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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 14 — Jul. 2, 2012
  • pp: 14754–14768
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Investigation of the effects of nonlinear optical gain and thermal carrier excitation on characteristics of self-assembled quantum-dot lasers

Davoud Ghodsi Nahri  »View Author Affiliations


Optics Express, Vol. 20, Issue 14, pp. 14754-14768 (2012)
http://dx.doi.org/10.1364/OE.20.014754


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Abstract

Comparing simulation results with experimental findings, it is found that considering nonlinear optical gain is quite essential to accurately obtain dynamic and static characteristics of self-assembled quantum-dot lasers (SAQDLs). In fact, the nonlinear optical gain prevents extreme decline or growth of photon population as the time increases and of output power as the injected current enhances. It also results in multi-mode lasing and increasing the number of lasing modes with elevation of the injected current. In addition, the best performance of SAQDLs, at a certain injected current, depends on homogeneous and inhomogeneous broadening. Thermal carrier excitation results in degradation of light-current characteristics. It also leads to a red shift in dominant lasing modes at low injected currents, the dominant lasing modes move toward higher energies as the current enhances until the most dominant mode becomes the central one.

© 2012 OSA

1. Introduction

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

6. M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, and A. Sakamoto, “Effect of homogeneous broadening of optical gain on lasing spectra in self-assembled InGaAs/GaAs quantum dot lasers,” Phys. Rev. B 61(11), 7595–7603 (2000). [CrossRef]

,23

23. K. Mukai, Y. Nakata, H. Shoji, M. Sugawara, K. Ohtsubo, N. Yokoyama, and H. Ishikawa, “Lasing with low threshold current and high output power from columnar-shaped InAs/GaAs quantum dots,” Electron. Lett. 34(16), 1588–1590 (1998). [CrossRef]

,24

24. H. Shoji, Y. Nakata, K. Mukai, Y. Sugiyama, M. Sugawara, N. Yokoyama, and H. Ishikawa, “Lasing characteristics of self-formed quantum-dot lasers with multistacked dot layer,” IEEE J. Sel. Top. Quantum Electron. 3(2), 188–195 (1997). [CrossRef]

], thus, only the transitions between ground states (GSs) of electrons and holes are dominant recombination processes [5

5. M. Sugawara, “Self -Assembled InGaAs/GaAs Quantum Dots,” (Academic Press, 60, 1999), Chap. 6.

,24

24. H. Shoji, Y. Nakata, K. Mukai, Y. Sugiyama, M. Sugawara, N. Yokoyama, and H. Ishikawa, “Lasing characteristics of self-formed quantum-dot lasers with multistacked dot layer,” IEEE J. Sel. Top. Quantum Electron. 3(2), 188–195 (1997). [CrossRef]

]. 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

6. M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, and A. Sakamoto, “Effect of homogeneous broadening of optical gain on lasing spectra in self-assembled InGaAs/GaAs quantum dot lasers,” Phys. Rev. B 61(11), 7595–7603 (2000). [CrossRef]

]
G(1)(E)=2πe2DQD3Dcnrε0m02c,v|Pcvσ|2Ecv(fc(E)fv(E))×LHom(1)(EE)GInh(EEcv)dE
(1)
wherenris the refractive index,DQD3D is the volumetric density of states (DOS) of QDs, |Pcvσ|2 is the transition matrix element,Ecvis the center of the energy distribution function of each interband transition,fc(E)is the electron occupation function of the conduction-band discrete state of the QDs with the interband transition energy ofE, andfv(E)is that of the valence band discrete state. The linear optical gain shows the HB of a Lorentz shape as LHom(1)(EE)=(γcv/π)/[(EE)2+(γcv)2] in which the full width at half maximum (FWHM) is given as Γcv=2γcv with the polarization dephasing or scattering rate γcv. The HB depends on temperature and various scattering mechanisms. The IHB which models shape, size, and composition fluctuations of QDs are represented byGInh(EEcv) that takes a Gaussian distribution function as
GInh(EEcv)=12πξ0exp[(EEcv)2/2ξ02]
(2)
whose FWHM is given byΓ0=2.35ξ0. Neglecting the optical-field polarization dependence, the transition matrix element is given by |Pcvσ|2=|Icv|2M2 whereIcvrepresents the overlap integral between the envelope functions of an electron and a hole, and M2=(m02/12me)(Eg(Eg+Δ)/(Eg+2Δ/3)) that is derived by the first-order k.p interaction between the conduction and valence band. Here,Egis the band gap of In(Ga)As bulk material,m0is the electron rest mass,meis the electron effective mass, and Δis the spin-orbit interaction energy of the QD material [6

6. M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, and A. Sakamoto, “Effect of homogeneous broadening of optical gain on lasing spectra in self-assembled InGaAs/GaAs quantum dot lasers,” Phys. Rev. B 61(11), 7595–7603 (2000). [CrossRef]

].

3. Derived total optical gain formula

G(3)(E)=πe4ξc2nr2ε02m04VQDΓ||(Ip(E)E2)c,v|Pcvσ|4Ecvγcv(fc(E)fv(E))×LHom(3)(EE)GInh(EEcv)dE
(4)

The total optical gain, including linear and nonlinear optical gain, is given by
Gtot(E)=G(1)(E)+G(3)(E)
(5)
after doing little mathematical calculation, the total optical gain of SAQDs is derived as
Gtot(E)=c,vdG(1)(E)[1+ΓSε(E)LHom(1)(EE)/Va]
(6)
wheredG(1)(E)is represented by
dG(1)(E)=2πe2DQD3Dcnrε0m02|Pcvσ|2Ecv(fc(E)fv(E))×LHom(1)(EE)GInh(EEcv)dE
(7)
and the third-order nonlinear gain coefficient is given as

ε(E)=πe2|Pcvσ|2ε0m02nr2Γ||(1E)
(8)

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
Fig. 1 Energy diagram of the laser-waveguide region and diffusion, recombination, relaxation, and escape processes.
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.,fc(E)=1fv(E).

(2K+1)=4Γ0/ΔE
(9)

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.

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

6. M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, and A. Sakamoto, “Effect of homogeneous broadening of optical gain on lasing spectra in self-assembled InGaAs/GaAs quantum dot lasers,” Phys. Rev. B 61(11), 7595–7603 (2000). [CrossRef]

] 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

6. M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, and A. Sakamoto, “Effect of homogeneous broadening of optical gain on lasing spectra in self-assembled InGaAs/GaAs quantum dot lasers,” Phys. Rev. B 61(11), 7595–7603 (2000). [CrossRef]

] 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.

Figure 3
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.
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.

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

22. D. Ghodsi Nahri, “Simulation of output power and optical gain characteristics of self-assembled quantum-dot lasers: Effects of homogeneous and inhomogeneous broadening, quantum dot coverage and phonon bottleneck,”Opt. Laser Technol. 44(8), 2436–2442 (2012), http://dx.doi.org/10.1016/j.optlastec.2012.04.002.

]. 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

22. D. Ghodsi Nahri, “Simulation of output power and optical gain characteristics of self-assembled quantum-dot lasers: Effects of homogeneous and inhomogeneous broadening, quantum dot coverage and phonon bottleneck,”Opt. Laser Technol. 44(8), 2436–2442 (2012), http://dx.doi.org/10.1016/j.optlastec.2012.04.002.

].

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
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.
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.

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

22. D. Ghodsi Nahri, “Simulation of output power and optical gain characteristics of self-assembled quantum-dot lasers: Effects of homogeneous and inhomogeneous broadening, quantum dot coverage and phonon bottleneck,”Opt. Laser Technol. 44(8), 2436–2442 (2012), http://dx.doi.org/10.1016/j.optlastec.2012.04.002.

].

Figure 5
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.
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.

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.

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.

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

6. M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, and A. Sakamoto, “Effect of homogeneous broadening of optical gain on lasing spectra in self-assembled InGaAs/GaAs quantum dot lasers,” Phys. Rev. B 61(11), 7595–7603 (2000). [CrossRef]

] in Fig. 6(e) and Fig. 6(f).

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
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.
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.

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.

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D. Ghodsi Nahri, “Simulation of output power and optical gain characteristics of self-assembled quantum-dot lasers: Effects of homogeneous and inhomogeneous broadening, quantum dot coverage and phonon bottleneck,”Opt. Laser Technol. 44(8), 2436–2442 (2012), http://dx.doi.org/10.1016/j.optlastec.2012.04.002.

23.

K. Mukai, Y. Nakata, H. Shoji, M. Sugawara, K. Ohtsubo, N. Yokoyama, and H. Ishikawa, “Lasing with low threshold current and high output power from columnar-shaped InAs/GaAs quantum dots,” Electron. Lett. 34(16), 1588–1590 (1998). [CrossRef]

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OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.5960) Lasers and laser optics : Semiconductor lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: August 4, 2011
Revised Manuscript: November 2, 2011
Manuscript Accepted: December 15, 2011
Published: June 18, 2012

Citation
Davoud Ghodsi Nahri, "Investigation of the effects of nonlinear optical gain and thermal carrier excitation on characteristics of self-assembled quantum-dot lasers," Opt. Express 20, 14754-14768 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-14754


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References

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