Electrodynamic linear response of a superconductor film located at the surface of semiconductor

The susceptibility of thin superconductor film was obtained. The absorption of energy of external electromagnetic field by superconductor film located at a semiconductor substrate was calculated. A new approach to calculation of the dispersion relation for eigenmodes of a strong inhomogeneous system...

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Datum:	2004
Hauptverfasser:	Lozovski, V., Reznik, D.
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Sprache:	English
Veröffentlicht:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2004
Schriftenreihe:	Semiconductor Physics Quantum Electronics & Optoelectronics
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spelling	irk-123456789-1181122017-05-29T03:03:54Z Electrodynamic linear response of a superconductor film located at the surface of semiconductor Lozovski, V. Reznik, D. The susceptibility of thin superconductor film was obtained. The absorption of energy of external electromagnetic field by superconductor film located at a semiconductor substrate was calculated. A new approach to calculation of the dispersion relation for eigenmodes of a strong inhomogeneous system was developed. The dispersion relations of eigenmodes of a superconductor thin film on the semiconductor substrate was calculated using the developed approach. 2004 Article Electrodynamic linear response of a superconductor film located at the surface of semiconductor / V. Lozovski, D. Reznik // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 1. — С. 36-42 — Бібліогр.: 24 назв. — англ. 1560-8034 PACS: 42.25.Dd http://dspace.nbuv.gov.ua/handle/123456789/118112 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The susceptibility of thin superconductor film was obtained. The absorption of energy of external electromagnetic field by superconductor film located at a semiconductor substrate was calculated. A new approach to calculation of the dispersion relation for eigenmodes of a strong inhomogeneous system was developed. The dispersion relations of eigenmodes of a superconductor thin film on the semiconductor substrate was calculated using the developed approach.
format	Article
author	Lozovski, V. Reznik, D.
spellingShingle	Lozovski, V. Reznik, D. Electrodynamic linear response of a superconductor film located at the surface of semiconductor Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet	Lozovski, V. Reznik, D.
author_sort	Lozovski, V.
title	Electrodynamic linear response of a superconductor film located at the surface of semiconductor
title_short	Electrodynamic linear response of a superconductor film located at the surface of semiconductor
title_full	Electrodynamic linear response of a superconductor film located at the surface of semiconductor
title_fullStr	Electrodynamic linear response of a superconductor film located at the surface of semiconductor
title_full_unstemmed	Electrodynamic linear response of a superconductor film located at the surface of semiconductor
title_sort	electrodynamic linear response of a superconductor film located at the surface of semiconductor
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate	2004
url	http://dspace.nbuv.gov.ua/handle/123456789/118112
citation_txt	Electrodynamic linear response of a superconductor film located at the surface of semiconductor / V. Lozovski, D. Reznik // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 1. — С. 36-42 — Бібліогр.: 24 назв. — англ.
series	Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv	AT lozovskiv electrodynamiclinearresponseofasuperconductorfilmlocatedatthesurfaceofsemiconductor AT reznikd electrodynamiclinearresponseofasuperconductorfilmlocatedatthesurfaceofsemiconductor
first_indexed	2025-07-08T13:23:11Z
last_indexed	2025-07-08T13:23:11Z
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fulltext	Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 1. P. 36-42. © 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine36 PACS: 42.25.Dd Electrodynamic linear response of a superconductor film located at the surface of semiconductor V. Lozovski, D. Reznik V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, 41, pr. Nauky, 03028 Kyiv, Ukraine E-mail: vlozovski@isp.kiev.ua Abstract. The susceptibility of thin superconductor film was obtained. The absorption of energy of external electromagnetic field by superconductor film located at a semiconductor substrate was calculated. A new approach to calculation of the dispersion relation for eigenmodes of a strong inhomogeneous system was developed. The dispersion relations of eigenmodes of a superconductor thin film on the semiconductor substrate was calculated using the developed approach. Keywords: linear response, superconductivity, local field, surface, energy absorption. Paper received 25.11.03; accepted for publication 30.03.04. 1. Introduction The superconductor-semiconductor systems are actual because the extremely low level of self-noises is the main characteristic of the systems. In this connection the elec- trodynamics of thin superconductor film at semiconduc- tor substrate is the actual problem. The study of super- conductor-semiconductor systems is of interest now. Nu- merous works are devoted to consideration of different features in behavior of the superconductor-semiconduc- tor systems. The works touch upon both fundamental and application aspects of the problem as well. In [1] the thin superconducting LaSrCuO film electrical and optical properties were studied. In [2] presented were some su- perconductor-semiconductor devices. It was, for exam- ple, the superconductor straight-line resonator that pos- sesses has an electromagnetic feed-through level below �65 dB up to 10 GHz. In this work, a semiconductor- superconductor microwave digital phase shifter based on an YbaCuO thin film was demonstrated. The organic layered superconductor composites are synthesized and studied (see, for example [3]). The electrodynamical prop- erties of various superconducting structures are inten- sively studied, too. Thus, in Ref.[4] the measurements of the emitted spectra for electromagnetic waves of the tera- Hertz range from tunnel-injected non-equilibrium YbaCuO superconductor were carried out. The observed spectra have a single broad-picked structure with the peak frequency about 5 THz. The linear and nonlinear electromagnetic properties of superconducting systems are widely studied (see, for example [5]). All these studies suggest that the study of interaction between external elec- tromagnetic field and the system containing supercon- ductor nano-systems (including thin films) is very impor- tant and actual problem. The characteristic dispersion properties for supercon- ductors obviously are in the range of frequencies close to superconducting energy gap, because the state density of superconductor quasi-particles is rather high just near the energy gap. For the most �classic� superconductors these frequencies are small (1011�1012 Hz) as comparable to characteristic frequencies of semiconductor (the opti- cal phonons frequencies, for example). One can think that more interesting could be the systems consisting of high-TC (HTSC) structures at a semiconductor substrate. These systems are characterized by the gap frequencies about 1013�1014 Hz that lie in the range of characteristic optical phonon frequencies of semiconductors. As a re- sult, for superconducting thin film with the thickness h ≈100 nm being comparable with the Pippard penetra- tion depth, the wavelength of electromagnetic waves cor- responding to the resonant range of the frequencies are much larger than h. Then, the near-field approximation can be used in the studies of these systems. On the other hand, at so small distances the local field are rapidly varied and these inhomogeneities have to be taken into V. Lozovski, D. Reznik: Electrodynamic linear response of a superconductor film ... 37SQO, 7(1), 2004 account. Then the non-local electrodynamical interac- tions inside the system can result in arising the local os- cillations. These oscillation processes are caused by in- teraction between electromagnetic field and local cur- rents inside the system. The local currents depend on elec- tron and phonon excitations. Then, the oscillating elec- tromagnetic field is characterized by the frequencies de- pendent on the electronic properties of the system as well as the shape and dimension of superconducting system. The generalized excitations that arise in the system are eigenmodes of the system. These excitations character- ize the system as a whole. The dispersion relation of these oscillations - the connection between the resonant fre- quencies and wave vector of the external field that causes the oscillations - are the important characteristic of the system. The dispersion relations define the response of the system on the external electromagnetic field and are the most important characteristic. This work is devoted to calculations of the non-local effective susceptibility of thin superconductor film at the semiconductor surface. The calculations are made in the frame of the near-field approximation [6] in the meso- scopic approach. Unlike most of works where the elec- trodynamical properties of thin films are considered (see, for example [7]), the developed approach is based on the solution of the Lippmann-Schwinger equation for self- consistent field [8], which enabled us to calculate the ef- fective susceptibility directly. The main idea of determi- nation of the dispersion relations in the inhomogeneous system is the calculation of the absorbed energy by the system of external electromagnetic field, because this absorption have to extremely increase when frequency and wave vector of external field correspond to eigenmodes of the system. 2. Dispersion relations The eigenmodes of a system are characterized by both time parameter (frequency � ω) and space propagation parameter (wave vector � k r ). Then, the eigenmode is com- pletely determined if the relation between the frequency and wave vector is known. This dispersion characteristic can be very simply determined for a homogeneous sys- tem. Indeed, in this system the Fourier transformation can be performed. In this case, the susceptibility of the system is dependent on ω and k r . As it is well known, the dispersion relations in this case can be determined by putting the pole part of susceptibility to zero [9, 10]. This point has very simple explanation. Namely, let one de- fines the transmitting function of the system that con- nects the linear response of the system with the external (exciting) field � the effective susceptibility. In the case of homogeneous system, this connection can be presented as ),(),(),( )( ωωω kEkkJ ext rrrtrt Χ= , (1) where ),( ωkJ rr is the response of the system to external field ),( ωkE ext rr , ),( ωk rt Χ is the effective susceptibility. The excitation of the eigenmodes means that action of infinitesimal field caused the nonzero response. It can be realized only if at 0),( →ωkE ext rr , ∞→Χ ),( ωk rt . Because the effective susceptibility is connected with an initial susceptibility � the linear response on the total (local) field ),( ωχ k rt � via self-energy part ),( ωkS rt as [ ] 11 ),(),(),( −− −=Χ ωωχω kSkk rtrtrt , (2) then the condition of arising the eigenmodes can be writ- ten in the form [ ] 0),(),(det 1 =−− ωωχ kSk rtrt . (3) Eq. (3) is the dispersion relation that gives us the set of the wave vectors and frequencies at which action of in- finitesimal external field generates nonzero local cur- rents. In the case of inhomogeneous systems, the effective susceptibility can not be a simple function of the wave vector. The effective susceptibility will depend on the coordinate of a point in which the local current is deter- mined. In other words, the problem is that the effective susceptibility is not the global characteristic of the sys- tem as it was in the homogeneous system case. The effec- tive susceptibility here only describes the response of the point in the system on the external excitation. Obviously, this characteristic can not give us the information about general oscillating processes in the system. As an ana- log, the local current in a single element of an oscillating contour does not characterize the oscillating process in the whole contour. Because the superconductor is characterized by a nonlocal response on the total field, the problem of deter- mining the dispersion relation for a thin superconductor film falls into the kind of the problems mentioned above. The way that can lead to solution of the problem is in the finding the global characteristic of the system. That char- acteristic, obviously, can be the energy of external field absorbed by the system. It is clear that the resonance condition � or excitation of an eigenmodes leads to ex- tremely strong absorption of energy of the external field by the system. The idea of calculation of the absorbed energy value to establish the eigenmode dispersion rela- tions lies in the base of the present consideration. 3. Linear response on the external field To solve of the problem of determining eigenmodes of the system under consideration, one needs to calculate the absorption of energy of external electromagnetic field acting on the system. For this purpose, one needs to in- troduce the linear response of the bulk superconductor on the local field ),( ωχ RR ′ rrt . It is clear that this charac- teristic is the characteristic of the material. This linear response function connects the local currents inside the superconductor medium with the local electromagnetic field. In a general case of nonlocality, this connection has the form 38 SQO, 7(1), 2004 V. Lozovski, D. Reznik: Electrodynamic linear response of a superconductor film ... ),(),,(),( ωωχωω RERRRdiRJ V ′′′−= ∫ rrrrtrrr , (4) where integration is performed over the particle volume. We shall represent the system under consideration as the superconductor film located in any medium. Then, for this system integration in Eq. (4) should be fulfilled over the film volume. For definiteness, we suppose that the me- dium consists of two semi-spaces with flat interface � the semiconductor substrate and external medium (Fig. 1). The electrodynamical properties of the medium are characterized by the Green function ( )ω,,rrG ′rrt . As it is well known, for the considered medium, the Green func- tion consists of two parts [11,12] ),(),,(),( ωωχωω RERRRdiRJ V ′′′−= ∫ rrrrtrrr . (5) The first of them named as the direct part of the Green function describes the photon propagating in the upper medium. The second part of the Green function is named as indirect part. In far zone, it describes the reflection and transmission processes caused by the interface. To calculate the energy of external field absorbed by the system under consideration, one needs to know the effective susceptibility that is the mean linear response on the external field. To perform this calculation one should consider the equation for self-consistent field. This equation is usually named as the Lippmann-Schwinger equation [6,12] and for the system under consideration has the form ( ) ( ) ( ) ( ) ( )ωωχω ω ε ωω ,,,,,,,, 1 ,,,, 00 2 2 0 )( zkEzzkzdzzkGzd c zkEzkE ljl h ij h ext ii ′′′′′′′′′− −= ∫∫ rrr rr , (6) with k r wave vector in the plane of substrate surface, and ),,,( ωωχ zzki jl ′′′− r bulk nonlocal tensor of conductivity. In Eq. (6), the fact that the system in the plane of the interface is homogeneous was used for performing the Fourier transformation in the plane of the film (XOY plane, Fig. 1). Integration is made over the thickness of the film. The conductivity tensor connects the local cur- rents with self-consistent local field ( ) ( ) ( )ωωχωω ,,,,,,, 0 zkEzzkzdizkJ lil h i ′′′′′′−= ∫ rrr . (7) The conductivity can be calculated in the microscopic approach. For example, this conductivity was calculated for BCS model of superconductivity in the frame of RPA in the work [13]. Using modified UV transformations method, the conductivity was calculated for supercon- ductors with a strong coupling [14]. The analytical solution of Eq. (6) can be written in the terms of effective susceptibility ),,,( ωχ zzkij ′ r or lin- ear response on external field that connects the local currents caused by external field with this external field ( ) ( ) ( )ωωωω ,,,,,,, )( 0 zkEzzkzdizkJ ext lil h i ′′′′Χ′′−= ∫ rrr . (8) The solution can be obtained in the framework of the method developed in the works [8,15] and has the form ),,(),,,(),,,( ωωχω zkQzzkzzk ljilij ′′=′Χ rrr , (9) with [ ] 1 ),,(),,( −′+=′ ωδω zkSzkQ jljllj rr . (10) The expression ),,( ωzkS jl ′ r has the meaning of self- energy part and describes the corrections of a pole part of effective susceptibility by electrodynamical interac- tions in the system. For monochromatic external field, the self-energy part can be written in the form ( ) ( ) ( )[ ],exp,,, ,,,),,( 0 0 2 0 2 ∫ ∫ ′′′−′′′′′′′′′× ×′′′′′=′ h Zkl h jkjl zzikzzkzd zzkGzd c zkS ωχ ω ε ωω r rr (11) where kz is z-component of the wave vector of external field. If the external field is the plane wave with the inci- dent angle ϑ, then ϑω cos)/( ckz = . Then, the linear response on the external field (the effective susceptibility) allows us to calculate the local fields and currents in the system. Namely, the self-con- sistent (local) field in arbitrary point in the zk −, r space can be calculated according to ( ) ( )ωωω ,,),,,(,, )( zkEzzkLzkE ext lili ′′= rrr , (12) with the local-field operator h Z X s w( ) PE SE Jp −2/ J k Y Fig. 1. Sketch of the system under consideration. V. Lozovski, D. Reznik: Electrodynamic linear response of a superconductor film ... 39SQO, 7(1), 2004 ( ) ( ) ( )...,,,,,, 1 ,,, 2 2 0 , ωχω ω ε δδω zzkzdzzkGzd c zzkL jl H ij H zzilil ′′′′′′′− −=′ ∫∫ ′ rr r . (13) One should note that representation of self-consistent field in the term of local-field operator instead of the stand- ard representation via the local-field factor [12] caused by nonlocal nature of superconductivity requires to use the integral nonlocal constitutive equations [Eq. (7)] 4. Absorption of external field energy Let us define the dissipative function as the energy of external electromagnetic field absorbed by the unit vol- ume of the system for unit time. Then, normalized dissipative function can be written as ( ) extI EEJJ k >++< =Ω ))(( , ** rrrr r ω , (13) where <�> means the averaging over the volume and (...) over time, and Iext is the intensity of external field. Substituting Eqs. (8) and (12) into this equation and ta- king into account that ( ))()( ext k ext j ext EEI = , one can ob- tain [ ] , ),(),,( ),(),,( )( )(* 00 kj zzik ijik zzik ikij hh Z Z ezLzz ezLzzzddz h i ττωω ωω ω ′−− ′− ′Χ− −′Χ′−=Ω ∫∫ (14) where τi is the unit vector of the external field polari- zation. It should be noted that in the case of a local constitu- tive equation (London-type superconductor) ),,()(),,( ωωωχω zkEizkJ lili rv −= (15) the connection between the current and external field has the form ( ) ( ) ( )ωωωω ,,,,,, )()( zkEzkXizkJ ext l loc ili rrr −= , (16) where ),,()(),,( )()( ωωχω zkQzk L klik loc il rr =Χ , (17) and 1 0 )( 2 0 2 )( )(),,,( ),,( − ′−         ′′+= = ∫ h zzik mllmlk L kl zezzkGzd c zkQ ωχω ε ω δ ω r r (18) Using Eqs.(15) and (16) one obtains the connection between local and external fields ( ) ),,(),,()(),,( )(1 ωωωχω zkEzkXzkE ext lilijj rrr −= . (19) Then, the dissipative function of the film of London- type superconductor is ( ) ( ) ] ( ) .),,(),,( ),,()(),,( )(),,(),,( 1 )()( 1 1** 0 ωω ωωχω ωχωωω zkEzkE zkzk zkzkdz h i ext j ext j kjkiji ikjkij h rr rr rr × ×ΧΧ− −  ΧΧ−=Ω − − ∫ (20) Taking into account Eq. (17) one obtains ( ) ( ) .),,( ),,()(Im 1 2),( *)( 0 )( lk L jl h L ikij zkQ zkQdz h k ττω ωωχωω r rr × ×=Ω ∫ (21) Because electrical susceptibility for superconductor is a real value, Eq. (21) shows us that superconductor film of London-type can not absorb energy of external field. This, at first view, strange result has very simple explanation. Namely, for the nano-scale objects, the lo- cal field is a very rapidly varying value. Then, all the interactions inside the object are nonlocal. In this con- nection, one can assert that only nonlocal models for su- perconductivity one has to consider for studying electro- dynamical interactions in the system containing the small superconducting particles (or thin films). To analyze the dispersion relations of the small object, one should to attentively consider the experimental set up to excite any eigenmode in the bounded system. Let us, for example, consider the experimental set up for surface wave excitations. The most widely used method of excitation of surface plasmon-polariton is the method of attenuated internal reflectivity (Fig. 2) in which the probing beam radiates the film placed at the surface of ATR-prism. The intensity of reflected radiation is measured as a function of the incidence angle ϑ. The excitation of the surface wave that occurs at the angle ϑR is characterized by abrupt decrease of the reflection signal at the detector. This surface plasmon-polariton resonance is character- ized by strong absorption of the external field energy by 1 surface wave 3 2 4 J Fig. 2. Set up for excitation of surface wave by the attenuated total internal reflectivity (ATR) method. 1 � light source (la- ser); 2 � ATR prism; 3 � detector; 4 � the film in which the surface wave is excited. 40 SQO, 7(1), 2004 V. Lozovski, D. Reznik: Electrodynamic linear response of a superconductor film ... the system under consideration. It means that the dissipative function has a maximum at any values of fre- quency ω and wave vector Rprism ck ϑωε sin)/(= . It is obvious that the set of pairs frequency-wave vector com- posite the curves that are the dispersion relations of eigenmodes. To demonstrate this, method we shall cal- culate the absorption of the external electrodynamic field energy for thin superconducting film placed at the sur- face of semiconductor. 5. Numerical calculations and discussion To calculate the energy absorbed by the film, one needs to know the effective susceptibility and local-field opera- tor [Eqs. (9) and (14)]. From the other side, to calculate these values, one should know the linear response to the local field ),,( ωχ RRij ′ rr . One should perform a calcula- tion of the function ),,( ωχ RRij ′ rr in the frame of any mi- croscopic model. There are a big number of models (see, for example, Refs [17�21]) in the frame of which the nu- merous attempts to describe the HTSC phenomena were performed. Despite this fact, most of the properties of HTSC materials can be rather adequately described us- ing the phonon model with strong coupling. Then, for simplicity, we shall use this model [22, 23]. Here, we shall use the approach proposed in [16] that is based on the modified u-v transformations. The linear response to the local field ),,( ωχ RRij ′ rr for strong-coupling supercon- ductor was calculated in the framework of RPA using modified u-v transformation method [16]. Using this non- local linear response function, one can calculate the en- ergy of external field absorbed by superconducting film at the semiconductor substrate. Then, we shall assume that thickness of the supercon- ductor film is much less then the wavelength of the exter- nal field. At the same time, to use mesoscopic approach to describe the electrodynamical properties of the sys- tem, one needs to suppose that film thickness is much larger than the lattice constant or Dkh /1>> (kD is the Debye wave vector). Moreover, the thickness h should be comparable to the Pippard penetration length ΛP. This requirement provides the penetration of the field into the whole thickness of the film. These requirements are rather good satisfied for HTSC, where the values of energy gap are rather great ( Hz1010/ 1513 ÷≈∆= hgω ) and the Cooper pair dimensions are about few nanometers. Then the films with the thickness m1010 97 −− ÷≈h are satisfy the abovementioned conditions. In the frame of random phase approximation (RPA), the linear response function can be written in the form [13] ( )                 ⊗ +    += 2 22 2 ),( k kk U k U m ne k rr ttrt ω ωαωχ , (22) where m and e are the mass and charge of the charge carriers, respectively. Symbol ⊗ means the direct tensor product. U t is the unit tensor. The calculations were made in the frame of the so-called method of the pseudo-vacuum Green function [12]. The main idea of the method con- sists in consideration of the object (here it is the film) situated in any medium (pseudo-vacuum), electrodynami- cal properties of which are supposed as known. In the case under consideration, this medium consists of two semi-spaces. Upper semi-space is vacuum and lower semi- space is semiconductor substrate characterized by dielec- tric function ε1(ω). For definiteness, we suppose that the dielectric function has the fallowing form : ( ) ωγωω ωω εωε iTO TOLO −− − = ∞ 22 22 1 , (23) where ε∞ is the high frequency dielectric constant; ωLO and ωTO are the frequencies of longitudinal and trans- verse optical phonons, respectively; γ is decay constant. As it was mentioned above, the electrodynamical Green function of the medium in which the film is situ- ated (two semi-spaces with flat interface) consists of two parts and has the form [Eq. (5)]. Because the system un- W w( , )q 1 2 w ( )q w ( )q w 1 2q q s Fig. 3. The method of dispersion relations determination. The profile of the dissipative function in the cross-section q = qs (top). The dissipative function image in Ω, q, ω � space (bottom). The projection of the local maxima points on the q, ω � plane are the functions ω1(q) and ω2(q) play the role of two branches of disper- sion relations of eigenmodes of the system. V. Lozovski, D. Reznik: Electrodynamic linear response of a superconductor film ... 41SQO, 7(1), 2004 der consideration is homogeneous in the plane of inter- face, it is convenient to use the so-called −− zk , r represen- tation. In this representation, direct and indirect parts of the Green function in the coordinate system where )0,( === yx kkkk r can be written in the form (k = k0sinϑ) ,)( ),,( ),,( 2 0 \|\|0 zz k uu ezzkgzzkD zz zzi ′− ⊗ − −′−=′− ′− δ ωω η rr rtrt (24) and . )/(0 0)/(0 0 2 ),,( 0 2 0 2 0 0 )(0           −⋅− ⋅ = =′+ ′+ pp s pp zzi ij RkRk Rk RkR ie zzkI η η η π ω η (25) In these equations, the next designations are used: , /0)( 0/0 )(0 2 ),,( 0 2 0 2 0 0 2 0           ′−⋅− ′−⋅− = =′− η η η ω kzzsignk k zzsignk k i zzkg rt (26) and 22 0 kk −= εη , 22 00 kk −=η . Moreover, 0 0 εηη εηη + − =pR , ηη ηη + − = 0 0 sR are the Fresnel coefficients for reflection of p- and s- polarized waves, respectively. The results of numerical calculations are shown in Figs. 4 and 5. In Fig. 5, the projections of the ridges of dissipative function �mountains� are shown. The lines of these projections are the dispersion relations of electro- magnetic waves localized at the superconductor film. We can see that three branches of plasmon oscillations caused by the complicated structure of the dispersion law of su- 2 6 10 14 w w ´/ , 10�3 H q p/( /2) 1E�6 1E�5 0.8 0.6 0.4 0.2 a b so p ti o n c o e ff ic ie n t 2 6 10 14 w w ´/ , 10�3 H q p/( /2) 1.0 1.1 0.8 0.6 0.4 0.2 a b so p ti o n c o e ff ic ie n t, 1 0 × � 6 Fig. 4. The view of the normalized dissipative function of super- conducting film at a semiconductor substrate for different polarizations of external field. The top pannel corresponds to absorption of p-polarized wave. The bottom panel corresponds to absorption of s-polarized wave. 0.2 0.4 0.6 0.8 2 4 6 8 10 12 14 w w ´ / , 1 0 � 3 H q p/( /2) Fig. 5. The dispersion curves of the waves p-polarized (top) and s-polarized (bottom) localized at the superconducting films, which are the eigenmodes of the system. 0.2 0.4 0.6 0.8 2 4 6 8 10 12 14 w w ´ / , 1 0 � 3 H q p/( /2) 42 SQO, 7(1), 2004 V. Lozovski, D. Reznik: Electrodynamic linear response of a superconductor film ... perconductor which are only slightly renormalized by electromagnetic interactions in the case of excitation of s-polarized waves, in the case of p-polarized eigenmodes change very strongly. Namely, the interaction with the external field (its dispersion law is shown by the dashed line) of the plasmon oscillations results in to splitting and occurring the energy gaps in the dispersion relations of eigenmodes. One should to note that the question of sta- bility of these excitations should be considered in a spe- cial study. References 1. M. Suzuki, Hall coefficients and optical properties of La2�xSrxCuO4 single-crystal thin films // Phys. Rev. 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Electrodynamic linear response of a superconductor film located at the surface of semiconductor

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