Investigation of the effects of nonlinear optical gain and thermal carrier excitation on characteristics of self-assembled quantum-dot lasers

Davoud Ghodsi Nahri

doi:10.1364/OE.20.014754

1. Introduction

Due to unique three-dimensional quantum confinement of carriers, quantum-dot (QD) lasers are expected to show many superior properties such as ultra-low and temperature-stable threshold current, high optical gain, high-speed operation, broad modulation bandwidth, and narrow spectrum linewidth [1–3]. However, actual self-assembled QDs (SAQDs) do not meet our expectation because of inhomogeneous broadening (IHB) and homogeneous broadening (HB) of the QDs’ energy levels, retarded carrier relaxation dynamics (phonon bottleneck) [4], nonlinear optical gain, and thermal carrier excitation. Thus, for an accurate and rigorous analysis of SAQD lasers (SAQDLs) performance, we need to take into account all of these actual aspects of QDs. The effects of linear optical gain, containing both of the HB and IHB, and of carrier relaxation dynamics on the performance of SAQDLs have been theoretically and experimentally modeled and investigated [3–22]. However, there is one related issue which to date has not been dealt with; the nonlinear optical gain of a set of QDs with inhomogeneously and homogeneously broadened energy levels.

The aim of this paper is to investigate the effects of nonlinear optical gain of inhomogeneously and homogeneously widen SAQDs and of thermal carrier excitation (escape) from QDs on dynamic behavior and static characteristics of the SAQDLs. I consider columnar-shaped SAQDs that have been grown using Stranski-Krastanov mode with dimensions less than 15 nm [6,23,24], thus, only the transitions between ground states (GSs) of electrons and holes are dominant recombination processes [5,24]. The paper is organized as follows: In section 2, the theory of linear optical gain of SAQDs is described. In section 3, I derive a formula containing both the linear and nonlinear optical gain in order to enter the total optical gain at the rate equations. In section 4, the multi-mode and multi-population rate equations (MPREs) model that is the analyzing theory of QD laser performance is brought. I also enter thermal carrier excitation rate at the MPREs and apply the derived MPREs for In(Ga)As/GaAs SAQDLs. In section 5, photon-time evolution (PTE) response and light-current (L-I) characteristics of the SAQDLs are simulated considering the linear optical gain at the MPREs. In section 6, I simulate PTE response and L-I characteristics again considering the total optical gain at the MPREs, I compare PTE response and L-I characteristics of the SAQDLs existing and without existing the nonlinear optical gain and, for the first time, show that considering the total optical gain at the MPREs is entirely necessary to attain more exact PTE response and L-I properties. Finally, in section 7, I investigate the effects of nonlinear optical gain and thermal carrier excitation on light-emission (L-E) characteristics and plot multi-mode PTE response of the SAQDLs. Section 8 is devoted for conclusion.

2. Linear optical gain theory

The density-matrix theory offers the linear optical gain of actual SAQDs by taking into account the IHB as [6]

\begin{matrix} G^{(1)} (E) = \frac{2 π e^{2} ℏ D_{Q D}^{3 D}}{c n_{r} ε_{0} m_{0}^{2}} \sum_{c, v} \frac{{| P_{c v}^{σ} |}^{2}}{E_{c v}} \int_{- \infty}^{\infty} (f_{c} (E^{'}) - f_{v} (E^{'})) \\ \times L_{H o m}^{(1)} (E - E^{'}) G_{I n h} (E^{'} - E_{c v}) d E^{'} \end{matrix}

where

n_{r}

is the refractive index,

D_{Q D}^{3 D}

is the volumetric density of states (DOS) of QDs,

{| P_{c v}^{σ} |}^{2}

is the transition matrix element,

E_{c v}

is the center of the energy distribution function of each interband transition,

f_{c} (E^{'})

is the electron occupation function of the conduction-band discrete state of the QDs with the interband transition energy of

E^{'}

, and

f_{v} (E^{'})

is that of the valence band discrete state. The linear optical gain shows the HB of a Lorentz shape as

L_{H o m}^{(1)} (E - E^{'}) = (ℏ γ_{c v} / π) / [{(E - E^{'})}^{2} + {(ℏ γ_{c v})}^{2}]

in which the full width at half maximum (FWHM) is given as

Γ_{c v} = 2 ℏ γ_{c v}

with the polarization dephasing or scattering rate

γ_{c v}

. The HB depends on temperature and various scattering mechanisms. The IHB which models shape, size, and composition fluctuations of QDs are represented by

G_{I n h} (E^{'} - E_{c v})

that takes a Gaussian distribution function as

G_{I n h} (E^{'} - E_{c v}) = \frac{1}{\sqrt{2 π} ξ_{0}} \exp [- {(E^{'} - E_{c v})}^{2} / 2 ξ_{0}^{2}]

whose FWHM is given by

Γ_{0} = 2.35 ξ_{0}

. Neglecting the optical-field polarization dependence, the transition matrix element is given by

{| P_{c v}^{σ} |}^{2} = {| I_{c v} |}^{2} M^{2}

where

I_{c v}

represents the overlap integral between the envelope functions of an electron and a hole, and

M^{2} = (m_{0}^{2} / 12 m_{e}^{*}) (E_{g} (E_{g} + Δ) / (E_{g} + 2 Δ / 3))

that is derived by the first-order k.p interaction between the conduction and valence band. Here,

E_{g}

is the band gap of In(Ga)As bulk material,

m_{0}

is the electron rest mass,

m_{e}^{*}

is the electron effective mass, and

Δ

is the spin-orbit interaction energy of the QD material [6].

3. Derived total optical gain formula

The third-order nonlinear optical gain of QDs just considering HB also is written as [7]

\begin{matrix} G^{(3)} (E) = \frac{- π e^{4} ℏ ξ}{c^{2} n_{r}^{2} ε_{0}^{2} E^{2} m_{0}^{4} V_{Q D}} \sum_{c, v} \frac{{| P_{c v}^{σ} |}^{4} (f_{c} - f_{v})}{E_{c v} γ_{c v} Γ_{| |}} \\ \times I_{p} (E) L_{H o m}^{(3)} (E - E_{c v}) \end{matrix}

where

Γ_{| |}

is the longitudinal relaxation rate of the electrons due to intraband scattering,

ξ

is the QD coverage which is related to QD-DOS and QD volume,

V_{Q D}

, as

ξ = D_{Q D}^{3 D} V_{Q D}

.

I_{p} (E)

that is the power density of the electromagnetic field is given as

I_{p} (E) = (c E S Γ / n_{r} V_{a})

where

V_{a}

is the active region volume,

Γ

is the optical confinement factor, and

S

is the number of cavity photons. The third-order nonlinear optical gain shows the HB as

L_{H o m}^{(3)} (E - E_{c v}) = {(2 {(ℏ γ_{c v})}^{3} / π) / [{(E - E_{c v})}^{2} + {(ℏ γ_{c v})}^{2}]}^{2}

. I revise the third-order nonlinear optical gain formula of Eq. (3) by taking into account the IHB of SAQDs in terms of a convolution integral as

\begin{matrix} G^{(3)} (E) = \frac{- π e^{4} ℏ ξ}{c^{2} n_{r}^{2} ε_{0}^{2} m_{0}^{4} V_{Q D} Γ_{| |}} (\frac{I_{p} (E)}{E^{2}}) \sum_{c, v} \frac{{| P_{c v}^{σ} |}^{4}}{E_{c v} γ_{c v}} \int_{- \infty}^{\infty} (f_{c} (E^{'}) - f_{v} (E^{'})) \\ \times L_{H o m}^{(3)} (E - E^{'}) G_{I n h} (E^{'} - E_{c v}) d E^{'} \end{matrix}

The total optical gain, including linear and nonlinear optical gain, is given by

G^{t o t} (E) = G^{(1)} (E) + G^{(3)} (E)

after doing little mathematical calculation, the total optical gain of SAQDs is derived as

G^{t o t} (E) = \sum_{c, v} \int_{- \infty}^{\infty} \frac{d G^{(1)} (E)}{[1 + Γ S ε (E) L_{H o m}^{(1)} (E - E^{'}) / V_{a}]}

where

d G^{(1)} (E)

is represented by

\begin{matrix} d G^{(1)} (E) = \frac{2 π e^{2} ℏ D_{Q D}^{3 D}}{c n_{r} ε_{0} m_{0}^{2}} \frac{{| P_{c v}^{σ} |}^{2}}{E_{c v}} (f_{c} (E^{'}) - f_{v} (E^{'})) \\ \times L_{H o m}^{(1)} (E - E^{'}) G_{I n h} (E^{'} - E_{c v}) d E^{'} \end{matrix}

and the third-order nonlinear gain coefficient is given as

ε (E) = \frac{π e^{2} ℏ {| P_{c v}^{σ} |}^{2}}{ε_{0} m_{0}^{2} n_{r}^{2} Γ_{| |}} (\frac{1}{E})

4. Rate equations

One of the best ways to deal with carrier and photon dynamics in lasers is to solve rate equations for carriers and photons. Figure 1 illustrates the energy diagram of the conduction band of the SAQDL-waveguide region and diffusion, relaxation, recombination, and excitation processes of carriers. Our model is an excitonic one, therefore, the relaxation means the process that both an electron and a hole relax into the GS simultaneously to form an exciton and the charge neutrality always holds in each QD, i.e., $f_{c} (E^{'}) = 1 - f_{v} (E^{'})$ .

Fig. 1 Energy diagram of the laser-waveguide region and diffusion, recombination, relaxation, and escape processes.

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In order to take into account the IHB of the SAQDs, we divide the QD ensemble into J = 1, 2,..., 2K+1 groups depending on their GS interband transition energies over the longitudinal modes of an edge-emitting cavity. Therefore, different QD groups have diverse GS interband transition energies and the energy width of each group is equal to the mode separation of the longitudinal modes, $Δ E = c h / 2 n_{r} L_{c a v}$ where $L_{c a v}$ is the cavity length. J = K corresponds to the central QD group which has the GS interband transition energy $E_{c v}$ . K is also related to the central lasing mode with the energy $E_{c v}$ . I also consider (2K + 1), which is the number of QD groups and of the longitudinal cavity modes, as a function of the IHB as

(2 K + 1) = 4 Γ_{0} / Δ E

In fact, we consider the contribution of QDs that are distributed in an energy span which is defined by $4 Γ_{0}$ around $E_{c v}$ . This span includes more than 99.7% of all QDs. The GS interband transition energy of the Jth group of QDs is represented by $E_{J} = E_{c v} - (K - J) Δ E$ . QD-DOS of the Jth group is given by $D_{Q D}^{3 D} G_{J}^{I n h} = D_{Q D}^{3 D} G_{I n h} (E_{J} - E_{c v}) Δ E$ where $G_{J}^{I n h}$ is the QD percentage of that group within the QD ensemble. Let $N_{J}$ be the carrier number in the Jth QD group, according to the Pauli’s exclusion principle, the occupation probability in the GS of that group is defined as $P_{J} = N_{J} / (2 D_{G S} D_{Q D}^{3 D} V_{a} G_{J}^{I n h})$ where $D_{G S}$ is the degeneracy of the QD-GS without spin. The MPREs are as follows [4-10, 17, 19–22]

d N_{S} / d t = I / e - N_{S} / τ_{S} - N_{S} / τ_{S r} + N_{W} / τ_{W e}

\begin{matrix} d N_{W} / d t = N_{S} / τ_{S} + \sum_{J} N_{J} / τ_{J}^{e x c} D_{G S} - N_{W} / τ_{W r} \\ - N_{W} / τ_{W e} - N_{W} / {\bar{τ}}^{r e l} \end{matrix}

\begin{matrix} d N_{J} / d t = N_{W} G_{J}^{I n h} / τ_{J}^{r e l} - N_{J} / τ_{r} - N_{J} / τ_{J}^{e x c} D_{G S} \\ - \frac{c Γ}{n_{r}} \sum_{M} G_{M J}^{t o t} S_{M} \end{matrix}

d S_{M} / d t = β N_{M} / τ_{r} + \frac{c Γ}{n_{r}} \sum_{J} G_{M J}^{t o t} S_{M} - S_{M} / τ_{p}

where

N_{S}

and

N_{W}

are the carrier numbers in separately confinement heterostructure layers (SCHL) and wetting layer (WL), respectively.

S_{M}

is the photon number of Mth mode, whereM = 1, 2,…, 2K+1.

I

is the injected current,

e

is the electron charge, and

β

is the spontaneous-emission coupling efficiency to a lasing mode. The related lifetime constants are;

τ_{S}

, and

τ_{S r}

, carrier diffusion and recombination in the SCHL,

τ_{W e}

, carrier excitation from the WL to the SCHL,

τ_{W r}

, and

τ_{r}

, carrier recombination in the WL and QDs. The term which describes thermal carrier excitation rate from the Jth QD group to the WL is given by [4,17]

{1 / τ}_{J}^{e x c} = (1 - (N_{W} / 2 D_{W}^{3 D} V_{W})) / τ_{J_{0}}^{e x c}

where

D_{W}^{3 D}

is the volumetric WL-DOS,

V_{W}

is the WL volume, and

τ_{J_{0}}^{e x c}

, which is the carrier excitation time when the WL occupation is zero, corresponds to temperature,

T

, as

τ_{J_{0}}^{e x c} = τ_{0} (D_{Q D}^{3 D} / D_{W}^{3 D}) \exp [(E_{W} - E_{J}^{G S}) / K_{B} T]

where

E_{W}

and

E_{J}^{G S}

are the interband transition energies of the WL and GS of the Jth QD group, respectively, and

τ_{0}

is the relaxation lifetime when the GS of QDs is unoccupied (the initial carrier relaxation lifetime). The average carrier relaxation rate is given as [6]

1 / {\bar{τ}}^{r e l} = \sum_{J} G_{J}^{I n h} / τ_{J}^{r e l} = \sum_{J} (1 - P_{J}) G_{J}^{I n h} / τ_{0}

where

τ_{J}^{r e l}

is the carrier relaxation lifetime into the Jth QD group. The photon lifetime in the cavity is given by

τ_{p}^{- 1} = (c / n_{r}) [α_{i} + L n (1 / R_{1} R_{2}) / 2 L_{c a v}]

where

R_{1}

and

R_{2}

are the cavity mirror reflectivities, and

α_{i}

is the internal loss. The total optical gain that photons of the Mth mode get from the QD ensemble is given as

\begin{matrix} G_{M}^{t o t} = \sum_{J} G_{M J}^{t o t} = \\ \sum_{J} \frac{G_{M J}^{(1)}}{[1 + Γ S_{M} ε (E_{M}) L_{H o m}^{(1)} (E_{M} - E_{J}) / V_{a}]} \end{matrix}

where

G_{M J}^{(1)}

is represented by

\begin{matrix} G_{M J}^{(1)} = \frac{2 π e^{2} ℏ D_{Q D}^{3 D}}{c n_{r} ε_{0} m_{0}^{2}} \frac{{| P_{c v}^{σ} |}^{2}}{E_{c v}} (2 P_{J} - 1) G_{J}^{I n h} \\ \times L_{H o m}^{(1)} (E_{M} - E_{J}) \end{matrix}

The output power of the Mth mode from one cavity mirror is given by [6]

P_{M}^{o u t} = c E_{M} S_{M} L n (1 / R) / 2 L_{c a v} n_{r}

where

E_{M}

is the photon energy of the Mth mode, and

R

is

R_{1}

or

R_{2}

. The main difference of this set of rate equations with the previous versions is that I have considered the total optical gain instead of the linear optical gain. It is also assumed that all the carriers are injected into the WL, i.e.,

τ_{S r} = τ_{W e} = \infty

. I have solved the coupled MPREs numerically using the fourth-order Runge-Kutta method to simulate dynamic and static characteristics of In(Ga)As/GaAs SAQDLs. The parameters that I used in the simulation have been taken from Ref [3,5–8, 22].

5. Derived PTE response and L-I characteristics solving the MPREs with the linear optical gain

First, I solved the coupled MPREs just considering the linear optical gain rather than the total optical gain. I also did not consider the term describing thermal carrier excitation from QDs.

Figure 2 shows calculated PTE response of the central lasing mode at the FWHM of IHB 20 meV for different injected currents 2, 2.5, 5, and 10 mA when the FWHM of HB is (a) 0.2 meV, (b) 2 meV, (c) 6 meV, (d) 10 meV, (e) 14 meV, and (f) 20 meV. Actually, Fig. 2 shows PTE response of the central lasing mode of the SAQDL that its L-E characteristics have been presented at Ref [6] in Fig. 6 .

Fig. 2 Simulated PTE response at the FWHM of IHB 20 meV for different injected currents 2, 2.5, 5, and 10 mA when FWHM of HB is (a) 0.2 meV, (b) 2 meV, (c) 6 meV, (d) 10 meV, (e) 14 meV, and (f) 20 meV considering the linear optical gain at the MPREs.

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Fig. 6 (a) L-I characteristics of a columnar quantum-dot laser [23] and (b) L-E properties, taking into account the total optical gain, for the injected currents 2.15 and 10 mA considering (sign 'TCER') and without considering thermal carrier excitation rate from QDs. The origin of the lasing mode is related to the central mode, K.

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As it is shown in Fig. 2(a) and 2(b), the SAQDL reaches the steady-state immediately after finishing dynamic response. In Fig. 2(c), the steady-state photon population at I = 2.5 mA is lesser than that of I = 2 mA (we can also see this phenomenon at Ref [6] in Fig. 6(c) for the central lasing mode). Lasing photon populations at I = 5 and 10 mA completely reach the steady state just after passing 80 and 40 ns, respectively. In fact, they do not reach the steady-state immediately after finishing the transient response. It is shown in Fig. 2(d) that the lasing photon populations at I = 5 and 10 mA decrease as the time increases and become even lesser than that of I = 2 mA after passing 45 ns (we can also see at Ref [6] in Fig. 6(d) for the central lasing mode that the output power at I = 10 mA is lesser than that of I = 2 mA), they do not completely reach the steady-state even after elapsing 100 ns. Lasing photon population at I = 10 mA also becomes lesser than that of I = 5 mA after passing 30 ns. Besides, the lasing photon population at I = 2.5 mA increases as the time enhances and does not reach the complete steady-state even after passing 100 ns. We can see in Fig. 2(e) that the lasing photon population at I = 2.5 mA reaches the complete steady-state after elapsing 80 ns but, those of I = 5 and 10 mA do not reach the complete steady-state and elevate as the time increases to more than 100 ns. As it is also shown in Fig. 2(f), the lasing photon population at I = 2.5 mA reaches the complete steady-state after elapsing 100 ns, while those of I = 5 and 10 mA reach the complete steady-state just after passing 400 and 600 ns, respectively.

We can conclude that the SAQDL reaches the complete steady-state immediately after finishing the transient response when the HB is small, in comparison to the IHB, because different QD lasing groups lase independently. While when the HB increases, especially when its FWHM becomes comparable, close, and equal to that of the IHB (Fig. 2(d), 2(e) and 2(f)), the SAQDL does not reach the steady-state after ending the dynamic response and the lasing photon populations continue to increase (or decrease) as the time enhances. These uncompleted steady-states may appear as a result of extreme emission of QD groups lying within the scope (FWHM) of HB of the central mode into it (or of extreme emission of these QD groups into other modes). On the other hand, they may not be reasonable and appear due to not considering the gain saturation effect. However, these uncompleted steady-states show that simulated static-characteristics of the SAQDLs presented at the previous papers, after passing small times like 8 ns or even larger times like 50 ns, have not been exact because the SAQDLs do not reach the complete steady-state after passing such times when the HB is comparable, close, or equal to the IHB.

Figure 3 shows L-I characteristics of the SAQDL calculated after passing the large time 100 ns for the central lasing mode at the FWHM of IHB and HB (a) 20 meV and 4, 9, 10, 11, and 13 meV, (b) 20 meV and 14, 16, 20, 40, and 90 meV, (c) 30 meV and 14, 20, 30, 40, and 80 meV, and (d) 60 meV and 30, 40, 50, and 60 meV.

Fig. 3 L-I characteristics for the FWHM of IHB and HB (a) 20 meV and 4, 9, 10, 11, and 13 meV, (b) 20 meV and 14, 16, 20, 40, and 90 meV, (c) 30 meV and 14, 20, 30, 40, and 80 meV, and (d) 60 meV and 30, 40, 50, and 60 meV considering the linear optical gain at the MPREs.

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As revealed in Fig. 3(a) to 3(c), nonlinearity appears in the L-I curves and continues until the HB becomes near to the IHB. For larger HBs, equal to and somewhat larger than the IHB, output power increases linearly with elevation of the injected current. Slope efficiency (external quantum differential efficiency) enhances as the HB increases from a special value up to the IHB. In addition, when the HB exceeds the IHB, L-I characteristics degrade (the output power and slope efficiency decrease and threshold current continues to increase). We can conclude that the SAQDL has the best L-I characteristics when the HB is equal to the IHB, in this case, the SAQDL has the largest slope efficiency and the highest output power for the injected currents more than 4 mA. For the low injected currents (0 to 4 mA), the highest output power corresponds to lower HBs [22]. When the FWHM of IHB is 60 meV (Fig. 3(d)), the L-I curves are linear for every HB. The highest output power corresponds to the FWHM of HB 30 meV for the intermediate currents of 3 to 6 mA and to the FWHM of HB 40 meV for the higher currents. The L-I curve vanishes when the HB is equal to the IHB.

Increasing the HB (equally, enhancing temperature), the threshold current increases (Fig. 3(b), 3(c) and 3(d)). We may phenomenologically attribute this result to elevating thermal carrier excitation from QDs to the WL that, in turn, leads to enhancement of the amount of the injected current required for establishing population inversion. For the HBs which are small in comparison or are comparable to the IHB, there is a small increase at the threshold current with enhancement of the HB (Fig. 3(a)) [22].

6. Calculated PTE response and L-I characteristics solving the MPREs with the total optical gain

In this section, I offer PTE response and L-I characteristics of the central lasing mode of the SAQDL derived solving the coupled MPREs considering the total optical gain.

Figure 4 shows calculated PTE response at the FWHM of IHB 20 meV for different injected currents 2, 2.5, 5, and 10 mA when the FWHM of HB is (a) 0.2 meV, (b) 2 meV, (c) 6 meV, (d) 10 meV, (e) 14 meV, and (f) 20 meV.

Fig. 4 PTE response at the FWHM of IHB 20 meV for different injected currents 2, 2.5, 5, and 10 mA when the FWHM of HB is (a) 0.2 meV, (b) 2 meV, (c) 6 meV, (d) 10 meV, (e) 14 meV, and (f) 20 meV considering the total optical gain at the MPREs.

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As we can see in Fig. 4(a) and 4(b), the lasing photon populations reach the steady-state quickly after finishing the relaxation oscillation similar to the previous case (Fig. 2(a) and 2(b)) because of independent lasing of diverse QD lasing groups. As it is shown in Fig. 4(c), the lasing photon populations at I = 5 and 10 mA completely reach the steady-state after passing 40 and 20 ns, respectively. As shown in Fig. 4(d), the lasing photon populations at I = 5 and 10 mA do not become lesser than that of I = 2 mA and reach the complete steady-state after elapsing 60 ns. Besides, the lasing photon population at I = 10 mA does not become lesser than that of I = 5 mA and the lasing photon population at I = 2.5 mA reaches the complete steady-state after passing 60 ns. It is clearly shown in Fig. 4(e) and 4(f) that the lasing photon populations at I = 5 and 10 mA reach the complete steady-state after passing 20 and 10 ns, respectively, and those of I = 2.5 mA reach the complete steady-state after elapsing 30 and 40 ns, respectively.

We infer that, for the FWHM of HBs that are equal to and more than 6 meV (especially for those that are comparable, close, and equal to the FWHM of IHB), considering the total optical gain at the MPREs causes the SAQDL to reach the complete steady-state quicker than the previous case (Fig. 2). In fact, the carriers whose QD lasing groups lie within the scope of HB of the central lasing mode, at the time of the relaxation oscillation and after ending it, do intra-subband scattering and emit into the central lasing mode until the gain saturation occurs at it after passing few nanoseconds (except to Fig. 4(c) at the current 2.5 mA and Fig. 4(d) at the currents 5 and 10 mA), therefore, the SAQDL reaches the complete steady-state faster. While as shown in Fig. 4(c) at the current 2.5 mA, emitting the carriers (whose QD lasing groups lie within the scope of HB of the central lasing mode), just at the time of the relaxation oscillation, within the other modes and as shown in Fig. 4(d) at the currents 5 and 10 mA, emitting those carriers, at the time of the relaxation oscillation and after ending it, within the other modes result in decreasing the semi steady-state photon populations of the central mode until the gain saturation happens at those modes after passing few nanoseconds. thus, the SAQDL again reaches the complete steady-state quicker. At the previous case (Fig. 2), since I did not consider the total optical gain at the MPREs, for the HBs that are comparable, close, and equal to the IHB (Fig. 2(d), Fig. 2(e), and Fig. 2(f)), there was nothing to inhibit the increase (or decrease) of the central lasing photon populations until large times passed.

These results have been achieved just because I considered the total optical gain (gain saturation effect) at the MPREs and are the same considering and not considering the term describing thermal carrier escape from QDs.

Figure 4 also reveals that with increase of the HB from (a) to (f), the steady-state photon population at the current 10 mA, except to Fig. 4(d), increases till a special HB (in Fig. 4(e)). After that, with further enhancement of the HB, a rollover happens and the steady-state photon population declines (Fig. 4(f)). We can deduce that for a specific current, enhancing the HB up to a special value (for example, for I = 2 mA, up to the FWHM of HB 10 meV in Fig. 4(d) and for I = 10 mA, up to the FWHM of HB 14 meV in Fig. 4(e)) leads to elevation of the steady-state photon population of the central lasing mode because of enhancement of the number of QD lasing groups lying within the scope (FWHM) of HB of the central lasing mode and as a result, increasing the carriers emitting within it. Further enhancement of the HB from that special value, on the other hand, results in further elevation of the thermal carrier escape while the carrier emission into the central mode does not increase significantly, thus, an inverse phenomenon that is, decreasing the steady-state photon population occurs [22].

Figure 5 shows simulated L-I characteristics for the central lasing mode at the FWHM of IHB and HB (a) 20 meV and 10, 11, 13, and 20 meV, (b) 20 meV and 20, 30, 40, 50, 60, 70, and 90 meV, (c) 30 meV and 14, 16, 30, and 34 meV, (d) 30 meV and 34, 40, 50, 60, 70, and 80 meV, (e) 60 meV and 14, 20, 30, and 34 meV, and (f) 60 meV and 34, 40, 50, 52, 54, and 60 meV.

Fig. 5 L-I characteristics for the FWHM of IHB and HB (a) 20 meV and 10, 11, 13, and 20 meV, (b) 20 meV and 20, 30, 40, 50, 60, 70, and 90 meV, (c) 30 meV and 14, 16, 30, and 34 meV, (d) 30 meV and 34, 40, 50, 60, 70, and 80 meV, (e) 60 meV and 14, 20, 30, and 34 meV, and (f) 60 meV and 34, 40, 50, 52, 54, and 60 meV considering the total optical gain at the MPREs.

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As shown in Fig. 5, in contrast with the previous case (Fig. 3), nonlinearity appears and remains in the L-I curves for every HB and fluctuations of the L-I curves are not as intensive as the previous case (for instance, compare Fig. 3(a) with Fig. 5(a) for the FWHM of HBs 10 and 11 meV) due to the gain saturation. Output power for every HB, at which the L-I curve just enhances as the injected current increases, is lesser than that of the previous case.

For the FWHM of IHBs 20 and 30 meV, as the HB increases up to the IHB, the slope efficiency and output power (for the injected currents more than 3 mA) enhance. Further enhancement of the HB up to a special value (in Fig. 5(b), up to the FWHM of HB around 30 meV, and in Fig. 5(c), up to the FWHM of HB 34 meV) leads to further increase of the slope efficiency and output power for the injected currents more than 7 mA. Comparing to the previous case, for the HBs which are near, equal to, and somewhat larger than the IHB, the output power does not grow linearly with increase of the injected current from a specific current. In Fig. 5(e), as the FWHM of HB increases up to 34 meV, the slope efficiency and output power (for the injected currents more than 4 mA) enhance but the output power does not grow linearly with elevation of the injected current in comparison with the previous case. The output power and slope efficiency elevate with further increase of the FWHM of HB up to 40 meV for the injected currents more than 5.5 mA (Fig. 5(f)). We deduce that, in contrast with the previous case, the best performance of the SAQDL occurs at a certain HB depending on the injected current and the IHB.

Further increase of the HB from that certain value (equally, further elevation of temperature) leads to degradation of L-I characteristics as finally the L-I curve vanishes. The HB, at which the L-I curve vanishes, is the same considering and not considering the nonlinear optical gain and the term describing thermal carrier excitation rate (compare Fig. (3) to Fig. (5)). Therefore, we may phenomenologically infer that declining and finally vanishing the L-I curve is due to enhancement of the thermal carrier escape to the WL. This implication is approved because the number of WL carriers, for example in Fig. 5(b), for the zero L-I curve elevates as much as 100 times more than that of the L-I curve which corresponds to the FWHM of HB 20 meV. Consequently, we find that the term describing thermal carrier escape from QDs to the WL is not sufficient to model the effects of temperature on the carrier escape dynamics. Besides, as the IHB increases, the zero L-I curve takes place at a smaller HB. Thus, it is acceptable to conclude that the (thermal) carrier excitation rate itself depends on the IHB directly. This also explains why the threshold current increases with enhancement of the IHB (refer to Fig. 5). This dependency means that changing the IHB, that is as a result of changing structural parameters like lattice mismatch, affects phonon energies of the structure and accordingly, carrier dynamics. In other word, the IHB variations affect the carrier dynamics similar to the temperature changes.

Experimental nonlinear L-I curves, at room-temperature (RT) and higher temperature operations, presented at Ref [23], are revealed in Fig. 6(a). Figure 6(a) is similar to our results in Fig. 5(b), at the FWHM of HB 30 meV and higher, and in Fig. 5(d) and Fig. 5(f), at the FWHM of HB 40 meV and higher. The experimental L-I curves approve that our simulated L-I curves are more accurate than the previous simulated ones (Fig. 3 and Ref [4–6].). Therefore, we find that it is completely necessary to take into account the total optical gain at the MPREs in order to precisely achieve dynamic and static behavior of the SAQDLs. Considering lasing of higher states in our model, the simulated L-I curves and the experimental ones offered in Fig. 6(a) would be exactly the same.

Comparing the HBs which have the highest L-I curves (in Fig. 5(b), 5(d), and 5(f)) with Fig. 6(a), we find that the HB which is related to RT operation varies depending on the IHB (assuming that the highest L-I curve corresponds to RT operation of the SAQDL at every IHB). This FWHM of HB is around 30 meV for the FWHM of IHB 20 meV and is around 40 meV for that of IHBs 30 and 60 meV. Consequently, I conclude that the relation between the HB and temperature may depend on the IHB. If the highest L-I curve, at different IHBs, occurs at different temperatures, it is also implied that the best HB, at which the maximum emission within the central mode happens, depends on temperature and the IHB.

In each case (considering and without considering the nonlinear optical gain at the MPREs), the threshold currents are the same. In fact, the nonlinear optical gain does not affect the threshold current.

I also obtained PTE response and L-I properties of the SAQDL considering the total optical gain and the statement describing thermal carrier excitation from QDs and found that the statement does not have a significant effect on PTE response and on L-I characteristics.

7. Simulation of L-E characteristics considering and not considering the total optical gain and thermal carrier excitation rate at the MPREs

In this section, I offer L-E properties of In(Ga)As/GaAs SAQDLs derived solving the MPREs considering and not considering the total optical gain and thermal carrier excitation and also multi-mode PTE response considering both of them.

Figure 6(b) shows L-E characteristics, at around RT, taking into account the total optical gain for the injected currents 2.15 and 10 mA considering (sign 'TCER') and without considering the term describing thermal carrier escape. The origin of the lasing mode is taken at the central mode, K.

As revealed in Fig. 6(b), considering the thermal carrier escape, the lasing mode corresponding to K-1 has the highest output power at the spectrum that is related to I = 2.15 mA, after that, K-2 and K are the dominant modes, respectively. While without considering the thermal carrier escape, the dominant lasing mode is K and after that, K-1 and K + 1 are the next ones, respectively. At the injected current 10 mA, considering and not considering the thermal carrier excitation, the dominant lasing modes are K, K-1, and K+1, respectively. We see that the thermal carrier escape shifts the lasing spectrum toward lower energies at low injected currents. This red shift becomes negligible for higher currents. In fact, considering the thermal carrier excitation rate, the dominant lasing modes are shifted toward higher energies as the injected current elevates till they become K, K-1, and K + 1, respectively (we will see this result clearly in Fig. 7(b) ). Movement of the dominant lasing modes toward higher energies as the current increases is observable at the experimental data at Ref [6] in Fig. 4(b) and 4(c) and at Ref [24] in Fig. 4, Fig. 7, and Fig. 11.

Fig. 7 L-E characteristics (a) not considering and (b) considering the nonlinear optical gain and thermal carrier excitation rate for different injected currents I = 2.1, 2.15, 2.2, 2.5, 3.5, 5, 10, and 20 mA. The origin of the lasing mode is related to the central mode and the FWHM of HB and IHB is 20 meV.

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Figure 7 shows L-E properties of the SAQDL for different injected currents I = 2.1, 2.15, 2.2, 2.5, 3.5, 5, 10, and 20 mA, at around RT, (a) not considering and (b) considering the nonlinear optical gain and thermal carrier excitation rate. The origin of the lasing mode corresponds to the central mode and the FWHM of IHB is 20 meV.

As shown in Fig. 7(a), there are 7 modes at the FWHM of the lasing spectrum corresponding to the injected current 2.1 mA. The number of lasing modes decreases with increase of the injected current as there are 3 dominant lasing modes at the lasing spectrums relating to I = 2.15 up to 5 mA and there is just one dominant lasing mode at those relating to the injected currents 10 and 20 mA. We can conclude that the lasing emission profile is narrowed as the injected current enhances and is dominated by the central lasing mode at currents that are near and higher than 10 mA. This result has been also obtained at the theoretical results presented at Ref [6] in Fig. 6(e) and Fig. 6(f).

As revealed in Fig. 7(b), there are 7 modes at the FWHM of the lasing spectrum relating to the injected current 2.1 mA. The lasing spectrum is narrowed with increase of the injected current up to 2.2 mA as there are 3 dominant lasing modes at the lasing spectrum of the current 2.2 mA. With further enhancement of the injected current, the lasing spectrum is broadened as there are 6 dominant lasing modes at the lasing spectrum of the injected current 5 mA, 9 dominant lasing modes at that of the current 10 mA, and 11 dominant lasing modes at that of the current 20 mA. It is clear that taking into account the nonlinear optical gain at the MPREs results in broadening the lasing spectrum (increasing the number of dominant lasing modes) as the injected current elevates from 2.2 mA, This is because, as the injected current increases from 2.2 mA, the QD ensemble emit into a more number of modes due to happening the gain saturation at the previous dominant lasing modes, as a result, the number of dominant lasing modes enhances. This result, that in reality, in contrast with Fig. 7(a), there is not a single mode lasing especially at higher currents like 10 mA, is also observable at the experimental findings at Ref [6]. in Fig. 4(c). In the figure, $Δ λ = 0.8 n m$ is the FWHM of the lasing spectrum of I = 10 mA. Calculating the number of dominant lasing modes (with cavity length 300μm as mentioned at the paper), we find that there are 24 dominant lasing modes at the spectrum. The lasing emission at I = 20 mA, presented here, is not the same as that of presented in Fig. 4(d) at Ref [6] because Fig. 4 corresponds to a SAQDL containing QDs with three dominant interband transition states; GS and two excited states, while here, I am investigating a SAQDL with QDs that just have dominant GS transitions.

I summarize that multi-mode lasing emission occurs, even at the HBs which are close and equal to the IHB and even at high injected currents, and the number of lasing modes enhances, even at (around) RT, as the injected current elevates from a special value due to the gain saturation effect.

Shift of the most dominant lasing modes (K-1, K-2, and K), toward higher energies (K, K-1, and K + 1) as the injected current heightens, is also clearly revealed in Fig. 7(b).

We also see similar results in Fig. 8 where the dynamic response of the SAQDL for the central lasing mode and some of neighboring modes has been calculated considering the nonlinear optical gain and thermal carrier escape rate. An important result is obtained from the figure; although the HB is equal to the IHB, for injected currents (like I = 2.1 mA) which are a bit larger than threshold current, QD lasing groups lase almost independently (Fig. 8(a)). While when the injected current heightens (I = 2.15 mA), the QD lasing groups lying within the scope of HB of three central modes begin to emit into them (especially into the central one) as the time elevates (Fig. 8(b)). It might be due to (repulsive) Coulomb interaction between electrons and holes, that enforces some of them to do intra-subband scattering within homogeneously broadened energy levels of the QD groups and emit into the central modes. Since our model is an excitonic one, similarly, I propose that the (repulsive) Coulomb interaction between excitons is responsible for the excitonic intra-subband scattering within homogeneously broadened energy levels of the QD groups.

Fig. 8 Multi-mode PTE response at the FWHM of IHB and HB 20 meV for the central lasing mode and some of neighboring modes for the injected currents (a) 2.1 mA, (b) 2.15 mA, (c) 5 mA, and (d) 10 mA considering the total optical gain and thermal carrier excitation at the MPREs.

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As the injected current enhances further (I = 5 and 10 mA), the number of dominant lasing modes elevates and also the SAQDL reaches the complete steady-state quicker. In addition, turn on delay decreases with increase of the injected current.

8. Conclusion

I investigated the effects of considering the nonlinear optical gain and thermal carrier excitation on the characteristics of In(Ga)As/GaAs SAQDLs. Comparing experimental findings with the simulation results, we found that it is quite crucial to consider the total optical gain at the MPREs in order to obtain more precise dynamic and static characteristics. For small HBs (in comparison to the IHB), considering or not considering the nonlinear optical gain, the lasing photon populations reach the complete steady-state immediately after finishing the relaxation oscillation. On the other hand, for the HBs that are comparable, close, equal to, and somewhat larger than the IHB, taking into account the nonlinear optical gain, the lasing photon populations do not decline or grow extremely as the time enhances and the output powers do not fluctuate intensely or increase linearly as the current elevates because the gain saturation occurs after few nanoseconds and the SAQDL reaches the complete steady-state sooner. Furthermore, the highest output power for the SAQDL, at a certain injected current, is a function of the HB and IHB. Taking into account the nonlinear optical gain, multi-mode lasing occurs, the number of dominant lasing modes increases, and the SAQDL reaches the complete steady-state faster as the injected current enhances. Elevating temperature from RT results in declining L-I characteristics due to enhancement of the thermal carrier escape. Thermal carrier excitation leads to a red shift in the dominant lasing modes at low injected currents, those modes are shifted toward higher energies as the injected current increases till the most dominant mode becomes the central one.

References and links

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Investigation of the effects of nonlinear optical gain and thermal carrier excitation on characteristics of self-assembled quantum-dot lasers

Abstract

1. Introduction

2. Linear optical gain theory

3. Derived total optical gain formula

4. Rate equations

5. Derived PTE response and L-I characteristics solving the MPREs with the linear optical gain

6. Calculated PTE response and L-I characteristics solving the MPREs with the total optical gain

7. Simulation of L-E characteristics considering and not considering the total optical gain and thermal carrier excitation rate at the MPREs

8. Conclusion

References and links

Cited By

Figures (8)

Equations (19)

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