**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].

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

#### **Figure 1.**

*Carrier relaxation in a quantum dot [1]. (A, Red Color) The relaxation from continuous levels. (B, Blue Color) The relaxation between discrete levels.*

**Figure 2.**

*Carrier capture and relaxation processes [4, 12]. (a) LO- and LA-phonons. (b) Auger processes.*

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 *I*/*e* where *I* is the injected current and *e* is the electron charge.

They can either relax with a time of *τ<sup>s</sup>* in the WL state or come back from the barrier with a time of *τqe*. These carriers can be captured by different dot sizes from the WL state. Assume that each dot has only two distinct energy states; GS and excited state (ES). In these levels, the captured carriers have a time of *τ<sup>c</sup>* from WL to ES and relaxed carriers have a time of *τ<sup>d</sup>* from ES to GS. They can come back in the reverse path at the times of *τeES* an *τeGS*, respectively. There are several radiative or nonradiative recombination times for carriers. *τSr* is carrier recombination in the SCH region and *τqr* is carrier recombination in the wetting layer (WL). *τ<sup>r</sup>* is recombination in the QD. Assume that an excited emission is only due to an electron hole recombination in ES and GS. The photons emitted from the laser cavity have a rate of *S*/*τp*, where *S* is the number of photons and *τ<sup>p</sup>* is the photon lifetime. The governing rate equations for various components of these carriers are explained as [14–16]:

$$\frac{dN\_s}{dt} = \frac{I}{e} - \frac{N\_s}{\tau\_s} - \frac{N\_s}{\tau\_{sr}} + \frac{N\_q}{\tau\_{qe}},\tag{1}$$

$$\frac{dN\_q}{dt} = \frac{N\_s}{\tau\_s} + \frac{N\_{ES}}{\tau\_{\varepsilon\_{ES}}} - \frac{N\_q}{\tau\_{qr}} - \frac{N\_q}{\tau\_{q\varepsilon}} - \frac{N\_q}{\tau\_{\varepsilon}},\tag{2}$$

$$\begin{split} \frac{dN\_{ES}}{dt} &= \frac{N\_q}{\tau\_c} + \frac{N\_{GS}(\mathbf{1} - P\_{ES})}{\tau\_{\epsilon\_{GS}}} - \frac{N\_{ES}}{\tau\_r} - \frac{N\_{ES}}{\tau\_{\epsilon\_{ES}}} - \frac{N\_{ES}}{\tau\_d} \\ &- \frac{(\boldsymbol{\varepsilon}\_{\boldsymbol{n}\_r}) \mathbf{g}\_{mES}^{(1)} \Gamma}{\mathbf{1} + \boldsymbol{\varepsilon}\_{mES} \Gamma^S \boldsymbol{\zeta}\_{V\_d}} S, \end{split} \tag{3}$$

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

$$\begin{split} \frac{d\mathbf{N}\_{\rm GS}}{dt} &= \frac{\mathbf{N}\_{\rm ES}}{\tau\_{d}} - \frac{\mathbf{N}\_{\rm GS}}{\tau\_{r}} - \frac{\mathbf{N}\_{\rm GS}(\mathbf{1} - P\_{\rm ES})}{\tau\_{\rm e\_{\rm GS}}} \\ &- \frac{(\boldsymbol{\epsilon}\_{\rm r})\mathbf{g}\_{m\rm GS}^{(1)}\boldsymbol{\Gamma}}{\mathbf{1} + \boldsymbol{\varepsilon}\_{m\rm GS}\boldsymbol{\Gamma}^{\rm S}\boldsymbol{\beta}\_{\rm V\_{s}}} \mathbf{S}, \end{split} \tag{4}$$
 
$$\begin{split} \frac{d\mathbf{S}}{dt} &= \frac{(\boldsymbol{\epsilon}\_{\rm r})\mathbf{g}\_{m\rm ES}^{(1)}\boldsymbol{\Gamma}}{\mathbf{1} + \boldsymbol{\varepsilon}\_{m\rm ES}\boldsymbol{\Gamma}^{\rm S}\boldsymbol{\beta}\_{\rm V\_{s}}} \mathbf{S} + \frac{(\boldsymbol{\epsilon}\_{\rm r})\mathbf{g}\_{m\rm GS}^{(1)}\boldsymbol{\Gamma}}{\mathbf{1} + \boldsymbol{\varepsilon}\_{m\rm GS}\boldsymbol{\Gamma}^{\rm S}\boldsymbol{\beta}\_{\rm V\_{s}}} \mathbf{S} \\ &- \frac{\mathbf{S}}{\tau\_{p}} + \frac{\boldsymbol{\beta}\mathbf{N}\_{\rm GS}}{\tau\_{r}} + \frac{\boldsymbol{\beta}\mathbf{N}\_{\rm ES}}{\tau\_{r}} \end{split} \tag{5}$$

With the following parameters: *Ns*: number of carriers in the SCH layer, *Nq*: number of carriers in the WL layer, *NES*: number of carriers in the ES layer, and *NGS*: number of carriers in the GS layer.

In the above equations, *Γ* is the optical confinement factor, *Γ*<sup>0</sup> is the inhomogeneous broadening of the optical gain, *c* is the speed of light, *β* is the spontaneous coupling efficiency, and the photon lifetime *τ<sup>p</sup>* can be found from [17]:

$$\tau\_p^{-1} = \left(\frac{c}{n\_r}\right) \left\{ a\_i + \frac{\ln\left[\mathbb{1}\_{\left(R\_1 R\_2\right)}\right]}{2L} \right\} \tag{6}$$

Where *R*<sup>1</sup> and *R*<sup>2</sup> are the facet reflectivity of the laser cavity with a length of *L* and the internal loss of *αi*, and *nr* is the cavity refractive index. Where the nonlinear gain coefficient (*εmGS* and *εmES*) are defined as [4, 16, 18]:

$$\varepsilon\_{m\text{GS}} = \frac{e^2}{2n\_r^2 \varepsilon\_0 m\_0^2} \frac{\left|P\_{c,v}^\sigma\right|^2}{E\_{\text{GS}}} \frac{1}{\Gamma\_{cv} \Gamma\_{\parallel}} \tag{7}$$

$$\varepsilon\_{m\to S} = \frac{e^2}{2n\_r^2 \varepsilon\_0 m\_0^2} \frac{\left|P\_{c,\nu}^{\sigma}\right|^2}{E\_{\text{ES}}} \frac{1}{\Gamma\_{cv} \Gamma\_{\parallel}} \tag{8}$$

and *Γ*|| � 1/*τ<sup>p</sup>* is the longitudinal relaxation constant and *Γcv* is the scattering or polarization de-phasing rate. Based on the density matrix theory the linear optical gain of the active region, with a dot density of *ND*, can be found as [10, 16, 19]:

$$\mathbf{g}\_{m\text{GS}}^{(1)} = \frac{2.35\sqrt{2\pi}e^2\hbar}{c n\_{r\text{e}0}m\_0^2} \frac{\left|P\_{cv}^{\sigma}\right|^2}{E\_{\text{GS}}} \frac{(2P\_{\text{GS}} - 1)}{\Gamma\_0} N\_D D\_{\text{GS}}\tag{9}$$

$$\mathcal{g}\_{m\text{ES}}^{(1)} = \frac{2.35\sqrt{2\pi}e^2\hbar}{c n\_r \varepsilon\_0 m\_0^2} \frac{\left|P\_{cv}^{\sigma}\right|^2}{E\_{\text{ES}}} \frac{(2P\_{\text{ES}} - 1)}{\Gamma\_0} N\_D D\_{\text{ES}}\tag{10}$$

Parameters *DGS* and *DES* in Eqs. (5) and (6), impose the GS and ES degeneracy which are considered to be 2 and 4, respectively [15]. *P<sup>σ</sup> c,v* is a matrix element, related to the overlap integral *I σ c,v* between the envelope functions of an electron and hole, defined as [19]:

$$\left|P\_{c,\nu}^{\sigma}\right|^2 = \left|I\_{c,\nu}\right|^2 \mathcal{M}^2 \tag{11}$$

Where *M*<sup>2</sup> is the first-order *K*-*P* interaction between conduction and valence bands, with a separation of *Eg* and is equal to:

$$M^2 = \frac{m\_0^2}{12m\_\epsilon^\*} \frac{E\_\text{g} \left(E\_\text{g} + \Delta\right)}{E\_\text{g} + \frac{2\Delta}{3}}\tag{12}$$

In this relation, *Eg* is band gap, *m*<sup>∗</sup> *<sup>e</sup>* is the effective mass for electrons and Δ is defined as the spin-orbit interaction energy for a QD material. Based on the Pauli's Exclusion Principle, the occupation probabilities at the ES and GS states of the QD are defined as [19]:

$$P\_{\rm GS} = \frac{N\_{\rm GS}}{2N\_D V\_a D\_{\rm GS}}\tag{13}$$

$$P\_{\rm ES} = \frac{N\_{\rm ES}}{2N\_D V\_a D\_{\rm ES}}\tag{14}$$

Where *Va* = *LWd* is the cavity volume with length *L*, width *W* and thickness *d*. Effects of the non-uniform dot size can be included in the relaxation, capture, and escape times with the following definitions [14–16, 18]:

$$
\pi\_{\mathfrak{c}} = \frac{\pi\_{\mathfrak{c}0}}{(\mathbbm{1} - P\_{\text{ES}})} \tag{15}
$$

$$
\pi\_d = \frac{\pi\_{d0}}{(1 - P\_{GS})} \tag{16}
$$

Where *τc*<sup>0</sup> is the capture time from WL to ES and *τd*<sup>0</sup> is the relaxation time from EL to GS with the assumption of an empty final state. Without the stimulated emission and at room temperature, the system must converge to a quasi-thermal equilibrium based on a Fermi distribution. To have this condition, the carrier capture time, *τc*0, and relaxation time, *τd*0, and the carrier escape times, *τeGS*, *τeES*, should be satisfied by the following relations [14, 15]:

$$
\pi\_{\epsilon\_{\rm GS}} = \pi\_{d0} \left(\frac{D\_{\rm GS}}{D\_{\rm ES}}\right) e^{\frac{E\_{\rm ES} - E\_{\rm GS}}{kT}} \tag{17}
$$

$$\tau\_{\varepsilon\_{\rm ES}} = \tau\_{c0} \left( \frac{D\_{\rm ES} N\_d}{\rho\_{\rm WL\_{\rm eff}}} \right) e^{\frac{E\_{\rm MZ} - E\_{\rm ES}}{kT}} \tag{18}$$

$$
\rho\_{\rm WL\_{eff}} \equiv \left(\frac{m\_{\rm e\_{WL}}kT}{\pi\hbar^2}\right) \tag{19}
$$

Where *ρWLeff* is the effective density of states per unit area of the WL and *Nd* is the QD density per unit area. Both phonon- and Auger-assisted capture and relaxation are taken into account phenomenologically through the relation [16, 20, 21]:

$$
\pi\_{c0}^{-1} = \pi\_{c01}^{-1} + C\_W N\_q \tag{20}
$$

$$
\tau\_{d0}^{-1} = \tau\_{d01}^{-1} + \mathcal{C}\_{\rm E} \mathcal{N}\_q \tag{21}
$$

Where *τ* �1 *<sup>c</sup>*01, *τ* �1 *<sup>d</sup>*<sup>01</sup> are characteristic rates of phonon-assisted capture and interlevel relaxation processes, respectively, and *CW* and *CE* are the coefficients for *Investigating the Role of Auger Recombination on the Performance of a Self-Assembled… DOI: http://dx.doi.org/10.5772/intechopen.102042*

Auger-assisted relaxation related to the WL and the ES, respectively. Assume that all of the carriers are injected into the WL layer or equivalently *τqe* = *τsr* ! ∞.

#### **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*<sup>0</sup>*

**Figure 4** shows the results of our simulations on the *L*-*I* curve of SAQD laser considering the effects of inhomogeneous broadening. Note that, the physical origin of this parameter is in the random size of the QDs [4]. As shown in the figure, for higher inhomogeneous broadening factor, more threshold current or equivalently more injection currents are needed but in this case, the external quantum efficiency does not have dominant changes. In other words, for higher *Γ*<sup>0</sup> the output power of the laser decreases. Physically speaking, for increased inhomogeneous broadening factors there is a higher occupation probabilities in the lasing action of the structure, so that the relaxation times increase and the output power decreases.

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, *τqr* which equals WL crystal quality, has been shown on *L*-*I* feature in **Figure 6**. As *τqr* degrades, the carriers would find more opportunities to recombine through non-radiative process out of the quantum dot which results in degradation of external quantum efficiency. The findings indicate that *τqr* degradation also increases the threshold current.

#### **Figure 4.**

*Effects of the inhomogeneous broadening variations on the* L*-*I *curve of SAQD laser considering the excited state with different values of the broadening parameter* Γ*<sup>0</sup> = 5, 20, and 40 meV.*

#### **Figure 5.**

*Effects of the inhomogeneous broadening variations on the frequency response of SAQD laser with different values of* Γ*<sup>0</sup> = 5, 20, and 40 meV.*

**Figure 6.** *The* L*-*I *curve of a SAQD laser for different values of carrier recombinations 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,* τ<sup>r</sup>

**Figure 8** shows the effect of carrier recombination inside quantum dot, *τr*, on the *L*-*I* curve. While the effect of *τ<sup>r</sup>* on threshold current is significant, it does not have a considerable effect on external quantum efficiency.

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

**Figure 7.** *The modulation response of a SAQD laser for different values of carrier recombination in WL.*

**Figure 8.** *The* L*-*I *curve of a SAQD laser for different values 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 *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).
