Formation of microand nanostructures on the surface of laser-created molten layer with inverted normal temperature gradient

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Emel'-yanov Vladimir I., the Doctor of Science in Physics and Mathematics, professor. Physics Faculty of Moscow State University, Moscow, Russia, emel@em. msk. ru
Abstract: A nonlinear two-dimensional hydrodynamic (HD) equation of Kuramoto-Sivashinsky (KS) type for the thickness of the laser pulse-induced viscous molten layer on solid base is derived in the long wavelength and weak nonlinearity approximation. Linear stability analysis shows that under the condition that the temperature gradient in the surface laser-melted layer is directed from the surface to the bulk, the thermocapillar or evaporative hydro-dynamic instability sets in, that leads to the formation of surface relief structures with dimensions proportional to the thickness of the molten layer. Computer simulations predict the successive formation, in linear and nonlinear regimes, of extended lamellar-like, disordered and hexagonal periodic structures of the surface relief when the time of irradiation is increased. Under tight laser light focusing, in the quasi-nonlinear regime, phase synchronization of Fourier harmonics of the surface relief, occurring due to the three HD mode interactions, is shown to lead to formation of holes or nanobumps (nanojets). Crown-like computer solutions of the HDKS equation are obtained in the case of Gaussian intensity distribution in the laser beam.
Index terms: pulsed laser irradiation of solids, laser-induced surface melt, inverted normal temperature gradient, thermocapillar and evaporative instability of surface relief, modified hydrodynamic Kuramoto-Sivashinsky equation, computer simulations of HDKS equation, lamellar, disordered, and hexagonal periodic surface structures, nanoholes, nanobumps (nanojets) and crowns
1. Introduction
The spontaneous formation of periodic micro-and nanostructures upon the action of energy or particle fluxes on the surface of solids has become the subject of intensive studies [1]. Оne of the main tools for the theoretical investigation of these phenomena is the nonlinear Kuramoto-Sivashinsky (KS) equation [2,3]. The two-dimensional KS equations with specific variables are used, for example, for the description of nanostructuring of solid surfaces using ion beams [4] or via electrochemical etching [5]. Two-dimensional KS equations were derived for the defect concentration in a defect-enriched solid layer formed by laser irradiation [6,7] and for surface defect (adatom) concentration [8]. The defect concentrations in both cases are coupled with the corresponding self-consistent surface deformation comprising surface defect-deformational (DD) fields. Numerical solutions of corresponding DDKS equations have been used in [7,8] for the interpretation of the selforganization of surface nano- and microstructures in the solid state.
Studies of formation of surface micro- and nanostructures occurring in laser-induced liquid state (surface melts) also pose a number of problems and questions regarding irradiation conditions, formation mechanisms and ways of controlling surface structure characteristics. Among them are establishing of necessary conditions of surface HD instability leading to structure formation and the dependence of the size and symmetry of fabricated surface structures on conditions of the laser irradiation.
In this work, we derive a new two-dimensional nonlinear equation for the perturbation h of the thickness hm
of the laser-induced molten layer. This Kuramoto-Sivashinsky (KS) type HD equation is derived in the weak nonlinearity approximation (h & lt-<- hm) from a general
equation for the thickness of molten layer bounded by a solid base from below [9]. The latter equation is derived in Ref. [9] for viscous liquid layer on solid base from the initial set of Navier-Stokes equations, the continuity equation, and boundary conditions at the interfaces in the long wavelength approximation, when the characteristic wavelength of the perturbation A & gt->- hm.
In this work, we apply solutions of the derived HDKS equation to the problem of description of formation of long-wavelength surface structures in thin surface laser-induced surface melts of semiconductors and metals (see Sec. 8). The weak nonlinearity condition is frequently met under multipulse laser irradiation since a seeding surface structure with small amplitude h & lt-<- hm, formed spontaneously at the initial stage of irradiation, is etched by subsequent laser pulses so that the wavelength and the symmetry of the small amplitude seeding structure is imposed on these characteristic of the final structure.
The weak nonlinearity approximation, exploited in this work, yields relatively simple modified two-dimensional nonlinear HDKS equation for h. The salient feature of
this equation is the occurrence of a new linear damping term specific to conditions of surface melting by laser radiation with Gaussian distribution of intensity in the beam, translating the derived HDKS equation in the class of stabilized KS equations. The derived HDKS equation is
analyzed analytically and numerically in this work. A broad variety of different solutions is obtained in dependence on irradiation conditions and types of irradiated materials (extended lamellar, disordered, and hexagonal periodic surface structures and solitary nanoholes, bumps, nanojets and crowns). The characteristic lateral scale of all these surface structures formed due to the HD instability is proportional to the thickness hm of the laser-induced surface molten layer.
The necessary condition of occurrence of the HD instability of the relief of laser-molten surface layer is the laser-induced creation of the inverted temperature gradient at the irradiated surface (see Fig. 1). Experimental situations in which this crucial condition can be met are considered in Sec.8.
We reveal other conditions of laser irradiation and materials for which the laser-induced creation of inverted temperature gradient is possible and briefly review relevant experimental results (Sec. 8).
2. Outline of the physical mechanism of hydrodynamic instability of the relief of laser-molten surface with inverted normal temperature gradient
Consider a solid (a semiconductor or a metal) irradiated by the laser pulse with the fluence exceeding the melting (or ablation) threshold, creating a surface molten layer of thickness hm with the inverted temperature gradient along the normal to the surface (Fig. 1).
Let the free surface of the melt be perturbed by a fluctuation harmonic h1 of the surface relief: h = hm + h,
where hm is independent of lateral coordinates x and y (Fig. 1).
Fig. 1. Laser pulse-created molten layer on solid base (hatched). The thickness of the melt hm is perturbed by the
surface relief modulation with the small amplitude h (
h & lt-<- hm) and the wavelength A. The inverted surface
normal temperature gradient VT can be created in a number of cases of pulsed laser irradiation of solids (see Sec. 8). The temperature throughout the text is assumed to be constant along the surface on much larger scale than the dimensions of the surface structures formed. But we take into account a slow radial drop of the surface temperature with the distance from the laser spot center due to the Gaussian distribution of laser intensity in the laser
pulse and possible lateral heat diffusion. On the contrary, the temperature gradient along the normal to the initially flat surface is assumed to be large due to intense laser heating and small melt thickness (Fig. 1).
The surface tension 7 is periodically modulated by the lateral gradient of and directions of surface liquid
flows induced by this modulation depend on the direction of temperature gradient at the surface along the surface normal. If this normal temperature gradient is directed from the surface to the bulk (Fig. 1), liquid flows, driven by thermocapillar forces (d (dT & lt- 0), are directed up hills of surface relief and the thermocapillar hydrodynamic (HD) instability of surface relief sets in,
with the perturbation amplitude h1 exponentially growing in time. At higher fluences, evaporative instability sets in due to higher rate of material evaporation from the valleys of surface relief occurring under condition of inverted temperature gradient (see Sec.8. 3).
The period of the perturbation harmonic of surface relief with the maximum growth rate determines, in the linear regime of the HD instability, the wavelength of the dominating surface relief modulation that is rapidly solidified after the termination of the laser pulse. This wavelength A is proportional to the thickness of the molten layer but may be several times larger than hm.
The above described physical picture of the HD molten surface instability driven by the thermocapillar force follows from the solutions of a modified two-dimensional nonlinear KS equation for the perturbation h1 of the
thickness hm of the laser-induced molten layer. We
briefly review the derivation of this two-dimensional thermocapillar HDKS equation in Sec.3. The modification of this two-dimensional HDKS equation by taking into account of laser-driven spatially nonuniform surface evaporation, dominating at high laser intensities, is performed in Sec.8.3. Also, the one-dimensional HDKS equation on the ring is obtained in Sec.8.3. Solutions of these equations are shown to describe the formation of extended lamellar, disordered, and hexagonal periodic surface structures, and solitary surface nanoholes, bumps, nanojets and crowns.
3. Derivation of the modified hydrodynamic Kuramo-to-Sivashinsky equation for the modulation of relief of the molten surface layer
We consider a laser pulse-induced viscous molten surface layer of thickness h = h (X, y, t) (the liquid film) bounded below by a horizontal plate of non-melted dielectric material (& quot-the substrate& quot-) and above by the interface between the molten layer and ambient gas (or water). The z -axis is directed from the substrate upward, the plane z = 0 coincides with the flat film-substrate interface (Fig. 1).
The molten film of thickness h is described by the set of three Navier-Stokes equations for the components of
liquid velocities Vx, Vy and Vz along x, y and z -axes, respectively, by the continuity equation, involving Vx, Vy and Vz, and boundary conditions at the interfaces [9]. In the long wavelength approximation, under the condition of zero velocity of the melt at the solid-melt interface (z=0), this set of equations can be reduced to the following equation for h (see Ref. 9]):
Ih = --L-dw[h2 (r + Vo)] --P-div[h3V (oAA -P)]
Here V = ex (d/dx) + ey (d/dy), ex and ey are unit vectors along x and y axes, respectively,
A=32/3×2 +32/dy2, p and n are the melt density and kinematic viscosity, T is the surface tangential stress at z = h, exerted by surrounding water or laser-induced plasma, 7 is the surface tension. The pressure P = p + pgh, where g is the gravity acceleration and
P is the vapor pressure. In this section, upon derivation of the thermocapillar HDKS, we will neglect pressure effects (P = 0) and tangential stress (T = 0). Spatially nonuniform pressure effects will be taking into account in Sec.8.3 upon derivation of evaporative HDKS.
We, thus, focus here on the contributions, stemming from 7 and take into account that
Va= (dap) (dTdz) Vh (2)
with da/dT & lt- 0. Possible laterally fast periodic change of T arising due to the HD instability development is neglected in Eq. (2). However, the slow monotonous lateral temperature (and, thus, the surface tension) variation arising due to the Gaussian intensity distribution in the laser beam will be taken into account. It will be shown to lead to occurrence of linear damping of the surface relief undulations arising as a result of the HD instability (see Eq. (3)). Besides, in Sec. 8.3 it is shown that lateral radial outward melt flows within molten pool, induced by Gaussian temperature distribution, lead to creation of the circular rim with inverted normal temperature gradient.
We use in Eq. (1) the representation h = hm + h, where h is a small perturbation of the flat surface z = hm. Introducing the small parameter 1 = 2nhJA & lt-<- 1, where A is the wavelength of the
perturbation of surface relief (Fig. 1) and retaining leading terms in? I, we obtain from Eq. (1), after the differentiation, the following equation:
oh 3
Ahl-°h^ A2h +
dh, idT^ h:
-L = -rh+l — I or--¦.,., dt 1 I dz Jv'- T| 2pn 3pt] 1 ^ dz
hl (Vh,)2 + V (Ah,)
where (t| = |a (3T|, (3^dz)hm ,(3^dz)z=hm •
The first term in the right hand side of Eq. (3) with
Y= hm A (pn (4)
takes into account a slow lateral variation of 7 stemming from the laser heating of the surface by the laser pulse with Gaussian intensity distribution (Aa& gt- 0). It leads to damping of the HD surface relief instability.
For generality and to establish the limits of applicability of the considered mechanism of surface relief instability, we take into account that, due to the possible laser-induced spatially uniform evaporation, the interface z = h moves towards the bottom of the molten layer with the velocity
V = C0expl-
0 -Ahl «V0 — V0-^-Ahl '-
where V0 = C0 exp (-UlnakBT), C0 is a constant,
U is the activation energy and na is the number of atoms in a unit volume. This leads to occurrence of additive term v 7
in the right hand side of Eq. (3).
Let us assume that we have the inverted temperature gradient at the interface (relevant experimental situations are discussed in Sec. 8):
I'- & lt- 0-
Then, from Eq. (3), we obtain the following HD equation:
dh = -rh — (D — R) DAK -Dihm2A2h -('-Vh)2 -(Ah)2,
dt hm 3
D = (|7T|hm2/2pn)|dT/ 3z|hm& gt- 0, R = V0(nkBT,
D =ahj3pn& gt- 0. As is seen from Eq. (7), the driving force of the instability is the thermal capillary force and necessary condition of the instability is the negative sign of the second term in the right-hand side of Eq. (7).
In comparison with the HDKS equation from Ref. [10], the modified HD equation (7) contains the additive nonlinear term in the right-hand side ~ (Ah1)2. This term
has been deliberately omitted in the preliminary derivation of the HDKS equation made in Ref. [10] since it represents the term of higher order in small parameter
1 = 2nhm/A compared to (Vh1)2 and does not play a
role in formation of extended surface structures. But as it is shown in Sec.8. 2, under conditions of tight axially
symmetrical focusing of laser pulse, the term (Ah1)2
dominates and determines the formation of localized holes and bumps (nanojets).
We take also into account that of interest here is the regime of moderate laser fluences, when R & lt- D, so we set R = 0 in the equation (7). At a very high laser flu-ence, when R & gt- D, the HD instability does not occur.
Thus, we confine below to studies of the following HD equation
% = -A -DAh-DhjAh -^(Vh)2 -IDhm (Ah)2 '- (8)
dt hm 3
Eq. (8) is the modified two-dimensional HDKS equation. If we neglect the last term in the right-hand side of Eq. (8), then it reduces to the general form of the KS equation [2,3] with account for damping, i.e. the term ~ y (the stabilized KS equation). Thus, Eq. (8) can be referred to as the modified stabilized HDKS equation describing ther-mocapillar instability of surface relief. Modification of Eq. (8) to cover the case of evaporative instability is performed in Sec. (8. 3). One-dimensional HDKS equation on the ring is derived also in Sec. 8.3.
4. The dimensionless HDKS equation in the mode representation. Three regimes of surface HD thermocapillar instability
Let us introduce the dimensionless surface layer thickness modulation variable H = 2h^/hm, the dimensionless time 0 = t (Djhm2), coordinates X = x / hm and Y = y / hm, the dimensionless decay parameter r = y (hm 2/D1) and the control parameter
? = D/Di & gt- 0. Then, from Eq. (8), we obtain the dimensionless thermocapillar HDKS equation:
= -ra-eAH-A2 H-e
A2 H-e[(VH)2 + (AH)2 ] -
dH do
where A and V are two-dimensional Laplace and the gradient operators, respectively, written in dimension-less coordinates. We seek a solution of Eq. (9) as the Fourier series
H (r, 0) = Z H (k, 0) exp (/kr), (10)
where r ={X, Y} and the dimensionless wave vector
k =Khm & gt- Qyhm}.
The substitution of Eq. (10) in Eq. (9) yields a set of coupled equations for Fourier-amplitudes
where the dimensionless growth rate
= (ek2 — k4)-r
surface relief. Below, we consider three regimes of the laser-induced surface HD instability.
a) The linear regime (LR). In the LR, we neglect altogether nonlinear terms in Eq. (11). In the superposition (10) Fourier-amplitudes h (k, 0) = |H (k, 0)| exp[^(k)]
evolve independently of each other. This evolution is governed by linearized Eq. (11) with a time independent random phases ^(k), equal to a random phases at the
initial moment 0 = 0. Results of computer simulations of the LR are given in Sec.5 and 7. It is shown that an extended lamellar-like surface relief structure is formed as a result of the HD instability in the LR regime.
b) Quasi-nonlinear regime (QNLR). In the QNLR, the rapid phase synchronization occurs, described by the nonlinear terms in Eq. (11) (see Sec. 6). Due to this, all Fourier-amplitudes in the superposition (10) acquire one and the same phase ^(k) = 0, so that in Eq. (10) we have
H (k, 0) = H (k, 0)|. In the QNLR, all Fourier-amplitudes
H (k, o)| evolves independently of each other according
to the linearized Eq. (11). It is shown that in the QNLR regime in axially symmetric case, solitary holes (or bumps) are formed (see Sec. 6).
c) Nonlinear regime (NLR). This regime is described by the nonlinear two-dimensional partial differential equation (9), which is studied numerically in Sec.7 neglecting the last term in the right-hand side. Computer simulations predict the successive formation, with increase of time of evolution, first, in the linear regime, of extended lamellar-like structures and then, in the nonlinear regime, of disordered and hexagonal periodic cluster structures of surface relief. In axially symmetric case upon tight focusing of laser radiation, the last nonlinear term
in the right-hand side of Eq.9 (~ (AH)2) dominates and
solutions of the HDKS equation (9) describes the formation of needle-like structures-nanojets. Micro- and nanocrowns as solutions of the nonlinear HDKS equation (9) on the ring are obtained in Sec.8.3.
5. The linear regime of HD instability. Formation of extended lamellar-like structures of surface relief
The growth rate, Eq. (11a), achieves a maximum at the dimensionless wavenumber
k = k™ = I ~2
The dimensionless growth rate of the dominant modulation is
L. =

V 4 y

The condition ^ & gt- 0 defines a band of unstable HD modes taking part in the thermocapillar HD instability of
At exceeding the threshold value of the control parameter:
?& gt-? = 2r12,
the surface HD instability sets in (Am & gt- 0) and the
Fourier-amplitude of the dominant surface relief modulation starts to grow in time. The dimensionless wavelength of this dominant modulation is
-n M !
Corresponding dimensional characteristics and numerical estimate are given in Sec.8.
The 2D surface relief is given by the superposition of solutions of the linear equation (11):
H (r, 0) = Z| H (k)| exp[Ak0+ikr+((k)], (15)
where|H (k)| is the modulus of an initial Fourieramplitude and ((q) is a random initial phase (see
Eq. (21a) for the condition of neglecting temporal phase evolution). For simplicity, we assume that the initial conditions are such that |h (k)| = const ^ A and initial phases
((k) are randomly distributed in the interval [0, 2n ]
(the white noise). Then, introducing the normalized relief height: z (r, 0) = H (r, 0)/2A, we obtain from Eq. (15)
Z (r, 0) = Z exp (^O)cos (kr+((k)), (16)
|k| & lt-k0
where k0 is found from the condition Ak =0. The surface relief, described by Eq. (16), is shown in Fig. 2.
Fig.2. Lamellar-like surface relief generated in the linear regime. This 3D image is constructed with the help of solution of
Eq. (16) with random phases ^(k) and growth rates A given
by Eq. (11a) at e = 0. 7875, r = 0. 15, T = 17.2. Computer simulations of Eq. (9) show that the extended lamellar-like structure is universal and appears in the linear regime of the HD instability as an intermediate state in sequence of surface relief transformations during laser irradiation (see Sec.7 and Sec.8. 1).
6. The quasi-nonlinear regime of HD instability. Phase synchronization and formation of solitary holes and bumps on the surface in axisymmetric case
In Sec. 5, considering the LR, we neglected possible temporal phase evolution. The condition of validity of
this regime can be obtained by investigation of the temporal evolution of phases.
Highlighting the phase, we represent Eq. (15) in the form
H (r, 0) = Z| H (k, 0)| exp (ikr+((k, 0)). (17)
Substituting Eq. (17) in Eq. (11) and separating real and imaginary parts, we obtain the following equation for the phase
Our aim is to reduce Eq. (18) to the relaxation equation for the phase ((k).
We consider below axially symmetric case realized in laser-induced melting experiments, in which the surface is melted by the laser pulse with Gaussian intensity distribution il ® = /"exp (-r2/rL2), where r is the distance from the center of the laser spot, rL is the radius
of the spot and I0 is the laser intensity in the center of
the spot. Thus, neglecting lateral thermal diffusion, we have for the temperature variation in the spot: T® = T0 exp (-r2/r, 2). This temperature distribution
creates a axisymmtrical bump (or a depression due to evaporation) on the surface with the spatial dimension of order of rL. This means, that a low spatial frequency
band of Fourier harmonics with wavenumbers kj & lt-<- km,
centered at k = 0, with the width of order of rL-1 is
created. Fourier amplitudes in this narrow spatial frequency band can be considered as given quantities determined by conditions of laser excitation: |h (kt)| ~ lL
and ((kj)= 0 (all low frequency harmonics are in phase).
Taking into account the contribution of these low spatial frequency harmonics in the sum in Eq. (18), we can use approximation
sin (((kj) + ((k — kj)-((k)) = ((k — kj)-((k). Besides,
in Eq. (18) we have ^-^-^(k-k,)1 *k1-k-ft1V'- |ff (k)|"|ff (k-kl)| and 1^(^)1 = 17/(^)1.
Then, from Eq. (18), we obtain the following phase relaxation equation
The steady state solution of Eq. (19) is ((k-kj) = ((k) = 0 (the phase synchronization). The
characteristic inverse phase synchronization time
The contribution with j^'-k ?n Eq. (20) cancels in considered axially-symmetric case upon integration over the polar angle. Note that in the case of molten stripe the term kj-k becomes prevailing and determines formation of grooves (ripples). The surface ripples formation described the corresponding HDKS equation in anisotropic case needs separate consideration.
According to Eq. (19), on times exceeding the phase synchronization time (0& gt-T<-), all high spatial frequency
harmonics k in the amplification band (Ak & gt- 0) have one and the same phase p (k) = 0, equal to the phase of & quot-injected"- signal (p (k 1) = 0), and the QNLR sets in. The condition of the QNLR realization,
Tp (k)-1 & gt->- Am = [4]-r, (21)
can be fulfilled in a vicinity of the HD instability threshold.
In the opposite limit,
T. (k)-1 & lt-<- Am =
V 4 y
we can neglect the phases evolution and use Eq. (15) which leads to formation of extended lamellar structure.
In the QNLR, we set in Eq. (15) p (k) = 0 and describe
the surface relief by the equation
H (r, 0) = ±Z| H (k)| exp[Ak^+ikr], (22)
where |h (k)| is an initial amplitude. To simplify the analysis, we set in Eq. (22) |h (q)| = A = const and introduce the normalized relief height: H (X, Y,0) = H (r, 0)/A. Then, from Eq. (22), we have, selecting the negative sign
H (X, Y, ff) = - J J exple (qX2 +qY2)-(qX2 +qY2) Iff
H (X,(c)) = - J exp[(eq*2 — qx4)(c)]cos (qxX)dqx and is shown in Fig. 4b.
cos (X +qyY)dq^ dqy
The surface 2D relief, given by Eq. (23) is shown in Fig. 3a. The cross section of the 2D relief along X-axis is given by the formula following from Eq. (23)
Fig. 3. a) The 2D surface relief (the hole) described by Eq. (23) at e = 1, © = 1. b) The cross section of the relief shown in Fig. 3a. Constructed with the help of Eq. (24) at e = 1 and © = 1. Selecting the positive sign in Eq. (23) leads to formation of a bump (see Fig. 8a).
Thus, in the QNLR, under the condition of rapid phase synchronization, the formation of either solitary holes (negative sign in Eq. (22), Fig. 3) or bumps (the positive sign in Eq. (22), Fig. 8a) occurs. The mechanism of selection of corresponding sign in Eq. (22) is not clear at the moment. In strongly nonlinear regime, bumps acquire the shape of jets (see Sec.8. 2). Corresponding experimental works of laser-induced holes, bumps and jets are briefly reviewed in Sec.8.
7. Computer simulations of nonlinear regimes of the surface HD instability described by the HDKS equation
To describe the formation of extended surface structures in the NLR, we address Eq. (9), where we neglect
the last term in the right-hand side ~ (AH)2 (it will be
taken into account in Sec.8. 2) and obtain the dimension-less HDKS equation in the form:
= -rH-eAH -A2H -e (VH)
In numerical studies of Eq. (25), we use the spatial region 100×100- temporal steps AT=0. 075 and coordinate steps AX =AY=1. The boundary conditions are periodic: H (-a, Y, T) = H (a, Y, T) and H (X, -a, T) = H (X, a, T), where 2a is the size of the region under study along
each axis. Thus, we consider the boundary-value problem (with respect to spatial variables) and assume that initial values H (X, Y, 0) are randomly distributed in the interval [0,0. 1] with the mean value & lt-H (X, Y, 0)& gt-X, Y =0. Solutions were obtained at different values of dimensionless time 0.
First, we present in Fig.4 results of numerical solution of Eq. (25) in the linear regime (on small times 0).
Fig.4. (a) Numerical solution of the 2D HDKS equation (25) with? = 0. 7875, r = 0. 15 at the dimensionless time 0= 17. 2- (b) corresponding 2D Fourier spectrum. © and (d): the 3D representation of the same solution
It seen that in this case, at the initial stage of the selforganization, the lamellar structure is formed. The Fourier-spectrum of the lamellar structure shown in Fig. 4b is the ring in the wavevector space. The 3D image of this lamellar-like structure, shown in Fig. 4c, is similar to the structure shown in Fig. 2, obtained simply by summation of Fourier harmonics of surface relief (16) with wavevectors k, the modules of which lie within the gain band Ak & gt- 0, and directions are randomly distributed in
the angle 2n (the ends of the wavevectors k lying on the ring shown in Fig. 4b).
The results of numerical solution of HDKS equation (25) at longer times of irradiation (in the nonlinear regime) are shown in Fig. 5.
Fig. 5. (a) Numerical solution of the 2D HDKS equation (25) with? = 1. 2325, r = 0.2 at the dimensionless time 0 = 54. 4- (b) corresponding 2D Fourier spectrum. © and (d): the 3D representation of the same solution It is seen from Fig.5 that, in the nonlinear regime (large 0), the HDKS equation (25), at slightly different values
of control parameters? and r compared to Fig. 4, generates specific & quot-cobblestone"- structure. The distinctive feature of the latter structure is the broad Fourier spectrum (Fig. 5b and c), extending from small values of the
wavenumber k to the limiting value k — k0, determined from the condition A — 0, where the growth
rate Ak is given by (11a).
At still longer times d, in the developed nonlinear regime, the HDKS equation (25) generates quasi-hexagonal disordered ensembles of clusters (dots) (see Fig. 6).
Fig. 6. (a) Numerical solution of the 2D HDKS equation (25) with e — 0. 7875, r — 0. 15 at the dimensionless time e — 330 — (b) corresponding 2D Fourier spectrum. © and (d): the 3D representation of the same solution
At the same values of control parameters? and r as in Fig. 6, but at even more long times e, the numerical solution of the HDKS equation (25) describes the formation of ordered hexagonal ensembles of dots (clus-ters)(see Fig. 7).
Fig.7. (a) Numerical solution of the 2D HDKS equation (25) with e — 0. 7875, r — 0. 15 at the dimensionless time e — 458 — (b) corresponding 2D Fourier spectrum- © and (d): the 3D representation of the same solution From this consideration, we see that the HDKS equation (25) generates at the same values of control parameters? and rthe following sequence of solutions when the time of selforganization e is increased: at first, lamellar structures (Fig. 4), then disordered quasi-hexagonal dot structures (Fig. 6), and, at last, ordered hexagonal ensemble of dots (Fig. 7). As an intermediate
type, at slightly different values of control parameters e and r, the cobblestone structures can be generated (Fig. 5).
To elucidate the salient feature of obtained results and for making numerical estimates in the following, we return in formulas (11a)-(14) to dimensional variables and obtain the growth rate in the form:
It achieves a maximum at q = qm = (d/2D1)12 hml ¦ The growth rate of dominant surface relief modulation is A = (d2?4D)h -2 — y. The wavelength of the dominant
modulation is
Aq = Dq2 — Dihm2q4 — Y.
Am = - = 2nhm
q V D
2D, Y2 4n,
rldT/dzh hm
estimate of A.
we use
To obtain a rough m
and |or| ~ o/ T and have from (26a)
i.e the characteristic lateral scale of
A m ~4nhJ 3~4hm structures is proportional to the melt thickness but can essentially exceeds it.
8. Review of solutions of the HDKS equation relevant for interpretation of experimental results
There are a number of experimental situations in which laser pulse irradiation can create a thin surface molten layer or circular rim with inverted normal gradient (directed from the irradiated surface of the layer or from the rim top to the bulk) leading to HD instability. Below, we consider three examples of such situation and discuss corresponding solutions of the HDKS equation.
8.1. The sequence of extended lamellar-like, disordered quasi-hexagonal microdot and ordered hexagonal dot solutions of HDKS equation
Upon laser pulse irradiation of the solid surface in water (liquid) confinement, the necessary inverted temperature gradient can be created due to the effective thermal energy transfer from the molten surface to the confining liquid (by thermal conduction, liquid evaporation and convection, and plasma and shock wave formation). In the case of multipulse irradiation, the second and subsequent pulses in the train melt the subsurface layer of thickness h again and again, but the surface relief is not smoothed out, if the period of oscillations in capillary
wave i & gt- t, where t is the melt duration. This
cap m m
condition can be fulfilled in the case of the long-
wavelength (A & gt->- hm) surface structures formation. In
this case, the amplitude of the surface relief undergoes the cumulative growth for a sequence of laser pulses, under the condition that the wave pattern solidifies quickly, so that the frozen pattern serves as the initial condition at the time the next pulse arrives. Thus, with increase of the number of laser pulses the surface HD
instability undergoes effectively continuous temporal evolution, first, in the linear and then in the nonlinear regimes. In doing so, the surface relief goes through several stages including three main ones: lamellar, disordered quasi-hexagonal microdot and ordered hexagonal dot structures. This picture is observed in experiments on the nanosecond laser pulse-induced formation of hierarchy of microstructures on silicon surface in water confinement [11].
8.2. Solitary bump, hole and jet-like solutions of the HDKS equation
In the case of ultrashort laser pulses, the inverted normal temperature gradient at the surface can be created due to the evaporation of surface atoms [12]. This effect can be most strong in the case of binary semiconductors with volatile elements in composition, intensive evaporation of which causes effective cooling of surface layer and thus creates inverted normal temperature gradient. As an example may serve the formation 200 nm in diameter (one quarter of the illumination wavelength) holes upon single femtosecondlaser pulse irradiation of AS2S3
observed in Ref. [13].
The condition of inverted normal temperature gradient at the free surface can be met also in the case of formation of micron scale periodic ripples upon backside irradiation of thin metal films on glass substrate by a train of nanosecond laser pulses [14] and on laser pulse-induced formation of nanoholes in Si, cupped with artificial metal structure [15]. In the latter case, the inverted normal temperature gradient at the surface can be created due to effective thermal conduction from the surface upward to the cup metal structure, mediated by plasma ablated from Si surface by the laser pulse in tight confinement conditions.
Inverted surface temperature gradient can be created also in the case of using ultrashort (ps and fs) laser pulses for irradiation of two-layer structure with up-per (irradiated) layer having relatively low melting point and the underlying layer (the substrate) having relatively high melting point. In doing so, the thickness of melted layer (the film) must be small (of order or less than 100 nm) compared to the thickness of the underlying substrate with high value of latent heat of solidification. Besides, the thermal diffusivity of the molten layer must be relatively large compared to the thermal diffusivity of the substrate. In this case, melting and subsequent solidification of upper part of the substrate leads to extraction of latent heat of solidification creating buried source of heat underneath of the still molten upper film. Thermal diffusion from this source upward creates inverted normal temperature gradient in the film. This mechanism of creation of inverted temperature gradient was proposed in Ref. 16] and used in studies of laser pulse-induced melting and subsequent solidification of thin amorphous Ge film on Si substrate.
We supposed in Ref. 17] that these conditions were also met in experiments carried out in [17], where the Au/Pd (80/20) film of thickness 60 nm on CaF2substrate was melted by a sequence of femtosecond (Tp = 200fs) laser pulses with X = 515nm focused in
the spot of radius ~ 1.5 [im and a solitary nanospike (nanojet) was formed in the center of laser spot (see Fig. 8c). The melting temperature of Au/Pd film (TmAu/Pd = 1490K) is less than the melting temperature of
CaF2substrate (t mCaF2 = 1620K) while its thermal dif-
• 1sm2 *s-1) is larger than that of the sub-
fusivity (XAl
strate (xCaF2 & lt- 0.1 sm2*s-1).
It was assumed, thus, that the formation and development of a nanojet are due to the thermocapillary instability of the melted film, described by Eq. (9). The low thermal conductivity and partial melting and solidification of the dielectric substrate under the film results in the appearance of the normal gradient of the temperature T directed from the irradiated surface into the film along the z-axis. Due to this, the relief of the surface of the melt becomes unstable: at a local increase in the thickness of the melt h (r, t) = hm + h^r, t), where hm = const, hj® & lt-<- hm, and r = {x, y}, thermocapillary forces generate rising flows enhancing fluctuations. Fluctuation h1(r, t) satisfies the two-dimensional HDKS equation (9), where H=2hx/ hm. Formation of nanojet occurs in two stages.
At the quasilinear stage (at small times d), under the condition of rapid synchronization of the surface relief harmonics (see Sec. 6), the solution of Eq. (9) is represented in the form of the superposition of harmonics similar to Eq. (23), yielding the bump shown in Fig. 8a. This solution of the HDKS equation describes qualitatively experimentally observed nanobump (Fig. 8c).
In the nonlinear regime (at large times d, or large values of the control parameter ?), in the case of strong axisymmetric focusing, the last nonlinear term in Eq. (9) ~ (AH)2 dominates.
With neglect of damping (r = 0), the solution of one-dimensional Eq. (9) is shown in Fig. 8b, which describes the subsequent growth of a peak. Note, that with neglect of both r and (vh)2terms, Eq. (9) coincides with the KS
equation for the interface between different solid phases, studied numerically in Ref. [18], where an & quot-explosive"- (a sharp peak-like) solution similar to Fig. 8b was obtained.
One can expect, thus, that the solution of corresponding two-dimensional HDKS equation (9) will give axisym-metrical peak (jet-solution). This question needs extra studies.
Fig.8. Calculated profiles of the nanojet at the (a) quasilinear stage of its formation: solution by Eq. (23) with e = & amp- = 1 and (b) nonlinear stage (numerical solution of one-dimensional equation (9) with r = 0), © Scanning electron microscopy image of the surface of the gold film irradiated by single ultrashort laser pulse[17].
Another possible case, covered by the thermocapillar HDKS equation (8) (or, in dimesionless form (9)), is the formation of nanoholes in thin Au film on thermally isolating glass substrate under the irradiation with tight-focused laser pulse [19]. The calculated shape of cross section of the hole (Fig. 3b) is strikingly similar to that observed in Ref. 19].
8.3. Crown-like solutions of the HDKS equation on the ring
A crown-like solutions of the HDKS equation (see Fig. 9) describing laser-induced formation of a nanocrown observed in [24] or microcrown [20] occur in the case of axi-symmetrical distribution of temperature field in laser spot on the irradiated surface with maximum at its center. This case is realized under Gaussian distribution of laser fluence.
Laser-induced formation of surface crowns are observed at relatively low laser intensity (a nanocrown, induced by a single nanosecond laser pulse [24]) and at relatively high laser intensity as well (a microcrown, induced by a train of
femtosecond laser pulses [20]). In the low intensity case, the crown formation is described by the thermocapillar HDKS equation (8). For the description of a high laser intensity case, it is necessary to modify this equation with taking into account of spatially non-uniform surface evaporation which dominates in this case over the thermo-capillar force and, thus, determines the HD surface relief instability.
Such HD instability is described by the evaporative two-dimensional HDKS equation for h^r, t) which is obtained from Eq. (8) by subsitution: D ^ Dp, 2 ^ 3 and neglecting of the last nonlinear term in the right-hand side. Thus, we obtain the two-dimensional evaporative HDKS equation in the form
= -DpAh, — Dh where
3D 2
2A2h — 3DP- (Vh)2 h
DP =
hj dT
3pn dz
hm nkBT0
& gt- 0 •
ten rim solidifies in the crown-like shape. Upon multishot laser irradiation, each laser pulse melts the surface near the crown, regenerating the modulated melt rim. Again, evaporation occurs more intensively between the ridges, and the normal temperature gradient is still directed inward. Thus, there is a cumulative development of the melt rim instability from pulse to pulse, resulting in the growth of the crown ridges.
Let us assume ARm & lt-<- Rm and consider a one dimensional evaporation instability on the circular rim of radius Rm. In this case, the rim height depends only on the angle p between the x or y axis and the vector r (see Fig. 9a): h = hm + h (p, t). Then, v2 =_L and Eq. (27)
takes the form
32 H
dH dt
, a4 h

Here, (dT/dz)h — (dT/dz=h & lt- 0 is the inverted
normal temperature gradient, P0 is the recoil vapor pressure- Lv is the vaporization heat- T0 is the surface temperature, n is the atom concentration in the melt.
We invoked the evaporative HDKS equation (27) for interpretation of the formation the microcrown on the silicon surface upon femtosecond laser ablation with varying number of laser pulses [20]. According to proposed in [20] interpretation, laser-induced formation of surface micro-crown (or nano-crown), proceeds in two stages.
At the first stage, a circular rim appeared at the edge of the melt bath with the radius Rm and thickness ARm (Fig. 9a), with the melt depth Hm being the radial decreasing function Hm = Hm (|r|) with Hm (Rm) = 0, following a Gaussian laser fluence distribution with the peak fluence F0 exceeding the ablation threshold. Since the surface temperature is higher at the center of the melt bath than at its periphery, the lateral surface tension gradient (dofdT & lt- 0) pushes the melt outward. In doing so, the viscous friction force in the melt increases radially versus its decreasing thickness and slows down its motion until its complete stop at the melt edge (r = Rm), where the thermocapillary pressure pushes the melt up, producing a thin circular rim of radius Rm and height h (Fig. 9a).
During the second stage, the dynamics of the rim height is assumed to be driven by an inverted normal temperature gradient (dT/dz & lt- 0, Fig. 9a), since the bottom part of the rim is produced by hotter melt flow from the central part of the molten bath, while the top part is formed by the cooler peripheral melt. At any fluctuation of h, evaporation occurs more intensely at the height minima. Such spatially inho-mogeneous evaporation yields the instability of the melt height along its rim, which is stabilized by the surface tension force. After the pulse termination, the modulated mol-
where H = hjhm, / = DB h:)DpRl, a = DjRm2.
To perform a linear analysis of the evaporation instability, we wright the solution of the linearized Eq. (29) in the form
H (c, t) =? H0(N)exp (/Nc+ANt), (30)
where H0(N) = const, N is an integer number, and AN is the instability growth rate,
xN =m2-a/N4, (31)
for the N-th azimuthal harmonic. The solution (30) satisfies the periodic condition on the ring H (p, t)= = H (q& gt- + 2n, t).
According to Eq. (31), the maximum growth rate is achieved at
*==m?=f f D T=f
Rm dT
2a dz
Po L
hm nkBT0
The value Nm determines the number of ridges in the crown (Fig. 9b). The wavelength of the dominating harmonic of the rim modulation is
= 2nh

= 2n
with the growth rate of this harmonic
^ =
Q D 2 p hj f dT Po Lv 1
4? 4 DA2 12pj]a V dz hm nkBT0 ^
For numerical estimates in the linear regime of instability, we use parameter values relevant to experiments on formation of microcrown on the Si surface, irradiated by a train of femtosecond laser pulses with fluence F~0.5 J*cm-2 focused in laser spot with radius rL ~15m [17]. Thus, T0 = 5500 K and the recoil vapor pressure P0(T0) =0. 54Pb = 5. 86×106 dyn/cm2,where n = 1023 cm-3, the normal atmospheric pressure Pb = 106 dyn/cm2, for Si melt Lv ~ 3.2×1011 erg/cm3 [12], and the normal boiling temperature Tb = 3514 K. The crown formation time t! orm = 10xhmax~ is estimated,
2nkBT0 a
using Eq. (8) for a = 750 erg/cm2 and pn =6.9×10−2 g/(cm*s) [21], yielding tform ~ 10−8 s. Upon each laser pulse, the molten rim solidifies during tfreez /x ~ 10−7 s for x ~ 0.1 cm2/s, preserving the evolving crown. Under the assumption that
dT/dz|A ~ T0/hm and |aT| ~ a/T, the varied param-
eter equals / = 0. 003.
In order to obtain a numerical solution in the nonlinear regime, we introduce the dimensionless time u = tQ and reduce Eq. (29) to the dimensionless form
dH _ d2H ff94H idH -=-rr
de dp2
The numerical solution of Eq. (35) at this value of ^ = 0. 003 and e = 1 is shown in Fig. 9b.
Fig. 9. (a) Sketch of a micro-crown as a circular molten rim with radius Rm and periodically modulated height h = hm + h1 (see fragment of the modulated rim in Fig. 1) — (b) numerical solution of Eq. (9) at /3 = 0. 003 and 6 = 1.
The comparison between the parameters of the experimentally observed in [20] crown and numerical calculation results (Fig. 9b) demonstrates their qualitative (crown formation) and quantitative agreement. According to the model estimates from Eq. (32) for hm ~10−4 cm and Rm ~10−3 cm, N = 13 is in reasonable agreement with the experimental values Nexp ~ 18 [20] and the numerical solution of Eq. (35) (Fig. 5b) at the same ?. Thus, the proposed hydro-dynamic model of evaporation instability of surface melt provides an explanation of the sharp edge formation (with a rim or multiple spikes) during ablation of Si and other materials [22, 23].
Similar mechanism of circular molten rim formation and similar HDKS equation on the ring but with the thermocapil-lar effect as a driving force of the HD instability of the molten bath rim was invoked in Ref. 24] for description of formation of nanocrown under single nanosecond (r = 7 ns)
pulse irradiation of a thin (60nm) Au/Pd film on Cu substrate
using tight focusing (rL ~ 0.3 ^m) of radiation with Gaussian distribution of intensity and F~0.2 J*cm-2.
9. Conclusion
This review shows that the HD instability of the surface relief of laser-molten layer occurs under the condition of creating of the inverted temperature gradient at the surface (either over the whole surface of the molten bath or at its outward rim). At relatively low fluences, the thermocapillar effect serves as a driving force of HD instability, while at higher fluences spatially nonuniform evaporation plays the dominant role. In both cases, the HD instability is described by the universal HDKS equation.
Numerical solutions of the HDKS equation describe the formation of the hierarchy surface micro- and nanostruc-tures upon transition from linear to nonlinear regimes of the HD surface instability (first, lamellar structures, then disordered quasi-hexagonal dot structures, and, at last, ordered hexagonal ensemble of dots). Sequential transitions from one type of structure to another occur at constant exceeding over threshold? with the increase of duration of the selforganization process (the time of irradiation). Another way to control the characteristics of fabricated structures is the variation of control parameters in Eq. (25): exceeding over threshold? and the damping rate constant r. The establishing of regions of occurrence of different structures in the phase plane of control parameters? and r needs more extensive studies of solutions of the HDKS equation (see such phase diagram for the solutions of nonlinear DDKS equation for solid state in [7] and [26]). Practically important conclusion following from the analysis of the HDKS equation, is the proportionality of the characteristic lateral size of fabricated structures to the thickness of laser-induced surface molten layer hm, which can be varied from nanometer to micrometer regions by proper choose materials and adjusting laser irradiation conditions.
The performed analysis reveals the new marginal quasi-nonlinear regime, in which the nonlinear intermode interactions ensure the fast phase synchronization of unstable modes with the growth rate Aq & gt- 0, so that their superposition yields, in the axially symmetric case, local surface structures (holes, bumps and nanojets). It is interesting to note that frequently observed extended lamellar-like structures are formed in the linear regime by the superposition of the same unstable HD modes but with random phases.
The considered in this work for the first time effect of synchronization of spatial phases of HD modes, leading to formation of solitary nanoholes or nanobumps (& quot-ultrashort spatial pulses& quot- of surface relief), can be referred to as & quot-active spatial phase locking& quot- by the analogy with & quot-active phase locking& quot- in laser physics, leading to generation of ultrashort (pico- and femtosecond) laser pulses [25].
Another important development is the new class of crown-like solutions of the HDKS describing the latest experimental results on laser-induced formation of surface nano-and microcrowns [24] and [20].
The modified HDKS equation (25), derived in this work, contains two new additive terms in comparison with the conventional KS equation [2,3]. The first one is the linear damping term ~r. The occurrence of this term in the HD equation for surface laser-induced melt is due to the Gaussian intensity distribution in the laser beam. Corresponding lateral temperature distribution and resulting slow radial variation of the surface tension in the laser spot on the surface give rise to slow convection of the melt from the center of the spot to its periphery which is modulated by the surface relief perturbation. This gives rise to the linear damping of the relief undulation. The linear damping term, translating the DDKS equation in the class of stabilized KS equations, plays the important role in occurrence of different regimes of HD structure selforganization.
The second new term in the HDKS equation (25) is the nonlinear term ~ (AH)2. It can be expected that taking into
account of this new nonlinear term in HD equation will lead, under proper conditions, to appearance of a new, physically important, class of its two-dimensional numerical solutions: solitary and periodic axially symmetric blow up solutions with singularities corresponding to the formation of solitary and periodic needle-like jets. As was noted before, this expectation is based on the obtained solution of the one-dimensional HDKS equation (see Fig. 8). Obtaining of corresponding two-dimensional axisymmetrical jet solution of Eq. (9) needs extra theoretical studies.
We note in conclusion, that the dimensionless HDKS equation in the form (25), describing HD instability of the surface
relief of the laser-melted layer of thickness hm, considered in this work, is similar to the dimensionless DDKS equation describing the surface relief instability of the laser-created defect-enriched layer of thickness hd [7, 26]. So, in both
cases the lateral spatial scale of the structures formed due to HD (liquid state) or DD (solid state) instabilities are proportional to the thickness of laser-modified surface layer (hm or hd, respectively). In the case of adatoms, the spatial
dimensions of surface DD structures is determined by the scaling parameter of the nonlocal elasticity theory with taking into account the influence of defects [27]. Similar also are symmetries of different surface structures obtained in computer simulations of HDKS equations, performed in this work, and of DDKS equations [7,8, 26].
The author is grateful to Dr. S.A. Kudryashov, professor O.B. Vitrik and S.V. Makarov for numerous discussions of their experimental results on laser-induced formation of surface nanojets, nano- and microcrowns and to D.I. Shikunov and A.S. Kuratov for simulating a number of solutions of HDKS equations.
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