// module M5 · scaling laws
Scaling Laws
Simon's Law of Crystal-Electromagnetic Scaling — projected against silicon, GAA, CNTs, 2D, photonic, and quantum trajectories.
§ governing principle
Simon's Law of Crystal-Electromagnetic Scaling
"As crystal lattice purity and electromagnetic coupling efficiency improve, transistor switching performance scales proportionally to the product of the piezoelectric coefficient and EM coupling factor of the crystal medium, independent of physical transistor size."
P(t) = P_Si(t) · CEFP_Si(t) = P₀ · 2^((t−t₀)/τ_eff) · CEF = (μ_crystal/μ_Si)·(1 + κ·η_EM)ECCF = κ · η_EM = clamp01(d₃₃·σ_max/V_th) · clamp01(Q/Q_critical)P_Si(t)
silicon trajectory · P₀ · 2^((t−t₀)/τ_eff)
μ_crystal/μ_Si
material multiplier
κ
piezo gating · clamp01(d₃₃·σ_max/V_th)
η_EM
EM coupling · clamp01(Q/Q_critical)
τ_eff
effective doubling period (yr)
d₃₃400 pC/N
μ_crystal/μ_Si0.00
κ1.00
η_EM1.00
ECCF1.000
CEF0.00
τ_eff3.5 y
P₀0.50 TOPS/mm²
Performance Projection · 2000–2040
TOPS/mm² · log
Parameter Controls
d₃₃ = 400 pC/N
100 MPa
200
3.5 y
0.50 TOPS/mm²
// Live derivation (V_th = 0.7 V)
κ = clamp01(400·100/70) = 1.000
η_EM = clamp01(200/100) = 1.000
ECCF = κ · η_EM = 1.000
μ_crystal/μ_Si = 0.1/1400 = 0.000
CEF = (μ_crystal/μ_Si)·(1 + ECCF) = 0.000
Breakeven Analysis
No breakeven in window
Crystal-EM Hybrid does not cross silicon within 2026–2040 at current parameters. Switch to a higher-d₃₃ crystal (PZT), raise σ_max or Q, or lower τ_eff.
Competitive Landscape · TRL & scaling outlook
| Technology | Status | Best Mobility (cm²/V·s) | Power Efficiency | Scalability | Heat Mgmt | Cost Trend | TRL (1–9) |
|---|---|---|---|---|---|---|---|
Silicon CMOS | Mature, plateauing | 1400 | Moderate | Limited <3nm | Critical | ↓ | 9 |
GAA Nanosheets | Production (Intel 18A, Samsung) | 1200 | Good | Good to 2nm | High | → | 8 |
FinFET | Mature | 1100 | Moderate | Plateaued | High | ↓ | 9 |
Carbon Nanotubes | Lab demos (MIT, Stanford) | 100,000 | Excellent | Yield-limited | Excellent | ↑↑ | 4 |
2D Materials (MoS₂) | Sub-nm prototypes (Berkeley) | 200 | Excellent | Excellent | Good | ↑ | 4 |
Photonic Computing | Niche commercial (Lightmatter) | n/a | Excellent | Wavelength-limited | Excellent | ↑ | 5 |
Quantum Computing | Research | n/a | Domain-specific | Specialized | Cryogenic | ↑↑↑ | 3 |
Crystal-EM Hybrid// thesis | Thesis (Simon's Law) | 1000 (GaN host) | Excellent | Geometry-independent | Excellent | ↑ | 2 |