(2\pi\sqrt{2})$, i.e., 
$r_s/\lambda_p=1.013$, $ 1.125$, $1.238$, and $1.35$ for 
$\omega_p=9$, $10$, $11$,  $12$ in this order.
The metallic filling fractions in the same order are
$f_m\approx 0.557$, $ 0.574$, $0.588$, and $0.6$.
The symbols  {\large$\circ$}, $\triangle$, and  {\large$\times$} are for the 
upper, middle and the lower CPBG, respectively (cf. Fig. \ref{gfretot}).
Data on the right y-axis correspond to the case when $n_h$
matches $n_c$. 


\vspace{0.6cm}


\noindent {\bf Fig. 3.-}  The ratio $\omega_c/\omega_p$, corresponding 
to the CPBG's shown in Fig. \ref{rgtot}, as a function of 
the host refractive index $n_h$.
With a high precision the curves can be described
by a simple linear relation [see Eq. (\ref{srel})].


\newpage

\begin{figure}[tbp]
\begin{center}
\epsfig{file=coated.ps,height=20cm,angle=-180}
\end{center}
\caption{}
\label{coated}
\end{figure}


\begin{figure}[tbp]
\begin{center}
\epsfig{file=rgtot.ps}
\end{center}
\caption{}
\label{rgtot}
\end{figure}

\begin{figure}[tbp]
\begin{center}
\epsfig{file=gfretot.ps}
\end{center}
\caption{}
\label{gfretot}
\end{figure}


\end{document}

cond-mat0003518.extracted_bib
\begin{thebibliography}{99}

\bibitem{OG}Graydon O., OLE June 1998, p. 31.

\bibitem{Y}Yablonovitch E.,
Phys. Rev. Lett. {\bf 58} (1987) 2059.

\bibitem{HCS}Ho K. M., Chan C. T. and Soukoulis C. M.,
Phys. Rev. Lett. {\bf 65} (1990) 3152.

\bibitem{YGL}Yablonovitch E., Gmitter T. J. and Leung K. M.,
Phys. Rev. Lett. {\bf 67} (1991) 2295.

\bibitem{AMe}Moroz A., Europhys. Lett. {\bf 46} (1999) 419.

\bibitem{Tip}Glauber R. J. and Lewenstein M.,  
Phys. Rev. A {\bf 43} (1991) 467,
Eq. (6.12); Tip A., Phys. Rev. A {\bf 56} (1997) 5022;
ibid. {\bf 57} (1998) 4818.

\bibitem{WW}Weisskopf V. and Wigner E., Z. Phys. {\bf 63} (1930) 54.

\bibitem{By}Bykov V. P., Sov. J. Quant. Electron. {\bf 4} (1975) 861.

\bibitem{LZM}Lewenstein M., Zakrzewski J., Mossberg T. W. and Mostowski J.,
J. Phys. B{\bf 21} (1988) L9.

\bibitem{RSS}Russel W. B., Saville D. A. and Schowalter W. R.,
{\em Colloidal Dispersions} (Cambridge University Press, Cambridge, 1995).

\bibitem{WV}Wijnhoven J. E. J. G. and Vos W. L.,
Science {\bf 281} (1998) 802.

\bibitem{BSS}Biswas R., Sigalas M. M., Subramania G. and Ho K.-M., 
Phys. Rev. B {\bf 57} (1998) 3701.

\bibitem{MS}Moroz A. and Sommers C., 
J. Phys.: Condens. Matter {\bf 11} (1999) 997. 

\bibitem{Lev}Levi B. G., Phys. Today, January 1999, p. 17.

\bibitem{Lin}Lin S. Y. et al., Nature {\bf 394} (1998)  251.

\bibitem{AlM}Moroz A., Phys. Rev. Lett. {\bf 83} (1999) 5274.

\bibitem{BH}Bohren C. F. and Huffman D. R., {\em Absorption
and Scattering of Light by Small Particles}
(Wiley, New York, 1984), Chap. 9, 12.

\bibitem{AsM}Ashcroft N. W. and Mermin N. D., {\em Solid State Physics}
(Saunders College, 1976) pp. 73-75.

\bibitem{PHS}Pendry J. B., Holden A. J., Stewart W. J. and Youngs I.,
Phys. Rev. Lett. {\bf 76} (1996) 4773.

\bibitem{JC}Johnson P. B. and Christy R. W.,
Phys. Rev. B {\bf 6} (1972) 4370; ibid. {\bf 9} (1974) 5056.

\bibitem{PKA}Pan G., Kesavamoorthy R. and Asher S. A.,
Phys. Rev. Lett. {\bf 78} (1997) 3860;
Grier D., Phys. World, July 1997, p. 24.

\bibitem{KML}Leung K. M., Phys. Rev. A {\bf 33} (1986) 2461;
Chemla D. S. and Miller D. A., Opt. Lett. {\bf 11} (1986) 522.

\bibitem{ULM}Ung T., Liz-Marzan L. M. and Mulvaney P., 
Langmuir {\bf 14} (1998) 3740.

\bibitem{Mo}Moroz A., 
Phys. Rev. B {\bf 51} (1995) 2068.

\bibitem{KKR}Korringa J.,
Physica {\bf 13} (1947) 392; Kohn W. and Rostoker N.,
Phys. Rev. {\bf 94} (1954) 1111. 

\bibitem{KMP1}Kuzmiak V. and Maradudin  A. A.,
Phys. Rev. B {\bf 55} (1997) 7427.

\bibitem{AvBV}van Blaaderen A. and Vrij A., 
Langmuir {\bf 8} (1992)  2921.

\bibitem{AvB}van Blaaderen A., MRS Bulletin {\bf 23}, October (1998) 39;
Dassanayake U., Fraden S., and  van Blaaderen A.,
J. Chem. Phys. {\bf 112} (2000) 3851.

\bibitem{LM}van der Lem H. and Moroz A., to appear in J. Opt. A: Pure Appl.

\bibitem{GMa}Goia D. V. and Matijecic E., 
New J. Chem. {\bf 146} (1999) 139.

\bibitem{KPV}Velikov K. P., private communication.


\end{thebibliography}