PL power series — Lorentzian 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), 1024 points each, 1.196–1.606 eV.

Preprocessing: cosmic-ray spikes removed (points more than 8 robust sigma above a 7-point running median were replaced by that median; 1–2 points in the 45, 50, 60, 80 and 100 uW spectra, e.g. a 2066-count spike at 1.605 eV in the 45 uW spectrum).

Lorentzian fit model (every spectrum): y = y0 + sum_i A_i / (1 + 4((x - xc_i)/w_i)^2), where A = peak height (intensity), xc = peak centre, w = FWHM.
Three Lorentzians: Peak 1 (main PL band, ~1.33 eV), Peak 2 (~1.54 eV), and a third term for the band cut off at the high-energy edge (>1.6 eV). That third term is only a nuisance term: its centre lies outside the data window, so its parameters are not reliable and it is not reported. R^2 was 0.92 at 5 uW and 0.97–0.994 at higher powers.

Fit results (intensity A in counts, xc in eV, FWHM w in eV):
P(uW)  A1      xc1     w1      A2     xc2     w2
  5    122.7   1.3180  0.0692   79.1  1.5363  0.1150
 10    226.5   1.3231  0.0707  108.8  1.5306  0.0902
 15    319.3   1.3252  0.0700  153.2  1.5321  0.0868
 20    409.8   1.3269  0.0675  198.6  1.5333  0.0815
 25    500.7   1.3281  0.0663  243.4  1.5345  0.0834
 30    593.8   1.3291  0.0667  279.7  1.5362  0.0870
 35    669.1   1.3295  0.0657  297.7  1.5352  0.0799
 40    792.6   1.3307  0.0646  351.9  1.5372  0.0850
 45    853.0   1.3310  0.0643  370.3  1.5358  0.0834
 50    947.7   1.3320  0.0647  412.7  1.5353  0.0837
 60   1131.8   1.3324  0.0636  489.7  1.5358  0.0811
 70   1321.5   1.3336  0.0628  544.2  1.5372  0.0851
 80   1502.1   1.3338  0.0621  617.4  1.5398  0.0904
 90   1723.3   1.3344  0.0613  659.0  1.5392  0.0856
100   1989.4   1.3351  0.0602  722.4  1.5392  0.0864

Allometric fit y = a * P^b (unweighted least squares, standard errors):
Intensity vs power:
  Peak 1: a = 20.1 ± 1.6,   b = 0.990 ± 0.019  (close to linear, as expected for exciton-like emission)
  Peak 2: a = 17.8 ± 0.9,   b = 0.804 ± 0.011  (sublinear)
Peak position vs power:
  Peak 1: a = 1.3103 ± 0.0005 eV, b = 0.00412 ± 0.00010  (blueshift of about 17 meV between 5 and 100 uW)
  Peak 2: a = 1.5284 ± 0.0021 eV, b = 0.0014 ± 0.0004   (nearly constant; shifts about 3 meV)
Checks: a fit weighted by the fit errors gives b = 0.954 ± 0.015 for the Peak 1 intensity. Using the integrated area (pi/2 · A · w) instead of the height gives b = 0.911 ± 0.014.

Plots: intensity_vs_power.svg (both peaks, with their allometric fits), position_vs_power_peak1.svg and position_vs_power_peak2.svg (position vs power, with their allometric fits). All files are in the run folder.

The reported exponent is the one for the main PL peak (Peak 1): intensity vs power.

RESULT: b = 0.990 ± 0.019
