**4. Inhomogeneous gain model with global quasi-fermi levels**

Actually, QDs have imperfections in shape and randomly distributed on the substrate. These QDs emit photons at slightly different energies which results in an inhomogeneous broadening [11]. In our model, a Gaussian distribution function is used to describe inhomogeneous broadening. Accordingly, the linear gain of QD structures can be written as [9, 12]

$$\log^{(1)}\langle hw\rangle = c\_0 \sum\_i \int\_{-\infty}^{\infty} dE \left| M\_{env} \right|^2 \left| \hat{e}^\dagger, p\_{cc} \right|^2 D(\vec{E}') L(\vec{E}', hw) [f\_c(\vec{E}', F\_c) - f\_v(\vec{E}', F\_v)] \tag{1}$$

where 2 0 2 0 0 *b e c m cn* , *m*<sup>0</sup> is the free electron mass, 0 is the permittivity of free space, *c* is

the speed of light in free space, *nb* is the background refractive index of the material, is the optical angular frequency of the injected optical signal and *E* is the optical transition energy. <sup>2</sup> *Menv* is the envelope function overlap between the QD electron and hole states. The term <sup>2</sup> ^ <sup>0</sup> <sup>3</sup> [. ( )] 2 6 *cv <sup>p</sup> <sup>m</sup> ep E* is the momentum matrix element for electron-heavy hole transition energy in TE polarization, *Ep* is the optical matrix energy parameter. The Lorentzian line shape function *L E*( ) for gain is defined by

$$L(\boldsymbol{E}^{\circ}, \hbar \boldsymbol{w}) = \frac{\frac{\hbar \boldsymbol{\gamma}\_{cv}}{\pi}}{\left(\boldsymbol{E}^{\circ} - \hbar \boldsymbol{o}\right)^{2} + \left(\hbar \boldsymbol{\gamma}\_{cv}\right)^{2}}\tag{2}$$

With ( 1/ ) *cv in* is the intraband scattering rate. *D E*( ) is the inhomogeneous density of states of the self-assembled QD and is expressed as [13]

$$D(E') = \frac{\mu\_i}{V\_{dot}} \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(\frac{-(E'-E\_{\text{max}}^i)}{2\sigma^2}^2\right) \tag{3}$$

where *<sup>i</sup>* is the degeneracy of the *ith* state of a QD, 2, 4 *GS ES* for the ground state and the excited state, respectively. *Vdot* is the effective volume of the QDs. is the spectral variance of the QD distribution and max *<sup>i</sup> E* is the transition energy at the maximum of QD

(NNRL) a series of studies for some of the characteristics of Sb-based QD devices [10]. Here, the carrier dynamics in III-Sb based QD-SOAs are considered using four-level REs system. QD, WL and SCH barrier compositions are examined to specify their characteristics. The results then, compared with those of three-level REs show the importance of including the barrier layer in the QD SOAs calculations. Because of the much larger effective mass of holes and lower quantization energies of the QD levels in the valence band, electrons behavior limits the carrier dynamics while holes in the valence band are assumed to be in quasithermal equilibrium at all times [11]. Thus, we determine carrier dynamics here by the

Actually, QDs have imperfections in shape and randomly distributed on the substrate. These QDs emit photons at slightly different energies which results in an inhomogeneous broadening [11]. In our model, a Gaussian distribution function is used to describe inhomogeneous broadening. Accordingly, the linear gain of QD structures can be written as

> <sup>2</sup> (1) ' ^ '' ' ' <sup>2</sup> <sup>0</sup> ( ) . ( ) ( , )[ ( , ) ( , )] *env cv c cv v*

the speed of light in free space, *nb* is the background refractive index of the material,

the optical angular frequency of the injected optical signal and *E* is the optical transition

transition energy in TE polarization, *Ep* is the optical matrix energy parameter. The

'2 2 (, ) ( )( )

max

<sup>1</sup> ( ) ( ) exp 2 2

*E E D E*

*E*

*g hw c dE M e p D E L E hw f E F f E F*

(1)

is the permittivity of free space, *c* is

(2)

(3)

is the spectral

for the ground state and

is

<sup>2</sup> *Menv* is the envelope function overlap between the QD electron and hole states.

*<sup>m</sup> ep E* is the momentum matrix element for electron-heavy hole

*cv*

 

*cv*

2 '

*i*

*GS ES* 

*<sup>i</sup> E* is the transition energy at the maximum of QD

 

 

is the intraband scattering rate. *D E*( ) is the inhomogeneous density of

2 2

relaxation of electrons in the conduction band only.

*i*

2

<sup>2</sup> ^ <sup>0</sup> <sup>3</sup> [. ( )] 2 6 *cv <sup>p</sup>*

variance of the QD distribution and max

0 0 *b e*

*m cn* 

0 2

*c*

, *m*<sup>0</sup> is the free electron mass, 0

Lorentzian line shape function *L E*( ) for gain is defined by

states of the self-assembled QD and is expressed as [13]

'

'

*LE w*

*i dot*

*<sup>i</sup>* is the degeneracy of the *ith* state of a QD, 2, 4

the excited state, respectively. *Vdot* is the effective volume of the QDs.

*V* 

[9, 12]

where

energy.

The term

With ( 1/ ) *cv in* 

where  **4. Inhomogeneous gain model with global quasi-fermi levels** 

distribution of the *th i* optical transition. The terms *fc* and *fv* are the respective quasi-Fermi level distribution functions for the conduction and valence bands, respectively. Recent researches [8] uses global states to describe the global quasi-Fermi levels *Fc* and *Fv* in the conduction and valence bands where the contributions to the Fermi-levels from the barrier layer and WL are included in addition to that from QDs. They are determined from the surface carrier density per QD layer by the following relations [13]

$$\begin{split} n\_{2D} &= N\_D \Sigma \frac{s^i}{i \sqrt{2\pi\sigma\_\varepsilon^2}} \mathbb{I}\_{\varepsilon} \Big( -E\_c^c - E\_{cl}^0 \Big) \Big/ 2\sigma\_\varepsilon^2 \Big( E\_c^r, F\_c \Big) dE\_c^r \\ &+ \frac{m\_E}{I} \frac{K\_B}{\pi\hbar^2} \mathbb{I} \Big/ \ln \left( 1 + \epsilon \Big( F\_c^c - E\_{cl}^{\text{pr}} \Big) \Big/ K\_B T \right) \\ &+ H\_B \Big/ \frac{1}{2\pi^2} \Big( \frac{2m\_E^B}{\hbar^2} \Big)^{3/2} \sqrt{E\_c^r - E\_{cl}^B} f\_c(E\_c^r, F\_c) dE\_c^r \\ &p\_{2D} = N\_D \Sigma \frac{s^i}{i \sqrt{2\pi\sigma\_h^2}} \Big\{ \epsilon^- \Big( E\_h^r - E\_{th}^D \Big) \Big/ 2\sigma\_h^2 \Big/ \\ &+ \sum\_m \frac{m\_E^m}{\pi\hbar^2} \Big/ \ln \left( 1 + \epsilon \Big( F\_v - E\_{\mu m}^D \Big) \Big/ K\_B T \right) \\ &+ H\_B \Big/ \frac{1}{2\pi^2} \Big/ \frac{2m\_E^B}{\hbar^2} \Big/ \frac{3}{4} \sqrt{E\_h^B - E\_{th}} f\_v(E\_h^r, F\_v) dE\_h \end{split} \tag{5}$$

where *n*2*<sup>D</sup>* and *p*2*<sup>D</sup>* are the surface densities of electrons and holes per QD layer, respectively. *DEci* and *DEhi* represents the respective confined QD states in the conduction and valence bands. *<sup>e</sup>* and *<sup>h</sup>* are the spectral variance of the QD electron and heavy-hole distributions, respectively. The term *Ec* is the energy in the conduction band and *Eh* is the energy in the valence band. Heavy hole subbands are included, only, in the calculations of valence subbands since light hole subbands are deep and can be neglected. *Bk* is the Boltzmann constant and *T* is the absolute temperature. The terms *<sup>W</sup> me* ( *<sup>W</sup> mh* ) and *WEel* ( *WEhm* ) are the effective electron (hole) mass and the subband edge energy of the conduction (valence) band of WL. The term *Hb* is the thickness of SCH barrier. The terms *<sup>B</sup> me* ( *<sup>B</sup> mh* ) and *<sup>B</sup> Ec* ( *<sup>B</sup> Eh* ) are the electron (hole) mass and the conduction (valence) band edge energy of barrier layer.
