Computational study of laser-produced plasma, EUV generation, and the impact of magnetic fields

The advancement of high-power lasers has spurred interest in investigating magnetic field effects on laser-driven plasma, given its strong connections to the research fields of photonics,¹ inertial confinement fusion,^2,3 and particle physics in plasmas.⁴ Especially, it provides proper conditions to conduct laboratory astrophysical research,^5–7 enabling a better understanding of the stability, dynamics, and heating of plasma flows in the presence of magnetic fields, which are ubiquitous throughout the universe.

However, there has been limited exploration of the impact of magnetic fields on the generation of Extreme Ultraviolet (EUV) from the laser-produced plasma (LPP).^8,9 Although magnetic fields were used to deflect Sn ion debris away from sensitive optical components inside EUV source systems and to modify the expansion of plasmas,^10–13 these approaches use constant/static magnetic fields, and their strength is limited to sub-Tesla levels. However, recent advancements in laser-related concepts and pulsed power drives have enabled the utilization of external magnetic fields with strengths reaching tens of Tesla.^14,15 Such strong magnetic fields can significantly alter plasma behavior and consequently impact EUV generation. Increasing EUV output power is critical for increasing wafer throughput and vital to the entire semiconductor industry.

Previous studies, particularly those focused on laboratory astrophysics, have shown that when axial magnetic fields are aligned with the plasma expansion direction, they act as guides for ion streams, resulting in the collimation of plasma into an axisymmetric jet flow.^5,11 In addition, studies employing LPP expanding into a transverse magnetic field showed that the plasma plume becomes confined within a slender, rapidly elongating slab. This slab structure is influenced by the magnetized Rayleigh–Taylor instability (MRTI), where a slab can develop kink-like motions that disrupt its propagation.^6,7,16 In addition to the study of plasma dynamics, significant efforts have been devoted to developing theories and validating thermal conductivity models in magnetized plasmas. While the heat flux in plasma is typically parallel to the temperature gradient, the presence of magnetic fields alters this relationship due to the cyclotron motion of electrons. As a consequence, anisotropic thermal conductivity arises in magnetized plasmas.^17,18

As noted earlier, prior research has primarily concentrated on plasma dynamics in the presence of magnetic fields, and the direct link between these changes in plasma behavior and the generation of EUV radiation remains largely unknown. Further comprehensive studies are required to establish a clearer understanding of this relationship and EUV generation in magnetized plasma environments.

In this paper, we focus on the parameter ranges of laser intensities of $<1011$ W/cm², and magnetic field strengths up to a few tens of Tesla to investigate the influence of magnetic fields on plasma dynamics, heating processes, and 13.5 nm radiation generation. To model all the components accurately, we employ a two-stage simulation approach: the radiation hydrodynamics code FLASH¹⁹ and the atomic code SPECT3D²⁰ to model laser-driven plasma and EUV generation, as well as radiation transport.

In FLASH simulations, the Sn target (spherical liquid droplet) has an initial density of 7 g/cm³ and a diameter of 100 μm and is set up in a simulation box of 300 × 300 μm² size in Cartesian coordinates. The laser pulse, with a temporal shape of 3 ns flat top +2 ns for rising and falling slopes, is incident from the bottom boundary of the simulation. Here, FLASH uses a ray-tracing algorithm to compute laser energy deposition using the inverse Bremsstrahlung mechanism. As FLASH uses a three-temperature fluid model, the tabulated equation of state and opacity,^21–23 and multigroup diffusion radiation transfer, it can accurately compute energy transfer between fluids and radiation throughout the plasma evolution. Figure 1(a) shows different time snap-shots of the hydrodynamic simulation of the Sn target (mass density). During the interaction with a laser pulse with a power of 5 × 10⁵ W, Sn plasma expands isotropically, with an equal expansion in different directions, as clearly seen at 4 ns. The detailed properties of the expansion at 6 ns [Fig. 1(c)] are found in the lineout (y direction) in Fig. 1(b). Sn plasma shock front (y ∼ −170 μm) has a density of 10⁻⁵ g/cm³, more than four orders of magnitude lower density than the initial target, and its velocity is $∼5×106$ cm/s, which is far higher than the ion sound speed; this velocity is on the same order of magnitude as the value derived from the simplified scaling,²⁴  v (cm s⁻¹) = 4.6 × 10⁷I^1/3λ^2/3, where I is the laser intensity in units of 10¹⁴ W/cm² and λ is the laser wavelength in μm. Another theoretical estimation,²⁵  v_front ∼ 2c_sln$(τ+τ2+1)$⁠, that includes charge separation effects, gives similar values. Here c_s is ion acoustic velocity and $τ=ωpit/2e$⁠, with ω_pi the plasma frequency and e the Euler number. The vacuum region in simulation is prefilled with background hydrogen with a density of 10⁻⁷ g/cm³, which is sufficiently low but may have a minor impact on the expansion velocity.

To model plasma formation and heat transfer with a magnetic field accurately, the simulation needs a description of particle motion affected by magnetic fields. The FLASH code includes magneto-hydrodynamic (MHD) models with different algorithms. In addition, it includes anisotropic thermal conduction with the Epperlein and Haines model,²⁶ which is important for modeling heat transport in magnetized plasma. For a comprehensive examination of the dynamics of laser-driven plasma depending on a magnetic field, a series of simulations are carried out, where detailed setup parameters are presented in Table I. As mentioned earlier, plasma expansion velocity depends on laser intensity. We primarily utilize laser intensities from 5 × 10⁸ to 5 × 10¹⁰ W/cm², which aligns with the laser parameters used in recent studies with 1 μm wavelength^27–29 and 10 μm wavelength.^8,9 In addition, two different wavelengths of 1 and 10 μm are compared. Magnetic fields, up to 50 T, are initially set up in two different directions: parallel to the laser direction (y) and transverse to the laser direction (x).

The notable result obtained from the comparison of simulations with B-fields is a clear dependence of plasma expansion on the direction of the applied magnetic field. Figures 1(c)–1(e) present the ion densities of Sn plasma in three different cases: (c) no B-field, (d) B-field in the x-direction, and (e) B-field in the y-direction. These cases correspond to scenarios a, c, and f, as listed in Table I. While plasma expansion is isotropic without B-field (c), plasma expansion is constrained in cases with B-field (d) and (e). When a B-field is in the x-direction, plasma is almost free expanding in the x-direction but decelerated in the y-direction. Here, thermal β (plasma pressure/magnetic pressure) is slightly less than unity and the dynamic β, β = ρv²/[B²/(2 μ₀)] (ram pressure/magnetic pressure), is about 1–2, meaning a pressure balance between plasma expansion pressure and ambient magnetic pressure, constraining diamagnetic cavity expansion [see Fig. 2(b)]. In this case, the Hall parameter for electrons (λ_mfp,e/r_L,e) is greater than unity, which also indicates that electrons are magnetized. It is not shown in the x − y plane in our two-dimensional simulation but previous studies^6,7 describe that the curved shock layer induced by the magnetic field plays a crucial role in redirecting the plasma flow toward the tip of the cavity. This redirection effectively collimates the plasma flow in the y-direction, resulting in a narrow slab-like plasma structure when viewed in the y − z plane. This slab shape is a characteristic signature of the confined plasma in the presence of transverse B-fields. When the B-field is aligned parallel to the plasma expansion (y-direction), the plasma experiences significant confinement in the x-direction, leading to the formation of a plasma jet shape [see Fig. 1(e)], which is consistent with the previous studies that have discussed plasma collimation in astrophysical models.^5–7  Figure 1(f) is the same case as Fig. 1(d) but with a stronger B-field strength of 50 T, where plasma (both electrons and ions) is fully magnetized and magnetic pressure dominantly alters plasma expansion.

One effective way to characterize plasma confinement due to a magnetic field is by comparing the expansion velocity in two different (longitudinal and transverse) directions, as direction-dependent velocity indicates how the shape of the magnetized plasma evolves. Figure 2(a) summarizes all simulation cases (presented in Table I). Here, the flow velocities are measured at the plasma shock front, where the velocity is slightly lower than the maximum expansion velocity, which occurs in the cavity region, on the inner side of the shock front. From plot (a), first, it is noticeable that both velocities (v_x and v_y) increase as laser intensity rises from 5 × 10⁸ to 5 × 10¹⁰ W/cm² [see case (b)–(d)]. Second, it is clear that the B_x constrains plasma expansion in y-direction as cases (b), (c), and (e) show higher v_x than v_y. Although the B-field is in the same direction, case (d) has higher v_y as high plasma thermal pressure with high laser intensity boosts the expansion. In contrast, the B_y confines plasma in the x-direction [cases (f) and (h)], forming a collimated plasma jet. Finally, a much higher expansion velocity (v_y) from 10 μm wavelength case (h) is seen compared to the case of 1 μm wavelength laser (f). This is associated with laser energy deposition into the Sn target where the critical density, n_c ≈ 1.1 × 10²¹/λ², for 10 μm wavelength is 100 times lower compared to 1 μm. Therefore, a higher wavelength laser deposits more energy into a lower density of expanding plasma, leading to increased heating and faster ablation velocity. Figure 2(b), a snapshot of the plasma at 6 ns, provides more detailed physical parameters, including the expanding velocity gradient corresponding to the electron temperature gradient. It is also seen from the B-field map that plasma at the shock front encounters the initial external B-field, while inside the cavity region, the plasma behaves as a conductive medium and a relatively lower B-field remains.

The data obtained from the FLASH simulation at defined time steps is exported to SPECT3D for further analysis and post-processing, where it calculates the emission, absorption, and ionization properties of the plasma using Collisional-Radiative (CR) kinetic models with frequency-dependent opacities based on non-local thermodynamic equilibrium (LTE) atomic level populations.²⁰ Then the data are converted into spectral fluxes by virtual detectors. This work stands out as the first to use these codes to simulate EUV generation from magnetized LPP.

Figure 3 presents a comparison of the radiation generated from a laser with a power of 5 × 10⁶ W in two different scenarios: with and without the applied B-field of 10 T. As described earlier, the presence of an external B-field (transverse direction) has a significant effect on the plasma expansion, resulting in confined expansion along the laser axis. This trend is also evident in the ion number lineout, depicted in Fig. 3(a). In Fig. 3(b), the intensity of radiation with a photon energy of 92 eV (corresponds to ∼13.5 nm wavelength) is plotted. The radiation intensity is measured by a virtual detector placed at a distance of 0.025 cm away from the initial Sn target. The radiation intensity varies as a function of distance from the initial Sn target. For instance, the intensity reaches its peak value at a position located ∼0.02 cm away from the detector, which corresponds to a distance of about 50 μm from the initial Sn target. As radiation passes through the plasma plume, it experiences attenuation, leading to a decrease in its intensity. The final intensity of the radiation measured at the detector is significantly lower than its peak value. It appears that the case without the B-field exhibits a higher peak intensity within the plasma plume. However, when the radiation reaches the detector, the intensity is lower compared to the simulation with the B-field. It can be inferred that the radiation interacts more with the plasma and undergoes more attenuation as plasma is not confined by a B-field. It is worth noting that within the plasma at a distance of 0.01 cm, the optical depth is ∼0.4, indicating that radiation is noticeably absorbed, while there is essentially no radiation loss in the case of magnetically confined plasma. This trend is also evident in the spectral emission characteristics shown in Fig. 3(c). This feature highlights the significance of plasma dynamics and the influence of magnetic fields on radiation generation and transport.

It is well known that EUV radiation from LPP exhibits angle dependence, which is important for characterizing EUV sources. Our simulations incorporating plasma dynamics, optical depth, and anisotropic thermal transport allow us to capture the trend of angular-dependent EUV modeling from magnetized LPP. As shown in Fig. 4, in-band EUV (2% bandwidth around 13.5 nm) is measured at different angles (10°, 30°, 50°, and 70° from the laser axis). In-band EUV from LPP with a B_x shows the highest value across all detection angles. On the other hand, the case of the B_y shows slightly higher values than the scenario with no B-field. Especially, the strong increase in EUV flux near the laser axis with the B_x (50% higher value than the case of no B-filed) corresponds to the trend observed in Fig. 3, which is associated with the confined plasma and radiation transport. The integrated in-band EUV flux, considering all angles, exhibits about a 40% increase when a magnetic field is applied in the x-direction, compared to scenarios where no magnetic field is present. The overall conversion efficiencies (integrated over time and angle) of in-band EUV are 1.6%, 1.3%, and 1.2% for the cases of Bx = 10 T, By = 10 T, and no magnetic field, respectively. This signifies a maximum 33% enhancement in conversion efficiency for LPP when a magnetic field is applied. It is worth noting that total conversion efficiencies may appear relatively modest. However, this is attributed to the LPP scheme employed in this study, which involves the interaction of a single main pulse with a spherical target. This approach contrasts with the conventional technique of using a prepulse followed by the main CO₂ pulse, hence influencing the observed conversion efficiency levels.

In summary, our parametric studies using simulations show the substantial influence of an applied magnetic field on LPP dynamics and radiation generation. The results show that the confinement of plasma expansion strongly depends on the magnetic field’s strength and direction. Furthermore, the transport of charged particles in this magnetically confined plasma favors effective plasma heating. For instance, in the case of a plasma with a 10 T B-field, the maximum temperature is ∼10%–20% higher than the case without B-field. In addition, a significant finding is that the EUV output undergoes notable alterations within magnetized plasma. The density profile of a confined plasma with effective heating, which in turn allows the radiation to escape more easily with reduced loss. The comparisons of angle-dependent in-band EUV flux further emphasize that applying the magnetic field in the transverse direction is beneficial for achieving high yield flux. The ability to control and optimize EUV radiation emission by applying a magnetic field opens up exciting possibilities for enhancing EUV lithography for semiconductor processing.

This work was supported by ASML, San Diego. The authors also thank M. Purvis for a helpful discussion.