Version: 1.0


· Introduction
· Case study
· How to use?
· Copyright
· Authors
· References

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GranFilm : Optical Properties of Granular Thin Films

by

Ingve Simonsen and Rémi Lazzari






R. Lazzari and I. Simonsen, Granfilm: a software for calculating thin-layer dielectric properties and fresnel coefficients, Thin Solid Films 419, 124 (2002).




Introduction

Since the pioneering work of Maxwell Garnett [1] and Mie [2] at the turn of the century, there has been large scientific interest in the optical properties of metallic clusters [3]. Their optical behavior are driven to a large extent by the Mie excitations [3] which can be viewed as surface plasmon-polaritons [4]. Nearly hundred years ago Mie [3] derived an exact theory for the scattering of light from a free standing spherical particle. If, for isolated clusters with simple shapes, such as spheres or spheroids in vacuum, the exact solution of the Maxwell equations is well known, the difficulty of a reliable description of the optical properties of particles dramatically increases for interacting particles of complex shapes, either in a matrix or on a surface. Even though the Maxwell Garnett effective medium theory and other such theories [5] have been quite successful in tackling these questions, an accurate description of the macroscopic optical properties of a collection of particles supported on a substrate, such as the Fresnel coefficients (absorption, reflection and transmission), requires a more sophisticated approach that accounts correctly for the break of symmetry brought by the substrate and for realistic cluster shape. Indeed for clusters deposited on a surface, a quantitative description of the optical properties of thin films is not only hampered by the interactions between aggregates but also by the image interactions between the latter and the substrate [6-9].

In order to handle this latter situation in an adequate way, Bedeaux and Vlieger [10-14] in the first half of the 1970's introduced a model valid for layers whose thickness are in the sub-wavelength regime. In this case, they introduced modified boundary conditions on the surface for the electromagnetic fields which depend on what they called integrated excess quantities. In the closure relation between these quantities, they introduced surface susceptibilities which govern completely the far field behavior of the electromagnetic fields and thus the Fresnel quantities. In this way, they get rid of the complex behavior of the field on the surface. This approach is really analogous to the model of Feibelmann [15] and Barrera [16] who applied these notions in the case of the electromagnetic jump at metallic surfaces. However, Bedeaux and Vlieger were mainly interested in the description of the optical properties of island layers. For such layer, the main quantity which is directly related to the surface susceptibilities is the island polarizability. For simple shape like truncated spheres or spheroids with an axis of revolution normal to the surface of the substrate, a model based on a multipole expansion of the electrical potential in the non retarded limit was developed [17-20]. A nice recent, detailed and pedagogical introduction to this fascinating field of optics can be found in the recent  book: Optical properties of Surfaces (Imperial College Press, 2001) by D. Bedeaux and J. Vlieger [21]. The aim of the present web page is to give access to the softwares which englobes most of the models developped in [21]. This software package we have named -- GranFilm.


A case study: Ag deposit on an MgO substrate

Fig 1: A scanning electron micrography of an Ag deposit on MgO obtained by vapor deposition (equivalent thickness 3nm).

Fig. 2 : Comparison of the experimental and theoretical differential reflectivity curves. Consult Ref. [22,23] for details.



To motivate the need for a software like GranFilm, a case study of silver (Ag) deposit on a substrate of magnesium oxide (MgO) is considered [22-23]. The layer was elaborated by vapor deposition in ultra-high vacuum conditions. Due to the poor adhesion energy of noble metal on wide band gap oxide surface, the growth mode proceeds by nucleation and growth of nanometric islands (see Fig. 1). Light emitted from a deuterium lamp impinges at an angle of incidence of 45 degrees onto such a granular thin metal film during its elaboration and the scattered p-polarized light is collected in the specular direction and analyzed in energy by a grating spectrograph in the UV-visible range. The recorded quantity is the Differential Reflectance i.e. the relative variation of the sample reflectivity (the reference is the bare substrate reflectivity). The incident electric field excites collective oscillation modes for the electronic gas in the metallic islands. These modes are damped by the imaginary part of the metal dielectric constant and leads to optical absorptions which appear as a peak and dip structure in the experimental spectra (open blue circles in Fig. 2).

The solid red curve in Fig. 2 is a simulation result, obtained by using the GranFilm software. In obtaining these results, the islands were assumed to be approximated by truncated spheres of the same size and located on square grid (see [23] for details ). The island polarizability, which is the main ingredient in the calculations, was derived from a multipolar expansion of the potential with the image charge technique which allowed to account correctly for the coupling with the substrate and for the island shape through the fulfillment of the boundary conditions. The two main resonances in Fig. 2 are associated, in a first approximation, to the excitation of dipoles perpendicular and parallel to the substrate by the two components of the electric field. These modes, are depicted in Fig. 3 (0-A and 1-A, respectivley) by the polarization charges computed in the limit of small damping through the equipotential lines. Moreover, other multipolar modes (essentially of quadrupolar order) can be excited by the incident light thank to the special considered geometry.

Fig 3 : Equipotential maps for the eigenmodes of the charge polarization for a truncated sphere. Notice the two types of modes which can be excited by the normal component of the electric field (left panel) and by the parallel one (right panel) (from [25])


Of course GranFilm is capable of computing all the Fresnel coefficients once the island polarizability is known. An example of this is given in Fig. 4.

Fig 4 : Computed Fresnel quantities (reflection, transmission, absorption and ellipsometric coefficients) for Ag islands on MgO (same parameters as for Fig. 2) (from [25]).



The software

GranFilm is a software package aimed at calculating the polarizability of truncated spheres or spheroids deposited on a substrate as well as to calculate all the derived Fresnel quantities for island layers. It is suited for performing simulations for most of the models described in depth in the book Optical properties of Surfaces [21]. The implemented model account for the multipolar interaction of the particle with the substrate and for the coupling between clusters up to quadrupolar order. It as well includes simple dipole theory known as Yamaguchi model [6,7]. The software is written in the Fortran 90 programming language and has been extensively tested under the Linux and Windows 95/98/2000/NT operating systems. It should probably also work under other flavors of Unix as well as on the Mac OS, but this has not been tested. To compile the software from the provided source code, one will need the following libraries: Lapack, Slatec, FMZM and Pgplot. These libraries can be downloaded for free from the web by following the links. However, precompiled executable are provided for Linux and Windows which have these libraries statically linked (i.e. you do not need to have any of these libraries installed to run the software).


How to use GranFilm?

The installation and user instructions for GranFilm is provided in the ReadMe file that come with the distribution.

In order to use GranFilm, the user must provide the dielectric functions of both the metallic islands as well as the substrate. This information is located in files island.nk and substrate.nk respectively. They contain the index of refraction (n,k) of the material in two columns. The header of the file (first line) gives information about the abscissa axis : Unit(1=Energy in eV,2=Wavelenght in m) Start value End value Number of points. A dielectric database is furnished with the program and is taken from the free distribution by SOPRA corporation.

Once launched the program automatically reads from the current directory a parameter file GranFilm.par which complies to your needs. For sphere and spheroid, they have slight different structures but contains more or less the same types of parameters. For example for a truncated sphere,

Parameters

Possible values

Path to the dielectric constants files

User defined (e.g. c: \Dielectric)

Geometry for the layer: either truncated sphere or thin continuous films


island, film, 2film, thin_cap

Films thickness (nm)

User defined

Radius of the sphere (nm)

Truncation value tr for the sphere

Position of the multipole expansion

User defined

User defined

User defined (=0 is a good choice)

Type of clusters-clustesr interaction

Type of lattice for the clusters (MFT=Mean Field Theory RPT=Renormalized Polarization theory)

Lattice parameter (nm)

Coverage (for MFT-RPT)

none, dipole, quadrupole


square, hexagonal, MFT, RPT

User defined

User defined

Computed Fresnel coefficient (Reflection, Differential Reflection, Transmission, Differential Transmission, Absorption)

Polarization of the incident beam

Incident angle (rad)

Type of formulae for computing the Fresnel coefficient from the island polarizability

R, DR, T, DT, A

p,s

User defined

constitutive, constitutive_all, invariants, aspnes

Start energy (eV)

End energy (eV)

Number of points

User defined

User defined

User defined

Island or first film dielectric file (*.nk)

Substrate dielectric file (*.nk)

Second film dielectric file (*.nk)

User defined

User defined

User defined

Type of correction for the dielectric constant

Correction for surface effect (Logical)

none, finite_size, tau, manual, A-parameter, s-only

T or F

Output for the polarizability (Logical)

Normalization of the polarizability by the island volume (Logical)

Output for the multipolar coefficients (Logical)

T or F

T or F

T or F

Type of calculation for the potential : none, the potential itself or fulfillment of the boundary condition on the sphere surface

Energy (eV) where calculation is performed

Type of normalization for the boundary conditions (BC)

None, Pot, BC

User defined

none, incident_field, mean_value

Number of multipoles in the expansion of the potential

User defined (less than 40 otherwise numerical limits are reached)

Name of the ASCII output file (*.dat)

User defined

Comparison with experimental data of differential reflectivity (Logical)

Name of the experimental file (*.dat)


T or F

User defined

 

For spheroid, the most different parameters are the following:

Parameters

Possible values

Type of calculation

island, yamaguchi, coated

Island radius parallel to the surface (nm)

User defined

Island radius perpendicular to the surface (nm)

User defined

Thickness of the coating (nm)

User defined

Coating dielectric constant

User defined



The coated geometry is that of a full spheroid coated by an overlayer and in dipolar interaction with the substrate [24].

To illustrate how the program works, the case described by the provided parameter file Sphere.inc corresponds to truncated silver spheres on-top of a dielectric (MgO) substrate. The main output from the simulations using this parameter file should be given as in the file SphereTest.dat. We have also provided as test case for the Spheroid program (Spheroid.inc and SpheroidTest.dat). Check that you can reproduce these results!

The authors are aware than these previous notice is succinct but with a close inspection of the literature, of the book Optical properties of Surfaces and of the program source codes, the user will undoubtedly be able to manage by himself. However, any comments, questions and suggestions are welcome.


Copyright

GranFilm was developed by  Ingve Simonsen and  Rémi Lazzari. It is being released under the GNU General Public License and is provided as-is without any warranty. Any publication resulting from this software should acknowledge its use.


Contact Auhtors

Ingve Simonsen

Department of Physics
Norwegian University of Science and Technology (NTNU)
NO-7491 Trondheim
NORWAY

Office : +47 73 59 34 17
Fax : +47 73 59 77
Email : ingve.simonsenntnu.no

Rémi Lazzari

Institut des NanoSciences de Paris
UMR7588, Boite Courrier 840
4 place Jussieu
75252 Paris Cedex 05
FRANCE

Office : +33 (0)1 44 27 46 28
Fax : +33 (0)1 44 27 39 82
Email : remi.lazzariinsp.jussieu.fr


Acknowledgment

The authors would like to warmly acknowledge Dick Bedeaux and Jan Vlieger for having shown such a great support and enthusiasm during the development of GranFilm.

References

[1] J. C. Maxwell Garnett, Phil. Trans. Roy. Soc. London 203A, 385 (1904)

[3] G. Mie, Ann. Phys. 25, 377 (1908).

[3] U. Kreibig, Optical properties of metal clusters (Springer Verlag, 1995)

[4] J.D. Jackson, Classical Electrodynamics (John wiley and Sons, New York 1975)

[5] R. Landauer, in Proceedings of the First Conference on the Electromagnetic and Optical Properties of Inhomogeneous Media (AIP, New-York 1978)

[6] T. Yamaguchi, S. Yoshida and A. Kinbara, Thin Solid Films 18, 63 (1973)

[7] T. Yamaguchi, S. Yoshida and A. Kinbara, Thin Solid Films 21, 173 (1974)

[8] R.G. Barrera, M. del Castillo-Mussot and G. Monsivais, Phys. Rev. B 43,13879 (1991)

[9] C. Noguez and R.G. Barrera, Phys. Rev. B 57, 302 (1998)

[10] D. Bedeaux and J. Vlieger, Physica A 67, 55 (1973)

[11] D. Bedeaux and J. Vlieger, Physica A 73, 287 (1974)

[12] J. Vlieger and D. Bedeaux, Physica A 82, 221 (1976)

[13] D. Bedeaux and J. Vlieger, Thin Solid Films 69, 107 (1980)

[14] J. Vlieger and D. Bedeaux, Thin Solid Films 102, 265 (1983)

[15] P.J. Feibelman, Progress in Surface Science 12, 287 (1982)

[16] A. Bagchi, R.G. Barrera and R. Del Sole, Phys. Rev. B 20, 4824 (1979)

[17] M.M. Wind, J. Vlieger and D. Bedeaux, Physica A 141,133 (1987)

[18] M.M. Wind, P.A. Bobbert, J. Vlieger and D. Bedeaux, Physica A 143,164 (1987)

[19] M.M. Wind, P.A. Bobbert and J. Vlieger, Thin Solid Films 164, 57 (1988)

[20] M.T. Haarmans and D. Bedeaux, Thin solid Films 224, 117 (1993)

[21] D. Bedeaux and J. Vlieger, Optical properties of Surfaces (Imperial College Press, 2001) (Description of the book and how to order it)

[22] R. Lazzari, J. Jupille. and Y. Borensztein., Appl. Surf. Sci. 142, 451 (1999)

[23] I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, Phys. Rev. B 61, 7722 (2000)

[24] C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, 1983)

[25] R. Lazzari, Vers la maîtrise de la croissance de couches minces : une étude par spectroscopie optique et d'eacute;lectrosn, PhD Thesis Paris XI - France






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