## 1 Introduction  
Quantum well (qw) LEDs and laser diodes are the primary photon generators in numerous modern lighting and photonics devices. For these applications, it is crucial to be familiar with the qw-spectra and the related physics. The wavelength of the spectral intensity maximum is controlled by adjusting the dimensions and material composition in the quantum well. However, for the spectral shape, it is necessary to know also the physical broadening mechanisms and their characteristic functions. They are examined here and compared against the spectral broadening in gases. This resemblance is useful for reviewing the homogeneous and inhomogeneous broadening effects and postulating a unified and pervasive qw-lineshape. Pressure-related lifetime broadening [1,2] is used to describe effects arising from the increased carrier density in the quantum well confinement. Physically relevant lineshape functions are convolved with the theoretically predicted unbroadened emission spectrum. Convolution theorem and fast Fourier transform (FFT) are applied to evaluate the convolution integrals [3]. The resulting lineshapes are compared against each other and to the experimental data from the literature. The postulated unified broadening function is concluded to be valid for both the homogeneously and inhomogeneously broadened qw-LED spectra. Optical emission and absorption lines are classified to be homogeneous if all active population members behave identically, which requires they have identical transition energies and probabilities. In gases, the active population consists of freely moving atoms or molecules. Their velocities follow the Maxwell-Boltzmann distribution determined by the temperature. The related Doppler-shifts disperse the population and make the transition energies inhomogeneously broadened. In normal conditions and low gas pressures, the Doppler broadened, Gaussian distribution represents the minimum line width at a given temperature:  
\[L_{ih}(\Delta E)=\frac{1}{\sigma\sqrt{2\pi}}\text{exp}(-\frac{\Delta E^{2}}{2 \sigma^{2}}). \tag{1}\]  
Here \(\Delta E\) is the energy deviation from the unbroadened spectral line and \(\sigma\) the line width parameter. With increasing pressure, the gas molecules collide more frequently with each other. Collisions decrease the lifetime of the optical energy levels and make them lifetime broadened. The average collision frequency depends on the gas molecule size, pressure, and temperature. Through lifetime broadening, it determines the average linewidth of the population members. At high enough pressures, the originally Gaussian lineshape gradually transforms to a Lorentzian line typical for the homogeneous broadening:  
\[L_{h}(\Delta E)=\frac{\tau/(\pi\hbar)}{1+\tau^{2}\Delta E^{2}/\hbar^{2}}, \tag{2}\]  
where \(\tau\) is the lifetime parameter. Between the two extremes of Equations (1) and (2) the spectrum follows the Voigt profile [2], obtained by convolving these two functions. The associativity of the convolution integral guarantees that the convolution with the Voigt function produces identical results as would be obtained by convolving with the Gaussian and Lorentzian functions one after another. In LEDs, the emission originates from the electron and hole pair recombinations, and the broadening is at the easiest observable just below the bandgap energy. Instead of perfect and abrupt bands, in measurements, semiconductors usually show more or less exponential band edges. Exponential Urbach tail is a usual absorption edge behavior in disordered solids and often met also in the emission and absorption spectra of semiconductor materials [4; 5; 6; 7; 8]. It is also common in the electroluminescent (EL) and photoluminescence (PL) spectra of quantum well LEDs [9; 10; 11; 12]. The Urbach effect reflects the tail in the joint density of states (JDOS), combining the broadening in the valence and conduction band energy states. Instead of the Doppler effect, it arises from the lattice potential disorder inside the material.