On quantum-dot lasing at gain peak with linewidth enhancement factor αH = 0

This paper describes an investigation of the linewidth enhancement factor αH in a semiconductor quantum-dot laser. Results are presented for active region parameters and laser configurations important for minimizing αH. In particular, the feasibility of lasing at the gain peak with αH = 0 is explored. The study uses a many-body theory with dephasing effects from carrier scattering treated at the level of quantum-kinetic equations. InAs quantum-dot lasers with different p-modulation doping densities are fabricated and measured to verify the calculated criteria on laser cavity design and epitaxial growth conditions.This paper describes an investigation of the linewidth enhancement factor αH in a semiconductor quantum-dot laser. Results are presented for active region parameters and laser configurations important for minimizing αH. In particular, the feasibility of lasing at the gain peak with αH = 0 is explored. The study uses a many-body theory with dephasing effects from carrier scattering treated at the level of quantum-kinetic equations. InAs quantum-dot lasers with different p-modulation doping densities are fabricated and measured to verify the calculated criteria on laser cavity design and epitaxial growth conditions.

Quantum well (QW) gain regions have replaced bulk ones for virtually all commercial applications. Further improvement in laser performance may have to come from an underlying physics level. A strong candidate is the class of lasers with quantumdot (QD) active regions, where quantum confinement increases from one-dimensional to three-dimensional, or equivalently, electronic density of states reduces from two-dimensional (2-d) to zerodimensional (0-d). 1 The atomic (0-d) nature of optical emission was demonstrated in the 1990s. 2,3 Predicted advantages of low lasing threshold, 4 high temperature operation, 5 tolerance to crystalline defects, and optical feedback 6,7 are being realized.
With successes in threshold performance, attention in QD lasers is shifting toward the above-threshold properties. Important for applications, ranging from datacom and telecom to chemical sensing and laser radar, are laser linewidth, 8 chirp during high-speed modulation, 9,10 and optical feedback sensitivity. 11 A critical gain-medium parameter is the linewidth enhancement factor, 12,13 where δn is the carrier-induced refractive index, G is the intensity gain, Ne is the carrier density, and K is the lasing wavevector. We investigated the minimization of αH, in particular, the feasibility of αH = 0 at the gain peak. This paper describes application of a many-body QD gain theory to identify relevant device parameters and desirable laser configurations. The study involves lasers, each consisting of multiple QWs embedding InAs QDs. The QWs are separated by GaAs barriers, and the entire gain region is cladded by graded-index AlGaAs layers. [14][15][16][17][18] The calculations are for the gain and carrier-induced refractive index from one of the QWs, specifically for a 7 nm In 0.15 Ga 0.85 As QW with InAs QD density N 2d QD = 5 × 10 10 cm −2 and with various p-modulation doping densities and inhomogeneous broadening. These parameters represent tunable criteria in laser cavity design and epitaxial growth conditions for engineering αH = 0 QD lasers.
From semiclassical laser theory, the intensity gain G and carrier-induced refraction index δn at laser frequency ν are 19,20

ARTICLE scitation.org/journal/app
where c and ϵ 0 are the vacuum speed of light and permittivity, nB is the background refractive index, h is the QW width, ε is a weak laser probe field for extracting the susceptibility, A is the QW area, and ℘n and ℘ k are the QD and QW dipole matrix elements. The subscripts n and k label the QD and QW optical transitions, pn,q and p k are the QD and QW polarizations, and n inh n,q is the density of QDs with electronic structure labeled q, contributing to the nth QD transition. The computed G and δn are useful in 2 ways. They give the gain parameters at saturated carrier density in the often used class B semiconductor laser model. 21 They are also used in predicting general laser performance (such as light-current characteristics) when the saturated gain and carrier density are clamped at the threshold values. 22 The calculations start with solving the equations of motion for electron-hole polarizations. For the n-th QD transition belonging to the q-th population group, In the first line are the single-particle contributions from frequency detuning and stimulated emission, where ω (0) n,q is the unrenormalized transition frequency and f σ n,q (σ = e or h) is the QD carrier population. The second line contains the many-body corrections due to exchange and excitonic effects, with dependences on the QW carrier population f σ k . These effects renormalize the transition frequency and light-matter interaction strength via the Coulomb potential energy matrix element Vn, k . In the third line, S c−p n and S c−c n represent the higher-order many-body effects describing dephasing and screening from carrier-phonon and carrier-carrier scattering. The expression for S c−c n comes from the continuation of the Cluster 37 expansion giving the renormalization contributions to the 2nd level in Coulomb correlations. 23 For S c−p n , we use a non-perturbative, quantum-kinetic approach adapted from treating carriers and phonons as composite polarons. 24 The corresponding equations of motion for the QW transitions are derived similarly, giving where V |k−k ′ | is the QW Coulomb potential matrix element. Many-body Coulomb effects are important for the QD αH because they influence the carrier density dependences of shift and broadening of QD resonances. The shift causes semiconductor QDs to deviate from the ideal αH = 0 of an atom. 25 The broadening modifies the shift effects. 20 Inhomogeneous broadening from QD dimension and composition variations are treated by grouping the QDs according to the electronic structure. For the QD density in each group, we assume a Gaussian distribution so that where ωn is the renormalized frequency of the nth QD transition in the q th group and ωn ,0 and Δ inh are the central transition frequency and standard deviation. We assume carrier populations defined by where ε σ n,q and ε σ k are the electron and hole (σ = e, h) energies, kB is the Boltzmann constant, and T is the active region temperature. The chemical potential μσ is determined from the total electron and hole densities, Nσ = N σ ′ + ∑ n ∑ q n inh n,q f σ n,q + ∑ k f σ k , where N σ ′ with σ ′ = d or p for σ = e or h, respectively, is the doping density. Figure 1 illustrates the inhomogeneous broadening model with the example of spontaneous emission. A laser sample has QD populations spreading over an energy range according to n inh n,q [dashed curve, Fig. 1(a)]. Within the distribution, each group of QDs with similar transition energy emits a homogeneously broadened spectrum (grey curve), which is calculated using Eqs. (1)-(4) and the Kubo-Martin-Schwinger transformation. 26 A luminescence measurement produces the inhomogeneously broadened  2. (a) Intensity gain and (b) linewidth enhancement factor spectra for the undoped InAs QD structure with 20 meV inhomogeneous width. The carrier densities are 3 × 10 11 cm −2 , 5 × 10 11 cm −2 , and 7 × 10 11 cm −2 (black, red, and blue curves, respectively). In Fig. 1(b), the dots indicate the ground-state gain peak locations.
spectrum (solid black curve), which is computed by summing the different homogeneously broadened spectra, each weighed by n inh 1,q . Since its measurement is relatively straightforward, the spontaneous emission linewidth Δsp is often used to gauge QD inhomogeneity. A more direct measure of QD uniformity is the inhomogeneous width, Δ inh , which gives the standard deviation in level energies. Figure 1(b) shows the Δsp to Δ inh conversion, using the calculated homogeneously broadened (intrinsic) spectrum. This paper considers active regions ranging from state-of-the-art to typical, with 10 meV ≤ Δ inh ≤ 20 meV, corresponding to 24 meV ≤ Δsp ≤ 48 meV. 2728 Figure 2(a) shows the gain spectra for an undoped 7 nm In 0.15 Ga 0.85 As QW with 5 × 10 10 cm −2 InAs QD density and 20 meV inhomogeneous width. The resonances are from one ground-state and two excited-state transitions (n = 1, 2, and 3 with degeneracies 1, 2, and 3, respectively). The absorption edge at 1.2 eV is from the GaAs QW exciton. Figure 2(b) shows the corresponding αH spectra. The points indicate the ground-state gain peak values, which are all positive, with αH(ν pk ) ≈ 2 prior to the onset of excited-state gain. Figure 3(a) shows narrower and more distinct QD resonances when inhomogeneous broadening reduces to 14 meV and with carrier densities chosen to produce similar peak gains from the ground-state QD transition. The spectra show that the 4 × 10 11 cm −2 p-doped density leads to similar peak gains with lower carrier densities. Figure 3(b) depicts an interesting feature involving αH at the gain peak. The points indicate αH(ν pk ) changing from negative to positive, suggesting that with proper laser design, αH(ν pk ) can vanish.
To further explore the vanishing of αH, we repeated the calculations for broader ranges of carrier and p-doped densities. Figure 4(a) shows that with sufficient p-doped density, αH(ν pk ) = 0 exists at specific carrier densities. Even for curves not crossing αH(ν pk ) = 0, a minimum αH(ν pk ) exists. Assuming laser operation with the saturated gain clamped at the threshold value, the desired carrier density may be achieved by cavity design via where Γ is the confinement factor involving the waveguide and the QW embedding the QDs, L is the cavity length, α abs is the intracavity absorption, and R 1 and R 2 are the facet reflectivities. Figure 4(b) shows plots of the calculated carrier density dependence of peak gain for the different p-doped densities.
To verify the calculations, we fabricated three laser batches with undoped, 5 × 10 11 cm −2 and 10 12 cm −2 p-doped active regions. Each laser is epitaxially grown on the Si substrate, with a 1.25 mm long, uncoated-facet, Fabry-Perot cavity, and 3.5 μm wide ridge. The active region has 5 QD layers, where each layer is as in the   FIG. 3. (a) Intensity gain and (b) linewidth enhancement factor spectra for the InAs QD structure with 4 × 10 11 cm −2 p-doped density and 14 meV inhomogeneous width. The carrier densities are 2.5 × 10 11 cm −2 , 3.0 × 10 11 cm −2 , and 3.5 × 10 11 cm −2 (black, red, and blue curves, respectively). In Fig. 3(b), the dots indicate the gain peak locations and only the ground-state transition is plotted.

ARTICLE
scitation.org/journal/ app   FIG. 4. (a) Linewidth enhancement factor at the gain peak and (b) peak gain per QD layer vs carrier density for groundstate transition and 10 meV inhomogeneous broadening. The calculated curves are labeled by p-doped density in units of 10 11 cm −2 . The diamonds are from measurements for p-doped densities Np = 0, 5 × 10 11 cm −2 , and 10 12 cm −2 (black, blue-green, and brown, respectively).
calculations. The mode-sum method 29 is used to obtain αH(ν pk ) from amplified spontaneous emission spectra measured at different pulsed currents. The current pulse duration is 500 ns, giving a 1% duty cycle. Figure 1(b) is used to extract Δ inh from the measured spontaneous emission spectra. 30,14 The material gain is that from a 7 nm In 0.15 Ga 0.85 As QW embedded with a density of 5 × 10 10 cm −2 InAs QDs. Total gain region thickness is 5 × 7 nm inside a 285 nm thick GaAs waveguide. 27 Table I lists the device parameters for one laser in each batch. Note that we extracted Δ inh = 10 meV for all samples. This does not contradict earlier reports on photoluminescence spectral broadening with p-doping 31-33 because Δ inh refers to the standard deviation in the electronic structure. Moreover, the earlier results are for p-doping densities an order of magnitude higher than those considered here.
The experimental results are plotted as diamonds in Fig. 4. First, we used Fig. 4(b) to connect the experimental material gain to carrier density. Then, we plot in Fig. 4(a) the measured αH(ν pk ) vs the obtained carrier densities. The closeness of the experimental points to the respective theoretical curves indicates good theoretical and experimental agreement. Figure 4 also indicates αH(ν pk ) < 0, as observed in QD laser experiments in the form of absence of filamentation. 34 While eliminating filamentation is useful for high-power single-mode performance, 35 we chose instead to concentrate on datacom and telecom applications, where there are more opportunities for QD lasers to contribute. There, the concerns are linewidth, chirp, and feedback sensitivity so that minimizing the absolute value of the gain-peak linewidth enhancement factor, |αH(ν pk )|min, is more important.
Further parametric study suggests that lasing at |αH(ν pk )|min depends on having certain combinations of Δ inh , Np, and G th . Figure 5 summarizes the results, with Fig. 5(a) giving the necessary combinations and Fig. 5(b) showing the resulting |αH(ν pk )|min with those combinations. Only ground-state lasing is presented because most experimental efforts for improving performance are concentrated there.
Of interest is the subset of Δ inh , Np, and G th giving αH(ν pk ) = 0. Figure 5(b) indicates a sizable region reachable with present QD lasers. Within this region, QD non-uniformity may be compensated by p-doping. For Δ inh < 8 meV (Δsp < 20 meV), p-doping is even unnecessary. When Δ inh > 16 meV (Δsp > 38 meV), αH(ν pk ) = 0 is unachievable regardless of pdoped density. The challenge is fabricating a resonator accurately for a prescribed G th . Fortunately, there are combinations that are less sensitive to G th , e.g., with Δ inh ≈ 10 meV, which is state-of-the-art QD uniformity 27 and Np > 2 × 10 11 cm −2 ; Fig. 5(a) indicates a wide range of 80 cm −1 < G th < 100 cm −1 . For an active region consisting of 5 QD layers in a 0.3 μm thick, 1.25 mm long waveguide with facet reflectivities R 1 = 0.32 and R 2 = 0.90, Eq. (6) gives G th = 86 cm −1 , assuming α abs = 7.9 cm −1 . For 1.3 μm lasing wavelength, this translates to a resonator Q factor of 4800. Not shown in the plots but relevant to modulation bandwidth and threshold current are differential gain and threshold carrier density ranges of 2 × 10 −16 cm 2 < dG/dN 3D e < 4.3 × 10 −16 cm 2 and 1.5  × 10 11 cm −2 < Ne < 2.8 × 10 11 cm −2 for lasing with αH(ν pk ) = 0. At the low Ne limit, the threshold current density is J th ∼ 320 A/cm 2 , ]}, where Jsp = 3.4 A/cm 2 is obtained from the gain calculation. We also use defect (Shockley-Read-Hall) loss A = 1.2 × 10 9 s −1 , injection efficiency η = 0.6, and Auger coefficient C = 10 −28 cm 6 s −1 , either reported in the literature or extracted from characterizing our QD active regions. 30,36 In summary, QD lasers may be configured to operate with linewidth enhancement factor αH = 0. The beneficial effects are the reduced linewidth, chirp elimination, and reduced optical feedback sensitivity. That αH = 0 can occur at the gain peak simplifies device design and minimizes power consumption, which are important considerations in datacom and telecom applications. Many-body renormalizations and dephasing significantly contributed to QD αH. The parametric study, supported with experiment, provides the necessary combinations of the inhomogeneous linewidth, p-doped density, and threshold gain that are reachable by the present QD lasers.