PL power series: Lorentzian peak fits and allometric (power-law) fits

Data: input.csv holds one merged spectrum per excitation power (15 powers: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90, 100 uW) on a common energy axis, 1.196 to 1.606 eV, 1024 points.

Spectra: the dominant emission peak sits near 1.32 to 1.34 eV. There is a weaker, broad band near 1.56 eV, and the signal rises at the high-energy edge (above 1.59 eV) from a feature whose centre lies outside the measured range, so that feature cannot be fitted reliably.

Lorentzian fit (per spectrum): y = y0 + (2A/pi) * w / (4(x - xc)^2 + w^2), intensity = peak height H = 2A/(pi w).
Main peak fitted over 1.196 to 1.46 eV; secondary band over 1.44 to 1.585 eV.

Main peak (P1):
Power(uW)  xc(eV)            H(counts)       A        w(eV)
  5        1.3195 ± 0.0004    115.5 ± 1.6     11.39   0.0628
 10        1.3246 ± 0.0003    217.4 ± 1.9     22.74   0.0666
 15        1.3266 ± 0.0003    307.4 ± 2.5     32.07   0.0664
 20        1.3280 ± 0.0002    397.4 ± 2.9     40.29   0.0645
 25        1.3292 ± 0.0002    487.2 ± 3.2     48.83   0.0638
 30        1.3302 ± 0.0002    578.8 ± 3.6     58.68   0.0645
 35        1.3306 ± 0.0002    653.8 ± 3.9     65.57   0.0638
 40        1.3317 ± 0.0002    776.5 ± 4.1     76.79   0.0630
 45        1.3319 ± 0.0002    836.0 ± 4.4     82.53   0.0628
 50        1.3329 ± 0.0002    928.9 ± 4.8     92.29   0.0632
 60        1.3333 ± 0.0001   1111.2 ± 5.6    108.66   0.0623
 70        1.3344 ± 0.0001   1301.2 ± 6.1    126.48   0.0619
 80        1.3346 ± 0.0001   1480.0 ± 6.8    142.37   0.0612
 90        1.3352 ± 0.0001   1700.8 ± 7.5    161.79   0.0606
100        1.3357 ± 0.0001   1966.3 ± 8.3    184.16   0.0596

Secondary band (P2, broad, about 0.15 eV wide): xc stays at 1.558 to 1.574 eV; H rises from 133 (5 uW) to 992 (100 uW).

Plots (no matplotlib on the server; the plotted data are the tables above):
 1) Peak intensity vs power, with allometric fit y = a * x^b
 2) Peak position vs power, with allometric fit y = a * x^b

Allometric fit results (y = a * P^b, P in uW):
 P1 intensity (height): a = 18.8 ± 1.5,         b = 1.003 ± 0.019,      R^2 = 0.9974
 P1 integrated area:    a = 2.31 ± 0.13,        b = 0.945 ± 0.014,      R^2 = 0.9984
 P1 position xc:        a = 1.3123 ± 0.0005 eV, b = 0.00390 ± 0.00010, R^2 = 0.991 (blueshift of about 16 meV from 5 to 100 uW)
 P2 intensity (height): a = 42.6 ± 2.4,         b = 0.684 ± 0.014,      R^2 = 0.997
 P2 position xc:        b = -0.0024 ± 0.0007 (essentially flat, R^2 = 0.48)

Interpretation: the intensity of the main peak grows linearly with power (b ≈ 1.00), which is typical of excitonic emission with no sign of saturation. The peak shifts slightly to higher energy as power increases.

RESULT: b = 1.003 ± 0.019
