Numerical analysis of the sub-wavelength fabrication of MTMO grayscale photomasks by direct laser writing

Feng Xia; Xinzheng Zhang; Meng Wang; Sanming Yi; Qian Liu; Jingjun Xu

doi:10.1364/OE.22.016889

1. Introduction

Grayscale photomasks with continuous-tone transmittance are widely applied in the fabrication of three dimensional micro-optical-elements (MOEs) and of micro-electro-mechanical systems (MEMS) [1, 2]. The mainstream techniques of making grayscale photomasks are based on chromium film on glass (COG) [3] and high-energy-beam-sensitive (HEBS) glass [4]. The former needs many steps such as film deposition, lithography, etching, resist striping, etc., while the latter processes a very complicated material system used for electron beam, both of which are expensive that limits their industrial applications. Metal-transparent-metallic-oxides (MTMO) were first proposed and experimentally demonstrated by Chapman et al. [5] and subsequently explored as grayscale masks [6–13]. In a parallel research Qian Liu et al. have been exploring grayscale implementations [14–18]. The continuously variable transmittance was achieved by changing the laser power to adjust the ratio between the transparent metallic oxides and the metal. MTMO grayscale photomasks have many advantages such as continuous-tone gray levels, high resolution, simple preparation techniques, simple and low cost material system and good photothermal stability. The fabrication principle is that the absorbed laser energies cause the metal oxidization. However, it is difficult to measure the oxidation process of a metallic nano-film from nanosecond to microsecond time scale. Therefore, numerical simulation of the absorbed laser power density and the temperature field is meaningful to understand the oxidation process and the mechanism of the fabrication of MTMO grayscale photomasks.

In this paper, a laser-induced melt-oxidization (LIMO) model by laser direct writing on an indium film with a glass substrate is proposed. The optical properties of the sample are calculated based on the Airy summation. Temperature fields of the sample are simulated by solving the heat transfer equations with the Finite-Difference Time-Domain (FDTD) method. The fabrication area of the indium film is studied based on the absorbed laser power density distribution in molten indium films. We show that sub-wavelength fabrication of the MTMO grayscale photomasks can be realized based on our LIMO model. The calculated fabrication size under a laser power of 6 - 8 mW is consistent with the experimental results.

2. Model of light-induced melt-oxidation

For an indium film on a hot plate in air, there is an upper limiting oxide thickness X_L below a critical temperature T_c which can be interpreted by the Mott-Cabrare theory based on the efficiency of tunneling and thermionic emission [15, 16]. The critical temperature was the melting point and the limiting thickness was about 4.0 nm [17]. However, with the laser irradiation, the electrons can be excited to a higher energy state to keep on the tunneling and thermionic emission effects, the oxidation process will be accelerated and the oxide thickness will be beyond the limit X_L.. According to the experiment results in Ref. [19], under a given high laser power, the laser-induced oxidation rate of metal films in molten state is significantly larger than that in solid state due to the increased diffusion of ions. In this paper, we study only the light-induced oxidation above melting temperature. In molten area, indium will be oxidized when the absorbed laser power density Q is above one threshold value Q_th, which can be calculated from experimental results. Therefore, the temperature fields decide the oxidation environment and the absorbed power density dominates the oxidation distribution.

The simulation is based on Ref. [16], a 20 nm-thick indium film was obtained by four-layer deposition with a single-layer thickness of 5.0 nm and with grain sizes around 10-20 nm. Therefore, we consider a thin indium film coated on a glass substrate as the sample. For preparing MTMO photomasks, a pulsed laser beam with a certain temporal and spatial distribution is incident normally on the sample from the metal film side, as shown in Fig. 1.For simplicity, the following assumptions about the metal film and the laser beam are made:

Fig. 1 Schematic view of the fabrication of the MTMO grayscale photomasks. The metal and substrate are both isotropic. The laser beam is normally incident onto the air-metal interface.

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The metal film surface is flat, metal and substrate are both isotropic.
The properties of solid and liquid metal films are both independent of temperature.
Chemical or other thermodynamic reactions are not taken into account.

2.1 The absorbed laser energy

Due to the inverse Bremsstrahlung effect, the laser energy will be absorbed by free electrons. Because the thickness of the metal film is 20 nm, the interference due to the multiple transmission and refraction at the interfaces of the metal-air and metal-substrate has been considered into our model. The reflectivity $R$ and the transmissivity $T^{'}$ of this two-layer system are calculated by Airy summation [5]. A precise energy distribution can be calculated from Poynting vector under the consideration of Airy summation [6]. Here we adopt a simple approximation of the energy distribution by the Beer–Lambert law, then the effective absorption coefficient α can be achieved:

α = - \frac{1}{d} \ln (\frac{T^{'}}{1 - R})

By using of the effective absorption coefficient, the total absorbed energy is consistent with that by the method in Ref. [6].

Usually, the spatial and temporal distributions of laser pulses can approximately be regarded to have a Gaussian shape, which can be expressed in a cylindrical coordinate system by the equation [20]:

I (r, t) = \frac{2 P}{π w^{2}} \exp (- 2 \frac{r^{2}}{w^{2}}) exp (- \frac{4 t^{2} l n 2}{τ_{p}^{2}})

where

P

is the peak power of the pulsed laser at the beam center, and

w

is the waist of the Gaussian beam.

In the experiment, the laser pulses were formed by an acousto-optic modulator, so the temporal distribution shape can be viewed as rectangular [21]:

I (r, t) = {\begin{cases} 0, t < 0 \\ \frac{2 P}{π w^{2}} \exp (- 2 \frac{r^{2}}{w^{2}}), 0 \leq t \leq τ_{p} \\ 0, t > τ_{p} \end{cases}

2.2 Heat transfer and boundary equations

The heat energy diffuses through the electron subsystem and transfers to the lattices by electron–lattice coupling, which is described by the famous two-temperature diffusion model. In this system, the pulse duration $τ_{p}$ is 1000 ns and the condition $τ_{e} ≪ τ_{i} ≪ τ_{p}$ is fulfilled, where $τ_{e}$ is the electron cooling time and $τ_{i}$ is the lattice heating time. Therefore, the electron temperature $T_{e}$ and the lattice temperature $T_{i}$ will be equal to the metal film temperature $T_{}$ . In this case, two-temperature diffusion model can be replaced by the typical heat conduction equation [20]:

ρ [c + Δ H_{m} δ [T - T_{m}]] \frac{\partial T}{\partial t} = \nabla (k \nabla T) + (1 - R) α I e^{- α z}

where

ρ

is the mass density,

c

is the specific heat capacity,

T_{m}

is the melting temperature,

Δ H_{m}

is the latent heat of fusion and

δ

is the Dirac delta function. The absorbed laser power density Q is defined as

(1 - R) α I e^{- α z}

.

Since there is axial symmetry along the light axis, the heat conduction equations are formulated in cylindrical coordinates:

{\begin{cases} ρ_{1} [c_{1} + Δ H_{m} δ [T - T_{m}]] \frac{\partial T_{1}}{\partial t} = k_{1} (\frac{\partial^{2} T_{1}}{\partial r^{2}} + \frac{1}{r} \frac{\partial T_{1}}{\partial r} + \frac{\partial^{2} T_{1}}{\partial z^{2}}) + (1 - R) α_{1} I (r, t) e^{- α_{1} z}, z \leq d \\ ρ_{2} c_{2} \frac{\partial T_{2}}{\partial t} = k_{1} (\frac{\partial^{2} T_{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial T_{2}}{\partial r} + \frac{\partial^{2} T_{2}}{\partial z^{2}}), z > d \end{cases}

where the subscript 1 refers to the metal film while the subscript 2 refers to the substrate.

Because the process takes only 1 μs, the energy losses by heat convection and thermal radiation are small enough to be neglected. Hence the outside boundaries can be simplified to be insulated which can be expressed as:

\begin{array}{l} k_{i} \frac{\partial T_{i}}{\partial r} = h_{i} (T_{i} - T_{0}) \approx 0 \\ k_{i} \frac{\partial T_{i}}{\partial z} = h_{i} (T_{i} - T_{0}) \approx 0 \end{array}

where

h_{i}

is the convective heat transfer coefficient.

During the thermal transmission, the transferred energy between the metal film and the substrate is conserved. Thus, the boundary conditions of this interface are:

\begin{array}{l} {k_{1} \frac{\partial T_{1}}{\partial z} |}_{z = d} = {k_{2} \frac{\partial T_{2}}{\partial z} |}_{z = d} \\ {T_{1} |}_{z = d} = {T_{2} |}_{z = d} \end{array}

The temporal-spatial distribution of temperature fields of the sample can be calculated based on this model.

3. Numerical results and discussions

3.1 Temporal and spatial temperature fields

To make a direct comparison with the MTMO grayscale photomask experiments which were conducted in Ref. [16], a 532 nm laser was assumed with a spot size of 350 nm and a pulse duration of 1 μs. The laser power varied from 2.5 to 10.0 mW. The thicknesses of the indium film and the substrate were 20 nm and 2480 nm, respectively.

The radius of the simulation model is 2500 nm. The thermal and optical parameters of the model are given in Table 1. The reflection coefficient R and absorption coefficient α calculated by Eq. (4) and Eq. (5) are 0.7902 and 5.6681 × 10⁷ m⁻¹, respectively. The melting point of the thin film is lower than the corresponding bulk, the melting temperature for a 20 nm In film is 419.6 K [22].

Table 1. Thermal and optical parameters adopted in the model [23–25]

View Table

Figure 2 shows the temperature fields after 1 μs heating time (i.e., the pulse duration) with a laser power of 5.0 mW and after another 1 μs self-cooling time, respectively. For visibility, the scale of the indium thin film is adjusted so that it is different from that of the substrate at z direction. The indium film and the substrate are illustrated in two separated sub-pictures. The energy concentrates in the irradiation area of the indium film when the sample is heated. But after self-cooling, the temperature of glass is higher than In film near the lower interface.

Fig. 2 Temperature fields of the sample after (a) 1 μs the heating time and (b) after another 1 μs self-cooling time with a laser power of 5.0 mW. For visibility, the indium film (upper pictures) and the substrate (lower pictures) are drawn separately with different scales at z direction.

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The temperature evolution of the indium film at the coordinate point zero with a laser power of 4.0 mW, 4.2 mW, 4.5 mW and 5.0 mW are shown in Fig. 3(a). The temperature changes drastically in the beginning of heating, just after fully melting and in the beginning of self-cooling, but it changes slowly during most of the laser pulse duration.

Fig. 3 (a) Temperature evolution at the coordinate of zero point with a laser power of 4.0, 4.2, 4.5, 5.0 mW. (b) Temperature distribution along the radial direction at the interface between the indium film and the substrate and the time of 1 μs with a laser power of 4.0, 5.0, 6.0, 8.0, 10.0 mW.

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3.2 Sub-wavelength fabrication based on the LIMO model

Figure 3(b) shows the temperature distribution along the radial direction at the interface between the indium film and the substrate and the time of 1 μs with the laser power of 4.0, 5.0, 6.0, 8.0, 10.0 mW, from which one can know the melting area. The temperature fields show that the indium film can be molten when the laser power is higher than 4.0 mW, and the oxidation environments should be similar by then [19]. The major factor that determines the oxidation rate is the laser power distribution which can be represented by the absorbed power density Q. Because the accurate laser-induced oxidation equations based on the oxidation theories has not been proposed, we can calculate the threshold value Q_th from experimental results.

The resolution is defined as the smallest spot with the maximum transparency for a single laser pulse shot. In our MTMO preparation, complex grayscale patterns could be fabricated by the raster scan mode. In order to get the maximum transparency for a relatively large area (e.g., 1 μm²) under the given maximum laser power, the scanning step d should be as small as possible. It was found that a scanning step of 200 nm can ensure the complete oxidation of the corresponding area under a laser power of 10.0 mW, then the resolution can be calculated as $\sqrt{2} d$ , i.e., 282 nm.

Figure 4 shows the distribution of absorbed power density in local indium films with a laser power of 4.0, 6.0, 8.0, 10.0 mW. As assumed that the indium film will be fully oxidized when Q is greater than Q_th, one can get Q_th is about 2.17 × 10¹⁷ W/m³ from Fig. 4(d) with the resolution is 282 nm under the power of 10.0 mW. The threshold is assumed to be same with different laser power when the laser power is higher than 4.0 mW, i.e., the temperature is above the melting temperature. The black curves represent the threshold contours under different laser powers, which show the boundary between indium and indium oxides.

Fig. 4 The absorbed laser power density in local indium films with a laser power of (a) 4.0 mW, (b) 6.0 mW, (c) 8.0 mW, (d) 10.0 mW. The black curves are the threshold of absorbed laser power density for oxidation. The threshold value is 2.17 × 10¹⁷ W/m³.

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According to the definition of the resolution, the fabrication size should be larger than the resolution at a given laser size. The diameter of oxide region at the half thickness of the film will be more appropriate to represent the fabrication size. The radii of the fabrication region with a laser power of 6.0 mW, 7.0 mW and 8.0 mW are 143 nm, 151 nm and 158 nm, respectively. The average radius of the fabrication region is 151 nm, which coincides with the experimental average diameter of the fabrication region of about 300 nm in Ref. [26].

From Fig. 4, one can see that the absolute change of the absorbed power density at the edge of the fabrication area is much smaller than that at the center while increasing the laser power, i.e., the absolute change of the oxidation rate at the edge is much smaller than that at the center. Under high laser power, the oxidation rate will increase dramatically from the edge to the center. Therefore, the resolution can be very small when the laser pulse is very short. As demonstrated by Chapman et al., 45 nm structures were created with high laser power and 4 ns pulse duration by interference lithography [11]. In our analysis, we have not considered the dynamic behavior of the oxidation. Actually, the oxidation process can be so rapid that the film may become transparent during the duration of a laser pulse with high laser power. Then the absorbed power goes down limiting the temperature even for long pulses, and thus limiting the final transparency. At lower power, the reduced resolution might be caused by a thermal resist effect.

According to our simulation, a threshold laser power of 4.0 mW is found, i.e., the indium film will not melt below this laser power. However, our experiments showed that the indium film could be oxidized even when the laser power was lower than 4.0 mW. That might be from the increase of upper limiting oxide thickness X_L due to the excitation of electrons, which is under our further study and does not affect the validity of our current model above threshold laser power. In other words, the indium film is not fully oxidized at a laser power lower than the threshold power, the oxidation layer will become thicker with the increase of the pulse duration. Moreover, a saturation point at a certain optical density was always produced for a given specific laser power even for a much longer pulse duration [19], which suggests the indium film will not be oxidized continuously if we further decrease the laser power. However, different oxidation findings were described in Ref. [6] that simply melting of the MTMO film, even for long time, did not generate oxidation, i.e., the oxidation occurred only when the laser melted the metal film. This might be due to the difference between their bimetallic film of Bi/In samples and our indium film samples.

4. Conclusions

In this paper, a LIMO model of In-glass MTMO grayscale photomasks is proposed. The mechanism of the sub-wavelength fabrication by the laser direct writing has been studied. The temperature fields of the model are calculated by solving the heat transfer equations with the FDTD method. The fabrication area is determined by the distribution of the laser power density. By analyzing the fabrication resolution at the laser power of 10.0 mW, the threshold of absorbed laser power density can be deduced as 2.17 × 10¹⁷ W/m³. The average fabrication size of MTMO grayscale photomask with a laser power of 6.0 - 8.0 mW is 302 nm, which is consistent with the experimental results.

Acknowledgments

We gratefully acknowledge financial support for this work by the National Basic Research Programs of China (2010CB934101, 2010CB934102), the National Science Foundation of China (11304162, 61205035), International S&T cooperation program of China (2011DFA52870), Oversea Famous Teacher Project (MS2010NKDX023) and the 111 Project (B07013).

References and links

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Parameter	Symbol	Value
Complex refractive index of In	${\overset{⌢}{n}}_{s}$	0.76 + 5.08i
Latent heat of melting (J/kg)	$Δ H_{m}$	28575.16
Refractive index of glass	$n_{g}$	1.5469
Density of solid In (kg/m⁻³)	$ρ_{s s}$	7310
Density of liquid In (kg/m⁻³)	$ρ_{s l}$	7020
Density of glass (kg/m⁻³)	$ρ_{g}$	2200
Thermal conductivity of solid In (W/m⁻¹﹒K⁻¹)	$k_{s s}$	81.8
Thermal conductivity of liquid In (W/m⁻¹﹒K⁻¹)	$k_{s l}$	42.0
Thermal conductivity of glass (W/m⁻¹﹒K⁻¹)	$k_{g}$	1.4
Specific heat of solid In (J/kg⁻¹﹒K⁻¹)	$c_{s s}$	243
Specific heat of liquid In (J/kg⁻¹﹒K⁻¹)	$c_{s l}$	259
Specific heat of glass (J/kg⁻¹﹒K⁻¹)	$c_{g}$	966

Numerical analysis of the sub-wavelength fabrication of MTMO grayscale photomasks by direct laser writing

Abstract

1. Introduction

2. Model of light-induced melt-oxidation

2.1 The absorbed laser energy

2.2 Heat transfer and boundary equations

3. Numerical results and discussions

3.1 Temporal and spatial temperature fields

3.2 Sub-wavelength fabrication based on the LIMO model

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (7)

Optics Express