Physical model for high indium content InGaN/GaN self-assembled quantum dot ridge-waveguide lasers emitting at red wavelengths (λ ~ 630 nm)

Guan-Lin Su; Thomas Frost; Pallab Bhattacharya; John M. Dallesasse

doi:10.1364/OE.23.012850

1. Introduction

Indium gallium nitride alloy (InGaN), having a direct band gap covering the entire visible spectrum, is an exceptional material for monolithically fabricating devices used in full-color displays, projectors and solid-state lighting [1–3]. In particular, to achieve the best color-rendering index (CRI) in display applications, quantum-well (QW) based laser diodes (LDs) of high brightness emitting in the blue and green spectral range have been realized [4,5]. Nevertheless, besides the polarization field along the [0001] axis, issues associated with indium incorporation, and producing laser mirrors via cleaving in wurtzite crystals, it is the large strain resulting from the lattice mismatch that hinders the development of InGaN QW-based LDs emitting at wavelengths longer than those in the green spectral region [6–8].

Self-assembled quantum dots (QDs) are a promising solution for addressing the limitations imposed by strain accumulation and pushing the emission wavelength of InGaN LDs towards the red portion of the visible spectrum [9, 10]. As a laser gain medium, QDs offer several advantages over QWs, such as a smaller effective volume that reduces threshold current, tighter carrier localization avoiding non-radiative recombination at defect centers, higher temperature stability, and most importantly reduced strain that decreases the piezoelectric field separating the electron and hole wavefunctions [11–14]. InGaN QD-based ridge-waveguide lasers emitting at blue, green and red wavelengths have been recently demonstrated [10, 15, 16]; however, a comprehensive theoretical understanding of the physics underlying the high indium content InGaN QD gain medium is still lacking. To properly design long-wavelength LDs adopting self-assembled QDs to realize blue, green and red emission on the same material platform it is crucial to build a detailed model with reasonable assumptions to extract and analyze the material properties and to explain and estimate the device performance from experimental data.

Despite significant effort put into the growth of InGaN QDs and the recent rapid development of theoretical models, the existing literature still focuses on the study of QDs with emission wavelengths in the blue region, and the assumptions used in those models are scattered [17–22]. Most importantly, reasonably good alignment between the theoretical and experimental work has been difficult to achieve to this point. We recently reported a model that fits the experimental data of an In₀_.₁₈Ga₀_.₈₂N/GaN QD-based ridge-waveguide laser emitting at 420 nm [27]. The spatial distribution of strain in the QD/matrix structure was calculated using linear elastic theory with a shrink-fit boundary condition at the material interface, and the strain-induced quantum-confined Stark effect (QCSE) was then properly reproduced. The electronic energy states were also predicted by solving the Schrödinger-Poisson equations self-consistently. Subsequent calculation showed that the emission wavelengths as well as the estimated threshold material gain agreed fairly well with the measured values. This study revealed that the linear elastic theory can not only legitimately approximate the strain distribution in the QD structure but also give the flexibility in mimicking the strain relaxation process in the Stranski-Krastanov growth mode. These insights pointed to several key questions that remain unanswered: (1) can this model be extended into high-indium content cases; (2) are there any physical mechanisms only existing in high-indium content QDs; and (3) what are the physical quantities that can be extracted from this model?

In this paper, we refine our previous truncated, hexagonal pyramid-shaped QD model and apply it to the study of an In₀_.₄Ga₀_.₆N/GaN QD-based ridge-waveguide laser emitting at 630 nm. Figure 1 details the modeling flow chart of this study, where the physical quantities output separately from the QD and cavity models are fed into the final rate equation model to generate a calculated P_out-I curve that is compared with and explains the measured data. The experimental I–V curve is used to provide a reference for estimating the active region temperature using the impact of thermal effects on device performance. Our theoretical results show very good agreements with the measured Hakki-Paoli gain spectrum and the P_out-I curve. The description of our model and information about the sample will be presented in the next section.

Fig. 1 Flow chart of our QD-based ridge-waveguide laser model.

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2. Structural model of InGaN/GaN QD structures and sample information

Though the growth, fabrication and characterization of the laser used in this study can be found elsewhere [16], several key experimental facts are briefly summarize here for completeness. The left part of Fig. 2 shows the schematic of the epitaxial structure of the sample. The laser structure is grown on a bulk HVPE-grown c-plane n-GaN substrate, followed by the growth of a GaN buffer layer. Additional lattice-matched In₀_.₁₈Al₀_.₈₂N layers are inserted between the Al₀_.₀₇Ga₀_.₉₃N claddings and In₀_.₀₂Ga₀_.₉₈N waveguide layers to enhance optical confinement in the visible range. Seven periods of In₀_.₄Ga₀_.₆N QDs, separated by 17-nm GaN barriers, are incorporated in the active region as the gain medium. The aerial density of the QD is estimated to be 3.9 × 10¹⁰ cm⁻² from the AFM image, and an average indium composition around 40% in the QD/wetting layer structure is read from the energy dispersive X-ray spectroscopy (EDX) measurement. The laser cavity is defined by coating two cleaved m-plane facets with SiO₂/TiO₂ distributed Bragg reflectors (DBRs) which provide reflectivities roughly 0.73 and 0.95 around the lasing wavelength of 634 nm. The ridge structure is formed by a shallow etch down to the Al₀_.₀₇Ga₀_.₉₃N/In₀_.₀₂Al₀_.₉₈N interface in order to minimize optical diffraction loss and parasitic leakage paths associated with the surface damages.

Fig. 2 Epitaxial layer structure (not to scale) of the In₀_.₄Ga₀_.₆N/GaN QD-based ridge-waveguide laser used in this study (left), together with an illustration of the side and top views of our truncated hexagonal pyramid QD model (right).

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The right part of Fig. 2 shows our QD model used for the active medium from different viewing angles. The shape of the QD is chosen to be a truncated hexagonal pyramid, in order to better reflect the hexagonal crystal symmetry in the c-plane of III-N semiconductors [21–24]. Due to the nature of the self-assembly process, the whole QD ensemble is characterized by a Gaussian distribution around the “dominant dots” [25]. The geometry parameters a, b, h and wetting layer thickness, as labeled in Fig. 2, are chosen to be 10 nm, 25 nm, 3.5 nm and 0.5 nm respectively for the dominant dots according to the AFM image. Choosing these geometry parameters is critical because it affects not only the energy levels of the electron and holes but the spatial distribution of the strain in the QD, which is the origin of QCSE in III-N semiconductor material systems.

The fundamental origin of the QCSE in III-N QDs is the piezoelectric effect arising from the strain distribution, which can be predicted using either linear elastic theory or the valance force field (VFF) model. In their detailed comparison between these two models for strained InGaAs QDs, Stier et al. found that one may require modified stiffness constants (C_ij’s) that are different from experimentally derived values in the VFF model [26]. We also expect such uncertainty will be further magnified by the less-symmetric wurtzite lattice. In contrast, even though the correctness of linear elastic theory breaks down on the atomic scale, the size of our InGaN QDs (tens of nanometers) allows reasonable use of this approach. As a result, because of its simplicity and flexibility, we apply linear elastic theory to the calculation of the strain distribution in our high indium content self-assembled InGaN/GaN QDs. Here we primarily discuss the physical significance and results of the linear elastic theory, as the mathematical formulation and its application to the truncated hexagonal pyramid QD can be found in our previous work [27].

Since the QD itself is a three dimensional structure formed by self-assembly process, the strain distribution is generally inhomogeneous and the amount of residual strain after relaxation is readily different from that in the QWs. The shrink-fit boundary condition provides an elegant way to address these issues [28]. At the material boundary, it is required that the stress field σ (r) to be continuous across the boundary and the discontinuity in the displacement field u(r) on both sides to be connected by the residual strain ε⁽⁰⁾. The residual strain ε⁽⁰⁾ is defined as

ε^{(0)} = s_{0} \frac{ε_{x x}^{(0)} + ε_{y y}^{(0)} + ε_{z z}^{(0)}}{3}

where the “relaxation factor” s₀ describes the fraction of the initial strain that remains after the QD formation, and

ε_{i i}^{(0)}

’s refer to the biaxial strain components resulting from the lattice mismatch between In₀_.₄Ga₀_.₆N and GaN [27].

Because the laser is biased with high injecting currents, the large amount of dissipated heat can potentially cause thermal expansion in the materials. In our refined model, such effect enters the strain calculation through

a_{III - N} (T) = a_{III - N} (298 K) [1 + α_{a} (T - 298)]

c_{III - N} (T) = c_{III - N} (298 K) [1 + α_{c} (T - 298)]

where α_a and α_c are the coefficients of thermal expansion (CTEs) of the c-plane wurtzite lattice in the in-plane and out-of-plane directions respectively [29–31].

The piezoelectric polarization is related to the strain components by $P_{pz} = \bar{\bar{e}} : ε$ [25, 32, 33]. In this work, while linear interpolation between GaN and InN is acceptable for the piezoelectric coefficients e_ij’s of InGaN alloy, the bowing effect needs to be taken into account when evaluating the spontaneous polarization P_sp [1]. The charge distribution produced by the polarization fields creates a polarization-induced electrostatic potential V_pol. (r) that bends the energy band. Note that the permittivity is anisotropic in III-N crystals [34].

Figure 3(a) and (b) show the spatial distribution of biaxial strain components ε_xx and ε_zz at the bottom of the wetting layer and at the side of the QD structure respectively. Though a small directionally-dependent numerical error may exist at the edge of the QD structure as can be seen in the plot of ε_xx, the spatial distribution of ε_yy basically has the same feature as ε_xx in the basal plane of wurtzite crystal. Also, it should be noted that the self assembly process renders the whole QD/matrix under a force-free condition, where the strain in the wetting layer must be of the opposite sign of the QD structure. Figure 3(c) and (d) depict the spatial distribution of the polarization charge density at the bottom of the wetting layer and at the side of the QD structure. Known as QCSE, the positive and negative charge accumulating at the top of the QD and the bottom of the wetting layer form an effective electric dipole that separates the electron and hole wavefunctions and decreases their recombination efficiency.

Fig. 3 Spatial distribution of biaxial strain (a) ε_xx and (b) ε_zz, and (c) and (d) polarization charge in the QD/matrix structure at different viewing angles. (a) and (c) are plotted in the x–y plane and (b) and (d) are plotted in the y–z plane. The QD region is specified by dashed black line. The temperature is assumed to be 25 °C.

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3. Effective mass Hamiltonian and electronic states in InGaN/GaN QD structures

Although the multi-band k · p method is usually applied to the calculation of electronic energy states in the QDs [20, 21, 23], it is worthwhile to consider whether the significantly increased computation resources required to effect such an analysis can be justified by the incremental improvement in the accuracy. Because the conduction-valance band coupling is weak in wide-band-gap semiconductor materials and because of the absence of momentum k in three-dimensional quantum confining structures the interaction between the hole states is minimized. In addition, as pointed out by Pryor, the computation error resulting from the uncertainty in QD size and shape is a dominant effect relative to that from the difference in the number of bands included in the k · p method [35]. Consequently, following the QD geometry revealed by AFM images, we adopt a single-band effective mass Hamiltonian for the calculation of electronic energy states in the InGaN/GaN QD structure [36].

However, to refine the effective mass entering our single-band Hamiltonian, a separate six-band k · p calculation is used to investigate the influence of strain [37]. We approximate the strain components ε_ij that can be unambiguously determined in bulk or QW structures by “averaged” strain components 〈ε_ij〉 in our QD structure:

〈 ε_{i j} 〉 = \frac{\int_{QD} ε_{i j} (r) d^{3} r}{QD volume}

Figure 4 shows the difference in the E(k) dispersion between strained and unstrained In₀_.₄Ga₀_.₆N material. Since the compressive strain pushes the crystal-field split-off hole (CH) band downward significantly, the nearly-200-meV energy separation between either heavy hole (HH) or light hole (LH) bands and CH band allows us to further simplify the calculation by only taking HH and LH bands into consideration. This argument is further supported by the fact that the heavy masses of both HH and LH bands generate a dense energy spectrum that keeps the valence quasi Fermi level away from the CH band. Table 1 emphasizes the difference in HH and LH zone-center effective masses, especially for the LH out-of-plane effective masses $m_{l h, z}^{*}$ , whose value nearly doubles after correction.

Fig. 4 Valance band structures of strained and unstrained In₀_.₄Ga₀_.₆N in the in-plane and out-of-plane directions. The difference in the curvatures at the zone center leads to different hole effective masses.

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Table 1. Heavy and light hole effective masses of strained and unstrained InGaN, in terms of free electron mass m₀.

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The energy states in either conduction or valance bands are calculated using single-band effective mass Hamiltonian:

- \frac{{\bar{h}}^{2}}{2} \nabla \cdot ({\bar{\bar{m}}}^{- 1} : \nabla ψ) + V (r) ψ = E ψ

where

\bar{\bar{m}}

is the anisotropic effective mass tensor. The potential operator V(r) consists of inherent band offsets in the heterostructure V₀(r), strain-induced deformation potential Vstrain(r), and polarization-induced electrostatic potential V_pol.(r). In this work, the inherent band offsets are evaluated using Anderson’s rule

Δ E_{g} (T) = Δ E_{c} (T) + Δ E_{v} (T)

Δ E_{v} (T) = \frac{900 meV}{Δ E_{g} (298 K)} Δ E_{g} (T)

and Varshni’s parameters to include temperature effects [38]:

E_{g} (T) = E_{g} (0) - \frac{α T^{2}}{β + T}

The valance band offset (VBO) at the InN/GaN interface is chosen to be 900 meV at room temperature [39]. However, due to the heavy effective masses of HH and LH bands and high indium content, the exact value of VBO does not seriously affect the calculation of hole states.

The “averaged strain components” are used to simplify the strain-induced deformation potential. It can be observed from [27] that the shear strains ε_ij (i ≠ j) are symmetric around the origin, which makes |〈ε_ij〉| ≪ |〈ε_ii〉|. Therefore, the strain-induced deformation potential can be written as

V_{strain} (r) = \sum_{i} v_{i} ε_{i i} (r)

where v_i’s are appropriate deformation potentials of either conduction or valance band [1].

The aforementioned potential term V(r) refers to the quantum barrier of an “empty dot,” where no additional charge is present to screen the polarization electric dipole. Upon injection, electrons and holes start to occupy states in the QD and another electric dipole is formed that repels the QCSE. Though in high indium content QDs the energy barrier is high and the number of bound states is large, the GaN matrix still needs to provide carriers to ensure a continuous change in the quasi Fermi level at a certain injection level n_2d:

n_{2 d} = 2 N_{2 D} \sum_{bound} \int_{- \infty}^{\infty} G_{c} (E - E_{i}) f_{c} (E, T) d E + L_{eff} \int_{E_{b}}^{\infty} ρ_{3 D} (E - E_{b}) f_{c} (E, T) d E

where N_2D is the aerial density of the QD, G_c(E – E_i) the Gaussian distribution describing the dot size inhomogeneity, f_c(E,T) the Fermi-Dirac distribution at temperature T, E_b the barrier height, and ρ_3D(E − E_b) the three-dimensional (3D) density of states of the GaN matrix. L_eff is the effective QD height describing the number of 3D carriers entering the QD:

L_{eff} = N_{2 D} \times QD volume

The valence band quasi Fermi level can be calculated using an equation similar to (8). Since physically there is no clear boundary between the QD and the wetting layer, they are essentially connected. Consequently, we solve (4) for the entire InGaN QD-wetting layer structure embedded in the GaN matrix and the resultant eigen-states, which include bound states in QD, wetting layer states, and even hybrids between them, are accounted as the bound states in (8).

The screening electric dipole formed by the electron-hole pairs is expressed as

ρ_{sc .} (r) = 2 q \sum_{bound} | ψ_{h, j} (r) |^{2} [1 - f_{v} (E_{h, j}, T)] - 2 q \sum_{bound} | ψ_{e, i} (r) |^{2} f_{c} (E_{e, i}, T)

which adds a screening potential V_sc.(r) to V(r) in (4). In this work, we solve Schrödinger and Poisson’s equations self-consistently until the C1-HH1 transition wavelength stabilizes. Figure 5(a) shows a significant screening effect at a high injection level n_2d = 37N_2D. The left panel plots both the conduction and valence band edges along the c-axis (x = y = 0) and the right panel plots the conduction band edge at the top of the QD and valence band edge at the bottom of the wetting layer (in both cases y = 0) in order to show the transverse potential profiles at locations where the electron and hole wavefucntions reside. The reduced polarization dipole allows larger electron-hole wavefunction overlaps thus a higher optical gain as shown in Fig. 5(b). The overlaps increases almost linearly initially but show a saturation behavior at high carrier densities. This can be attributed to effect of the larger density of states in the surrounding GaN matrix, which effectively suppresses the growing rate of quasi Fermi levels.

Fig. 5 (a) Band edges before (dashed black) and after (solid red) including screening effect at different viewing angles. The injection level is n_2d = 37N_2D and T = 120 °C. (b) The wavefunction overlap of the C1-HH1 and C2-HH2 transitions as functions of injection level.

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At first glance the QD structure seems to confine the carriers in the same manner as the QW does in the z direction and it barely provide any extra benefit in enhancing electron-hole wavefunction overlap. However, it is the three-dimensional structure of the QD that actually allows the incorporation of a high percentage of indium and strain relaxation to decrease the piezoelectric polarization field, and prevents the carriers from diffusing to defects in the x–y plane. These features do not present in QW structures in general.

4. Optical gain in InGaN/GaN QD structures

Since we have neglected the influence of CH states in the calculation of energy levels, the optical gain is mostly contributed by the TE components (the electric field lies in the epitaxial plane.) This fact, occurring in InGaN/GaN QD-based lasers emitting at all visible wavelengths, has been confirmed by measuring the polarization of the light output [15,16].

The optical material gain in the InGaN/GaN QD structure can be derived from the calculated electronic states:

g^{TE} (\bar{h} ω) = \frac{2 π q^{2} N_{2 D} | 〈 i S | p_{x} | X 〉 |^{2}}{n_{r, t} ε_{0} c_{0} ω m_{0}^{2} L_{D}} \sum_{bound} P_{i j} (\bar{h} ω)

where |〈iS|p_x|X〉| is the optical matrix element [37], n_r,t the in-plane index of refraction, L_D the sum of the dot height and wetting layer thickness and

P_{i j} (\bar{h} ω)

a function that accounts for the optical transition probability:

P_{i j} (\bar{h} ω) = | I_{h, j}^{e, i} |^{2} | w_{h, j} |^{2} \int_{- \infty}^{\infty} G_{c v} (E - E_{h, j}^{e, i}) [f_{c} (E_{e, i}, T) - f_{v} (E_{h, j}, T)] L (E - \bar{h} ω) d E

where

| I_{h, j}^{e, i} |

is the wavefunction overlap of i-th electron to j-th hole state transition, |w_h,j| the in-plane component of the hole wavefunction,

G_{c v} (E - E_{h, j}^{e, i})

the Gaussian distribution for dot size fluctuation,

L (E - \bar{h} ω)

the Lorentzian distribution describing the homogeneous broadening around the photon energy

\bar{h} ω

.

While the inhomogeneous broadening in InGaN QWs is generally caused by fluctuations in both alloy composition and well width, and its value ranges from 8 to 48 meV [40, 41], the inhomogeneous broadening in QD systems is mainly due to the variation in dot size. However, because GaN-based materials generally have larger effective masses and higher band offsets than traditional III-V semiconductors, we argue that the effect of size inhomogeneity tends to have less impact on the electron and hole energy states in the QD structures. Given inhomogeneous broadening values generally around 30 meV in InAs QDs [43, 44], an inhomogeneous broadening FWHM in our InGaN/GaN QD system is assumed to be 15 meV to reflect the less variation in the energy spectrum.

The modal gain for a certain cavity mode is obtained by multiplying (11) with the modal confinement factor Γ, which is defined as the fraction of the modal field overlapping with the QD active material. Figure 6 shows a comparison between our theoretical calculation and the measured Hakki-Paoli modal gain at threshold. The model agrees with the experimental data fairly well in the high energy region; however, the measured modal gain is slightly higher than the calculated results in the low energy region. This is probably due to either the indium content fluctuation in the QD from layer to layer, or the distribution of the QD size inhomogeneity is other than Gaussian-like. The homogeneous broadening, which is half of the FWHM of the Lorentzian function in (12), is found to be around 58 meV, which is considerably larger than that in InGaAs QD active materials [44–46]. Limited data have been reported on the direct measurement of homogeneous broadening in InGaN alloys, nevertheless it can be inferred that the temperature sensitivity, or more specifically the occurrence of phonon scattering events, is more significant in such material. In fact, the large optical phonon energy in the GaN material system (∼90 meV) may facilitate carrier scattering between multiple energy states and render a wider gain spectrum. Under such high injection levels, an active region temperature of around 120 °C at threshold is estimated by adjusting the fitting parameter T in our model. Though being high, it renders a derived thermal impedance of 43 °C/W, which falls between a value of 30 °C/W in a blue QW-based ridge-waveguide laser grown on a GaN substrate and a value of 60 °C/W in another blue laser of the same structure but on a sapphire substrate [47]. The high indium content in our laser structure is presumably the factor that increases the thermal impedance. The high injection level required to reach threshold is because of the need for charge screening to cancel the polarization electric dipole and to regain the wavefunction overlap of the optical transition. Moreover, a differential gain of 4.4 × 10⁻¹⁷ cm² at 634 nm is derived from our model, which is between a value of 3.8×10⁻¹⁷ cm² obtained from measuring the threshold current densities in LDs with different cavity lengths and a value of 5.3 × 10⁻¹⁷ cm² measured from the small-signal modulation response [16, 48]. The fitting parameters of our InGaN/GaN self-assembled QD model can be found in Table 2 in the Appendix.

Fig. 6 Measured (squares) and calculated (solid lines) modal gain spectra of the In-GaN/GaN QD-based ridge-waveguide laser. The active region temperature increases from 65 °C up to around 120 °C at threshold.

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Table 2. Material parameters for the In₀_.₄Ga₀_.₆N/GaN self-assembled QD model.

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5. Device-level modeling of InGaN/GaN QD-based ridge-waveguide lasers

We further incorporate our QD gain model into the study of our In₀_.₄Ga₀_.₆N/GaN QD-based ridge-waveguide lasers. The device-level study consists of a current-voltage (I–V) simulation based upon a combined diode-resistor model and a light-current (P_out-I) curve simulation based upon phenomenological rate equations. Several important quantities, including diode ideality factor, current leakage paths, external series resistance and recombination lifetimes can be extracted from our model.

Figure 7 shows the power flow profile of the fundamental mode P_z(x,y) in the cross section of the ridge-waveguide structure, and a visualization of the combined diode-resistor model. In the model, the diode is characterized by reverse saturation current I₀, ideality factor n_d, and diode turn-on voltage V_d. The two resistors R_S and R_P account for the series resistance external to the LD and the shunt current leakage path through the device sidewalls, respectively. A transcendental equation relating the measured current through the device I and voltage across the device V is given by such combination:

I = \frac{V - I R_{S}}{R_{P}} + I_{0} {\exp [\frac{q (V - I R_{S} - V_{d})}{n_{d} k_{B} T}] - 1}

Fig. 7 Power flow profile P_z(x,y) of the fundamental mode together with the combined diode-resistor model.

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where k_B is Boltzmann constant. The five parameters I₀, n_d, V_d, R_S and R_P can be extracted from the experimental I-V curve in different intervals: the turn-on behavior gives information about the diode ideality factor and turn-on voltage, and the slopes of the I-V curve at low and high voltages determine the shunt and series resistors. The device current at low-voltages is actually controlled by either the shunt leakage or reverse saturation current, depending on which is larger. Parameters in the diode model are listed in Table 3 in the Appendix.

Table 3. Fitting parameters used in I-V modeling.

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The light-current characteristics are described by two phenomenological rate equations governing the time variation of carrier and photon densities. Since the 3D carrier densities cannot be rigorously defined in QD systems due to the complex dot and wetting layer geometry, both carrier and photon are expressed in aerial densities in our model:

\frac{d n_{2 d}}{d t} = η_{i} \frac{J}{q} - \frac{n_{2 d}}{τ_{nr}} - \frac{n_{2 d}}{τ_{r}} - g^{TE} (n_{2 d}) v_{g} s_{2 d}

\frac{d s_{2 d}}{d t} = Γ g^{TE} (n_{2 d}) v_{g} s_{2 d} - \frac{s_{2 d}}{τ_{ph}} + Γ β_{sp} \frac{n_{2 d}}{τ_{r}}

where η_i is the injection efficiency describing the fraction of current density entering the QD active layers, τ_nr and τ_r the non-radiative and radiative recombination lifetime, g^TE(n_2d) the material gain as defined in (11), v_g the group velocity of light, τ_ph the cavity photon lifetime and β_sp the spontaneous emission coupling factor. Note that the carrier recombination lifetimes and material gain are implicit functions of active region temperature, which is determined by the thermal impedance and experimental I-V and P_out-I curves:

T = T_{sub} + Z_{th} (I V - P_{out})

and the current density J in (14a) refers to the current passing through the contact W × L in dimension:

J = \frac{1}{W L} (I - \frac{V - I R_{s}}{R_{P}})

The radiative recombination lifetime is calculated using our QD model and the non-radiative recombination lifetime is assumed to be solely related to carrier loss at the defect centers and characterized by an activation energy ΔE_a:

\frac{1}{τ_{nr}} = \frac{1}{τ_{0}} \exp (- \frac{Δ E_{a}}{k_{B} T})

The light output P_out is calculated using both spontaneous and stimulated emission power:

P_{out} = (β \frac{n_{2 d}}{τ_{r}} + \frac{α_{m} v_{g}}{Γ} s_{2 d}) W L \bar{h} ω

where β is the collection efficiency of the spontaneous emission light.

Figure 8(a) shows a comparison between the P_out-I-V curves calculated using our model (13)–(15) and the experimental curves measured from a 5 μm × 1 mm, continuous-wave (CW) biased laser with a substrate temperature of 15 °C. The fitting parameters are given in Table 4 in the Appendix. Figure 8(b) shows the effect of active region temperature on the modal gain spectrum calculated using (11) while all the other input parameters are kept the same. The great amount of dissipated Ohmic power, which reaches almost 6 W particularly at the maximum measured output power, raises the active region temperature to around 276 °C assuming a thermal impedance of 43 °C/W. Such high temperature was also reported in a simulation of high-power GaN LDs emitting at 400 nm [49]. Even though the effective density of states is not a function of active region temperature in QD systems, the Fermi-Dirac distribution is significantly affected by the large temperature change and the charge screening effect is also degraded. A very high diode ideality factor n_d = 38.5 found by our combined diode-resistor model suggests a large resistance causing the self-heating in the LD. In fact the unoptimized p-doping in GaN material system is mainly responsible for the low wall-plug efficiency, which is only slight above 0.1 %. To maintain laser operation, more carriers are needed to bring the modal gain back to balance the total loss. The high injection level also strengthens the screening effect, increases the peak gain and blue-shifts the peak wavelength. The lasing wavelength at peak output power is estimated to be around 649 nm, indicating a 15-nm red shift. Since the SiO₂/TiO₂ DBR has a broad bandwidth, we expect a less significant change in the mirror loss within the calculated red shift range. However, the total loss seems to be dominated by the intrinsic loss, which is higher than 30 cm⁻¹. This is probably due to both the free-carrier absorption process at high injection levels and the scattering of light at sidewall interfaces or due to the complex QD structure itself.

Fig. 8 (a) Calculated P_out-I-V curves and the comparison with experimental data of a 5 μm × 1 mm, CW-biased laser at a substrate temperature of 15 °C. (b) Modal gain spectra as functions of active region temperature under an injection level of n_2d = 40N_2D. An active region as high as 276 °C substantially degrades the peak gains and red shifts the spectra.

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Table 4. Fitting parameters used in P_out-I modeling. Symbol Value

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Our model also reveals the details of device performance and suggests possible ways to obtain better devices. The shunt resistance, representing parasitic leakage through the sidewall imperfections left by the dry etching process, is found to be around 6 kΩ, meaning the shunt leakage component is much less significant than the current entering the LD. Indeed, the shunt leakage remains as low as a few mA over the entire operating range. The shallow etch only going down to the waveguide layer instead of passing through the active region explains the low leakage current. Figure 9 plots the injection rate, non-radiative carrier recombination rate, radiative recombination rate and stimulated emission rate in the modeled device. Among the three competing rates, it can be seen that non-radiative recombination is actually comparable to the stimulated emission process even above the threshold. This result implies that though QDs may quickly catch the carriers entering the active layers and prevent them from undergoing non-radiative recombination process, the not completely screened polarization-induced electric field along the c-axis still limits efficient radiative recombination process even at high injection levels. The 3D carriers supplied by the GaN matrix can also diffuse and recombine at defect centers. Above threshold, the high active region temperature not only facilitates the non-radiative recombination process but degrades the material gain as indicated in the Fig. 8(b). It should be noted that the self-heating effect also causes significant carrier unpinning due to the reduction of the material gain. The excess carrier density required to produce the lasing action, on the other hand, contributes to heat dissipation and further raises the active region temperature. In spite of the fact that the band offset at the In₀_.₄Ga₀_.₆N/GaN heterostructure is relatively high as shown in Fig. 5(a), hot carriers with sufficient kinetic energy could surpass the barriers and recombine elsewhere. This is a possible reason for the low value of measured injection efficient η_i = 0.3. As a result, the optimization of the p-GaN contact and the incorporation of heat sinking structures are of top priority in designing better high indium content devices, as the thermal cycle is inevitable in every laser. The remedy to thermal issues can not only avoid the high carrier density to reach threshold but also reduce the free-carrier absorption, which partly accounts for the high intrinsic loss in this specific device.

Fig. 9 Simulated injection (black), non-radiative recombination (red) and spontaneous emission (blue) rates, together with stimulated emission (magenta) rate in the device modeled in Fig. 8.

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6. Summary

We have developed a detailed physical model for high indium content QD-based ridge-waveguide lasers emitting at red wavelengths. The QD model uses linear elastic theory for strain calculations and a single-band effective mass Hamiltonian for electronic energy state calculations to produce gain spectra that show good agreement with the measured Hakki-Paoli gain. Combined with our previous work, this provides strong evidence that these two major approximations can precisely describe the physical mechanisms in In_xGa₁₋_xN/GaN QD active materials having emission wavelengths covering the whole visible spectrum (x = 0.18, blue, to x = 0.40, red). A derived differential gain of 4.4 × 10⁻¹⁷ cm² falls between the values 3.8 × 10⁻¹⁷ cm² and 5.3 × 10⁻¹⁷ cm², obtained from two independent measurements on LDs incorporating the same kind of QD. An estimated thermal impedance of 43 °C/W is derived for our high indium content LD grown on a GaN substrate, while values of 30 °C/W and 60 °C/W are reported for low indium content LD grown on GaN and sapphire substrates respectively. Our device-level simulation, consisting of a combined diode-resistor model for the I–V curve and phenomenological laser rate equations for the P_out-I curve fits the experimental data fairly well. The model reveals very good sidewall passivation but weak diode electrical properties, as well as breaks down several key competing rates in the LD. The thermal performance is the key factor limiting good device performance. Possible improvements, such as optimizing p-GaN contacts and facilitating heat dissipation, are suggested based upon our simulation results.

Appendix

Acknowledgments

This work is supported by the National Science Foundation (MRSEC program), under grant number DMR-1120923.

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	$m_{h h, t}^{*}$	$m_{l h, t}^{*}$	$m_{h h, z}^{*}$	$m_{l h, z}^{*}$
Strained	0.209	0.217	1.742	1.720
unstrained	0.213	0.228	1.742	0.884

	In₀ _. ₄Ga₀ _. ₆N	GaN
Lattice constants [1]
a(T = 298 K) (Å)	3.331	3.189
c(T = 298 K) (Å)	5.392	5.185
CTEs [29–31]
α_a (10⁻⁶/K)	3.90	3.95
α_c(10⁻⁶/K)	2.78	2.80
Band parameters [1]
E_g(T = 0 K) (eV)	1.91 ^a	3.51
α (meV/K)	0.634	0.909
β (K)	748	830
Δ_cr (meV)	22	10
Δ_so (meV)	12.2	17.0
Stiffness constants [1]
C₁₁ (GPa)	323.2	390
C₁₂ (GPa)	133.0	145
C₁₃ (GPa)	100.4	106
C₃₃ (GPa)	328.4	398
C₄₄ (GPa)	48	105
Dielectric constants [34]
ε_z (ε₀)	12	10.4
ε_t (ε₀)	10.94	9.5
Relaxation factor ^*
s ₀	0.3	0.3
Deformation potentials [1]
a_z (eV)	−4.34	−4.9
a_t(eV)	−8.18	−11.3
D₁ (eV)	−3.7	−3.7
D₂ (eV)	4.5	4.5
D₃ (eV)	8.2	8.2
D₄ (eV)	−4.1	−4.1
Mass parameters [1]
$m_{e}^{∥} (m_{0})$	0.148	0.2
$m_{e}^{⊥} (m_{0})$	0.148	0.2
A ₁	−7.61	−7.21
A ₂	−0.54	−0.44
a ₃	7.04	6.68
A ₄	−4.17	−3.46
A ₅	−4.08	−3.40
A ₆	−5.32	−4.90
Polarization [32,33]
P_sp (C/m²)	−0.029 ^a	−0.034
e₁₅ (C/m²)	−0.301	−0.326
e₃₁ (C/m²)	−0.522	−0.490
e₃₃ (C/m²)	0.826	0.730

	Symbol	Value
Reverse saturation current	I ₀	42 μA
Ideality factor	n_d	38.5
Turn-on voltage	V ₀	4.3 V
Shunt resistor (leakage path)	R _P	6 kΩ
Series resistor (external resistance)	R _S	2 Ω

	Symbol	Value
Injection efficiency	η_i	0.3
Non-radiative recombination lifetime parameter	τ ₀	1.2 ns
Non-radiative recombination activation energy	ΔE_a	30 meV
Radiative recombination lifetime	τ _r	calculated
Group index	n_g	3.08
Optical confinement factor	Γ	0.148
Spontaneous emission coupling factor	β _sp	0.00404
Spontaneous emission collection efficiency	β	0.016

	$m_{h h, t}^{*}$	$m_{l h, t}^{*}$	$m_{h h, z}^{*}$	$m_{l h, z}^{*}$
Strained	0.209	0.217	1.742	1.720
unstrained	0.213	0.228	1.742	0.884

Physical model for high indium content InGaN/GaN self-assembled quantum dot ridge-waveguide lasers emitting at red wavelengths (λ ~ 630 nm)

Abstract

1. Introduction

2. Structural model of InGaN/GaN QD structures and sample information

3. Effective mass Hamiltonian and electronic states in InGaN/GaN QD structures

4. Optical gain in InGaN/GaN QD structures

5. Device-level modeling of InGaN/GaN QD-based ridge-waveguide lasers

6. Summary

Appendix

Acknowledgments

References and links

Cited By

Figures (9)

Tables (4)

Equations (21)

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