THE differential gain of a semiconductor laser relates the

Transcription

1 26 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 1, JANUARY 2005 The Impact of Energy Band Diagram and Inhomogeneous Broadening on the Optical Differential Gain in Nanostructure Lasers Hanan Dery and Gadi Eisenstein, Fellow, IEEE Abstract We present a general theoretical model for the optical differential gain in semiconductor lasers. The model describes self assembly quantum dots (QDs), self assembly quantum wires (QWRs) and single quantum-well lasers. We have introduced the inhomogeneous broadening due to size fluctuations in the assembly cases. At each dimensionality, we have considered the carrier populations in the excited states and in the reservoirs, where conduction and valence bands are treated separately. We show that for room temperature operation the differential gain reduction due to increased size inhomogeneity is more pronounced in QDs than in QWRs. We show this reduction to be smaller than the one-order reduction attributed to state filling in conventional dot and wire assemblies operating at room temperature. The integration prefactor coefficient of the differential gain in zero-dimensional cases exceed one- and two-dimensional coefficients only for low temperatures where the homogenous broadening is considerably smaller than the thermal energy. The differential gain of QDs, QWRs, and compressively strained single quantum-well lasers operating at room temperature and close to equilibrium is nearly the same. Index Terms Modulation, nonhomogeneous media, quantum dots (QDs), quantum wires (QWRs), semiconductor lasers. I. INTRODUCTION THE differential gain of a semiconductor laser relates the gain to the carrier density by the simple phenomenological equation [1] where is the transparency carrier density. In lasers based on simple bulk gain media, fast intraband transitions ensure that any perturbation is translated to a gain change at the laser transition energy. is used then to describe the rate of change in gain at the lasing wavelength to changes in the carrier density occupying states at the corresponding energy,. Naturally, plays a key role in the formulation of diode laser modulation response where the resonance frequency is known [1] to vary as. The situation is more complicated in advanced lasers based on nanostructures, such as multiple or single quantum wells (SQWs) [2], [3], self-assembled quantum dots (SAQDs) [4], [5] Manuscript received March 25, 2004; revised September 1, This work was supported in part by the BigBand project of the EC and by a grant from the Ministry of Science, Israel. The work of H. Dery was supported in part by the Vatat graduate student fellowships. H. Dery was with the Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel. He is now with the Physics Department, University of California at San Diego, La Jolla, CA USA. G. Eisenstein is with the Department of Electrical Engineering, Technion- Israel Institute of Technology, Haifa 32000, Israel ( gad@ee.technion. ac.il). Digital Object Identifier /JQE or quantum dashes [6] [8]. The latter have been shown to behave as a self-assembly quantum wires (SAQWRs) [9]. All such nanostructure gain media comprise a reservoir through which carriers are fed to the gain region via complex carrier relaxation mechanisms [10]. There is a finite steady-state carrier population of reservoir states so that induced carrier perturbations are distributed over a range of energies and contribute only partially to the gain at the lasing energy. This amounts to an effective reduction in the differential gain. In lasers based on SAQD and SAQWR, there are three additional problems. The energy separation between reservoir and lasing state is often large complicating the coupling process between the two regions. Moreover, the spectra of such lasers is inhomogeneously broadened [11] [16] and these structures may also have a measurable carrier population of excited states. Any carrier density perturbation is fed now to several states other than the lasing state (which is most often the ground state) resulting effectively in a reduced differential gain. Documented experimental results suggest that SAQD exhibit rather moderate modulation bandwidths [17] [22]. Examination of the details of these papers highlights a very large range of extracted differential gain values, from 10 cm in [19] to 2 10 cm in [21]. These deviations originate from different evaluation techniques as well as from very large discrepancies in the total confinement factor value (heterostructure confinement factor times the dots filling factors), from in [23] to 3 10 in [24] per dot layer. Standing out among reported experimental results is the work of Bhattacharya et. al. [25] who used a tunnelling injection scheme to demonstrate a bandwidth of 22 GHz. The principle of tunnelling injection was previously used [26] to widen the bandwidth of QW lasers. Similarly, n-type -doped regions enhanced the bandwidth of QW lasers [27]. A recent paper [28] employing a thin InGaP matrix buffer beneath the QDs also achieved a respectable bandwidth of 12 GHz. All those special laser structures circumvent limitation imposed by the differential gain and the nonlinear gain coefficient. In this paper, we present a general theoretical model for the differential gain in lasers based on nanostructure gain media. The formalism has a common framework for the three nanostructures we consider. The SAQD, SAQWR, and SQW gain media differ only in their respective density of states (DOS). In the two-dimensional (2-D) case, we derive the well known analytical expression where we show results for a compressively strained separate confinement QW heterostructure. A model was used to calculate the energy placement of the excited /$ IEEE

2 DERY AND EISENSTEIN: THE IMPACT OF ENERGY BAND DIAGRAM AND INHOMOGENEOUS BROADENING 27 subbands and the reservoirs. The expression we derive for the zero-dimensional (0-D) case is the same as the one derived previously by Shchekin and Deppe [29] except that the confinement factor is formulated differently here so as to enable a clear comparison with the other dimensionalities. Finally, we present here the first calculation for the differential gain of a SAQWR structure. The calculation employs an analytical approximation for the Fermi energy of an ideal one-dimensional (1-D) Fermi gas under the condition of carrier densities which are relevant for lasing conditions. We point out three factors which affect the differential gain, all of which are manifestations of the different DOS functional forms. First, the state filling factor resulting from carrier energy distribution. We address issues related to the reservoir and to excited states. We use realistic energetic structures for the different geometries and highlight the reasons for hindered differential gain in SAQDs. Second, we introduce the inhomogeneously broadened nature of the gain in SAQD and SAQW structures. Finally, we address the effects of the homogeneous broadening, the temperature and the optical dipole matrix element. We consider operation at room temperature and use Fermi Dirac statistics which implies equilibrium conditions. The equilibrium condition is unquestionably valid below and near threshold. Far above threshold, where the largest modulation bandwidths are achieved, there may be a measurable deviation from equilibrium due to intraband saturation processes. The differential gain reduces however in the high power regime. Deviations from equilibrium in SAQDs result in multiline lasing [11] [16] from different dot size populations or by simultaneous lasing from the ground and excited states. Recalling the physical interpretation of the optical differential gain, the perturbation in the reservoir is spread now over different spectral regions which are separated by more than the homogeneous line width hence reducing the differential gain. The differential gain near threshold is, therefore, an upper limit. II. THEORETICAL MODEL The linear optical gain is derived from the linear susceptibility using the density matrix formalism for a two-level system as in [30] where are the velocity of light, the refractive index, and the angular photon frequency, respectively. is the quantum number enumerating states (including spin) where for one and two dimensional structures it corresponds to the wave vector. is an index counting the gain element number (well, wire or dot), with being the th element volume, the optical dipole matrix element and the inverse dephasing time. and are, respectively, electron and hole distribution functions. We aim to formulate a general expression for the ground-state gain which can serve to analyze all nanostructures. We assume parabolic bands and a wavevector independent matrix element [31]. Doubly degenerate spin is included and the distribution functions are assumed to vary slow (1) on the scale of the Lorentzian shaped homogeneously broadened gain function. The summation in (1) is replaced by a continuous integration and becomes dot wire well where denotes the well width and is the th wire crosssection area. and are the th element energy gap and the reduced mass. We have also considered the homogeneous broadening to be independent of the element index (this assumption is not essential and is used for simplicity). We define the differential gain as where is an index counting confined states, and denote, respectively, the th element particle number of subband in the band and the th element volume. and describe the corresponding values in the reservoir. In the following we consider SAQD, SAQWR, and SQW, all operating under equilibrium conditions. The equilibrium assumption is valid below and slightly above threshold where carrier escape [3], [10], [32], [33] is fast compared with the spontaneous and stimulated emission rates. These conditions are easily obtainable in SAQD and SAQWR operating near room temperature [34]. We use the chain rule to rewrite (4) where relates to the chemical potential and will be described later. is the ground-state carrier density in the th element. For the simple case of infinite confinement and parabolic dispersion in the free dimensions, the state energies are proportional to the inverse value of the mass, and the distribution function is where is a reference energy which, for the SQW, denotes the gap energy and, hence, the chemical potential is calculated from the edge energy of each band. For the lower dimensional structures, denotes the (2) (3) (4) (5) (6)

3 28 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 1, JANUARY 2005 average ground-state transition energy of the various elements. Using (6) and (2), the first term of (5) becomes with being the coefficient of the phase filling factor according to (2) where denotes the corresponding dimensionality. The second term in (5) involves the ground-state carrier density which is given by (7) A. Self Assembly Quantum Dots The reservoir in the SAQD structure is assumed to be a two dimensional wetting layer characterized by a single level which is energetically placed above the th dot ground state of band. We consider two distributions for each band with one having an intifisimal increase at the energy of the chemical potential. A simple algebric manipulation leads to dot wire well. The solution is analytical for the 0- and 2-D cases. The quantum-wire case is more complicated. We solve it by modifying the Joyce and Dixon [35] or Aguilera-Navaro et al. [36] procedures. For symmetrical bands, one can make use of the similarity between the wire energy integral and the bulk density integral and use the well known expression [36]. However, in cases where the masses and total densities within each wire are not equal (only the total assembly density is), the entire analysis must be repeated. We define (8) (12) where is the width of the reservoir well, are, respectively, the occupation probabilities of the th state and edge energy of the band reservoir. denotes the degeneracy ratio between the th state and the ground state. Considering the SAQD to have a Gaussian-shaped inhomogeneously broadened function and inserting the dot terms into the differential gain equation leads to (9) Using series inversion techniques [37] leads to (10) where the logarithmic term describes the classical result and where. Retention of terms through yields a good approximation with an accuracy of better than one percent up to. The standard Padé approximation technique for the Fermi Dirac integral function yields poor results for the low order polynomial ratios. The second term of (5) can now be written as dot wire well. (11) The third part of (5) which is referred to as the state filling reduction term is calculated separately for each dimension. (13) The square brackets term in (13) describes the state filling reduction. It is obvious that the differential gain degrades when the ground-state population increases and for small energetic spacing between the various transition. This is simply explained by state filling and off-resonance state filling ( as ). For an ideal assembly with identical dots, the Gaussian function is replaced by a delta function yielding an analytical expression. Similarly, for a narrow homogeneous broadening the Lorentzian function reduces to a delta function. Fig. 1 shows the state filling reduction of the conduction term at room temperatures. The solid lines represent the ground state and two excited states with equal spacing, which describe the case of a parabolic potential. This choice agrees very well with many published photo and electro luminescence measurements of GaAs InAs QDs grown by molecular beam epitaxy [24], [38] [42] where the growth conditions determine the wavelength and the state spacing. The separation between the

4 DERY AND EISENSTEIN: THE IMPACT OF ENERGY BAND DIAGRAM AND INHOMOGENEOUS BROADENING 29 Fig. 1. Differential gain reduction in SAQD due to state filling versus the electronic ground-state occupation probability. Each of the solid lines denote a different energetic spacing between the states characterized in units of room temperature thermal energy. These are considering two excited states and a wetting layer. The dashed line denotes the case of a single excited state and a wetting layer (see text). wetting layer and the second excited state was chosen to be the same which fits reasonably well to luminescence measurements in cases where the wetting layer emission is clearly detected. A6-Å wetting layer and excited-state degeneracy ratios of were chosen according to the parabolic potential approximation where the uniaxial nature is considered due to the known low height to diameter aspect ratio of conventional QDs. The electron effective mass in the reservoir is and the dot volume is cm (which describes a pyramidal dot with 100-Å base length and 60-Å height). Fig. 1 shows five such energy separations. For twice the room temperature thermal energy ( 50 mev) and ground-state occupation of 0.85, the differential gain drops by one order of magnitude. In InGaAs InAs QDs, whose confinement is shallower than in GaAs InAs QDs, photoluminescence experiments [33], [43] [46] reveal emission from a single excited state above which the wetting layer emits. The energy separation between the ground state and wetting layer transitions varies in those structures in the range of mev. Here, we describe the state filling reduction by approximating the two conduction band separations to be 50 mev each. This case, presented by the dashed line in Fig. 1, is nearly identical to the case of two excited states and a wetting layer with twice the room temperature energy spacing. The valance band state spectrum in SAQDs grown by the Stranski Krastanow (SK) technique, is very dense (with separations of the order of 10 mev). The conduction band states, on the other hand, are separated by more than 50 mev. This was demonstrated theoretically for pyramidal dots using an empirical pseudopotential analysis [47] and by an eight-band model [48]. The same conclusions are drawn from simple parabolic confinement potential models that fit photoluminescence experiments [38], [49] and which qualitatively reflect the shallow cross sectional shape of the QDs. The various calculations reveal the dense valence band spectrum with state separations of approximately 10 mev. Considering the first five hole states along with their degeneracies as given by parabolic potential for shallow dots, the state filling reduction for holes is approximately 0.09, 0.06, 0.04 for ground-state populations of 0.25, 0.5, and 0.75, respectively. Considering the degeneracies of the hole states to be s-, p-, d-like, etc. yields basically the same results and trends. The degeneracies of the holes is most likely more complex due to strong interband mixing. However, the large number of discrete states keeps the state filling reduction in the valence band considerably lower than one. Large differential gain in room temperature conventional SAQD lasers may be predicted based on the conduction band filling reduction whose peak values may exceed considerably the valence band peaks as seen in the large spacing cases of Fig. 1. However, the increase in differential gain is hindered by neutrality conditions which are only satisfied for high ground-state occupation of electrons. The electron-state filling reduction term tends to zero as so that in order to maximize the electron-state filling term, the laser should operate close to transparency. This can be achieved by increasing the waveguide confinement factor and/or the QD coverage. Alternatively, reduced inversion levels may be achieved by long cavities but this increases the photon lifetime which is detrimental for fast modulation. Furthermore, high-energy spacings in the conduction band slow the carrier capture [10], [50] [53] also affecting the modulation capabilities. Equation (13) can be modified to partially incorporate excitonic populations in QDs at elevated temperatures. Considering the binding energy of a pyramidal InAs QDs to be of the order of 20 mev [54], room temperature exciton population becomes non negligible. Dikshit and Pikal [55] have found the exciton population to be one fifth of the total population at room temperature by modifying the Saha equation following [56]. For the exciton population, we replace the free carrier phase filling term with the corresponding excitonic term. Considering the higher energetic spacing in the conduction band, the exciton distribution can be approximated by the electrons distribution and hence increasing the state filling reduction term by a factor of two relative to the free carrier case. Two additional modifications are needed. One is factorization of (13) by the fraction of the exciton population within the total population. The second modification is due to the increased dipole matrix element [57], [58] caused by the strong confinement in the excitonic regime. Introducing p-doping modulation in the barriers was also suggested as a means to overcome the imbalance between the valence and conduction bands. Vastly improved static characteristics such as threshold current and characteristic temperature were indeed demonstrated by Shchekin and Deppe [59]. The barriers in these structures are partially ionized and hence the electrons are rejected by Coulomb forces and tend to localize in the QDs. This rejection reduces thermal escape from the QDs and diminishes electron diffusion in the barriers leading to less trapping in nonradiative centers. The predicted enhancement of dynamical properties [60] has, on the other hand, not been proven experimentally. A recent report [61] of such a p-doped laser shows but a mild enhancement

5 30 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 1, JANUARY 2005 compared with regular structures. We postulate that the discrepancy is related to spatial effects which are governed by the neutrality conditions. Coulumb forces between the ionized acceptors and the holes modify the energy band structure and keep the acceptors only partially ionized. QDs are, therefore, populated by a small number of holes. In order to properly model this effect, the Poisson equation has to be solved self consistently with (at least) a simplified 1-D Schrödinger equation. The overall system neutrality can be derived then from the carrier densities by using the probability of the convergent wave function. Such a calculation was reported for elevated injection levels in intrinsic multiple QW structures [3] and doped single QW structures [27]. Finally, even in extreme cases where imposing dozens of holes within each QD is a constraint, the differential gain of SAQDs does not exceeds that of tunneling barrier QW structures as can be viewed from the resonance peak positions of the modulation response ([60] compared with [62]). B. SAQWR In the case of SAQWR structures, we consider once more a 2 D reservoir. Repeating the analysis with the corresponding adjustments to the QWR configuration results in where (14) (15) Gathering all of the wire terms leads to the SAQWR differential gain (16) where and are taken from (3), (9), and (15). We have simulated the state filling reduction, described by the square bracket term in (16), by following the energy diagram of a self assembled InP InAs quantum dash structure as calculated by Miska et al. [8]. These calculations fit well to photoluminescence experiments and present a rich and dense spectrum in both bands with a similar energy spacing. The similarity stems from the flat cross section of quantum dashes having an aspect ratio Fig. 2. Differential gain reduction in SAQWR due to state filling versus the conduction ground subband population at the average transition energy of the various elements. The lower graph refers to InP InAs quantum dashes. We have considered four excited states as in [8]. The upper graph denotes a quantum dash with a wetting layer placed 100 mev above a single subband (see text). of about 0.1. The quantum dash resembles a highly biaxially strained thin QW with a distinct light-hole heavy-hole decoupling where the heavy hole and electron have almost identical masses in the well plane. Due to the similarities of the masses and band diagrams, the hole contribution to the state filling reduction almost equals the electron contribution, where the subband degeneracy is lifted by the highly asymmetrical geometry. The large number of subbands has the same effect as the degeneracy in the SAQD cases. The dense conduction band spectrum is similar in nature to that of V-groove GaAs AlGaAs QWRs [63], [64]. A second class of quantum dashes are the InAlGaAs InAs [6], [7] where the quaternary material is lattice matched to InP. In this category, the dashes have considerably lower barriers ( 1.06 ev compared with 1.35 ev in the InP dashes, where the wire luminescence in both is centered around 0.8 ev). Since no rich luminescence spectra was observed [9], [65], we assume a single subband with a wetting layer for the conduction band. No sound analysis of the energy diagram in these structures was carried out. We, therefore, do not refer to the holes contribution but rather postulate that their contribution is limited compared to the InP quantum dash case. This is supported by some indirect (calculated) evidence for additional hole subbands [9]. As no clear wetting layer emission was detected [9] for a wide range of driving currents, we assume that the energy spacing between the ground state and the reservoir is large. In addition, considering the approximately 60/40 band offset for the conduction valence band in this material compositions [66], and the fact that the total ground state to barrier offset is 250 mev, we use in our calculation a 100-meV spacing between the conduction band ground state and the wetting layer for this wire class. The state filling reduction of the conduction band for both quantum dash families is presented in Fig. 2. The abscissa denotes the conduction ground subband population at the average ground-state transition energy of the various elements. These

6 DERY AND EISENSTEIN: THE IMPACT OF ENERGY BAND DIAGRAM AND INHOMOGENEOUS BROADENING 31 are also the energies for which the differential gain reduction is calculated. The lower graph denotes the InP InAs structure where we have used the parameters calculated in [8]. The electron and reduced masses in the wire are, respectively, and the excited states are 15, 30, 55, and 80 mev above the ground state. Considering more states has but a minor effect on the results and due to the strong confinement, the reservoir contribution to the reduction is also negligible. The upper graph denotes the InAlGaAs InAs structure where we have used a 6-Å wetting layer with an electron mass of, the wire reduced and electron masses are and, respectively. For this wire class, we have calculated the masses by using an eight band model following [67] for a biaxially strained quantum-wire configuration [9]. C. SQWs The reservoir in the SQW case is considered to have a 3-D configuration. Repeating the analysis with some mathematical manipulations yields the differential gain Fig. 3. Differential gain reduction in a compressively strained AlGaAs GaAs InGaAs SQW due to state filling versus the ground subband population at the edge energy. The lower graph refers to holes and the upper to electrons. Two hole subbands, one electron subband and bulk reservoirs are used. (17) with and denoting the distributions of the band edge and reservoir energies (the chemical potential is measured from the ground-state edge). is the well width. We have assumed the reservoir population to be low so we invoke the Maxwell Boltzmann distribution. Using compressive strain QWs increases the differential gain as the ratio between the reduced and heavy hole mass increases. This enhances the weight of hole contribution to the differential gain. We simulate a typical separate confinement heterostructure QW laser with the following parameters. The total width of the heterostructure is 985 Å comprising a single In Ga As 85-Å well with 100 ÅGaAs barriers on each side. The rest of the structure comprises Al Ga As layers on both sides. The temperature was fixed at 300 K and strain effect were included. The diagrams and the confined wave functions were computed following [67]. The particle masses and the energies at the point are ev ev ev ev. The reservoir we consider is the GaAs separate confinement region and the masses are. The filling reduction is presented in Fig. 3. D. Inhomogeneous Broadening The inhomogeneous broadening due to size fluctuations in SAQDs and SAQWRs also introduces a reduction of the optical gain and the differential gain. For SAQDs we make use of the fact that the state filling reduction term, denoted by the square bracket in (13) varies slowly within the Lorentzian-shaped homogeneous broadening function. Moreover, for the ground state central transition frequency, taking the state filling term out of the integral in (13) amounts to a second-order correction with a nearly identical reduction of the optical gain and the differential gain. The differential gain becomes (18) Fig. 4 shows the differential gain as a function of the inhomogeneous broadening in SAQDs. The energy spacing in the conduction band is twice the room temperature thermal energy. Other parameters we use are: the dipole matrix element C M [68], [69]. The central transition energy ev, the refractive index and the dephasing energy mev [70], [71]. The other parameters are the same as in Fig. 1. Considering the state filling reduction as a constant prefactor has a negligible effect on the results. Changing the assembly full width half maximum from 5 to 100 mev leads to a reduction by a factor of four in both the gain and the differential gain. This specific reduction stems from the fact that for a wide inhomogeneous broadening, the homogeneous width can be viewed as a delta function and hence the differential gain scales as. This behavior dominates the

7 32 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 1, JANUARY 2005 Fig. 4. Differential gain versus inhomogeneous broadening in SAQDs with conduction band energy spacing of twice the room temperature thermal energy. Upper, middle, and lower sets denote different populations for the central transition dots. Solid and dashed lines denote, respectively, results with the state filling term kept inside and taken out of the integral. right side of Fig. 4. For the opposite extreme case, where hypothetical identical dots are considered, the gain inhomogeneity is basically a delta function and the differential gain scales as. This behavior dominates the left-hand side of Fig. 4 and explains the fact that the two edges differ by a factor of four. Applying other energy spacings between the states were found to yield the same results (with the only differences being vertical shifts of the lines) as the state reduction term is slowly varying within the Lorentzian shape. Fig. 5 shows the results for SAQWR based on an InP InAs structure whose state filling reduction is shown in the lower curve of Fig. 2. The central transition energy was ev and all other parameters were unchanged. Placing the state filling term outside the integral is somewhat less accurate in this case compared with the SAQD case. This is related to the highly nonsymmetrical integrand expression below and above the photon frequency which describes a broadened wire DOS function due to the Lorentzian shape [9]. Changing the assembly FWHM from 5 to 100 mev reduces the gain and the differential gain by a factor of two. Reaping the analysis as for the SAQDs reveals that the differential gain scales as for ideal uniform wire assemblies (the left side of Fig. 5). For the opposite case where the homogeneous broadening is considerably narrower than the inhomogeneous broadening, one can not replace the functional form which involves the expression in (16) with a delta function. In this regime, the complex expression can be replaced (in the limiting case where )by for most of the integration interval where the Gaussian shape is nonnegligible. The integration interval is restricted to positive values (as can be seen from the highly nonsymmetrical integrand). Using these approximations (with the assumption of a slowly varying filling term) the outcome is a numerical constant times which describes the behavior on the right side of Fig. 5. Therefore, the QD case reduction in both gain and differential Fig. 5. Differential gain versus inhomogeneous broadening in InP InAs SAQWR. Upper, middle, and lower sets denote different populations for the central transition wires. Solid and dashed lines denote, respectively, results with the state filling term kept inside and taken out of the integral. gain is proportional to where an ideal assembly is replaced by a wide assembly (with being a measure of the size fluctuations). The wire proportionality scales as the square root of this value. In this context SAQWRs are less sensitive for size fluctuations. This behavior is essentially not influence by the excited state and reservoir positions as the filling term corrections to the integral are of lower order (first and second-order for 1- and 0-D). The results for wires with different energetic diagrams introduce minor modifications. The physical interpretation for the above discussion is simple since different dot sized populations do not simultaneously contribute to the optical gain due to their discrete DOS shape. On the other hand, different sized wires share optical transitions on the high energy side and hence the effect of inhomogeneous broadening is smaller. Self-assembly laser structures with wide inhomogeneous lines require higher threshold inversion levels and hence their overall reduction (compared with similar structures having narrow lines) is the product of the state filling ratio and the inhomogeneous linewidth ratio. E. Integration Prefactor The last difference between the three geometries relates to the differential gain integration prefactor in (13), (16), and (17). The SAQD, SAQWR, and SQW prefactors are, respectively, proportional to and. If the dephasing energy varies as [72], the differential gain scales, respectively, as, and. Considering the dephasing to be 10 mev at room temperature and at elevated carriers densities, we note that the SAQD coefficient is larger than the other coefficients, only at low temperatures where the homogeneous broadening is determined by the excitons lifetime which yields a dephasing energy that is considerably smaller than the thermal energy. At low temperatures, one should also consider the random dot population effect [73] as the Fermi model is invalid. The advantage of low temperatures may be

8 DERY AND EISENSTEIN: THE IMPACT OF ENERGY BAND DIAGRAM AND INHOMOGENEOUS BROADENING 33 somewhat diminished as the revised state filling reduction term may not be ideal due to the random capture process as in higher dimensional equilibrium systems where there is no reduction when. Using the fact that the optical dipole matrix elements is approximately the same for all nanostructures near room temperature ( C M [68], [69]), we find that the optimal achievable differential gain is 2 10 cm for lasing in the m regime (in the ideal case of no state filling reduction and no size inhomogeneity). Even this idealized case yields a differential gain value which is one order of magnitude lower than the reported value of 2 10 cm of Klotzkin and Bhattacharya [21]. We believe this discrepancy to result from the extracting technique as indicated by Riedl et al. [19]. Our argument is also supported by the narrow bandwidth and low resonance frequencies reported in the experiment. The optimal value for the differential gain of SAQDs would not increase if nonequilibrium distributions would have been considered. This is based on the fact that the change in the phase filling term in (2) with a change in the density is by definition bounded by unity (transforming the total perturbation into a mere change of the ground-state populations). III. CONCLUSION This paper described a theoretical model which compares the optical differential gain for practical 2-, 1-, and 0-D semiconductor lasers. The model assumes room temperature operation and equilibrium conditions. We present a new expression for the 1-D case and use a unified formalism for the differential gain in the three structures. The differential gain has three main factors. First, the state filling term which considers the excited states and subbands as well as the reservoirs populations. Second, the inhomogeneous broadening which affects 1- and 0-D assemblies. Third, the integration pre factor which includes the homogenous broadening and the temperature. Our calculations estimate the differential gain of the different geometries by using energy schemes which rely on published luminescence experiments and their corresponding theoretical modeling. For the SAQDs, our calculations yield conservative estimates for the differential gain. We postulate that exaggerated published values stem from the extracting technique. We find that the differential gain of SAQD, SAQWR, and compressively strained QW structures are of the same order ( 10 cm to 10 cm ). We believe that our analysis clarifies the disagreements in reported differential gains and previous predictions. In devices with low modal gain, where a high inversion population is imperative to reach threshold, the differential gain can be reduced by more than one order of magnitude from its optimal value. This effect is highly important in conventional SAQD and SAQWR structures. In SAQDs, the imbalance between the bands energy spectra imposes neutrality constraints which lead to strong deviations in the electron and hole ground state populations, an effect that lowers the differential gain appreciably and also reduces the modal gain. The inhomogeneous broadening was shown to have a larger effect on conventional SAQDs as compared with SAQWRs. The difference stems from the respective shapes of the DOS. The overlap of high energy transitions between different populations in a SAQWR assembly reduces the impact of the size inhomogeneity, whereas in SAQDs the atom like energy spectra inhibit light amplification from different dot size populations. Considering the FWHM of present assemblies to be 20 mev ( 8 mev) and a room-temperature homogeneous broadening of 10 mev, leads to the conclusion that even if uniform dots could be realized, the differential gain improvement would be small. With respect to that, we emphasize that imbalanced spectra are the critical issue for lowering the differential gain in SAQD-based devices. Another important parameter, the linewidth enhancement factor ( parameter) is predicted to be small in SAQD lasers. Indeed, small parameter values have been observed in several experiments. The parameter is defined as the ratio of the refractive index derivative with respect to the carrier density to the differential gain. The low parameter in SAQD lasers is solely due to the small influence the carrier density has on the refractive index in delta-like energy spectra (which in principle can be zero). Reported parameter values are finite (although very low) due to contributions from excited state populations. In conventional SAQD lasers a differential gain which is larger than that of lasers based on higher dimensional gain media is not attainable at room temperature. 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Gadi Eisenstein (S 80 M 80 SM 90 F 99) received the B.Sc. degree from the University of Santa Clara, Santa Clara, CA, in 1975 and the M.Sc. and Ph.D. degrees from the University of Minnesota, Minneapolis, in 1978 and 1980, respectively. In 1980, he joined AT&T Bell Laboratories, Holmdel, NJ, where he was a member of the Technical Staff in the Photonic Circuits Research Department. His research at AT&T Bell Laboratories was in the fields of diode laser dynamics, high-speed optoelectronic devices, optical amplification, optical communication systems, and thin film technology. In 1989, he joined the faculty of the Technion Israel Institute of Technology, Haifa, Israel, where he holds the Dianne and Mark Seiden Chair of Electro-Optics in Electrical Engineering and serves as the head of the Barbara and Norman Seiden Advanced Optoelectronics Center. His current activities are in the fields of quantum-dot lasers and amplifiers, nonlinear optical amplifiers, compact short-pulse generators, bipolar heterojunction photo transistors, wideband fiber amplifiers, and broadband fiber optics systems. He has published over 250 journal and conference papers, lectures regulary in all major fiber optics and diode laser conferences, and serves on numerous technical program committees. Prof. Eisenstein is an Associate Editor of the IEEE JOURNAL OF QUANTUM ELECTRONICS.