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

Data: input.csv. There is one energy axis (1.196 to 1.606 eV, 1024 points) and 15 spectra, one per excitation power: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90 and 100 uW. The merged data set is the 15 columns on the shared energy axis.

Preprocessing: I removed single-pixel cosmic-ray spikes. A point was replaced by its 9-point running median when it differed from it by more than 6 robust sigma. This affected 0 to 5 points per spectrum, for example a 2066-count spike in the 45 uW spectrum.

Model: each spectrum was fitted with a constant offset y0 >= 0 plus three Lorentzians (Origin "Lorentz" form):
  y = y0 + sum_i (2*A_i/pi) * w_i / (4*(x - xc_i)^2 + w_i^2)
The intensity of a peak is its fitted height, H = 2A/(pi*w). Area A and FWHM w were also recorded.
  P1: the main PL peak, about 1.318 to 1.335 eV
  P2: a weaker, broad peak, about 1.534 to 1.540 eV
  P3: a peak at the high-energy edge of the window, about 1.613 eV. Only its low-energy side is inside the data range, so its parameters are poorly constrained. Its centre was bounded to <= 1.625 eV.

Fit results per power (xc in eV, H in counts):
P(uW)  P1 xc    P1 H    | P2 xc    P2 H   | P3 xc    P3 H
  5   1.31848   113.4  | 1.53377   64.8  | 1.61199   172.2
 10   1.32364   212.2  | 1.53352  110.4  | 1.61515   314.9
 15   1.32567   300.3  | 1.53481  157.9  | 1.61482   395.0
 20   1.32722   389.6  | 1.53560  202.9  | 1.61575   529.4
 25   1.32847   477.0  | 1.53612  242.2  | 1.61573   639.4
 30   1.32944   564.3  | 1.53721  270.7  | 1.61476   682.6
 35   1.32991   638.4  | 1.53730  306.9  | 1.61359   725.2
 40   1.33105   756.9  | 1.53856  342.4  | 1.61562   902.8
 45   1.33132   813.9  | 1.53775  367.3  | 1.61520   945.5
 50   1.33233   902.9  | 1.53718  403.1  | 1.61379   969.0
 60   1.33275  1080.3  | 1.53764  476.3  | 1.61362  1147.2
 70   1.33393  1261.9  | 1.53843  514.8  | 1.61296  1244.5
 80   1.33412  1435.1  | 1.54062  566.9  | 1.61178  1329.6
 90   1.33478  1648.7  | 1.54035  620.6  | 1.61237  1561.2
100   1.33536  1905.5  | 1.54028  657.1  | 1.61237  1673.1
(The fitted xc uncertainties are about 1-4e-4 eV for P1, 1-2e-3 eV for P2 and 3-5e-3 eV for P3. The full table, including area and FWHM, is in peaks_vs_power.csv.)

Allometric fits, y = a * P^b (unweighted least squares, 1-sigma errors):
Plot 1, peak intensity vs power (intensity_vs_power.svg):
  P1: a = 18.7 +- 1.5,  b = 0.996 +- 0.019  (R^2 = 0.9974)
  P2: a = 21.8 +- 0.9,  b = 0.744 +- 0.010  (R^2 = 0.9985)
  P3: a = 58.5 +- 4.8,  b = 0.725 +- 0.019  (R^2 = 0.9941; edge peak, less reliable)
  (Check using integrated area instead of height: b = 0.928 +- 0.014 for P1, 0.690 +- 0.013 for P2, 0.646 +- 0.020 for P3.)
Plot 2, peak position vs power (position_vs_power.svg):
  P1: a = 1.31082 +- 0.00046 eV, b = 0.00409 +- 0.00010  (R^2 = 0.993). This is a blue shift of about 17 meV from 5 to 100 uW.
  P2: a = 1.52859 +- 0.00089 eV, b = 0.00160 +- 0.00016  (R^2 = 0.884). This is a blue shift of about 6 meV.
  P3: a = 1.6158 +- 0.0016 eV,  b = -0.0003 +- 0.0003   (R^2 = 0.09). The position does not change measurably with power.

Interpretation:
The main PL peak P1 grows almost linearly with power (b close to 1, slightly sublinear by integrated area). That fits excitonic or free-carrier recombination well below saturation. Its position blue-shifts weakly with power. P2 and P3 are clearly sublinear (b about 0.7), which points to saturation, as is typical for defect- or band-tail-related emission.

The headline exponent is the one for the main peak's intensity versus power.

RESULT: b = 0.996 ± 0.019
