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The main purpose of the work is to clarify the physical mechanisms which leads to the specific spectrums of the structures [1-3]. The work is based on the ideas of the effective susceptibility. The effective susceptibility of cone-shaped nanohills located at the semiconductor surface is obtained in the frame of local-field approach. The knowledge of the effective susceptibility allows to calculate the optical absorption profiles. Using the approach similar to Levshin rule the photoluminescence spectra were calculated. Obtained results were compared with experimental luminescence spectra obtained earlier. The significant machanisms, that define the peculiarities of the spectra, is the shape of the nanoclasters and the inhomogeneity of the nanohills array are justificated. The main issue of the work is that the cause of the luminescence spectrums has electodynamical nature without spatial quantisation effects.

Nowadays the experiments of luminescence of the nanohills fabricated by laser beam scanning along the silicon surface were reported [1-3]. The effect of luminescence by the nanohills in the works [1-3] was explained by existing varyband structure of the nano-rods. The spatial quantization is the cause of luminescence effect in the nanohills. This explanation seems to us rather not wellgrounded and, of course, it is only for qualitative. Indeed the electron state of the quantum dot is formed in the particle as a whole. It means one should obtain the electron eigenstates inside the cone-like quantum dot and, then, calculate the transition currents, effective susceptibilities and luminescence spectra. The developing of correct microscopic model for explanation of discussed results becomes very hard problem. However, one can get round the difficulties using the mesoscopic approach using the local-field method. In the frame of so-called effective susceptibility concept [

The scanning by the laser beam of the semiconductor surface can lead to fabrication of the systems of the nanohills shaped as a cone, which was reported in [1-3]. It is well known, that the mechanical stresses appear when the nano-objects are fabricated at the surface [8-11]. The stress fields lead to distortion of the electron and optical properties of the nano-systems [12,13].

Specifically, these distortions become apparent in shifts of the absorption edge and transform the indirect band energy structure to the direct one [12,13]. This fact leads, in part, to ability of the luminescence observation in Si nano-particles [

The paper is organized as follows. In Section 3 we give a brief description of the local field method and obtain the single meso-particle effective susceptibility. In Section 4 we obtain the formula of the effective susceptibility of the inhomogeneous two-dimensional array of the particles in the frame of Green function formalism. And in Section 5 the results of our calculations by semi analytical method are presented and compared with the experimental data.

For solving the problem of the calculation of the linear response to the external field (so-called, effective susceptibility) of the single meso-particle at the surface of a solid (

where is an electric component of external field, is the photon propagator, describes the electromagnetic field propagating in the medium in which the particle is embedded. This medium is the substrate with flat surface in the case under consideration.

Because of connection between the local current and the local field which usually named as constitutive equation,

where is susceptibility of the bulk material from

which the particle is fabricated, Equation (1) transforms to self-consistent equation for local-field which usually named as Lippmann-Schwinger equation [5,14,15],

The introduction of the effective susceptibility which connects the local current with the external field,

allows us to write the solution of Lippmann-Schwinger equation in the form,

with the dimensionless effective susceptibility:. The effective susceptibility of the single meso-particle at the surface of the solid can be calculated according to the equation obtained in [

where photon propagator corresponds to a two semi-spaces with the ideal flat interface. This expression will be initial one for calculations provided in the present work. The detail analysis of the effective susceptibility in form Equation (6) was provided in [

determinant, and equation

defines the eigenmodes in the system. It is clear that because integral over the particle volume in Equation (6) the behavior of the effective susceptibility of the particle will be strongly depended on the particle shape and size. The behavior of the effective susceptibility can be studied numerically. For example, the imaginary part of averaged over the volume of the particle for the single cone-like particle is depicted in the

It is clear that experimentally obtained meso-particle arrays are characterized by the particles with the different

shapes and dimensions. Then, using the result of previous section, one should calculate the effective susceptibility of the meso-particle array. To calculate the effective susceptibility of the meso-particles array one should consider the self-consistent equation,

where the summation is over the different particles.

Let us suppose that the meso-particles form the rare array, which means that average distances between the particles are larger than linear dimensions of them. Taking into account the results of the works [14,19,20], one can consider the meso-particles as the point-like dipoles which polarises as the nonpointness object with the defined shape and dimension. Thus the second term in the right part of Equation (8) can be reduced to

with

is the averaged polarizability of the single particle of α-th kind at the surface. In general, the mesoparticle arrays consist of the particles having different dimensions and differently shapes. It means that the sum in Equation (9) splits to the double sum, internal term of which is the sum of particles with the same parameters (named by the α-th subsystem),

where is a number of particles in the α-th subsystem, and s is a number of the subsystems. It is naturally to suppose that the particles are distributed homogeneously along the surface of the substrate. Then, one should average Equation (9) over the particles positions. For this purpose one should, first, to perform Fourier transformation of Equation (8) taking into account Equation (9), due to lateral homogeneity. Let one average out Equation (9) by the particle positions, using Equation (10). Then, taking into account that occupied position can’t be occupied by the other particle, one should use the uniform distribution [14,19]. Because the meso-particle array can be considered as quasi-uniform spatial system in the plane of the substrate (XOY plane), one can transmit to so-called k-z representation [

where r and k are the coordinate and wave vector in the XOY plane, respectively, S is the area of the substrate. To calculate this expression one should to consider summand of the sum in right part of Equation (11), which has a form,

As a result, the right part of Equation (9) is written as

Then, the equation of the self-consistent field written for the system under consideration has form

where

is concentration of the particles of α-th type. The dependence between the induced subsystem dipole moment and the external field,

Substituting Equation (15) in Equation (14) one can obtain the polarization of the β-th subsystem

Then, the polarizability (the effective susceptibility) of the total system has a form,

This equation will be used for further calculations of the absorption and luminescent spectra of the system under consideration. Present paper calculation based on the method developed in [

All characteristics that include Equation (18) are volume averaged already, and due to time averaging,

Let consider expressions,

where

is local field factor.

Formulas above are written in representation, they must be convenient to use in Equation (19). Thus it is necessary to perform inverse Fourier transformation. Due to suppositions considered above they has simple view,

Substituting Equations (22) and (23) into Equation (19) one obtains the dissipative function of nanoparticle array in the form,

Include into consideration distribution function

where is some parameter of the particle.

Final view of absorption spectra formula

where is effective susceptibility of the single particle with the parameter, which founded in previous section.

Numerical calculations were performed in the frames of the model of the bulk material susceptibility with corrections discussed above. The susceptibility of the bulk silicon and germanium were taken from [13,16]. As could be seen from the experimental data (green colored lines at Figures 4 and 5) the range of luminescence spectrum is about 1 eV near the absorption edge from 1.2 eV to 2.3 eV. That is why in the theoretical calculations only absorption edge were taken into consideration with the frequency shift about 0.2 eV and grows of the intensity about 2 units. In the model we use the different distribution functions which fulfill different models of growing of the structures to find the most suitable. The averaged dimension of the particles were chosen as the next: Radius of the base of the cone was equal to, the averaged height of the cone-like particles was equal to. It was supposed that heights of the cone-like particles are varies from to. There different shape distributions of the particles were used in the present work. Gauss distribution function has form

with the constant being founded from the relation

, where is the total number of the particles in the array. Lifshits-Slezov-Wagner distribution function can be represented in the form [

where, is the maximal dimension of the particle in the array, x is the parameter of the model depends on the kinetic properties of the different materials () [

The luminescence spectra for structures 1) SiO_{2} (substrate)/Si(particle) and 2) Si_{0.7}Ge_{0.3}/Ge are depicted in Figures 4 and 5, green colored lines are experimental data. For estimation the luminescence spectra via absorption profile we use so-called Levshin rule [6,7,21],

where is the fenomenological constant. From the form of the structures obtained by the laser beam scanning along the silicon surface [1-3] one can see that interparticle distances are comparable with wave length. As we can see from the experimental spectra wave length at the left side of the spectra two times larger than at the right side. Moreover intensity of the diffuse scattering for the small wave length is larger. That is why the effective intensity of radiation at the detector must be smaller. For the estimation of the theoretical curve one uses the phenomenological function which describes intensity at the detector. The function must satisfy the next conditions: 1) The scattering processes are not appreciable for the long wave length photons. It means the function must be constant in the long wavelength limit; 2) When the wavelength is comparable with inhomogeneties of the structure the function must abate,

where is the phenomenological energy which corresponds to the critical wavelength, and which corresponds to range of the frequencies of the effective scattering.

It could be easily seen (_{0.7}Ge_{0.3}/Ge) we can see the same features at the low frequency range and it is comparable with experimental data. It is easy to see that both SiO_{2}/Si and Si_{0.7}Ge_{0.3}/Ge have similar peculiarities in the same region of the photon energies. It allows us to think that the nature of the features observed above are similar.

The model of two dimensional array of cone-shaped meso-particles for explanation of the experimental data reported earlier by the authors in [1-3] is proposed. As opposed to [1-3] where obtained the results were explained by the spatial quantization effects here one proposed the additional approach based on taking into account the local-field interactions inside the particles and the interparticle interactions. The local-field interactions inside the particles were considered in the frame of effective susceptibility concept in the near-field approximation. The interparticle electro dynamical interactions were considered rather strongly by solution of Lippmann-Schwinger equation for the system under consideration with what taken into account as nearas middleand far-field interactions. Different models of the particle distributions—Gaussian-like distribution and Lifshits-Slezov-Wagner distribution were used. The both models of the particle distribution give qualitatively correct results. But the LifshitsSlezov-Wagner distribution model is the most suitable. This fact allows us to conclude that particle distribution obtained by laser heating [1-3] of SiO_{2} surface is described by Lifshits-Slezov-Wagner distribution with x = 1. In the case of Si_{0.7}Ge_{0.3} surface the luminescence spectrum is described by Wagner distribution. Comparison of calculated and experimentally obtained luminescence spectra allows us to think that the luminescence of the nanohills can be explained by the local-field effects.