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. Builds on #233 (contains its commits; this PR shrinks to the feature commit once #233 merges).

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

Built-in voltage:

$$V_{bi} = \frac{k_B T}{q},\ln!\left(\frac{N_A N_D}{n_i^2}\right)$$

Abrupt-junction depletion width under reverse bias $V_R$:

$$W = \sqrt{\frac{2,\varepsilon_s,(V_{bi}+V_R)}{q}\cdot\frac{N_A+N_D}{N_A,N_D}}$$

Asymmetric split of the depletion into the two sides:

$$x_p = W,\frac{N_D}{N_A+N_D} \quad \text{(into P)}, \qquad x_n = W,\frac{N_A}{N_A+N_D} \quad \text{(into N)}$$

Linearly graded junction (grade constant $a = |dN/dx|$):

$$W = \left[\frac{12,\varepsilon_s,(V_{bi}+V_R)}{q,a}\right]^{1/3}$$

Depletion capacitance:

$$C_j = \frac{\varepsilon_s}{W} ;;\text{(per unit area)}, \qquad C = \frac{\varepsilon_s A}{W}$$

Doping concentrations use the industry-standard $\mathrm{cm^{-3}}$; 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 $W \ge \tfrac{1}{5}\min(\text{P flank},\text{N flank})$, 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 $\varepsilon_s$, no carriers); no lumped boundary

The profile builder returns the doped regions, the P/N regions, and the 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: the default junction ($N_A=N_D=10^{18},\mathrm{cm^{-3}}$, $V_R=0$, with $T$ and $n_i$ exposed as user parameters) gives $W \approx 50$ nm and auto-selects high-res mode — the depletion strip is drawn on the actual mesh. A doping sweep in the demo shows $W$ crossing back below the threshold into capacitance mode at heavier doping.

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 $C_s$ value in capacitance mode; junction dielectric domain group with $\varepsilon = 11.9$/no conductivity in high-res mode).

[codecov]

Codecov Report

❌ Patch coverage is 80.49536% with 63 lines in your changes missing coverage. Please review.
✅ Project coverage is 61.99%. Comparing base (c3f26a4) to head (a152263).
⚠️ Report is 39 commits behind head on main.

Files with missing lines	Patch %	Lines
src/gsim/palace/runtime.py	67.37%	34 Missing and 12 partials ⚠️
src/gsim/common/stack/junction.py	88.28%	7 Missing and 6 partials ⚠️
src/gsim/common/stack/doping.py	96.55%	1 Missing and 1 partial ⚠️
src/gsim/palace/base.py	75.00%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #242      +/-   ##
==========================================
+ Coverage   59.98%   61.99%   +2.01%     
==========================================
  Files          94      106      +12     
  Lines       13334    15382    +2048     
  Branches     2703     3028     +325     
==========================================
+ Hits         7998     9536    +1538     
- Misses       4439     4869     +430     
- Partials      897      977      +80     

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