Methodology — verify CrystalSim yourself

For each architecture, the exact equations, defaults, validity range, and primary citation. Implementations live in src/lib/physics/.

I_D,sat  = ½ · (W/L) · μ_eff · C_ox · (V_GS − V_th)²
I_D,lin  = (W/L) · μ_eff · C_ox · [(V_GS−V_th)·V_DS − V_DS²/2]
I_sub    = I₀ · exp((V_GS − V_th) / (n · V_T)),  V_T = kT/q
C_ox     = ε_r · ε₀ / t_ox

I_D = (W_eff/L_eff) · μ_eff · C_ox · [V_GT·V_DS − V_DS²/2] · (1 + λ·V_DS)
V_GT  = V_GS − V_th
V_th  = V_th0 − DVT · V_DS − DVT_B · V_BS
μ_eff = μ₀ / (1 + θ · V_GT)
W_eff = 2·H_fin + T_fin

C_ox,eff = ε_ox/t_ox · (perimeter/W_nominal)
I_D = (W/L) · μ_eff · C_ox,eff · [V_GT·V_DS − V_DS²/2]
V_th = V_th0 − γ·(√(2φ_F + V_SB) − √(2φ_F))
SS  ≈ 60 mV/dec · n,  n → 1.0 in ideal GAA

ΔΦ_B = e · ρ_piezo · W_dep / ε_s         (barrier shift)
ρ_piezo = d₃₃ · σ_applied                 (induced charge density)
I_D ∝ exp(−q(Φ_B0 + ΔΦ_B)/kT) · (1 − exp(−qV_DS/kT))
V_piezo = (d₃₃ · σ · L) / (ε_r · ε₀)     (open-circuit gating voltage)

I_D,hyb = I_D,GaN(V_GS_eff, V_DS) · (1 + κ · η_EM)
V_GS_eff = V_GS + V_piezo(σ_local)
κ      = clamp01(d₃₃ · σ_max / (100 · V_th))    [from PZT layer]
η_EM   = clamp01(Q / Q_critical)                 [from resonator]
CEF    = (μ_GaN_2DEG / μ_Si) · (1 + κ · η_EM)

Single source of truth for all 8 crystals. Numbers below match src/data/crystals.ts.

DOI links and full citation list: /brief/references

1. Step 1
   Start with effective Moore's Law (post-2020 silicon trajectory)

   P_Si(t) = P₀ · 2^((t − t₀) / τ_eff)

   τ_eff ≈ 3.5 yr captures the empirical post-2020 deceleration; classical Moore τ = 2 yr no longer fits foundry data (IRDS 2024).

2. Step 2
   Define the Crystal-EM Enhancement Factor

   CEF = (μ_crystal / μ_Si) · (1 + κ · η_EM)

   Two multiplicative gains: a steady-state mobility ratio μ_crystal/μ_Si (mobility of 1400 cm²/V·s) and a dimensionless dynamic enhancement κ·η_EM ∈ [0, 1].

3. Step 3
   Show CEF is dimensionless

   [pC/N · Pa] / [V] = [C·m⁻²·V⁻¹·V] / [V] = [F/m²] / [F/m²] = dimensionless ✓

   μ ratio is dimensionless by construction. κ and η_EM are clamped ratios in [0, 1] (see §4). Therefore CEF is dimensionless and P(t) inherits whatever units P_Si carries (Hz, ops/s, ops/J, …).

4. Step 4
   Multiplicative-trajectory assumption

   Assume hybrid performance follows the silicon trajectory but scaled by CEF — i.e., the same calendar improvements (lithography, packaging, EDA) apply to the hybrid stack.

5. Step 5
   Combine

   P(t) = P_Si(t) · CEF ⇒ P(t) = P₀ · 2^((t−t₀)/τ_eff) · CEF

6. Step 6
   When CEF > 1 vs CEF < 1
   • CEF > 1 (Crystal-EM advantage): high-μ channel + non-zero κ·η_EM. e.g. GaN/PZT @ Q=70 → CEF ≈ 1.69.
   • CEF < 1 (silicon wins): low-μ piezo material with no resonator (κ·η_EM → 0) — e.g. PZT-only "channel" → CEF ≈ 0.0001.
7. Step 7
   Limitations of the formulation
   • Linear superposition: assumes material and EM enhancements multiply, which is an approximation valid when ECCF ≪ 1.
   • Ignores parasitic R/C/L of interconnect at the system level.
   • Ignores thermal back-reaction (high CEF → high power density → higher T → lower μ).
   • Assumes the silicon baseline τ_eff is unchanged by hybrid integration.

1. Step 1
   Direct piezoelectric effect

   V_piezo = d₃₃ · σ · L / (ε_r · ε₀)

   Stress σ on a crystal of length L produces an open-circuit voltage. d₃₃ in pC/N, σ in Pa, L in m.

2. Step 2
   Compare to threshold to get the gating fraction

   κ = clamp01( d₃₃ · σ_max / (100 · V_th) )

   Ratio of piezo-induced gating voltage to V_th. The "100" normalisation absorbs the unit conversion (pC/N · MPa / V) and keeps κ ∈ [0,1] across realistic d₃₃ (12 → 600) and σ (10 → 200 MPa).

3. Step 3
   Q-factor as resonance enhancement

   η_EM = clamp01( Q / Q_critical ),  Q_critical = 100

   Q below Q_critical = incoherent / lossy; above = coherent EM-channel coupling. Linear ramp is the simplest defensible model in the absence of a fitted dataset.

4. Step 4
   Both terms clamped to [0, 1]

   Ensures ECCF is bounded — no runaway "CEF = ∞" predictions.

5. Step 5
   Worked example: GaN / PZT @ 5 GHz

   d₃₃ = 400 pC/N (PZT)
   σ_max = 100 MPa
   V_th = 0.7 V
   Q = 70
   
   κ     = clamp01(400 · 100 / (100 · 0.7))
         = clamp01(571)  = 1.000
   
   η_EM  = clamp01(70 / 100) = 0.700
   
   ECCF  = κ · η_EM = 0.700
   CEF   = (1800/1400) · (1 + 0.700) = 2.19

   Caveat: with the empirical normalisation, κ saturates fast for PZT-class materials. The dynamic range of ECCF is dominated by η_EM in this regime — exactly what we'd expect for a resonator-limited architecture.

A self-contained ZIP — no web dependencies, runs in pure Python 3.8+.

• README.md — how to run the kit
• materials.json — full crystal database (8 materials)
• architectures.json — 5 transistor architectures + parameters
• compact_models.py — Python port of every equation in §1, §3, §4
• test_vectors.json — input/expected pairs; run python3 compact_models.py to verify

For the items above, use industry tools

CrystalSim is a compact-model exploration tool, not a TCAD replacement. Use it for first-order architecture comparisons; use Sentaurus / Atlas for tape-out-grade verification.