Lithography §

A lens is a finite-aperture system. When a point source is imaged through a circular aperture, the image is not a point but an Airy disk: a bright central spot surrounded by concentric rings of decreasing intensity. The radius of the first dark ring is

Two point sources are resolvable only when the first minimum of one Airy disk falls at or beyond the center of the other (the Rayleigh criterion). This gives a minimum separation of $\Delta x \approx \lambda / \mathrm{NA}$.

The underlying reason is that the finite aperture of the lens truncates the spatial frequency spectrum of the image. A perfect image of an arbitrary pattern would require all spatial frequencies up to infinity; a lens with numerical aperture NA can only transmit frequencies up to $f_{\max} = \mathrm{NA} / \lambda$. Features with spatial frequency above this cutoff are simply lost, setting a hard lower bound on the smallest printable feature.

This is the Abbe diffraction limit: no optical system can focus light to a spot smaller than approximately $\lambda / (2 \, \mathrm{NA})$. It is not an engineering limitation that better lenses can overcome; it is a consequence of the wave nature of light.

• Wavelengths and generations:

• EUV lithography: 13.5 nm light is generated by laser-ablated tin droplets in a vacuum chamber and focused by a multi-layer Mo/Si Bragg reflector (no glass optics exist at this wavelength). Enables 7 nm and below nodes.
• High-NA EUV: The next generation uses a numerical aperture of 0.55 (versus 0.33 for standard EUV), improving resolution by a factor of $0.55/0.33 \approx 1.67$. Each tool costs approximately $380 M. High-NA EUV is targeted for sub-2 nm logic nodes (expected deployment 2025–2027).
• Immersion: In ArF immersion, a layer of ultrapure water (refractive index 1.44) fills the gap between the final lens element and the wafer, effectively increasing NA from 0.75 to 1.35 without changing the wavelength.
• Multi-patterning: At nodes below the single-exposure resolution limit, the same area is exposed multiple times with different masks to create features smaller than the diffraction limit. See the diagram below.

• Resolution: < 10 nm (down to ~1 nm with advanced tools). Used for mask fabrication and R&D.
• Throughput: very slow (serial writing). A single 300 mm mask can take hours to days.
• Proximity effect: forward- and back-scattered electrons expose neighboring areas; correction algorithms (e.g., PROX) are required.
• Use: mask blanks for photolithography, nanophotonic structures, quantum devices, research prototyping.
• Variants: variable-shaped beam (VSB) EBL writes rectangles at high speed; multi-beam EBL (e.g., Vistec, Elionix) uses hundreds of beams in parallel to improve throughput.

• Wavelength: ~0.5 nm (soft X-rays), far shorter than EUV.
• Principle: X-rays pass through a phase mask (no absorption mask needed at this wavelength) and expose a resist on a wafer in close proximity to the mask (no projection lens; demagnification is achieved geometrically).
• Status: Demonstrated at IBM and others in the 1990s–2000s; not adopted for high-volume manufacturing because EUV proved more practical. Remains of interest for mask inspection and specialized applications.

• Principle: A focused ion beam (typically Ga⁺ or He⁺) writes the pattern directly, analogous to EBL but with ions instead of electrons. Ions have negligible de Broglie wavelength, so there is no diffraction limit.
• Resolution: ~1–5 nm (limited by beam spot and ion-induced damage).
• Advantage: Ions implant into the substrate, enabling direct material modification without a separate etch step.
• Use: Focused ion beam (FIB) systems are used for mask repair, cross-sectioning, and nanoscale prototyping. Not used for high-volume IC patterning due to low throughput.

• Principle: A physical stamp (mold) with the inverse pattern is pressed into a thermoplastic or UV-curable resist, mechanically transferring the pattern. No light, no electrons, no ions.
• Resolution: < 10 nm demonstrated; limited by stamp fabrication and alignment.
• Advantage: No diffraction limit; very low cost per wafer once the stamp is made.
• Limitations: Alignment for multi-layer stacking is challenging; defect control at high volume is difficult; throughput is lower than optical scanners.
• Use: Research, some display backplane manufacturing, and potential future high-volume applications.

• Optical Proximity Correction (OPC): The mask geometry is deliberately distorted (serifs, hammerheads, line-end extensions) so that the image on the wafer matches the intended pattern. OPC compensates for the fact that isolated lines, dense lines, and corners all diffract differently. Computed by rigorous electromagnetic simulation of the optical system.

• Phase-Shift Masks (PSM): Adjacent transparent regions of the mask are given a 180° phase difference. The resulting destructive interference at the boundary sharpens the image edge, effectively improving resolution by ~20–30% without changing λ or NA. Types: attenuated PSM (aTSM) and alternating PSM (aPSM).

• Off-Axis Illumination (OAI): The illumination is shifted or shaped (annular, quadrupole, dipole) so that only specific diffraction orders reach the lens. This improves image contrast for particular pattern types (e.g., dense line/space) at the cost of other pattern types. Often combined with OPC.

• Inverse Lithography Technology (ILT): The mask is treated as a grid of pixels (or sub-pixels), and an optimization algorithm (often gradient-based or adjoint-method) computes the pixel intensities that produce the target wafer image. ILT subsumes OPC and can achieve the best possible image fidelity for a given optical system, at the cost of enormous computational resources and mask complexity.

• Hybrid approaches: Modern leading-edge layers typically combine OAI + ILT + advanced resist + multi-patterning. The choice of RET stack is a co-optimization problem involving the optical system, the resist, and the downstream Etching process.

• Positive resist: exposed regions become soluble in developer and are washed away. Most common in IC manufacturing (e.g., chemically amplified resists based on polycaprolactone).
• Negative resist: exposed regions cross-link and become insoluble. Used for high-resolution or high-aspect-ratio structures (e.g., SU-8 for MEMS).
• Thickness: 0.1–2 µm for IC; up to 100 µm for thick-film or MEMS.
• Coating: spin-coating at 2000–5000 rpm produces a uniform film; post-apply bake (PAB) removes residual solvent.
• EUV resists: require new chemistries because 13.5 nm photons are absorbed in the top ~50 nm of the resist, generating secondary electrons that do the actual exposure. Stochastic effects (shot noise) become significant at the smallest features.
• Role in the fab flow: After development, the resist pattern serves as a mask for Etching of the underlying Deposition layer. The resist is then stripped and the cycle repeats for the next layer.

ArF immersion:

• Single-exposure resolution: $R = k_1 \cdot \lambda / \mathrm{NA} = 0.25 \times 193 / 1.35 \approx 35.7$ nm.
• Required half-pitch: 21 nm. Ratio: $35.7 / 21 \approx 1.7$.
• Since the ratio is between 1 and 2, double patterning (2×) is needed: two masks, two exposures, plus a spacer etch step to create the intermediate lines. Total: 2 litho exposures + 1 spacer deposition + 1 spacer etch + 2 alignment steps.
• If the required pitch were tighter (e.g., 15 nm), triple patterning (3×) would be needed: 3 masks, 3 exposures, 3 alignment steps.

EUV low-NA:

• Single-exposure resolution: $R = 0.30 \times 13.5 / 0.33 \approx 12.3$ nm.
• Required half-pitch: 21 nm. Since 21 nm > 12.3 nm, a single exposure suffices with a comfortable margin ($k_1^{\text{effective}} = 12.3/21 \approx 0.59$, well within the 0.25–0.35 design target when accounting for process margin).
• Total: 1 mask, 1 exposure, 1 alignment step.

Comparison:

Conclusion: EUV single-exposure is strongly preferred for the 21 nm half-pitch gate layer: it halves the mask count, eliminates the spacer process, and reduces alignment error accumulation. ArF immersion multi-patterning remains viable for less-critical layers where EUV is not yet cost-effective or where the required pitch is above ~25 nm.