Implementación de la política monetaria en casos de dolarización financiera

The interaction of laser ablation plumes with a background gas is of importance in PLD, nanoparticle formation and growth [51]. Compared with expansion into a vacuum, the interaction of the plume with an ambient gas is a far more complex gas dynamic process due to the appearance of new physical processes such as deceleration, thermalisation of the ablated species, interpenetration, recombination, formation of shock waves and clustering [52][53][54]. Numerical investigations of PLD have been performed both experimentally and theoretically for decades. Several diagnostic tools have been used to study plume formation and content: they mainly include time-resolved optical spec-troscopy and time-of-flight mass spectrometry [55]. Different models describing the expansion of the plume were then developed according to the background gas pressure

2.3. Pulsed Laser Deposition 17

range used. Shock wave models are commonly used for the case of ablation in a dense background gas when typically pressures of above a few Pa are used. Nevertheless, a number of gas dynamic aspects of the expansion are not yet fully understood [56].

Optimal properties of films deposited by PLD require the use of a background gas to obtain high-quality ITO films, with both high conductivity and transparency. Background gas influences the kinetic energy distribution of plume constituents when they impact on the substrate. Moreover, oxygen reacts with the ablated metal species of the plume leading to the formation of oxides and oxygen-containing clusters [55]. These reactions play an important role in preserving film stoichiometry and ensuring good functional properties. It was found in [57] that high quality as-deposited superconducting films could be obtained at only a given target-substrate distance (7 cm) and oxygen background pressure (5 mT). Authors related this observation to the velocity distribution uniformity of the various species, ensuring homogeneous substrate surface densities of the constituents and identical coverage rates. As a consequence, there is a better chance for the atoms to rearrange amongst themsleves to form a stoichiometric film. There are narrow ranges of parameters for which a uniform velocity distribution occurs. It was shown in [58] that the velocity distribution of each atomic species could be described very accurately by the following equation (theory of supersonic molecular beams):

f (v) = Av³× e^m(v−v0)

2

2kTs (2.15)

where A is normalization constant, v₀ is the so-called stream velocity, m is the mass of the ejected atom or ion under consideration, k is Boltzmann’s constant and T_s is a temperature parameter describing the velocity spread.

For laser pulses of some tens of nanoseconds duration, the expansion of the evaporated cloud into the ambient gas starts right after the laser pulse termination. Thus, the evaporation and expansion can be considered separately. The expansion starts from a cloud with the density and temperature uniformly distributed within a half sphere spreading above the surface exposed to the laser beam. The particles in the cloud have a Maxwell-Boltzmann distribution with the temperature [56]. After its formation and until the end of the laser pulse, it can be considered isothermal, with the temperature well exceeding 1000 K. Immediately after irradiation, the plasma will expand away from the target because it is the direction of the greatest density gradient. This expansion pushes any existing gases away from the target, setting up a pressure wave: this is

2.3. Pulsed Laser Deposition 18

a shock-wave. There is a threshold pressure above which shock-wave formation can occur [59]. During propagation through a gas, the presence of this shock-wave front causes spatial confinement resulting in a reduced cooling rate of the plume and high ionisation degree of its species. Under such conditions mutual aggregation of plume constituents occurs [60] leading to the formation of clusters which can grow on the substrates if they are larger than a critical size. Expansion of the saturated vapour, combined with heat transfer at the contact surface between the plume and the ambient gas, leads to very rapid cooling. This cooling can occur more rapidly than condensation, leading to exceptionally high saturation ratios (S).

After evaporation from the target, material from the plume is then allowed to recondense on a substrate, where film growth occurs.

2.3.4 Film growth

2.3.4.1 Adsorption and diffusion

After an atom or an ion is adsorbed on a surface, it has two possibilities: it might diffuse across the surface and then escape to the vacuum or it might bond. The diffusion rate of an adatom across a surface at a temperature T is given by a Maxwell-Boltzmann relation:

Ds = D0× e^−DkT (2.16)

where D is the activation energy for diffusion (cal/mole) and D0 is a constant. In order to grow high quality crystalline film, there must be sufficient surface diffusion to allow adatoms to migrate to thermodynamically stable sites and minimize their surface energy within the time needed to deposit a monolayer of atoms. There are two ways to enhance this surface diffusion: increasing the substrate temperature and/or the kinetic energy of the ablated species. One problem with increasing the surface temperature is the increase in surface-to-bulk diffusion and bulk interdiffusion. This leads to a smearing out of planes so that it can lower the limit to device size [61]. This problem can be overcome by enhancing surface mobility by energy transfer from the species impinging from the plume to the surface. A compromise needs to be found between promoting surface mobility and avoiding bulk displacement phenomena. The arrangement and growth of indium oxide and ITO thin films can be simplified in terms of the stacking of MO6 coordination groups. Amorphous indium oxide is probably formed during physical vapour deposition processing when MO₆ coordination units, which evolve from the target or are formed while chemisorbed on the growth surface, are incorrectly oriented

2.3. Pulsed Laser Deposition 19

when they are incorporated into the growing film. For example, at low temperature, the restricted mobility of the indium oxygen clusters preserves the misorientation of the coordination units and consequent bond distortion [62].

2.3.4.2 Nucleation and growth

2.3.4.2.1 Energetic considerations and supersaturation When a cluster is allowed to diffuse on a given substrate, it will try to minimize its free energy G. G is the free energy function and is a convenient measure of the feasibility of reaction. It is defined as:

G = H − T S (2.17)

where H is the enthalpy, S the entropy and T the absolute temperature. Thus, if a system changes from some initial (i) to a final (f) state at constant temperature due to a chemical reaction or physical process, a free-energy change ∆G=G_f-G_i occurs given by:

∆G = ∆H − T ∆S (2.18)

where ∆H and ∆S are the corresponding entahalpy and entropy changes. Systems naturally tend to minimize their free energy and successively proceed from a value G , to a still lower, more negative value G, until it is no longer possible to reduce G further.

When this happens, ∆G =0 . The system is said to have achieved equilibrium and there is no longer a driving force for change. It is convenient to work with the change in chemical free energy per unit volume ∆Gv.

For most deposition processes including pulsed laser deposition, the two main thermo-dynamic parameters that determine to a great extent the growth mechanism are the substrate temperature T_s and the supersaturation S. Both parameters are involved in the expression for ∆Gv which can be written as [63]:

∆G_v = −kT_s Ω ln(P

P_e) = −kT_s

Ω ln(S) (2.19)

P is the pressure of the arriving atoms and P_e is the equlibrium vapor pressure at the substrate temperature T_s. S is the vapour supersaturation defined by S=P/P_e. Ω is the atomic volume. Without supersaturation, S=0 and hence ∆G_v=0 so that nucleation and growth are impossible. Therefore, any level of gas-phase supersaturation generates a

2.3. Pulsed Laser Deposition 20

negative ∆G_v, which makes nucleation possible. A supersaturated solution/vapour is not at equilibrium (∆Gv 6= 0). In order to relieve the supersaturation and move towards equilibrium (∆G_v = 0), the solution or the thin film crystallises. Thus, a supersaturated solution/vapour is required for crystallisation to occur. In PLD thin film growth, the degree of supersaturation of the plume (vapour) is mainly controlled by the energy density (E_d) falling on the target. Therefore, it is a critical parameter to control.

On the other hand, the growth orientation is determined by the surface free energy G_s given by:

G_s = (γ_f − γ_s) + γ_i− γ_e (2.20)

where γf, γs, γi and γe are respectively the free surface energy for the film-air, substrate-air, film-substrate interfaces and the epitaxial energy gain [64]. The film actually grows along the orientation leading to a minimum value of G_s. The crystal plane with the smallest surface energy tends to be exposed at the surface of the crystal. Crystallisation includes nucleation and growth; the ratio of nucleation to growth controls the grain size distribution. The next paragraph gives more details about these steps.

2.3.4.2.2 Details of the processes All phase transformations, including thin film formation, involve the processes of nucleation and growth. During the earliest stages of film formation, a sufficient number of vapour atoms condense and establish a permanent residence on the substrate. Depending on their size, they will either grow in size or dissociate into smaller entities. After repeated exposures of the substrate to the incident vapour, a uniform distribution of small but highly mobile clusters or islands is observed [65]. If the nucleus size is greater than the critical size, the nuclei incorporate impinging atoms and subcritical clusters and grow in size while the island density rapidly saturates. The next stage involves merging of these islands by a coalescence phenomenon.

Then, a second layer can be formed. This the “layer by layer growth” (or Frank-van der Merwe growth) mechanism which is clearly favoured in PLD given the small size of critical nuclei (practically one atom) owing to the high degree of supersaturation. This results in two-dimensional nucleation of mono atomic height islands. Fig. 2.4 illustrates the three main basic modes of thin film growth.