Effect of the number of stacking layers on the characteristics of quantum-dash lasers

A theoretical model is evaluated to investigate the characteristics of InAs/InP quantum dash (Qdash) lasers as a function of the stack number. The model is based on multimode carrier-photon rate equations and accounts for both inhomogeneous and homogeneous broadenings of the optical gain. The numerical results show a non monotonic increase in the threshold current density and a red shift in the lasing wavelength on increasing the stack number, which agrees well with reported experimental results. This observation may partly be attributed to an increase of inhomogeneity in the active region. ©2011 Optical Society of America OCIS codes: (140.5960) Semiconductor lasers; (250.5590) Quantum-well, -wire and -dot devices. References and links 1. F. Lelarge, B. Dagens, J. Renaudier, R. Brenot, A. Accard, F. van Dijk, D. Make, O. Le Gouezigou, J. Provost, and F. Poingt, “Recent advances on InAs/InP quantum dash based semiconductor lasers and optical amplifiers operating at 1.55 m,” IEEE J. Sel. Top. Quantum Electron. 13(1), 111–124 (2007). 2. J. Reithmaier, A. Somers, S. Deubert, R. Schwertberger, W. Kaiser, A. Forchel, M. Calligaro, P. Resneau, O. Parillaud, S. Bansropun, M. Krakowski, R. Alizon, D. Hadass, A. Bilenca, H. Dery, V. Mikhelashvili, G. Eisenstein, M. Gioannini, I. Montrosset, T. W. Berg, M. Poel, J. Mørk, and B. Tromborg, “InP based lasers and optical amplifiers with wire-/dot-like active regions,” J. Phys. D Appl. Phys. 38(13), 2088–2102 (2005). 3. C. Tan, H. Djie, Y. Wang, C. Dimas, V. Hongpinyo, Y. Ding, and B. Ooi, “Wavelength tuning and emission width widening of ultrabroad quantum dash interband laser,” Appl. Phys. Lett. 93(11), 111101 (2008). 4. D. Zhou, R. Piron, M. Dontabactouny, O. Dehaese, F. Grillot, T. Batte, K. Tavernier, J. Even, and S. Loualiche, “Low threshold current density of InAs quantum dash laser on InP (100) through optimizing double cap technique,” Appl. Phys. Lett. 94(8), 081107 (2009). 5. D. Zhou, R. Piron, M. Dontabactouny, E. Homeyer, O. Dehaese, T. Batte, M. Gicquel, F. Grillot, K. Tavernier, J. Even, and S. Loualiche, “Effect of stack number on the threshold current density and emission wavelength in quantum dash/dot lasers,” Phys. Status Solidi 6(10), 2217–2221 (2009) (c). 6. D. Zhou, R. Piron, F. Grillot, O. Dehaese, E. Homeyer, M. Dontabactouny, T. Batte, K. Tavernier, J. Even, and S. Loualiche, “Study of the characteristics of 1.55 m quantum dash/dot semiconductor lasers on InP substrate,” Appl. Phys. Lett. 93(16), 161104 (2008). 7. C. Tan, H. Djie, Y. Wang, C. Dimas, V. Hongpinyo, Y. Ding, and B. Ooi, “The influence of nonequilibrium distribution on room-temperature lasing spectra in quantum-dash lasers,” IEEE Photon. Technol. Lett. 21(1), 30– 32 (2009). 8. M. Gioannini, “Numerical modeling of the emission characteristics of semiconductor quantum dash materials for lasers and optical amplifiers,” IEEE J. Quantum Electron. 40(4), 364–373 (2004). 9. Z. Mi, and P. Bhattacharya, “DC and dynamic characteristics of P-doped and tunnel injection 1.65m InAs quantum-dash lasers grown on InP (001),” IEEE J. Quantum Electron. 42(11–12), 1224–1232 (2006). 10. H. Dery, and G. Eisenstein, “Self-consistent rate equations of self-assembly quantum wire lasers,” IEEE J. Quantum Electron. 40(10), 1398–1409 (2004). 11. M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, and A. Sakamoto, “Effect of homogeneous broadening of optical gain on lasing spectra in self-assembled In_ {x} Ga_ {1-x} As/GaAs quantum dot lasers,” Phys. Rev. B 61(11), 7595–7603 (2000). 12. K. Veselinov, F. Grillot, C. Cornet, J. Even, A. Bekiarski, M. Gioannini, and S. Loualiche, “Analysis of the double laser emission occurring in 1.55μm InAs–InP (113) B quantum-dot Lasers,” IEEE J. Quantum Electron. 43(9), 810–816 (2007). 13. F. Grillot, K. Veselinov, M. Gioannini, I. Montrosset, J. Even, R. Piron, E. Homeyer, and S. Loualiche, “Spectral analysis of 1.55 μm InAs–InP (113) B quantum-dot lasers based on a multipopulation rate equations model,” IEEE J. Quantum Electron. 45(7), 872–878 (2009). #143370 $15.00 USD Received 1 Mar 2011; revised 12 May 2011; accepted 17 May 2011; published 27 Jun 2011 (C) 2011 OSA 4 July 2011 / Vol. 19, No. 14 / OPTICS EXPRESS 13378 14. D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, “Gain and noise saturation of wide-band InAs-InP quantum dash optical amplifiers: model and experiments,” IEEE J. Sel. Top. Quantum Electron. 11(5), 1015–1026 (2005). 15. R. Schwertberger, D. Gold, J. Reithmaier, and A. Forchel, “Long-wavelength InP-based quantum-dash lasers,” IEEE Photon. Technol. Lett. 14(6), 735–737 (2002). 16. T. Amano, S. Aoki, T. Sugaya, K. Komori, and Y. Okada, “Laser characteristics of 1.3-μm quantum dots laser with high-density quantum dots,” IEEE J. Sel. Top. Quantum Electron. 13(5), 1273–1278 (2007). 17. N. Nuntawong, Y. Xin, S. Birudavolu, P. Wong, S. Huang, C. Hains, and D. Huffaker, “Quantum dot lasers based on a stacked and strain-compensated active region grown by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 86(19), 193115 (2005). 18. L. Asryan, and R. Suris, “Inhomogeneous line broadening and the threshold current density of a semiconductor quantum dot laser,” Semicond. Sci. Technol. 11(4), 554–567 (1996).


Introduction
Tuning the emission wavelength of Qdash and quantum dot (Qdot) lasers depends primarily on the size of the nano structures and particularly the height [1,2].This feature together with the InP technology and various process techniques has enabled the demonstration of Qdash/Qdot lasers in the c-band telecommunication window with improved performances [3,4].On the other hand, recent experimental studies on InAs/InP Qdash lasers by different research groups have reported the dependence of the lasing wavelength on laser structural parameters [5][6][7].They have observed a red shift in the lasing wavelength on increasing the number of stack layers or the cavity length.This observation is ascribed to the unique characteristics of Qdashes determined by their density of states (DOS), in general [5,7].However, the phenomenon needs more physical insight, particularly the stacking layer effect as this is a fundamental technique to improve the characteristics of Qdash lasers.Therefore, a theoretical assessment is essential to understand the phenomenon in a more comprehensive manner.In addition, one may expect this observation as a result of alteration to other laser parameters that change with the stack number.Our results are important as this is directly related to the spectral characteristics and, in general, the performance characteristics of the device.
In this work, a simulation model is considered to compare the characteristics of InAs/InP Qdash lasers as a function of the number of stacking layers.We theoretically verify the red shift in the central lasing wavelength (calculated by identifying the central wavelength at full width at half maximum (FWHM) of the lasing spectra) and non monotonic increase in the threshold current density on increasing the stack number.The numerical simulations show a good agreement with experimental observations.By the simulations we find that the phenomenon is partly due to the inherent change in the laser parameters particularly the active region inhomogeneity.

Theoretical model
The numerical model, applicable to InAs/InP Qdash lasers, is evaluated from the basic coupled rate equations of each Qdash ensemble incorporating the carrier and photon dynamics at each energy level.The technique is based on the density matrix formulation of the Qdash gain media [8,9] where the quantum wire like nature has a large influence on the gain properties of the laser [10].The formulation is similar to the one reported for the analysis of both InGaAs/GaAs [11] and InAs/InP [12,13] Qdot lasers, and InAs/InP Qdash semiconductor optical amplifier [14], following identical assumptions., jk E represents a generic energy level of the system.We consider a three level energy system [11] consisting of the separate confinement heterostructure (SCH), the wetting layer (WL) and the GS energy levels of the dashes with carrier populations B E E  with FWHM hom  and inhomogeneous Gaussian broadening, of the optical gain is considered in the formulation.Therefore, the fraction of energy states available at the energy level , jk E is given by [14]: The first term of Eq. ( 1) is the inhomogeneous term with FWHM of the second term is the ratio of two integrals [14].Note that , jk G is normalized as , , . The rate equations are as follows: , .
Equations ( 2), ( 3) and ( 4) refer to the carrier dynamics in the SCH, WL and dash GS energy levels.I is the current injection, i  is the internal quantum efficiency, ( , ) is the carrier relaxation lifetime from SCH(WL) to WL(GS) with the bar denoting an average lifetime [11,14], ()


is the excitation lifetime from WL(GS) to SCH(WL), and , p Sp  are the lifetimes of spontaneous emission and photons, respectively [11].Note that DW  is calculated through the condition of detailed balance [14] and p  according to [11].The multi-mode photon rate equation is given by Eq. ( 5) where m S is the photon population of the th m mode.Moreover, represents the linear optical gain [8,9] of the , jk E dash group contributing to the th m mode photons, where is the transition matrix and , , , is the carrier occupational probability (including the spin) [11,14].The Qdash laser considered for the analysis is obtained from [15] and is based on the InAs/InP material system.Four stacks of InAs Qdashes with an average height of 1.5 nm and width of 20 nm constitutes the active region with volume .The steady state lasing spectra are calculated [11] using the above set of rate equations with the fourth-order Runge-Kutta numerical method.We adopt an initial carrier relaxation lifetime of The degeneracies of the WL and GS is taken as , and 21 d M  varies from 201 to 401 based on the convergence achieved at each inh  value once stabilizing the lasing spectrum.

Numerical results
Figure 1 illustrates the effect of the number of stacking layers ( lyr N ) on the threshold current density ( th J ) of Qdash lasers.The experimental data is obtained from [5,6] and is plotted in Fig. 1(a) for comparison.Figure 1(b) corresponds to the simulation results at various values of the inhomogeneous broadening ( inh  ).A non monotonic increase in the threshold current density is observed on increasing the stack number, which is in good agreement with the experimental data except that the threshold current density values are different in Figs.1(a) and (b).This is an anticipated discrepancy owing to the fact that the two laser structures considered are rather different.Our aim here is to numerically address the trend of th J and the central lasing wavelength ( c  ) as a function of the number of stacking layers and explain the behavior qualitatively.Therefore, we model the dash DOS high energy tail by a stair case approximation utilizing Eq. ( 1) with N = 50 rather than calculating the accurate energy states of the Qdashes.An almost linear increase in th J is seen when 2 lyr N  as show in Fig. 1(b), at all inhomogeneity values.This may be attributed to the active region volume A V , the optical confinement factor  , and the inhomogeneous broadening.Increase in A V due to stacking of the dash layers may enhance internal absorptions, thus affecting the threshold current density.On the other hand, a lower  (due to increase in A V as a result of increase in lyr N ) probably assists in attaining the modal gain ( th g  ) relatively fast, thereby decreasing the threshold current density [6,16].However, the results show an increase in th J for 2 lyr N  which suggests that A V dominates.Moreover, increase in th J could be due to an increase in modal gain as a result of stacking of the dash layers which has been reported experimentally [1,5,16].Nevertheless, our model considers a fixed modal gain ( ) independent of the number of stacking layers in spite of varying  accordingly (0.009 per layer) and therefore, probably, does not affect the threshold current density numerically.
Increase in th J due to lyr N has also been attributed to the increase in inh  by various experimental work as a result of change in dash sizes and density due to subsequent growth of dash layers [1,16,17].However, in our analysis we assume an identical dash density per layer and fixed the inhomogeneous broadening for all the values of This observation is relatively consistent with the results from literature that are based on Qdot model [18] since Qdashes may be thought of an elongated Qdots with quasi zero dimensional DOS.Here, we make an effort to explain this observation qualitatively in a generalized manner.Increase in th J may be ascribed to the increase in size dispersion of the dashes, particularly the height, which possibly increases the internal absorptions (due to dispersion in energy states of dashes resulting in overlapping states).Higher energy photons from smaller dashes which acquire lasing conditions first (due to their dot like features) get absorbed by the longer dashes (with relatively smaller band transition energies) which eventually dominate due to their higher modal gain and DOS.
For a Qdash laser with a single stack layer, no lasing is observed experimentally and th J reaches an infinite value, as depicted in Fig. 1(a).This has been attributed to the very small  and low dash density [5].However, the numerical results show that besides the above mentioned parameters inh  strongly affects the lasing condition and is an important parameter A cm ).This observation may again be ascribed to reduced internal absorptions due to relatively similar energy states of dashes in the less inhomogeneous system, thus being able to attain lasing from the low density single dash layer with small  .Our model also predicts a minimum of th J for the two and three layer stack structures irrespective of the active region inhomogeneity values.This supports the experimental observation of Fig. 1(a) and also the numerical study of the Qdots [18].In general, based on our observation we may write a relation for th J in a similar manner reported for Qdots [16], as / th inh A JV    , where  , inh  , and A V dominates at the two extreme values of stack number (1 and 8, respectively).However, for 2 lyr N  and 3 these parameters probably balance each other thus attaining a relatively small value of threshold current density.The experimental results [5] of lasing spectra as a function of the stack number are shown in Fig. 2(a) and the results obtained from the model in Fig. 2

(b). A red shift trend in c
 is observed experimentally on increasing lyr N , which is well reflected by our calculation, thus showing the effectiveness of our model.The behavior is seen to be consistent with increasing inhomogeneity.A total red shift of ~7.5 nm is observed on increasing the stack number from 2 to 8, corresponding to 15 inh meV


. Since, the model does not take into consideration the growth and processing parameters that affect the lasing wavelength, we may attribute this observation to the optical confinement factor (lowers with increase in lyr N ) which probably assists in achieving the th g  rather early.Therefore the lasing condition might be achieved with occupation of carriers in relatively lower energy states, thereby lasing at longer wavelength.Note that the emission at shorter wavelength due to increase in th J with lyr N is suppressed.
Interestingly, our results show that the red shift phenomenon may also be attributed to active medium inhomogeneity.To support this statement we compare the lasing spectra of Qdash lasers at various explicit values of Moreover, a total red shift of ~26 nm is observed for 45 inh meV  on increasing lyr N from 2 to 8, which is more than three times the value of less inhomogeneous system.This unique observation is a consequence of a quasi zero dimensional DOS of the Qdashes which exploits the increase in the higher energy tail states due to dispersion in dash sizes as a result of increase in inh  .In general, dispersive dash sizes result in overlapping DOS which probably increases the states in the high energy tail of the dashes DOS.Therefore, higher energy photons from shorter dashes (smaller height and larger band transition energies) which lase first probably get absorbed in the high energy tails (now incorporating more electronic states due to overlap) of longer dashes (larger height and smaller band transition energies) which lase later and eventually dominate.Therefore, lasing occurs at longer wavelengths due to the small energy photons of the longer dashes.As inh  increases, the dashes with least band transition energy subsequently dominate and the lasing shifts to longer wavelengths (red shift of c  ).In general, we may then deduce the relationship c inh     .).Hence, further shift to the longer wavelengths is predicted theoretically.However, an increase of th J and a reduction of red shift phenomenon is observed at large i  ( 1 14 cm  ) as illustrated in Fig. 3.This is again consistent with the number of stacking layers.The observation may again be related to the practical case of a large stacking layer structure which improves the modal gain (

Conclusion
In conclusion, we have demonstrated theoretically the effect of the number of stacking layers on the characteristics of Qdash lasers.Our model predicts an increase in threshold current density as a result of an increase in the stack number, inhomogeneous broadening and modal gain.We have shown that the red shift phenomenon is a result of the optical confinement factor and, more significantly, of the inhomogeneous broadening.We have qualitatively explained the enhanced red shift phenomenon due to the active medium inhomogeneity by considering the unique DOS of Qdashes.

Fig. 1 .
Fig. 1.(a) Experimental[5,6] and (b) Calculated threshold current density versus the number of stacking layers for the Qdash lasers reported in[5] and[15], respectively.The three curves in (b) correspond to various inhomogeneous broadening values.
is 1 nm thick with a cross section dash density of dash GS, while the carrier relaxation to and re-excitation from SCH are 0The recombination lifetimes within SCH, WL and Qdash GS are, respectively, S

lyrN
. Therefore, to understand the effect of inh  we plot the trend of th J at different inh  in Fig. 1(b).A further increase in th J for the entire values of lyr N is observed when inh  increases from 15 meV to 45 meV .

Fig. 2 .
Fig. 2. (a) Experimental [5] and (b) calculated lasing wavelength as a function of the stack number for the Qdash lasers reported in [5] and [15], respectively.The three curves in (b) correspond to various inhomogeneous broadening values calculated at low current injection ( 1.1 th J ).
to a red shift of ~16.5 nm when inh  is increased three times.

Fig. 3 .( 1 7
Fig. 3. Calculated (a) threshold current density and (b) central lasing wavelength (at 1.1 , Fig.3(b)  show that inh  is still the dominant parameter of the red shift phenomenon.
The dashes are grouped into 21 d M  groups according to their central transition wavelength