Size Effects on Surface Elastic Waves in a Semi-Infinite Medium with Atomic Defect Generation

Abstract

The paper investigates small-scale effects on the Rayleigh-type surface wave propagation in an isotopic elastic half-space upon laser irradiation. Based on Eringen’s theory of nonlocal continuum mechanics, the basic equations of wave motion and laser-induced atomic defect dynamics are derived. Dispersion equation that governs the Rayleigh surface waves in the considered medium is derived and analyzed. Explicit expressions for phase velocity and attenuation (amplification) coefficients which characterize surface waves are obtained. It is shown that if the generation rate is above the critical value, due to concentration-elastic instability, nanometer sized ordered concentration-strain structures on the surface or volume of solids arise. The spatial scale of these structures is proportional to the characteristic length of defect-atom interaction and increases with the increase of the temperature of the medium. The critical value of the pump parameter is directly proportional to recombination rate and inversely proportional to deformational potentials of defects.

1. Introduction

During the last decades elastic wave propagation in solids (metals and semiconductors) with nonequilibrium atomic defects has received a lot of attention [1–4]. Various types of lattice defects (vacancies and interstitials), produced from the lattice site atoms due to intense external energy fluxes (laser and corpuscular radiations), introduce a significant strain of the medium as a result of the difference between the radii of lattice atoms and defects [5] and play an important role in surface modification of solids exposed to laser radiation [1, 5, 6]. The formation of atomic defects may occur also in a number of other technologies processes: in the laser fast recrystallization, laser annealing, multipulse laser etching, pulsed laser-assisted thin-film deposition, and so forth. Strains in an elastic wave cause a strain-induced drift of defects, whereas the strains and a variation in the temperature in the wave modulate the rates of generation and recombination of defects of the thermal-fluctuation origin (via variations in the energies of the defect formation and migration) [5, 6].

Several mathematical models have been considered to study the self-organization of two-dimensional (2D) ordered microscale concentration-strain (CS) structures (at high concentrations of atomic defects) on the surface of the solid half-space [7, 8] and in an isotropic solid elastic layer [9, 10] under the action of laser irradiation. In these studies, attention was also focused on the study of an influence of the strain-induced diffusion, generation, and annihilation of defects on the propagation characteristics of strain waves. In both [1–4] and [7–10] the analysis was based on coupled evolution equations for the atomic defect density fields and the classical local elasticity equations for the elastic displacement of the medium.

Classical continuum theory is characterized by the local character of stress (stress at a point depends only on the strain at that point) and does not contain internal length scale. The absence of the length scale creates several discrepancies in the predictions of mechanical responses, for example, nondispersive wave behavior (constant wave velocity, independent of frequency). For example, according to the classical elasticity, Rayleigh waves propagating on the surface of a semi-infinite isotropic elastic space are not dispersive at any frequency, whereas experiments and the atomic theory of lattice predict otherwise.

Limitations of the classical elasticity theory are also demonstrated in the study of the formation of coupled strain-defect nanometer sized ordered structures (short-range CS structures) on the surface of laser-irradiated solids and mechanical behavior of microstructured materials, because their behavior is characterized by nonlocal stresses and the existence of an internal length scale [11, 12]. The accurate analysis of dynamic behaviors of these structures cannot be correctly described by classical local elasticity theory. Conducting experiments with nanoscale size materials is found to be difficult and expensive. Therefore, development of appropriate nonlocal elasticity mathematical models for nanostructure formation and nanomaterials is of great importance.

In contrast to local approach of zero-range internal interactions, the nonlocal elasticity theory, originated and developed in the last four decades, postulates that the stress at a point 𝑥 in a body depends not only on the strain at point 𝑥 but also on those at all other points of the body. Various nonlocal theories of linear elasticity have been proposed to describe the scale effects on the characteristics of the vibration and elastic wave propagation in the above-mentioned submicro- or nanosized structures. The basic constitutive equations and governing equations of linear nonlocal elasticity were derived by Eringen [11] and Kunin [12]. Later the nonlocal theories have been applied for the analysis of micro- and nanoscale plate-like structures, in which the small-scale effects become significant. In [13] Eringen considered vibration behavior of a nanoplate by using linear theory of nonlocal continuum mechanics. By using [11], Reddy [14] reformulated the classical and shear deformation beam and plate theories. The nonlocal scale influences on the wave dispersion properties of the nanoplates are discussed in detail in [15]. Vibration characteristics of nanoplates, based on three-dimensional theory of elasticity employing nonlocal continuum mechanics, have been discussed in [15]. A review of some other applications of nonlocal elasticity theories for nanostructures can be found in [16]. Propagation of longitudinal elastic and thermoelastic waves in an isotropic, homogeneous infinite medium with long-rang interactions has been studied by Nowinski [17, 18]. Other advances have been made by the application of nonlocal elasticity to such fields as the solid defects [19, 20] and fracture mechanics [21].

The present paper is concerned with the nonlocal elasticity theory of laser-excited solid half-space with mobile atomic point defects. We summarise the theory formulated in [8] to analyze the effects of nonlocal atom-atom and atom-defect interactions on the surface wave propagation in solids with defect generation. The changes in defect concentration are controlled by the (i) generation of defects by laser irradiation, (ii) their diffusion, and (iii) strain-induced diffusion. Dispersion equation that governs the propagation of elastic-concentration waves has been derived by solving a system of coupled partial differential equations. Some limiting cases of the dispersion equations are considered. For a small value of coupled parameter obvious expressions for the phase velocity and the attenuation (amplification) factor of surface elastic-concentration waves are determined. It is found that both phase velocity and attenuation coefficient are to be influenced by the presence of defect generation. We also obtain that at certain conditions concentration-elastic instabilities with the formation of regular nanosized structures in a system of atomic defects on the surface of the solids can be developed. We demonstrate that due to the nonlocal character of the defect-atom and atom-atom interactions the dispersive curve of the instability has two maxima. As a result, size distribution functions of the surface nanoscale nonhomogeneities having two maxima can be formed. The results of some earlier works are also deduced from the present formulation. To our knowledge, the problem in its present form was not investigated before.

2. Governing Equations

In this section we give basic 2D equations governing the defect density and elastic displacement fields based on the nonlocal constitutive relations of the elasticity theory. Consider an isotopic elastic semi-infinite medium occupying a region −∞<𝑥<∞, 0<𝑧<∞ in a rectangular Cartesian coordinate system 𝑂𝑥,𝑦,𝑧, where the origin 𝑂 is situated at any point on the plane boundary and 𝑂𝑧 points vertically downwards, that is, towards the bulk of the medium. The surface 𝑧=0 is supposed to be free from stresses. Let a plane elastic wave propagate along the 𝑥-axis. Denoting 𝑢 and 𝑣 as the nonzero components of the displacement vector 𝑢, we set 𝑢=𝑢(𝑥,𝑧,𝑡), 𝑣=𝑣(𝑥,𝑧,𝑡), and 𝑤=0 and consequently write for the strain tensor (𝑒_𝑖𝑗): 𝑒_𝑥𝑥=𝜕𝑢𝜕𝑥,𝑒_𝑧𝑧=𝜕𝑣𝜕𝑧,2𝑒_𝑥𝑧=𝜕𝑢+𝜕𝑧𝜕𝑣𝜕𝑥. (1)

Let us assume that an external energy flux (e.g., laser radiation) creates mobile atomic defects in a surface layer. The corresponding defect density profile results in a force that may induce strain field in medium. Let 𝑛_𝑖(𝑥,𝑧,𝑡) be the density of these defects of the 𝑗th-type (𝑖=𝑉 for vacancies (𝑉-defects) and 𝑗=𝐼 for interstitials (𝐼-defects)). We limit our consideration to the case of only one type of atomic defects (for definiteness, 𝑉-type defects).

The concentration field of atomic defects is dependent on temperature of the medium. One thus needs to know how the laser irradiation affects the local temperature of the surface at the laser spot. We will consider here situations where the laser irradiation only heats the solid (the light energy absorbed by the medium is transformed into heat) and that an equilibrium between laser radiation and the temperature field (𝑇) is reached on time scales much shorter than the characteristic time scale of defect density evolution. Typically, the time scale for equilibration between photon absorption and defect generation is on the order of picoseconds, while that for defect diffusion is of the order of microseconds. We also assume that the contribution of thermal strains to deformation fields is negligible compared to lattice dilatation due to atomic defects and the phase changes and chemical reactions in the medium are absent.

In this paper, we will consider the problem of the wave propagation in an elastic solid irradiated over a large area by CW or pulsed lasers. Furthermore, we will assume that the temperature field has reached its equilibrium value. Its evolution is sufficiently slow compared to atomic defect generation and can be considered as quasistationary. The solution of the heat conduction equation for this case is given by Duley [22].

Taking into account the defect generation, the constitutive equations of an isotropic nonlocal elastic solid are 𝜎_𝑖𝑘(𝐱,𝑡)=_Ω𝛼𝐱^′−𝐱𝑠_𝑖𝑘𝐱^′,𝑡𝑑Ω𝐱^′−_Ω𝛽𝐱^′−𝐱𝑛𝐱^′,𝑡𝛿_𝑖𝑘𝑑Ω𝐱^′,𝑠_𝑖𝑘𝐱^′,𝑡=𝜆₀𝑒𝐱^′,𝑡𝛿_𝑖𝑘+2𝜇₀𝑒_𝑖𝑘𝐱^′,𝑡, (2) where 𝜎_𝑖𝑘 is the nonlocal stress tensor, 𝑠_𝑖𝑘 is the classical stress tensor, 𝜆₀ and 𝜇₀ are Lamé’s constants [23], 𝑒_𝑖𝑘=(𝑢_𝑖,𝑘+𝑢_𝑘,𝑖)/2 is the strain tensor, 𝑒=𝑢_𝑘,𝑘 is the dilatation, and 𝑑Ω(𝐱^′)=𝑑𝑥^′₁𝑑𝑥^′₂. The functions 𝛼 and 𝛽 are known as atom-atom (short-range) and atom-defect (long-range) interaction kernels or moduli of nonlocality, which decay smoothly with distance. They reflect the influence of an independent constitutive variable at a point 𝐱^′ on a dependent constitutive variable at 𝑥. Nonlocal moduli are physical properties of materials like other physical constants and need to be determined experimentally.

The basic difference between classical and nonlocal elasticity is in the presence of the volume integrals in (2) which indicates that the stress at (𝐱,𝑡) depends on the strain and defect density at all other points {𝐱^′} of the body, at time 𝑡. This signifies that the distant neighbors of a point 𝐱 have a role to play in the propagation of waves.

In absence of body forces the equations of motion and the equations of defect density dynamics have the following form: 𝜌𝜕²𝑢_𝑖𝜕𝑡²+∇_𝑘𝜎_𝑖𝑘=0,𝜕𝑛𝜕𝑡=𝐺+𝐷∇²𝐷𝑛−𝑘_𝐵𝑇∇⋅(𝑛𝐟)−𝑛𝜏⁻¹, (3)(4) where 𝜌 is the density of the medium, 𝐺=𝐺₀exp(−𝑤_𝑔/𝑘_𝐵𝑇) is the thermal-fluctuation generation rate of atomic defects at sites (𝐺₀ is the constant and 𝑘_𝐵 is the Boltzmann constant), and 𝜏=𝜏₀exp(𝑤_𝑚/𝑘_𝐵𝑇) is the relaxation time of defects in the absence; 𝑤_𝑔 and 𝑤_𝑚 are the formation and migration energies for the defects in crystals.

The first term in the right-hand side of (4) takes into account laser-induced generation of defects, the second term represents diffusion with a coefficient 𝐷, the third term corresponds to the drift of defects under the influence of the force 𝐟=−∇𝑈_int resulting from the nonlocal interaction of defects with an inhomogeneous strain field, and the fourth term describes the rate of their disappearance due to recombination processes. It is assumed that the generation rate (𝐺) is spatially uniform.

The expression for interaction energy 𝑈_int of a defect with the strain field 𝑒_𝑚𝑚 in a nonlocal elastic medium is given by the formula 𝑈_int=−_Ω𝛽𝐱^′−𝐱𝑒𝐱^′𝑑Ω𝐱^′. (5) In the limit, when the kernel functions 𝛼 and 𝛽 become Dirac-delta functions, (2) reduces to the classical constitutive equations of local elasticity theory.

Thus the final dynamic field equations obtained are integrodifferential equations for the functions 𝑢(𝑥,𝑡) and 𝑛(𝑥,𝑡). It seems to be obvious that a rigorous solution of such equations encounters serious but not insurmountable mathematical difficulties. However, these equations can be reduced to the partial differential forms under certain conditions with physical admissible kernels. We consider that the long-range internal influences of particles of the body are rather rapidly with increasing distance from the particle. Thus the 2D-kernel functions that characterize the nonlocal interaction in the 𝑧-direction may be approximated in terms of delta-like functions as 𝛼=𝛼𝑥^′−𝑥,𝑧^′−𝑧=𝛼𝑥^′−𝑥𝛿_𝑛𝑧^′,𝑥−𝑧𝛽=𝛽^′−𝑥,𝑧^′−𝑧=𝛽𝑥^′−𝑥𝛿_𝑛𝑧^′−𝑧. (6)

Besides the nonlocal kernels 𝛼 and 𝛽 are symmetric with respect to the interchange of the arguments 𝝃=𝐱^′−𝐱. Then expanding in (3) the functions 𝑒(𝐱+𝝃) and 𝑛(𝐱+𝝃) in Taylor series around 𝝃=0 and neglecting the terms higher than second-order gradients, we can obtain the following expression for the nonlocal constitutive relation with strain and defect concentration gradients: 𝜎_𝑖𝑘(𝐱)=1+𝑔²∇²𝜆₀𝛿_𝑖𝑘𝑒(𝐱)+2𝜇₀𝑒_𝑖𝑘(𝐱−)1+ℎ²∇²𝛽₀𝛿_𝑖𝑘𝑛(𝐱), (7) where ℎ and 𝑔 are the intrinsic characteristic length scale parameters, characterizing the defect-atom and atom-atom interactions, respectively; ∇²=𝜕²/𝜕𝑥²; 𝛽₀=𝐾Ω_𝑑 is deformation potential of the defect (Ω_𝑑 is the change of the volume of the medium under formation of a single defect and 𝐾 is the bulk modulus). The gradient coefficients 𝑔 and ℎ are dependent on the lattice or atomic chain model used and the interatomic potentials assumed. The values of these coefficients are taken to be order of interatomic distance. For the interaction energy 𝑈_int we have from (5) 𝑈_int=−(1+ℎ²∇²)𝛽₀𝑒.

To obtain the nonzero stress components in terms of the displacement components we substitute (1) into (7): 𝜎_𝑥𝑥=1+𝑔²∇²𝜆₀+2𝜇₀𝜕𝑢𝜕𝑥+𝜆₀𝜕𝑣−𝜕𝑧1+ℎ²∇²𝛽₀𝜎𝑛,_𝑧𝑧=1+𝑔²∇²𝜆₀+2𝜇₀𝜕𝑣𝜕𝑧+𝜆₀𝜕𝑢−𝜕𝑥1+ℎ²∇²𝛽₀𝜎𝑛,_𝑥𝑧=1+𝑔²∇²𝜇₀𝜕𝑢+𝜕𝑧𝜕𝑣.𝜕𝑥 (8)

Then from (4), two nonzero equations of motion can be obtained: 𝜌𝜕²𝑢𝜕𝑡²=𝜕𝜎_𝑥𝑥+𝜕𝑥𝜕𝜎_𝑥𝑧,𝜌𝜕𝜕𝑧₂𝑣𝜕𝑡²=𝜕𝜎_𝑧𝑥+𝜕𝑥𝜕𝜎_𝑧𝑧.𝜕𝑧 (9)

Suppose that the stress components and defect density field satisfy the boundary conditions 𝜎_𝑧𝑧=𝜎_𝑧𝑥=0,𝜕𝑛𝜕𝑧=0,for𝑧=0;𝑢,𝑣,𝑛⟶0as𝑧⟶∞. (10)

We can express the defect density field as 𝑛=𝑛₀+𝑛₁ (𝑛₀=𝐺𝜏 is a spatially homogeneous solution; 𝑛₁ is small nonhomogeneous perturbations). Inserting in (4) the nonlinear terms, we get the linearised equation as 𝜕𝑛₁+𝑛𝜕𝑡𝜏¹𝜕−𝐷²𝑛₁𝜕𝑥²+𝜕²𝑛₁𝜕𝑧²𝑛=−⁰𝛽₀𝐷𝑘_𝐵𝑇𝜕²𝑒𝜕𝑥²+𝜕²𝑒𝜕𝑧²+𝑔²𝜕⁴𝑒𝜕𝑥⁴+𝜕²𝑒𝜕𝑥²𝜕𝑧². (11)

The system of (9) and (11) is closely coupled. 𝜕²𝑢/𝜕𝑡² in (9) depends on the defect density field (𝑛₁) and 𝜕𝑛₁/𝜕𝑡 in (11) depends on elastic displacement field (𝑢). The system of equations thus becomes highly nonlinear.

3. Solution of the Problem

To solve the above problem, we apply the Fourier integral transform in the following form: 𝑛₁()=1𝑥,𝑧,𝑡2𝜋^∞_−∞𝑛{)=1𝑞,𝑧,𝜔exp−𝑖𝑞𝑥+𝜔𝑡𝑑𝑞𝑑𝜔,𝑢,𝑣}(𝑥,𝑧,𝑡2𝜋^∞_−∞{𝑢,𝑣}̃𝑞,𝑧,𝜔exp−𝑖𝑞𝑥+𝜔𝑡𝑑𝑞𝑑𝜔. (12) Substitution of this into (9) gives 𝜕𝜎_𝑧𝑥𝜎𝜕𝑧−𝑖𝑞_𝑥𝑥+𝜌𝜔²𝜕𝜎𝑢=0,_𝑧𝑧𝜎𝜕𝑧−𝑖𝑞_𝑥𝑧+𝜌𝜔²̃𝜕𝑣=0,₂𝑛𝜕𝑧²−𝑞²+𝑖𝜔𝜏+1𝑙²𝑛=𝑛₀𝛽𝑘_𝐵𝑇−𝑖𝑞³𝑢+𝑞²𝜕²𝑣̃𝜕𝑧²𝜕+𝑖𝑞²𝑢𝜕𝑧²−𝜕³𝑣̃𝜕𝑧³, (13) where 𝜎_𝑥𝑥𝜕𝑣̃=−𝑖𝑞𝜆+2𝜇𝑢+𝜆𝜎𝜕𝑧−𝛽𝑛,_𝑧𝑧=𝜕𝑣̃𝜆+2𝜇𝜎𝜕𝑧−𝑖𝑞𝜆𝑢−𝛽𝑛,_𝑥𝑧𝜕𝑢=𝜇𝑣̃,𝜕𝑧−𝑖𝑞 (14) where 𝜔 is the frequency of wave propagation, 𝑞 is the wave number, and 𝑙=√𝐷𝜏; the phase velocity is given by 𝑐=𝜔_𝑟/𝑞 and attenuation constant by Γ=−𝜔_𝑖, where 𝜔_𝑟=Re(𝜔) and 𝜔_𝑖=Im(𝜔) mean, respectively, the real and imaginary parts of 𝜔. 𝜆=𝜆(𝑞)=𝜆₀(1−𝑔²𝑞²) and 𝜇=𝜇(𝑞)=𝜇₀(1−𝑔²𝑞²) are the nonlocal elastic moduli; 𝛽=𝛽(𝑞)=𝛽₀(1−ℎ²𝑞²) is the nonlocal constant characterizing lattice deformation due to atomic defects; 𝑢(𝑞,𝑧),𝑣(̃𝑞,𝑧), and 𝑛(𝑞,𝑧) are unknown functions (amplitude functions).

The general solutions to (13) are 𝑢=𝑎₁exp−𝜂₁𝑧+𝑎₂exp−𝜂₂𝑧+𝑎₃exp−𝜂₃𝑧,𝑣̃=−𝑖𝑞𝜂₁𝑎₁exp−𝜂₁𝑧+𝑚₂𝑎₂exp−𝜂₂𝑧+𝑚₃𝑎₃exp−𝜂₃𝑧,𝑛=𝑑₂𝑎₂exp−𝜂₂𝑧+𝑑₃𝑎₃exp−𝜂₃𝑧, (15) where 𝜂²₁=𝑞²−𝜌𝜔²𝜇. (16)𝜂²₂ and 𝜂²₃ are the roots of the equation (1−𝛿)𝜂⁴−𝜂²2𝑞²()𝜔1−𝛿−𝜌²+𝜆+2𝜇𝑖𝜔𝜏+1𝑙²+𝑞²𝜔−𝜌²𝜆+2𝜇𝑖𝜔𝜏+1𝑙²+𝑞²−𝛿𝑞⁴=0, (17) where 𝑎_𝑖, 𝑖=1,2,3 are arbitrary constants, 𝛿=𝑛₀𝛽²/(𝜆+2𝜇)𝑘_𝐵𝑇 is the coupling constant of nonlocal defect-strain interaction, and 𝑚_𝑗=𝜂_𝑗𝑖𝑞,𝑑_𝑗=𝜆+2𝜇𝜂𝛽𝑖𝑞²¹−𝜂²_𝑗,𝑗=2,3. (18)

Then 𝜂²₂ and 𝜂²₃ are defined from (17) as follows: 𝜂²₂+𝜂²₃=2𝑞²()𝜔1−𝛿−𝜌²+𝜆+2𝜇𝑖𝜔𝜏+1𝑙²×(1−𝛿)⁻¹,𝜂²₂𝜂²₃=𝑞²𝜔−𝜌²𝑞𝜆+2𝜇²+𝑖𝜔𝜏+1𝑙²−𝛿𝑞⁴×(1−𝛿)⁻¹. (19)

Stress components are obtained by substituting (15) into (8): 𝜎_𝑥𝑥𝑞,𝑧=−2𝑖𝑞𝜆+𝜇𝑎₁exp−𝜂₁𝑧+−𝑖𝑞𝜆+2𝜇+𝜆𝜂₂𝑚₂+𝛽𝑑₂𝑎₂exp−𝜂₂𝑧+−𝑖𝑞𝜆+2𝜇+𝜆𝜂₃𝑚₃+𝛽𝑑₃𝑎₃exp−𝜂₃𝑧,𝜎_𝑧𝑧𝑞,𝑧=2𝑖𝑞𝜇𝑎₁exp−𝜂₁𝑧+𝑖𝑞𝜆−𝜆+2𝜇𝜂₂𝑚₂+𝛽𝑑₂𝑎₂exp−𝜂₂𝑧+−𝑖𝑞𝜆−𝜆+2𝜇𝜂₃𝑚₃+𝛽𝑑₃𝑎₃exp−𝜂₃𝑧,𝜎_𝑥𝑧𝑞,𝑧=𝜂₁𝑞1+²𝜂²₁𝑎₁exp−𝜂₁𝑧+𝜂₂+𝑖𝑞𝑚₂𝑎₂exp−𝜂₂𝑧+𝜂₃+𝑖𝑞𝑚₃𝑎₃exp−𝜂₃𝑧. (20)

4. Dispersion Equations of the Wave in an Infinitive Medium

In this section as a special case we consider the generation of plane harmonic structures in unbounded nonlocal medium. Setting 𝜂_𝑗=0,𝑗=1,2,3 in (16) and (17) we obtain 𝜌𝜔²=𝜇𝑞𝑞²,𝑞²+𝑖𝜔𝜏+1𝑙²𝑞²−𝜔²𝜌𝜆𝑞𝑞𝑞𝑞+2𝜇−𝛿⁴=0. (21)(22)

Suppose now that one ignores the influence of the non-local properties of the medium and sets 𝜆(𝑞)=𝜆₀=const, 𝜇(𝑞)=𝜇₀=const, and 𝛿(𝑞)=𝛿₀=const. In this case (21) and (22) reduce to frequency equations𝜔²=𝑐²_𝑇0𝑞²𝑞²−𝜔²𝑐⁻²_𝐿0𝑞²+(𝑖𝜔𝜏+1)𝑙⁻²−𝛿₀𝑞⁴=0, (23a)(23b) (𝑐_𝐿0=[(2𝜇₀+𝜆₀)/𝜌]^1/2 and 𝑐_𝑇0=(𝜇₀/𝜌)^1/2 are the classical velocities of the longitudinal and shear waves, resp.), coinciding with the equations derived in the conventional local theory (see, e.g., [8]). In the nonlocal case, the frequency equation has, of course, a more complex structure.

To explore and delineate the strain and defect generation effects, we will seek solutions of (22) for small values of 𝛿. For 𝛿=0, (22) admits the following solutions 𝜔⁽⁰⁾_1,2=±𝑐_𝐿𝑞 (acoustical mode) and 𝜔₃⁽⁰⁾=𝑖𝜏⁻¹(𝑙²𝑞²+1) (diffusion mode). Now, for small 𝛿≪1, we may write 𝜔_1,2=𝜔⁽⁰⁾_1,2+𝛿𝜔⁽¹⁾_1,2+𝛿²𝜔⁽²⁾_1,2𝜔+⋯,₃=𝜔₃⁽⁰⁾+𝛿𝜔₃⁽¹⁾+𝛿²𝜔₃⁽²⁾+⋯. (24) Substituting (24) into (22), and, equating the coefficients of like powers of 𝛿, we finally arrive to 𝜔_1,2=𝜔⁽⁰⁾_1,2−𝛿2𝜔⁽⁰⁾_1,2𝑞²𝑙²𝑞²𝑙²+1−𝑖𝜔⁽⁰⁾_1,2𝜏𝑞²𝑙²+1²+𝜔⁽⁰⁾_1,2𝜏²+𝑂𝛿²,𝜔₃=𝜔₃⁽⁰⁾−𝑖𝛿𝑙²𝑐²_𝐿𝑞⁴𝜏𝑐²_𝐿𝑞²𝜏²+𝑞²𝑙²+1²+𝑂𝛿². (25)(26) From (25) we obtain Re𝜔_1,2=𝜔⁽⁰⁾_1,2𝛿1−2𝑞²𝑙²𝑞²𝑙²+1𝑞²𝑙²+1²+𝑐_𝐿𝑞𝜏²,Im𝜔_1,2=𝛿𝜏2𝑐²_𝐿𝑞⁴𝑙²𝑞²𝑙²+1²+𝑐_𝐿𝑞𝜏². (27)(28)

If 𝑐_𝐿𝑞≫𝜏⁻¹(𝑙²𝑞²+1) and the viscosity is taken into account [by adding in (9) the terms 𝜂_𝑇Δ𝑢̇ and 𝜂_𝐿∇(div𝑢̇), where 𝜂_𝑇=𝜂; 𝜂_𝐿=4𝜂/3+𝜁; 𝜂 and 𝜁 are the first and second viscosity coefficients] dispersion (28) describes attenuation of the amplitude of acoustic waves Im𝜔_1,2=12𝜌𝜂𝑞²+𝛿𝑙2𝜏²𝑞²+1. (29) For the frequency spectrum of acoustic wave we have Re𝜔_1,2=𝜔⁽⁰⁾_1,2𝛿1−𝑙2𝜏²𝑞²+1𝑐_𝐿𝑞≈𝜔⁽⁰⁾_1,2. (30)

Note that the frequency of acoustic waves is hardly changed. But the additional contribution to attenuation coefficient of waves arises. As attenuation decrement of acoustic waves is proportional to 𝑞², according to (29), short waves fade fast.

If 𝑐_𝐿𝑞≪𝜏⁻¹(𝑙²𝑞²+1)Re𝜔_1,2=𝜔⁽⁰⁾_1,21−𝛿𝑞²𝑙²2𝑞²𝑙²,𝜔+1Im_1,2=12𝜌𝜂𝑞²+𝛿𝜏2𝑐²_𝐿𝑞⁴𝑙²𝑞²𝑙²+1². (31)(32) In this case there is a softening of frequencies of acoustic waves (instability of frequencies Re(𝜔_1,2)→0,Im(𝜔_1,2)>0) and this is related to taking into account the generation of atomic defects. It is necessary to notice that reduction of frequency occurs not up to zero but up to value 𝜔_𝜂=𝜂𝑞²/2𝜌≪𝑐_𝐿𝑞. Equation (32) describes attenuation of the amplitudes of acoustic waves.

At 𝜔_𝑖≪𝑐_𝐿𝑞 from (22) we obtain dispersion relation   (growth rate) Γ=−Im(𝜔) of the volume concentration-elastic instability: Γ𝜏=𝑙²𝑞²(𝛿−1)−1 (33) or Γ=𝐷𝑞²𝛿₀1−𝑔²𝑞²²1−ℎ²𝑞²−1−1𝜏, (34) where 𝛿₀=𝑛₀𝛽²₀/(𝜆₀+2𝜇₀)𝑘_𝐵𝑇.

Let us introduce the dimensionless growth rate Γ=Γ𝑔²/𝐷, the wave number 𝑞=𝑞𝑔, the life-time of defects 𝜏=𝜏𝐷/𝑔², and parameters 𝛾=𝑔²/ℎ². Dependence (34) in dimensionless variables takes the form 𝑞Γ=²1−𝑞²𝑞²+𝛿₀1−𝛾𝑞²²−1−1𝜏. (35)

We consider two situations that differ from each other by the relation (𝛾) of the squares of the characteristic lengths 𝑔 and ℎ. Figure 1 demonstrates the graphs of the growth rate calculated using (35) for two values of the ratio parameter 𝛾=8 and 10, respectively. The following typical values of parameters were used in the calculations: ℎ=0.6 nm, 𝛿₀=10, and 𝐷=4×10⁻¹¹ cm² s⁻¹. Comparison of the plots presented in Figure 1 shows that the dependences of the growth rates of the wave number for the two cases qualitatively differ from each other. It follows from Figure 1 that the growth rate exhibits a single maximum at 𝑞_𝑚=3×10⁶ for small values of 𝛾 and an additional maximum in the short-wavelength region at 𝑞>6×10⁶ cm⁻¹. Therefore, depending on the relation between the characteristic lengths 𝑔 and ℎ, elastic-concentration structures have one or two maximum growth rates.

Figure 1 

Dependences of the growth rates on the wave number for two values of the ratio parameter 𝛾: 8 (solid line); 10 (dot line). Control parameter: 𝛿0=10.

Figure 2 shows the dependences Γ(𝑞) of the increment on the wave number for two values of the control parameter 𝛿₀=5 and 15 when the ratio parameter is 𝛾=8. It follows from this figure that the growth rate has a single maximum not far above the threshold. However, at sufficiently high concentrations of the defects exceeding the threshold appears an additional maximum in the short-wavelength range. Therefore, as in the previous case, two gratings have the maximum growth rates.

Figure 2 

Dependences of the growth rate on the wave number for two values of the control parameter 𝛿0: 5 (solid line), 15 (dot line). Ratio parameter: 𝛾=8.0.

The analytical expression for the maximum of the growth rates can be obtained provided that the nonlocal parameters are of the same order in (34) (i.e., 𝑔≈ℎ). Then, using (34), we have Γ𝑞=𝐷𝑞²𝐺𝐺_𝑐1−𝑔²𝑞²−1−1𝜏, (36) where 𝐺_𝑐=𝑘_𝐵𝑇𝜌𝑐²_𝐿𝜏𝛽²₀⁻¹, (37) is the critical value of the control parameter (defect generation rate). So, putting values into (37) for molybdenum: 𝛽₀=20 eV, 𝜌𝑐²_𝐿=10¹² erg cm⁻³, 𝑇=10³ K, and 𝜏=10⁻³ s, one can get the estimate 𝐺_𝑐=10²⁴ s⁻¹ cm⁻³.

The dispersion curve Γ(𝑞) has a maximum Γ_𝑚 at 𝑞=𝑞_𝑚, which determines the spatial scale 𝑑_latt=2𝜋/𝑞_𝑚 of the dominant CS structures: 𝑑_latt=√8𝜋𝑔𝐺𝐺−𝐺_𝑐. (38)

The maximum growth rate is Γ_𝑚=𝐷4𝑔²1−𝐺_𝑐/𝐺²𝐺/𝐺_𝑐−𝜏⁻¹. (39)

It is seen from (39) that the development of the elastic-concentration instability with the self-organization of CS structures in the volume of the solids occurs in a threshold manner when the defect generation rate exceeds the critical value: 𝐺>𝐺_𝑐. At the elastic-concentration instability threshold (𝐺→𝐺_𝑐), the period tends to infinity 𝑑_latt→∞. At high values of generation rates 𝐺≫𝐺_𝑐, the period asymptotically tends to its minimal value 𝑑_latt=√8𝜋𝑔.

Near the threshold 𝐺→𝐺_𝑐 the time of formation of the CS structures Γ_𝑚⁻¹→∞ that is critical slowing down takes place, which is characteristic of the phase transitions of the second kind.

The critical value of the generation rate derived here is directly proportional to recombination rate (𝜏⁻¹) and inversely proportional to deformational potentials (𝛽₀) of defects. Putting 𝜏=𝜏₀exp(𝑤_𝑚/𝑘_𝐵𝑇) into (37) for the generation rate, we get 𝐺_𝑐(𝑇)=𝜌𝑐²_𝐿𝑘_𝐵𝑇𝜏₀𝛽₀²−𝑤exp^𝑚𝑘_𝐵𝑇. (40) One notes that this dependence is very rapid, which is largely due to variation of the defect relaxation time with temperature.

From (38), it follows that the period of the volume ordered CS structures is proportional to the characteristic length ℎ of the defect interaction with the crystal-lattice atoms, which lies in the obtained nanometer range. As temperature is increased, the period grows for a particular value of ℎ. For a particular value of 𝑇, the period decreases with an increasing generation rate (𝐺).

A periodic structure of vacancy clusters in the bulk of a solid irradiated with nitrogen ions was first observed in 1971 by Evans in pure molybdenum [24]. The medium temperature was 𝑇= 1040°K. Subsequently, the same effect had been observed in other metals such as aluminum, tungsten, nickel, and niobium [25, 26]. Vacancy cluster sizes in the lattice were 2–4 nm, and values of the lattice parameter change in the range of (20–60) nm. At 𝑔=1.5 nm (for 𝐺=1.1𝐺_𝑐) from (38) we obtain an estimate for the period of the volume structure 𝑑_latt=44 nm. This is in qualitative agreement with the above-mentioned experimentally observed interval.

Note that the linear theory considered in this section describes early stage of the development of an elastic-concentration instability only. However, the nature of generated ordered volume structures (due to an instability) and the amplitudes of these structures as functions on material and irradiation conditions can only be determined by considering the influence of elastic nonlinear effects in the model.

5. Dispersion Equations of the Surface Concentration-Strain Structures

Substituting expressions (20) into boundary conditions (10) and taking into account expressions (18), we find a set of three linear algebraic equations: 2𝑞²𝑎₁+𝜂²₁+𝑞²𝑎₂−𝜂²₁+𝑞²𝑎₃𝜂=0,₂₁+𝑞²𝑎₁+2𝜂₂𝜂₁𝑎₂+2𝜂₃𝜂₁𝑎₃𝜂=0,₂𝜂²₂−𝑞²+𝜌𝜔²𝑎𝜆+2𝜇²+𝜂₃𝜂²₃−𝑞²+𝜌𝜔²𝑎𝜆+2𝜇³=0. (41)

The condition of existence of a nontrivial solution of this system yields 𝜂²₁+𝑞²²=4𝜂₁𝜂₂𝜂₃𝑞²𝜂₂+𝜂₃𝜂²₂+𝜂²₃+𝜂₂𝜂₃+𝜌𝜔²𝜆+2𝜇⁻¹−𝑞², (42) where 𝜂₁,𝜂₂, and 𝜂₃ are given by (16) and (19).

Equation (42) shows dispersive character of the Rayleigh waves propagating in an elastic half-space. This dispersive character of these waves propagation arises due to atomic defect generation in the medium with long-range interactions.

Note that the form of this dispersion equation is identical to the corresponding equation obtained in the context of concentration-elastic coupled theory, using classical constitutive equations of motion, though each of 𝜂₁,𝜂₂, and 𝜂₃ represents different expressions for classical elasticity and nonlocal elasticity.

Equation (21) is generalized form of the classical local Raleigh equations. The assumption of vanishing defect generation in this equation gives us exactly the same results as those obtained by Nowiński [17] for solids with long-range interactions; that is, 4𝑞²𝜂₁𝜂₂−𝜂²₁+𝑞²²=0,𝜂²₁=𝑞²−𝜌𝜔²𝜇,𝜂²₂=𝑞²−𝜌𝜔².𝜆+2𝜇 (43)(44)

6. Solution of Dispersion Equation

A general discussion of (42) presents considerable difficulties. The structure of (42) is identical to the structure of the dispersion equation for DEI considered in [8]. This equation can be readily solved numerically or graphically, but in some cases of interest it can also be solved analytically. We shall now consider the results for some limiting cases. We can rewrite (42) as 𝜂1+²¹𝑞²²=4𝜂₁𝐵(𝐴+2𝐵)^1/2𝑞²𝐴+𝐵−𝜂²₂, (45) where 𝐴=𝜂²₂+𝜂²₃ and 𝐵=𝜂₂𝜂₃.

Equation (45) has solutions describing qualitatively different types of instability: (1) instability of frequencies of surface acoustic waves and (2) generation of ordered surface (static) concentration-strain structures.

If the coupling term is small (𝛿≪1) the expressions that appear in the dispersion equation (25), to a first approximation, then read 𝐵=𝜂₂𝜂₃1+𝛿𝜂²₂𝜂²₃−𝑞⁴2𝜂²₂𝜂²₃,𝐴=𝜂²₂+𝜂²₃+𝛿𝜂²₂+𝜂²₃−2𝑞²,𝜂²₃=𝑞²+(𝑖𝜔𝜏+1)𝑙². (46) Then we may express the dispersion equation in the following form: 𝑅𝜔,𝑞−𝛿𝑍𝜔,𝑞=0. (47) Here 𝑅𝜔,𝑞=𝜂²₁+𝑞²²−4𝑞²𝜂₁𝜂₂,𝑍𝜔,𝑞=2𝑞²𝜂₁2𝜂₂+𝜂₃𝜂²₂−𝑞²²𝜂₃𝜂₂𝜂₂+𝜂₃². (48) For 𝛿=0 we obtain the nonlocal elastic solution given by Nowiński [17]. Introducing the dimensionless variable 𝜉=(𝜔/𝑐₁𝑞)² we may write the dispersion equation (47) in the following form: 𝑅(𝜉)−𝛿𝑍(𝜉)=0. (49) Now we may consider that 𝜀 is the increment of the value of 𝜉 due to 𝛿≠0. Then (49) can be rewritten in the form of 𝑅𝜉_𝑅+𝜀−𝑍𝜉_𝑅+𝜀𝛿=0, (50) where 𝜉_𝑅 is a root of the equation 𝑅(𝜉_𝑅)=0.

Assuming that 𝜀≪1, we may expand both sides of (50) into a Taylor series in the vicinity of the point 𝜉=𝜉_𝑅. Retaining only the first two terms, we obtain 𝜀𝜉_𝑅𝑍𝜉,𝛿=−^𝑅(𝜕𝑅/𝜕𝜉)_{𝜉=𝜉𝑅}𝛿=𝜀_𝑟+𝑖𝜀_𝑖, (51) where 𝜀_𝑟=Re(𝜀) and 𝜀_𝑖=Im(𝜀).

The real part of 𝜀 characterizes the change of the phase velocity and its imaginary part (𝜀_𝑖) defines the attenuation constant. We have 1𝜉=𝑐²₁𝜔𝑞²=𝜉_𝑅+𝜀,𝜔=𝑞𝑐₁𝜀_𝑟+𝜉_𝑅+𝑖𝜀_𝑖^1/2. (52) This equation gives the following expressions:𝜔_𝑟=𝑐₁𝑞𝜉_𝑅1−𝛿𝑍_𝑟2𝜉_𝑅𝑀,𝜔_𝑖=𝑐₁𝑞𝛿𝑍_𝑖𝜉_𝑅𝑀, (53a)(53b)where 𝑍_𝑟=Re(Ζ), 𝑍_𝑖=Im(Ζ), and 𝑀=(𝜕𝑅/𝜕𝜉)|_{𝜉=𝜉𝑅}.

Then for the phase velocity we have 𝑐=𝑐₁𝜉_𝑅𝛿𝑍1−^𝑟2𝜉_𝑅𝑀. (54) Since 𝛿>0, in this case there is a softening of frequencies of acoustic waves (instability of frequencies 𝜔_𝑟→0 and 𝜔_𝑖>0) and this is related to taking into account the generation of atomic defects. Equation (53b) describes attenuation of the amplitudes of acoustic waves.

Expanding 𝜂₁ and 𝜂₂ in powers of the small parameter 𝜔_𝑖/𝑐_𝑇⋅𝐿≪1 (𝜔_𝑟=0), we obtain from (21) the expression for the growth rate Γ=−𝜔_𝑖 of the surface periodical static structures in the form Γ=𝐷𝑞²𝛿1−𝜁⁻¹1−ℎ²𝑞²²1−𝑔²𝑞²−1−𝜏⁻¹, (55) where 𝜁=𝑐²_𝐿𝑐_𝑇⁻².

Introducing the dimensionless growth rate Γ=Γ𝑔²/𝐷, the wave number 𝑞=𝑞𝑔, the life-time of defects 𝜏=𝜏𝐷/𝑔², and parameters 𝜙=𝛿/(1−𝜁⁻¹), 𝛾=𝑔²/ℎ², dependence (34) can be represented as 𝑞Γ=²1−𝑞²𝑞²+𝜙1−𝛾𝑞²²−1−𝜏⁻¹. (56)

The analysis shows that the growth rate exhibits a single maximum at 𝑞=𝑞_𝑚 for small values of ratio parameter (𝛾) and an additional maximum (𝑞_𝑐) in the short-wavelength region for large values of 𝛾. Thus depending on the relation between the characteristic lengths 𝑔 and ℎ, elastic-concentration structures can have one or two (with 𝑞=𝑞_𝑚 and 𝑞=𝑞_𝑐 ) maximum growth rates.

Besides the growth rate has a single maximum not far above the threshold. However, at sufficiently high concentrations of the defects exceeding the threshold appears an additional maximum in the short-wavelength range. Therefore, as in the previous case, two gratings have the maximum growth rates.

The obvious expression for the maximum of the growth rates can be obtained if the scale coefficients (𝑔,ℎ) are of the same order in (34): Γ=𝑞²𝜙1−𝑞²−1−𝜏⁻¹. (57)

The growth rate Γ(𝑞) reaches maximum Γ_𝑚 at 𝑞=𝑞_𝑚=1√2𝐺1−^𝑐(𝑠)𝐺, (58) where 𝐺>𝐺_𝑐^(𝑠)=𝑘_𝐵𝑇𝜌𝑐²_𝐿1−𝑐²_𝑇𝑐_𝐿⁻²𝜏𝛽²₀ (59) is the critical value of the control parameter, at exceeding of which the self-organization of surface-ordered CS structures becomes possible due to development of the elastic-concentration instability. As a result, the surface deformation lattice and the corresponding relief modulation with the period 𝑑^(𝑠)_latt grow in time on the surface, as well as the defect concentration lattice (defect accumulations at the extrema of the surface relief).

The spatial scale of surface SC structures with wave number 𝑞_𝑚 is given by 𝑑^(𝑠)_latt=2𝜋𝑞_𝑚=√8𝜋𝑔𝐺𝐺−𝐺_𝑐^(𝑠). (60)

The maximum growth rate is Γ_𝑚=𝐷4𝑔²𝐺𝐺_𝑐^(𝑠)𝐺1−𝐺_𝑐^(𝑠)2−𝜏⁻¹. (61)Near the threshold 𝐺→𝐺_𝑐 the time of formation of the CS structures Γ_𝑚⁻¹→∞ that is critical slowing down takes place, which is characteristic of the phase transitions of the second kind.

The period of the ordered CS structures formed on the surface is proportional to the characteristic length ℎ of the nonlocal defect-atom interaction, which lies in the obtained nanometre range. As temperature is increased, the period grows for a particular value of ℎ. For a particular value of 𝑇, the period decreases with an increasing generation rate (𝐺).

Thus at certain conditions concentration-elastic instabilities in a system of atomic defects in unbounded condensed media as well as on the free surface of a solid half-space can be developed. We have observed that if the pump parameter is above the critical value, due to concentration-elastic instability, coupled strain-defect periodic nanometer sized structures on the surface of solids arise. A mechanism on the development of the instability is due to the coupling between defect dynamics and the elastic field of the solids. Laser radiation (or, in general, a flux of particles) generates high concentrations of mobile atomic defects in the surface layer of the irradiated material. When a fluctuation harmonic of the elastic deformation field appears in a medium because of the generation of atomic defects, the strain-induced drift of atomic defects occurs. This is a consequence of nonlocal defect-strain interaction. The strain-induced flux of defects gives rise to periodic spatial-temporal fields of the defect concentration. The redistribution of defects creates forces proportional to their gradients. These forces lead in turn to additional growth of strain fluctuations. When a certain critical value of the defect density or rate of defect generation is exceeded, diffusion-elastic instabilities develop as a result of positive feedback, which result in the formation of ordered CS structures.

As an example let us consider the formation of periodic surface CS structures in laser-irradiated semiconductors (in particular, CdTe). To evaluate the concentration of generated lattice defects (𝑛₀), we consider here conditions when the duration of a laser pulse (𝜏_Las) exceeds the defect-relaxation time (𝜏). In this case the density of defects on the surface of the solid reaches a steady-state value 𝑛₀=𝐺₀𝜏=𝐺₀𝜏₀exp−𝑤_𝑑0𝑘_𝐵𝑇₀, (62) where 𝑤_𝑑0=𝑤_𝑓0−𝑤_𝑚0,𝐺₀=𝐺𝑇₀ (63) is the defect generation rate, 𝑇₀ is the steady-state value of the temperature field on the surface, and 𝐺⁽⁰⁾ and 𝜏₀ are constants. If 𝐼_Las=const (uniform irradiation) and the optical absorption length (𝛼⁻¹_Las) is sufficiently less than heat diffusion length 𝑙_𝑇=(𝜒𝜏_Las)^1/2, the maximum temperature rise at the substrate surface owing to the laser pulse action may be evaluated [22] as 𝑇₀=2(1−𝑅)𝐼₀𝜆_𝑇𝜒𝜏_Las𝜋, (64) where 𝑅 is the reflectivity coefficient, 𝜆_𝑇 is the thermal conductivity coefficient, and 𝜒 is the thermal diffusion coefficient. So putting values into (64) for CdTe (𝜆_𝑇=0.05 W/cm°K, 𝜒=3×10⁻³cm²/s, 𝐼₀=1.2×10⁷ W/cm², pulse duration 2×10⁻⁸ s, and 𝑅=0.4) one can get 𝑇₀=1.2×10³ °K. Then, taking Ω_𝑑=10⁻²² cm³, 𝐾=5.10¹¹ erg/cm³, 𝜁=2.44, and 𝑤_𝑑0=1 eV, a value of 2×10¹⁹ 1/cm³ may be estimated for the critical density of defect concentration (𝑛_𝑐=𝐺_𝑐^(𝑠)𝜏), which is several orders of magnitude less than the concentration of the host atoms and shows that this mechanism of the formation of ordered structures may be realized on practice. For the period (𝑑_latt) of the resultant surface CS structure we have an estimate of 71 nm, which follows from expression (60) (for typical values of parameters 𝜌=5.85 g/cm³, 𝛽₀ = 30 eV, 𝑐_𝐿=2.4×10⁵ cm/s, 𝐺=1.1𝐺_𝑐, 2𝜋𝑔=15 nm, 𝐷=10⁻⁶ cm²/s, and 𝜏=3×10⁻⁹ s). The maximum growth increment of the instability is Γ_𝑚=6×10⁸s⁻¹; the instability increment exceeds the reciprocal of the duration of a laser pulse acting on the surface of a solid (Γ_𝑚𝜏_Las=12≫1).

The appearance of periodic nanostructures of the surface relief was detected experimentally [27] when the surface of the p-CdTe semiconductor crystal was irradiated by single pulses of the multimode ruby laser (pulse duration 𝜏_Las=2×10⁻⁸ s) with a homogeneous distribution of intensity. The period of the relief structure (with lateral size of the nanoclusters ∝ (19–23) nm) was 74 nm, which is in agreement with the predictions of our theory.

7. Conclusions

Based on the nonlocal elastic theory, which accounts for the small-scale effect by assuming the stress at a reference point as a function of the strain at every point in the body, the governing equations for the elastic displacement vector and defect density fields in an isotropic laser-excited solid have been derived. The defect-density dynamics are controlled by the generation and annihilation processes and also the strain-stimulated transport of defects. We have provided exact formal solutions for the displacements and defect density fields in an elastic solid half-space. A linear analysis is used to obtain dispersion equations of surface waves through the solution of three coupled integropartial differential equations, by assuming that the surface of the solids is stress-free. A procedure for determining the phase velocity and the attenuation (amplification) factor has been discussed. The proposed analysis is applied for the special case of very short wavelengths. The phase velocity and attenuation constants of the waves get modified due to the defect subsystem of the solid. We have shown that the dispersion equations predict two types of the instability: (1) instability of frequencies of acoustic waves and (2) generation of ordered (surface or volume) structures.

We observed that the small-scale effect plays an important role in the dispersion behaviors for larger wave numbers. We have demonstrated that due to the nonlocal character of the defect-atom and atom-atom interactions the dispersion curves of the instability have two maxima sufficiently far above the instability threshold. When the regular surface relief is generated, two maxima of the growth rate must give rise to two scales of the surface relief. These two scales of the relief are the characteristic of the surface nanostructure self-organization upon both laser and ion-beam irradiation of semiconductors.

In the case of the laser molecular-beam epitaxy (MBE) and the laser-controlled deposition process of atoms on substrates, the extrema (minima or maxima) of the surface relief formed by the CS structure serve as the nucleation and growth centers of the nanoparticles. Similarly, in the processes of multipulse laser etching, the rapid material removal also takes place on the extremes of the CS field, where the removal rate takes maximum values. As a result, on the surface of substrate, size distribution functions of the nanoscale nonhomogeneities having two maxima can be formed. A transformation of the unimodal distribution to the bimodal (with two maxima) distribution upon a variation in the irradiation regime (control parameter) is typical of both the MBE and the laser-induced surface nanostructuring of metals and semiconductors. Formation of bimodal metal (Au) nanoparticle size distribution function upon the pulsed laser deposition was detected experimentally in [28].

The results obtained in this work are expected to be helpful in designing the nanostructures in small-scale physical devices.