Abstract
We first examine the relaxation dynamics inside quantum dot structures. After presenting the rate equations model, we investigate the effect of some parameters introduced in the rate equation on the performance of quantum dot lasers. The effects of QDs coverage factor, inhomogeneous broadening, which its physical source is the size fluctuation of quantum dot in forming self-assembled quantum dots, as well as cavity length, on SAQD laser have been analyzed. Then, based on the rate equations, a circuit model will be introduced. Finally the effect of phonon bottleneck and Auger recombination on the performance of quantum dot lasing, are examined. It is shown that, there is more output power and quantum efficiency, and higher modulation bandwidths when Auger recombination is considered for these lasers.
Keywords
- self-assembled quantum-dot laser (SAQDL)
- inhomogeneous broadening
- phonon bottleneck
- coverage factor
- relaxation dynamic
1. Introduction
The reduction in dimensionality caused by confining electrons (or holes) to a thin semiconductor layer leads to a dramatic change in their behavior. This principle can be developed by further reducing the dimensionality of the electron’s environment from a two-dimensional quantum well to a one-dimensional quantum wire and eventually to a zero-dimensional quantum dot. In this context, however, dimensionality refers to the degree of electron momentum freedom. In fact, within a quantum wire, unlike the quantum well where the electron is confined in just one dimension, it is confined in two dimensions and thus, the freedom degree is reduced to one. In a quantum dot, the electron is confined in all three-dimensions, hence reducing the degree of freedom to zero. Under certain growth conditions, when a thin semiconductor layer grows on a substrate having a completely different lattice constant, the thin layer is spontaneously arranged or changes into quantum dots through self-assembles while attempting to minimize the total strain energy between the bonds. Microscopy can show quantum dots that are in the shapes of pyramids, square based, and tetrahedron [1]. The performance expected from quantum dot lasers is often due to the density of their quasi-atomic states. Using the quantum structures confined in some dimensions will reduce the momentum freedom of the carrier in a certain direction. Ideally, carriers are completely enclosed in quantum dots. Therefore, the density of quantum dot states, that is, the number of states per volume unit and per energy unit, is expressed by the delta function. The gain spectrum amplitude is determined only by homogeneous broadening due to intraband relaxation at the quantum dot. These gain properties are the basis of the features that give the quantum dot laser some advantages over conventional lasers [2, 3]. The effect of carrier dynamics on the performance of quantum dot laser and the possibility of bi-exciton lasing have been studied. Bi-excitons are achieved if the increase in ground state of the quantum dot reaches the laser threshold, and if the carrier relaxation is rapidly below 100 ps, the laser will be observed [4, 5]. Carrier relaxation in quantum dots (QDs) is studied widely when applications of these devices are reported for optical communications [6]. The problem of
2. Theoretical background
2.1 Carrier capture and relaxation dynamics into a quantum dots
The carrier relaxation process within quantum dots is actually two steps, as shown in the Figure 1.
One is the carrier relaxation from continuous energy levels within the discrete levels of the quantum dot (A, Red Color). Another is the relaxation between discrete levels within dots (B, Blue Color). In many light experiments, as well as in quantum dot lasers, carriers go through these two steps unless they are brought directly into discrete levels by excitation resonant or tunneling. Since the principle of energy conservation must be satisfied for the carrier to relax, relaxing carriers transfer the corresponding energy to other particles (such as phonons) and to other carriers in the bulk. Therefore, “the relaxation rate strongly depends on the density of final levels and on the number of particles other than the transition matrix elements” [4]. Introducing the LO- and LA-phonons, it is possible to satisfy the energy conservation rule [11], is shown in Figure 2a [4, 12]. Such a two-phonon process decreases the lifetimes severely [4], but it cannot be adequate to relax back the carriers inside the dots deeply. Carrier trapping into the dot energy levels and hence energy conservation rule, are satisfied whenever the number of carriers outside the dots are increased. So, Auger process can be proper phenomenon due to increase of the captured carriers that occupy the QD energy levels [13]. Therefore, with more injected currents or equivalently more injected carriers, the relaxation lifetime increases and Auger process can be effective due to carrier relaxation into deep lower levels by a step-like energy decrement. Illustration of the carrier relaxation processes, considering the Auger effect, is shown in Figure 2b. Auger scattering is more important when the density of excited carriers is high. As illustrated in Figure 2b, one of the two electrons in the wetting layer (WL) is captured by the Auger mechanism in the excited state (ES) of QD, while the other electron is emitted upward in the WL or separate confinement heterostructure (SCH). Then, the electron which is captured in ES transfers its energy to the third electron of the barrier or even more probably to another electron in the excited state (ES) of QD and relaxes downward in the QD ground state (GS).
2.2 Rate equation model
Usually, the carrier and photon behaviors in SAQD semiconductor lasers are expressed by a set of coupled differential equations called rate equations [14].
In Figure 3, there is a simple energy band diagram to explain different levels, ground state (GS), excited state (ES), and wetting layer (WL) state, in these structures [14]. In this model, there are some injected carriers into a SCH barrier with a rate of
They can either relax with a time of
With the following parameters:
In the above equations,
Where
and
Parameters
Where
In this relation,
Where
Where
Where
Where
2.3 Effect of parameter variations
Section 2.3 covers investigation of the effects of some parameters expressed in the rate equations on the performance of quantum dot laser. This section is based on the results obtained the reference [22].
2.3.1 Inhomogeneous broadening Γ 0
Figure 4 shows the results of our simulations on the
This effect can be studied in the dynamic response of a SAQD laser, too. The results of such variations for the frequency responses are plotted in Figure 5. The simulation results show that for higher inhomogeneous broadening factors the frequency responses of the laser deteriorates.
2.3.2 Carrier recombination time τqr in a WL
The effect of carrier recombination in WL,
The effect of carrier recombination in the WL state on the small-signal frequency response of laser is shown in Figure 7. As shown, the carrier recombination in the WL state has no considerable effect on modulation response.
2.3.3 Carrier recombination inside quantum dot, τr
Figure 8 shows the effect of carrier recombination inside quantum dot,
The effect of carrier recombination inside quantum dot on frequency response has been shown in Figure 9. What is important here is that, as
2.3.4 Carrier escape time from ground state to excited τeGS state and from excited state to WL state τeES
The effect of carrier escape time from ground state to excited state and from excited state to WL state on
Figure 11 also shows that, as the carrier escape time degrades from ground state to excited state and from excited state to wetting layer, the frequency response degrades as well.
2.3.5 Coverage factor ξ
Figure 12 show the frequency response for different amounts of QDs coverage factor
Figure 13 shows the simulation results for the effects of coverage factor
2.3.6 Cavity lengths L
Figure 14 shows the
2.3.7 QD height
Figure 15 shows the simulation results of modulation response for different quantities of QD height. As the QD height degrades, the modulation band width improves. The reason for modulation band width improvement while the QD height degrades can be caused by increasing carrier confinement within quantum dot in growth direction (
2.3.8 Stripe width of the laser cavity
Figure 16 shows the effect of stripe width of the laser cavity on frequency response. As the stripe width of the laser cavity degrades, the modulation band width improves. It is inferred from the figure that degradation of the stripe width of the laser cavity and therefore the degradation of the active region can provide a higher total capture rate. Hence, it results in a greater modulation band width.
3. Circuit model implementation
To solve the rate equations of a SAQD laser, considering the excited state and Auger effect, a conceptual equivalent electrical circuit is proposed is shown in Figure 17. The aforementioned equations convert to some simple electrical circuit equations and then, the resulting circuit is simulated by a circuit simulator such as HSPICE [23]. The corresponding parameters for the equivalent circuit model of SAQD lasers are described in detail in Ref. [24].
4. Simulation results
In this simulation, typical parameters, which are shown in Table 1, are used.
Quantity | Value |
---|---|
Inhomogeneous broadening, | 20 meV |
Diffusion in SCH, | 6 ns |
SCH recombination, | 4.5 ns |
WL recombination, | 3 ns |
Capture from WL to ES, | 1 ps |
Capture from ES to GS, | 7 ps |
ES and GS recombination, | 2.8 ns |
Energy separation SCH and WL, state | 84 meV |
Average energy separation WL and ES | 100 meV |
Average energy separation ES and GS | 80 meV |
Average recombination energy from GS, EGS | 0.96 eV |
Average recombination energy from ES, EES | 1.04 eV |
Spin-orbit interaction energy of QD material, Δ | 0.35 eV |
Facet reflectivity, | 30%, 90% |
Active region length, | 900 μm |
SCH thickness, | 90 nm |
WL thickness, | 1 nm |
Active region width, | 10 μm |
Active region volume, | 2.2 × 10−16 m3 |
QD density of QD per layer, | 5 × 1010 cm−2 |
QD density per unit volume, | 6.3 × 1022 m−3 |
QD optical confinement factor, | 0.06 |
Intrinsic absorption coefficient, | 1 cm−1 |
Spontaneous emission coupling efficiency, | 10−4 |
Output power coupling coefficient, | 0.449 |
SCH, WL, and QD (GS, ES) diode ideality factors, | 2 |
Figure 18 illustrates the output power as a function of injected currents neglecting the Auger effect for
This phenomenon can be considered in dynamic response of the laser, too. Application of the proposed model to SAQD lasers considering this effect for dynamic response of the laser are shown in Figure 19. With higher relaxation times
Figure 20(a) and (b) illustrate the output power as a function of the injected currents considering the Auger effect for
Figure 21 shows the effect of Auger coefficient increment on the small-signal frequency response of the laser for different values of
5. Conclusions
The Auger effect can be overcome in the phonon bottleneck problem in SAQD lasers propose a modified equivalent circuit for simulation of the rate equations, considering this effect for the first time. The static and dynamic behavior of these lasers were studied. It is found that, if we want to solve the rate equations of a SAQD laser, it is possible to compensate the effect of phonon bottleneck by introducing the Auger effect. This can decrease the threshold current and increase the quantum efficiency of the SAQD laser, so the output power can be raised. It is shown that the Auger effect can increase the modulation bandwidth of a SAQD laser. Quantum dot lasers have been more favored for applications in optical communications, silicon photonics, lightwave telecommunication systems, high power lasers, Q-switched or mode locked lasers for short pulse generation and broad band light sources. The future implications of quantum dot lasers are issues such as quantum dot lasers and integration with Si photonic integrated circuits and colloidal quantum dot lasers.
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