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A modelling routine has been developed to quantify effects present in p-modulation doped 1.3 μm InAs/InGaAs quantum dot laser and modulator devices. Utilising experimentally verified parameters, calculated modal absorption is compared to measurements, prior to simulation of structures under reverse and forward bias. Observed broadening and a reduction of absorption in p-doped structures is attributed primarily to increased carrier scattering rates and can bring benefit when structures are configured as optical modulators with enhancements in the figure of merit. However, increased carrier scattering limits the maximum modal gain that can be achieved for lasers. The state filling caused by p-doping only marginally reduces absorption but assists laser operation with increased differential gain and gain magnitude at lower current densities.
Quantum dots (QDs) are promising candidates as an alternative to quantum wells (QWs) in laser sources for photonic integrated circuits (PICs) (Smowton and Blood 2010). This is due to lower threshold current densities (Dikshit and Pikal 2004), lower temperature sensitivity (Arakawa and Sakaki 1982), and greater tolerance to threading dislocation defects incurred during epitaxy on silicon substrates (Zhang et al. 2018). Additionally, III-V compound semiconductors offer high electron mobility and direct bandgaps at telecom wavelengths (Zhao et al. 2021), with InAs/InGaAs QDs demonstrating high performance in lasers at 1.3 μm (Qiu et al. 2001). These properties make III-V QDs attractive for use in silicon photonics, creating opportunities for mass producing low-cost PICs, provided remaining challenges can be overcome (Norman et al. 2019).
Full device band diagram calculated in Nextnano for undoped structure including corrections, with active region shown in inlay. Conduction band Ec, valence band, Ev, and overlapping electron and hole quasi-Fermi levels, Efe and Efh respectively
A one-dimensional geometry was employed, modelling the band structure as a function of the growth direction (along the z-axis), hence only the layer thickness and average dot height were required as structural inputs. This removed quadratic or even cubic dependencies associated with two- and three-dimensional computation times. QD regions were initially predicted with QW-like properties, with only one dimension of confinement available. Further approximations were input via the QD layer material parameters to correct the effective density of states, effective mass, and band gap energy to reflect experimental measurements. The effective density of states, used to calculate band edge densities and hence critical to the magnitudes of the generation/recombination processes involved in the current continuity equations, was initially predicted as a QW, following Eq. 1.
\(m_DOS^*\) described the average of the effective mass tensor relating the effective mass \(m^*\) along each axis, in each band. In the EMA, bands are considered parabolic and isotropic near the band edge, with \(m_x^*=m_y^*=m_z^*\). Although only the z-dimension was evaluated by the Schrdinger-Poisson equations, the current continuity equations required an average of all dimensions via \(m_DOS^*\). The in-plane effective masses \(m_x^*, m_y^*\) were reduced to alter the effective density of states while not impacting calculations of band alignment and quantization. Therefore, an effective mass tensor ellipsoid was employed, shown in Fig. 4, equating the effective density of states in the QD regions to twice that of the dot density observed during epitaxy, accounting for spin degeneracy. This reduction corrected the recombination/generation magnitudes in the QD regions and provided realistic calculation of the movement of the global quasi-Fermi levels across the full structure.
Undoped and p-doped structures were simulated under reverse, zero, and forward bias to predict the QCSE, modal absorption, and peak gain respectively, with the central dot layer shown in figs. 5 and 6.
Optical confinement factor weightings at each layer of dots were determined using an eigenmode expansion solver from Lumerical, with the mode profile and weightings as percentages shown in Fig. 7. This permitted calculation of modal absorption and gain, which was directly comparable to experimental measurements. A layer thickness of 3MLs was used for the QD layers, equal to the quantity of InAs deposited during epitaxy. Note, this differed from the 5 nm thickness used in the full device band structure calculations, which referred to the mean height of the QDs observed during AFM. InAs dot layers, shown in the table of Fig. 7, were labelled as deposited, 1 through to 7.
Fitted experimental data for modal absorption coefficient against photon energy for undoped structure. LD ground and excited state, and SD ground state fitting functions represented by solid and dashed lines respectively
Fitted experimental data for modal absorption coefficient against photon energy for p-modulation doped structure. LD ground and excited state, and SD ground state fitting functions represented by solid and dashed lines respectively
Preceding data from Nextnano, Lumerical, combined with parameters extracted from fitted absorption data were used with an in-house program for calculating absorption and gain, similarly to (Dikshit and Pikal 2004), using Eq. 3. Reverse and forward bias conditions employed in the band structure calculations were used to predict the QCSE and gain spectra respectively.
The calculated modal absorption under zero bias was compared to experimental data to verify the modelling procedure prior to simulating structures under applied reverse and forward bias to predict the QCSE and gain. The inhomogeneous broadening and dot density was assumed to be constant in these calculations. Appropriate replication of the band edge absorption, though with some departure at the first excited state was observed in figs. 8 and 9.
However, this difference was only a fraction of the total difference between p-doped and undoped absorption spectra and we therefore attributed the larger part of the effect to greater homogeneous broadening caused by increased carrier scattering rates. A 41% increase in the carrier scattering rate, calculated from the values of τ, used in the homogeneous broadening term was necessary to describe the remaining difference between p and undoped structures (Figs. 10 and 11).
When reverse bias was applied, we calculated further depletion of the QD layers in the p-doped structure. This led to an increase in the effective absorption with applied reverse bias, an uncommon but beneficial phenomenon for modulating devices. The band diagrams, shown in figs. 12 and 13, demonstrated increased deformation in the active region of the p-doped structure, signifying the non-uniformity of carrier depletion across the QD layers.
Qualitatively, under a reverse bias of 5 V, the relevant quasi-Fermi levels in the undoped structure were always distant from the dot potential signifying minimum carrier-blocking. Whereas, the hole quasi-Fermi level in the p-doped structure remained very close to two dot layers under a 5 V reverse bias. This signifies appreciable population of holes in valence states, with continued blocking of absorption compared to the undoped structure.
The calculated absorption spectra for undoped and p-doped structures are shown in Fig. 14 under a zero and 5 V reverse bias respectively. Stark shifts of 7.5 nm and 5.4 nm are observed in undoped and p-doped structures respectively, suggesting enhanced ER in the undoped structure. The reduced broadening and higher magnitude in absorption coefficient in the undoped structure shown in Fig. 14 gives a sharper absorption edge and is indicative of potentially higher peak ERs. The p-doped data derived from an absorption spectrum that is significantly broader than the undoped spectrum and increases in magnitude under bias, resulting from growing depletion in the conduction band, reducing carrier blocking, may have benefits for modulating devices, with a low IL and improved temperature stability as suggested by (Mahoney et al. 2021).
Although there is a reduction in the peak of the IL, this is not necessarily an improvement to the modulator performance, as the increased broadening caused protrusion of the absorption edge at higher wavelengths where the QCSE is most pronounced.
The laser performance of the structures was predicted under forward bias to calculate modal gain and peak modal gain with the resulting band diagrams shown in figs. 18 and 19, and calculated modal gain spectra in figs. 20 and 21. A linearly increasing carrier scattering rate was included to account for the predicted increase with carrier injection, following (Zhang et al. 2018). Initial values were derived from the fitted absorption spectra increasing to a value around three times that in the undoped structure for the given p-dopant concentration.
Conduction band occupation for the first bound state in each dot layer for undoped and p-doped structures. Solid and translucent lines represent the mean, and the minimum to maximum between the seven layers respectively
Valence band occupation for the first bound state in each dot layer for undoped and p-doped structures. Solid and translucent lines represent the mean, and the minimum to maximum between the seven layers respectively
Degree of inversion for the first bound state in each dot layer for undoped and p-doped structures. Solid and translucent lines represent the mean, and the minimum to maximum between the seven layers respectively
Figure 23 exhibits the effect of state filling for holes in the ground state of the valence band as a result of p-doping. The significance of this reduction can clearly be linked to potential decreases in the threshold current density of p-doped laser diodes. Figure 24 further supports these claims, with a significantly reduced transparency current density shown by the degree of inversion in the p-doped structure, with a reduced variance across all dot layers. This is contrary to several examples (Deng et al. 2022; Korenev et al. 2017a; Dong et al. 2020) of experimental measurements observing increased threshold current densities with p-doping. This discrepancy is attributed to an optimal range of p-doping concentration which is difficult to measure and to appropriately incorporate in real devices. This is corroborated by (Zhang et al. 2018) with calculations of threshold current density versus increasing p-doping concentration and experimentally verified reductions in transparency at appropriate p-doping levels. P-type modulation doping has been found to improve gain but also increase internal optical loss (Shchekin and Deppe 2002) and non-radiative recombination (Jang et al. 2008) with reduced threshold current typically observed under high gain operation.
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