The linewidth enhancement factor α of quantum dot semiconductor lasers

The linewidth enhancement factorα ofquantum dot semiconductor lasers

Sergey Melnik and Guillaume HuyetTyndall National Institute and Physics Department, Lee Maltings, National University of

Ireland, University College, Cork, Ireland.

[email protected], [email protected]

http://www.tyndall.ie/research/optoelectronics

Alexander V. UskovP. N. Lebedev Physical Institute, Leninsky pr. 53, 117924 Moscow, Russia

[email protected]

Abstract: We show that the various techniques commonly used tomeasure the linewidth enhancement factor can lead to different valueswhen applied to quantum dot semiconductor lasers. Such behaviour is adirect consequence of the intrinsic capture/escape dynamics of quantum dotmaterials and of the free carrier plasma effects. This provides an explanationfor the wide range of values experimentally measured and thelinewidthre-broadening recently measured.

© 2006 Optical Society of AmericaOCIS codes: (250.0250) Optoelectronics; (140.0140) Lasers and laser optics; (140.5960)Semiconductor lasers

References and links1. D. Bimberg, M. Grundmann, and N. N. Ledentsov,Quantum Dot Heterostructures (John Wiley & Sons, Chich-

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Zhukov, S. S. Mikhrin, A. R. Kovsh, “Feedback sensitivity of1.3 µm InAs/GaAs quantum dot lasers,” Electron.Lett. 39, 1819-1820 (2003).

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factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527-1529 (1999).7. C. Harder, K. Vahala, and A. Yariv, “Measurement of the linewidth enhancement factorα of semiconductor

lasers,” Appl. Phys. Lett.42, 328-330 (1983).8. R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, “Linewidth

broadening factor of a microcavity semiconductor laser,” Appl. Phys. Lett.61, 1883-1885 (1992).9. Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, “Newlinewidth enhancement determination method in

semiconductor lasers based on spectrum analysis above and below threshold,” Electron. Lett.28, 9-11 (1992).10. K. Petermann,Laser Diode Modulation and Noise (Kluwer, Dordrecht, 1991).11. S. Schneider, P. Borri, W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Linewidth enhance-

ment factor in InGaAs quantum-dot lasers,” IEEE J. Quantum Electron.40, 1423-1429 (2004).12. J. Muszalski, J. Houlihan, G. Huyet, and B. Corbett, “Measurement of linewidth enhancement factor in self-

assembled quantum dot semiconductor lasers emitting at 1310nm,” Electron. Lett.40, 428-430 (2004).13. A. A. Ukhanov, A. Stintz, P. G. Eliseev, and K. J. Malloy, “Comparison of the carrier induced refractive index,

gain, and linewidth enhancement factor in quantum dot and quantum well lasers,” Appl. Phys. Lett.84, 1058-1060 (2004).

#67519 - $15.00 USD Received 6 February 2006; revised 20 March 2006; accepted 20 March 2006

(C) 2006 OSA 3 April 2006 / Vol. 14, No. 7 / OPTICS EXPRESS 2950

mailto:[email protected]



http://www.tyndall.ie/research/optoelectronics

14. A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoen and A. Fiore, “Impact of intrabandrelaxation on the performance of a quantum-dot laser,” IEEE J.Select. Topics Quantum Electron.9, 1308-1314(2003).

15. B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le Gouezigou, J. Landreau, A. Fiore and B.Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dotlaser,” Electron. Lett.41, 323-324 (2005).

16. A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, G. Huyet, “Carrier-induced refractiveindex in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl.Phys. Lett.84, 272-274 (2004).

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19. A. V. Uskov, Y. Boucher, J. Le Bihan, and J. McInerney, “Theory of a self-assembled quantum-dot semiconductorlaser with Auger carrier capture: quantum efficiency and nonlinear gain,” Appl. Phys. Lett.71, 1499-1501 (1998).

20. S. P. Hegarty, B. Corbett, J. G. McInerney and G. Huyet, “Free-carrier effect on index change in 1.3µm quantum-dot lasers,” Electron. Lett.41, 416-418 (2005).

21. H. C. Schneider, W. W. Chow and S. W. Koch, “Anomalous carrier-induced dispersion in quantum-dot activemedia,” Phys. Rev. B66, 041310 (2002).

22. H. C. Wong, G. B. Ren, J. M. Rorison. ”Mode amplification in inhomogeneous QD semiconductor optical am-plifiers,” Opt. Quant. Electron. (to be published).

23. H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, “Linewidth study of InAs-InGaAs quantumdot distributed feedback lasers,” IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).

1. Introduction

Quantum dot (QD) semiconductor lasers have already demonstrated many interesting proper-ties such as temperature insensitive low threshold currentdensities [1], high modulation band-widths [2, 3], and a strong resistance to optical feedback [4], therefore showing great prospectstowards realizing uncooled, isolator-free, directly modulated semiconductor lasers. All of thesefeatures originate from the quantum confinement that usually characterizes atoms or moleculesin contrast to semiconductor materials. Indeed bulk and quantum well semiconductor materialshave a large density of states at high energy and, as a result,the maximum gain suffers a blueshift with increasing carrier density.

To quantify the effect of the change of refractive index withcarrier density on the dynamicalproperties of semiconductor lasers, Henry introduced the linewidth enhancement factorα [5],

α = −4πλ

dn/dNdg/dN

, (1)

wheren is the refractive index,g is the gain per unit length,λ is the wavelength andN is thecarrier density. The introduction of theα-factor in rate equations not only explained the exper-imentally observed broad linewidth and large chirp, but also the onset of coherence collapseinduced by optical feedback and filamentation in broad area lasers and amplifiers.

Since its definition, researchers have developed various experimental techniques to measurethe α-factor. The most commonly used techniques are based on the analysis of the amplifiedspontaneous emission (ASE) [6], on the FM/AM response ratiounder small signal currentmodulation [7], linewidth measurements [8, 9, 10], and pump-probe experiments [11].

Simple analytical calculations based on rate equations forbulk and quantum well semicon-ductor lasers show that all of these measurements should theoretically give the same valuefor the α-factor. For this reason, experimentalists have used thesetechniques to measure theα-factor of bulk, quantum well and, more recently, of QD semiconductor lasers.

For QD lasers, simple models predict a small if not zeroα-factor due to the discrete densityof states. However, experiments have reported a great variety of values for theα-factor rangingfrom 0 [2, 6], to values of the order of those commonly observed in bulk and quantum well



materials (1.5−3) [12, 13]. Larger values for theα-factor than those published for quantumwell devices have also been reported (4−11) [14], and a value forα as high as 60 has beenpublished [15]. It is also worth noticing that while the values for α reported from FM/AMmeasurements are much larger than those obtained from the analysis of the ASE, both tech-niques have shown an increase inα with injection [12, 14, 15]. The non-zeroα-factor could beexplained as a direct consequence of the plasma effect [16],or dispersion of the dot sizes [17].However, the large variance in the values forα suggests that the unique carrier dynamics needsto be taken into account to explain these experimental results.

The aim of this paper is to compare three techniques commonlyused to measure theα-factor of QD lasers, namely, analysis of ASE, small signal FM/AM response and linewidthmeasurements. Our analysis is based on QD rate equations incorporating both the finite capturetime of carriers into the dots and plasma effects [16]. It is found that each of the three techniquesgives a different value for the measuredα-factor, which increases with injection current due tothe plasma effect. These results are in qualitative agreement with recent experimental resultsand suggest that theα-factor might not be a well-defined parameter to characterise QD devices.

2. Quantum dot rate equation model

In QD semiconductor devices, the carriers are first injectedinto a wetting layer be-fore being captured into a dot at a capture rate that depends strongly on the dot pop-ulation. Thus, rate equations that commonly describe carrier dynamics of QD materialsread [18, 19],

N = −γnN +Jq−2CN (1−ρ) , (2)

ρ = −γdρ +CN (1−ρ)− vgσ (2ρ −1) |E|2, (3)

whereN is the number of carriers in the wetting layer per dot andρ is the occupation probabilityof a dot in the laser pumped with the current per dotJ. The parametersγn andγd are the non-radiative decay rates,C is the capture rate from the wetting layer into an empty dot,vg is thegroup velocity andσ is the cross section of interaction of the carriers in a dot with the electricfield. The constantq is the elementary charge andE is the complex amplitude of the electricfield in the cavity, normalized to the photon densityS: |E|2 = S.

The gaing0 (2ρ −1) is defined by the dot population. Experimental [20], and theoreticalanalyses [21, 16], have shown the refractive index of quantum dot materials can depend on theground state populationρ and on the surrounding carriersN. Thus, the temporal evolution forthe electric field reads

E = −12

γsE +12

vgg0(2ρ −1)(1+ iαd)E + iαnrvgg0NE, (4)

whereγs is the photon decay rate in the cavity andg0 is the differential gain. The coefficientsαd andαnr describe the change of refractive index with the change of dot carrier populationand with non-resonant population respectively. In real QD materials the non-resonant carriersare described by more than one quantity, as carriers can be present in the wetting layers or invarious excited states of the dots. The size dispersion of the dots and thermalisation through awetting layer also lead to a non-zeroα-factor which is here described by the parameterαd . Ouranalysis could also be carried out using a model describing the evolution for each dot size asin [22]. However, it would include several other unknown parameters and make the derivationof analytical results much more difficult. Here, for simplicity, only a single non-resonant carrierdensityN and a single dot populationρ are taken into account.



0 Jth

Injection Current

-10-505

101520

α-fa

ctor

0 Jth

Injection Current

05

1015202530

α-fa

ctor

(a)

(b)

Fig. 1. Values of theα-factor calculated by the following techniques: solid line is theASE method, dashed line is the FM/AM method, dotted line is the linewidth method. (a)Laser operates for non-saturated dots (ρth = 0.6). (b) Laser operates close to full inversion(ρth = 0.9). For the non-saturated operating point (a) theα-factor does not show strong de-pendence on the injection current. This is not the case for operation closeto full inversion(b).

3. Analysis of the amplified spontaneous emission

The first technique commonly used to measure theα-factor is based on the analysis of the ASEof a laser biased below threshold. In this regime, a change ofthe injection currentδJ induceschanges in the carrier densitiesδN andδρ. This in turns leads to changes in the gainδg =2g0δρ and laser frequencyδω = vgg0(αdδρ + αnrδN). From (1), it follows thatα = 2

vg

δωδg

and

α = αd +αnrδNδρ

. (5)

This shows that theα-factor depends on the non-resonant carrier densityN and that thiscontribution becomes more important near full inversion asshown in Fig. 1. Close to full in-version the dot occupation probability saturates to 1, but the carrier density in the wetting layercontinues to increase with current, leading to a dramatic increase inα. From Eqs. (2) and (3) itfollows thatδN/δρ = γd/(C(1−ρss)

2), whereρss is a steady state value, which increases withinjection current up to the threshold currentJth, and then clamps at a valueρth.

In [20], it was demonstrated that the non-resonant carrierscontribute approximately half thetotal index change. For this reasonαd andαnr were chosen to be 1.2 and 130.6 respectively.In this case the first and the second terms in Eq. (5) are equal to 60% and 40% of totalα = 2whenρss = 0.6. The following values for other parameters were used in thecalculations:γn =γd = 10−3 (ps)−1, γs = 0.3 (ps)−1, vg = 167·106 m/s andC = 1ps. Values ofg0 = 90(cm)−1,σ = 7.5·10−7(µm)2 are implied forρth = 0.6; g0 = 36(cm)−1, σ = 3·10−7(µm)2 are impliedfor ρth = 0.75 and correspondinglyg0 = 22.5(cm)−1, σ = 1.88· 10−7(µm)2 are implied forρth = 0.9.

This technique can indeed give a lowα-factor if the device operates with a low carrier den-sity. However, if the carrier density is increased and the dots are almost fully occupied then theα-factor increases [12].



0 10 20 30 40 50 60 70 80Modulation frequency (GHz)

0

5

10

15

20

25

FM

/AM

res

pons

e F(ω

)

0 0.2 0.4 0.6 0.8 1Laser intensity (a.u.)

0

0.2

0.4

0.6

0.8

1

Lase

r lin

ewid

th (

a.u.

)

(a) (b)

Fig. 2. (a) FM/AM response under small signal current modulation as defined by Eq. (6).The minimum of this function corresponds toαFM/AM as defined in Eq. (7). Dotted line isthe value ofα calculated just below threshold by the ASE method. Solid line is the FM/AMresponse just above threshold. Dashed line is the FM/AM response at twice threshold.Hereρth = 0.6. (b) Linewidth due toαlw. Solid line is for non-zero values ofαd andαnr.Dashed line is forαnr = 0. Here the current ranges from threshold to 8 times threshold andρth = 0.75.

4. Small signal response

The second technique [7], commonly used to measure theα-factor is based on the FM/AMresponse of a laser biased above threshold to small modulation of the injection currentJ =Jdc + δJ cos(ωt). In this case, the electric field can be written asE =

√Seiϕ with S = Sdc +

δScos(ωt +φ1) andϕ = ϕdc + δϕ cos(ωt +φ2). The values ofδS andδϕ can be calculatedby solving the small signal response. The FM/AM response is defined by

F (ω) = 2SdcδϕδS

. (6)

A typical curve of the FM/AM response calculated for Eqs. (2), (3), (4) is depicted in Fig. 2 (a).This function first decreases, reaches a minimum atωmin =

√

2Sdcγsvgσ and increases linearlyat high frequency. This situation is very different from conventional rate equations where theFM/AM response converges toward theα-factor at high frequencies. In this technique,α isdefined asF(ωmin)

αFM/AM ≡ F(ωmin) = αd +αnrγd

C(1−ρth)2

ρth −1+ρthJdcJth

2ρth −1(7)

It increases linearly with the injection currentJdc and, at threshold currentJth, is equal to theα-factor as measured from the ASE for the same injection current. Notice that the slope ofF (ω) at high frequency is given bydF/dω ≃ αnrτcap and thus is directly proportional to theplasma contribution to the linewidth enhancement factor and to the effective dot carrier capturetime τcap = 1/(C(1− ρth)). When this technique is applied to QD devices to extract anα-factor, extremely large values for theα above the lasing threshold can occur, as experimentallyobserved in reference [15].

5. Linewidth measurements

Since theα-factor was originally introduced to explain the broad linewidth of semiconductorlasers, it is worthwhile to calculate the value of this parameter based on a measurement of the



laser linewidth. This can be performed by adding a white noise term to Eq. (4) and calculatingthe root mean square of the phase fluctuations [10]. Such a calculation gives

αlw = αd +αnr2CNss

γn +2C(1−ρss). (8)

This yields a different value forα than those calculated using any of the previous two tech-niques as shown in Fig. 1. This means that a prediction of the laser linewidth using theα-factormeasured by either the ASE or FM/AM technique is not straightforward. However,αlw in-creases linearly with injection current asNss ∝ Jdc, as in the case forαFM/AM. As a result, thelinewidth, which varies as∆ν ∝ (1+ α2

lw)/Sdc, increases linearly at high injection currents asshown in Fig. 2 (b). This is confirmed experimentally in [23].

6. Conclusion

In conclusion, we have shown that various experimental techniques commonly used to measuretheα-factor of semiconductor lasers lead to different values when applied to QD semiconductorlasers, depending on device design. For devices operating at low carrier densities or ifαnr = 0,the various measurement techniques will give similar values for theα-factor. However, forsaturated dots and ifαnr 6= 0, the various measurements may give great discrepancies since theplasma effect plays an important role in this regime. Indeed, the ASE measurements, carried outat lower injection current, will give small values for theα-factor, but the other two techniques,carried out well above threshold, will give extremely largevalues forα. For example, Fig. 1shows that analysis of the ASE can giveα = 1.5 at J = 0.4Jth, while the FM/AM techniquecarried out on the same device could giveα = 28 atJ = 2Jth. These results are consistent withexperimental results for the ASE method [12] and the FM/AM technique [15].

This analysis also suggests that one should deal carefully with the α-factor of QD materi-als and devices. For some applications, such as direct modulation, it is clear that the FM/AMtechnique may give the best indication about device performance but for others, such as fila-mentation in broad area devices or feedback sensitivity, the ASE technique may also be a goodindicator of the device behaviour. Since such behaviour is directly linked to the finite capturetime and the plasma effect, which are characteristic of QD materials, it might be useful tocompute these parameters from experimental measurements.

Acknowledgments

This work was supported by Science Foundation Ireland undercontract numberSFI/01/F.1/CO13, the EU IST projejct ”DOTCOM”, and the Irish HEA under the PRTLI pro-gram.



The linewidth enhancement factor α of quantum dot semiconductor lasers

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