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Abstract
An axisymmetric model of 50 nm titanium thin film on glass substrate is proposed to study the mechanism of pulsed-laser-induced oxidation of titanium. The oxidation rate is determined by the oxygen ions migration rate, which is significantly influenced by the laser-induced Mott potential and temperature. The oxidation processes are calculated by finite-difference time-domain method. The simulation results are in good agreement with experiment results, which verify that the laser-induced Cabrera-Mott oxidation theory is the mechanism of laser-induced oxidation of titanium. This work is beneficial to study the improvements for fabricating TiO$_{2}$ nanostructured materials on the resolution and efficiency.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Titanium dioxides(TiO$_{2}$) have been widely applied in industry for glass, ceramic, electronic components, cosmetics, painting dye, medicine and food over the past several decades [1]. In recent years, thanks to the unique physicochemical properties, TiO$_{2}$ nanostructured materials have been extensively studied on the fields of photocatalytic degradation of pollutants [2], photocatalytic hydrogen generation from water splitting [3], photocatalytic sterilization [4], photo-anodes in dye-sensitized solar cells (DSSCs) [5], gas sensors, chemical oxygen demand sensors and biosensors [6]. TiO$_{2}$ nanostructured materials can be fabricated by many methods: magnetron sputtering, plasma deposition, laser deposition, vapor deposition, thermal oxidation and laser-induced oxidation. In particular, the laser-induced oxidation method attracted lots of attentions for some prominent advantages: significant cost reduction, high fabrication resolution with wavelength scale, high precision spatial-temporal control, easy to control the oxidation degree and high repeatability. The laser-induced surface oxidation of thick titanium film for color marking were studied over many years [7–10]. The resolution of laser-induced coloring could be increased by single spot oxidation instead of by laser scanning oxidation [10]. TiO$_{2}$ line structures written on an ultrathin titanium films (6-15 nm) by CW-laser were narrower than the laser spots [11]. A simple two-step method with laser direct writing of titanium thin film and then etching it was proposed to fabricate TiO$_{2}$ micro-devices [12]. Complex patterns of TiO$_{2}$ nanostructures such as Fresnel lens, gear structure, suspended beam were fabricated. Highly ordered TiO$_{2}$ nanoribbons with seamless connection were also fabricated by laser-induced oxidation and etching [13]. The ribbon width can be precisely controlled by adjusting laser power, the minimum width can reach to 150 nm. Mostly the fabrication mechanisms are explained as the thermal oxidation by the absorbed laser energies [14,15]. L. Lavisse et al. used the pulsed laser with duration of 50 ns and wavelength of 1.06 µm to study the very early stage of laser-induced oxidation [14]. They think that the thermal induced by the pulsed laser can lead to a molten pool on titanium where the oxide layer was formed due to the oxygen diffusion. E. A. Shakhno et al. take the Wager’s theory to explain the pulsed laser induced titanium oxidation that allowing for the change of the film’s absorptivity during irradiation [15]. However, the thermal oxidation rate is not enough to cause the titanium thin film to be full oxidized within a laser pulse on microsecond time scale [16]. Recently, the laser-induced Cabrera-Mott oxidation theory was proposed to analyze the fabrication mechanism of In-MTMOs by laser directing writing [17]. In this paper, an axial-symmetric model of 50 nm titanium thin film on glass substrate is proposed to study the mechanism of the laser-induced fabrication of TiO$_{2}$ nanostructured materials. The simulations of oxidation processes realized by the boundaries moving are calculated based on the Transfer Matrix Method (TMM) and Finite-Difference Time-Domain (FDTD) method. The simulation results are in good agreement with the experimental results.
2. Model of laser-induced titanium oxidation
The simulation model is based on the experiment in Ref. [12]. A 50 nm-thick titanium film was deposited on glass substrates with thickness of 0.17 mm by electron beam evaporation. The 532 nm-wavelength pulsed laser beam formed by an acousto-optic modulator is normal incidence on the sample to write the structures by raster-scan with a typical scan speed of 50 µm $\cdot$ s$^{-1}$ and a repetition rate of 250 Hz. The laser duration is 1 ms and the focus point diameter is about 350 nm, so the laser energy densities are ranging from 0 to 150 J $\cdot$ mm$^{-2}$ with the laser power changing from 0 to 15 mW. The sample are fabricated in air with the ambient temperature of 24 $^{\circ }$C and normal atmospheric pressure. The accurate calculations of temperature distribution and oxidation process of each grains in titanium film are complicated and time-consuming. For simplicity, we build a two-layer model of TiO$_{x}$/Ti thin film on glass substrate to describe the actual sample, as shown in Fig. 1. Both the metal and oxide layers are assumed to be isotropic in simulation. The increase of oxide layer thickness is realized by the moving of TiO$_{x}$/Ti boundary. Because the system is axial-symmetric, we apply the cylindrical coordinate to simulate the oxide process. The ordinate origin is set at the central of the laser on the surface of film.
Fig. 1. Schematic view of the simulation model of the laser induced oxidation of titanium. The laser beam is normal incidence on the titanium film surface. The oxide layer and titanium layer are assumed to be isotropic.
Based on the Cabrera-Mott theory, the oxidation of thin titanium film is determined by the interstitial ions migration [18,19]. A few angstroms-thick TiO$_{2}$ layer will generate once the titanium film is exposed to the air. For further oxidation, the electrons will jump from titanium to the oxide surface to ionize the adsorbed oxygen molecules by tunneling and thermionic emission effects until an equal Fermi level in titanium layer, TiO$_{2}$ layer and film surface is set up, as shown in Fig. 2(b). As a result, a uniform electric field is created in the TiO$_{2}$ layer that drives the oxygen ions migration and causes the TiO$_{2}$ layers thickness increasing. However, the natural oxidation rate will decrease sharply because of the rapidly declining electric field as the TiO$_{2}$ layer growing. The oxide thickness no longer increases once it reaches a threshold and the Mott potential will disappear. So there is a 3-5 nm thick TiO$_{2}$ layer on the film surface for natural oxidation [20].
Fig. 2. (a) The electron energy-level diagrams of the Ti/TiO$_{2}$/TiO$_{2}$ surface; (b) An equilibrium energy-level diagrams with the natural Mott potential by the electrons jump from titanium to the TiO$_{2}$ surface by tunneling and thermionic emission effects; (c) An equilibrium energy-level diagrams with laser-induced Mott potential under the laser irradiation.
Under the irradiation of light, the electrons are excited to higher energy states to keep jumping. In addition, the surface adsorbed oxygen molecules are more probably to decompose into oxygen ions [21]. The Mott potential will be regenerated and the oxide layer continues to grow again. For the diffusion rate of oxygen ions is far less than the migration rate induced by the laser-induced Mott potential, the oxide growth rate is approximately equal to the oxygen ions migration rate:
where $\nu$ is the vibrational frequency of ions, $k$ is Boltzmann’s constant, $q$ is the electric charge of oxygen ions. The surface jump distance of oxygen ions $a'$ is about 1 Å [22], the activation energy for hopping from TiO$_{2}$ surface to subsurface $W$ is 1.2 eV [23], $T$ is the temperature and $V_{M}$ is the Mott potential across the oxide layer. The natural Mott potential $V_{0}$ is the difference between the work function of titanium and TiO$_{2}$ surface, as shown in Fig. 2. The work function is 4.33 eV for titanium [24] and 4.82 eV for TiO$_{2}$ surface [25], so the value of $V_{0}$ is 0.49 eV. The laser-induced Mott potential $V'$ is determined by the density of surface charges $\sigma$ which is proportional to the concentration of surface oxygen ions. Because the surface oxygen ions are reproduced by the laser irradiation, $\sigma$ is also proportional to the laser intensity $I$ that can be express as: where $P$ is the pulsed laser power, $\omega$ is the waist of the Gaussian beam, $r$ is the radial coordinate. Therefore, $V'$ should also be proportional to $I$.The proportional coefficient can be analyzed from the experiment. For the laser-induced oxidation of titanium film, there is a threshold laser power that the oxidation process will not happen when the laser power is lower than it. In Ref. [12], the threshold laser power is measured to be 3 mW. This means to that the maximum laser-induced Mott potential $V'$ under the threshold laser power should be equal to natural Mott potential $V_{0}$. So, the proportional coefficient can be written as $V_{0}/I_{0}$, where $I_{0}$ is the maximum laser intensity for the threshold laser power. The laser-induced Mott potential $V'$ can be expressed as $V'=(V_{0}/I_{0}) \cdot I$.
3. Simulation results and discussions
3.1 Temperature fields
The oxidation rate is related closely with the temperature fields in the film which are induced by the absorbed light due to the inverse Bremsstrahlung effect. For the laser heating time, i.e. the laser pulse duration, is 1 ms which is much larger than electron-lattice relaxation time, so a typical heat conduction equation is applied to calculate the temperature. The interference due to the multiple transmission and refraction at the interfaces should be considered for the film thickness is only 50 nm. The transfer matrix methods (TMM) can be used to calculate the distribution of electromagnetic fields $E$ in the film [26], the equations are programmed in MATLAB. The absorbed laser power density $Q=1/2\sigma \left | E \right |^{2}$. The temperature field distributions in the thin film are calculated by using the time-domain Finite-Difference Time-Domain (FDTD) method in COMSOL Multiphysics based on the typical heat conduction difference equation:
where $T$ is the temperature, $\rho$ is the mass density, $c$ is the specific heat capacity, $Q$ is the absorbed laser power density. The energy losses come mainly from the heat convection and thermal radiation which are calculated by setting the boundary conditions in COMSOL Multiphysics. The thermal properties of materials are shown in Table 1.
Table 1. Thermal properties of materials adopted in model$^a$
3.2 Simulation of oxidation process
The oxidation processes of titanium thin film by laser direct writing are simulated through the growth of oxide layer which can be realized by moving the boundaries of oxide layer. The lower surface of oxide layer refers to the oxidation degree of titanium whose moving rate is determined by the Eq. (1). The upper surface of oxide layer refers to the thickness increase of the film due to the oxidation. The titanium oxidation kinetics will go through the three-layer process: TiO layer, TiO$_{x}$ layer and TiO$_{2}$ layer [26]. In experiment, the 50-nm-thick titanium film will begin to be oxidized at the laser power of 3 mW. However, the heights of grating lines will increase by 10 nm with laser power from 3 mW to 4 mW. This abrupt change should be attributed to that the titanium film has been oxidized into the three-layer titanium oxide in the region of laser irradiation. Here, the TiO should be in the majority of the three-layer titanium oxide because that the Raman spectra peak of TiO$_{2}$ just appear at 4 mW. The height increment is 10 nm when the 50-nm-thick film is mostly oxidized into TiO, so the moving rate of the upper surface of TiO layer is set to be 0.2 times of Eq. (1). Figure 3(a) (b) and (c) show the simulation results of the laser-induced oxidation degree at the time of 1 ms with the laser power of 2 mW, 3 mW and 4 mW, respectively. It can be seen that the titanium film has no change at the laser power of 2 mW. The oxide layer begins to grow at the laser power of 3 mW, the radius of oxide region is about 150 nm and the maximum thickness of oxide layer is about 8 nm. When the laser power is 4 mW, the radius of oxide layer which is about 250 nm in experiment reaches to about 260 nm. The simulation results are consistent with the experiment results which provides a well support to our theoretical analysis.
Fig. 3. The simulation results of laser-induced oxidation degrees of titanium films with the laser power changes from 2 mW to 15 mW at the time of 1 ms. The simulation applied the cylindrical coordinate system. The colors refer to the values of temperature fields. The oxidation region grows as the laser power increases.
In Ref. [12], the Raman spectra peak of TiO$_{2}$ increases as the laser power increases and reaches the maximum at the laser power of 15 mW. It means that the TiO layer will be oxidized into TiO$_{2}$ layer with the increase of laser power. This transformation rate is also determined by the oxygen ions migration rate of Eq. (1). Considering the lattice constant $a$ is 0.22 nm for TiO and 0.234 nm for TiO$_{2}$ [27], the heights increase rate for the process of TiO layer to TiO$_{2}$ layer is assumed to be 0.014 times of Eq. (1). Figure 3(d)-(n) show the laser-induced oxidation degrees of titanium film at the time of 1 ms with the laser power changes from 5 mW to 15 mW. The background colors represent the values of temperature fields. It can be seen that the value of temperature field decreases along the radial direction due to the Gaussian pulsed laser beam and the heat transfer inside the film. As the laser power increases, the completely oxidized regions are enlarged. This is attributed to that the laser-induced Mott potentials and the temperature fields which are the significant influencing factors for oxygen ions migration rate increase with the laser power. For a more direct comparison between our simulation results with the experiment results, the heights and widths of the grating lines for laser power of 4-15 mW are shown in Fig. 4.
Fig. 4. The widths (hollow blue) and heights (solid red) of the lines written by laser-induced oxidation for the simulation results (square) and experiment results (circle) in Ref. [12]. The simulation results are in agreement with the experiment results.
The blue hollow squares represent the simulation results of widths, the blue hollow circles represent the experiment results of widths in Ref. 12, the red solid squares represent the simulation results of heights and the red solid circles represent the experiment results of heights. The widths and heights of grating lines in experiment are measured by atomic force microscopy. It can be seen that the simulation results are in good agreement with the experiment results except that there are relatively large differences for the heights at the laser powers of 14 mW and 15 mW. It is mainly due to the saltation of heights in experiment between 13 mW and 14 mW. This saltation maybe attributed to the molten and aggregation of the titanium film when the laser power is larger than 14 mW. But the consistencies of the most simulation results and experiment results verify that the laser-induced Cabrera-Mott oxidation theory is the fabrication mechanism of the laser-induced TiO$_{2}$ nanostructured materials.
4. Conclusion
In conclusion, we proposed an axisymmetric model of 50 nm titanium film on glass substrate to study the mechanism of pulsed-laser-induced oxidation of titanium. The oxidation rate is determined by the oxygen ions migration rate which is influenced significantly by the laser-induced Mott potential and temperature. The consistencies of the simulation results and experiment results verify that the laser-induced Cabrera-Mott oxidation theory is the mechanism of laser-induced oxidation of titanium. Our work is beneficial to study the improvements for fabricating TiO$_{2}$ nanostructured materials on the resolution and efficiency.
Funding
China Postdoctoral Science Foundation (2017M612182); National Natural Science Foundation of China (11144007, 11274188, 51472174); Natural Science Foundation of Shandong Province (ZR2017MF059).
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