3.1.1  Toward Inp-Based Quantum Dot Lasers for Otical Telecommunications

Semiconductor lasers play a crucial role in optical data communication and telecommunication applications. Nowadays, high-definition television, video on demand, broadband internet, and mobile phones are available all around the globe. The exponential rise in cost-effective information transmission would not have been possible without the introduction of optical transmission systems, which in turn are enabled by semiconductor lasers. A hundred million new semiconductor lasers are deployed in communication systems every year, generating several billion dollars of annual revenue at the component level [1]. Higher performance semiconductor laser sources are in strong demand because of the increasing data traffic in the WAN (wide area network), MAN (metropolitan area network), and LAN (local area network), which drives the development of novel semiconductor laser technologies.

Since the first demonstration of a semiconductor laser in 1962, the field has been witnessing the development from bulk-structure lasers and quantum-well (Qwell) to the advanced nanostructure quantum wire (Qwire), quantum dash (Qdash), and quantum dot (Qdot) lasers. In the 1960s, the bulk laser was developed on the basis of semiconductor heterostructures, which provided efficient confinement of charged carriers in the active region [2]. In particular, the double heterostructure (DH), which also yields optical confinement, has transformed semiconductor lasers from the laboratory to industry [3]. Quantum confinement occurs when one or more spatial dimensions of a nanocrystal approach the de Broglie wavelength of the carrier (on the order of 10 nm). The confinement of carriers leads to the quantization of the density of states, and splits the energy band of bulk semiconductors into energetic subbands [4]. In the 1970s, the first Qwell laser, in which carriers are confined in one dimension, was demonstrated [5]. Its recognized advantages over DH lasers were the reduced threshold current by decreasing the thickness of the active layer and the tunability of the wavelength via changing the Qwell thickness.

Increase of the confinement dimension leads to Qwire (2D confinement), Qdash (quasi-3D confinement), or Qdot structures (3D confinement). The 3D spatial confinement of Qdots results in an atomic-like density of states. The concept of Qdot semiconductor was proposed by Arakawa and Sakaki in 1982 [6], predicting temperature independence of the threshold current. Thereafter, reduction in threshold current density, high spectral purity, enhancement of differential gain, and chirp-free properties were theoretically discussed in the 1980s [7]. The most straightforward technique to produce an array of Qdots is to fabricate suitably sized mesa-etched quantum wells grown by metal-organic chemical vapor deposition (MOCVD) or molecular beam epitaxy (MBE). However, nonradiative defects produced during the etching procedure leads to a degradation of the material quality, which results in unsuitable structures for lasers. In the 1990s, both selective growth and self-assembled growth technique, which can avoid nonradiative defects, were well developed. Particularly, the Stranski-Krastanov growth mode turned out to be very successful for InGaAs/GaAs systems [8,9]. The strain-induced self-organization of InGaAs/GaAs quantum dots [10,11] yields threshold current densities as low as ~60 A cm^-2 at room temperature [12]. Extensive work on the GaAs-based Qdot system has been carried out, which resulted in tremendous improvement in laser performance [13]. An ultralow threshold current density of 17 A cm^-2 and a high output power of 7 W were achieved in InAs/GaAs lasers [14,15]. Nowadays, the InAs/GaAs Qdot products have already become commercially available in the market [8,16]. Meanwhile, several self-assembled growth techniques, such as solid-state MBE [17], gas-source MBE [18], MOCVD [19], and chemical-beam epitaxy (CBE) [20], have been improved and successfully used to grow Qdot materials.

Nevertheless, the GaAs-based Qdot laser devices emit usually in the O band (1260-1360 nm) of the telecommunication windows and are hardly able to reach the desirable long-haul communication window of ~1.55 μm. Instead, the InAs active region grown on InP substrates allows the realization of laser devices working in the C band window (1530-1565 nm) because of the smaller lattice mismatch. In the current market, InP-based 1.55 μm Qwell laser devices have shown substantial improvement in the optical characteristics in comparison with their DH counterparts. In order to improve the Qwell laser performance, InAs dots grown on InP substrates have attracted much attention in the research field. However, the formation of nanostructures on InP is much more challenging than on the GaAs substrate [21]. Although the InAs/InP and the InAs/GaAs systems have the same material in the dots, they differ in three aspects: (a) lattice mismatch in InAs/InP (~3%) is smaller than that in InAs/GaAs (~7%); (b) InAs/InP dots exhibit less confinement potential for electrons, but a stronger confinement for holes; (c) InAs/InP material shares the same cation (In), while the InAs/GaAs shares the same anion (As) at the interface [22]. The small lattice mismatch and the complex strain distribution can result in the formation of a new class of self-assembled Qdash nanostructures instead of Qdots. These are elongated, dot-like structures exhibiting interesting mixed characteristics in between those of the Qwell and the Qdot [23]. Strongly anisotropic QDash nanostructures are even closer to Qwire in their electronic properties [24]. Therefore, realization of real InAs/InP Qdots requires specific epitaxial growth procedures. This is done by employing conventional or miscut (100) InP substrates as well as (311) InP misoriented substrates along with various innovations in the growth process [25]. Nevertheless, reduction of the dot size and suppression of the size dispersion due to the self-assembly growth procedure are still challenges in achieving high-quality epitaxial material and hence better device performance. Currently, the size dispersions, characterized by the photoluminescence (PL) line-width in terms of the full-width at half-maximum (FWHM), are ~20 meV at 10 K for Qdots and ~50 meV at room temperature for Qdashes [26-29] and Qdots [30]. Further improvements in material quality are still required for them to be competitive with the mature InAs/GaAs Qdot systems. InAs dots formed on the (100)InP substrate usually have a low dot density (on the order of 10⁹-10¹⁰ cm^-2) [31]. These laser structures usually require multiple stacked layers for sufficient material gain. In contrast, dots grown on high-index (311)B substrates can lead to a large increase of the Qdots density, commonly in the 5 × 10¹⁰-10¹¹ cm^-2 range [32,33]. Figure 3.1 shows the atomic force microscopy (AFM) images of (a) InAs/InP Qdash, (b) Qdots on a (311)B substrate, and (c) Qdots on a (100) substrate. The first InAs/InP Qdot laser was reported by Ustinov et al. in 1998 [34,35], which emitted at ~1.9 μm at 77 K. Room-temperature operation was realized soon [36]. In the following, we review the development of InAs/InP Qdot lasers on both (311)B and (100)InP substrates employing the InAs/InGaAsP or InAs/InGaAlAs active material regions.

3.1.2  Lasers on (311)B Inp Substrate

High-indexed (311)B InP substrates can offer a high density of nucleation points for the Qdot islands, which strongly reduces surface migration effects and leads to the formation of more symmetric Qdots in the planar direction. As a result, high density and uniform distribution of Qdots can be obtained. Many studies have been carried out on the InP (311)B substrate, and a relatively high density of 5-13 × 10¹⁰ cm^-2 has been realized [36,38]. The inhomogeneous line-width broadening due to Qdot size dispersion can be constrained within 50 meV by using the double-capping-layer technique [30,39]. Figure 3.2a presents the AFM image of a typical InAs/InP dot on a (311)B substrate. The Qdot base resembles a circle rather than a square with a typical size of 30-50 nm in diameter, and the height for this kind of dot with cylindrical symmetry is usually several nanometers [33]. The cross-sectional scanning tunneling microscopy (X-STM) image of the Qdot structure in Figure 3.2b shows a truncated, faceted profile of the Qdot.

Using the InAs/InGaAsP active region, Nishi et al. demonstrated a Qdot laser with seven stacking layers grown by MBE [36]. The laser device had a dot density of 2 × 10¹⁰ cm^-2 and a threshold current density of 4.8 kA cm^-2, and emitted at 1.4 μm at room temperature. By using a double-capping technique, Caroff et al. achieved a high dot density of 1.1 × 10¹¹ cm^-2, emitting close to 1.55 μm at room temperature [40]. The laser had a high modal gain of 7 cm^-1 per dot layer. The threshold current density and transparency current density were 190 A cm^-2 (63 A cm^-2 per layer) and 68 A cm^-2 (23 A cm^-2 per layer), respectively. These are the best threshold performances reported in this material system. Homeyer et al. reported a broad-area laser emitting at 1.54 μm at room temperature with a record internal quantum efficiency of 62% [41]. The laser consisted of only two stack layers with a modal gain of 8 cm^-1 per layer. The dot density was also as high as ~1.0 × 10¹¹ cm^-2. A single-active-layer laser device with a gain of 13 cm^-1 was demonstrated in 2007, but it lased only up to 295 K [32]. Recently, a low-internal-loss laser of 6 cm^-1 was reported, which included nine stack layers with a total modal gain of 25 cm^-1 [42]. However, the InAs/InGaAsP material system usually has poor temperature stability with a characteristic temperature of only 25-50 K at room temperature due to the low conduction band offset [25].

By employing the InAs/InGaAlAs active region, Saito et al. first demonstrated a Qdot laser at the wavelength of 1.63 μm; the threshold current density was 660 A cm^-2 (132 A cm^-2 per layer) [43]. The extracted internal loss was 3.6 cm^-1, which is the lowest reported value for any InAs/InP system. Six years later, by exploiting the Al atoms in the spacer layers and employing the strain compensation technique, Akahane et al. demonstrated a 30-stack-layer laser with a threshold current density of 2.7 kA cm^-2 (90 A cm^-2 per layer) [44]. Subsequently, with improvement in the material growth quality, the threshold current density of the 30-stack laser was reduced to 1.72 kA cm^-2 (57.4 A cm^-2 per layer) with extremely high temperature stability. The characteristic temperature (T₀) was 114 K (20°C-75°C), which further improved to a record 148 K (25°C-80°C) in a 20-stack laser [45]. However, the laser had a high internal loss of ~26 cm^-1 due to the imperfect coupling of the optical mode with the multistack gain medium.

3.1.3  Lasers on (100)lnP Substrate

The formation of self-assembled Qdot on the (100)InP substrate is more complicated. The formation of the dot or dash strongly depends on the growth conditions and the thickness of InAs layers. The major problem in obtaining good performance of the laser is the low dot density on this kind of substrate.

In the InAs/InGaAsP material system, the growth of dots is mostly based on CBE or MOCVD techniques. Allen et al. reported a Qdot laser using the CBE technique, which was operated in pulsed mode at room temperature [46]. The pulsed threshold current density was 3.56 kA cm^-2 (713 A cm^-2 per layer), where a dot height trimming procedure with growth interruptions was employed. The dot density was increased by using the higher energy barrier to 3-6 × 10¹⁰ cm^-2. On the other hand, Lelarge et al. used a hybrid growth technique with an MBE-grown active region in conjunction with MOCVD grown p-doped cladding and contact layers [47]. The continuous wave (CW) threshold current density was 1.4 kA cm^-2 (240 A cm^-2 per layer) and T₀ was 56 K at room temperature. Particularly, the laser showed a very high modal gain of 64 cm^-1 (10.7 cm^-1 per layer). The laser grown by MOCVD showed a reduced threshold current density of 615 A cm^-2 (123 A cm^-2 per layer) [48]. The transparency current density was as low as 30 A cm^-2 and the internal loss was only 4.2 cm^-1.

In the InAs/InGaAlAs material system, Kim et al. first demonstrated a Qdot laser using an assisted growth technique of a thin gas underlying layer before the growth of InAs dots in the InGaAlAs matrix [49,50]. The achieved dot density was 6.0 × 10¹⁰ cm^-2, and the laser exhibited a threshold current density of 2.8 kA cm^-2 (400 A cm^-2 per layer). The measured T₀ was 377 K for temperatures up to 200 K, and 138 K above 200 K. More recently, Gilfert et al. reported a high-gain Qdot laser using the MBE method [51]. A low internal loss of 4 cm^-1 and a high gain of 15 cm^-1 per layer were reported. The lasing threshold current density was 1.95 kA cm^-2 (325 A cm^-2 per layer).

Very recently, Mollet et al. reported a very high modal gain of 97 cm^-1 for an InAs/InP(100) Qdash laser. However, the internal loss was also as high as 23 cm^-1 [52]. Generally, the performance of InAs/InP lasers has improved significantly since its first demonstration. However, it still needs improvement to be comparable with InAs/GaAs laser devices.

3.1.4  Current Status of the Dynamic performance of Inp-Based Qdot Laser

In fiber-optic links, the laser transmitter may be either directly modulated, known as directly modulated (DM) laser, or externally modulated using a modulator. In the direct modulation scheme, the driving current carries the transmitted data and is directly applied to the laser. In the external modulation scheme, the laser, which is subjected to a constant bias current, emits a continuous wave, while an external modulator switches the optical power on or off according to the data stream. In the external modulation scheme, electrooptic (EO) or electroabsorption (EA) modulators are commonly used [53]. EO modulators, such as the Mach-Zehnder modulator, utilizes a signal-controlled crystalline material exhibiting the EO effect (Pockels effect) to modulate the CW laser light. The EO effect is the change in the refractive index of the material resulting from the application of a DC or low-frequency electric field. The EA modulator controls the intensity of a laser beam via an electric voltage. Its operation can be based on the Franz-Keldysh effect [54], that is, a change in the absorption spectrum caused by an applied electric field, which changes the bandgap energy but usually does not involve the excitation of carriers by the electric field. However, most EA modulators are made in the form of a waveguide with electrodes for applying an electric field in a direction perpendicular to the modulated light beam. For achieving a high extinction ratio, one usually exploits the quantum-confined Stark effect, which describes the effect of an external electric field upon the light absorption spectrum or emission spectrum of a quantum-well structure. Both EO and EA modulators are operated at a few volts (below 10 V). In comparison with EO modulators, EA modulators have the convenient feature that they can be integrated with the laser on a single chip to create a data transmitter in the form of a photonic integrated circuit [55,56].

In contrast, DM lasers are the most common, particularly for short-reach systems. They have the lowest cost and least energy consumption, but they usually suffer the chirp characteristics [68]. DM lasers generally produce more chirps for higher extinction ratios, leading to an optimum setting for trading off the signal-to-noise ratio and chirp penalty. Aiming to realize chirp-free DM laser devices, many attempts have been made to develop nanostructured semiconductor lasers. Table 3.1 summarizes the reported dynamic performance of InP-based Qdot and Qdash lasers in the literature, including the modulation bandwidth and the α-factor, which is linked to the frequency chirp under direct modulation.

Martinez et al. reported a Qdot laser grown on the (311)B InP substrate with a modulation bandwidth of 4.8 GHz. The above-threshold α-factor was high (~6) and did not significantly depend on the bias current [57]. However, most reported dynamics of 1.55 μm lasers are for nanostructures grown on (100)InP substrates. Gready et al. demonstrated an InAs/InGaAlAs/InP Qdot laser with a 3 dB bandwidth of 5 GHz [58]. Interestingly, the laser showed a much larger signal modulation capability: 15 Gbps at 4 dB extinction ratio. The discrepancy between the small- and large-signal performances was attributed to the large nonlinear gain compression effect [69]. By optimizing the barrier width and the number of stack layers, the performance of the structure was improved by the same group. The small-signal modulation bandwidth was increased to 9 GHz, and the large signal modulation was operated up to 22 Gbps with an extinction ratio of 3 dB [59]. The best performance of InAs/InP Qdot laser was achieved by Bhowmick et al. recently [60]. The laser material system was InAs/InGaAlAs, and the active zone was grown on the (100)InP substrate. Tunnel injection and p-doping techniques enhanced the modulation bandwidth up to 14.4 GHz, and a near-zero α-factor was realized in this structure. For Qdash lasers on the (100)InP substrate, most works employed the p-doping technique to improve the modulation bandwidth and to reduce the α-factor [52,62,67,70]. Mi et al. showed that the modulation bandwidth could be increased from 6 GHz for the undoped laser to 8 GHz with the p-doped laser [62]. Mollet et al. demonstrated that the α-factor could be reduced from 5 down to 2.7 by p-doping [52]. On the other hand, the tunnel injection technique further increased the modulation bandwidth to more than 10 GHz while reducing the α-factor to near zero [60,62]. Consequently, a low chirp of 0.06 nm at a modulation frequency of 8 GHz was achieved in the Qdash laser and the same chirp level at 10 GHz in the Qdot laser.

3.2.1  Quantum Dot Electronic Structure

Schematically, the active region of a Qdot laser often consists of a 3D separate confinement heterostructure (SCH, also known as barrier), a 2D carrier reservoir (RS, roughly corresponding to the wetting layer), and dots spatially confined in three dimensions. Figure 3.3 shows an illustration of the electronic structure in a Qdot laser.

Carriers in the barrier and the wetting layer can be treated as quasi-free particles. Thus, quasi-continuum electronic states are formed in the SCH and the RS. The total densities of states for the SCH (per volume) and that for the RS (per area) are, respectively, given by [71]

with m* the effective mass of either electrons or holes. The quasi-continuum carrier reservoir coupling with the localized energy states of the dots results in smaller energy separations and thus overlapping states at higher energies [72]. The discrete states lying at lower energies are separated by a few tens of millielectronvolts in the conduction band (CB), while it is smaller in the valence band (VB) due to higher effective hole mass.

Accurate simulation of the electronic structure of an InAs/InP quantum dot can be performed using multiband k · p theory including strain and piezoelectric effects [73]. Based on the eight-band k · p theory, Figure 3.4a through c shows the confinement potential of the InAs/InP Qdot in comparison with the InAs/GaAs Qdot system [33]. In the absence of strain (Figure 3.4b), the band edge of InP is different from that of GaAs relative to the active material InAs. InP confines the holes more strongly, whereas GaAs does the same for the electrons. In the presence of strain, the band-edge energies are altered as shown in Figure 3.4a for InAs/InP and in Figure 3.4c for InAs/GaAs, by mainly hydrostatic strain in the conduction band and biaxial strain in the valence band. Consequently, the heavy/light hole degeneracy is lifted at the Γ point. The change of band edge for InAs embedded in GaAs is stronger because of the larger lattice mismatch (6.6% compared to only 3.1% for InP). The most striking feature is the smaller (strained) bandgap of InAs in InP than in GaAs, which enables the former to reach the 1.55 μm emission, which is hard to achieve for InAs/GaAs Qdots. Figure 3.4a and c shows that the depth of the electron confinement potential is similar in both systems, and therefore one can expect a comparable spectrum for confined electron states provided the Qdots share the same morphological properties. However, this does not hold for hole states since the confinement potential for InAs/InP Qdots is much deeper and the heavy-light hole splitting is smaller. Figure 3.4d and e illustrates the first three electron and hole wave functions (70% isosurface) of a single particle for both (100) (Figure 3.4d) and (311)B InP (Figure 3.4e) substrates, which is obtained by solving the Schrödinger equation. The symbols e0 and h0 stand for 1Se and 1Sh states, and e1, e2, and h1, h2 stand for 1Pe and 1Ph states. The single-particle states provide a basis for the configuration interaction model, which can be applied to calculate the excitonic properties, including correlation and exchange. Finally, the excitonic optical absorption spectra can be computed; the details can be found in [33].

Because of the Coulomb interaction, electrons and holes in the semiconductor can be bound into electron-hole pairs, known as excitons. The distance between the electron and the hole within an exciton is called the Bohr radius of the exciton. Typical exciton Bohr radius of semiconductors is a few nanometers [74]. The exciton’s nature can be modified by the confinement structure, and thus it can exhibit different optical properties. The assumption holds as long as the electron and hole populations do not show significant deviations. Such a simplified picture was capable of describing the basic optical properties.

3.2.2  Carrier Scattering Processes in Quantum Dot

In a Qdot device, once the current injection generates charge carriers in the 3D barrier, the carriers will be transported into the 2D RS, which acts as a carrier reservoir for the localized discrete Qdot states. The carrier capture process refers to the subsequent carrier capture from the RS to the ESs of the dots. In the dots, the carriers relax from high energetic ESs down to the GS level. Finally, radiative recombination of electrons and holes takes place and lasing occurs often on the GS. These processes are well reflected by the time-resolved photoluminescence (TRPL) of an InAs/InP Qdot device, as shown in Figure 3.5a. The spectra are fitted with three Gaussian peaks, which, respectively, correspond to the Qdot excitonic GS at 0.94 eV, the first Qdot ES at 0.99 eV, and the RS ~1.05 eV. The peak of the spectra shifts from RS at 10 ps via ES at 600 ps toward GS at 1500 ps with time evolution after the optical excitation [75]. In addition, carrier capture directly from the RS into the GS is also possible. This direct channel accelerates the carrier indirect process (via ES) to the lasing GS, and plays an important role on the dual (GS and ES) lasing process, as pointed out in InAs/InP Qdot lasers [76]. Moreover, those electronic states also exhibit inter-dot electronic coupling [72]. As in Qwell lasers, the carrier transport process plays an important role in determining the Qdot laser’s dynamics as well, which induces a parasitic-like roll-off that is indistinguishable from an RC roll-off in the modulation response, and thus limits the modulation bandwidth [77,78]. TPRL shows that the carrier transport time across barrier to the RS is several picoseconds (1-5 ps) depending on the thickness of the SCH layer [79,80]. The carrier capture and relaxation transition processes are supported mainly by two physical mechanisms: Coulomb-interaction-induced carrier-carrier scattering (Auger process), and carrier-LO phonon scattering. The scattering behavior is different at low and high excitation carrier densities. At low excitation density, the carrier interaction with LO phonons can provide efficient scattering channels provided that energy conservation is fulfilled. While the energy separation of Qdot states typically does not match the LO phonon energy, this scattering mechanism is often possible for holes due to their dense states [81]. The carrier-LO phonon scattering process is found to be temperature-dependent: high temperature accelerates this scattering rate [82,83]. When a high-density carrier plasma is created in the carrier reservoir, carrier-carrier scattering accounts for the efficient capture from the RS into the localized Qdot states as well as the relaxation between the discrete Qdot states [84]. Auger scattering can be categorized into three types according to the initial electronic states of the carriers involved. One type involves two carriers in the RS states [85]; another type is with one carrier in an ES state while the other in the RS [86]; and the third type has both carriers occupied by the ES [87]. In contrast to the carrier-phonon interaction process, the Auger process is carrier-density-dependent, as shown in Figure 3.5b [88]. At 10 K, the carrier relaxation time into the GS of the Qdots reduces from ~75 ps under an optical excitation of 1 W cm^-2 down to about 10 ps for an excitation density of 2000 W cm^-2.

From the analysis of TRPL rise time of Qdot devices, both the capture and relaxation times are found to vary over a wide range from 1 up to 100 ps depending on the excitation intensity [82,89]. However, for a moderate RS carrier density of 10¹¹-10¹² cm^-2, the typical carrier scattering times are on the order of 1-10 ps [79,90]. For the processes related to Coulomb many-body interactions, relaxation within the Qdot is typically on a faster timescale than the carrier capture from the RS into the Qdot. Processes involving holes are typically faster than the corresponding processes involving electrons, and capture to the excited states is faster than capture to the ground states. Hence, in a dynamical scenario, first the holes are captured to the excited Qdot states and immediately scattered via relaxation to the Qdot ground states. Capture of electrons is somewhat slower; the subsequent relaxation for electrons is only slightly slower than for holes [81].

From the aspect of rate equation modeling of Qdot lasers, carrier distributions in all states are usually assumed to be under the quasi-equilibrium condition with Fermi-Dirac distribution, which is suitable for sufficiently rapid intraband relaxation processes [91]. Carrier occupations, at least in the GS and in the RS, must be modeled in order to distinguish the Qdot laser from the Qwell laser. However, for achieving moderate accuracy and correlating with experimental data in the InAs/InP Qdot system, it is necessary to consider the population in the first ES, which can have significant influence on the laser’s static and dynamic characteristics. Inclusion of more states would be more accurate but at a price of losing simplicity as well as the intuitive physical image.

Rigorous calculation of carrier scattering rates is a stiff task, which requires sophisticated many-body quantum theory that treats intraband collision processes. In [91-93], a phenomenological formula was proposed to take into account the carrier-dependent capture and relaxation time in a semiempirical model:

where

• i denotes the capture or relaxation process
• A_i is the phonon-assisted scattering rate
• C_i is the coefficient determining the Auger-assisted scattering by carriers in the RS (N_RS)

Although this expression leads to good agreement with the TRPL experiments in [75], parameters A_i and C_i can be quite different from device to device [76], thereby limiting the applicability of this expression. Because of the fact that once the laser is operated above threshold there is a large density of carriers in the RS, which does not vary much with the bias current, it is a reasonable approximation to assume the carrier scattering time as constant, which simplifies the rate equation model for the study of Qdot laser dynamics.

3.2.3  Gain, Refractive Index, and Line-width Enhancement Factor

The laser field and the semiconductor gain medium are coupled by the gain and the carrier-induced refractive index, or equivalently, by the complex optical susceptibility. To determine these quantities, it is necessary to solve the quantum mechanical gain medium equations of motion for the microscopic polarization. In principle, these dynamical equations should be derived using the full system Hamiltonian, which includes contributions from the kinetic energies, the many-body Coulomb interactions, the electric-dipole interaction between the carriers and the laser field, as well as the interactions between the carriers and phonons. The effects of injection current pumping should also be included [92].

The connection between the classical electrodynamics and quantum mechanics is effected through the macroscopic polarization P and the microscopic polarization p_α:

In Qdot lasers, the processes include the GS, ES, RS, and barrier transitions. V is the active region volume, and the polarization summation is performed over all interband optical transitions. The complex optical susceptibility is connected with the polarization via

where ε₀ and E are the vacuum permittivity of light and the amplitude of the electric field, respectively. The gain g and the carrier-induced refractive index δn in the model are defined by

where

• ω and c are the laser frequency and the velocity of light, respectively
• n_b is the refractive index
• Γ_P is the optical confinement factor

We can obtain the following relation between the gain, refractive index, susceptibility, and the polarization [94]:

Introducing the differential gain a, the phenomenological gain can be expressed as

with N and N_tr being the injected carrier density and transparency carrier density of zero gain, respectively.

In semiconductor lasers, the nonlinear gain phenomenon plays an important role in both static and dynamic characteristics such as spectral properties, modulation bandwidth, and frequency chirping [95]. The main physical mechanisms behind nonlinear gain are attributed to the spectral hole-burning, spatial hole-burning, and carrier heating [96,97]. Furthermore, the gain nonlinearity was found to enhance the quantum confinement of carriers and carrier relaxation processes [98]. Spectral hole-burning is the formation of a dip in the gain spectrum due to stimulated emission. The dip occurs by the recombination of electrons and holes at a specific energy and the subsequent redistribution of carrier energies due to carrier-carrier scattering. The scattering process takes place on the timescale of the order of 50-100 fs and leads to a dip width of about 20-40 meV. It also ensures that temperature equilibrium is established among the carriers within the same timescale. Carrier heating is related to the fact that the carrier temperatures can be different from the lattice temperature due to the stimulated emission and free-carrier absorption [96]. The carrier temperatures relax toward the lattice temperature by electron-phonon scattering processes within a timescale of 0.5-1 ps. The nonlinear gain effect is usually characterized by a phenomenological gain compression factor, as

with S being the photon density. This expression shows that the linear gain g is reduced at high power density.

The refractive index change related to the optical interband transition as expressed in Equation 3.8 is known as anomalous dispersion [99]. Another important contribution to the index change is the free-carrier plasma originating from intraband transitions [100]. In Qwell lasers, this contribution to the differential index is well described by the Drude model [101]:

with e the electronic charge. Analogous transitions in Qdot lasers can be envisaged between bound Qdot states and the continuum levels of the RS and the barrier. It has been shown that the Drude formula can also be applied to the case of Qdot lasers when the Qdot carriers are not tightly confined and when working at photon energies in the 0.8-1.0 eV region [100].

In semiconductor lasers, it is well known that any change in the imaginary part of the susceptibility (gain) will be accompanied by a corresponding change in its real part (refractive index) via the Kramers-Kronig relations. The line-width enhancement factor (or α-factor) describes the coupling between the carrier-induced variation of real and imaginary parts of susceptibility, and is defined as [102]

Employing Equations 3.7 and 3.8, the above definition is equivalent to the following often-used expression:

In a practical case, the variation of the carrier concentration is usually small, which justifies taking the derivatives at the operating point and assuming a linear dependence of g(N) and n(N). Equation 3.13 can therefore be written as

Through the relation between the refractive index and frequency variation, we can obtain the following equivalent formula:

The α-factor plays a crucial role in driving fundamental features of semiconductor lasers such as the spectral line-width broadening [103], frequency chirp [104], mode stability [105], and nonlinear dynamics subject to optical injection [106] or optical feedback [107-109]. Typical Qwell lasers often exhibit α-factor values in the order of 2-5 [78]. For Qdot lasers, earlier analyses have suggested a zero or near-zero α-factor due to the delta-function-like discrete density of states. A symmetrical gain curve indeed leads to a dispersive curve of the refractive index with a zero value at the gain peak. However, experimental α-factor values in Qdot lasers vary over a wide range from zero up to more than 10, particularly a giant value (as high as 60) was also reported [110-112]. The nonzero α-factor is attributed to the large inhomogeneous broadening, the off-resonant bound and continuum states, as well as the free-carrier plasma effect.

3.3.1  Amplitude Modulation Response

High-speed, energy-effective, and low-cost optical communication networks primarily require semiconductor laser sources of broad modulation bandwidth. In order to theoretically discuss the amplitude modulation (AM) performance of Qdot lasers, we employ a semiclassical rate equation model [76]. The model will be analyzed in a semianalytical approach, which has the merit of giving an intuitive physical image. This numerical model of the Qdot laser holds under the assumption that the active region consists of only one Qdot ensemble; that is, the inhomogeneous broadening due to the dot size fluctuation is not considered. The electrons and holes are treated as electron-hole (e-h) pairs, meaning that the system is in excitonic energy states. Two discrete states in Qdots are taken into account: a twofold degenerate GS and a fourfold degenerate first ES. The Qdots are interconnected by the 2D RS. This simplified picture corresponds to the TRPL experimental observations in Figure 3.5 [75]. Carriers are supposed to be injected directly from the contacts into the RS, so the carrier dynamics in the 3D barrier are not taken into account in the model.

Figure 3.6 shows the schematic of the carrier dynamics in the exiton framework. First, the externally injected carrier fills directly the RS reservoir; some of the carriers are then either captured into the ES within time ${\tau }_{\mathit{\text{ES}}}^{\mathit{\text{RS}}}$

or directly into the GS within time ${\tau }_{\mathit{\text{GS}}}^{\mathit{\text{RS}}}$ , and some of them recombine spontaneously with a spontaneous emission time ${\tau }_{\mathit{\text{spon}}}^{\mathit{\text{RS}}}$ . Once in the ES, carriers can relax into the GS within time ${\tau }_{\mathit{\text{GS}}}^{\mathit{\text{ES}}}$ or recombine spontaneously. On the other hand, carriers can also be thermally re-emitted from the ES to the RS with an escape time ${\tau }_{\mathit{\text{RS}}}^{\mathit{\text{ES}}}$ , which is governed by the Fermi distribution for the quasi-thermal equilibrium without external excitation [113]. A similar dynamic behavior is followed for the carrier population on the GS level with regard to the ES. Direct injection from the RS to the GS was introduced to reproduce the experimental results [76]. Stimulated emission occurs from the GS when the threshold is reached, and that from the ES is not taken into account in the model. Following Figure 3.6, the four coupled rate equations on carrier and photon densities are described as follows:

where

• N_RS, N_ES, and N_GS are the carrier densities in the RS, ES, GS, respectively
• S_GS is the photon density in the cavity with GS resonance energy
• β_SP is the spontaneous emission factor
• Γ_P is the optical confinement factor
• τ_p is the photon lifetime
• v_g is the group velocity
• V_g is the volume of the laser’s active region

The GS gain is given by

where

• a_GS is the differential gain
• N_B is the total Qdot surface density
• H_B is the height of the dots.

In what follows, it is important to stress that the effects of gain compression are not taken into account. In Equations 3.16 through 3.18, ρ_GS,ES are the carrier occupation probabilities in the GS and the ES, respectively:

Since the carrier escape from the GS to the RS has little effect on lasing properties [76], the ${\mathit{\text{N}}}_{\mathit{\text{GS}}}\text{/}{\tau }_{\mathit{\text{RS}}}^{\mathit{\text{GS}}}$

term in Equations 3.16 and 3.18 can be neglected.

The rate equations can be linearized through a small-signal analysis. Assuming a sinusoidal current modulation dI = I₁ejωt with modulation frequency ω, the corresponding carrier and photon variations are of the form

Substituting the above formulas into the rate equations (Equations 3.16 through 3.18), we obtain the linearized differential rate equation in matrix form:

with the following elements:

Finally, the AM or intensity modulation response of the Qdot laser is calculated by

This equation is also known as the modulation transfer function. Through proper approximation, the AM response is given by [114]

where the first part, H₁(ω), is dominated by the carrier-photon interaction processes. The resonance frequency ω_R and the damping factor Γ of the Qdot laser are, respectively, expressed as

where the steady-state relationship $1\text{/}{\tau }_{\mathit{\text{p}}}-{\mathrm{\Gamma }}_{\mathit{\text{p}}}{\mathit{\text{v}}}_{\mathit{\text{g}}}{\mathit{\text{g}}}_{\mathit{\text{GS}}}={\mathrm{\Gamma }}_{\mathit{\text{p}}}{\mathrm{\beta }}_{\text{S}\mathit{\text{g}}}{\mathit{\text{N}}}_{\mathit{\text{GS}}}\text{/}\left({\tau }_{\mathit{\text{GS}}}^{\mathit{\text{spon}}}{\mathit{\text{S}}}_{\mathit{\text{GS}}}\right)$

has been used. In comparison with Qwell lasers [78], both expressions have an additional term $\left({\mathit{\text{H}}}_{\mathit{\text{B}}}{\mathit{\text{N}}}_{\mathit{\text{ES}}}\text{/}\left(2{\mathit{\text{N}}}_{\mathit{\text{B}}}{\tau }_{\mathit{\text{GS}}}^{\mathit{\text{ES}}}\right)+(1-{\mathrm{\rho }}_{\mathit{\text{ES}}})\text{/}{\tau }_{\mathit{\text{ES}}}^{\mathit{\text{GS}}}\right)$ , which describes the effective carrier scattering rate into and out of the GS. The second part, H₀(ω), is dominated by the carrier capture and relaxation processes, and the introduced two parameters ω_R0 and Γ₀ are given by

Figure 3.7 presents an example of the calculated AM response of a Qdot laser. The solid curve H_Qdot(ω) shows that the response is strongly damped as usually observed in experiments [115-117], which can be attributed to the carrier relaxation and escape process of the GS as described in Equation 3.28. On the other hand, H₀((o) (dash-dot) exhibits a much smaller bandwidth than H₁(ω) (dash), which demonstrates that the finite carrier capture and relaxation times limit the modulation bandwidth of Qdot lasers.

3.3.2  Line-Width Enhancement Factor (α-FAcroR)

The line-width enhancement factor (α-factor) plays an important role in determining fundamental features of semiconductor lasers. Although the Qdot laser is predicted to have a delta-function-like discrete density of states, the measured α-factor values in experiments vary over a wide range from zero up to >10. The nonzero α-factor in Qdot lasers is partly attributed to the asymmetric gain spectrum because of the inhomogeneous broadening [118] and the carrier population in off-resonant states [119]. On the other hand, the free carrier plasma effect in the barrier and in the RS is reported to contribute almost half of the total refractive index change [100,101]. In this section, we describe an improved electric field model for Qdot lasers taking into account the contribution of off-resonant states (ES and RS) on the refractive index change, which allows a semianalytical study of the α-factor features in the Qdots. The model is capable of exploring the crucial physical mechanisms driving the Qdot laser’s α-factor.

The conventional model describing the complex electric field of semiconductor lasers is given by

where the first term on the right-hand side gives the gain and the photon loss of the laser cavity. The second term, Δω_N, describes the carrier-induced frequency shift of the laser field with respect to the frequency at the lasing threshold. Δω_N is usually expressed by the α-factor α_H as

This model is able to study the impacts of a nonzero α-factor on the modulation dynamics and nonlinear dynamics of Qdot lasers [120] but does not allow the study of α_H itself under different operating conditions. In order to investigate the α-factor, we need to obtain the expression of the gain and the refractive index separately. In the semiclassical theory, the semiconductor laser system can be fully described by the optical Bloch equations together with Maxell’s equations [121]. The RS is treated as a discrete energy state of degeneracy D_RS = k_BTm* A_RS/(πħ²), with m* the reduced carrier mass and A_RS the RS surface area (see Section 3.1) [122]. The active region consists of only one Qdot ensemble. In addition, the electrons and holes are treated as neutral pairs (excitons). Two discrete states—the ground state (GS) and the first excited state (ES)—are considered in the dots. With these assumptions, the slowly varying electric field E(t) is given by [123]

where

• ${\mathrm{\text{ω}}}_{\mathit{\text{LS}}}^0$
  is the lasing frequency in the cold cavity
• ε_bg and ε₀ are the background and vacuum permittivity, respectively
• Γ_P is the optical confinement factor
• H_B is the height (equal to the dot’s height)

The sum over X (X = GS, ES, RS) includes all possible optical transitions, with μ_x being the corresponding dipole transition matrix element and P_X the microscopic polarization. τ_p is the photon lifetime in the laser cavity. Assuming a sufficiently short dephasing time T_D and adiabatically eliminating the interband polarization yields the quasi-static relation

where

• ρ_x denotes the carrier occupation probability
• ħω_x gives the transition energy of each state

Inserting Equation 3.34 into Equation 3.33, we obtain the complex gain

The optical susceptibility can be derived from this complex gain expression through the relationship $\chi ({\mathrm{\text{ω}}}_{\mathit{\text{LS}}}^0,\mathit{\text{t}})=2{\varepsilon }_{\mathit{\text{bg}}}\widetilde{\mathit{\text{G}}}({\mathrm{\text{ω}}}_{\mathit{\text{LS}}}^0,\mathit{\text{t}})\text{/}\left(\mathit{\text{j}}\mathrm{\text{ω}}{\mathrm{\Gamma }}_{\mathit{\text{P}}}\right)$

. The real part of Equation 3.35 is related to the laser gain, while the imaginary part gives the instantaneous frequency shift of the electric field. The three terms on the right-hand side give contributions of the GS, ES, and RS, respectively. Introducing the differential gain (a_X), we have

with v_g being the group velocity of the light. The material gain of each state is then given by

where

• N_X is the carrier density in each state
• S_GS is the photon density in the GS
• ξ denotes the gain compression factor

Because the real part of the complex gain approaches zero very quickly when off resonance, the field gain originates mainly from the resonant state. Considering the lasing emission in resonance with the GS transition ${\mathrm{\text{ω}}}_{\mathit{\text{LS}}}^0={\mathrm{\text{ω}}}_{\mathit{\text{GS}}}$

, we obtain

In contrast, the imaginary part of the complex gain decays slowly for off-resonant frequencies. Thus, the off-resonant states can significantly influence the refractive index change, even though their gain contribution to the GS lasing is almost nil. Carrier populations in the off-resonant ES and RS induced frequency shifts of the laser field respectively are

with coefficients

From Equation 3.35, it is seen that the resonant GS has no contribution to the refractive index change, which is the case when the laser is operated at the gain peak with a symmetric gain distribution. Nevertheless, as mentioned in the introduction, due to the asymmetric Qdot size dispersion, the resonant state induces a finite α-factor ${\alpha }_{\mathit{\text{H}}}^{\mathit{\text{GS}}}$

, and the corresponding frequency shift with respect to the cold cavity can be expressed by

Employing Equations 3.37 through 3.42, the electric field (Equation 3.33) is re-expressed as

With carrier injection, the lasing frequency becomes ${\mathrm{\text{ω}}}_{\mathit{\text{LS}}}={\mathrm{\text{ω}}}_{\mathit{\text{th}}}^{\mathit{\text{LS}}}+\mathrm{\Delta }{\mathrm{\text{ω}}}_{\mathit{\text{N}}}^{\mathit{\text{LS}}}$

, where $\mathrm{\Delta }{\mathrm{\text{ω}}}_{\mathit{\text{N}}}^{\mathit{\text{LS}}}=\mathrm{\Delta }{\mathrm{\text{ω}}}_{\mathit{\text{N}}}^{\mathit{\text{GS}}}+\mathrm{\Delta }{\mathrm{\text{ω}}}_{\mathit{\text{N}}}^{\mathit{\text{ES}}}+\mathrm{\Delta }{\mathrm{\text{ω}}}_{\mathit{\text{N}}}^{\mathit{\text{RS}}}$ gives the total frequency shift of the electric field from its threshold value $\left({\mathrm{\text{ω}}}_{\mathit{\text{th}}}^{\mathit{\text{LS}}}\right)$ . Through the $\mathit{\text{E}}\left(\mathit{\text{t}}\right)=\sqrt{\mathit{\text{S}}\left(\mathit{\text{t}}\right)\mathit{\text{V}}/{\mathrm{\Gamma }}_{\mathit{\text{P}}}}{\mathit{\text{e}}}^{\mathit{\text{j}}\phi \left(\mathit{\text{t}}\right)}$ relationship, the photon density S(t) and the phase ϕ(t) can be separately described. Combining with the equations describing the carrier dynamics in Qdot lasers, the laser system is finally given by

where

• ${\tau }_{\mathit{\text{GS}}}^{\mathit{\text{spon}}}$
  is the spontaneous emission time
• β_SP is the spontaneous emission factor

Carriers in the RS are scattered into the dots through phonon-assisted and Auger-assisted processes [82,83]. The latter makes the scattering rates nonlinearly depend on the carrier density in the RS. However, for the sake of simplicity, the carrier capture time ${\tau }_{\mathit{\text{ES}}}^{\mathit{\text{RS}}}$

and the relaxation time ${\tau }_{\mathit{\text{ES}}}^{\mathit{\text{GS}}}$ are both treated as constants in this work. On the other hand, the carrier escape times $({\tau }_{\mathit{\text{RS}}}^{\mathit{\text{ES}}},{\tau }_{\mathit{\text{ES}}}^{\mathit{\text{GS}}})$ are governed by the Fermi distribution for a quasi-thermal equilibrium system [124]. For semiconductor lasers operating under small-signal modulation with frequency ω, the bias current change δI induces variations of the carriersδN_X the photon δS_GS, and the phase δϕ. In order to perform the analyses, the differential rate equations are derived as follows:

where

with

Based on the above differential rate equations, the α-factor of the Qdot laser is derived as

Following the definition in Equation 3.13, it is noted that only the carrier contribution (δN) is included in the above equation, while the photon contribution (δS) is excluded. In the following, it will be shown that the α-factor of Qdot lasers presents peculiar characteristics under direct modulation.

Over the last decades, various techniques have been proposed for the measurement of the α-factor. In this work, we employ the well-known “FM/AM” technique for the above-threshold analysis and the widely used “Hakki-Paoli” method for the below-threshold analysis [102]. The FM/AM technique relies on the direct current modulation of the laser, which generates both the optical frequency (FM) and amplitude (AM) modulations [125]. With respect to the linearized rate equations, the ratio of the FM/AM index is derived as

where the relation δω_LS = jωδϕ is used in the above derivation. In this approach, the laser’s α-factor is usually extracted through the formula ${\alpha }_{\mathit{\text{H}},\mathit{\text{QD}}}^{\mathit{\text{FM/AM}}}=\text{min}\left\{2\mathrm{\beta }\left(\mathrm{\text{ω}}\right)\text{/}\mathit{\text{m}}\left(\mathrm{\text{ω}}\right)\right\}$

. This is indeed true for Qwell or bulk lasers; however, we show that the α-factor of Qdot lasers is dependent on the modulation frequency but we still take the minimum value to characterize the Qdot laser as reported in [119].

For semiconductor lasers operating below threshold, the Hakki-Paoli method relies on the direct measurement of the optical spectra of amplified spontaneous emission (ASE) in the laser cavity. Tuning the pump current slightly step by step (ΔI), the gain change can be extracted by the Hakki-Paoli method, and the wavelength variation can be directly recorded using an optical spectrum analyzer. Correspondingly, the below-threshold α-factor is calculated as

The laser parameters used in the simulation are listed in Table 3.2 [57,75,126]. It is noted that the carrier occupation in the GS has a small contribution to the α-factor (<1) [127]; hence we assume the value ${\alpha }_{\mathit{\text{H}}}^{\mathit{\text{GS}}}=0.5$

in the simulation. Figure 3.8a depicts the carrier density variations in the three states under small-signal modulation. For low frequencies (smaller than 0.1 GHz), all the carrier density variations remain almost constant, but the variations of the ES (δN_ES) and RS (δN_RS) populations are 15 dB larger than that of the GS (δN_RS) one. The small variation of the GS carrier population is associated with the gain-clamping above the threshold. Both δN_GS and δN_ES exhibit resonances at ~7 GHz. Beyond the resonance frequency, δN_GS decays faster than δN_ES and δN_RS These features significantly impact the behavior of the α-factor, as described in Equation 3.52.

Figure 3.8b compares the difference between the α-factor $\left|{\alpha }_{\mathit{\text{H}},\mathit{\text{QD}}}^{\mathit{\text{GS}}}\left(\mathrm{\text{ω}}\right)\right|$

and the ratio |2β(ω)m(ω)| as a function of the modulation frequency. At low frequencies (<0.1 GHz), there is a large discrepancy between the two parameters. As expected, |2β(ω)/m(ω)| exhibits large values due to the gain compression and the large carrier variations in the ES and in the RS. Nevertheless, $\left|{\alpha }_{\mathit{\text{H}},\mathit{\text{QD}}}^{\mathit{\text{GS}}}\left(\mathrm{\text{ω}}\right)\right|$ remains constant. On increasing the modulation frequency beyond several gigahertz, the two values of both parameters decrease down to a plateau, which gives the conventional α-factor, indicated by the horizontal line. As can be seen, ${\alpha }_{\mathit{\text{H}},\mathit{\text{QD}}}^{\mathit{\text{FM/AM}}}$ is almost the same as $|{\alpha }_{\mathit{\text{H}},\mathit{\text{QD}}}^{\mathit{\text{GS}}}|$ , which indicates that the FM/AM method is a reliable technique for the measurement of Qdot laser’s α-factor. Further increase of the modulation frequency raises again both the values, as observed experimentally in a Qdot laser (inset of Figure 3.8b) [129]. It is emphasized that such a situation is not encountered in Qwell lasers [125]. This behavior is attributed to the different decay rates (versus modulation frequency) of carrier variations in each state, as shown in Figure 3.8a. In addition, Figure 3.8b shows that the ES (dash-dot curves) contributes more to the α-factor $|{\alpha }_{\mathit{\text{H}},\mathit{\text{QD}}}^{\mathit{\text{GS}}}|$ than the RS due to its smaller energy separation with the resonant GS.

Based on the ASE and the FM/AM methods, Figure 3.9 illustrates the α-factor as a function of the normalized pump current I/I_th. Below threshold, carrier populations in both the resonant and off-resonant states increase with the pump current. In consequence, the α-factor increases nonlinearly. Above threshold, the carrier population in the GS is clamped, while the off-resonant state populations keep increasing. Thus, the α-factor varies almost linearly above threshold as usually measured in experiments [57,71]. At threshold, the α-factor extracted from the ASE method is similar to that using the FM/AM technique. In addition, the α-factor is larger than the sole GS-induced value of ${\alpha }_H^{\mathit{\text{GS}}}=0.5$

both below and above threshold, which means the off-resonant ES and RS contribute to the increase of the α-factor in the Qdot laser. This is explained by the fact that the coefficients $F_{\mathit{\text{ES}}}^{\mathit{\text{GS}}}$ and $F_{\mathit{\text{RS}}}^{\mathit{\text{GS}}}$ are both positive since the ES and RS have higher energies than the GS (see Equation 3.41).