File structure is clear: single workbook `Book1` with one column per power level (5–100 µW, 15 powers), x-axis in eV, peak around 1.23–1.45 eV. No separate Si/reference folder exists, so no merge or background subtraction is needed — the data is already organized by power.

Despiking is mandatory before any fit (cosmic rays destroy Lorentzian fits):



Despiking complete — 130 cosmic-ray spikes removed across all 15 power columns and written back. Clean spectra show the main PL peak near ~1.33 eV.

Now running the power-dependent analysis: Lorentzian fit on every power column, extracting intensity and xc, building both panels (intensity vs power, xc vs power), and the allometric fit:



![Power Fit]([file link removed])

All 15 power spectra (5–100 µW) were despiked, Lorentzian-fitted (window 1.23–1.45 eV), and the peak intensity + center (xc) extracted per power. Both panels above carry their fits: the **intensity panel** has the allometric fit $I = a\cdot P^{b}$, and the **xc panel** shows the peak-position trend.

**Allometric fit — intensity vs power ($I = a\cdot P^{b}$)**
- $a = 16.28 \pm 1.81$
- $b = 1.0048 \pm 0.0261$
- $R^2 = 0.995$, n = 15 → essentially linear (excitonic, radiative-dominated)

**Peak position vs power (xc)**
- Monotonic **blueshift**: 1.3206 eV (5 µW) → 1.3356 eV (100 µW), ≈ 5 meV/decade
- FWHM narrows 62.4 → 54.4 meV over the same range

Per-power extraction (Lorentzian, wing-baseline intensity):

| P (µW) | Intensity | xc (eV) | FWHM (meV) | R² |
|---|---|---|---|---|
| 5 | 98.8 | 1.3206 | 62.4 | 0.887 |
| 10 | 189.9 | 1.3244 | 64.4 | 0.950 |
| 15 | 267.9 | 1.3264 | 64.1 | 0.958 |
| 20 | 353.8 | 1.3278 | 62.7 | 0.968 |
| 25 | 425.2 | 1.3290 | 61.7 | 0.973 |
| 30 | 516.0 | 1.3300 | 61.3 | 0.975 |
| 35 | 564.8 | 1.3304 | 60.8 | 0.978 |
| 40 | 694.6 | 1.3315 | 59.3 | 0.981 |
| 45 | 738.5 | 1.3318 | 59.1 | 0.981 |
| 50 | 818.3 | 1.3328 | 59.0 | 0.981 |
| 60 | 935.9 | 1.3332 | 58.0 | 0.982 |
| 70 | 1140.4 | 1.3343 | 57.1 | 0.983 |
| 80 | 1264.7 | 1.3345 | 56.3 | 0.983 |
| 90 | 1538.0 | 1.3351 | 55.3 | 0.984 |
| 100 | 1714.6 | 1.3356 | 54.4 | 0.984 |

All 15 columns passed at the HIGH tier (R² 0.887–0.984, SNR 7.6–24). Full table: [power_fit_results.csv]([file link removed])

Note: the allometric (power-law) form is well-defined for intensity (which passes through the origin). For xc, a raw $y=a\cdot x^b$ is not physical (xc must stay ≈1.32 eV at zero power), so that panel is reported as its fitted blueshift trend rather than a forced power-law exponent.

RESULT: b = 1.0048 ± 0.0261