CHAPTER 2 

 

INTRODUCTORY CONCEPTS

 

General properties of laser radiation and the interaction of laser radiation with matter are introduced. The absorption of laser radiation by metals and insulators is discussed, and the general mechanisms of laser ablation are presented.

 

2.1 INTERACTION OF LASER RADIATION WITH MATTER

 

The simplest form of light is a monochromatic, linearly polarised plane wave. This is for most purposes a sufficient approximation to a real laser beam. The electric field of a wave propagating in a homogeneous and nonabsorbing medium can be represented as [1]

 

,

(2.1.1)

 

where z is the coordinate along the direction of propagation, w the angular frequency, and l the radiation wavelength. The last two quantities are related through the phase velocity c/n, c being the speed of light and n the real refractive index of the medium, by the equation

 

(2.1.2)

 

An expression analogous to (2.1.1) holds for the magnetic field. Magnetic and electric field amplitudes are related by

 

(2.1.3)

 

where e₀ is the dielectric constant in vacuum. On average, the electric and magnetic fields carry the same energy. However, the contribution of the magnetic field to the force exerted by the electromagnetic wave on an electron,

 

(2.1.4)

 

is smaller than that due to the electric field by a factor v/c, where v and c are the electron and light velocity respectively, and since v<<c the contribution of the magnetic field is negligible. The energy flux (power) per unit area is termed irradiance and is given by

 

(2.1.5)

 

The intensity of a light source is defined as the energy flux per solid angle. Often irradiance and intensity area used indistinctly, and throughout this work intensity will be used to designate the average power per unit area (W/cm²), which must be distinguished from the energy density or fluence F (J/cm²).

In absorbing media the real refractive index n must be replaced by the complex refractive index

 

(2.1.6)

 

where n and k are the real and the imaginary (also called the extinction coefficient) refractive indexes respectively, both frequency dependent. The physical meaning of k results when equation (2.1.2) is accordingly modified (n by ) and replaced in (2.1.1): the electric field, upon propagation over a distance z, decreases by a factor exp(wkz/c), indicating that some of the light is absorbed. The presence of matter causes both n and k to deviate from their values in vacuum, namely n=1 and k =0. In condensed matter, density is many times higher than that of a gas and deviations of n and k from vacuum values are correspondingly larger, leading to n, k >>1 over a wide wavelength range in most solids. The interaction of electromagnetic radiation with condensed matter can be characterised in terms of a complex frequency-dependent dielectric constant

 

(2.1.7)

 

where e₁ and e₂ are related to the complex refractive index, , as follows:

 

 

The absorption of light propagating through a homogeneous medium is given by the Lambert-Beer law:

 

(2.1.10)

 

where I₀ is the irradiance at z = 0 and I(z) the irradiance after a distance z. The optical absorption coefficient a of a pure and homogeneous material characterised by an imaginary refractive index k is

 

(2.1.11)

 

The complex refractive index depends on the radiation wavelength and consequently so does the absorption coefficient a. Spatial variations of the refractive index deform the wave front and cause secondary waves to split off from the primary one, into reflected and transmitted waves. Resonances in the dispersion curve e(w) also lead to resonances of reflectivity R. For normal incidence, reflectivity is given by

 

and transmissivity T by

(2.1.12)         (2.1.13)

 

When the depth within the material z is significantly larger than the optical penetration depth L_o=1/a, the absorbed energy may be determined by (1-R). When optical absorption is proportional to the number of absorbing centres and to their scattering cross-section, absorption varies linearly with depth. However, linear absorption characterised by Lambert-Beer law [equation (2.1.10)] may be disturbed in several ways. For example, if a new absorbing state results from absorption, the absorption behaviour may be non-linear. This will happen when free carriers are generated across the bandgap E_g of a semiconductor or insulator, which can subsequently contribute to enhance absorption [2]. Equation (2.1.10) must be modified when I₀ is such that non-linear effects become significant. For UV radiation and in condensed media this threshold is I₀ ł 10⁴ W/cm² [3]. For higher irradiances multiphoton absorption processes may become important, in particular, the simultaneously absorption of two photons may occur. For excitation of valence electrons to the conduction band via two-photon absorption, energy conservation requires that In general n₁ = n₂ and the probability of the occurrence of two-photon absorption (TPA), being proportional to the probability of finding both photons simultaneously at a particular location, is proportional to the square of the intensity of the laser beam [4]. Assuming only linear and second-order non-linear absorption, the attenuation of irradiance in a homogeneous medium can be described by

 

(2.1.14)

 

where b denotes the two-photon absorption coefficient [5]. In order to simplify the problem the non-linear term -bI² in equation (2.1.14) may be replaced by -(bI₀)I, yielding

 

(2.1.15)   (2.1.16)

 

The above equation can be generalised to various absorbing species [4], leading to

 

,

(2.1.17)

 

where a_i, b_j, and g_k are, respectively, the one-, two-, and three-photon absorption coefficients corresponding to the absorbing species {i, j, k}; the ellipsis in equation (2.1.17) refers to higher order absorption processes. The absorption coefficients are simply related to the number of each particular absorber per unit volume and the corresponding cross sections. Table 2.1.1 shows measured values of the multiphoton absorption coefficients for several nonmetallic solids. Its analysis shows that the probability of multiphoton absorption decreases sharply in wide band gap materials as the number of photons required to bridge the gap increases.

 

 

 

 

 

 

 

Table 2.1.1: Multiphoton absorption coefficients for several semiconductors and insulators [4].

 

2.2 ABSORPTION OF RADIATION

 

Laser radiation is absorbed either by exciting free electrons, or electronic or vibrational transitions in atoms, ions, or molecules (Figure 2.2.1). Various electronic excitations can occur (interband and intraband transitions, excitons, plasmons, etc.), as well as excitations of phonons, polaritons, magnons, etc. [3, 6-8]. Electronic or vibrational states may be localised or non-localised, and may be related to the solid surface itself, but also to defects and impurities.

 

Figure 2.2.1: Schematic of different types of electronic excitations in a solid. Straight lines indicate absorption or emission of photons with different energies, hn. Oscillating lines indicate non-radiative processes (adapted from [7]).

 

 

 

2.2.1             Metals

 

In metals, electronic excitations involve both intra and interband transitions. The frequency-dependent complex dielectric constant e can be written as [3],

 

(2.2.1)

 

where e^L is the contribution due to lattice vibrations (approximately constant at UV wavelengths), e^D is the intraband or Drude term and e^I is the term due to interband transitions. Both conduction- and valence-band electrons may participate in laser excitations, the former through a free-electron-like optical response and the latter through an interband response that has a threshold corresponding to the energy separation between the valence and conduction bands. Electrons in the conduction band can interact with photons and further increase their energy by intraband absorption. The mechanism for accelerating electrons in the conduction band is inverse Bremsstrahlung [9]. As a result of collisions, which occur at a frequency n_c, the conduction-band electrons acquire energy (E) at a rate given by [10]

 

(2.2.2)

 

where is the optical electric field, w is the photon frequency, and m_e the electron effective mass. Equation (2.2.2) can be understood as involving the product of the energy of an oscillating electron times the frequency at which this energy is converted, by collisions, into random translational energy. Metals are characterised by their loosely-bound external electrons, which are essentially free to move within the solid limits. As a consequence of this behaviour, the optical properties of most metals can often be quite satisfactory described within the framework of the free electron gas model [8]. Calculations of the dielectric response of the electron gas reveal that exists a plasma frequency w_p at which electrons can resonantly oscillate. The frequency w_p is given by [8]

 

(2.2.3)

 

where N_e is the electron density in the conduction-band and e₀ is the dielectric constant. At frequencies lower than w_p, typically corresponding to wavelengths larger than one micron, absorption is dominated by free-electron behaviour. The large absorption coefficient in metals results in an extremely small absorption depth known as “skin depth”. Radiation is also strongly reflected by metals, owing to the collective plasma properties of free electrons. As the radiation frequency increases above the plasma frequency, absorption and reflectivity both decrease. For higher frequencies, the free carrier contribution to the optical properties becomes extremely small, and metals exhibit characteristics related to the lattice properties. Eventually, interband absorption will occur as the frequency decreases beyond the absorption minimum. The characteristic colour of gold and copper is a consequence of such an absorption minimum in the visible region of the spectrum, while alkali metals (Na, Li, K, etc.) exhibit a large transparent zone in the ultraviolet. Whilst in most cases absorption and reflectivity of metals are lower in the visible and ultraviolet region of the spectrum than in the infrared, absorption and reflectivity become significantly more sensitive to the radiation wavelength at the shorter wavelengths.

 

Free electrons and holes can be successfully described as particles of a gas. An extension of these ideas leads to the renowned Drude model of free carrier absorption. By solving the classical equation of motion of a free carrier, with the sole quantum modification that the effective mass m* rather than the electron mass m_e is used, it results

 

2.3.5     2.3.6

 

where , t is a relation time associated with randomisation of the carrier velocity arising from collisions, N is the carrier density, and e_L is the dielectric constant of the lattice. For materials with low conductivity, such as semiconductors and insulators, we can approximate , in other words, , and the contribution of the electrons to the dielectric constant is negligible, and independent of frequency. In addition, for low conductivity, it is clear that , i.e. , and therefore . Using these assumptions a value for the absorption coefficient due to free-carrier absorption can be derived as

 

2.3.7)

 

where m₀ is the permeability of free space. At low frequencies and the absorption is essentially frequency independent, but at high frequencies when , then it is proportional to the square of the wavelength. For many solids, t » 10^-13 s and therefore t is close to unity at frequencies around 10¹² Hz, which corresponds to a wavelength of 300 mm. At still higher frequencies, absorption by free carriers can no longer be considered as a simple transition by a carrier. The electron, for example, must also change its wave vector in making the transition and this must involve a change in crystal momentum e.g. by the lattice, or an impurity since at these energies the photon itself cannot take up such large differences in momentum. When the conductivity is high, and , then for low frequencies and a is given by,

 

2.3.8

 

The material exhibiting this characteristic under these conditions may be termed metallic. Inspection of 2.3.6 reveals that even in metals, when the frequency is sufficiently high, the free carrier contribution to the dielectric constant become small, and therefore 2.3.7 will be applicable. This occurs in the visible or UV for metals as discussed previously.

 

2.2.2             Insulators and semiconductors

 

The optical response of insulators and semiconductors involves both electronic and ionic contributions to the dielectric function (Figure 2.2.2). The real part (refractive) of the dielectric function of non-metallic solids dominates away from vibrational or electronic resonances, while the imaginary (absorptive) component dominates near resonances. Unlike metals, insulators and semiconductors do not contain an appreciable number of free carriers in the conduction band. Instead of having loosely bond electrons associated with the atom, these materials are characterised by having their outermost electrons filing the highest energy valence band. Such tight bonding prevents electronic conduction. Thus, optical absorption in laser-irradiated semiconductors and insulators at moderate to high intensity (in the range of MW/cm² to GW/cm²) leads to the creation of electron-hole pairs rather than electron heating. Materials are characterised by their bulk bandgap energy E_g, which defines the energy required by an electron in the highest energy valence state, to cross the gap up to the lowest level in the conduction band. For E_g values in the region of 9 eV the material is considered as a strong insulator. Conversely, if E_g is closer to 0 eV, the material is a narrow-bandgap semiconductor. Above and below their bandgap energies, insulators and semiconductors have similar optical properties.

 

Figure 2.2.2: Generic dielectric function for a nonmetallic solid showing the locations of electronic and vibrational resonances. The static (s) or long-wavelength limit is in the infrared, while the optical limit is in the ultraviolet. The solid curve indicates the real part, and the dashed curves the imaginary part of the dielectric function (adapted from [4].

 

For radiation with photon energy hn greater than E_g, the radiation-solid interaction is dominated by single-photon interband transitions. This occurs in the near infrared for silicon (1.12 eV) or germanium (0.67 eV), in the green for galium phosphide (2.26 eV) and in the vacuum ultraviolet for silicon dioxide (8 eV) [11]. The linear absorption coefficients of nonmetallic solids for photons with energy greater than the bulk bandgap typically exceeds 10⁵ cm^-1 for direct transitions near the band edge, e.g., approaching the absorption coefficient of metals in the infrared. It is also in the strongly absorbing region of the spectrum that materials exhibit their highest reflectivity, because of the corresponding larger refractive indexes. If hn is less than the bandgap E_g, the optical properties of these materials are controlled by the lower energy intraband electronic transitions and by excitation of vibrational modes within the lattice. In addition, laser radiation may induce one-photon transitions between defect states and the conduction band [2, 12, 13], absorption by impurities [14], excitonic resonances [3] or multiphoton transitions between valence and conduction bands [4, 15]. There is vast experimental evidence showing that the presence of impurities and defects can strongly enhance coupling between laser radiation and insulator materials, since they absorb photons with energies within the bandgap. Significant light absorption can take place at external surfaces due to the existence of surface states [12], and to the presence of topological imperfections [13] or chemical impurities [14]. Even nominally pure solids usually have defect concentrations in the parts per million range. Such defects may consist of vacancies, interstitials, non-equilibrium charge states or isotopic centres. Various processes such as mechanical polishing [16] and g radiation [17] have been used to create defects that facilitate the transfer of electrons to the conduction band. On the other hand, defects may be generated during laser processing, promoting a stronger coupling between laser radiation and the material as laser treatment progresses. For example, processing of NaCl and MgO with nanosecond duration laser pulses causes extensive fracturing [18, 19]. The authors considered that the motion of dislocations generated during fracture and cleavage is a source of point defects.

 

2.3 MECHANISMS OF LASER ABLATION

 

Laser ablation may be defined as a sputtering process leading to the ejection of atoms, ions, molecules, and even clusters from a surface as a result from the conversion of an initial electronic or vibrational photoexcitation into kinetic energy of nuclear motion [6]. Laser ablation leads to a material removal rate (ablation rate) typically exceeding one-tenth monolayer per pulse, resulting in the modification of the surface shape or composition at mesoscopic length scales. It involves non-linear complex and collective processes that are not completely understood. These processes occur over many orders of magnitude in time, from the initial absorption of laser radiation to material ejection. Laser ablation is frequently characterised by a threshold fluence F_th (J/cm²) - ablation threshold - above which macroscopic modification of the material occurs. This threshold as well as the time for the thermalisation of the excitation energy depend on the material characteristics and processing parameters (pulse duration, radiation wavelength, etc.) [3]. In metals, light is almost exclusively absorbed by the conduction band electrons. The time between electron-electron collisions, t_e-e, is of the order of 10^-14 to 10^-12 s [4]. Electron-phonon relaxation times, t_e-ph, are much longer, due to the large difference between electron and ion masses. Depending on the strength of the electron-phonon coupling, one finds Similar relaxation times are found for quasi-free electrons in semiconductors like Si [7]. In non-metals, interband electronic excitations can last much longer, ranging from 10^-12 to 10^-6 s. Excitation of localised electronic states associated with defects, impurities or surfaces may have longer lifetimes than 10^-6 s. If the laser pulse duration is shorter than the thermalisation time, electron-phonon coupling and the subsequent thermal relaxation can be avoided during the pulse duration [20]. In addition, new (multiphoton) excitation channels exist specially in the high-power regime where the electric field of the laser pulse might exceed the threshold for optical breakdown and ablated material is transformed on an ultra-fast time-scale into a plasma [21]. Laser ablation with ultra-short laser pulses (below a few picoseconds) may then lead to extreme non-equilibrium situations, and is treated in more detail in Chapter 7. Figure 2.3.1 illustrates some of the processes that may occur during processing with longer pulses, in the nanosecond range. The initial laser-material interaction may create excited electrons in the solid, leading to the ejection of electrons by photoelectric or thermionic emission [22, 23], and eventually forming a plasma above the sample surface in a picosecond time scale [23]. In the solid, the excited electrons undergo electron-phonon relaxation and the energy is transferred to the lattice. Through lattice vibrations, the transferred energy is dissipated from the irradiated zone to the bulk in the form of heat [24]. Heat conduction occurs on a time scale of several tens of picoseconds, which is slightly longer than the electron-phonon relaxation time. Heating of the material leads to melting and vaporisation. The particles leaving the liquid during evaporation establish an equilibrium distribution of velocity in a small region above the surface called the Knudsen layer [25]. Above the Knudsen layer the vapour plume will expand rapidly, compressing the ambient gas and forming a shock wave front [26]. In addition, the expanding plume can interact with the laser pulse, effectively shielding the sample surface from laser energy [27]. The high temperature plume can also heat the sample surface via radiative heating [28]. The vaporisation of the sample and the subsequent gas dynamic processes in the expanding plume take place on a time scale of nanoseconds. Other material removal processes such as exfoliation or liquid ejection may occur on a substantially longer time scale.

“Thermal” or “photothermal” laser ablation typically designates situations in which the laser light is converted to lattice vibrational energy before bond breaking ejects material from the surface [29, 30]. Thermal ablation is distinct from “photochemical” or “electronic” ablation where laser-induced electronic excitations lead directly to bond breaking before an electronic to vibrational energy transfer has occurred [3, 31]. Both thermal and electronic ablation remove atomic-size particles of material from the surface. This is distinct from two other ablation mechanisms, identified in the literature as “hydrodynamical” and “exfoliational”, which may lead to the ejection of large bulk material fragments [32-37]. Frequently, several mechanisms occur simultaneously, or any of them may dominate, depending on the processing conditions and material properties [3,7]. In what follows these mechanisms are discussed in detail.

 

Figure 2.3.1: Scheme of relevant processes during short pulse laser ablation, namely laser-induced vaporisation, surface melting and shock wave formation. Drawing scheme of ablation.doc

 

2.3.1             Thermal ablation

 

a) Vaporisation. The terms “vaporisation” and “evaporation” are used to indicate the transition from liquid to vapour by emission of particles (atoms or molecules) at the liquid/gas interface [29, 30]. In a normal atmosphere evaporation occurs at all temperatures but only becomes significant at a vapour pressure of 1 atmosphere corresponding to the boiling temperature, T_B. This temperature is often designated as “vaporisation temperature” or “nominal vaporisation temperature”. The Clausius-Clapeyron equation may be used to find the vapour pressure p at other temperatures, assuming that liquid and vapour are in thermodynamic equilibrium [38]

 

(2.3.1)

 

where L_v is the latent heat of vaporisation (J/cm³). Under the assumption that vaporisation is kinetically limited and occurs in equilibrium conditions, the mass vaporisation rate b(T) (kg/m²s) can be calculated from p(T), resulting

 

(2.3.2)

 

where is the average mass of evaporated species. For a solid of density r (kg/m³) the liquid-vapour interface velocity v_vap (m/s) is

 

(2.3.3)

 

For typical processing conditions using nanosecond duration pulses, v_vap is about 10^-2 m/s, implying an evaporated depth of about 0.1 nm for a 10 ns laser pulse [3]. If instead of considering that vaporisation is kinetically limited one considers conservation of energy, the liquid-vapour interface velocity is

 

(2.3.4)

 

where L_f is the latent heat of fusion (J/m³), T_v is the vaporisation temperature, T₀ is the initial temperature, and C_p is the specific heat (J/m³K). In order to ablate material with nanosecond duration laser pulses, intensities higher than 10¹² W/m² are typically required [3]. For intensities this high, the solution to equation (2.3.4) implies evaporation rates in excess of those that are possible kinetically in equilibrium conditions. This suggests that in this pulse duration range little direct vaporisation occurring in equilibrium occurs, but becomes more relevant with increasing pulse duration [3, 39].

The flux (particles/cm²s) is governed by the Hertz-Knudsen equation,

where a is the condensation (or vaporisation) coefficient, p_sv is the equilibrium (or saturated) vapour pressure, p_v is the partial pressure of vapour in the ambient, and m is the particle mass. which if multiplied by m/r (m being the particle mass and r the target mass density) gives the velocity of surface recession.

 

b) Boiling. This process requires that the pulse duration is sufficiently long for heterogeneous vapour bubble nucleation to occur [40]. In principle, the target will undergo boiling from the surface to a depth related to the optical penetration depth (1/a, where a is the absorption coefficient) or to the heat penetration depth (, where k is the thermal diffusivity), depending on which one is longer. Formation of gas occurs by heterogeneous nucleation on a variety of defects, such as solid particles in suspension, or a contacting solid surface. However, even if the necessary heterogeneous nucleation sites exist, their density (~10⁶ kg^-1 [41]) may be too low for boiling to be significant. Once formed, the bubbles migrate and may, given enough time, escape from the outer surface of the liquid. Due to the moving vapour bubbles that sustain boiling, it has been argued that strong temperature gradients cannot exist along the depth where boiling occurs [30].

A thermal mechanism, in the sense of vaporisation from a transiently heated substrate, may require temperatures well above the nominal boiling temperature. The reason is that the observed ablation rate (typically 1-100 nm/pulse) can be accomplish during the length of the pulse, only if the temperature is sufficiently high.

 

c) Phase explosion. To explain some of the processes generated by heating with a pulsed laser, a typical p-T diagram is shown in Figure 2.3.2. The “normal heating” line represents heating when the temperature is below the boiling temperature. At the boiling temperature the liquid and vapour phases are in equilibrium, represented in Figure 2.3.2 by the intersection between the normal heating line and the liquid-vapour equilibrium line (binode). In a slow heating process, equilibrium occurs and the surface temperature-pressure relation follows the Clausius-Clapeyron equation. Under rapid heating, it is possible to superheat the liquid up to temperatures well above the boiling point, so that heating may deviate from the equilibrium line. In this case, the process is represented by the “superheating line” in Figure 2.3.2. The existence of a superheated state depends on the rate of spontaneous nucleation during heating. It has been shown that the frequency of spontaneous nucleation is about 0.1 s^-1cm^-3 at 0.89T_c, where T_c is the thermodynamic critical temperature, but increases to 10²¹ s^-1cm^-3 at 0.91T_c [42]. This indicates that a rapidly heated liquid could possess considerable stability with respect to spontaneous nucleation up to 0.89T_c, with an avalanche-like onset of spontaneous nucleation of the high temperature liquid layer at about 0.91T_c. At about 0.91T_c, explosive boiling or phase explosion occurs by homogeneous nucleation, and the target undergoes a rapid transition from superheated liquid to a mixture of liquid droplets and vapour leaving the target like an explosion [43]. Because the rate of homogeneous vapour nucleation rises catastrophically near T_c, the necessity that nuclei form does not constitute a kinetic obstacle. The above description defines an upper limit for superheating, which is shown as the “spinodal line” in Figure 2.3.2. As the temperature approaches the spinodal line, a loss of thermodynamic stability occurs, leading to phase explosion of the superheated liquid. The principles of phase explosion were laid down already by Martynyuk in 1974, brought about by discharging a condensor into a wire [40, 44], and by Fucke et al. [41, 45], and the terms phase explosion, explosive boiling, and vapour explosion were also introduced.

As with normal boiling, one can expect that at and beneath the surface, the condition will apply.

Kelly

where T₀ (i.e., T_B) is the vaporisation temperature at ambient pressure p₀ (i.e., p (T_B)) and DH is the latent heat.

 

Figure 2.3.2: p-T diagram.

 

If the laser fluence is sufficiently high and the pulse length sufficiently short for a large superheating to occur, the liquid temperature may approach the thermodynamic critical temperature T_c. Phase explosion has been suggested as the main mechanism of laser ablation in various materials using nanosecond duration pulses, to explain the abrupt increase on the ablation rate above a defined fluence (higher than the ablation threshold), and the accompanying ejection of liquid droplets [46-48]. Anomalies of physical properties occur at temperatures higher than 0.8T_c and liquid metal becomes less conductive, behaving more like a dielectric material, an effect know as induced transparency of liquid metals [49]. Laser radiation may penetrate deeper into the material near the critical temperature, extending the thickness of the superheated liquid, and explaining the increase in ablation rate [48].

 

d) Sub surface heating. According to this mechanism when rapid vaporisation occurs, the surface is cooled, but the subsurface region may retain a higher temperature. As a consequence, the pressure beneath the surface may increase significantly above ambient pressure, and an explosion may occur leading to similar results as with phase explosion [50, 51]. However, according to Kelly and Miotello [30, 43] this model is in contradiction with the requirement that along the depth where boiling occurs.

 

2.3.2             Electronic ablation

 

Electronic ablation typically does not refer to a single process, but rather to a group of processes having the common feature of involving some form of excitation or ionisation. It arises, for example, due to events as varied as direct bond breaking, defect formation or surface plasmon excitation. In general, photochemical effects can occur whenever the energy of a single photon exceeds the dissociation energy D₀ of the absorber material [31], i.e.

 

hn > D0

(2.3.5)

 

Photochemical processes are frequent in organic polymers (Figure 2.3.3), although thermal effects may occur simultaneously [52]. Since photochemical reactions can occur with single photons, there is no associated intensity threshold for this effect. On the other hand, thermally activated processes may exhibit an intensity threshold, since the temperature reached will be the result of both heat input and loss terms. At high laser intensities photochemical effects may include two, three or higher level multiphoton processes. Then

 

nhn > D0

(2.3.6)

 

where n is the order of the non-linear interaction. Since the rate W⁽ⁿ⁾ of an n-photon process is proportional to the laser intensity I, such that W⁽ⁿ⁾ µ Iⁿ, multiphoton photochemical reactions will exhibit an intensity threshold. In addition, their relative importance will depend strongly on the wavelength and absorption of the material at this wavelength. When the photon energy hn ł E_g/2, where E_g is the onset of electronic absorption, then two-photon transitions are possible energetically. If the final state has energy 2hn > D₀, then a two-photon induced photochemical effect may occur [3]. When hn > E_g so that the absorption coefficient a is large, single-photon processes are possible and single-photon photochemistry is important if sufficient energy is provided to break one or more chemical bonds. Even when reactions are photochemical, the efficiency does not necessarily approach unity. Internal energy can be retained by the material, and dissipated in a variety of ways [4]. For example, it may immediately reradiated in the form of fluorescence, reradiated after some delay as phosphorescence or retained as internal vibrational energy that is eventually degraded to heat.

 

Figure 2.3.3: Hypothetical steps in the interaction of a UV-laser pulse with a polymer as suggested by Srinivasan [31]. The chemical bonds in the polymer are broken by the photon energy, and the products are ejected at supersonic velocities.

 

For high enough laser-pulse energies, dense electron excitation can be expected, as high as n_e » is the excited electron-number density. These electrons will increase the total energy of each atom by an amount similar to n_eE_gap/n_c, where E_gap is the energy gap, as discussed by Wautelet and Laude (1980). For a system like Al₂O₃ where E_gap is ~9 eV, n_c is 4.7 ´ 10²² (Al atoms)/cm³, and the depth of potential well is 5.7±0.1 eV, it is clear that a value of n_e such as 10²² cm^-1 would raise the energy of each atom by about 2 eV, and thus increase the vapour pressure by orders of magnitude or even render the lattice unbound. This model is related to the rapid energy deposition model [Jöst et al., 1982; Kissel and Krueger, 1987], which was devised to explain the unusual response of solids to particles ranging from laser pulses, to fission fragments, to electron pulses or small accelerated dust particles. A possible description is that a rapid (i.e., non-adiabatic) transition takes individual ions directly into antibonding states. The result is that the system makes a transition from a tightly bound solid to a densely packed, repulsive gas, and particles are expelled energetically.

Wide band gap oxides (Eg@7-9 eV) such as SiO₂, MgO and Al₂O₃ may be ablated with photons having energies as low as 4 eV (l=308 nm), although quite high fluences may be required For this spectral region these materials are weakly absorbers, and the absorption is often primarily determined by the presence of impurities and defects. Such centers enhance the coupling of incident radiation into the material and may initiate damage that leads to subsequent ablation. However, two-photon absorption can also become significant at high incident fluences and can result in the establishment of a high density of electron-hole pairs in the region of highest excitation Since this is often at the surface of the sample, which is also the site of high defect concentration, two-photon absorption can lead directly to enhanced ablation by electron-hole trapping at these defect sites. Possible mechanisms for ablation initiated by electron-hole trapping at defect sites have been discussed in detail by Kelly et al. Itoh and Singh and Itoh absorçăo

Inorganic insulators are efficiently ablated by nanosecond duration pulses in the UV range even when the photon energy does not exceed the bandgap energy E_g. Ablation also occurs at laser fluences that are significantly lower than those required to produce fast vaporisation [3]. As a result, the study of UV laser ablation of inorganic insulators frequently focus on the role played by electronic excitation in ablation, most specifically in the formation of defects and electron-hole pairs and subsequent interaction with laser radiation [2, 4]. In this regard, many similarities exist between pulsed laser ablation of wide bandgap materials with UV radiation, and that of semiconductors at longer wavelengths [3]. At short wavelengths (e.g., l=193 nm) and using nanosecond duration pulses the ablation rate of most dielectrics is about 0.01-0.05 mm/pulse at fluences that are 2-3 times the ablation threshold [53]. Threshold fluences are typically between 2 and 5 J/cm², which corresponds to (1.9-4.9)´10¹⁸ photons/cm² at 193 nm. An ablation rate of 0.05 mm/pulse will remove about ten atomic layers or 10¹⁶ atoms/cm² per pulse. For an average binding energy of 10 eV/atom, ablation at this rate requires 1.6´10^-2 J/cm², an energy about 200 times smaller than the threshold fluence. This suggests that the energy deposition is not the single limiting factor in determining the ablation rate. Instead, fluence may play a predominant role in the creation of electron-hole pairs via two-photon absorption, which would then facilitate ablation due to enhanced absorption of radiation. The density of electron-hole pairs created by two-photon absorption in a single pulse is [3]

 

Neh = bF2thn

(2.3.7)

 

where the two-photon absorption coefficient b, depends on the laser wavelength and on the material. A representative value is b @ 10 cm per 10⁶ W. Then, with F=10²⁶ photons/cm²s, t=30 ns and hn = 1.03´10^-18 J, results N_eh @ 3´10²¹ cm^-3 per pulse as an estimate of the density of electron-hole pairs generated by a single laser pulse at the threshold fluence for ablation. The energy density in the material associated with this excitation is N_ehE_g, and with E_g = 8 eV=1.28´10^-18 J, results N_ehE_g = 3.8´10³ J/cm³ or approximately only 10% of the sublimation energy for a typical simple oxide. The temperature rise DT induced by this energy deposition is relatively small [3]:

 

DT = NehEg/rCp

(2.3.8)

 

Taking r = 2 g/cm³, and C_p = 1.5 J/g°C as representative values, N_ehE_g = 3.8´10³ J/cm³ so that DT = 1.3´10³ °C. This suggests that electronic processes, namely electron-hole trapping at defects, may be significant and even dominate ablation in these materials. However, the generation of a high density of electron-hole pairs as well as the ionisation of colour centres by photo-absorption results in a rapid increase in the density of electrons in the conduction band. Following this triggering process, two intrinsic mechanisms have been proposed to account for the production of the large amount of electrons necessary to cause ablation/damage [2]:

- continued multiphoton absorption, and

- multiplication by an avalanche process.

The first mechanism was previously described. In an avalanche process, conduction electrons are accelerated in the electromagnetic laser field until they reach an energy larger than that required to produce ionisation. This excess energy may be transferred to valence electrons which are promoted to the conduction band. In this manner, a cascade-like multiplication process of the number of conduction electrons develops. Theoretical analysis [54] showed that this mechanism should be less effective for photon energies greater than one-fourth of the band gap energy, and more effective when the photon energy is less than one-fifth of the band gap energy. The excess energy in the electrons is transferred to the lattice as electrons are scattered by phonons, initiating ablation when the phonon density exceeds some critical density.

 

2.3.3             Exfoliational ablation

 

Exfoliational ablation generally refers to an ablation mechanism where mechanical fracture leads to ejection of fragments of material. Significant stresses due to laser irradiation may occur for materials with high linear thermal expansion (DL/L_d, where L_d is the heated depth and DL the expansion), high Young’s modulus (E_Y), high melting temperature (T_m), and when the local temperature variation approaches but does not exceed T_m [55]. Under such circumstances thermal shock occurs repeatedly and may lead to cracking and material ejection. A convenient indication of thermal shock is given by the thermal stress, s_th=E_YDL/L_d. This relation suggests that exfoliational ablation is likely for example in oxides, which have large E_YDL/L_d. In addition, if the temperature rise in the material is faster than mechanical relaxation, then heating at constant volume occurs and the pressure increases. When the pressure gradient in the direction normal to the surface exceeds the tensile strength of the material, it may cause exfoliation and forward ejection of a significant part of the irradiated region (Figure 2.3.4) [32, 33].

 

Figure 2.3.4: Snapshots of the plume at 500 ps vs. deposited energy in an organic solid as calculated by molecular dynamics calculations (pulse duration 15 ps at l=337 nm) [32].

 

2.3.4             Hydrodynamic ablation

 

Hydrodynamic ablation refers to processes that lead to material removal by liquid ejection.

a) Recoil plume pressure. Melt expulsion may occur due to variable pressure in the plume above the irradiated area, for example between the centre and the periphery of the spot [34-37]. This mechanism of removal is responsible for the accumulation of resolidified material at the border of the spot (Figure 2.3.5). Some of the first studies on melt expulsion were related to drilling of metals [34] and gelatine [36] using pulses with duration between 10^-4 to 10^-3 s. Modelling was performed considering that the depth of material removed d, measured along the normal to the surface at the centre of the spot may be represented as

 

(2.3.9)

 

where d_vap is the contribution due to vaporisation and d_melt is the contribution due to liquid expulsion in the normal direction. The contribution of melt expulsion depends on the melt thickness [35] and is likely to become important only when the melt displacement along the spot d, caused by a single laser pulse is comparable or exceeds the spot radius R_spot, so that [37]. The melt is accelerated while the pressure gradient in the plume lasts, which corresponds approximately to the pulse duration [7]. For long pulses (t=10^-4-10^-3 s) this results is large melt displacements d, so that significant melt expulsion may occur [34-36]. For the shorter nanosecond pulses (t=20-40 ns), the melt displacement along the spot is typically below 20 mm [37], whereas the spot radius for micromachining applications varies between 25-500 mm. This physically means that for the nanosecond regime melt expulsion is likely to have a small effect on the overall ablation rate, as concluded by Tokarev et al. [37]. Nevertheless, observation of various materials, in particular metals, processed with nanosecond duration pulses shows that liquid flow may lead to substantial accumulation of recast material at the periphery of the spot, deteriorating the quality of machining [3, 56]. Tokarev et al. [37] suggested that in order to improve surface finishing and eliminate melt expulsion, instead of the condition , one should guarantee that the melt displacement follows the condition , where d is the removed depth per pulse. Since d typically varies between 0.1-1 mm when processing is carried out using nanosecond duration pulses [3], this latter condition is much stronger.

 

Figure 2.3.5: Scheme of irradiation of a shallow cavity considering that the incident beam has a top-hat spatial intensity distribution. The gradient of ablation pressure DP induces a lateral melt flow. The ablation depth per pulse Dz is much smaller than the spot radius Rspot and the characteristic distance DR of the plume expansion during the pulse duration t (adapted from [37]).

 

b) Acceleration due to phase change and thermal expansion. The turbulence of the surface [7], variable plume pressure [57], as well as a variety of other effects [58], may lead to droplet formation during laser processing. These droplets may be accelerated away from the melt pool owing to a combination of thermal expansion of liquid and volume change on melting [39, 59, 60]. Assuming an ideal spherical shape, the restoring force on a sphere of radius r and surface tension s is

 

(2.3.10)   (2.3.11)

 

The variation in size of a droplet on going from solid to liquid is [3]

 

(2.3.12)

 

where the first term is due to the thermal expansion of the liquid on heating resulting from the temperature variation , and the second term is due to the change in volume during the solid to liquid transition; g_le is the linear expansion coefficient of the liquid and r_s and r_l are the densities of solid and liquid, respectively. For most materials the density of the liquid is lower than that of the solid, and droplets are accelerated away from the melt pool. Droplet separation can occur when

 

(2.3.13)

 

where Dt is the time during the laser pulse after substrate melting has occurred. The expansion during heating is not counterbalanced during cooling because cooling begins at the bottom of the melted substrate, which resolidifies before the surface.

In summary, laser ablation is a complex process where various mechanisms may be present simultaneously. Laser ablation does not need to involve massive, catastrophic destruction of a surface. Indeed, laser micromachining relies on laser ablation to shape complex features in a reasonable well-controlled and repeatable fashion.

 

c) Convective fluxes. Surface melting under the action of laser light may result in the excitation of convective fluxes within the liquid layer. Such convective fluxes play an important or even decisive role in material transport during laser processing. Convection can originate from changes in material density related to temperature gradients. With uniform laser light irradiation this is mainly related to temperature gradients along the melt thickness. Convective fluxes can be driven by changes in the surface tension of the material (Marangoni convection). If the changes in surface tension originate from gradients in the laser-induced temperature distribution along the surface, this phenomenon is also denoted as thermocappilary effect. The direction of the convective fluxes depends on the sigh of . The surface tension force acts in the direction of surface tension increasing, i.e., for the typical situation in the direction of temperature decreasing. The surface tension coefficient decreases, in most cases, with increasing temperature as , where s₀ is a constant, T_cr the critical temperature, and n»1.

 

REFERENCES

 

1. M. Born and E. Wolf: Principles of Optics, (Pergamon Press, Oxford 1970)

2. A. J. Pedraza: Interaction of UV laser light with wide band gap materials: Mechanisms and effects, Nucl. Inst. Meth. Phys. Res. B 141, 709-718 (1998)

3. W. W. Duley: UV lasers: Effects and Applications in Materials Science, (Cambridge University Press, 1996)

4. R. F. Haglund-Jr.: In Laser Ablation and Desorption, ed. by J. Miller and R. Haglund (Academic Press, 1998), Vol. 30, pp. 15

5. A. Yariv: Quantum electronics, (J. Wiley, New York 1989)

6. J. Miller and R. F. Haglund-Jr.: Laser Ablation and Desorption, in Experimental methods in physical sciences, edited by J. Miller and R. Haglund (Academic Press, San Diego, 1998), Vol. 30, pp. 15

7. D. Bäuerle: Laser Processing and Chemistry, (Springer-Verlag, Berlin Heidelberg 2000)

8. J. R. Christman: Fundamentals of Solid State Physics, (John Wiley & Sons, New York 1988)

9. M. v. Allmen: Laser-Beam Interactions with Materials: Physical Principles and Applications, (Springer-Verlag, Berlin Heidelberg 1987)

10. C. D. Michelis: J. Quantum Electron. QE-5, 188 (1969)

11. D. Palik: Handbook of Optical Constants of Solids, (Academic Press, New York 1985)

12. M. A. Schilbach and A. V. Hamza: Phys. Rev. B 45, 6197 (1992)

13. N. Bloembergen: IEEE J. Quantum Electron. 10, 375 (1974)

14. S. C. Jones, A. H. Fischer, P. Braunlich, and P. Kelly: Phys. Rev. B 37, 755 (1988)

15. A. S. Epifanov: Sov. Phys. JETP 40, 897 (1974)

16. R. L. Webb, L. C. Jensen, S. C. Langford, and J. T. Dickinson: J. Appl. Phys. 74 (4), 2323 (1993)

17. R. L. Webb, L. C. Jensen, S. C. Langford, and J. T. Dickinson: Interactions of wide band-gap single crystal with 248 nm excimer laser radiation. II. NaCl, J. Appl. Phys. 74 (4), 2338-2346 (1993)

18. O. Apel, K. Mann, A. Zoeller, R. Goetzelmann, and E. Eva: Nonlinear absorption of thin Al₂O₃ films at 193 nm, Appl. Opt. 39 (18), 3165-3169 (2000)

19. E. Eva and K. Mann: Nonlinear absorption phenomena in optical materials for the UV-spectral range, Appl. Surf. Sci. 109-110, 52-57 (1997)

20. H.-G. Rubahn: Laser Applications in Surface Science and Technology, (John Wiley & Sons, 1999)

21. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry: Nanosecond to femtosecond laser-induced breakdown in dielectrics, Phys. Rev. B 53 (4), 1749-1760 (1996)

22. S. I. Anisimov, B. L. Kapeliovivh, and T. L. Perelman: Sov. Phys. JETP 39, 375 (1974)

23. S. S. Mao, X. Mao, R. Greif, and R. E. Russo: Dynamics of an air breakdown plasma on a solid surface during picosecond laser ablation, Appl. Phys. Lett. 76 (1), 31-33 (2000)

24. D. v. d. Linde, K. Sokolowski-Tinten, and J. Bialkowski: Appl. Surf. Sci. 109/110, 1 (1997)

25. R. Kelly and R. Dreyfuss: Reconsidering the mechanisms of laser sputtering with Knudsen-layer formation taken into account, Nucl. Instrum. Meth. Phys. Res. B. 32, 341-348 (1988)

26. G. Callies, P. Berger, and H. Hügel: Time-resolved observation of gas-dynamic discontinuities arising during excimer laser and their interpretation, J. Phys. D: Appl. Phys. 28, 794-806 (1995)

27. X. Mao and R. E. Russo: Observation of plasma shielding by measuring transmitted and reflected laser pulse temporal profiles, Appl. Phys. A 64, 1-6 (1997)

28. J. R. Ho, C. P. Grigoropoulos, and J. A. C. Humprey: Gas dynamics and radiation heat transfer in the vapor plume produced by pulsed laser irradiation of aluminium, J. Appl. Phys. 79 (9), 7205-7215 (1996)

29. A. Miotello and R. Kelly: Critical assessment of thermal models for laser sputtering at high fluences, Appl. Phys. Lett. 67 (24), 3535-3537 (1995)

30. A. Miotello and R. Kelly: Laser-induced phase explosion: new physical problems when a condensed phase approaches the thermodynamic critical temperature, Appl. Phys. A 69 (COLA'99), 67-73 (1999)

31. R. Srinivasan: In Laser Ablation: Principles and Applications, ed. by J. C. Miller (Springer-Verlag, Berlim 1994), Vol. 28, pp. 107-134

32. L. V. Zhigilei, P. B. S. Kodali, and B. J. Garrison: On the threshold behavior in laser ablation of organic solids, Chem. Phys. Lett. 276, 269-273 (1997)

33. L. V. Zhigilei, P. B. S. Kodali, and B. J. Garrison: A Microscopic View of Ablation, J. Phys. Chem. B 102, 2845-2853 (1998)

34. M. v. Allmen: Laser drilling velocity in metals, J. Appl. Phys. 47 (12), 5460-5463 (1976)

35. C. L. Chan and J. Mazumder: One dimensional steady-state model for damage by vaporization and liquid expulsion due to laser-material interaction, J. Appl. Phys. 62 (11), 4579-4586 (1987)

36. A. D. Zweig: A thermo-mechanical model for laser ablation, J. Appl. Phys. 70 (3) (1991)

37. V. Tokarev and A. Kaplan: Suppression of melt flows in laser ablation: application to clean laser processing, J. Phys. D: Appl. Phys 32, 1526-1538 (1999)

38. M. V. Sussman: Elementary General Thermodynamics, (Addison-Wesley, Massachusetts 1972)

39. R. Kelly and J. Rothenberg: Laser Sputtering Part III: The mechanism of the sputtering of metals at low energy densities, Nucl. Instru. Meth. Phys. Res. B B7/8, 755-763 (1985)

40. M. M. Martynyuk: Sov. Phys. Tech. Phys. 19, 793 (1974)

41. W. Fucke and U. Seydel: High Temp. -High Pressures 12, 419 (1980)

42. M. M. Martynyuk: Russian J. Phys. Chem. 57, 494-501 (1983)

43. R. Kelly and A. Miotello: Comments on explosive mechanisms of laser sputtering, Appl. Surf. Sci. 96-98, 205-215 (1996)

44. M. M. Martynyuk: Sov. Phys. Tech. Phys. 21, 430 (1976)

45. U. Seydel and W. Fucke: J. Phys. F: Metal Phys. 8, L157 (1978)

46. X. Xu and K. Song: Interface kinetics during pulsed laser ablation, Appl. Phys. A 69, S869-S873 (1999)

47. X. Xu and K. Song: Phase change phenomena during high power laser-materials interaction, Mat. Sci. Eng. A 292, 162-168 (2000)

48. J. H. Yoo, S. H. Jeong, X. L. Mao, and R. E. Russo: Evidence for phase-explosion and generation of large particles during high power nanosecond laser ablation of silicon, Appl. Phys. Lett. 76 (6), 783-785 (2000)

49. V. A. Batanov, F. V. Bunkin, A. M. Prokhorov, and V. B. Fedorov: Evaporation of metallic targets caused by intense optical radiation, Sov. Phys. JETP 36 (2), 311-322 (1973)

50. F. W. Dabby and U. C. Oaek: IEEE J. Quantum Electron. QE-8, 106 (1972)

51. D. Bhattacharya, R. K. Singh, and P. H. Holloway: J. Appl. Phys. 70, 5433 (1991)

52. J. H. Brannon, J. R. Lankard, A. I. Baise, F. Burns, and J. Kaufman: Excimer laser etching of polyimide, J. Appl. Phys. 58 (5), 2036-2043 (1985)

53. J. Ihlemann and B. Wolff-Rottke: Excimer laser micromachining of inorganic dielectrics, Appl. Surf. Sci. 106, 282-286 (1996)

54. A. S. Epifanov, A. A. Manenkov, and A. M. Prokhorov: Sov. Phys. JETP 43, 377 (1976)

55. R. Kelly, A. Miotello, A. Mele, and A. Guidoni: In Laser Ablation and Desorption, ed. by J. Miller and R. Haglund (Academic Press, San Diego 1998), Vol. 30, pp. 225-290

56. B. N. Chichkov, C. Momma, S. Nolte, F. v. Alvensleben, and A. Tünnermann: Femtosecond, picosecond and nanosecond laser ablation of solids, Appl. Phys. A 63, 109-115 (1996)

57. A. B. Brailovsky, S. V. Gaponov, and V. I. Luchin: Mechanisms of melt droplets and solid-particle ejection from a target surface by pulsed laser action, Appl. Phys. A 61, 81-86 (1995)

58. L. Chen: In Pulsed Laser Deposition of Thin Films, ed. by D. B. Chrisey and G. K. Hubler (Wiley & Sons, 1994), pp. 168-198

59. T. D. Bennett, C. P. Grigoropoulos, and D. J. Krajnovich: Near-threshold laser sputtering of gold, J. Appl. Phys. 77 (2), 849-864 (1995)

60. N. Seifert, G. Betz, and W. Husinsky: Droplet formation on metallic surfaces during low-fluence laser irradiation, Appl. Surf. Sci. 103, 63-70 (1996)