## 4 Discussion  
### Quantum Dot (QD) Modeling  
To explain the resonant absorption around the 1550 nm wavelength, we calculate the energy levels and dipole matrix elements by solving the Schrodinger equation with four essential assumptions: (1) that the quantum dots are perfect spheres, (2) that the binding energy of electrons and holes is determined by the known ErAs-GaAs band offsets, (3) that ErAs and GaAs are exactly lattice matched, and (4) there are enough ErAs unit cells in each dot that we can apply the effective-mass approximation (EMA). The ErAs-GaAs band offsets are illustrated in Figure 7a [10]. Note that ErAs is not a semiconductor in bulk form, but rather a semi-metal lacking a direct band gap with the \(\Gamma\) and X band overlapping in energy. We have calculated the bound energy levels for the X- and \(\Gamma\)-point envelope functions vs. the quantum-dot diameter d. The most probable optical dipole transition is between a \(\Gamma\)-point S state (quantum numbers \(l=0\), n = 1), and an X-point P state (\(l=1\), n = 2, m = -1, 0 or 1). The resulting spontaneous emission coefficient A(\(\lambda\)) for this transition has a maximum \(\lambda_{\rm p}=1476\) nm (Figure 7b) which is equivalent to a photon energy of 0.84 eV, for d = 2.6 nm. Generally speaking, the absorption coefficient \(\alpha(\lambda)\) is proportional to the product of N \(\times\) A(\(\lambda\)) \(\times\) g(\(\lambda\)), where N is the population difference between two energy levels or approximately by the number of QDs, and g(\(\lambda\)) is the normalized lineshape function with a fullwidth at half-maximum determined by the homogeneous and inhomogeneous broadening of the QD array [6]. Therefore, the size distribution of QD diameters can broaden the absorption spectrum around the peak wavelength \(\lambda_{\rm p}\). Nevertheless, our calculations reveal that the optical transition between two quantum-dot energy levels can explain the resonant behavior around \(\lambda=1550\) nm. Furthermore, we estimate that the spontaneous lifetime is on the order of \(\rm T_{\rm sp}=1/[A(\lambda_{\rm p})]=60\) ns, which is comparable to the spontaneous lifetime for QDs reported in the literature e.g., References [5; 6], and much greater than the spontaneous lifetime in typical bulk direct-bandgap semiconductors. Figure 6: Power spectrum of 1550 nm photomixing. Next we show that the photon-excited level(s) of our QDs can be resilient to thermal de-population effects such as phonon scattering. The key ingredient is excitonic binding. Figure 7a shows the \(\Gamma\)-point ground level E\({}_{1}\), the X-point upper level E\({}_{2}\) and a bound-exciton level E\({}_{3}\) just below it. The bound exciton should be very stable and have a large oscillator strength. It is analogous to the indirect \(\Gamma\)-X exciton in bulk silicon, but with a much higher binding energy and, therefore, temperature stability. As is well known in many-body theory, the energy levels obtained above through the Schrodinger equation and the EMA is not a complete picture. There is a correlative effect between an excited electron in a quantum dot and the hole that remains in the ground level that the electron started in levels. In semiconductors this correlation is associated with the fundamental quasiparticle called an exciton. The exciton binding energy in the limit of small diameter quantum dots is estimated by the famous Brus formula [11]. Strictly speaking, this is valid only for the ground state, (\(\Gamma\)-point) n = 1, l = 0, to (X-point) n = 1, l = 0 transition in the quantum dots. Evaluation for the 2.6 nm quantum dot (supporting the 1550 nm optical transition) yields E\({}_{\text{ex,dd}}\) = 0.132 eV, \(\approx\)18 times larger than the bulk binding energy. And remarkably, it is much greater than the thermal energy k\({}_{\text{B}}\)T = 0.025 eV (k\({}_{\text{B}}\) being Boltzmann's constant) at T = 300 K, meaning that if such excitons were excited optically, they would be quite stable thermally even in the presence of electron-photon scattering.