Surface charging under pulsed laser ablation of

Invited Paper

Surface charging under pulsed laser ablation of solids and its consequences: Studies with a continuum approach a

N.M. Bulgakovaa, R. Stoianb,c, A. Rosenfeldb, I.V. Hertelb, E.E.B. Campbelld Institute of Thermophysics SB RAS, 1 Lavrentyev Ave., 630090 Novosibirsk, Russia b Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born Str. 2a, D-12489 Berlin, Germany c Laboratoire TSI (UMR 5516 CNRS), Université Jean Monnet, 10 rue Barrouin, 42000 Saint Etienne, France d Department of Physics, Göteborg University, SE-41296 Göteborg, Sweden ABSTRACT

Dynamics of electronic excitation, heating and charge-carrier transport in different materials (metals, semiconductors, and dielectrics) under femtosecond pulsed laser irradiation is studied based on a unified continuum model. A simplified drift-diffusion approach is used to model the energy flow into the sample in the first hundreds of femtoseconds of the interaction. The laser-induced charging of the targets is investigated at laser intensities slightly above the material removal threshold. It is demonstrated that, under near-infrared femtosecond irradiation regimes, charging of dielectric surfaces causes a sub-picosecond electrostatic rupture of the superficial layers, alternatively called Coulomb explosion (CE), while this effect is strongly inhibited for metals and semiconductors as a consequence of superior carrier transport properties. Various related aspects concerning the possibility of CE for different irradiation parameters (fluence, wavelength and pulse duration) as well as the limitations of the model are discussed. These include the temporal and spatial dynamics of charge-carrier generation in non-metallic targets and evolution of the optical (reflection and absorption) characteristics. A controversial topic concerning CE probability in laser irradiated semiconductor targets is also a subject of this work.

1. INTRODUCTION During the past decade, femtosecond laser ablation has been demonstrated to be a powerful technique for both fundamental studies and potential technologies.1 High-precision processing of any kind of material is possible due to the remarkable features of the ultrashort laser pulse interaction with matter such as minimizing the laser-plume interaction and reducing the heat-affected zones. Very promising applications of pulsed laser ablation can also be found in the generation of high-energetic ion and cluster beams by controlling the thermodynamic paths of the materials under laser exposure.2,3 However, for further development of the present pulsed laser technologies, a detailed understanding of the dynamics and mechanisms of particle desorption and ablation from the irradiated surfaces is of crucial importance. The mechanisms of laser ablation of different materials are still debated, especially for ultrashort, femtosecond laser pulses. Understanding the mechanisms requires extensive knowledge of the fundamental physics and chemistry of energy absorption and bond breaking in the irradiated targets. Due to rapid energy delivery, material removal is strongly localized with minimal residual damage and requires less energy compared to longer laser pulses.1,4 However, the laserinduced processes leading to material ablation are strongly dependent on the material properties and can be sufficiently different or proceed differently for materials of different classes.5 In metals, laser energy is absorbed by free electrons via the inverse Bremsstrahlung process in a skin layer of order of 10 nm. The electron-electron relaxation occurs rapidly with the result that the two-temperature model is valid on the 100-fs timescale.6-8 Electron-lattice relaxation requires longer times, in the range of 10 - 100 ps for different types of metals8 and this is much shorter than the characteristic time for establishing stationary distributions of the parameters in superheated liquids (teq ~10-9 s).9 Even for nanosecond laser pulses, heating can be extremely rapid at high laser fluences so that the melted material experiences superheating that results in phase explosion as the main mechanism of ablation.10,11 At shorter, picosecond and femtosecond laser pulse durations, a fast solid – vapor/plasma transition may take place2 with the possibility to drive the superheated material into a metastable state culminating with phase explosion.12,13 Laser ablation of dielectrics and semiconductors proceeds through a whole sequence of physical processes such as nonlinear absorption, non-equilibrium effects related to electronic and vibrational excitations and avalanche breakdown, giving rise to generation of dense, overcritical electron-hole plasmas.14 One of the most intriguing and complicated features of laser heating and ablation of dielectrics is the strong temporal and spatial variation of the optical properties,

Commercial and Biomedical Applications of Ultrafast Lasers V, edited by Joseph Neev, Christopher B. Schaffer, Andreas Ostendorf, Stefan Nolte, Proceedings of SPIE Vol. 5714 (SPIE, Bellingham, WA, 2005) · 0277-786X/05/$15 · doi: 10.1117/12.594047

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ranging from a fully transparent, non-reflecting substance to a metal-like state. It must be emphasized that, in dielectrics, metal-like behavior develops in a narrow surface layer. This results in strong localization of energy coupling since the electron thermal conductivity is restricted basically to the high-density region. This aspect accounts for one of the main distinctions between metals and dielectrics under pulsed laser irradiation, the possibility of Coulomb explosion (CE) as an ablation mechanism alternative or additional to phase explosion. Coulomb explosion is one of the electronic mechanisms of laser ablation which was widely discussed for different materials during the last decade.14-20 It has been perceived that CE can be exploited in different applications such as surface nanostructuring15,21 and nanoparticle formation.22 Addressing the subject of Coulomb explosion, we imply removal of at least several monolayers14,15 but not electrostatic desorption, which is of stochastic nature and results from localization of electronic excitation energy at specific atomic sites.23 The main features of CE are the following: The energetic ions of different species (e.g., aluminum and oxygen in the case of ultrafast laser ablation of sapphire15) observed in time-of-flight signals show the same momenta and not the same energy. Also, doubly-charged ions have velocities twice as high as singly-charged ones. This indicates that the ions could be extracted and accelerated in the electric field generated in the target. The electric field can be generated due to intensive electron photoemission, leading to accumulation of positive charge in a superficial target layer. Thus, the basic concept of Coulomb explosion is based on the fact that, due to photoemission, the irradiated surface gains a high positive charge, so that the repulsion force between ions exceeds the lattice binding strength, resulting in surface layer disintegration. Coulomb explosion has been proven, both experimentally15-18 and theoretically,14,17,19 to occur in dielectrics under femtosecond laser irradiation with 800-nm laser wavelength. It should be underlined that modeling of the charging dynamics14,17,19 was performed for laser fluences slightly above the experimental ion emission thresholds namely 4 J/cm2 for Al2O3, 0.8 J/cm2 for Si, and 1.2 J/cm2 for Au, respectively. It was demonstrated that CE was strongly inhibited for metals and semiconductors as a consequence of superior carrier transport properties. It is intriguing that the theoretical study and numerical simulation for the experimental conditions involving nanosecond laser pulses have indicated the possibility of CE for silicon,20 the fact seeming to be in contradiction to previous studies.14,17,19 However, the contradiction has been resolved24 by the fact that both the theoretical and experimental studies20 have been performed for ultraviolet excimer laser irradiation (also for laser fluences slightly above the experimental ion emission threshold) that implies changes in electron excitation dynamics and charge-carrier transport. There are some indications of the possibility of CE at higher laser fluences that those studied in [14,17,19]. Generation of extremely high electric fields on the order of 1010 V/m was achieved recently even in thin metal targets under ultraintense laser irradiation (~1019 W/cm2).25,26 Theoretical considerations27 are also in favor of CE in metals under such extreme conditions. Irradiation of silicon with fluences ≥ 3 J/cm2 (wavelength of 800 nm, pulse duration of 160 fs) results in bimodal structures of the time-of-flight ion signals28 that has been attributed to CE. However, because of the necessity to separate the ions originated from CE from the thermal ones, including those accelerated by the plume double layer,29 this question requires a subtle analysis and necessitates further study. In this paper we present a unified model to describe the processes that can influence the development and decay of charging accumulated on the surfaces of laser-irradiated materials of different classes. As we are interested in macroscopic CE, a macroscopic drift-diffusion approach is used as the basic concept for modeling. The modeling results have been obtained for gold, silicon, sapphire as typical representatives of metals, semiconductors and dielectrics respectively. Whenever possible, the model parameters follow mainly the general characteristics of different material classes rather than underlining particular properties of the selected materials. The irradiation conditions considered in the modeling correspond to the experimental studies:14,17 ultrashort laser pulse of 100 fs duration at 800 nm wavelength with laser fluences slightly above the experimental ion emission thresholds. It is shown that, under such conditions, laserinduced charging of dielectric surfaces should cause the electrostatic rupture of the superficial layers whereas in metals and semiconductors CE is strongly inhibited. Separately we analyze a specific case of silicon irradiated with higher laser fluences, above the ultrafast (UF) melting threshold. The paper is organized as follows. In Section 2 we give a short overview of the main modeling approaches developed to understand the dynamics of free carrier generation in the dielectric and semiconductor targets and describe the details of our model, emphasizing special features for metals, dielectrics and semiconductors. In Section 3 we introduce a criterion of Coulomb explosion and present the results of modeling. Finally, in Section 4 we discuss the results of modeling of silicon breakdown with analysis of ultrafast melting and its possible interplay with CE. We consider also the question of the impact of charge separation on homogeneous melting of semiconductors.

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2. FORMULATION OF THE MODEL 2.1 Theoretical considerations of carrier dynamics under pulsed laser ablation The importance of carrier dynamics under pulsed laser ablation (PLA) conditions, especially for ultrashort laser pulses, has been demonstrated in numerous studies.30-32 Electronic transport has been shown to differ strongly in various classes of materials irradiated under similar conditions at moderate energy densities.14,17 It is clear that mobility of the charge carriers, electrons and holes, is the main parameter influencing the rate of charge redistribution in self-consistent electric fields generated under the action of pulsed laser radiation. Mobility determines both drift and diffusion of charge carriers, including thermal diffusion in the presence of temperature gradients. Even in the absence of photoemission i.e. preserving the target neutrality, the internal electric field produced by the diffusion of charge carriers due to strong gradients of density and temperature can lead to CE.33 The theoretical description of carrier dynamics in dielectric and semiconductor targets under short pulsed laser irradiation or electron beam bombardment is a complicated task which implies consideration of a broad spectrum of interrelated processes. A simplified scheme of the processes triggered by ultrafast laser irradiation in dielectrics and semiconductors is shown in Figure 1. A laser pulse impinging on a target causes photoionization (with the order of the process depending on the relation between the photon energy and the material bandgap) and photo-emission (with the order of nonlinearity determined by the ratio of the work function to the photon energy). The electrons excited to the conduction band absorb laser radiation and can produce secondary electrons by collisional ionization. In addition, they change the dielectric response of the material, thus influencing explicitly the optical properties. As soon as electronic avalanche is developed, absorption and reflection dynamics vary dramatically. Photo-emission depletes a superficial target layer breaking further development of collisional multiplication (which depends on the presence of initial electrons to initiate the process) and, in turn, avalanche regulates photo-emission through its influence on the available freeelectron density. Another drastic consequence of electron photo-emission is the violation of the target quasi-neutrality that results in positive charging of the target as a whole and thus in generation of an ambipolar electric field. The electric field forces the charge carriers to relocate in order to neutralize the excess positive charge. The electric current is a superposition of two components: charge-carrier drift in the electric field and diffusion under the action of the gradients, both in density and temperature. Redistribution of free electrons due to the electric current strongly influences both electron photo-emission and the avalanche multiplication process. Existing models treat most, or at least some of the processes described above and may, by convention, be divided into three groups. One of the dominating approaches to describe carrier dynamics in silicon targets is based on ambipolar diffusion with an implicit assumption of an equal number of electrons and holes in the solid and the preservation of local quasi-neutrality of the sample.34-36 Another group of models, developed for semiconductors irradiated by laser pulses37 and dielectrics under the action of electron beams,38,39 takes into account the generation of local electric fields inside the target with the assumption that the target remains neutral as a whole. This implies the absence of electron photoemission37 or relies on secondary electron emission equal to the absorbed electron flux.38,39 In this approach the concept of a double layer can be applied to describe processes involving particle beam–matter interactions including the spatial charge arrangement in the bulk of the irradiated materials.38 It should be underlined that the irradiation regimes considered in Refs. [37-39] are far below material breakdown. Fig. 1. Modeling scheme. Major interconnections between different processes are shown by the arrows. Direct effects are indicated by the arrows with hard lines and backward influences are shown with broken lines.



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A third approach40,41 proposed for the case of a dielectric target (MgO) irradiated by a laser pulse of nanosecond duration may be labeled as the drift-diffusion approach. The authors studied the self-consistent generation of an electric field as a result of laser heating and thermionic emission of the electrons excited to the conduction band and their diffusion and drift in the locally established fields. It was found that the self-consistent electric field could reach values exceeding 108 V/m under normal ablation conditions. In several works14,17,19,24 we used the same simplified approach to model charging on different types of materials under ultrashort pulsed laser irradiation, taking into account specific properties of the materials. This implies calculations of energy deposition into the electronic subsystem and subsequent heating of the lattice through electron-lattice energy exchange. For the latter task we used a two-temperature model which is mostly utilized to describe metals heated by ultrashort laser pulses.6-8 Absorption of the laser energy is considered as a volume process governed by the Beer–Lambert law for metals whereas the energy source terms for the other types of materials are accurately calculated from the kinetics of electron generation, recombination and photoemission. 2.2 A unified approach Under high power ultrafast laser irradiation, dense plasmas are generated in dielectrics and semiconductors,42 thus justifying a unified approach based on a intrinsic or a laser-induced metallic behavior for the three classes of materials investigated. The following equations are used as the building blocks for a common simplified frame applicable to different kinds of materials: (1) The continuity equations for the evolution of the laser-generated charge carriers:

∂n x 1 ∂J x + = S x + Lx , ∂t e ∂x

(1)

where Sx and Lx are the source and loss terms describing the free carrier populations, nx denotes the carrier densities with subscript x = e, i representing electrons and ions, respectively; (2) The expression for the electric current density Jx includes drift and diffusion terms41 and can be considered as the equation of motion: (2) J x = e n x µ x E − eD x ∇n x . Here the time and space dependent diffusion coefficient Dx is calculated according to the Einstein relation as Dx = kTxµx/e, where Tx represents the carrier temperature, k is the Boltzmann constant, and µx is the carrier mobility. We assume that the charge-carrier flows are caused by quasi-neutrality violation on and below the target surface due to electron photoemission and strong density and temperature gradients. (3) Poisson equation to calculate the electric field generated as a result of breaking quasi-neutrality in the irradiated target: ∂E e (3) = (ni − ne ) ∂x εε 0 (4) Energy conservation equations to account for the heating of electronic and lattice subsystems. We assume that laserexcited metals and strongly ionized insulators and semiconductors can be considered as dense plasmas so that the twotemperature model6-8 may be applied for an energy balance description:  ∂T ∂T J ∂Te  ∂ (4) = C e  e + K e e − g (Te − Tl ) + Σ( x, t )  ∂x ene ∂x  ∂x  ∂t

∂Tl ∂ ∂T (5) = K l l + g (Te − Tl ) . ∂t ∂x ∂x Even if complete equilibration does not take place in the subsystems, the values Te and Ti can be considered as measures for the average energies of electrons and lattice. In the Equations (4) and (5), indexes e, l refer to the electron and lattice parameters; Ce, Cl, Ke, Kl are the heat capacities and the thermal conductivities of electrons and lattice respectively; g is the electron-lattice coupling constant; Σ(x,t) is the energy source term. All these parameters will be specified below for every type of material. The term Lmδ(T - Tm) allows calculations of the liquid–solid interface,43 having the temperature Tm. Lm is the latent heat of fusion. The δ-function was approximated as43 2  (T − Tm )  1 (6) δ (T − Tm , ∆) = exp −  2 2∆ 2π ∆   (Cl + Lmδ (Tl − Tm ))

with ∆ in the range of 10 - 100 K depending on the temperature gradient (the defined domain of the δ-function should be bigger than 3 computational cells). The sign of the δ-function depends on the direction of motion of the solid - liquid

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interface (i.e., whether melting or solidification takes place in the target). The electron energy equation accounts for both heat conductivity and direct, bulk or across the vacuum interface electronic transport (e.g. due to photoemission). The system of equations (1)–(5) was solved for targets irradiated with a laser pulse of a Gaussian temporal profile: I (t ) = (1 − R (t ))

2 F0

ln 2

τL

π





 

τ L

exp  − 4 ln 2

t

   

2

 .  

(7)

Here F0 is the incident laser fluence, the laser pulse duration is τL = 100 fs (FWHM), and R is the time dependent reflection coefficient. The calculations have been performed for laser radiation wavelength λ = 800 nm.15,17 A onedimensional model justified by enhanced transversal lateral dimensions for the laser spot with respect to the absorption depth has been employed. Every set of equations has its own features dictated by the material properties. Particular situations for each class of the materials will be discussed below. The modeling has been performed using an explicit numerical scheme for the one-dimensional case. The targets (gold, silicon, and sapphire as model systems for different material classes) are divided into regular layers (numerical cells). The electric current density was calculated at the cell boundaries, whereas the other parameters were calculated in the center of cells. The depth of the numerical region was chosen in such a way that further increase did not influence the numerical results, preserving it below the limit where a one-dimensional approach was not valid anymore. In the remote boundary (bulk) with respect to the target surface, the condition of free flow for electrons was set. A cell was added above the target surface to simulate the vacuum conditions, and the electric field at the target surface is calculated using the Gauss law. 2.3 Metals The continuity equation for free electrons (Eq. (1)) in a metal target (gold in our case) has no source terms. Photoemission was considered in the form of a surface boundary condition for the electron current density,44,45 describing electron flow into the vacuum. The three-photon photoemission cross-section c, containing information about the electron escape probability, escape depth, and energy gain, was determined empirically44 for nanosecond laser pulses by measuring the total electronic charge emitted during irradiation and corrected here for the absorption changes corresponding to the present irradiation wavelength. In contrast to nanosecond laser pulses where electron and lattice temperatures can be considered almost equal, ultrafast, sub-ps irradiation of metals can induce extremely high electronic temperatures, ~1 eV, while the lattice remains cold during the irradiation time. Assuming temperature dependent effects, we corrected the photoemission cross-section, based on the generalized Fowler-DuBridge theory for multiphoton photoemission at high temperatures.14 The thermal emission of high-temperature electrons can play a significant role in the tail of the laser pulse where the energy stored into the electronic system reaches a maximum value. Thus, the photoemission term is written in the form:  3hω − ϕ   eϕ  2(kTe ) 3 3 2 (1 − R ) I + A0Te exp  −  F  2   kT  (3hω − ϕ )  kTe  e   2

Js = c

(8)

Here A is the theoretical Richardson coefficient (120 A/cm2K2), ϕ is work function (4.25 eV 46), ħω is the incident photon energy (1.55 eV) and F represents the Fowler function.45 The parameters for the energy equations (4)-(5) for gold are as follows.8 The heat capacities are Ce = AeTe with Ae = 71 J/(m3⋅K2) and Cl = 2.5⋅106 J/(m3⋅K); the thermal conductivity of the electrons and lattice are Ke = Ke,0Te/Tl with Ke,0 = 318 W/(m⋅K)8 and Kl = 317 W/(m⋅K).47 The latter is of minor importance for Te >> Tl, a common situation under ultrashort laser irradiation, until electron-lattice relaxation occurs (10-100 ps depending on the metal). The electron-lattice relaxation is determined by the coupling constant g [2.1⋅1016 W/(m3⋅K) for gold].8 The energy source is written taking into account ballistic effects for electronic energy transport (λball =105 nm) as8 exp(− x /(λ0 + λball )) (9) Σ( x, t ) = I 0 (t )(1 − R ) (λ0 + λball ) where the reflection coefficient R = 0.949 46 and λ0 = 1.44⋅10-8 m 48 (optical properties are taken for 800 nm wavelength). Electron mobility was estimated from data on gold conductivity47 and found to be 5.17⋅10-3 m2/(V⋅s). 2.4 Dielectrics In dielectric materials, free electrons are generated in the processes of multiphoton (MPI) and collisional ionization and their recombination proceeds usually through a trapping-like phenomenon. For sapphire as an example, the source and loss terms in Eq. (1) for electrons can be written as:



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na , (10) Le = − Re − PE . na + ni Here Wmph=σ6I6 is the rate of a 6-photon ionization process corresponding to an energy forbidden band Eg of approximately 9 eV, na is the density of neutral atoms, Qav=αIne is the avalanche term49, Re represents the decay term taken in the form ne/τ (in the relaxation time approximation) with τ = 1 ps, and PE denotes the photoelectron emission process. The source terms are corrected for the reduction in the density of ionization centers (neutral atoms providing the electronic reservoir of the valence band). To compensate for the mobility decay with temperature, we use a simplified, time-independent diffusion coefficient, assuming that the electron average energy in the conduction band is ~5 eV.49,50 The MPI cross-section σ6 = 8⋅109 cm-3ps-1(cm2/TW)6 and avalanche coefficient α = 6 cm2/J were estimated by fitting the experimental results for the optical damage threshold at different pulse durations51 following a similar approach to Ref. [49], taking into account the observed decay in the threshold electron density at longer pulse durations.52 Attenuation of the laser power inside the dielectric target is determined by loss mechanisms involving free electron generation and by the optical response of a collisional free-electron plasma and the vacuum-plasma interface through the Fresnel formulas. The complex dielectric function can be seen as a mutual contribution of the unexcited solid and the response of the laser–induced free electron gas and reads as:53,54  n  n 1 (11) ε * (ne ) ≅ 1 + (ε g − 1)1 − e  − e n n 1 + i / ωτ  0  cr S e = (Wmph + Qav )

Here εg is the dielectric constant of the unexcited material (εg = n2 = 3.0983),54 ncr = ε0meω2/e2, and n0 are the critical electron density in vacuum (1.74⋅1021 cm-3 at 800 nm wavelength), and the valence band electron density respectively. The effective electronic mass was taken to be equal to the mass of the free electron in vacuum. To take into account reflection, a multilayered, step-like, inhomogeneous electron density profile53,55 was considered. The local intensity I(x,t) is given by the superposition of the direct irradiation and back-scattered radiation: na ∂ (12) I ( x, t ) = −Wmph hωn ph − a ab ( x, t ) I ( x, t ) ∂x ( n a + ni ) Here nph is the number of photons required for an MPI event (nph = 6). The free electron absorption coefficient aab(x,t) is calculated from the complex dielectric response49 [Eq. (11)] considering a damping term ωτ = 3 to match the observed reflectivities56 of ~70% for supercritical electron densities generated by high intensity radiation. For a quantitative estimation of the photoemission term, we assume a statistical distribution of free electronic momenta in a wide-band gap dielectric where the vacuum level lies close to the conduction band minimum, and only electrons with a momentum component normal to and in the direction of the surface can escape into the vacuum. Thus, we assume that, on average, half of the electrons which appear in the processes of MPI and avalanche are immediately emitted from the surface and below surface region, taking into account an instantaneously established angular distribution for the carrier momenta. Maximum photoemission occurs from the surface with an exponential decreasing to the bulk depth:14 PE ( x, t ) =

1 (Wmph + Qav ) na exp(− x / lPE ) 2 na + ni

(13)

where the electronic escape depth lPE was taken as 1 nm.57 The integral photoemitted charge calculated as above is in good agreement with reported experimental values for dielectric materials.58 The ion density is calculated based on Eq. (1) disregarding photoemission and neglecting hole transport in the bulk. The electron mobility is taken as µe = 3⋅10-5 m2/(V⋅s) (ten times lower than reported in Ref. [59]) for a better match of the observed diffusivities39 in laser induced dense plasmas. Since we consider electronic transport in the local electric field, calculated from the Poisson equation, plasma-screening effects within the bulk material are automatically taken into account. The source term in the energy equation (4) was constructed to account for the energy of the electrons generated via photoionization, energy expenses used for the development of avalanche multiplication, free electron absorption, and the energy localized in the strained lattice via trapping processes. Thus, the change in the energy of the electronic subsystem Ef due to the above-mentioned processes is ∂E f na na n (14) = (nph hω − E g )Wmph − E g Qav + α ab I (x, t ) − Ee e − Ee PE ( x, t ) ∂t na + ni na + ni τt

14



Here Ee is the average energy per electron (Ee = 3kTe/2) and τt is characteristic trapping time (τt = 1 ps). Following the source term for the temperature which is a characteristic of the average energy of the particles, it is easy to show that Σ( x, t ) = ne

∂n ∂Ee ∂E f 3 = − kTe e . 2 ∂t ∂t ∂t

(15)

The thermodynamic parameters for the electronic subsystem were taken as:35 Ce = 3kne/2, Ke = 2k2µeneTe/e. For ultrashort times of order 1 ps, lattice heat conduction can be neglected and lattice heat capacity is taken as Cl = 3kn0. The value g can be defined via the characteristic electron-lattice relaxation time τr as g = Ce/τr (τr ~ 1 ps). 2.5 Semiconductors The model for silicon is based on a similar approach as for sapphire. The reflection of radiation at the vacuum interface and light absorption inside the bulk of a strongly excited semiconductor are described based on the mutual contribution of the unexcited material and the Drude response of the laser generated free electron gas53 (εg = 13.46 + i0.048)60. In Eq. (1) to account for the evolution of the electron density, one and two-photon ionization terms and avalanche were considered:34, 53   na 1  I , (16) S e =  σ 1 + σ 2 I  + δ ne  2 h ω     na + ni (17) Le = − Re − PE where σ1 and σ2 are one- and two-photon ionization cross sections (σ1 = 1021 cm-1,61 σ2 = 10 cm/GW 62), the total atomic density is n0 = (na + ni) = 5⋅1022 cm-3, and δ is the avalanche coefficient.35 The loss term at low electronic densities is mainly determined by Auger recombination which saturates at electronic densities approaching 1021 cm-3.63 Consequently, the decay rate can be written as: ne (18) Re = τ 0 + 1 / Cne ni with τ0 = 6⋅10-12 s and C = 3.8⋅10-31 cm6/s.35 The equation for hole generation takes into account the hole transport process. The current densities for electrons and holes are based on the following parameters:64,65 µe = 0.015 m2/(V⋅s) and µh = 0.0045 m2/(V⋅s), ten times reduced compared to the low electron-hole density values.64 Diffusion coefficients for electrons and holes are calculated according to the Einstein relation as defined above. Electron photoemission66 was considered in analogy with a gold sample [Eq. (8)] taking into account a three-photon photoemission from the conduction band with a coefficient corrected for the wavelength and the accessible density of states.14 Instead of the Si work function (ϕ = 4.6 eV), an effective potential barrier ϕeff = 4.05 eV measured from the bottom of the conduction band67 was introduced as the relevant parameter for the Fowler function. The spatial and temporal distribution of the laser intensity in the sample was calculated as   na na ∂ (19) I ( x, t ) = −σ 1 +σ 2 I ( x, t ) + aab ( x, t )  I ( x, t ) ∂x n + n n + n a i a i   taking into account one-, two-photon ionization, and free-electron absorption of the light. The source term for the heat transport equation (4) was calculated using Eq. (15) considering the balance of the laser energy deposited in the free electronic system: ∂E f   na σ I σ I2 (20) =  (hω − E g ) 1 + (2hω − E g ) 2 − E g δne  + α ab ( x, t ) I ( x, t ) + E g Re − Ee PE ( x, t ). ∂t ω 2 ω h h   na + ni Note that, in analogy with a gold sample, the PE term has a non-zero value only at x = 0 (on the target surface). The thermodynamic parameters of the electronic subsystem were calculated according to Ref. [35]. The electron-lattice relaxation time was taken as τ = τ0(1 + (ne/ncr)2) with τ0 = 240 fs.62 Absorption and reflection coefficients were calculated by a Drude-type scheme for a collisionally damped electron-hole plasma induced by an ultrashort laser pulse (100 fs) at normal incidence [R(t = -∞) = 0.34]54 in a similar manner to the approach used for dielectric materials. 2.6 Criterion for Coulomb explosion The system of equations (1)-(5) together with special features described in Parts 2.3-2.5 allows to perform modeling of surface charging and electric field generation due to photoemission and charge carrier transport in the different materials. Under specific conditions, the electric field generated in the irradiated target can reach extremely high values so that the atomic bonds are broken and a surface layer of the material is disintegrated, thus resulting in the occurrence of Coulomb explosion. The estimation of the threshold electric field with respect to CE can be made under the following



15

assumptions. The cubic energy density of the electric field is expressed as w = εε0E2/2. The value Wat = εε0E2/(2n0) is the energy of the electric field corresponding to one atom in the crystal. The energy necessary to remove an atom from the target can be roughly estimated from the latent heat of sublimation calculated for a single atom. For sapphire this is Λsub = 485.7 kJ/mol,68 which corresponds to approximately 5 eV. Thus, the threshold electric field can be approximated as: 2Λ at n0 . (21) Eth | x = 0 = εε 0 For sapphire we obtain Eth ~ 5⋅1010 V/m whereas for gold and silicon the estimated threshold electric fields are smaller, 2.76⋅1010 and 2.65⋅1010 V/m, respectively. This assumption is valid for the case of a cold lattice, roughly during femtosecond laser exposure. For longer pulses or longer times, one should take into account that the heated surface atoms having a definite energy of vibrations can escape from the surface with higher probability than the cold ones, a fact which should decrease the threshold value. Thus, we can write approximately: εε 0 Eth2 2( Λ at − 3kTl ) n0 . or (22) + 3kTl = Λ at Eth x=0 = 2 n0 εε 0 Thus, lattice heating can distinctly reduce the threshold field.

3. RESULTS AND DISCUSSION To test the model, the damage thresholds for Au, Si and Al2O3 were firstly calculated and compared with previously published data. The test for Au was performed for a 1 µm thick film.8 Melting is reached at an incident fluence of 0.93 J/cm2 (equivalent to 0.047 J/cm2 absorbed fluence) that is in good agreement with experimental observations.8,17 The calculated damage thresholds for Si and sapphire at 800 nm wavelength, disregarding for the moment eventual thermal consequences, take place when an electronic density in excess of the critical value (1.74⋅1021 cm-3) is reached, at 0.3 and 2.7 J/cm2 respectively, also in good agreement with the experimental studies.31,51 Considerable electron depletion of the surface layer of the sapphire target leads only to a slight shift of the breakdown region toward the bulk (on the order of the electronic escape depth). According to the calculations for 800 nm and 100 fs laser pulse duration, the Si sample reaches the melting temperature also at around 0.3 J/cm2. For complete melting and further heating above the melting temperature, the material needs more energy to compensate the heat of fusion. This, according to the modeling results, takes place in the fluence range of 0.3 – 0.5 J/cm2 when homogeneous melting proceeds in a superficial layer of the target. In this fluence range, both absorption and reflection coefficients have to change from the transient values for solid excited silicon to that of molten silicon [ α abliquid ~108 m-1 69, Rliquid = 0.8]. Further heating of molten silicon is observed in calculations only starting from ~0.5 J/cm2, a value higher than those experimentally reported,70 since the hole contribution to phonon generation is not explicitly accounted for.71 In addition, melting may also be initiated inhomogeneously at the surface or at intrinsic defects, that is not taken into account in our ideal model. 3.1 Sapphire, gold, silicon: laser intensities slightly above the material removal threshold In Figure 2 the temporal profiles of the net positive charge resident on the surface for samples representative of different classes of materials (Al2O3, Si, and Au) are plotted as a function of time for laser pulses of 100 fs duration at 800 nm

1 µm

-3

ni-ne [10 cm ]

40 30

Al2O3

Fig. 2. Temporal behavior of the net surface charge density (the difference between the hole and electron populations) for different classes of materials. Laser fluences are chosen to be above the ion emission threshold for each material (4 J/cm2, 0.8 J/cm2, and 1.2 J/cm2 for Al2O3, Si, and Au, respectively). The laser pulse of 100 fs duration is centered at t = 0.

Silicon x 20

20

10 µm

20 Gold x 20 10 0 0,0

0,1

0,2

0,3

Time [ps] 16



wavelength. The laser fluences are chosen to be slightly above the experimental ion emission thresholds,8,31,51-53 namely 4, 0.8, and 1.2 J/cm2 for sapphire, silicon and gold respectively. Under these specific irradiation conditions the electronic temperature reaches considerable values ranging from around 1 eV in gold to approximately 6 eV in silicon and more than 10 eV in sapphire. It is obvious that the net charge is significantly higher for the dielectric target than for the metal or for the semiconductor target. For sapphire the results are given for two numerical regions of 1 and 10 µm to illustrate the importance of the position of the remote boundary which allows electron supply. The position of the remote boundary is important also for silicon while for metals the depth of the numerical region can be as small as 2-3 skin-layer widths. The size of 10 µm corresponds to the irradiated spot radius in the experiments15,17 and indicates the upper limit of the one-dimensionality assumption. Increasing the numerical region from 1 to 10 µm results in an insignificant reduction of the maximum charging (about 10%) with an approximately double increase of the charging period. The results presented in further Figures for sapphire have been obtained with the numerical region of 1 µm thickness. A comment should be made about the numerical stability and approximation of the original equations. Reaching stability does not necessarily imply a good approximation of the physical processes. The conservation of the number of electrons (and holes in the case of silicon) should be controlled thoroughly by balancing the generated, recombined, and photoemitted electrons, taking into account electron inflow across the remote boundary. Charge-carrier conservation may be reached by varying the time step in the calculations. It should be particularly emphasized that strong charging of sapphire (Fig. 2) is not a result of higher photoemission as compared to silicon and gold. According to modeling, the photoemission yield is approximately 6.8⋅108 electrons for the sapphire target over the whole laser pulse duration from an irradiated spot of 470 µm2, whereas 6.2⋅1011 and 3.5⋅1011 electrons are removed for Si and Au targets, respectively. At the same time, the largest electric field is generated in the sapphire target. Figure 3 shows the temporal behavior of the electric field values developed at the sapphire surface in comparison with the field values induced in other types of materials. The negative value implies that the field is directed away from the target, streaming from the sub-surface layers to the vacuum. It can be observed that the electric field exceeds the critical value and reaches a value of 8.4⋅1010 V/m beneath the surface. The above-threshold electric field exists for a few tens of fs. The spatial distribution of the electric field in the near-surface layers is given in Fig. 4 for time moments when the electrostatic field has reached its top value. For sapphire, the layer with overcritical electric field where electrostatic disintegration of the lattice should occur is approximately 40 Å wide, in excellent agreement with the experimental estimation of the Coulomb exploded region.15

0

10

ELECTRIC FIELD [10 V/m]

As discussed, the electric field in the first cell below the surface reaches the value of ~8.4×1010 V/m. An external field Eex=εEin is established in front of the surface. The accumulated electrostatic stress determines the surface disruption and the emitted Al+ ions will be driven by the field for about a few tens of fs (characteristic time of the electric field “pulse” in Fig. 3) and subsequently accelerated. The final momentum obtained by the ion subject to the action of the electric field Eex during time τ is written as mAl⋅v = eEexτ where mAl is the ion mass. This gives an estimate of the maximum velocity acquired by the ion of v ≥ 104 m/s that closely agrees with the value detected in the time-of-flight experiments.15 During the time to reach considerable surface charging, the ions travel a distance of the order of a few tens of Å. Thus, the charged surface layer is destroyed within an interval of several tens of femtoseconds.

-2 -4

Au x 100

Fig. 3. Temporal profiles of the laserinduced electric field in the surface region of the targets. Irradiation conditions as in Fig. 2.

Si x 100

-6 Al2O3

-8 CRITICAL ELECTRIC FIELD

-10 -0,1

0,0

0,1

0,2

TIME [ps]



17

2

SURFACE Au x 100

0 -3

ni-ne [10 cm ]

Si x 100

20

10

ELECTRIC FIELD [10 V/m]

With semiconductors and metals, the higher electron mobility and higher density of available free electrons ensure effective screening and a much smaller net positive charge accumulated during the laser pulse, in spite of the fact that for the Si sample supercritical carrier densities are reached. This is not sufficient to induce a macroscopic electrostatic breakup of the outer layers of the substrate. The maximum values of the electric field are only 4.1×107 V/m and 3.4×108 V/m for gold and silicon respectively (Figs. 3 and 4). The charge dynamics are strongly correlated with the absorption characteristics for each material (see Fig.2). In the case of sapphire, the charge maximum is strongly retarded, instead of roughly following the laser pulse envelope, as for metals and semiconductors. The effect is explained by the fact that electron heating and collisional multiplication take place during the tail of the laser pulse, thus being directly linked to the number of free electrons in the surface layers. As follows from the calculations, a sapphire sample irradiated by a laser pulse with a fluence of 4 J/cm2, accumulates multiphoton-generated seed electrons during the first half of the pulse. Only when the electron density reaches a value of order of 1017–1018 cm-3, does avalanche start to dominate over MPI that leads to the subsequent dielectric breakdown. At this point, very efficient electron heating and photoemission occurs resulting in supercritical surface charging. An additional factor for efficient electron heating at the end of the laser pulse is the drastic reduction of the available number of neutral atoms that results in decreasing energy losses for the electrons [see Eq. (14)]. Spatial distributions of the accumulated charge in the near-surface regions for the sapphire target are presented in Fig. 5 for different times. At t = 0 fs, corresponding to the peak of the laser pulse, breakdown conditions have not yet been reached so the laser-induced surface charging is weak and only a 50–60 Å thick surface layer is positively charged. The critical electron density in the surface layer is reached about 25 fs after the pulse maximum. The ionization process develops further in a skin layer where 20% ionization is reached approximately 20 fs later and the magnitude of the surface charge levels out (see Fig. 5, the curve corresponding to a delay of 60 fs). Considerable positive charging (from 3% to 12% of atomic number density in the lattice) arises in a superficial layer which is only 60–80 Å thick. Approximately 100 fs after the laser pulse, the process reaches a quasi-equilibrium between the opponent drift and diffusion terms, with a slight variation in time due to possible electronic decay channels (recombination at traps, selftrapping, Auger recombination) and electron supply from the bulk. The solid line for 500 fs in Fig. 5 corresponds to such a quasi-stationary picture. Since the direct photoemission process has stopped, the electron current from the bulk is gradually closing the charged gap near the surface. Despite the negative surface layer, a certain quantity of net positive charge still exists in the sub-surface region. Due to the attraction generated by the narrow positive sub-surface layer, a region with electron excess arises in the less-ionized interface region, so the picture of a classic double layer develops similar to that generated at the boundary of expending plasmas.29

-5 CRITICAL ELECTRIC FIELD Al2O3

-10 0

2

4

6

8

10

10 1 10 0 10 -1 10 -2 10 -3 10 -4 10 -5 10 1 2 3

0 fs 60 fs 500 fs 0

10

DISTANCE [nm] Fig. 4. Spatial bulk profiles of the electric field induced by ultrafast laser radiation in metals, semiconductors, and dielectrics at time moments of reaching the maximum values of the electric field for every material.

18

1

10

2

1x10

DEPTH [nm] Fig. 5. Spatial depth profiles of the induced charge density in sapphire at different time moments.



3

1x10

3.2 Silicon at laser intensities above UF melting threshold We have demonstrated above that, for ultrashort laser pulses with 800 nm wavelength at intensities slightly above the material removal threshold, CE is strongly inhibited for silicon as a consequence of superior carrier transport properties. However, in a number of studies performed for silicon irradiated by IR femtosecond laser pulses with intensities below the melting threshold,72 electronically-induced ion emission has been found to occur with the specific features inherent to CE (the same momenta of ionized clusters with different mass) that can be explained by desorption resulting from localization of electronic excitation energy at specific atomic sites.23 More intriguing are indications of CE possibility at high laser fluences, well above the ablation and plasma formation thresholds.28 The authors28 have shown clearly that doubly-charged ions have velocities twice as high as singly-charged ones that can yet be caused by a number of reasons73 such as double layer effects and charge acceleration in the plasma plume.29 In addition, under regimes where ablation is fully developed, the ions originated from CE are hardly distinguishable from ones belonging to the high-energetic tail of the thermal distribution. However, here we shall try to discuss some consequences of target charging under IR femtosecond irradiation of silicon with high-power laser pulses, where ultrafast phase transitions may occur within the laser pulse envelope. In particular, we raise the question about the influence of the band gap collapse (induced by a high level of electronic excitation) on laser-induced homogeneous nucleation of the liquid phase in a bulk semiconductor through a collective collapse of the lattice. We will discuss as well the potential consequences on the ablation process. It is generally believed that ultrafast excitation of semiconductors at fluences above the damage threshold proceeds through a highly reflective, metal-like phase upon the development of a dense, overcritical electron-hole plasma53 with the optical response of a free electron gas. Strong electron damping ensures efficient coupling of the incoming laser radiation. If a significant percent (~10-15%) of the valence electrons is pumped in the conduction band, the strong electronic excitation is almost immediately followed by premature bond softening and a sharp reduction of the average bonding-antibonding splitting, lattice destabilization, and the appearance of non-thermal phase transitions on a sub-ps time scale.74-77 For lower energy inputs, screened nonradiative recombination, delayed carrier-lattice thermalization, together with thermal transport restricted by the sound velocity will force the time for thermal melting on a few picoseconds timescale. A fast phase transition, implying a sharp change in the free electron population and carrier transport properties in a confined semiconductor surface layer or, in case of homogeneous nucleation, in islets78 of liquid phase, results in extremely steep gradients of the free electron density. This is an insurmountable obstacle for the direct application of the unified continuum approach proposed here. However, we undertook a simplified qualitative modeling, disregarding electron photoemission and electron and hole transport in Eq. (1). The goal is to obtain information concerning the possibility of CE whenever ultrafast (UF) melting of silicon irradiated by intense laser pulses occurs within the intensity envelope. The assumptions are partially justified by the following reasons: the laser-induced disordered phase has a metal-like nature, with high electronic densities, while the carrier mobility (~1cm2/Vs) lies well below the usual carrier mobility in ductile metals. The parameters of irradiation were the following: 800 nm laser pulse of 100-fs duration, with the fluence ranging from 0.6 to 5 J/cm2. The parameters of interest were the thickness of the UF melted layer and the average electron energy when the band gap collapse occurs. UF melting was assumed to be triggered in a numerical cell (5 Ǻ thick) when approximately 10% of the valence electrons (2·1022 cm-3) have been excited to antibonding states in the conduction band. During the next 400 fs 77 (fixed for every numerical cell) required for the completion of the ultrafast order-disorder phase transition, the silicon behavior follows the excited solid dynamics [Eqs. (16)-(20)] and then the properties of liquid silicon are switched instantaneously (with all the initial valence electrons belonging now to the conduction band). The results of these simplified calculations are presented in Fig. 6. According to our modeling, ultrafast melting starts at ~0.65 J/cm2, 1.3 times higher than the thermal melting threshold (according to our calculations, partial melting takes place in the fluence range of 0.3 – 0.5 J/cm2 and further heating of molten silicon is found to occur only starting from ~0.5 J/cm2) that is in reasonable agreement with the experimental observations.77 The depth of the non-thermally melted layer is 20–40 nm in the fluence range of 1–5 J/cm2 (Fig. 6,a) that is also consistent qualitatively with observations involving time-resolved X-ray diffraction on the melting dynamics.77 It should be emphasized that thermal melting which inevitably takes place after UF disordering following the dissipation of the energy confined in the electronic system occurs much faster in the high fluence regime than at low fluences, below the UF melting threshold, and a ten times thicker molten layer develops below the target surface. The temperature of the electrons immediately after band gap collapse (that here is a measure of the average electron energy) increases from ~0.5 to more than 2 eV in the fluence range of 0.6–5 J/cm2 (Fig. 6,b). Under these conditions, the Debye length is of the order of one monolayer (~2 Ǻ). However, in view of intensive photoemission which is a multiphoton process and thus becomes more efficient at higher



19

Melt depth [nm] Temperature [K]

40

(a)

30 20

Fig. 6. Numerical results on the depth of

10

the non-thermally melted layer (a) and the temperature of the electrons immediately after band gap collapse (b).

0 25000

(b)

20000 15000 10000 5000

1

2

3

4

5

2

Laser fluence [J/cm ]

fluences, breaking of the quasineutrality at the Si surface can take place in a wider region. In addition, electron thermalization takes additional time. One can expect that, immediately after the band gap collapse, the distribution function of the electrons consists of a cold electron core and a high-energy tail (>10% of electrons with Ee ≥ 10 eV). Such a situation should lead to a strong double layer effect29 at the semiconductor surface and may, presumably cause Coulomb explosion. The same situation (strong gradients of electron density, non-equilibrium distribution function) should exist in the bulk depth during the homogeneous nucleation of the liquid phase. At the boundaries of the liquid phase nuclei, in very narrow envelopes around the liquid islets, a double layer effect can influence strongly the motion of the liquid phase boundary. It should be underlined that this effect is expected to be much stronger for ultrafast excitations when electron energies are much higher than for longer pulses, when the electron and lattice subsystems can be considered to be in equilibrium. Our model, operating in terms of the temperatures of the electron and lattice subsystems, allows snapshots into the complexity of the phenomena with respect the average characteristics of the heated and melted substances. A more detailed picture of heating, melting and ablation of semiconductors can be obtained by application of molecular dynamic techniques which can provide insight into the fluctuating nature of ultrafast phase transformations including homogeneous nucleation of the liquid phase in solids78 and vapor phase in liquids79,80 and, probably, double layer effects.

CONCLUSION We have presented a unified continuum approach for modeling the dynamics of electronic excitation, heating and charge-carrier transport in different materials (metals, semiconductors, and dielectrics) under femtosecond pulsed laser irradiation. Using the model, the laser-induced charging of the targets has been studied at laser intensities above the material removal thresholds. Under such conditions, charging of dielectric surfaces is shown to cause Coulomb explosion, while this effect is strongly inhibited for metals and semiconductors. Questions about the probability of Coulomb explosion under high laser intensities in plasma plume regimes and about charge-assisted melting processes in semiconductors brought up in Part 3.2 invite a large body of further investigations.

ACKNOWLEDGEMENTS Financial support from the ISTC (Grant #2310) and the Wissenschaftlich-Technologischen Zusammenarbeit (WTZ) project RUS01/224 is gratefully acknowledged.

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