THREE-dimensional electronic confinement in semiconductor

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1 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 43, NO. 3, MARCH Differential Gain and Gain Compression in Quantum-Dot Lasers Andrea Fiore and Alexander Markus Abstract The dynamics of optical gain in semiconductor quantum dots (QDs) is investigated. Simple analytical expressions are derived, which directly connect the laser dynamical response to capture and intradot relaxation rates. The effect of hole spreading in the valence band and spectral hole burning in the QD ensemble is also quantitatively assessed. The analysis shows that intradot relaxation constitutes the main limitation in the dynamics and points to possible routes towards the improvement of QD lasers. Index Terms Differential gain, gain compression, quantum dots (QDs), semiconductor lasers. I. INTRODUCTION THREE-dimensional electronic confinement in semiconductor nanostructures [quantum dots (QDs)] gives rise to a completely discrete energy spectrum, allowing the investigation of effects typical of atomic physics in a solid-state environment. Many efforts have been dedicated to this system, both for its fundamental interest and for possible applications in optical sources. For example, QD lasers operating in the telecommunication bands are presently being explored as an alternative to quantum wells (QWs). The performance of QD lasers has largely surpassed that of QWs, at least in the most optimized material system InAs GaAs, for several device parameters: threshold current density, linewidth-enhancement factor, and temperature- and feedback- insensitivity. However, QD lasers have not fulfilled the initial expectation of improved modulation dynamics. The maximum modulation bandwidth has remained limited to below GHz for lasers operating in the nm bands [1] [6], much below the best values reported for QW lasers only lasers operating at 1100 nm and exploiting tunnel injection have been modulated at frequencies 20 GHz [7]. Strong damping of the modulation response is also commonly observed (large K-factor), pointing to the role of gain compression. Several physical mechanisms have been proposed to be at the origin of this bandwidth limitations: carrier capture from the two-dimensional wetting layer (WL) into the QDs [8], [9], state filling and consequent slow relaxation between the quantized energy states in the QD [10], limited differential gain due to hole spreading in the valence band [11], Manuscript received July 19, 2006; revised November 8, This research was supported by the EU integrated project ZODIAC, in part by the SER- COST project n , and in part by the Swiss National Science Foundation. The authors are with Institute of Quantum Electronics and Photonics, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland ( andrea.fiore@epfl.ch). Color versions of one or more figures in this paper are available online at Digital Object Identifier /JQE electron transport in the WL [12]. A quantitative assessment of the impact of these different mechanisms on key laser parameters, such as the differential gain and the gain compression factor, is essential to a thorough understanding and proper design of QD lasers. The key questions are, for example, whether the bottleneck in carrier transport can be overcome by the use of tunnel-injection (into the QD ground- or excited-states, depending on whether capture or relaxation is the limiting factor) or by the engineering of QD electronic spectrum, and what improvement may be expected from the reduction of the inhomogeneous broadening. Explicit forms for the differential gain and gain compression factors have been derived in [8], [9], and [12]; however, with the approximations of neglecting the excited states and thus the intradot relaxation process in the QDs [8], [9], neglecting the inhomogeneous distribution [9] or treating it in a simplified way [8], [12], neglecting the nonequilibrium dynamics within the QD [12], and neglecting the separate dynamics of electrons and holes [8], [9]. On the other hand, a microscopic theory [13] accounting for most of the effects (except the existence of excited states) has recently provided the time evolution and modulation response of laser output, but the complexity of the approach does not provide a direct relation between material characteristics and device parameters. The goal of this paper is to derive consistent values of the differential gain and gain compression factor from a simple rate-equation model but still including all the relevant physical mechanisms, such as capture, relaxation between QD electronic states, separate electron hole (e h) dynamics and inhomogeneous and homogeneous broadening. For clarity, we will start with the most simplified model of a single QD with excitonic energy states, for which we can derive analytical solutions that provide useful physical insight, and then add the more complex features of separate e h dynamics and inhomogeneous broadening. Our main conclusion is that, for experimentally reported values of physical parameters (capture times, etc.), the most relevant process is the intradot relaxation, which determines both the differential gain and the gain compression factor at high injection, and thus sets an ultimate limit to the maximum modulation bandwidth. Carrier hopping processes in the QD ensemble further increase the gain compression factor, but are significant only for small homogeneous broadening and at low injection. The derived frequency characteristics are in excellent agreement with a range of recent experiments using a limited set of parameters, most of which are extracted from the dc laser characteristics. II. MODEL DESCRIPTION Our model for QD energy spectrum is sketched in the inset of Fig. 1(a). Two bound states [an s-like ground state (GS) and /$ IEEE

2 288 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 43, NO. 3, MARCH 2007 described in terms of e h pairs, assuming the dynamics of the two types of carriers mirror each other this assumption will be lifted later on. The occupation probabilities for the GS and ES are defined as and, where is the number of e h pairs in the GS (ES), and is the total number of QDs, while the WL population and the photon number are described by the normalized terms:,. The temporal evolution of the occupation probabilities is written, in a rate-equation model [10], as (1a) (1b) (1c) (1d) Fig. 1. (a) Light current characteristics of the laser described in [2], using a single-mode excitonic model (SM exc) (continuous lines), a single-mode e h model (SM eh) (dashed lines), and a multimode excitonic model (dotted line) Inset: Schematics of the conduction band of a QD and corresponding electronic processes. (b) Small-signal modulation response (10 log jh (f )j ) calculated for the same laser, by using the quasi-static approximation (continuous lines) and the exact numerical solution in the Fourier domain (dashed lines), for two different injection currents. a p-like excited state (ES)] are assumed for both electrons and holes in the QD, with degeneracies of 2 and 4, respectively, and a continuum level associated to the wetting layer (WL), with a density of states supposed much larger than the QD areal density (i.e., state-filling in the WL is neglected). The higher ES degeneracy reflects the existence of two closely spaced p-levels [14] and is clearly observed in the QD emission at high excitation [15]. The consideration of two bound states is sufficient to describe most of the effects at typical laser population levels, however the neglect of higher energy bound states has some consequences, which will be commented later. The energy levels are derived from the experimental emission spectra of InAs QD lasers operating around 1300 nm at room temperature [16]. We note that the WL in practice presents compositional inhomogeneities which limit carrier diffusion our assumption of a QW-like WL may lead to an overestimation of carrier hopping processes. As it will be shown below (Section V), such processes do not have a large impact on the laser dynamics, so that the main conclusions of the following analysis are not affected by the specific model used for the WL. In the simplest approach ( single-mode excitonic model ), we consider an ensemble of identical, noninteracting QDs, whose populations are where is the injection rate per QD per unit time, and are the capture and intradot relaxation times (see Fig. 1), and are the escape times from the GS to the ES and from the ES to the WL, and are the total and radiative carrier lifetimes, and is related to the absorption/stimulated emission cross section. Capture and relaxation occur through phonon or carrier-carrier scattering, and have been theoretically [17] and experimentally [18] [24] investigated, with typical values in the range ps. In particular, high concentration of carriers in the QDs and neighboring WL allows large Auger scattering rates, as demonstrated experimentally [18], circumventing the so-called phonon bottleneck issue [25]. Note that our model assumes that WL carriers are first captured into the ES, then relax from ES to GS this cascaded-relaxation model has been experimentally confirmed in [26], and is essential to explain the phenomenology of two-state lasing [16]. In the following, is set to 1 ns (the typical low-temperature recombination time for 1300-nm QDs), is fixed to give the maximum modal gain reported in the experiments (10 30 cm, as discussed later), and are calculated from and by imposing a thermal distribution at equilibrium, and the spontaneous emission coupling factor is set to 10. In this paper we will neglect the ES photon number, and thus the possibility of ES lasing, since we assume that the laser is operated far enough from GS saturation, or with energy-selective cavity loss [distributed-feedback (DFB) or distributed Bragg reflector (DBR) laser]. While the numerical solution of (1) directly provides the laser temporal evolution [10], we aim at obtaining deeper physical insight by extracting the key parameters for highfrequency operation the differential gain and gain compression factor which will also allow a direct comparison with QW

3 FIORE AND MARKUS: DIFFERENTIAL GAIN AND GAIN COMPRESSION IN QD LASERS 289 lasers. To this aim, we need to recast the QD rate-equations in the standard form for semiconductor lasers, which is taken as [27] (2a) (2b) where and are carrier and photon volume densities, is the injection efficiency, is the current, is the electron charge, and are the active and the cavity volumes, and arethe radiative and nonradiative recombination rates, is the group velocity, is the material gain, and is the confinement factor (needed since the volume densities are used). Note that is the total carrier density (in QDs: ), which includes both carriers contributing to gain and lasing (i.e., GS population) and higher energy, spectator carriers (ES and WL population, and also GS population of nonlasing dots in a QD ensemble). For this reason, the gain becomes a function of both and :. The variation of g with the photon density, usually defined gain compression, physically comes from the redistribution of carriers among lasing and spectator states, following a change in the photon number. The small-signal modulation characteristics, and particularly the relaxation oscillation frequency and the damping rate can be derived [27] from (2) once the gain derivatives with respect to the carrier and photon populations are defined: ( differential gain ),. Instead of, the gain compression factor is often used to describe the gain dependence on the photon density, by assuming the functional dependence:. is thus related to by. We note that the modal gain, the differential gain, and the gain compression factor are well-defined physical quantities, in that their values can be measured directly or extracted from device characteristics. In contrast, the material gain and the gain derivative versus photon number are not uniquely defined in QDs since their values have to be calculated from measured quantities assuming a given active volume. The latter is not well defined in quantum-confined active materials, as pointed out by Blood [28] rather than material gain, a transition cross section per QD should be defined, in analogy to gas lasers. We note that no result in the following calculations depends on the active volume. Our approach is to cast (1) into the form (2) and derive the parameters and once this is done, the particular electronic spectrum of QDs can be neglected and standard results from semiconductor laser theory can be applied. To this aim, we simply define as, and sum up (1a) (1c) to obtain an equation of the form (2a). In this case the modal gain is expressed as. The gain derivatives and are then easily calculated around any bias point by imposing a small-signal variation (at constant )or (at constant N) and calculating (analytically or numerically) the corresponding from the steady-state solution of (1a) to (1c), thus freezing the interband dynamics. This is of course equivalent to a quasi-static approximation where intraband and interband dynamics are decoupled (the intraband being supposed much faster) in particular, we are neglecting additional poles which may be introduced in the response by e.g., the capture terms. The relaxation oscillation frequency is then calculated [27] as, the K-factor as, and the damping rate as. III. RESULTS: EXCITONIC ENERGY STATES We analyze the room-temperature static and dynamic characteristics of the laser described in [2] (700 m-long cavity with high-reflectivity coating on one facet, internal loss cm, 10 QD layers with a maximum modal gain of 30 cm ). For this calculation we assume ns (to reproduce the experimental threshold current), ps (to give the experimental K-factor, as discussed below), ps (as derived by us from the static laser characteristics [29]), and a gain cross section corresponding to the measured maximum modal gain. Fig. 1(a) reports the calculated light current characteristics (continuous line) in the single-mode excitonic (SM exc) model described above. The square modulus of the small-signal modulation response is plotted in log scale (continuous lines) in Fig. 1(b) for two bias points, using and values calculated from the gain derivatives and. For comparison, the direct small-signal solution of the rate (1) in the Fourier domain is also plotted (dashed lines). The close correspondence of the two sets of curves proves that the quasi-static approximation used in the derivation of a and is adequate for these parameters. At high bias currents, the quasi-static approximation slightly overestimates the damping rate, since the decoupling of intraband and interband dynamics is less accurate at high interband recombination rates. We verified that the quasi-static model can be applied for all experimentally relevant values of, in the range ps. The calculated K-factor matches the experimental value ns, with a corresponding maximum 3 db modulation bandwidth GHz. We then set to investigate the physical origin of the bandwidth limitation. Fig. 2(a) reports the calculated differential gain and gain compression factor as a function of current for the same laser. The differential gain at low injection 10 cm corresponds well to the modulation efficiencies 1 GHz ma measured in this and similar lasers [2], [30], and is of the same order as in QWs. In fact, differential gain is the incremental gain per e h pair injected in the active region, and is thus independent of the nature of the active region, as long as all injected carriers relax to the lasing state. However, in QDs the differential gain strongly decreases with the injection current above threshold, which is a consequence of gain compression related to the intraband dynamics. As the photon density and thus the stimulated emission rate increase, the relaxation rate from WL to the GS must also increase proportionately, which requires a redistribution of the ES and WL populations. This in turn reduces the variation of GS population with total carrier population, hence the differential gain. This decrease in the differential gain results in a saturation of the vs output power curve, as experimentally observed [1], [3]. In

4 290 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 43, NO. 3, MARCH 2007 Fig. 2. (a) Differential gain a (continuous lines) and gain compression factor " (dashed lines) as a function of current for the same laser as in Fig. 1. (b) Relaxationand capture-limited contributions to differential gain, a, a, as a function of current. (c) a as a function of the intradot relaxation time (for capture time =1ps) and a as a function of (for =1ps). (d) K-factor (K ) as a function of current, and its contributions from relaxation (K ) and capture (K ). order to separate the effects of finite capture and intradot relaxation times, we write the differential gain as where depends only on the transition cross section, is the relaxation-limited differential gain (relative change in GS population with respect to total QD population), and is the capture-limited differential gain (relative change in QD population with respect to total population). These definitions are general and apply to any quantum-confined active material, however they are particularly relevant for QDs, as their discrete energy structure reduces the capture and relaxation rates. Fig. 2(b) shows the calculated current dependence of, for the same laser. For all bias points,, showing that intradot relaxation is the bottleneck in carrier supply to the GS, thus limiting the differential gain. While the, values depend on and, holds true for any in the ps range, as shown in Fig. 2(c). In this figure, is calculated (at ma) for different fixing ps, and is calculated for different fixing ps. Even when intradot relaxation is faster than capture, the differential gain is limited by the intradot dynamics. This counterintuitive result derives from the different ES and WL steady-state populations. In fact, neglecting interband recombination terms at low injection levels in (1c), can be approximated by, where, are the steady-state occupations of GS and ES (a similar expression holds for, when the different degeneracies are taken into account). is fixed by the threshold condition, while has a minimum value determined by quasi-thermal equilibrium at low injection levels, resulting in even at low bias. (3) In contrast, the steady-state WL population is very low due to the large energy separation from the GS, so that at low injection for any value of. Note that quasi-thermal carrier distribution affects the laser dynamics only through the steady-state values,,, while the populations under modulation are not in quasi-thermal equilibrium, due to the relatively slow relaxation rate. We also underline that the laser examined [2] was operated far from gain saturation, due to a relatively large number of dot layers. Operating closer to gain saturation will result in an increased, and thus in a much smaller differential gain. The identification of intradot relaxation as the bottleneck to fast laser modulation is important as it points at the possible routes for improvement, such as tunnel-injection from a QW into the QD GS, as demonstrated experimentally in QD lasers emitting around 1.1 m [7] (while tunnel-injection into the ES leads to marginal improvements [4]). Alternatively, the intradot relaxation rate may be increased, e.g., through engineering of the QD electronic states (to match phonon energies) or increasing the Auger scattering rate. A similar analysis can be applied to the gain compression. In fact, the -factor can be written as where we have used the implicit function theorem to write and defined two effective times for relaxation and capture,, (a similar approach was taken in [8], [9], but there the distinction between capture and relaxation was not made). Neglecting thermal escape and spontaneous emission rates, approximate expressions for and can be derived from (1b) and (1c) and (4)

5 FIORE AND MARKUS: DIFFERENTIAL GAIN AND GAIN COMPRESSION IN QD LASERS 291. The physical meaning of (4) is clear: The damping is determined by the combination of the photon lifetime and the time needed for re-establishment of intraband steady-state populations, limited by capture and intradot relaxation rates, taking into account the Pauli-blocking effect. The larger GS population makes intradot relaxation the limiting process, for comparable and. Fig. 2(d) shows the relaxation and capture contributions to the -factor, and, calculated from the and values derived from the steady-state solution of (1b) and (1c), together with the -factor calculated directly from the and values. The laser maximum modulation frequency can be directly related to the effective relaxation times. This simple analytical form provides a direct physical insight on the bandwidth-limiting processes in a QD laser and allows an estimation of the bandwidth from the steady-state conditions. Optimizing the bandwidth indeed involves a compromise between the photon lifetime (reduced by increasing mirror loss) and the effective relaxation and capture times (reduced by reducing mirror loss), as pointed out in [31]. IV. SEPARATE ELECTRON HOLE DYNAMICS The calculations above neglect the separate dynamics of electrons and holes, which was predicted to play an important role in the modulation bandwidth [11]. In fact, the hole thermal distribution over closely spaced valence band states is expected to reduce both the modal and the differential gain. In order to check whether this effect constitutes an important limitation to the bandwidth, we lift the assumption of excitonic energy states by writing rate equations for both electrons and holes, ( electron-hole model ) assuming a 70% 30% splitting of the transition energies in the conduction and valence band, respectively. The e h rate equations are (5a) (5b) (5c) (5d) where, represent electron and hole populations, respectively,, the electron relaxation and capture times, and, the corresponding hole times, and is an effective WL lifetime taking into account WL degeneracy (here is set equal to ). The intraband processes in (5) have exactly the same form as in the exciton model of (1), while interband processes now explicitly involve both electrons and holes. Bimolecular recombination is assumed for all interband recombination processes (including the nonradiative ones), in order to maintain the total charge in the system. We first assume hole capture and relaxation times (, ps) much shorter than the corresponding conduction-band values ( ps, ps), consistently with the closer energy spacing of hole states, which allows efficient relaxation by acoustic phonon scattering, as shown experimentally [19]. The light current curves calculated with this model, shown in Fig. 1(a) (dashed line, SM eh ), are almost identical to those calculated with the excitonic model of (1), indicating a limited impact of hole spreading on the static characteristics, for this laser operated far from gain saturation. The differential gain and gain compression factor are calculated numerically by solving the intraband dynamics after a small variation of the injection or photon number, and the corresponding modulation curves are checked by a direct solution of the full rate equations. Fig. 3(a) shows the differential gain (continuous lines, left axis) and the -factor (dashed lines, right axis) calculated in e h model (red dots). The corresponding data (blue squares) calculated in the exciton model as described above is shown for comparison. The differential gain is strongly reduced at low bias, due to hole spreading among the closely spaced valence band states [11]. The consideration of higher energy levels in the valence band would further lower the differential gain value. We note that the effect of hole spreading strongly depends on the assumed band offsets in the conduction and valence bands, and on the valence band s WL density of states, which is difficult to estimate (here we assume parabolic bands with a hole effective mass, where is the electron mass). However, the -factor and thus the maximum modulation bandwidth are not strongly affected by the hole spreading. We investigated the role of the electron and hole relaxation and capture times. As in the excitonic model, both electron and hole capture times do not have a significant impact on the dynamic properties, except for a small contribution to the -factor, as long as they are smaller or comparable to relaxation times. Moreover, interchanging the electron and hole capture time has no effect. We therefore focus on the impact of the relative values of the electron and hole relaxation times. Fig. 3(b) shows the differential gain (continuous lines, left axis) and -factor (dashed lines, right axis) calculated as a function of the hole relaxation time, fixing the electron relaxation time to ps (squares), and as a function of the electron relaxation time, fixing the hole relaxation time to ps (circles). All values have been calculated at a fixed current of 94 ma, assuming ps and ps. A very similar dependence is observed for the two types of carriers, despite the different energy spacing between states in the two bands. In particular, the contribution of each carrier type to the -factor is directly related to its relaxation time. When the two relaxation times are very different, the dynamics of the slowest carrier limits the laser modulation dynamics. In the InAs GaAs system, the dynamics is likely limited by electron relaxation.

6 292 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 43, NO. 3, MARCH 2007 Fig. 3. (a) Differential gain (continuous lines, left axis) and K-factor (dashed lines, right axis) as a function of current for the same laser as in Fig. 1, for excitonic energy levels (squares), independent electrons-holes without doping (dots), and independent electrons-holes with p-doping (triangles). (b) Differential gain (continuous lines, left axis) and K-factor (dashed lines, right axis) calculated at fixed I =94mA as a function of the hole relaxation time, fixing the electron relaxation time to = 3:3 ps (squares), and as a function of the electron relaxation time, fixing the hole relaxation time to =3:3ps (circles). The effect of hole spreading can be counteracted by p-doping the QDs, as proposed in [11]. To simulate this situation, we assume an initial hole population in the QDs in the rate equations. The calculated differential gain and -factor are shown in Fig. 3(a) (triangles), for a doping level corresponding to 10 holes/qd. The differential gain increases back towards the value predicted by the exciton model, as expected since in a QD filled with holes the effect of hole spreading is less significant. The -factor is also reduced by the p-doping, which is consistent with the 30% increase in bandwidth observed experimentally [32]. However it is clear that the main bandwith limitation will not be lifted by p-doping, as it had been initially hoped, since it is in fact related to intraband relaxation in the conduction band. We nevertheless note that the increased hole population in p-doped QDs may further improve the response through faster capture and relaxation times due to increased Auger scattering rates [22] [24] an effect not taken into account in these calculations. V. INHOMOGENEOUS BROADENING We finally analyze the role of inhomogeneous broadening and carrier hopping in the QD ensemble. Inhomogeneous broadening obviously decreases the peak modal gain, however this is already taken into account in the single-mode model above by fitting the gain cross section per dot to the measured modal gain. We wish to consider the additional gain compression, which may be produced by spectral hole burning in the QD ensemble [8], [12]. We calculate the modulation response of the QD ensemble by discretizing the QD energy spectrum in 1 mev intervals and writing multimode rate equations for each QD family (using excitonic energy states), as described in [29]. As in the single-mode model, we set the gain cross section so that the saturated peak gain equals the experimental value of 30 cm. In our previous work [29], the role of carrier hopping and homogeneous broadening on the static characteristics of a Fabry Pérot laser has been studied in particular, the homogeneous linewidth and the capture time were shown to determine the lasing linewidth, the position of the lasing peak with respect to the spontaneous emission peak, and the population clamping at threshold. This allowed us to estimate a homogeneous linewidth mev (which also corresponds to the measured room-temperature dephasing time [21]). Here, lasing on a single photon mode is imposed by a frequency-dependent cavity loss, and the interaction of each QD with the optical field is weighted by a Lorentzian homogeneous broadening lineshape, with a full-width half-maximum mev The light current curves calculated with this model are shown in Fig. 1(a) (dotted line, multimode ) and are again very similar to those calculated with single-mode models, indicating the consistency of the two models. While it is possible to derive the differential gain and gain compression factor by solving the intraband dynamics similarly to the single-mode model above, we found that the resulting values are not consistent with the direct time-domain calculation of the laser response. Indeed, for the carrier transport mechanism considered here (thermal hopping through the WL), the quasi-static approximation is not applicable, since thermal hopping processes are much slower than typical interband recombination times. As a consequence, differential gain and gain compression are not well defined and their use leads to an overestimation of the laser damping. Instead, we numerically calculate the temporal response to a small current step by solving the full multimode rate equations and then fit the result with the theoretical step response of a laser [27] to derive the relaxation oscillation frequency and damping rate. Fig. 4 shows the and calculated for the same laser considered above, in the multimode (10 mev) model (red circles), and, for comparison, in the single-mode (SM) model of (1) (blue squares). The gain cross section is chosen so that the saturated GS gain and the light current curves are identical in the two simulations. Both the differential gain and the gain compression factor are slightly reduced by the same factor in the multimode model, resulting in a 10% reduction in the relaxation oscillation frequency and damping rate while the -factor and the modulation bandwidth are identical to the SM model. Indeed, although carrier hopping times are slow for these strongly confined QDs, the different QD families are effectively connected through the photon reservoir, as the homogeneous linewidth (10 mev) is comparable to the inhomogeneous broadening (35 mev). In contrast, when the homogeneous linewidth is reduced to 10 mev, spectral

7 FIORE AND MARKUS: DIFFERENTIAL GAIN AND GAIN COMPRESSION IN QD LASERS 293 REFERENCES Fig. 4. Relaxation oscillation frequency (continuous lines, left axis) and damping rate (dashed lines, right axis), as a function of current, for the same laser as in Fig. 1, in three different models: single-mode (squares), multimode with 0 =10meV (dots), and multimode with 0 = 1meV (triangles). The temporal dynamics in the 0 = 1 mev case cannot be fitted with a two-pole response for I < 45 ma. hole burning occurs, producing gain compression and a strong increase of the -factor over the SM value. As an example, Fig. 4 shows (triangles) the and values for a homogeneous linewidth of 1 mev (again keeping the saturated gain constant): The relaxation oscillation frequency is reduced to 6 GHz and the -factor increased to 2 ns. This conclusion is similar to the one of [12], where lasing and nonlasing QDs were treated as noninteracting. However, this additional gain compression will only be relevant at low temperatures ( 150 K), where the contribution of phonon scattering to homogeneous broadening is much smaller. In summary, homogeneous broadening contributes to establishing a single population in the QD ensemble at room temperature and thus plays an important role in the dynamical properties, as pointed out earlier [33]. VI. CONCLUSION In summary, we have derived analytical expressions for the differential gain and -factor of QD lasers, which relate the bandwidth limitations to the underlying carrier dynamics, and investigated the role of inhomogeneous broadening and independent e h dynamics. Intradot relaxation is shown to determine both the differential gain and the gain compression factor, and thus sets an ultimate limit to the maximum modulation bandwidth. Hole spreading due to closely spaced valence band states reduces the differential gain (which can be counteracted by p-doping), but does not significantly affect the maximum bandwidth. Carrier hopping processes in the QD ensemble further increase the gain compression factor, but are significant only for small homogeneous broadening and at low injection. The resulting picture of QD laser dynamics qualitatively and quantitatively explains a considerable body of recent experimental results and points to possible routes for improvement of laser characteristics. Specifically, increasing the intradot relaxation rate or injecting resonantly into the QD GS will be needed to increase the laser bandwidth well beyond 10 GHz. ACKNOWLEDGMENT The authors would like to acknowledge useful discussion with Dr. Marco Rossetti, EPFL, Lausanne, Switzerland. [1] R. Krebs et al., High frequency characteristics of InAs /GaInAs quantum dot distributed feedback lasers operating at 1.3 m, Electron. 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8 294 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 43, NO. 3, MARCH 2007 [25] H. Benisty, C. M. Sotomayortorres, and C. Weisbuch, Intrinsic mechanism for the poor luminescence properties of quantum-box systems, Phys. Rev. B, vol. 44, no. 19, pp , [26] T. Muller et al., Ultrafast intraband spectroscopy of electron capture and relaxation in InAs/GaAs quantum dots, Appl. Phys. Lett., vol. 83, no. 17, pp , [27] L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, K. Chang, Ed. New York: Wiley, [28] P. Blood, On the dimensionality of optical absorption, gain, and recombination in quantum-confined structures, IEEE J. Quantum Electron., vol. 36, no. 3, pp , Mar [29] A. Markus et al., Role of thermal hopping and homogeneous broadening on the spectral characteristics of quantum dot lasers, J. Appl. Phys., vol. 98, no. 10, p , [30] M. Kuntz et al., Spectrotemporal response of 1.3 mu m quantum-dot lasers, Appl. Phys. Lett., vol. 81, no. 20, pp , [31] M. Sugawara et al., Recent progress in self-assembled quantum-dot optical devices for optical telecommunication: temperature-insensitive 10 Gbs( 01) directly modulated lasers and 40 Gbs( 01) signal-regenerative amplifiers, J. Phys. D-Appl. Phys., vol. 38, no. 13, pp , [32] S. Fathpour, Z. Mi, and P. Bhattacharya, Small-signal modulation characteristics of p-doped 1.1 and 1.3 mm quantum dot lasers, IEEE Photon. Technol. Lett., vol. 17, no. 11, pp , Nov [33] M. Grundmann, How a quantum-dot laser turns on, Appl. Phys. Lett., vol. 77, no. 10, pp , Andrea Fiore received the M.Sc degrees in electronic engineering and in physics from the University of Rome La Sapienza, Italy, in 1994 and 1996, respectively, and defended the Ph.D. dissertation on nonlinear frequency conversion in semiconductor waveguides at Thomson CSF Central Research Laboratory, Orsay, France, in He worked at University of California at Santa Barbara ( ) on multiple-wavelength arrays of vertical-cavity surface-emitting lasers, at the Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switerland ( ) on quantum-dot lasers, and at the Institute of Photonics and Nanotechnology of the Italian CNR ( ) on nanostructured photonic devices. Since september 2002, he has been leading the activity on quantum devices at EPFL as Assistant Professor of the Swiss National Science Foundation. He has co-authored over 65 journal publications, 35 contributions to conferences, and given 18 invited talks. Alexander Markus was born in Feuchtwangen, Germany, in He received the M.S. degree in physics from the University of Erlangen-Nürnberg, Germany, in 2000, and the Ph.D. degree from the Institute of Quantum Electronics and Photonics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, in At present, he is with Lichti+Partners GbR, Karlsruhe, Germany, where his activities are focused on intellectual property rights. His current interests include optoelectronic device applications.