*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 *L*-*I* feature has been shown in **Figure 10**. As the carrier

**Figure 9.** *The modulation response of a SAQD laser for different values of carrier recombination inside quantum dot.*

#### **Figure 10.**

*The* L*-*I *curve of a SAQD laser for different values of carrier escape time from ground state to excited state and from excited state to wetting layer.*

escape time degrades, the number of carriers in WL states (*Nq*) increases. These carriers are generally used due to carrier non-radiative recombination (with *τqr* lifetime) and this leads to the increase of the threshold current. Generally, for more decrement in *τqr*, there is more increment in the threshold current.

**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 *ξ* = 0.1, 0.2, 0.4, relaxation time 100 ps and inhomogeneous broadening of *Г*<sup>0</sup> = 20 meV. As shown in **Figure 12**, the increase in coverage factor due to the

*Investigating the Role of Auger Recombination on the Performance of a Self-Assembled… DOI: http://dx.doi.org/10.5772/intechopen.102042*

#### **Figure 11.**

*The modulation response of a SAQD laser for different values of carrier escape time from ground state to excited state and from excited state to wetting layer.*

#### **Figure 12.**

*Modulation response of a SAQD laser at different coverage factors;* Γ*<sup>0</sup> = 20 meV and the relaxation time is 100 ps.*

increase in the volumetric density of QDs (*ND*) leads to the decrease in *PGS* and *PES*, that is, filling probability of the GS and ES. As filling probability of the GS and ES decreases, the relaxation rate increases for the carrier inside the GS and ES, and this leads to the increase in 3 dB bandwidth.

**Figure 13** shows the simulation results for the effects of coverage factor *ξ* = 0.1, 0.2, 0.4, inhomogeneous broadening *Г*<sup>0</sup> = 5 meV and relaxation time 1 ps on the frequency response band width. To achieve a high-speed modulation, higher than 10 GHz, not only the relaxation lifetime should be decreased to about 1 ps but also the QD coverage factor should also be increased and the inhomogeneous broadening should be decreased.

#### *2.3.6 Cavity lengths* L

**Figure 14** shows the *L*-*I* characteristics for different cavity lengths. The increase in cavity length leads to loss degradation and output power increase.

#### **Figure 13.**

*The modulation response of a SAQD laser for different values of coverage factor and* Г*<sup>0</sup> = 5 meV and relaxation time 1 ps.*

**Figure 14.** L*-*I *curve of a SAQD laser at different cavity lengths.*

#### *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 (*z*-) which leads to increase excited stimulated emission rate, respectively.

#### *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 *Investigating the Role of Auger Recombination on the Performance of a Self-Assembled… DOI: http://dx.doi.org/10.5772/intechopen.102042*

**Figure 15.** *Modulation response of a SAQD laser at different QD heights.*

**Figure 16.** *Modulation response of a SAQD laser at different stripe widths of the laser cavity.*

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.

**Figure 17.** *A modified equivalent circuit model of SAQD lasers, considering the Auger effect.*


*Investigating the Role of Auger Recombination on the Performance of a Self-Assembled… DOI: http://dx.doi.org/10.5772/intechopen.102042*


#### **Table 1.**

*Typical parameters used in the simulation [14, 15, 23].*

#### **Figure 18.**

*The* L*-*I *curve of a SAQD laser neglecting the Auger effect. (a) Various capture times* τc*<sup>0</sup> = 1, 10, 50, 100, 300, and 500 ps at a fixed relaxation time of* τd*<sup>0</sup> = 7 ps; (b) various relaxation times* τd*<sup>0</sup> = 1, 10, 50, 100, 300, and 500 ps at a fixed capture time of* τC*<sup>0</sup> = 1 ps.*

**Figure 18** illustrates the output power as a function of injected currents neglecting the Auger effect for *Cw* = *CE* = 0, *τ<sup>r</sup>* = 2.8 ns and *τqr* = 0.5 ns, when the carrier relaxation times *τd*<sup>0</sup> and capture times *τc*0, changes from 1 to 500 ps. As seen, there are more threshold currents for longer lifetimes. This is along with lower slopes for the output power curves which mean lower quantum efficiencies for increased lifetimes. Physically, such effect is due to the phonon bottleneck in QD lasers [4].

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 *τd*<sup>0</sup> and capture times *τc*0, the frequency response deteriorates and SAQD laser would have lower bandwidths.

**Figure 20(a)** and **(b)** illustrate the output power as a function of the injected currents considering the Auger effect for *CW* = 1 <sup>10</sup><sup>14</sup> <sup>m</sup><sup>3</sup> /s, *CE* = 1 <sup>10</sup><sup>12</sup> <sup>m</sup><sup>3</sup> /s, and *CW* = *CE* = 0 with *τC*<sup>01</sup> = *τd*<sup>01</sup> = 100 ps. Inhomogeneous broadening *Г*<sup>0</sup> = 20 meV and recombination lifetimes of *τ<sup>r</sup>* = 2.8 ns, *τqr* = 3 ns and *τ<sup>r</sup>* = 2.8 ns, *τqr* = 0.5 ns, have been used for **Figure 20(a)** and **(b)**, respectively. It is obvious that the Auger process increases the efficiency, degrades the threshold current, and provides a high output power. This is because, by taking the Auger effect into account, the capture times and carrier relaxation lifetime decreases from WL to ES and from ES to GS. This means the degradation of phonon bottleneck effect. In other words, in QD elements

**Figure 19.**

*Modulation response of a SAQD laser neglecting the Auger effect. (a) Various capture times* τc*<sup>0</sup> = 1, 10, 50, 100, 300, and 500 ps at a fixed relaxation time of* τd*<sup>0</sup> = 7 ps; (b) various relaxation times* τd*<sup>0</sup> = 1, 10, 50, 100, 300, and 500 ps at a fixed capture time of* τC*<sup>0</sup> = 1 ps.*

#### **Figure 20.**

*The* L*-*I *curve of a SAQD laser considering the Auger effect. Different values of recombination lifetimes (a)* τ<sup>r</sup> *= 2.8 ns,* τqr *= 3 ns,* Γ*<sup>0</sup> = 20 meV; (b)* τ<sup>r</sup> *= 2.8 ns,* τqr *= 0.5 ns,* Γ*<sup>0</sup> = 20 meV; and different values of inhomogeneous (c)* Γ*<sup>0</sup> = 30 meV; (d)* Γ*<sup>0</sup> = 40 meV.*

with a low density of the carriers in wetting layer (*Nq*), both relaxation and capture processes are mainly phonon assisted. However, as the QD laser pumping increases, the density of the wetting layer increases, too. This process is first slow but then it

*Investigating the Role of Auger Recombination on the Performance of a Self-Assembled… DOI: http://dx.doi.org/10.5772/intechopen.102042*

occurs in a more pronounced way. This is because, if the pumping increases from zero, most of the carriers get captured into the QDs. On the other hand, based on Pauli Exclusion Principle in QDs, the number of electron state in QD active layer, that is, GS and ES, has been limited. Therefore, if at first the GS and then the ES are filled with electrons, relaxation and capture in these states will get saturated as it is shown by factors (1-*PES*) and (1-*PGS*) in Eqs. (11) and (12), respectively. Thus, the drain of carriers from the WL towards the dots slows down and this leads to the formation of the WL carrier reservoir. The carriers in the reservoir increase the Auger assisted capture speed as shown in Eqs. (16) and (17). On the other hand, by decreasing *τqr* from 3 to 0.5 ns, the carriers find more chances for recombining via non-radiative process outside the QDs. This results in the degradation of quantum efficiency and increase of threshold current simultaneously. **Figure 20(c)** and **(d)** illustrate the output power as a function of the injected currents for inhomogeneous broadenings *Г*<sup>0</sup> = 30 meV and *Г*<sup>0</sup> = 40 meV, respectively. As the inhomogeneous broadening increases, the optical gain decreases [see Eqs. (5) and (6)], and this results in larger occupation probabilities in leasing, and longer relaxation lifetime. Therefore, the quantum efficiency and output power decrease and threshold current increases.

**Figure 21** shows the effect of Auger coefficient increment on the small-signal frequency response of the laser for different values of *CW* = 1 <sup>10</sup><sup>14</sup> <sup>m</sup><sup>3</sup> /s, *CE* = 7 <sup>10</sup><sup>12</sup> <sup>m</sup><sup>3</sup> /s, and *CW* = *CE* = 0. The inhomogeneous broadening is *Γ*<sup>0</sup> = 20 meV. The recombination lifetimes of *τ<sup>r</sup>* = 2.8 ns and *τqr* = 3 ns have been used. The simulation reveals that when the Auger coefficient increases, the relaxation lifetime decreases. It means that the phonon bottleneck effect is degraded and as a result, the modulation bandwidth increases. On the other hand, as Auger coefficient increases and carrier relaxation times *τd*<sup>0</sup> and capture times *τc*<sup>0</sup> decreases, the 3 dB frequency increases. What is important here is that, as *τ<sup>r</sup>* decreases from 2.8 to 0.5 ns, the frequency response degrades. Therefore, to prevent the effect of phonon bottleneck on frequency response, the recombination lifetime within quantum dots *τ<sup>r</sup>* must be much longer than the carrier relaxation time (the carrier relaxation time is about a few pico-seconds) [4]. Results obtained by using the presented circuit model are in good agreement with the results calculated by solving the rate equations numerically and also with experiments reported so far [4, 7].

#### **Figure 21.**

*Modulation response of a SAQD laser considering the Auger effect for different values of carrier recombination inside quantum dot (a)* τ<sup>r</sup> *= 2.8 ns and (b)* τ<sup>r</sup> *= 0.5 ns.*
