Four strands of lithography numbers, moved on one board: the linewidth and depth of focus set by the Rayleigh criterion, the photons that run short as the price of a shorter wavelength, the yield that multiplies in series over hundreds of steps, and the size of the field that reduction projection allows in a single exposure.
① Resolution and depth of focus
CD = k₁ · λ / NA · DOF = k₂ · λ / NA² · back-calculation: k₁ = CD · NA / λ
Light source and optics (the three generations of Exercise 1)
How to set k₁
Resolution stated by the tool
CD at current setting
Resolution vs. ArF immersion
Relative DOF (ArF immersion = 1.00)
DOF = k₂λ/NA²
DOF vs. EUV (NA 0.33)
EUV gains 14.3× from wavelength and loses 4.1× from NA, for a net gain of 3.5×. Because its NA is lower, its depth of focus is actually 17% more generous than that of ArF immersion. The real problem is High-NA. DOF shrinks to 0.36× that of EUV and 0.42× that of ArF immersion, about 22 nm. A 50 nm photoresist is already thicker than the DOF.
② Same dose, fewer photons
E = hc/λ · EEUV/EArF = 193/13.5 · fluctuation ratio = √((193/13.5)/m) · texp ∝ dose/source power
EUV dose multiple m (same dose as ArF = 1)
Photon energy EUV · ArF
Photon count (vs. ArF)
Relative fluctuation (vs. ArF)
Exposure time (source power unchanged)
Source power needed for the same exposure time
Throughput limit
"Limit" is the throughput when exposure time dominates the total. With a fixed time T₀ for wafer transfer and alignment, the drop is smaller, and how much smaller depends on how large T₀ is relative to the original exposure time (if T₀ equals the exposure time, throughput falls to 13%; if it is ten times as long, to 45%). So in practice the dose is not raised 14.3-fold to match the photon count. ASML rates EUV tool throughput at 30 mJ/cm² (NXE:3600D, 3800E) and 50 mJ/cm² (High-NA EXE:5000), and better resist sensitivity, design workarounds and metrology make up the rest.
③ Multiplying 800 steps
Per-step yield y = Y1/n · allowed defect rate = 1 − y · inverse: line yield = yn ≈ e−(1−y)n
Inverse: per-step yield
Per-step yield y
Allowed defect rate per step
When the step count grows 1.5×, the allowed defect rate falls to 2/3. A per-step yield of 99% sounds excellent, yet after 800 steps almost nothing comes out. Hundreds of steps multiply in series, so each step must be perfect to the ppm, not "nearly perfect". A decision that adds steps (multi-patterning) adds yield risk on top of cost.
④ Reduction projection and the largest die
Wafer field = pattern area on mask (104 × 132 mm) ÷ reduction ratio · largest-die yield = e−A·D₀
Optics
Field area = largest die
Largest-die yield · 0.33 NA (858 mm²)
Largest-die yield · High-NA (429 mm²)
Area at 50% yield A₅₀ = ln2/D₀
The field is halved so that the 6-inch mask can stay. The 104 × 132 mm area on the mask is unchanged; only the direction reduced 8× becomes 132 ÷ 8 = 16.5 mm. High-NA draws finer patterns but halves the area drawn in one exposure. An 800 mm²-class die cannot be printed in one High-NA field, so it needs stitching or chiplets.
Check against the solutions ()
Values from Session 5, Section 4 and the solutions to Exercises 1–5, recomputed with this figure's formulas and rounded to the digits the solutions give. The "Initial state" rows read the figure's starting values, not the values you have moved.
Item
Computed
Solution
What this calculation leaves out
k₁ = 0.3 is near the practical floor for single exposure (theoretical floor about 0.25). Finer patterns are split over several exposures with multi-patterning. That goes around k₁ rather than lowering it.
k₂ is held the same for all three generations. This is the assumption of Exercise 2.
Throughput is not plotted. The text gives throughput = 1/(T₀ + texp), but no value for the fixed time T₀. So only the exposure-time ratio and the required source power move, and throughput appears only as its limit when exposure time dominates everything (1/m), marked "limit".
Photon arrival is treated as a Poisson process (relative fluctuation 1/√N). Resist improvements that let one photon trigger more reactions (chemical amplification, metal-oxide resists) are left out. Dot counts in the photon figure show ratios only.
Line yield assumes total yield is the product of step yields. The inverse curve and readouts use the exact formula yn, as the table in the text does. The small-defect-rate approximation e−(1−y)n is noted in small type in the readout.
Largest-die yield uses the Poisson yield formula of Session 4. Stochastic defects add directly to this D₀. A mask defect repeats at the same spot in every field printed with that mask, so it behaves differently from random defects.