1. Introduction

An efficient and powerful extreme ultraviolet (EUV) light source is crucial for semiconductor nanolithography technology to produce finer line-width semiconductors. With the rapid development of information technology, the demand for faster and lower-power consumption semiconductors with finer line widths has increased, particularly for cell phones, computer servers at data centers for high-speed and high-volume data communication, high-performance computing, artificial intelligence, and autonomous driving cars [1]. Following Moore’s Law [2–4], nanolithography technologies with more powerful light sources and shorter wavelengths have been developed [5]. Nowadays, semiconductors with line widths of less than 7 nm nodes have entered into production, and 2 nm node chip technologies have been unveiled [6,7], in which the 13.5 nm wavelength EUV light with a 2${\% }$ bandwidth, emitted from laser-produced plasmas (LPP) of tin, is used as a light source for nanolithography [8,9]. However, a more efficient EUV light source is strongly desired as the required EUV light power for producing finer line-width chips with higher throughput increases [10]. So far, many trials have been conducted to achieve an efficient and powerful EUV light source for over three decades. O’Sullivan investigated the 4d-4f emission from various target materials irradiated by a laser [11]. Spitzer et al. obtained an experimental database of EUV conversion efficiency (CE) from 1.06 $\mu$m and 0.53 $\mu$m wavelength lasers to EUV light [12]. Based on their database, tin has been the most attractive material for efficiently generating EUV light of 13.5 nm, matching with the high reflectivity of Mo/Si mirror. Shimada et al. demonstrated CE of 3${\% }$ with a 1.06 $\mu$m wavelength laser uniformly irradiating a solid tin sphere [13]. Then, Tanaka et al. experimentally showed the high potential of 10.6 $\mu$m wavelength CO$_2$ laser irradiation on tin targets to generate EUV light more efficiently than 1.06 $\mu$m laser irradiation [14], which is theoretically confirmed by Nishihara et al. [15], and high CE was experimentally demonstrated [16,17]. Sasaki et al. showed that relatively lower-density tin LPP from CO$_2$ laser irradiation has an advantage in producing high EUV spectral purity (SP), defined as the ratio of EUV radiant energy of the 13.5 nm with a 2${\% }$ bandwidth to total radiation energy integrated over wavelengths [18,19]. Nishihara et al. showed that CO$_2$ laser irradiation needs a long-density scale pre-formed plasma to boost its relatively low laser absorption fraction [20]. Therefore, the current EUV light source adopts multi-pulse irradiation to the tin droplet, which comprises pre-pulse generating a tin pre-formed plasma and main laser pulse irradiation on the pre-formed plasma to emit EUV light. It can achieve CE of up to 6${\% }$ from the incident laser to EUV light, where the CO$_2$ laser is utilized as the main laser [21]. However, realizing a more efficient high-power EUV light source is strongly desired as the required EUV light power for producing finer line-width chips with higher throughput increases.

To improve the efficiency of the EUV light source system, some attention has recently turned to solid-state laser irradiation on the tin with a wavelength of 2 $\mu$m or longer, taking advantage of the high electrical plug-in efficiency of solid-state lasers [22–26]. Sizyuk and Hassanein, using their comprehensive 3D HEIGHTS simulation package, showed that 2 $\mu$m laser irradiation with longer pulse duration leads to an increase in CE and a reduction in the maximum energy of ion debris [23]. Tamer et al. reported that their LLNL group is developing a 1.9 $\mu$m-wavelength Tm:YLF laser as a Big Aperture Thulium (BAT) laser for the EUV light source [27]. Furthermore, Hemminga conducted a numerical survey on CE by irradiating various laser wavelengths on the tin droplet without pre-formed plasmas between a 1.06 $\mu$m Nd:YAG and a 10.6 $\mu$m CO$_2$ laser using 2D radiation hydrodynamic simulations [28]. He shows that 4-$\mu$m wavelength laser irradiation on tin has a high potential to provide a CE of 6${\% }$, comparable to the maximum CE obtained with the current EUV light source using a CO$_2$ laser. The Tm:YLF laser has a plug-in efficiency from electricity to laser light of 19${\% }$ [29], much larger than the 5.2${\% }$ of the CO$_2$ laser [30]. Thus, the entire EUV light system with Tm:YLF laser could have relatively higher efficiency from electricity to EUV light than the current CO$_2$ laser-based EUV light source. Such a laser system point of view enables us to consider improving the electrical efficiency of the EUV light source system. Nevertheless, the optimization of LPP conditions is still highly required to achieve more efficient high-power EUV and less ion debris, enabling a longer lifetime of the collector mirror. Although much work on modeling the dynamics of a laser-irradiated tin droplet and following EUV emission was significantly studied [31–42], there are still many unknowns about the optimized laser irradiation and plasma condition efficiently emitting powerful EUV light.

Our current study aims to better understand the parameter dependence of EUV emission on pre-formed plasma and laser conditions and to represent the solution of optimized parameters for CO$_2$ laser irradiation on pre-formed plasma to achieve more efficient and powerful EUV emission. The 3D HEIGHTS comprehensive and integrated simulation package at our CMUXE has been used to simulate LPP [23,43,44], discharged plasmas [45], and plasma-solid interactions, such as the plasma-facing materials in the fusion reactor with real experimental configuration [46,47]. However, the 3D physics integrated simulation requires more powerful computers to execute. Therefore, we conducted many radiation hydrodynamic simulations using the two-dimensional radiation hydrodynamic code Star2D [48,49].

According to our simulation results, we can obtain higher CE from CO$_2$ laser-irradiated tin plasmas than the current CE. In addition, the physical insight obtained from the simulations would be useful not only for CO$_2$ laser irradiation but also for other lasers’ wavelengths. This paper is organized as follows: Section 2 describes the computational method for our simulations of CO$_2$ laser irradiation on pre-formed tin plasma. Section 3 presents the calculated plasma dynamics and EUV emission properties. We investigated the dependence of EUV emission on the pre-formed plasma and laser conditions. In Section 4, we discuss our physical insights obtained from the simulations and the applicability of our findings. Finally, Section 5 summarizes our conclusions.

2. Simulation method

We conducted the radiation hydrodynamic simulations with various CO$_2$ laser parameters and pre-formed tin plasma conditions using Star2D code. The fundamentals of the code are already described in the Ref. [48,49]. Briefly, Star2D is the 2D radiation hydrodynamic code based on one-fluid two-temperature model, comprised of subroutines of hydrodynamics, radiation transport, thermal conduction, temperature relaxation between electrons and ions, and laser propagation and absorption. In the hydrodynamics part, the Harten-Lax-van Leer-contact (HLLC) scheme [50,51] is adopted to calculate the numerical fluxes of conservative two-dimensional Euler equations at each cell boundary with a realistic tin equation-of-state (EOS), calculated by More [52]. We adopted van Leer’s slope limiter in the spatial profile to implement second-order accuracy in space [53]. Also, the second-order Runge-Kutta is adopted for the temporal integration to obtain second-order accuracy in time [54]. The laser propagation is calculated as a straight line and incident normal to the target surface, and the laser absorption is calculated as the inverse-bremsstrahlung process [55]. In addition, we consider the Langdon effect [56,57], which is the reduction of laser absorptivity due to the non-Maxwellian electron velocity distribution function of laser-irradiated plasmas is considered, which isn’t negligible for the CO$_2$ laser irradiation. This is because CO$_2$ laser has a relatively larger quiver velocity of the electron in the laser electric field in the plasma, deforming the electron velocity distribution function apart from Maxwell distribution. The thermal conduction is calculated using the flux-limited diffusion model with different thermal conductivity for electrons [58] and ions [59]. The temperature relaxation between electrons and ions is calculated by the Spitzer relaxation coefficient [60]. The radiation transport is calculated by a multi-group flux-limited model with radiation bins of 40 groups covering the radiation energy ranging from 0 to 2 keV. The radiative emissivity and opacity refer to the data tables calculated by Sasaki [61], where both tables of the local thermal equilibrium (LTE) and collisional radiative steady state (CRSS) are interpolated to calculate emissivity and opacity, considering photo-excitation based on Novikov method [62].

In the simulation, meshes of $100 \times 200$ were assigned in cylindrical coordinates ($r, z$) with unequal spacing, which well resolves the plasma profile around the critical density. The simulation box size was 1200 $\mu$m $\times$ 2700 $\mu$m. To mimic the pre-formed plasma generated by pre-pulse before the main CO$_2$ pulse irradiation, we consider an initial ion density profile with a density gradient scale length $L_\mathrm {init}$ of [100, 200, 300] $\mu$m, respectively in the $z$-direction and uniform in the $r$-direction. We set the initial ion density profiles to be $n_i(z) = \max \{ n_{i, \mathrm {floor}}, n_\mathrm {i0} \exp [ -(z-z_0)/L_\mathrm {init} ] \}$. Here, $z_0$ is the position of the critical density and $n_{i0}$ is ion density at the critical density. $n_{i, \mathrm {floor}}$ is the minimum ion density value required for the simulation with Eulerian mesh, including the vacuum region. We set $n_{i, \mathrm {floor}}$ to be $3.7 \times 10^{15}$ cm$^{-3}$. Such a simplified setting of the initial ion profile was inferred from the experimentally observed plasma profile just after the main laser irradiation [36,63,64] because the density distribution of the pre-formed tin plasma is not yet well investigated quantitatively. The initial values of electron and ion temperatures were set to room temperature of $T_e = T_i = 0.025$ eV. Thus, the initial electron density profile was set to be $n_e(z) =\langle Z \rangle n_{i0} \, \exp [ -(z-z_0)/L_\mathrm {init} ]$, where $n_e(z)$ is the electron density profile and $\langle Z \rangle$ is the average ionization degree as a function of ($n_i,T_e$).

We carried out the simulations with different laser spot diameters and pulse duration. We varied the simulation conditions of the laser spot diameter $\phi _L$ to [100, 200, 300] $\mu$m (Full Width at Half Maximum: FWHM), respectively, of the Gaussian intensity profile of the laser, $I_L(r,t)=I_L(r=0, t) \exp \{ -\ln (2) [ (t-t_\mathrm {peak}) / (\phi _L/2) ]^2 \}$. Here, $I_L(r, t)$ is laser intensity at time $t$ and position $r$. $t_\mathrm {peak}$ is laser peak timing. In time, the laser pulse duration $\tau _L$ was set to [5, 10, 15] ns (FWHM) for the Gaussian pulse, $I_L(r=0,t)= I_{L, \mathrm {peak}} \exp \{ -\ln (2) \ [t/(\tau _L/2) ]^2 \}$, and corresponding $t_\mathrm {peak}$ was set to 7, 14, and 20 ns, respectively, to start the calculation at an early in the pulse. Here, the peak laser intensity $I_{L, \mathrm {peak}}$ is set to be $2 \times 10^{10}$ W/cm$^2$ until stated otherwise when investigating the dependence of the CE on the laser intensity. It should be noted that the parameters described above have been selected to provide that the density and temperature of their plasma conditions are to be close to the optimal point for the double-pulse CO$_2$ irradiation scheme proposed by Nishihara et al. [20]. They suggested the optimum plasma condition of $n_i = 5 \times 10^{17}$ cm$^{-3}$ and $T_e =$ 45 eV for the efficient EUV emission based on their 1D power-balance model. At this $n_i$ and $T_e$, the critical criteria of high EUV spectral purity, large laser absorption, and appropriate EUV optical depth can be simultaneously optimized for efficient EUV emission. However, the applicability of their 1D power balance model is restricted to the one-dimensional isothermal plasma expansion but not to the actual multi-dimensional plasma dynamics. Our simulation aims to find optimum and concrete parameters of CO$_2$ laser and pre-formed plasma conditions in actual plasma conditions.

3. Calculated results

3.1 Plasma dynamics

Figures 1(a)–1(i) show the ion density below $n_i <$ 10$^{20}$ cm$^{-3}$ at the laser peak timing for total $3 \times 3$ cases, calculated with different conditions of laser pulse width $\tau _L =$ [5, 10, 15] ns and laser spot diameter $\phi _L =$ [100, 200, 300] $\mu$m, respectively, for initial density gradient scale of pre-formed plasma $L_\mathrm {init}$ of 100 $\mu$m. In Figs. 1(a)–1(c), 1(d)–1(f), and 1(g)–1(i), respectively, where $\phi _L$ is fixed and $\tau _L$ is varied, three figures can be considered the temporal evolution of $n_i(r,z)$ due to laser irradiation. For example, the 300 $\mu$m laser spot diameter shown in Figs. 1(a)–1(c), the plasma is heated near the ion density region from $n_i = 3 \times 10^{17}$ cm$^{-3}$ to $1 \times 10^{18}$ cm$^{-3}$. As the laser irradiates the pre-formed plasma, the heated plasma expands against the cold plasma surrounding it in Fig. 1(a). In Fig. 1(b), plasma expansion spreads beyond the laser spot and sweeps the plasma outward. The density increase due to compression is particularly noticeable near the ablation front in the high-density region. In Fig. 1(c), plasma expansion develops further, and the plasma density scale length below $n_i = 3 \times 10^{17}$ cm$^{-3}$ in the laser spot becomes longer than the initial density scale length. On the other hand, the density region above $n_i = 3 \times 10^{17}$ cm$^{-3}$ becomes steeper and evolves into a much smaller scale length than the initial plasma scale length of 100 $\mu$m. This density steepening near the ablation front is qualitatively similar to the findings of Basko et al. [65,66].

 

Fig. 1. Ion density profiles $n_i(r,z)$ at the laser peak timing on the ($\tau _L,\phi _L$) plane for $L_\mathrm {init}$ = 100 $\mu$m. The horizontal axis represents the laser pulse duration $\tau _L$: (a), (d), and (g) correspond to 5 ns, (b), (e), and (h) to 10 ns, and (c), (f), and (i) to 15 ns, respectively. The vertical axis represents the laser spot diameter $\phi _L$: (g)-(i) correspond to 100 $\mu$m, (d)-(f) to 200 $\mu$m, and (a)-(c) to 300 $\mu$m, respectively.

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To compare the temporal evolution of the $n_i(r,z)$ concerning different laser spot sizes, we focus on Figs. 1(g)–1(i), which have $\phi _L$ of 100 $\mu$m. Compared to Figs. 1(a)–1(c) of $\phi _L = 300$ $\mu$m, the smaller $\phi _L = 100$ $\mu$m of Figs. 1(g)–1(i) show slower temporal increases of the plasma scale length below $n_i = 10^{17}$ cm$^{-3}$, while the plasma scale length above $n_i = 10^{17}$ cm$^{-3}$ decreases faster than these in Figs. 1(a)–1(c) due to the compression. Comparing Fig. 1(c) and Fig. 1(i), the difference in the spatial distribution of the steepening density structure is clear between different $\phi _L$. The density steepening spreads around the ablation front and the sides of the laser spot and surrounds the high-density side of the laser spot in Fig. 1(i). This comparison shows that for a smaller $\phi _L$, the lateral sweep motion of the expanding plasma relative to the laser axis more strongly governs the temporal evolution of an ion density profile. Consequently, the temporal increases of the plasma density scale length in the lower density region are suppressed compared to larger laser spot cases.

The corresponding electron temperature profiles $T_e(r,z)$ are shown in Figs. 2(a)–2(i). In comparing the different $\tau _L$ at the same $\phi _L$, the maximum $T_e$ at the laser peak timing remains almost similar, even with increased $\tau _L$. For example, in Figs. 2(a)–2(c) with $\phi _L=$ 300 $\mu$m cases, the spatial spread above $T_e =$ 20 eV along the direction of the laser axis increases from 400 $\mu$m at $\tau _L =$ 5 ns to approximately 1000 $\mu$m at $\tau _L=$ 15 ns. Nevertheless, the maximum $T_e$ value remains around 70 eV regardless of the pulse width. Also, the spatial spread of higher $T_e$ above $T_e =$ 60 eV remains temporarily constant at approximately 300 $\mu$m in space, even when the pulse width is elongated from 5 ns to 15 ns. A similar tendency can be seen for other $\phi _L$.

 

Fig. 2. Electron temperature profiles $T_e(r,z)$ at the laser peak timing on the ($\tau _L,\phi _L$) plane for $L_\mathrm {init}$ = 100 $\mu$m. The horizontal axis represents the laser pulse duration $\tau _L$: (a), (d), and (g) correspond to 5 ns, (b), (e), and (h) to 10 ns, and (c), (f), and (i) to 15 ns, respectively. The vertical axis represents the laser spot diameter $\phi _L$: (g)-(i) correspond to 100 $\mu$m, (d)-(f) to 200 $\mu$m, and (a)-(c) to 300 $\mu$m, respectively.

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On the other hand, the maximum value of $T_e$ changes significantly depending on the laser spot size $\phi _L$. In Figs. 2(a)–2(c) with $\phi _L=$ 300 $\mu$m, the $T_e$ reaches around 70 eV, while it remains at approximately 50 eV in Figs. 2(g)–2(i) with a smaller spot of $\phi _L=$ 100 $\mu$m. This dependence of the maximum electron temperature on the laser spot size is determined by the ratio of the laser spot size and the absorption length of radiation. Assuming that the dominant radiation loss from the heating region is EUV light of 13.5 nm wavelength, the EUV absorption length corresponding to the ion density of $n_i = 10^{18}$ cm$^{-3}$ at the position of maximum $T_e$ is approximately $100-200$ $\mu$m, as shown by Nishihara et al. [15]. Thus, at a spot size of 100 $\mu$m, the absorption length of radiation is longer than the spot radius ($\phi _L/2$), and the radiation escapes from the high-temperature region easily to the lateral direction relative to the laser axis. In contrast, with $\phi _L =$ 300 $\mu$m, the spot radius ($\phi _L/2$) and the absorption length of radiation are comparable and result in the relative suppression of the lateral radiation loss. Consequently, the maximum $T_e$ shows a relatively higher value than that for $\phi _L =$ 100 $\mu$m.

The corresponding electron density $n_e(r,z)$ profiles are shown in Figs. 3(a)–3(i). Comparing Figs. 3(a)–3(c), 3(d)–3(f), and 3(g)–3(i), respectively, where the $\tau _L$ is varied at the same $\phi _L$, we see that the electron density gradient scale length below $n_e = 10^{18}$ cm$^{-3}$ increases as $\tau _L$ increases for all $\phi _L$ cases. However, if we focus on $n_e(r, z)$ profile just below the critical density of $n_e = 10^{19}$ cm$^{-3}$, which is important for CO$_2$ laser absorption, we find that the density gradient scale length between $n_e = 10^{19}$ cm$^{-3}$ and $n_e = 3 \times 10^{18}$ cm$^{-3}$, corresponding to the orange region in Figs. 3(a)–3(i), decreases as $\tau _L$ increases. For example, in Figs. 3(a)–3(c) with $\phi _L=$ 300 $\mu$m for $L_\mathrm {init}=$ 100 $\mu$m, the density gradient scale length of that region along the laser axis is approximately 190 $\mu$m at $\tau _L =$ 5 ns, as shown in Fig. 3(a). However, it decreases to 90 $\mu$m with $\tau _L =$ 10 ns in Fig. 3(b) and 85 $\mu$m with $\tau _L =$ 15 ns in Fig. 3(c). Similarly, the density gradient scale length along the laser incident direction at ($r = \phi _L/2$), away from the laser axis, is greater for shorter $\tau _L$ and decreases with increasing $\tau _L$. This reduction is due to the transverse plasma motion to the laser axis, which sweeps the plasma from the laser spot. In comparing the different laser spot diameters $\phi _L$, at the same pulse duration, the electron density scale length is longer for larger laser spot diameters. Comparing Fig. 3(a) and Fig. 3(g), we find that for the smaller spot size in Fig. 3(g), the density gradient scale length in the region from $n_e = 10^{19}$ cm$^{-3}$ to $n_e = 3 \times 10^{18}$ cm$^{-3}$ shown as orange is 40 $\mu$m, which is significantly smaller than 190 $\mu$m in Fig. 3(a). Because a larger aspect ratio of plasma expansion scale to laser spot diameter leads to a larger effect of transverse motion of plasma relative to the laser axis in plasma expansion, it causes the plasma to expand three-dimensionally, promoting the temporal decrease of the density scale length along the laser axis and the density steepening around the ablation front. This shortened scale length reduces the laser absorption.

 

Fig. 3. Electron density profiles $n_e(r,z)$ at the laser peak timing on the ($\tau _L,\phi _L$) plane for $L_\mathrm {init}$ = 100 $\mu$m. The horizontal axis represents the laser pulse duration $\tau _L$: (a), (d), and (g) correspond to 5 ns, (b), (e), and (h) to 10 ns, and (c), (f), and (i) to 15 ns, respectively. The vertical axis represents the laser spot diameter $\phi _L$: (g)-(i) correspond to 100 $\mu$m, (d)-(f) to 200 $\mu$m, and (a)-(c) to 300 $\mu$m, respectively.

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3.2 EUV emission properties

We investigated the CE of EUV light from CO$_2$ laser-irradiated tin plasmas with different conditions of laser pulse width $\tau _L$ of [5, 10, 15] ns and initial density gradient scale of pre-formed plasma $L_\mathrm {init}$ of [100, 200, 300] $\mu$m, respectively, and for laser spot diameter $\phi _L$ of 100 $\mu$m in Fig. 4(a), 200 $\mu$m in Fig. 4(b), and 300 $\mu$m in Fig. 4(c), respectively. Regardless of $\phi _L$, the CE is relatively high for a short pulse duration $\tau _L$ of 5 ns, and inversely the CE decreases as $\tau _L$ increases. Also, as $L_\mathrm {init}$ increases, the CE increases for all $\phi _L$ cases. Comparing CE between different laser spots $\phi _L$, for $\phi _L$ = 100 $\mu$m, the CE ranges from 3.3% to 4% at $\tau _L =$ 15 ns depending on $L_\mathrm {init}$, and it reaches $5{\% }-6.7{\% }$ for $\tau _L =$ 5 ns, as shown in Fig. 4(a). For $\phi _L =$ 300 $\mu$m, the CE ranges from 5.6% to $7.1{\% }$ at $\tau _L$ of 15 ns depending on $L_\mathrm {init}$, and it reaches $8.2{\% }-10.2{\% }$ at $\tau _L =$ 5 ns, as shown in Fig. 4(c). For $\phi _L =$ 200 $\mu$m, as shown in Fig. 4(b), the CE shows the values in between Fig. 4(a) and Fig. 4(c). The condition with $\tau _L =$ 15 ns and $\phi _L =$ 100 $\mu$m is close to the parameters of one EUV light source [21], and the calculated CE reasonably reproduces 5% of that EUV light source. It is noteworthy that when $\tau _L$ is shortened to 5 ns, $\phi _L$ is increased to 300 $\mu$m, and $L_\mathrm {init}$ is set to be 300 $\mu$m, the CE exceeds 10%, as shown in Fig. 4(c), which is roughly twice the CE of the current mass production light source system. This condition is the most optimized point giving the highest CE among simulations with different parameter sets of [$\tau _L$, $\phi _L$, $L_\mathrm {init}$] in this paper. It represents significant potential to improve the efficiency of EUV light sources. From the comparison of Figs. 4(a)–4(c), we conclude that pulse shortening, increasing the laser spot size, and increasing the initial density scale length are crucial for improving the CE.

 

Fig. 4. CE, in function of $\tau _L$ with different laser spot diameters $\phi _L$: (a) 100 $\mu$m, (b) 200 $\mu$m, (c) 300 $\mu$m, respectively. Red circles, yellow triangles, and blue rectangles show $L_\mathrm {init}$ of 100, 200, and 300 $\mu$m, respectively.

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We examine the detailed physical mechanisms that control the CE in Figs. 4(a)–4(c). By defining the laser absorption fraction $f_L$, the radiation conversion ratio $C_R$, defined as the ratio from absorbed laser energy to radiation energy emitted to 2$\pi$ sr solid angle to the laser irradiation side, and the proportion of EUV light with a 2% bandwidth at 13.5 nm in the total radiation energy (SP), respectively, we can express (CE) as Eq. (1):

In Figs. 5(a)–5(c), we show the laser absorption fractions $f_L$ corresponding to each data point of Figs. 4(a)–4(c). We see that $f_L$ increases as $\tau _L$ becomes shorter and $L_\mathrm {init}$ becomes longer for each laser spot diameter $\phi _L$, as shown in Figs. 5(a)–5(c), respectively. Furthermore, when comparing different laser spot diameters $\phi _L$, $f_L$ increases as $\phi _L$ increases. Consequently, the value of $f_L$ has 57% with $\phi _L=$100 $\mu$m, $\tau _L =$ 15 ns, and $L_\mathrm {init}=$ 100 $\mu$m as shown in Fig. 5(a), and showing close to nearly 100% absorption with $\phi _L=$ 300 $\mu$m, $\tau _L =$ 5 ns, and $L_\mathrm {init} =$ 300 $\mu$m as shown in Fig. 5(c). These laser absorption characteristics can be understood from the analysis of plasma dynamics presented in the previous section. When laser irradiation is applied to the pre-formed plasma, the electron density gradient scale length just under the critical density decreases due to the steepening of density and transverse motion of the expanding plasma to the laser axis. Since laser absorption by inverse bremsstrahlung depends on the plasma density gradient scale length just under the critical density [67], $f_L$ decreases due to the shortening of the density scale length, as shown in Figs. 3(a)–3(i), as $\tau _L$ becomes longer and $\phi _L$ becomes smaller. The trends of $f_L$ for different parameter sets of [$\tau _L$, $\phi _L$, $L_\mathrm {init}$] show similar trends to those observed for the CE shown in Figs. 4(a)–4(c). Thus, it is obvious that laser absorption fractions are a crucial factor in determining the value of CE.

 

Fig. 5. Laser absorption fraction $f_L$ in function of $\tau _L$ with different laser spot diameters $\phi _L$: (a) 100 $\mu$m, (b) 200 $\mu$m, (c) 300 $\mu$m, respectively. Red circles, yellow triangles, and blue rectangles show $L_\mathrm {init}$ of 100, 200, and 300 $\mu$m, respectively.

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Next, we investigate the radiation conversion ratio $C_R$. In Figs. 6(a)–6(c), we show $C_R$ corresponding to each data point of Figs. 4(a)–4(c). Compared at the same $\phi _L$, $C_R$ weakly depends on $\tau _L$ slightly decreasing as $\tau _L$ increases for all laser spot $\phi _L$ cases. This reflects that the portion of the absorbed laser energy converted into the kinetic energy of the expanding plasma increases as $\tau _L$ increases. Furthermore, compared between different $\phi _L$ with the same $L_\mathrm {init}$ as shown in Figs. 6(a)–6(c), $C_R$ increases as $\phi _L$ increases. To understand the change in $C_R$ due to differences in laser spot diameter $\phi _L$, we show the ion density $n_i(z)$, electron density $n_e(z)$, and electron temperature $T_e(z)$ along the laser axis ($r = 0$) at the peak timing of the radiation emission power in Fig. 7(a). Also, the net EUV emission energy density $S^{*}(z) = \frac {1}{4}\;S(z)\cdot T(z)$ and the EUV transmittance $T(z)$ are shown in Fig. 7(b), in which $S(z)$ is the EUV emission energy density at $z$, and $T(z)$ is EUV transmittance from $z$ to plasma edge $z_\mathrm {max}$. $S(z)$ and $T(z)$ are given by $S(z)=(4\pi \;\eta _\mathrm {EUV}-c\;\chi _\mathrm {EUV}\;E_\mathrm {EUV})$ and $T(z) =$ exp[$-\mathrm {\mathrm {(O.D.)}}_\mathrm {EUV}$], respectively, in which, $\mathrm {\mathrm {(O.D.)}}_\mathrm {EUV}$ is EUV optical depth given by $\mathrm {\mathrm {(O.D.)}}_\mathrm {EUV}(z) = \int _z^{z_\mathrm {max}} \chi _\mathrm {EUV}(z^{'}) dz^{'}$. Here, $\eta _\mathrm {EUV}$, $\chi _\mathrm {EUV}$, $E_\mathrm {EUV}$, and $c$ are EUV emissivity, EUV absorption coefficient, EUV radiation energy density, and the speed of light, respectively.

 

Fig. 6. Radiation conversion ratio $C_R$ in function of $\tau _L$ with different laser spot diameters $\phi _L$: (a) 100 $\mu$m, (b) 200 $\mu$m, (c) 300 $\mu$m, respectively. Red circles, yellow triangles, and blue rectangles show $L_\mathrm {init}$ of 100, 200, and 300 $\mu$m, respectively.

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Fig. 7. (a) and (c) Plasma profiles of $n_i$, $n_e$, $T_e$ at the radiation peak timing with $\tau _L of$ 5 ns and various $\phi _L$ for $L_\mathrm {init}$ of (a) 100 $\mu$m, (c) 300 $\mu$m, respectively. (b) and (d) are corresponding EUV properties of $S^{*}(z)$ and $T(z)$ to (a) and (c), respectively. Solid, dashed, and dotted lines indicate $\phi _L$ of 100, 200, and 300 $\mu$m, respectively.

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In Fig. 7(a), the larger spot diameter $\phi _L$ shows higher $n_i(z)$, $n_e(z)$, and $T_e(z)$ of the low-density expanding plasma with a larger density gradient scale length. The corresponding EUV effective emission energy density $S^{*}(z)$ and EUV transmittance $T(z)$ are shown in Fig. 7(b), in which $S^{*}(z)$ consists of two components, one at the emission peak near the critical density and the other is the emission from the low-density expanding plasma region with $n_i = 10^{17}$ cm$^{-3}$. In comparing with different $\phi _L$ in Fig. 7(b), as $\phi _L$ increases, $S^{*}(z)$ shows that the contribution from the low-density plasma to the EUV output relatively increases, compared to that of the emission peak near the critical density.

The contribution to the EUV output from the low-density plasma becomes even more pronounced as $L_\mathrm {init}$ increases. Figure 7(c) shows the plasma profiles with the same parameter sets of [$\tau _L$, $\phi _L$] as in Fig. 7(a) except for setting $L_\mathrm {init}$ = 300 $\mu$m, and corresponding $S^{*}(z)$ and $T(z)$ are shown in Fig. 7(d). Comparing Fig. 7(a) and Fig. 7(c), it can be seen that increasing $L_\mathrm {init}$ from 100 $\mu$m to 300 $\mu$m leads to an increase in the spatial scale of the low-density expanding plasma near $n_i = 10^{17}$ cm$^{-3}$. Note that the horizontal scale of Fig. 7(a) and Fig. 7(c) differs by a factor of 2. Correspondingly, the spatial spread of the high-temperature region with $T_e(z) >$ 30 eV increases from approximately 400 $\mu$m to 800 $\mu$m. Consequently, it can be observed that the contribution of EUV emission from the low-density expanding plasma region dramatically increases with an increase in $L_\mathrm {init}$, in comparing Fig. 7(b) and Fig. 7(d). Therefore, increasing the initial plasma density scale length can relatively increase the contribution of EUV emission from the low-density expanding plasma region, which is mostly under the condition with $T(z) > T[\mathrm {(O.D.)}_\mathrm {EUV} = 1.0 - 1.5] = 0.37 - 0.22$ in Fig. 7(d), which leads to an increase in the radiation conversion ratio $C_R$. It should be noted that an increase in the initial plasma density scale length $L_\mathrm {init}$ slightly decreases the maximum $T_e$ value, as we can see from the comparison between Fig. 7(a) and Fig. 7(c). So, for relatively small laser spot diameters such as $\phi _L=$ 100 $\mu$m, showing relatively lower $T_e(z)$, an increase in $L_\mathrm {init}$ does not significantly affect the spatial spread of the high-temperature region $T_e(z) >$ 30 eV. Thus, an increase in $L_\mathrm {init}$ negatively affects $C_R$ for a laser spot diameter of 100 $\mu$m as seen in Fig. 6(a).

Next, we investigate the EUV spectral purity (SP), which gives the radiative energy proportion of EUV 2% bandwidth to the radiative energy integrated over total wavelengths corresponding to each data point set. The SP was first calculated for tin plasma by Sasaki et al. [19]. He showed that the maximum SP of 35% can be obtained, assuming an optically thin tin plasma which has a relatively low density and high temperature below an ion density of $n_i = 5 \times 10^{17}$ cm$^{-3}$ and above an electron temperature of $T_e =$ 45 eV. In achieving high efficiency and high output EUV light sources, finite optical depth $\mathrm {(O.D.)}$ has to be properly optimized [68]. If the plasma emitting EUV radiation is optically thin, the SP approaches the limit of 35%. Still, such a condition cannot produce high-power EUV light due to insufficient plasma generation. Conversely, if the plasma is optically thick, the power radiated as EUV light increases until the limit, which is Planckian for LTE plasmas. Still, the SP decreases, resulting in a decrease in CE and inefficient EUV light production [69]. Therefore, we assume the$\mathrm {(O.D.)}_\mathrm {EUV}$ to be 1, suitable for the efficient and powerful EUV light emission as the transition value between optically thin and thick states. Then, the value of SP corresponding to $\mathrm {(O.D.)}_\mathrm {EUV}= 1$ is estimated to be $\mathrm {SP} = 0.35 \times \exp (-1) = 0.13$.

In Figs. 8(a)–8(c), we show the calculated SP for different parameter sets of [$\tau _L$, $L_\mathrm {init}$] for each $\phi _L$ of 100, 200, and 300 $\mu$m, respectively. The SP decreases as $\tau _L$ increases for all laser spot $\phi _L$ cases. This is because the net EUV emission $S^{*}_\mathrm {EUV}$ near the critical density more dominantly contributes to the EUV output than $S^{*}_\mathrm {EUV}$ from the low-density expanding plasma as $\tau _L$ increases. Conversely, for a relatively short pulse width of 5 ns, the SP increases because $S^{*}_\mathrm {EUV}$ out of the low-density expansion plasma is larger and contributes more to the EUV output, decreasing the effective plasma density emitting EUV light. Compared with different laser spots $\phi _L$ between Figs. 8(a)–8(c), we see that as the $\phi _L$ increases, the SP increases. This is due to more increase in the effective temperature of EUV emission for larger laser spot $\phi _L$.

 

Fig. 8. SP in function of $\tau _L$ with different laser spot diameters $\phi _L$: (a) 100 $\mu$m, (b) 200 $\mu$m, (c) 300 $\mu$m, respectively. Red circles, yellow triangles, and blue rectangles show $L_\mathrm {init}$ of 100, 200, and 300 $\mu$m, respectively.

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We calculated the effective ion density $\langle n_i \rangle _\mathrm {EUV}$ and the effective electron temperature $\langle T_e\rangle _\mathrm {EUV}$ as the weighted average with $S^{*}_\mathrm {EUV}$ over time and space as;

4. Discussions

Next, we discuss the peak intensity of the CO$_2$ laser. So far, we have simulated assuming a fixed CO$_2$ laser peak intensity $I_{L,peak}$ of $2 \times 10^{10}$ W/cm$^2$. To investigate the validity of setting this peak intensity and its acceptable range for high CE, we additionally conducted simulations with each laser peak intensities of $1 \times 10^{10}$ and $4 \times 10^{10}$ W/cm$^2$ and parameter sets of [$\tau _L$, $\phi _L$] for each $L_\mathrm {init}$ of 100 and 300 $\mu$m, respectively, as shown in Figs. 9(a)–9(i). The highest CE was mostly measured at $I_{L,peak}$ of $2 \times 10^{10}$ W/cm$^2$. This result confirms the validity of using this peak intensity for our calculations. More importantly, for relatively long pulse widths of 15 ns and relatively small laser spot sizes of 100 $\mu$m, the peak laser intensity dependence of the CE is almost the same concerning different $L_\mathrm {init}$ of 100 $\mu$m and 300 $\mu$m, as shown in Fig. 9(i). However, as the pulse width becomes shorter and the laser spot size becomes larger, the peak laser intensity dependence of the CE becomes increasingly different for $L_\mathrm {init}$ of 100 $\mu$m and 300 $\mu$m. As shown in Fig. 9(a), for $\tau _L$ of 5 ns and $\phi _L$ of 300 $\mu$m, the CE gives the highest value at the laser intensities of $1 \times 10^{10}$ W/cm$^2$ for $L_\mathrm {init} =$ 100 $\mu$m and decreases as the intensity increases. On the other hand, for the relatively long initial scale length of $L_\mathrm {init} =$ 300 $\mu$m, the laser peak that gives high CE shifts to higher intensity. While the peak intensity of $2 \times 10^{10}$ W/cm$^2$ gives the highest CE, even at $4 \times 10^{10}$ W/cm$^2$, the decrease in the CE is minimal, and the CE remains relatively high. This result represents that the peak laser intensity range that gives high CE for the shorter pulse and the larger laser spot diameters is relatively wide and extends to higher intensities than $2 \times \; 10^{10}$ W/cm$^2$. This is very beneficial for achieving a powerful EUV light source because increasing the laser peak intensity can compensate for the decrease in EUV output per pulse due to the shortening of the CO$_2$ laser pulse [70]. Furthermore, it should be noted about the etendue restriction. In this paper, we varied the laser spot diameter from 100 to 300 $\mu$m. The etendue of the exposure apparatus, which limits the spot size of EUV emission, is $1 - 3.3$ mm$^2$sr [71]. Assuming isotropic EUV radiation in the $2\pi$ direction, the allowable spot size diameter is at least 400 $\mu$m. Thus, the maximum spot size of 300 $\mu$m adopted in this paper gives the etendue of 0.44 mm$^2$sr for an isotropic emission, which is still within the allowable range.

 

Fig. 9. The peak laser intensity $I_{L,peak}$ dependence of CE with $L_\mathrm {init}$ of 100 $\mu$m and 300 $\mu$m cases, respectively, on the ($\tau _L,\phi _L$) plane. The horizontal axis represents the laser pulse duration $\tau _L$: (a), (d), and (g) correspond to 5 ns, (b), (e), and (h) to 10 ns, and (c), (f), and (i) to 15 ns, respectively. The vertical axis represents the laser spot diameter $\phi _L$: (g)-(i) correspond to 100 $\mu$m, (d)-(f) to 200 $\mu$m, and (a)-(c) to 300 $\mu$m, respectively.

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5. Conclusion

We investigated the EUV emission characteristics of CO$_2$ laser-irradiated pre-formed tin plasmas by 2D radiation hydrodynamic simulations to achieve an efficient and powerful EUV light source for the lithography of next-generation semiconductors. We found that when the CO$_2$ laser is irradiated onto tin pre-formed plasma, the heated plasma expands into the surrounding plasma, resulting in the density steeping at the ablation front, and density decreases near the laser axis due to the transverse motion of the plasma. As a result, the laser absorption fraction decreases, and the contribution from the ablation front to EUV output becomes dominant compared to that from the low-density plasmas. These properties showed that longer laser pulse widths and smaller laser spots decrease CE and all of the important parameters that physically govern CE, namely, $f_L$, $C_R$, and SP. Therefore, it was confirmed that to achieve higher CE, shorter pulses, larger laser spot diameters, and longer initial density scale lengths are necessary. Specifically, with a laser spot size of 300 $\mu$m, a pulse width of 5 ns of the peak intensity $2 \times 10^{10}$ W/cm$^2$ CO$_2$ pulse, and a pre-formed plasma density scale length $L_\mathrm {init}$ of 300 $\mu$m, it was estimated that the CE of 10% could be achieved. Furthermore, under the conditions of a large laser spot diameter, short pulse, and sizeable initial density scale length, it was found that the laser peak intensity that gives high CE spreads to the high-intensity side. In the optimization strategy for achieving high CE and EUV power per pulse, it is possible to raise the laser peak intensity compensating for the reduced EUV power due to shortening the pulse width. Our computational results showed one solution for optimizing the conditions necessary for achieving a high efficiency and high output EUV light source required to produce next-generation semiconductors.