Modelling a Quantum Dot Ensemble
Contents
1.1 Spatially Resolved Description of an Ensemble of Quantum Dots
1.1 Material Gain of an Inhomogeneously Broadened QD Ensemble
1.1 Carrier Induced Refractive Index of QD Media
1 Modelling an Ensemble of Quantum Dots
Fig. 1.1: Animation showing the growth of an In_{0.6}Ga_{0.4}As QD on an In_{0.1}Ga_{0.9}As substrate. The QD is overgrown with a layer of In_{0.1}Ga_{0.9}As forming a quantum well with a height of 6.5 nm. Note: To start the animation hover the mouse pointer over the figure content.
The size of SOQDs is remarkably uniform, the fluctuations being in the range of only 10 % [3]. The SK growth method has been successfully applied using molecular beam epitaxy and metalorganic chemical vapour deposition.In general it is desired to produce QD arrays with high dot density in the range of 10^{11}/cm^{2} . A further increase of the number of QDs can be achieved by stacking layers of QDs. The thickness of the barriers between the layers determines the degree of vertical alignment and electronic coupling [5]. Due to the strain field induced by the QDs in the first layer the islands in the next layer tend to form just above the QDs in the bottom layer [3].
An optoelectronic device based on QDs (leaving out microcavity structures with a low number of QDs) has a size of the order of 10^{4} μm^{2} and contains a large number of QDs (of the order of 10^{6} − 10^{8}) that are excited simultaneously [8]. Equations (1.13), (1.14), and (1.15) describing the time evolution of the occupation probability of the bound QD states and the microscopic polarisation refer to a single QD. In order to simulate a QDSOA, it is therefore essential to make suitable approximations.
1.1 Spatially Resolved Description of an Ensemble of Quantum Dots
The principle of sampling is demonstrated in Fig. 1.2. For this purpose, we randomly draw values from a Gaussian distribution centered at 0 and with a standard deviation of 2. Each random value is represented by a square (shown in the inset). We start with a sampling size of 100 points and increase the sampling set up to 400000 points. As the number of points in the sampling set increases, the corresponding histogram (red curve) approaches the statistical distribution (green curve) the values are following.
Fig 1.2: The red graph represents a histogram (normalised frequency count) of values randomly drawn from a set of points that are normally distributed. The green curve is a plot of the Gaussian distribution centered at 0 and with standard deviation 2. The inset shows the set of values that have been drawn, each square representing one specific value (visualised as color shade).
To obtain a statistically representative subgroup of QDs, we assume that the mean of the Gaussian distribution is given by the centre energy of the homogeneously broadened spectral lines. A typical gain spectrum of a single InGaAs QD is represented by the green curve in the Fig. 1.3. For a single QD, the transitions are clearly visible (ground state transition at 1.09meV) and the broadening of the lines is given by ∆E_{hom}^{FWHM}. (FWHM stands for Full Width at Half Measure.)
An animation can be started by hovering the mouse cursor over the figure. It shows the average material gain of an ensemble of QDs as a function of energy. The inhomogeneously broadened gain profile is obtained by successively averaging over an increasing number of nonidentical QDs. The averaging leads to a broadening of the gain profile due to the inhomogeneous broadening. We notice that the averaged gain profile does not change significantly once the sampling rate exceeds approximately 10^{3} QDs.
Fig. 1.3: Material gain of an inhomogeneously broadened ensemble of InGaAs QDs. Inhomogeneous broadening QD ground state: 30meV. Inhomogeneous broadening QD excited states: 55meV. Homogeneous broadening QD ground state: 8meV, Homogeneous broadening QD excited states: 12meV.
The standard deviation of the normal distribution describing the inhomongeneous broadening is: σ=∆E_{FWHM}·[2√{2ln2}]^{−1}, where ∆E_{FWHM} is the FWHM of the inhomogeneously broadened spectral line.In order to integrate the two dimensional waveequation (1.34) numerically, the area of the device is discretised in approximately 10^{4} gridpoints (depending on the dimensions of the device and the wavelength of the emitted radiation). Assigning one representative QD per numerical gridpoint is a way to obtain the statistically representative subgroup of QDs. In this way a group of e.g. 10^{8} QDs is represented by a subgroup of 10^{4} QDs. Physical properties described by parameters that vary from dot to dot (like confinement energies, scattering rates, and the coefficients describing spectral broadening and shift) are calculated and assigned for each representative QD before starting the numerical integration. In this way, the inhomogeneous broadening, a characteristic property of an ensemble of QDs with varying sizes, is incorporated into the numerical model.
1.2 Material Gain of an Inhomogeneously Broadenend QD Ensemble

The index ξ runs over all QDs in the statistical ensemble, whereas i and j labels QD electron and hole levels, respectively. We have assumed that the transverse electric field is polarised in xdirection (compare with Fig. ). As a result the dipole matrix element tensor is multiplied with the unit vector in xdirection e_{x}. Additionally, the following definitions have been used:

Figure 1.4 shows the transient gain and emission spectrum recorded during the first 15ps after the startup of a QD laser. At timepoint 0ps the gain is negative (absorbtive medium) due to the low carrier occupation probability of the QD states. The material gain then gradually increases due to carrier capture into the QDs. Note that the gain spectrum differs from the one presented in Fig. 1.3 since in this case only the ground state and the first excited state have been included. Moreover, the gain is negative at energies around 1.12eV indicating that the excited state has carrier occupation probability below 0.5.
Fig.1.4: Transient gain and emission spectrum recorded during the startup of a QD laser with a cavity length of 600μm and an inhomogeneous broadening of 30meV. Recording time 15ps.
1.3 Carrier Induced Refractive Index of QD media

The carrier induced refractive index during the startup of a QDSOA is shown in Fig. 1.5. The QDSOA contains QDs with an inhomogeneous broadening of 30meV and and areal density of 10^{11}cm^{2}. After applying a voltage of 1.20V the carrier concentration in the 2D layer adjacent to the QDs increases. This leads to a increased scattering of carriers into the QDs inducing a change of the refractive index.
Fig. 1.5: Carrier induced refractive index during the startup of a QDSOA. A voltage is applied at timepoint 0 ps. The concentration of electrons and holes in the QDs increases gradually leading to a change of the refractive index.
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