PN Junction model from Sze

[mdmaas]

Summary

Implements a PN-junction depletion model for the 2D TW-MZM workflow, replacing the hand-picked junction capacitance with textbook physics.

Formulas (Sze & Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, ch. 2)

• Built-in voltage: V_bi = (k_B T / q) ln(Na·Nd / ni²)
• Abrupt-junction depletion width: W = sqrt(2·eps_s·(V_bi + V_R)/q · (Na+Nd)/(Na·Nd)), split asymmetrically x_p = W·Nd/(Na+Nd) into P and x_n = W·Na/(Na+Nd) into N
• Linearly graded junction: W = [12·eps_s·(V_bi+V_R)/(q·a)]^(1/3) with grade constant a = |dN/dx|
• Depletion capacitance: C_j = eps_s / W (per area), C = eps_s·A/W absolute

Doping concentrations use the industry-standard cm⁻³; lengths in µm.

User API

from gsim.common.stack.junction import PNJunctionConfig

junc = PNJunctionConfig(na_cm3=1e19, nd_cm3=1e19, v_reverse=0.0)
junc.w_um, junc.xp_um, junc.xn_um, junc.capacitance(length_um, height_um)

Two representation modes, auto-selected from W vs the flanking doped sections (high-res when W >= ~1/5 of the smaller flank, capacitance otherwise; manual override via mode=):

• capacitance: geometry unchanged (adjacent P/N); computed C applied as a lumped Impedance boundary — one call: sim.set_pn_junction(junc, layer_p="p_rib", layer_n="n_rib", length_um=10.0, height_um=0.22)
• high_res: contiguous depleted-Si dielectric strip of width W drawn between P and N on its own GDS layer and resolved on the actual mesh (pure real eps, no carriers); no lumped boundary

The profile builder returns doped regions, P/N regions, junction metadata (widths/capacitance/chosen mode):

pn = make_pn_junction_profile(comp, ..., junction=junc, p_region=("p_rib", (21,0), sigma),
                              n_region=("n_rib", (20,0), sigma),
                              junction_region=("junction", (22,0)), mode="auto")
pn["junction"]["mode"], pn["junction"]["c_f"]

Demo

The 2d_twmzm notebook now illustrates this: default doping (~1e19 cm⁻³, W≈17 nm) auto-selects capacitance mode (13.9 fF vs the previous hardcoded 1 fF), and a sweep shows W crossing the threshold into high-res mode.

Also fixed

• build_doped_cross_section() never registered the merged doping/rib materials on stack.materials, so doped domains silently resolved to eps=1.0 without conductivity in generated Palace configs (visible in prior demo artifacts). Now registered, with regression test.

Tests

41 new tests: physics values/limits/scalings and error paths, profile geometry/materials/auto-mode boundary cases, and end-to-end Palace checks (Impedance Cs value in capacitance mode; junction dielectric domain group with eps=11.9/no conductivity in high-res mode).