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Abstract
Recently, metal-transparent-metallic-oxides (MTMO) grayscale photomasks fabricated by laser direct writing technology have been widely considered for its advantages, such as a low cost system, simple preparation techniques, high resolution, and good photo-thermal stability. Compared with In-MTMO, Sn-MTMO grayscale photomasks are more stable and cheaper, but its oxide components are more complex. In this paper, we proposed a simple Sn/SnOx multilayer oxidation model to simulate the laser-induced oxidation processes of fabricating Sn-MTMO grayscale photomasks based on the laser-induced Cabrera-Mott oxidation theory. Simulation results are consistent with the experimental results. This work will pave the way to study the fabrication improvements of Sn-MTMO grayscale photomasks on resolution and efficiency.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, the laser direct writing technology, which has been widely used to ablate, modify or synthesize materials as metal, semiconductor and organics, was applied to fabricate the Metal-Transparent-Metallic-Oxides (MTMO) grayscale photomasks based on the transparent conducting oxides (TCOs), such as In$_{2}$O$_{3}$, SnO$_{2}$ and TiO$_{2}$ [1–4]. The continuously variable transmittance is achieved by changing the laser power to adjust the proportion between oxides and metals. The simple and low-cost MTMO grayscale photomasks can be prepared by only two simple steps: metal film deposition and laser direct writing. In addition, the MTMO grayscale photomasks have good photo-thermal stability and high resolution [2]. Some diffractive optical elements (DOEs) like as micro-lens arrays [5] have been fabricated by using the MTMO grayscale photomasks photolithography. Compared to the In-MTMO grayscale photomasks, Sn-MTMO grayscale photomasks are more stable and cheaper. However, the multivalent oxidation products of tin may lead to a complex nonstoichiometric compound of SnO$_{x}$ including stable SnO, SnO$_{2}$ and unstable Sn$_{3}$O$_{4}$ and Sn$_{2}$O$_{3}$. So, the laser-induced oxidation processes of fabricating Sn-MTMO grayscale photomasks are more complicated than that of In-MTMO grayscale photomasks. In our previous research, the oxidation processes of In-MTMO grayscale photomasks have been studied and the fabrication mechanism has been interpreted by the laser-induced Cabrera-Mott oxidation theory [6]. But whether this theory is appropriate to the MTMO grayscale photomask with other metals are uncertain. In this paper, a laser-induced Sn/SnO$_{x}$ multilayer film on glass substrate model is built to calculate the oxidation processes of Sn-MTMO grayscale photomask and to verify the laser-induced Cabrera-Mott oxidation theory. The electric field distributions in the film are calculated by transfer-matrix-method (TMM), the real-time temperature fields are calculated by solving the heat transfer equations by using the finite element method (FEM), the laser-induced oxidation processes are simulated.
2. Laser-induced Sn/SnO$_{x}$ multilayer oxidation model
The complex tin oxides include tin monoxide (SnO), tin dioxide (SnO$_{2}$) and unstable nonstoichiometric Sn$_{3}$O$_{4}$ and Sn$_{2}$O$_{3}$. The formation energies of SnO and SnO$_{2}$ calculated from first principles are −3.21 eV and −6.29 eV, respectively [7]. It is obvious that SnO will be formed firstly when the tin film was oxidized. But SnO is unstable and will transform into more stable SnO$_{2}$ at the higher oxygen pressure or higher temperatures. Experimental results have shown that the SnO and SnO$_{2}$ coexist in the tin amorphous oxide coat [1,8]. Recently, first principle calculations have shown that SnO (001) plane and SnO$_{2}$ (101) plane remain parallel during the transformation process of SnO to SnO$_{2}$, and Sn$_{2}$O$_{3}$ and Sn$_{3}$O$_{4}$ are an interface combination between SnO and SnO$_{2}$ [9,10]. The tin oxides contain only two and four valence Sn ions [10]. However, it is difficult to simulate the complicate phase transformation for SnO to SnO$_{2}$ during the laser-induced oxidation processes. Here we take the SnO$_{x}$ to describe the complex tin oxide layers in realizing the simulation of laser-induced oxidation processes.
In experiment, the laser direct writer adopted a 532 nm laser (Spectra Physics, Millennia Pro 2i) was applied to write patterns onto the films by raster-scan, the typical scan speed is 50 $\mu m\cdot s^{-1}$ with a repetition rate of 250 Hz. In the process of mask fabrication, each processing points just adopt the single pulse exposure with a pulse width of 1 ms and powers ranging from 0 to 4 mW controlled by an acousto-optic modulator. The optical micrographics on the masks were taken using an Olympus BX-51 microscope. In the process of mask fabrication, the film sample was firstly placed in an X-Y-Z sample stage (PI, precision 2 nm) at the focal plane of the objective lens (Nikon, NA 0.90, 100$\times$) [1]. The refined tin films in experiment are realized by ventilating oxygen in the deposition chamber during the sputtering intervals [8], there will be a tin oxide shell on each tin grain to prevent the grain growing up, as shown in Fig. 1(a). So a 20 nm-thick tin film for preparing Sn-MTMO grayscale photomasks can be fabricated by four times deposition with the thickness of a single layer is controlled to be about 5 nm, and each layer contains 1-nm-thick oxide coating on the upper and lower surfaces. However, it is not feasible to simulate the oxidation processes of each grain, a simplified four-layer Sn/SnO$_{x}$ thin film model is built to calculate the laser-induced oxidation processes, as shown in Fig. 1(b). The polycrystalline Sn film are composed by the random orientation single crystal grains, the diameter of grains can be controlled to be about 20 nm as shown in Ref. [1] which is far less than the radius of the model, isotropic behavior can be obtained when the individual crystallites are oriented throughout the film plane with equal directional probability, so the film can be considered as a unified whole and the properties of the metal and oxide layers are assumed to be isotropic in the plane of film. The grain boundaries effect for the thermal transferring process is mainly from the boundary scattering for the phonon. The four-layer deposition film is very impact that the grains are interacting with its surrounding grains directly, the affect by grain boundaries to thermal conductivity should be small to be neglected. Both the tin and oxide layers are assumed to be isotropic in model for that the random orientation of grains in samples.
Fig. 1. (a) The schematic of a four-layer Sn/SnO$_{x}$ thin film on glass substrate. The pulsed laser wavelength is 532 nm, the laser-induced oxygen ions are adsorbed on the surface of grains; (b) The simplified calculation model of the four-layer Sn/SnO$_{x}$ thin film. The oxygen ions can diffuse into the surface of each layer almost simultaneously due to the grain boundary effect.
The migration rate of oxygen ions at the grain boundaries is far larger than that in grains for the grain boundary effect [11], so the oxygen ions can diffuse into the surfaces of each layer almost instantaneously and the oxidation processes in the four layers should work simultaneously. Under the laser irradiation, a laser-induced Mott potential will be generated along the oxide layers which urges the oxygen ions to migrate through the oxide layers from the surfaces into the tin layers, and leads to the rapid growth of oxide layers [6]. So, the growth rate of oxide layers is determined by the oxygen ions migration rate which can be described by the laser-induced Cabrera-Mott oxide theory as:
where $L$ is the thickness of the oxide, $a$ is the ionic jump distance with a value of 4.8 Å [9], $\nu$ is the vibrational frequency of the ions with a value of 10$^{12}$ s$^{-1}$ [12], $W$ is action energy for the jump with a value between 0.4 eV and 1.4 eV [13], $k$ is the Boltzmann’s constant, $T$ is the temperature, $q$ is the charge of oxygen ions, $V'$ is the laser-induced Mott potential. The Mott potential can be achieved through the energy level of Sn/SnO$_{x}$ system, as shown in Fig. 2.Fig. 2. Schematic of the energy level diagram of Sn/SnO$_{x}$ system. The work function of Sn is 4.42 eV, and that of SnOx surface is 5.7 eV. So the natural Mott potential across the SnO$_{x}$ is 1.28 eV.
The initial natural Mott potential is the difference between the work function of Sn and oxide surface as defined by the famous Cabrera-Mott theory [14]. The work function of Sn is 4.42eV [15] and that of the stoichiometric surface of SnO$_{x}$ is 5.7 eV [16]. So the initial natural Mott potential $V_{0}$ across the SnO$_{x}$ is 1.28 eV. But this natural Mott potential will disappear once the oxide thickness is large enough to prevent the electrons jumping along the oxide layers [17]. With the help of laser irradiation, the electrons can be excited to jump again and in addition the energetic electrons can transfer into the surface adsorbed oxygen molecules to form the oxygen ions [12]. Thus the Mott potential will appear again which is called the laser-induced Mott potential $V'$ . The surface charge density $\sigma$ is proportional to the laser intensity irradiated on the sample surface $I$ for the surface oxygen ions are reproduced by the laser irradiation. So the $V'$ is also supposed to be proportional to $I$. The proportional coefficient is the ratio of the natural Mott potential $V_{0}$ to the maximum laser intensity $I_{0}$ at the threshold laser power. So, the laser induced Mott potential $V'$ can be written as:
The spatial distributions of light intensity for Gaussian laser pulses in a cylindrical coordinate system can approximately be regarded to: where $P$ is the pulsed laser power, $\omega$ is the waist of the Gaussian beam which is 350 nm in Ref. [8], $r$ is the radial coordinate. In experiment, the threshold laser power $P_0$ for fabricating the Sn-MTMO grayscale photomasks is about 1.0 mW.3. Simulation results and discussions
3.1 Temperature fields
From the Eq. (1), we can see that the oxidation rate is also related to the temperature $T$ which can be solved by the typical heat conduction equation because that the laser pulse duration is much longer than the electron-lattice coupling time. During the thermal transmission, the transferred energy between the metal film and the substrate is conserved. For the energy losses by heat convection and thermal radiation are small enough to be neglected, the outside boundaries can be simplified to be insulated. The partial differential equation is:
where $\rho$ is the mass density, $c$ is the specific heat capacity, $Q$ is the absorbed laser power density which is the function of electric fields $E$ by: where $\sigma$ is the conductivity of tin. For the thickness of Sn/SnO$_{x}$ film is 20 nm, there will be multiple transmission and refraction at the interfaces. The interference electric field distributions in the film can be achieved by the transfer matrix method (TMM) [18]. Depending on the Eq. 3–5, the temperature field distributions in the thin film can be calculated by using the time-domain Finite-Difference Time-Domain (FDTD) method in COMSOL Multiphysics software. The thermal properties of materials are shown in Table 1.
Table 1. Thermal properties of materials in model [19]
3.2 Simulation results of the laser-induced oxidation processes
The pulsed laser-induced oxidation processes are simulated with cylindrical coordinates in our 2D axisymmetric model. The thickness increases of the oxide layers are realized by the moving of boundaries and the rates are determined by the Eq. (1). Figure 3 shows the laser-induced oxidation conditions of Sn/SnO$_{x}$ film at the laser power of 1 mW to 4 mW for the time of 80 ns and 1 ms, respectively. The background colors represent the temperature fields. Grayscale bars with different gray levels were fabricated in experiments. For the grayscale bars were fabricated by a raster scan mode with a scanning step of 150 nm at the same laser power, the oxidation degrees in the region with a diameter of 150 nm can reflect the oxidation degree of the grayscale bars. Therefore, Fig. 3 just shows the simulation results of the laser-induced oxidation within the radius of 75 nm.
Fig. 3. Simulation results of the laser-induced oxidation conditions of Sn/SnO$_{x}$ film at the laser power of 1 mW to 4 mW for the pulse duration of 80 ns and 1 ms, respectively. The background colors represent the temperature fields. The SnO$_{x}$ layers grow up as the laser power and pulse duration increase.
Figure 3(a)-(d) show the laser-induced oxidation degrees of Sn/SnO$_{x}$ film with the laser power changes from 1 mW to 4 mW for the pulse duration of 80 ns while the situation for 1 ms are shown in Fig. 3(e)-(f). It can be seen that the thicknesses of SnO$_{x}$ layers increase as the laser power increases. But the oxidation degrees are significant different for 80 ns and 1 ms with the same laser power. From Fig. 3(a) and (e), we can see that the thicknesses of SnO$_{x}$ layers do not increase at the laser power of 1 mW for the pulse duration of 80 ns, but become thicker for the pulse duration of 1 ms. The reason of this phenomenon is that the absorbed laser energies will lead to the temperatures of film increase which cause a thermal oxidation process. However, the thermal oxidation rate is far slower than the laser-induced Mott potential oxidation rate, so the thicknesses of SnO$_{x}$ layers are almost unchanged in Fig. 3(a) for the short duration of 80 ns and increase slightly in Fig. 3(e) for the long duration of 1 ms. The SnO$_{x}$ layers grow up slowly for pulse duration of 80 ns and quickly for the pulse duration of 1 ms at the laser power of 2 mW, as shown in Fig. 3(b) and (f). This is attributed to that when the laser power beyond the threshold of 1 mW, 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. The laser-induced oxidation effect dominate the oxidation process that causes a greatly increase of oxidation degrees as laser power from 1 mW to 2 mW for the pulse duration of 1 ms. However, the SnO$_{x}$ layers have no apparent change for too short oxidation time of 80 ns, as shown Fig. 3(b). The Sn films are almost oxidized into SnO$_{x}$ when the laser power comes to 3 mW and completely oxidized at 4 mW for pulse duration of 1 ms, as shown in Fig. 3(g) and (h). But meanwhile, for the pulse duration of 80 ns, the changes of oxidation degrees are mainly occur at the laser power of 2-4 mW. For a direct comparison between our simulation results with the experiment results in Ref. [7], the oxidation degrees at laser power of 1-4 mW for pulse duration of 80 ns and 1 ms are shown in Fig. 4. The oxidation degree is described by the ratio of the oxide volumes to the whole film volumes in each grayscale bar that are fabricated by a scanning step of 150 nm. Therefore, the oxidation degrees in simulation can be the ratio of oxide region to the film region with the radius of 75nm.
Fig. 4. The oxidation degrees in experiment and simulation at laser power of 1-4 mW for pulse duration of 80 ns and 1 ms. The oxidation degrees are the ratio of oxide region to the film with the radius of 75nm. The simulation results are in agreement with the experiment results.
The red diamond refers to the oxidation degrees by simulation with pulse duration of 1 ms while the blue triangle refers to that of 80 ns. The red square refers to the experimental oxidation degrees derived from Ref. [7] with pulse duration of 1 ms while the blue circle refers to that of 80 ns. We can see that the simulation results are in good agreement with the experiment results, especially for the pulse duration of 1 ms. But there are some differences for pulse duration of 80 ns during the laser power from 1.5 mW to 2.5 mW. This is mainly due to that the evolution processes of SnO to SnO$_{2}$ are not considered in simulation. For the low laser power and short oxidation time, the amorphous SnO should be the main oxide in the SnO$_{x}$ layers that were not detected in experiment. So the oxidation degrees have no change before the laser power 2 mW for pulse duration of 80 ns in experiment. As the laser power increase, the laser-induced oxidation effect not only increases the oxidation rate, but also accelerates the transformation of nonstoichiometric SnO phase to SnO$_{2}$ phase. So the oxidation degree has a significant change in experiment around the laser power of 2.5 mW for pulse duration of 80 ns. Besides, most simulation results are consistent with the experiment results indicate that our laser-induced oxidation model can simulate the laser-induced oxidation processes in fabricating Sn-MTMO grayscale photomasks.
4. Conclusion
In conclusion, we proposed a simplified Sn-SnO$_{x}$ multilayer oxidation model to simulate the laser-induced oxidation process in fabricating Sn-MTMO grayscale photomasks. The oxidation processes are calculated based on the laser-induced Cabrera-Mott oxidation theory. Simulation results are consistent with the experimental results indicate that our laser-induced oxidation model can simulate the laser-induced oxidation processes in fabricating Sn-MTMO grayscale photomasks. This work will pave the way to study the fabrication improvements of Sn-MTMO grayscale photomasks on the resolution and efficiency and verify the validity of the laser-induced Cabrera-Mott oxidation theory.
Funding
Natural Science Foundation of Shandong Province (ZR2017MF059); National Natural Science Foundation of China (11144007, 11274188, 51472174); China Postdoctoral Science Foundation (2017M612182).
Disclosures
The authors declare no conflicts of interest.
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