| Symmetry of two-dimensional hybrid metal-dielectric photonic crystal within MAPLE © 2013 Prof. Gennady P. Chuiko and Olga V. Dvornik Petro Mohyla BlackSea State University, Dep. of Medical Devices and Systems, Mykolayiv, Ukraine, gp47@mail.ru | | 1. Introduction | | Hybrid structures were made by assembling monolayers (MLs) of closely packed colloidal microspheres on a metal-coated glass substrate . In fact, this architecture is one of several realizations of hybrid plasmonic-photonic crystals (PHs), which differ in photonic crystals dimensionality and metal film corrugation [1,2]. Table 1. Artificial two-dimensional plasmonic-photonic crystals | Figure 1. (a) Preparation of a corrugated metal film on a ML of spheres. (b) Top view of the Ag-coated PMMA spheres of D =431 nm. Scale bar is 1 μm. Inset: schematics of a metal-capped sphere. Note that the metal coating applies to the upper half of the sphere only. Metal also penetrates the interstitials and forms metal islands on the glass substrate (not shown here). |  | | | The basic hypothesis is that considered only the single hexagonal two-dimensional dense [acking of identical spheres irrespective of materials either of substrates or of shperes itself. Structures are considered as infinite two-periodical in two dimensions that lie in a plane (x,y). Axis z-normal to the plane of the layer. Basic vectors of translates are not normal, because with angle -  , but same for norm. Thus 2D elementary cell is correct hexagon (see fig. 1 b). The main challenge to us were exploring of those properties of structures which are caused by their space symmetry. In particular, it was necessary to establish the so-called "rules of selection", i.e. the list of the allowed transitions between electronic states of different symmetry and energy that can be induced by light of varying polarization. Additional interest for us was to demonstrate the possibilities of MAPLE within this specific field. | | | 2. The main symmetry properties and matrcies of symmetry elements | | The same norms of both translational vectors (  = LinearAlgebra[Norm](tau[2])](/view.aspx?si=151383/1687aa8b665504948440308cef34f41d.gif) ) and the angle between them ( ) define unambiguously a two-dimensional hexagonal primitive lattice. Hence the two-dimensional group of symmetry is fixed as  , because just five kinds of two-dimensional lattices are known and only one of them is hexagonal moreover [3,4].
In general, there are only 17 of two-dimensional symmetry groups, and 13 of those are symmorphic. It is important for us that  group is among those symmophic groups [3]. Such a property of two-dimensional group  , among other things, means the presence into this group a point subgroup, which is isomorphic to factor-group of  symmetry group by own translations subgroup. Elements of symmetry of this point subgroup are shown on fig. 2 [3] Table 2. Elements of point group of symmetry | Figure 2. The basic elements of symmetry for ![6*mm = C[6*v]](/view.aspx?si=151383/4611537e3864037b87d603e8ab0ec968.gif) point group (generators): main one-sided axis of sixth order, normal to the plane of the structure, and and two types of vertical planes of symmetry ( ![sigma[v], (diff(sigma(x), x))[v]](/view.aspx?si=151383/bb5b26979fe53459c0868e13ccdb3b5f.gif) ) , displayed by different way. The group generally consist of 12 elements of symmetry, divided into 6 classes of conjugated elements [4] |  | | | Main axis is one-sided, because rhe symmetry center is absent into  group of symmetry as an independent element of symmetry. | > | restart: with(LinearAlgebra): | We need only two matricies of group generators for the matrix presentation of all its elements of symmetry. Let it be such two matricies | > | C6:=Matrix([[ 1/2 , -sqrt(3)/2 , 0 ],[ sqrt(3)/2 , 1/2 , 0 ],[ 0 , 0 , 1 ]]);# The counterclockwise rotation around main axis on angle Pi/6
| |  | (2.1) |
| > | sigma1_1:=Matrix([[ -1 , 0 , 0 ],[ 0 , 1 , 0 ],[ 0 , 0 , 1 ]]);# A mirror reflection plane of first kind | |  | (2.2) |
Really we need just 2D-reduced submatricies of those generators and their traces, because we deal with 2D group of symmetry | > | C6:=SubMatrix(C6,1..2,1..2);Trace(C6);
sigma1_1:=SubMatrix(sigma1_1,1..2,1..2);Trace(sigma1_1); | |  | (2.3) |
Now we can obtain all another matricies and traces | > | C6_5:=C6^(-1);Trace(%); #The clockwise rotation around main axis on angle Pi/6 | |  | (2.4) |
| > | C3_1:=C6^2;C3_2:=C6^(4);
Trace(C3_1); # Two rotations around main axis on angle Pi/3 | |  | (2.5) |
| > | C2:=C6^3;
Trace(C2);# The rotation around main axis on angle Pi/2 | |  | (2.6) |
| > | sigma1_2:=C6^2 .sigma1_1 ;
sigma1_3:=C6^4 .sigma1_1 ;
Trace(%); # Two additional to (2.3) planes of mirror reflection of first kind | |  | (2.7) |
| > | sigma2_1:=C6 .sigma1_1 ;
sigma2_2:=C6^3 .sigma1_1 ;
sigma2_3:=C6^5 .sigma1_1 ;
Trace(%);# Three vertical planes of mirror reflection of sacond kind | |  | (2.8) |
| > | E:=IdentityMatrix(2);
Trace(%);# And identity element, of course | |  | (2.9) |
| | | 3. Irreducible representations characters as vectors for p6mm group | | We could to obtain the number of irreducible representations (irreps) and their characters by standard methods of group theory from above data. However, this is allready done before us [4]. The group has four 1D irreducible representations ( ) and two 2D irreps ( ) according to the well-known Bernside theorem (1+1+1+1+4+4=12). Therefore, the energy spectrum of p6mm structure consist of energy levels of six above mentioned types of symmetry, and moreover, two of them are twice degenerated by symmetry. Their characters are well-known too [4]. We only allowed ourselves to present them in a bit unusual, but convenient for our calculations of vector forms. | > | A1:=<1|1|1|1|1|1|1|1|1|1|1|1>; # According to this 1D irrep transforms z-components of polar vectors(pulse for instance or normal to plane and longitudinal polarized regarding to the main axis light) | |  = 1, (2) = 1, (3) = 1, (4) = 1, (5) = 1, (6) = 1, (7) = 1, (8) = 1, (9) = 1, (10) = 1, (11) = 1, (12) = 1})](/view.aspx?si=151383/b7094919f0db53386fe3c0298b5c20aa.gif) | (3.1) |
| > | A2:=<1|1|1|1|1|1|-1|-1|-1|-1|-1|-1>;# a 1D irrep
| |  = 1, (2) = 1, (3) = 1, (4) = 1, (5) = 1, (6) = 1, (7) = -1, (8) = -1, (9) = -1, (10) = -1, (11) = -1, (12) = -1})](/view.aspx?si=151383/e2fb00b5197568c36e2c0920894b5329.gif) | (3.2) |
| > | A3:=<1|1|1|-1|-1|-1|1|1|1|-1|-1|-1>;# According to this 1D irrep transforms z-componenets of axial vectors (pulse momentum for instance) | |  = 1, (2) = 1, (3) = 1, (4) = -1, (5) = -1, (6) = -1, (7) = 1, (8) = 1, (9) = 1, (10) = -1, (11) = -1, (12) = -1})](/view.aspx?si=151383/350a748d5077c357a7beb970ea5ad1c7.gif) | (3.3) |
| > | A4:=<1|1|1|-1|-1|-1|-1|-1|-1|1|1|1>; # a 1D irrep | |  = 1, (2) = 1, (3) = 1, (4) = -1, (5) = -1, (6) = -1, (7) = -1, (8) = -1, (9) = -1, (10) = 1, (11) = 1, (12) = 1})](/view.aspx?si=151383/8bc2d465f186da162fdb231b9d895aa2.gif) | (3.4) |
| > | E1:=<2|-1|-1|0|0|0|2|-1|-1|0|0|0>;# a 2D irrep | |  = 2, (2) = -1, (3) = -1, (4) = 0, (5) = 0, (6) = 0, (7) = 2, (8) = -1, (9) = -1, (10) = 0, (11) = 0, (12) = 0})](/view.aspx?si=151383/2c28780fcf5b3fafb8e105f7e1ddecb3.gif) | (3.5) |
| > | E2:=<2|-1|-1|0|0|0|-2|1|1|0|0|0>;# Another 2D irrep according to wich transforms (x,y)-componenets of polar and axial vectors, especially transversal regarding to the main axis polarized light | |  = 2, (2) = -1, (3) = -1, (4) = 0, (5) = 0, (6) = 0, (7) = -2, (8) = 1, (9) = 1, (10) = 0, (11) = 0, (12) = 0})](/view.aspx?si=151383/5933bd0b26fcbaaeaf3f2b4296063c69.gif) | (3.6) |
| | | 4. Selection rules | | | | 4.1Transitions stimulated by longitudinal polarized light | | Let the light rays are polarized normal to the the plane of structure i.e. longitudinal with respect to main axis of symmetry. Then the perturbation operator transforms according to 1D irrep A1 ( . Let  are characters of irrep according to which transforms the states belonging to two of six above mentioned energy levels (  and ![X[i], Y[i]](/view.aspx?si=151383/fb19ef75433f53c97b3a01c453bc4545.gif) - are components of these vectors ( . Then the transitions  are allowed at stimulation by operator, wich transforms according to the irrep W only under the condition, that the sum ![(1/(12)*∑)X[i]*W[i]*Y[i]](/view.aspx?si=151383/86ddc1bef137cf1c3bb428b6adbd3995.gif) is non-zero. Let us demonstrate, that the transitions between states of same symmetry but different energies are allways allowed in this case | > | [seq(A1[i]*A1[i]*A1[i],i=1..12)]:
sum(%[n],n=1..12)/12;# A1->A1 transitions are allowed | |  | (4.1.1) |
| > | [seq(A2[i]*A1[i]*A2[i],i=1..12)]:
sum(%[n],n=1..12)/12;# A2->A2 transitions are allowed | |  | (4.1.2) |
| > | [seq(A3[i]*A1[i]*A3[i],i=1..12)]:
sum(%[n],n=1..12)/12;# A3->A3 transitions are allowed | |  | (4.1.3) |
| > | [seq(A4[i]*A1[i]*A4[i],i=1..12)]:
sum(%[n],n=1..12)/12;# A4->A4 transitions are allowed | |  | (4.1.4) |
| > | [seq(E1[i]*A1[i]*E1[i],i=1..12)]:
sum(%[n],n=1..12)/12;# E1->E1 transitions are allowed | |  | (4.1.5) |
| > | [seq(E2[i]*A1[i]*E2[i],i=1..12)]:
sum(%[n],n=1..12)/12;# E2->E2 transitions are allowed | |  | (4.1.6) |
Any another transitions are obviously prohibited.
One of them for example: | > | [seq(E1[i]*A1[i]*A3[i],i=1..12)]: | | > | sum(%[n],n=1..12)/12;# E1->A3 transitions are prohibited | |  | (4.1.7) |
Thus the normal to the surface and parallel to the main optical axis polarization of light allowed only transitions between states of the same symmetry but varying energy. The selection rule is extremely simple at the case. | | | 4.2 Transitions stimulated by transversal polarized light | | The perturbation operator  transforms according to the 2D irrep E2 at this case ( . Since the selection rules are more complicated. The transitions between states with same symmetry are prohibited at first. There is one of them for instance | > | [seq(A1[i]*E2[i]*A1[i],i=1..12)]:
sum(%[n],n=1..12)/12; # The A1->A1 type transitions are now prohibited | |  | (4.2.1) |
The list of allowed transitions are following at second: | > | [seq(A1[i]*E2[i]*E2[i],i=1..12)]:
sum(%[n],n=1..12)/12;# A1->E2 | |  | (4.2.2) |
| > | [seq(A2[i]*E2[i]*E1[i],i=1..12)]:
sum(%[n],n=1..12)/12;#A2->E1 | |  | (4.2.3) |
| > | [seq(A3[i]*E2[i]*E2[i],i=1..12)]:
sum(%[n],n=1..12)/12;#A3->E2 | |  | (4.2.4) |
| > | [seq(A4[i]*E2[i]*E1[i],i=1..12)]:
sum(%[n],n=1..12)/12;#A4->E1 | |  | (4.2.5) |
| > | [seq(E1[i]*E2[i]*A2[i],i=1..12)]:
sum(%[n],n=1..12)/12;# E1->E2 | |  | (4.2.6) |
| > | [seq(E2[i]*E2[i]*A1[i],i=1..12)]:
sum(%[n],n=1..12)/12;# E2->A1 | |  | (4.2.7) |
| > | [seq(E1[i]*E2[i]*E2[i],i=1..12)]:
sum(%[n],n=1..12)/12;# E1->E2 | |  | (4.2.8) |
Thus in such polarization is allowed some transitions between states of different symmetry and energies in both possible directions. Especially, the transitions are allowed among states with such symmetries  e  | All conclusions obtained above (see 4.1, 4.2) were proved experimentally [1,2]. | | | References | | [1] . Romanov S.G. et al. Probing guided modes in a monolayer colloidal crystal on a flat metal film.// Phys. Rev. B 86, #19, 195145 – Published 30 November 2012. [2] Romanov S.G., . Korovin A.V, Regensburger A. and Peschel.U. Hybrid Colloidal Plasmonic-Photonic Crystals.//Adv. Mater. 2011, v.XX, зpp.1–19 [3] Yegorov-Tysmenko Y.K., Litvinskaya G.P. Theory of crystal symmetry. Textbook. Moskow: GEOS,2000, -410 pp. (In Russian) [4] Weinstein B.K. Modern Crystallography (in 4 volumes),vol.1. Crystals Symmetry. Methods of Structural Crystallography. Moskow: Nauka,1979,-392 pp. (In Russian) Legal Notice: Maplesoft, a division of Waterloo Maple Inc. 2009. Maplesoft and Maple are trademarks of Waterloo Maple Inc. Neither Maplesoft nor the authors are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the authors for permission if you wish to use this application in for-profit activities |  |
