Optical properties of metal-composite-based thin films

In the first part of this work we study the effect of a semi-infinite matrix dispersed system (MDS) on the external electromagnetic radiation in the electrostatic approximation. With the help of our previous technique, we obtain general expressions for the multipole expansion coefficients of the ele...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Veröffentlicht in:	Поверхность
Datum:	2002
Hauptverfasser:	Gozhenko, V.V., Grechko, L.G., Goncharuk, Yu.S., Ogenko, V.M.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут хімії поверхні ім. О.О. Чуйка НАН України 2002
Schlagworte:	Surface properties of inorganic materials
Online Zugang:	https://nasplib.isofts.kiev.ua/handle/123456789/126368
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Optical properties of metal-composite-based thin films / V.V. Gozhenko, L.G. Grechko, Yu.S. Goncharuk, V.M. Ogenko // Поверхность. — 2002. — Вип. 7-8. — С. 221-230. — Бібліогр.: 14 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-126368
record_format	dspace
spelling	Gozhenko, V.V. Grechko, L.G. Goncharuk, Yu.S. Ogenko, V.M. 2017-11-20T18:59:03Z 2017-11-20T18:59:03Z 2002 Optical properties of metal-composite-based thin films / V.V. Gozhenko, L.G. Grechko, Yu.S. Goncharuk, V.M. Ogenko // Поверхность. — 2002. — Вип. 7-8. — С. 221-230. — Бібліогр.: 14 назв. — англ. XXXX-0106 https://nasplib.isofts.kiev.ua/handle/123456789/126368 In the first part of this work we study the effect of a semi-infinite matrix dispersed system (MDS) on the external electromagnetic radiation in the electrostatic approximation. With the help of our previous technique, we obtain general expressions for the multipole expansion coefficients of the electric potential for a sphere accounting for the interaction between ambient particles and the substrate. The polarizability tensor and resonant frequencies of a single sphere show anisotropy due to the influence of a substrate. In the second part electrodynamical properties of thin percolating layers manufactured on the basis of the MDS are considered. Transition from 3-D to 2-D behavior, which is observed near the percolation threshold and shows itself as changing of some parameters (in comparison with those for 3-D percolating system) like the values of percolation threshold, critical indices of conductivity and permittivity, were studied. The authors acknowledge financial support from the National Science Foundation through the Faculty Early Career Development (CAREER) Award ECS-9624486 and the Eastern Europe Program Supplement. en Інститут хімії поверхні ім. О.О. Чуйка НАН України Поверхность Surface properties of inorganic materials Optical properties of metal-composite-based thin films Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Optical properties of metal-composite-based thin films
spellingShingle	Optical properties of metal-composite-based thin films Gozhenko, V.V. Grechko, L.G. Goncharuk, Yu.S. Ogenko, V.M. Surface properties of inorganic materials
title_short	Optical properties of metal-composite-based thin films
title_full	Optical properties of metal-composite-based thin films
title_fullStr	Optical properties of metal-composite-based thin films
title_full_unstemmed	Optical properties of metal-composite-based thin films
title_sort	optical properties of metal-composite-based thin films
author	Gozhenko, V.V. Grechko, L.G. Goncharuk, Yu.S. Ogenko, V.M.
author_facet	Gozhenko, V.V. Grechko, L.G. Goncharuk, Yu.S. Ogenko, V.M.
topic	Surface properties of inorganic materials
topic_facet	Surface properties of inorganic materials
publishDate	2002
language	English
container_title	Поверхность
publisher	Інститут хімії поверхні ім. О.О. Чуйка НАН України
format	Article
description	In the first part of this work we study the effect of a semi-infinite matrix dispersed system (MDS) on the external electromagnetic radiation in the electrostatic approximation. With the help of our previous technique, we obtain general expressions for the multipole expansion coefficients of the electric potential for a sphere accounting for the interaction between ambient particles and the substrate. The polarizability tensor and resonant frequencies of a single sphere show anisotropy due to the influence of a substrate. In the second part electrodynamical properties of thin percolating layers manufactured on the basis of the MDS are considered. Transition from 3-D to 2-D behavior, which is observed near the percolation threshold and shows itself as changing of some parameters (in comparison with those for 3-D percolating system) like the values of percolation threshold, critical indices of conductivity and permittivity, were studied.
issn	XXXX-0106
url	https://nasplib.isofts.kiev.ua/handle/123456789/126368
citation_txt	Optical properties of metal-composite-based thin films / V.V. Gozhenko, L.G. Grechko, Yu.S. Goncharuk, V.M. Ogenko // Поверхность. — 2002. — Вип. 7-8. — С. 221-230. — Бібліогр.: 14 назв. — англ.
work_keys_str_mv	AT gozhenkovv opticalpropertiesofmetalcompositebasedthinfilms AT grechkolg opticalpropertiesofmetalcompositebasedthinfilms AT goncharukyus opticalpropertiesofmetalcompositebasedthinfilms AT ogenkovm opticalpropertiesofmetalcompositebasedthinfilms
first_indexed	2025-11-26T06:27:18Z
last_indexed	2025-11-26T06:27:18Z
_version_	1850615461721407488
fulltext	221 OPTICAL PROPERTIES OF METAL-COMPOSITE-BASED THIN FILMS V.V. Gozhenko, L.G. Grechko, Yu.S. Goncharuk, and V.M. Ogenko Institute of Surface Chemistry, National Academy of Sciences General Naumov Str. 17, 03680 Kyiv-164, UKRAINE Abstract In the first part of this work we study the effect of a semi-infinite matrix dispersed system (MDS) on the external electromagnetic radiation in the electrostatic approximation. With the help of our previous technique, we obtain general expressions for the multipole expansion coefficients of the electric potential for a sphere accounting for the interaction between ambient particles and the substrate. The polarizability tensor and resonant frequencies of a single sphere show anisotropy due to the influence of a substrate. In the second part electrodynamical properties of thin percolating layers manufactured on the basis of the MDS are considered. Transition from 3-D to 2-D behavior, which is observed near the percolation threshold and shows itself as changing of some parameters (in comparison with those for 3-D percolating system) like the values of percolation threshold, critical indices of conductivity and permittivity, were studied. Introduction Interest in matrix dispersed systems is stimulated, first of all, by the possibility of manufacturing materials with predicted optical properties. At the same time, the properties of MDS may strongly differ from those of the materials used for the formation of MDS [1]. In the theoretical studies, MDS are usually considered as infinite systems. In the first part of this work, we take into consideration the effects of an MDS interface. Namely, the MDS is considered as a half space dielectric matrix with a plane interface separating it from another half space of homogeneous dielectric. The matrix is filled with spherical inclusions of different diameters that are randomly located. The results [2] obtained for the system of spheres on a dielectric substrate can be obtained from our model as a particular case. Basically, this part is a generalization of [3, 4]. In the second part we consider percolating layer of sizes HLL ´´ (L<<H), which is a useful model of disordered composite film, particularly, of metal-dielectric layer deposited on a substrate. In such a system near the percolation threshold a transition from 3-D to 2-D behavior is observed, that manifests itself in changing (in comparison with 3-D system) the values of percolation threshold, critical indices of conductivity and permittivity, frequency dependence of dielectric response and other parameters. Below scaling expressions for electrodynamics parameters of percolating layer near the threshold are obtained by method of percolation renormalization group. Spheres near a substrate Basic equations. We consider a semi-infinite MDS consisting of dielectric spheres of different diameters embedded in a homogeneous dielectric (ambient) as shown in Fig. 1. Another half space is filled with another homogeneous dielectric (substrate). The system is placed in the electric 222 field proportional to e i tw . Let ( ) ( )e w e wa s, and ( )e wi be the dielectric functions of the ambient, substrate and the ith sphere, respectively, and Ri be the radius of the ith sphere. Fig. 1. Geometry of the semi-infinite matrix dispersed system. Let the wavelength of the external electromagnetic field be much larger than radii of the spheres and the distances between them. In other words, we use the electrostatic approximation. In such a case resulting electric field is caused by the interaction of the external field with the MDS and the substrate and its potential satisfies the Laplace equation ( )Dy rr = 0 (1) in the regions I - inside MDS (out of spheres), II - inside the spheres, III - inside the substrate, and does the standard boundary conditions ijji syy )( = , ij j j j i i i nn s y e y e ÷ ÷ ø ö ç ç è æ ¶ ¶ = ¶ ¶ , (2) where ie is dielectric function of the matter filling out the ith region (i=I, II, III), iy is the resulting field potential in the ith region, ijs denotes the common bound surface of the regions i and j. Using ideas of the image and multipole expansion methods of solving of electrostatic problems we seek a solution of the problem (1, 2) in the following form: ( ) ( )ååå ¢¢++-=++= - ilm ilmilmi ilm lmilmo I substrate i I sperethi I ext I FAFArE rryyyy rrrr (3) ( )å= lm ilmilm II i GB ry r ; (4) ( )å ¢++= ilm ilmilm IIIIII ext III FC ryyy r 0 ; (5) ( ) ( )zcEybExaErE zEyExErE ozoyox III ext ozoyox I ext ++-=¢-= ++-=-= rr rr 0 0 y y (6) where ( ) ( )rYrrF lm l lm rr 1--º ; ( ) ( )G r r Y rlm l lm r r º ; r r r ri ir rº - ; r r r ¢º - ¢ri ir r ; rri is a radius-vector of the center of the ith sphere ; r¢ri is a radius-vector of the ith sphere center image and III 0y is a constant contribution to the potential IIIy related with a choice of radius-vector origin point. Note, that all the individual terms in (3, 4, 5) automatically satisfy 223 equation (1), and (6) expresses the idea of force lines refraction on the boundary of different media. The unknown coefficients cbaCBAA lmilmilmilmi ,,,,,, ¢ are obtained after applying the boundary conditions (2) to the expansions (3, 4, 5). Boundary conditions on the substrate surface. 1. Potential continuity condition on the surface IIII -s takes the form ( ) ( ) ( ){ } 0)( 000 IIIIilm ilmilmilmilmilmilm III FCFAFArEE - =-¢¢++--¢ å s rrry rrrrrr . Different terms here have different arguments. It proves to be more convenient to reduce all the terms to a common argument, e.g. to ir r . Using the fact, that for any point at the boundary surface IIII -s ( )iiii jqrr ,,= r ( ) ( )iiiiiii jqprjqrr ,,,, -=¢¢¢=¢ r and using the relation [5] ( ) ( ) ( )jqjqp ,1, lm ml lm YY +-=- we obtain ( ){ } ( ) 0100 //// 0 IIIIilm ilmilmilm ml ilm III FCAArErE - =-¢-++-D+D å +^^ s ry rrrrr , where we have used decomposition ^+= rrr rrr // and analogous to it for 00 EEE rrr -¢ºD . Obtained equation is equivalent to the set ( )ï î ï í ì =-¢-+ =-×D =×D + ^^ 01 0 0 00 //// 0 ilmilm ml ilm III CAA rE rE y rr rr (7) 2. Potential derivative continuity condition on the surface IIII -s in view of zn ¶ ¶ = ¶ ¶ takes the form ( ) ( ) ( ) 0)( IIIIilm ilm ilmilmsi ilm lmilmailmilmaozas F z CF z AF z AEc - = ¶ ¶ -¢ ¶ ¶¢+ ¶ ¶ +- å åå s rerereee rrr . Again, reducing all the terms to argument ir r and using relation ( ) ( ) ( )jqjqp ,1, 1 lm ml lm F z F z ¶ ¶ -=- ¶ ¶ -+ , which can be seen from [5] [ ] +÷ ø ö ç è æ - ¶ ¶ ú û ù ê ë é ++ -+ = ¶ ¶ + ),( )32)(12( )1(),()( ,1 2 1 22 jqjq mllm Yf r l r f ll mlYrf z ),(1 )12)(12( ,1 2 1 22 jqmlYf r l r f ll ml -÷ ø ö ç è æ + + ¶ ¶ ú û ù ê ë é +- - + , we obtain equation ( ) ( ) 0]1[)( 1 IIIIilm ilmilmsilm ml ailmaozas F z CAAEc - = ¶ ¶ -¢-++- å -+ s reeeee r or equivalent set 224 ( )î í ì =-¢-+ =- -+ 01 0 1 ilmsilm ml ailma as CAA c eee ee (8) 3. The solution of eq. (7, 8) is ( ) ï ï ï ï ï ï î ïï ï ï ï ï í ì + = + - -=¢ ÷÷ ø ö çç è æ -= = = = + ilm sa a ilm ilm sa saml ilm z s aIII s a AC AA hE c b a ee e ee ee e e y e e 2 1 1 1 1 000 (9) where 0h is the height of the global origin over the substrate. Boundary conditions on the sphere surface and equation for Ailm 1. On the surface of jth sphere the potential continuity condition takes the form ( ) ( ) ( )å å å - =-¢¢++×- ilm ilm lm jlmjlmilmilmilmilm jIII GBFAFArE 00 s rrr rrrrr . Applying representations jjrr r rrr += and ( )jiji rr rrrr --= rr , well-known addition theorem [6] for spherical harmonics ( ) ( ) ( )rGRFTRrF ml ml LM ml lmlm rrrr 11 11 11 å=- , ( Rr< ) where 2 1 11111 )!()!()!()!( )!()!( )12)(12( )12(4)1( 111 ú û ù ê ë é -+-+ -+ × ++ + -º + mlmlmlml MLML Ll lT mlml lm p , 1llL +º , 1mmM -º , and taking into account that ( )jjjj R jIII jqr s ,,= - r , we obtain equation ( ) ( ) ( )[ ] jIIIjj ml mjl ilm jiLMilmjiLMilm ml lm l jmjl l jjml ErEBrrFArrFATRARY -× -- +×= þ ý ü î í ì --¢¢+-¢+Wå å sr )( 00 12 11 11 111 11 1 11 rrrrrrrr where ( ) ( ) î í ì = ¹- º-¢ ji jirrF rrF jilm jilm ,0 ,rr rr Interpreting this equation as multipole expansion, we can obtain expression for the coefficients { }... by using standard procedure ò ××W * ...lmYd , that leads to ( ) ( )[ ] ( )[ ]åå -= -- += þ ý ü î í ì --¢¢+-¢+ 1 1 1010000 12 11111 11 111 11 3 44 m ml mmj ml j l jmjl ilm jiLMilmjiLMilm ml lm l jmjl EYRErERBrrFArrFATRA dpdp rrrrrrr While deriving last expression we have used relations [5] ( ) ( ) ml lmmmllmllm dYY ¢¢ ¢¢ ¢¢ º=WWWò ddd , ( )å -= * Ù ÷ ø öç è æ=÷÷ ø ö çç è æ ××=× 1 1 11 €€ 3 4cos m mm bYaYabbaabba rrrrrr p . 225 2. Potential derivative continuity condition on the surface of jth sphere in view of jjIII n rs ¶ ¶= ¶ ¶ - and jj R jIII = -s r takes the form ( ) ( )[ ] ( ) )€( 1 0 1 1 1 1 1 11 1 11 11 1111 jajml l j ml ilm jiLMilmjiLMilm ml lmamjljmjlja EYRlrrFArrFATBAR l l reeee rrrrrr ×-=W þ ý ü î í ì -¢¢+-¢-+ + --å å Applying to this expression the same procedure as earlier, we obtain relation ( ) ( ) ( )[ ] ( )[ ]åå -= --- -= þ ý ü î í ì -¢¢+-¢-+ + 1 1 1 010 1 1 12 1 1 1 111 1111 1 3 41 m m mlma l j ilm jiLMilmjiLMilm ml lmamjljmjl l ja EYERlrrFArrFATBAR l l dpeeee rrrrr . Two equations obtained from the boundary conditions on the surface of jth sphere form the full set defining unknown coefficients ilmA and ilmB (note, that explicit form of ilmA¢ as function of ilmA was found earlier, see eq. (9)). After some transformations it can be reduced to the form ( ) { } ïî ï í ì =+ = å ¹0 111111 l ilm mjlilm ilm mjl ilm mjl ilmilm VAК AfB d , (10) where ( ) ( ) ( ) þ ý ü î í ì -¢ + - -+-¢º + jiLM sa saml jiLM lm mljl ilm mjl rrFrrFTК rrrr ee ee a 1 11111 , ( ) 12 11 1 1 1 )1( + ++ - º l j aj aj jl R ll l ee ee a , å -= ÷ ø öç è æ= 1 1 1010 1 1111 € 3 4 m m mmljlmjl EYEV ddpa r , ( ) 000000 €,, EEEEEE zyx rr ×== . The explicit form of the function f in (10) is not needed for further consideration. Second equation of (10) can be written in the matrix form [ ] VAK €€€1€ =+ or [ ] VKA €€1€€ 1- += , that allows us to interpret the matrix [ ] 1€1€€ - +º KM , which connects external potential matrix Vilm and multipole coefficients Ailm, as the multipole polarizability matrix of the MDS spheres. A single sphere near a substrate. The resonant frequencies. For a single sphere near a substrate, we can obtain the polarizability tensor in the dipole-dipole approximation by using (10): ( ) ÷ ÷ ÷ ø ö ç ç ç è æ -= ^a a a eeepa 00 00 00 3 4€ 3 II II aaR , (11) where ( )[ ]a e e ei a i aL i II= + - = ^ -1 ; ( , ) ; L li i a s a s = + - + æ è ç ö ø ÷ 1 3 1 e e e e ; ï î ï í ì ^= = ×= )(, 4 1 //)(, 8 1 i i h Rli h is the distance between the sphere’s center and the substrate. 226 Let us consider the case of Lorentz’s dielectric functions and ea = 1 (vacuum): ( )e w w w w gw = + - - 1 2 0 2 2 p i ; ( )e w w w w g ws ps s si = + - - 1 2 0 2 2 . The resonant frequency is obtained by using the condition ( )a wi res = ¥ . In our case it reduces to the following algebraic equation with respect to the frequency: 001 2 2 3 3 4 =++++ aaaa wwww , (12) where ( )3 sa i g g= + ( ) 2 2 2 2 2 0 0 2 2 2 2 1 0 0 2 2 2 2 2 2 2 2 0 0 0 0 0 1 1 3 2 1 1 3 2 1 1 1 1 3 2 6 s p ps s s s s p ps s s p ps i p ps a a i a l w w w w gg g w gw g w gw w w w w w w w w æ ö= - + + + +ç ÷ è ø æ ö= - + + +ç ÷ è ø = + + + - A solution to (12) neglecting damping ( )g g= =s 0 is ( ) ( ){ }3 2 2121 2 2,1 4 2 1 ylyyyy i i +-±+=w (13) where 23 ; 2 ; 3 22 3 2 2 2 2 2 01 pspps os p yyy vvv v v v ×=+=+= . Particularly, for a metallic sphere on the dielectric substrate from (13), using the inequality 1ps pw w << , we obtain the following approximate expressions ï ï î ïï í ì -+= += 2 )1()( 23 )( 2 2 0 2)2( 22 2)1( ps isres ps i p res l l v ww vv w (14) for the four (i=//,^) resonant frequencies. Note that 3 pv is well-known surface plasmon frequency of a sphere and 2 psv is one of a substrate. As we see, substrate changes the dipole moment of a sphere in such a way, that the four resonant frequencies arise in the absorption spectrum of a sphere. What causes arising of such a number of the resonant frequencies? First, one pair of the frequencies is observed when the field direction is parallel to the substrate, while another one – when perpendicular, and these two pairs don’t coincide in addition. In general case field has both the components and absorption spectrum has the four resonant frequencies respectively. Second, under certain field direction (// or ^ to the substrate) the pair of frequencies arises due to an interaction between surface plasmons of the sphere and of the substrate. Under increasing the distance between sphere and substrate this interaction vanishes and we obtain well-known result: a single sphere and a single half-infinite substrate absorb radiation at the frequencies 3 pv and 2 psv respectively. 227 Electrodynamical properties of percolating layers based on metal-dielectric composites In this part we consider electrophysical and optical properties of percolating system like a layer consisting of conducting and non-conducting inclusions of typical size a, at that its volume fractions are p and 1-p respectively. The layer size is supposed to be infinite in the longitudinal directions, and of thickness H in the transversal direction. The layer is bound by the planes z=0 and z=H and has cubic packing consisted of randomly arranged conducting (black) and non-conducting (white) cubes. Such a system structure leads to no loss of generality of further obtained results, because inclusion shape is insufficient near metal-dielectric percolation phase transition. The system has anisotropical electrodynamics properties from the very beginning, because it always has finite conducting cluster that connects z=0 and z=H planes, when H<<¥ and p is close to a critical value [7]. It is clear, that percolation threshold for longitudinal direction // cp is dependent from H and ( )3// cc pp ® when ¥®H , where ( )3 cp is percolation threshold for 3-D packing. In the case H=a we have )2(// cc pp = , where )2( cp is percolation threshold for 2-D packing. In our case, as it follows from [7], ( ) 3117.03 »cp (percolation threshold of site problem for cubic lattice) and 59275.0)2( »cp (percolation threshold of site problem for square lattice). At finite layer thickness a<<H<¥ the system properties are defined by the relation between H and the correlation length of 3-D percolating system 3 3 nx --» cppa , (15) where 9.03 »n is the critical index of the correlation length [7]. If 3x <<H, then 3-D system is isotropic and its electrodynamics parameters are independent from H. In this case evaluation of the parameters should be performed by using the mean field approximation [9] at a»3x and the scaling relations at a>>3x . Particularly, the layer conductivity s at )3( cpp > is given by 3)( )3( 1// t cpp -==^ sss , (16) where 26.13 ¸»t [7], and s1 is the specific conductivity of respective (black) cubes. At H>x the layer behaves itself like a 2-D system consisting of effective HHH ´´ blocks. Characteristics of the blocks can be calculated by the method of percolation renormalization group transformation (PRGT) [13, 14]. This transformation fulfils transit ion from percolating system consisting of elements of size a to the system, which is equivalent to that on macroscopical properties but consists of effective elements (blocks) of magnified n times sizes. The effective element involves nd elements of original system (d is the space dimension; in our case d=3). Applying of PRGT is able under condition 3x<<an . In the transformed system the fraction of effective conductors р' and its specific conductivity * 1s are equal to [13] ),( )3(/1)3(* 3 c v c ppnpp -+= (17) .33 / 11 * vtn-=ss (18) Note, that relations (17, 18) reflect supposition that percolation threshold )3( cp is the fixed point of PRGT and that correlation length x as well effective conductivity s (16) are conserving under PRGT. 228 Let’s now apply PRGT to the system and put aHn /= . Then we transit from percolating layer of thickness H=na to 2-D mosaic formed by effective blocks of size HHH ´´ (Fig. 2, b). The mosaic properties are defined by the relations (17, 18). Fig. 2. (a) - Percolating layer; (b) - 2-D percolating system obtained as result of PRGT. Because the percolation threshold )2( cp for 2-D mosaic is larger, than that for respective 3-D packing, percolation breakdown in the longitudinal direction occurs when the effective conductor fraction р* becomes equal to )2( cp . Therefore the percolation threshold of percolating layer is defined by the condition )()/( 3,\|\|, 3/1 3,2, cccc ppaHpp -+= n (19) and is equal to 3/1 3,2,3,\|\|, )/)(( n--+= aHpppp cccc (20) Dependence (20) of percolation threshold of layer on its thickness for cubic packing is shown on Fig. 2. It consents qualitatively with experimental results [14] obtained by studying conductivity of a layer of conducting and nonconducting spheres and with results [13] obtained in studying of metal-ceramic films Au-Al2O3. Fig. 3. Percolation threshold for longitudinal direction as a function of film thickness. a b . . a H L H H 229 Near the percolation threshold at H>3x transversal conductivity of the layer is defined by the effective conductivity s of the conducting blocks accounting for its fraction p: ( ) 33 / 1 )3( 1 )/( nsss t c aHppH -** ^ »= (21) This formula corresponds to fractal law of conductivity of isotropic percolating system of finite size. Longitudinal characteristics of the layer at H>3x are defined by the critical indices of 2-D systems. In particular, the correlation length is 2322 ///1)2( 2 )/( nnnn x --- -=-= cc ppaHappH (22) where the critical index 3/42 =n [14] and longitudinal conductivity at // cpp > is 2332 )()/()( \|\| /)( 12, * 1\|\| ttt c ppaHpp -=-= - nsss (23) where critical index 3,12 »t [7]. Above used PRGT method allows us, in our opinion, to consider more complicated problems too. Evaluation of effective permittivity of the layer is performed in the low frequency limit. In this case, starting with expression for permittivity of conducting conclusions [1] ( ) ( )nww w ewe i p + -= ¥ 2 11 , (24) that at wn > takes the form ( ) ( ) w wps ewe 1 11 4i+» ¥ , (25) where ( ) pn w ws 4 2 1 pº is conductivity of conclusions near the percolation transition, we obtain for effective values of ( )we1 the same relations as (25), but only with ¥1 ~e instead of ¥1e , and ( ) ( )wsws º1 ~ instead of ( )ws1 . The form of dependency ( )ws is evaluated earlier, and that of ¥1 ~e can be easily evaluated from the well-known relations [1]. Conclusions We obtained the general expression for the resonant frequency of the model system, which is a dielectric sphere in vacuum on a dielectric substrate. The latter results in splitting and shifting of the resonant frequency depending on a direction of the external field according to (13). This allows one to suggest that layers of small particles on a substrate possess anisotropic electrodynamical properties. Using PRGT method, we have developed the theory on calculation of conductivity and permittivity of metal-dielectric films of any thickness near the percolation transition. The percolation threshold was found as well as its dependency on film thickness and scaling dependencies (at H>3x ) of both the longitudinal and transversal conductivities on the thickness. Obtained theoretical results consent qualitatively with experimental data [12, 13]. Acknowledgement The authors acknowledge financial support from the National Science Foundation through the Faculty Early Career Development (CAREER) Award ECS-9624486 and the Eastern Europe Program Supplement. 230 References 1. Bohren C.F. and Huffman P.R. Absorption and Scattering of Light by Small Particles. New York: John Wiley & Sons, 1983. 2. Haarmans M.T. and Bedeaux D. The polarizability and the optical properties of lattices and random distributions of small metal spheres on a substrate // Thin Solid Films. – 1993. - V.224. - P.117-131. 3. Grechko L.G., Blank A.Ya., Motrich V.V., Pinchuk A.O., and Garanina L.V. Dielectric function of matrix dispersed systems with metallic inclusions: account of multipole interaction between inclusions // Radio Physics and Radio Astronomy. – 1997. - V.2. - P.19-27. 4. Grechko L.G., Pustovit VN., and Whites K.W. Dielectric function of aggregates of small metallic particles embedded in host insulating matrix // Appl. Phys. Lett. – 2000. - V.76. - P.1854-1856. 5. Varshalovich D.A., Moskalyov A.N., and Khersonsky V.K. Quantum Theory of Angular Momentum. Leningrad: Nauka, 1975 (in Russian). 6. Cruzan O.R. Translational addition theorems for spherical wave functions // Quart. Appl. Math. – 1962. - V.20. - P.33-40. 7. Stauffer D. and Aharony A. Introduction to Percolation Theory. Philadelphia, London: Taylor & Francis Inc., 1998. 8. Dubrov V.E., Levinshtein M.E., and Shur M.S. Anomaly of dielectric permittivity at metal-dielectric transition. Theory and modeling // ZhETF. - 1976. - V.70. - P.2014-2024. 9. Neimark A.V. Bidispersed percolating system // J. Tech. Phys. – 1989. - V.59. - P.22-26. 10. Straley J.P. Critical exponents for the conductivity of random resistor lattices // Phys. Rev. B. - 1977. - V.15. - P.5733-5737. 11. Neimark A.V. Electrophysical properties of a percolating layer of finite thickness // ZhETF .- 1990. - V.98. - P.611-627. 12. Clerc J.P., Giraud G., Alexander S., and Guyon E. Conductivity of a mixture of conducting grains: Dimensionality effects // Phys. Rev. B. - 1980. - V.22. - P.2489-2494. 13. Wilkinson D., Langer J.S., and Sen P.N. Enhancement of the dielectric constant near a percolation threshold // Phys. Rev. B. – 1983. - V.28. - P.1081-1087. 14. Zhang X. and Stroud D. Optical and electrical properties of thin films // Phys. Rev. B. – 1987. - V.20. - P.2131-2137.

Optical properties of metal-composite-based thin films

Institution

Ähnliche Einträge