Self-consistent Rate Equation Research Articles

Island density scaling in reversible submonolayer growth with attachment barriers.

Island nucleation and growth play an important role in thin-film growth. One quantity of particular interest is the exponent χ, which describes the dependence of the saturation island density N_{sat}∼(D_{h}/F)^{-χ} on the ratio D_{h}/F of the monomer hopping rate D_{h} to the deposition rate F. While standard rate equation(RE) theory predicts that χ=i/(i+2) (where i is the critical island size), more recently it has been predicted that in the presence of a strong barrier to the attachment of monomers to islands, a significantly larger value χ=2i/(i+3) may be observed. While this prediction has recently been tested using kinetic Monte Carlo simulations for the case of irreversible growth corresponding to i=1, it has not been tested for the case of reversible island growth corresponding to i>1. Here we present a mean-field self-consistent RE method which we have used to study the dependence of the effective value of χ on D_{h}/F and barrier-strength for i=1,2,3, and 6. Both the no-nucleation-barrier case in which there exists a barrier for monomers to attach to islands larger than the critical island size (but not to smaller islands) and the nucleation-barrier case in which there is a barrier for monomers to attach to islands of all sizes are studied. In all cases, we find that the existence of attachment barriers significantly increases the effective value of χ for a given barrier strength. In addition, for i=1 we find good agreement between our extrapolated asymptotic value of χ and the theoretical strong-barrier prediction both with and without a nucleation barrier. In contrast, for i>1 the value of χ is significantly larger in the presence of a nucleation barrier than in its absence. In particular, while an asymptotic analysis of our results for i>1 also leads to excellent agreement with the strong barrier prediction in the presence of a nucleation barrier, in the absence of a nucleation barrier the asymptotic values are significantly lower.

Early stage nucleation mechanism for SiC(0001) surface epitaxial growth

Due to the complexity of the binary component system and the difficulty of tracing individual particles in experiments, it is highly desirable to develop simulation tools and models to further reveal the microscopic nucleation behavior of Si and C atoms. In this article, self-consistent rate equations (SCRE) theory combined with the Kinetic Monte Carlo (KMC) model are used to study the nucleation mechanism in the early stage of SiC(0001) surface epitaxial growth under a constant deposition flux. A set of rate equations describing the time evolution of the density of monomers, dimers, and islands are established. By introducing the effective absorption length, the rate equations can be solved self-consistently. In the KMC model, a set of crystal lattices of SiC, including the coordinates of individual Si and C particles and the bond indication, are established. In this model, deposition, adatom diffusion, attachment to and detachment from the clusters of Si or C, and their diffusion along the edge of clusters are considered, and the Hoshen–Kopelman algorithm is implemented to identify and label the clusters. The results show that the time evolution of the density of monomers, dimers, and islands and their dependences on the deposition flux, which are obtained from the SCRE theory, are consistent with the results of the KMC model. The nucleation rate of the dimer increases with the increase in the deposition flux until the number of islands becomes saturated, which leads to a higher density of stable clusters at higher flux. Two regimes governing the absorption length of the monomer are presented. First, before the appearance of the nucleus, the absorption length of the monomer is mainly dominated by monomer density and the diffusion coefficient of adatom. Second, with the growth process, the total capture rate of stable islands gradually plays a dominant role, which is responsible for the further reduction in absorption length and the equal absorption length of Si and C.

Effects of short-range attachment barriers on submonolayer growth

A variety of effects can lead to short-range attachment barriers in thin-film growth. While it has been predicted that the exponent χ which describes the dependence of the island density N on deposition rate F in the submonolayer regime (where N∼Fχ) crosses over from the diffusion-limited value i/(i+2) (where i is the critical island size) in the absence of an attachment barrier to the attachment-limited value 2i/(i+3) for a strong attachment barrier, this prediction has not been confirmed. Furthermore, the dependence of the effective value of χ on the barrier strength and ratio R=D/F (where D is the monomer hopping rate) has not been studied. Here we consider the effects of attachment barriers in irreversible growth (i=1) for both the case of a barrier to island nucleation and attachment as well as that of an island attachment barrier but no nucleation barrier. Our results indicate that in both cases the effective value of χ increases with increasing R to a maximum value χmax(Rmax) which depends on barrier strength before decreasing very slowly toward the diffusion-limited value. In addition, both χmax and Rmax increase as the barrier strength increases. The results of self-consistent rate-equation calculations are also presented and good agreement is found with our simulations. We also present a scaling analysis for the dependence of Rmax on the barrier strength for arbitrary critical island-size i and good agreement is found with our simulation results for the case in which there is both a nucleation barrier and a barrier to island attachment.

Application of self-consistent rate equations approach for SiC (0001) surface epitaxial growth

A self-consistent rate equations method is applied to investigate the nucleation process in the early stage of SiC (0001) surface epitaxial growth. Si-Si, C–C and Si-C are considered to be unstable dimers and the dissociation rates are determined by their binding energies, substrate temperature and the geometry of adsorption sites of Si and C on the surface. The time evolution of monomer, dimer and stable cluster density as well as their dependences on temperature are in good agreement with KMC simulation results. A transition zone in the profiles of stable cluster density for different temperatures is observed, and this is caused by the effect of temperature on net nucleation rate of dimer, which will determine the number of cores for stable cluster. Si-Si dimer can hardly exist due to its much smaller binding energy. The formation of Si-C is prior to the formation of C–C dimer, which indicates that the stable clusters are mainly formed by the Si-C nucleation until the density of C–C is higher than that of Si-C dimer. Furthermore, for higher coverage, the density of dimer decreases with the increase of temperature, leading to the smaller saturation value of stable clusters density at higher temperature.

GaInAs/AlInAs quantum cascade laser design based on optimized second harmonic generation

In this work, we present an innovative procedure for the design and optimization of GaInAs/AlInAs quantum cascade laser (QCL) structures based on the use of the genetic algorithm. The purpose of the algorithm is to determine the set of design parameters that would enable the maximization of the second order nonlinear susceptibility, thus facilitating significant optical nonlinearities to take place. In our optimization model, we start from the existing design in which the active region consists of two coupled InGaAs quantum wells separated by an AlInAs barrier, and the active region levels form double resonant nonlinear cascades. Upon obtaining the optimized structure and evaluating its energies and wave functions, the output characteristics are calculated by applying the full self-consistent rate equation modeling of the electron transport in a periodic QCL structure. The results of the calculations predict a noticeable improvement of targeted properties of the optimized design, while at the same time the original design calculations show excellent agreement with experimental results. The described procedure is applicable to various active region designs and can be used for other wavelength ranges.

Modeling techniques for quantum cascade lasers

Quantum cascade lasers are unipolar semiconductor lasers covering a wide range of the infrared and terahertz spectrum. Lasing action is achieved by using optical intersubband transitions between quantized states in specifically designed multiple-quantum-well heterostructures. A systematic improvement of quantum cascade lasers with respect to operating temperature, efficiency, and spectral range requires detailed modeling of the underlying physical processes in these structures. Moreover, the quantum cascade laser constitutes a versatile model device for the development and improvement of simulation techniques in nano- and optoelectronics. This review provides a comprehensive survey and discussion of the modeling techniques used for the simulation of quantum cascade lasers. The main focus is on the modeling of carrier transport in the nanostructured gain medium, while the simulation of the optical cavity is covered at a more basic level. Specifically, the transfer matrix and finite difference methods for solving the one-dimensional Schrödinger equation and Schrödinger-Poisson system are discussed, providing the quantized states in the multiple-quantum-well active region. The modeling of the optical cavity is covered with a focus on basic waveguide resonator structures. Furthermore, various carrier transport simulation methods are discussed, ranging from basic empirical approaches to advanced self-consistent techniques. The methods include empirical rate equation and related Maxwell-Bloch equation approaches, self-consistent rate equation and ensemble Monte Carlo methods, as well as quantum transport approaches, in particular the density matrix and non-equilibrium Green's function formalism. The derived scattering rates and self-energies are generally valid for n-type devices based on one-dimensional quantum confinement, such as quantum well structures.

Modeling pulsed-laser melting of embedded semiconductor nanoparticles

Pulsed-laser melting (PLM) is commonly used to achieve a fast quench rate in both thin films and nanoparticles. A model for the size evolution during PLM of nanoparticles confined in a transparent matrix, such as those created by ion-beam synthesis, is presented. A self-consistent mean-field rate equations approach that has been used successfully to model ion beam synthesis of germanium nanoparticles in silica is extended to include the PLM process. The PLM model includes classical optical absorption, multiscale heat transport by both analytical and finite difference methods, and melting kinetics for confined nanoparticles. The treatment of nucleation and coarsening behavior developed for the ion beam synthesis model is modified to allow for a nonuniform temperature gradient and for interacting liquid and solid particles with different properties. The model allows prediction of the particle size distribution after PLM under various laser fluences, starting from any particle size distribution including as-implanted or annealed simulated samples. A route for narrowing the size distribution of embedded nanoparticles is suggested, with simulated distribution widths as low as 15% of the average size.

Rate-equation approach to irreversible island growth with cluster diffusion

A self-consistent rate-equation (RE) approach to irreversible island growth and nucleation is presented which takes into account cluster mobility. As a first application, we consider the irreversible growth of compact islands on a two-dimensional surface in the presence of monomer deposition (with rate F) and monomer diffusion (with rate D(1)) while the mobility of an island of size s is assumed to satisfy D(s)=D(1)s(-μ) where μ>0. Results are obtained for the dependence of the island-density and island-size distribution (ISD) on the parameters D(1)/F, μ, and coverage θ. For all values of μ, we find excellent agreement between our self-consistent RE results and simulation results for the island and monomer densities, up to and even somewhat beyond the coverage corresponding to the peak island density. We also find good agreement between our self-consistent RE and simulation results for the portion of the ISD corresponding to island sizes less than the average island-size S. However, for larger island sizes the effects of correlations become important and as a result the agreement is not as good. Using our self-consistent RE approach we also demonstrate that the discrepancies between simulations and recent mean-field predictions for the exponent τ(μ) describing the power-law size dependence of the ISD for μ<1 can be explained almost entirely by geometric effects. Our results are also compared with those obtained using a simpler mean-field Smoluchowski approach. In general, we find that, except for the case μ=1/2 (for which the island and monomer densities are reasonably well predicted), such an approach leads to results which are in poor agreement with the simulations.

Dependence of Threshold Current Density on the Waveguide Ridge Width in Quantum-Cascade Lasers

We make a systematic study to find the dependence of the threshold current density of a mid-infrared quantum-cascade laser on the waveguide ridge width. The optical and thermal parameters of the waveguide are calculated with a finite-element method, and gain is found from a self-consistent rate equation model with energy-balance. The results show the existence of an optimal ridge width that minimizes the threshold current density of the laser. The influence of the interface roughness parameters on laser performance is also discussed.

Scaling of the island density and island-size distribution in irreversible submonolayer growth of three-dimensional islands

Motivated by observations of three-dimensional (3D) island formation in a variety of experiments, we have carried out kinetic Monte Carlo simulations of a simplified model of 3D island growth with critical island size $i=1$ (irreversible growth). Of particular interest are the exponents ${y}_{1}$ and ${y}_{1}^{\ensuremath{'}}$ describing the dependence of the submonolayer island density in the precoalescence regime on dose and coverage respectively, as well as the exponent ${\ensuremath{\chi}}_{1}^{\ensuremath{'}}$ describing the dependence of the peak island density on the ratio $D/F$, where $D$ is the monomer hopping rate and $F$ is the (per site) deposition rate. We find that the values of ${y}_{1}$ and ${y}_{1}^{\ensuremath{'}}$ (e.g., ${y}_{1}\ensuremath{\simeq}0.24,\text{ }{y}_{1}^{\ensuremath{'}}\ensuremath{\simeq}0.32$) are significantly lower than the standard rate-equation (RE) predictions $({y}_{1}=1/3,\text{ }{y}_{1}^{\ensuremath{'}}=3/7\ensuremath{\simeq}0.43)$. This may be explained by the fact that, in contrast to the standard RE assumption of size-independent capture numbers, for 3D islands the island radius increases with the number of atoms in an island. Accordingly, the average capture number increases with coverage and as a result, the coverage dependence of the island density for 3D islands is intermediate between that for two-dimensional (2D) islands $({y}_{1}={y}_{1}^{\ensuremath{'}}=0)$ and the standard RE prediction. As a result, the measured value of ${\ensuremath{\chi}}_{1}^{\ensuremath{'}}\text{ }({\ensuremath{\chi}}_{1}^{\ensuremath{'}}\ensuremath{\simeq}0.30)$ is slightly larger than the standard RE prediction $({\ensuremath{\chi}}_{1}^{\ensuremath{'}}=2/7\ensuremath{\simeq}0.29)$ but still lower than the value $({\ensuremath{\chi}}_{1}=1/3)$ for irreversible 2D island growth. For comparison with our simulations we also present self-consistent RE results for the island and monomer densities as a function of coverage and excellent agreement is obtained without any adjustable parameters. Results for the scaled island-size distribution (ISD) for 2D and 3D irreversible island growth are also presented. Somewhat surprisingly, we find that there is very little difference between the scaled ISD for 3D islands and that for 2D islands. In addition to our simulation results, the scaling behavior of the 3D island density for general critical island size $i$ is also discussed. Our results suggest that while the exponent ${\ensuremath{\chi}}_{i}^{\ensuremath{'}}$ describing the flux dependence of the peak island density is the most accurate indicator, the exponent ${y}_{i}$ may also be useful to estimate the critical island size $i$ in submonolayer 3D growth.

Carrier Relaxation and Modulation Response of 1.3-$\mu$m InAs–GaAs Quantum Dot Lasers

A self-consistent rate equation model is presented to investigate the influence of carrier relaxation on the modulation response of 1.3 mum InAs-GaAs quantum dot lasers. In this model, the carrier dynamics in GaAs barrier, relaxation pathways, and the phonon- and Auger-assisted relaxation are considered. The dependence of 3 dB bandwidth on the relaxation time and relaxation pathway is discussed. It is shown that carrier relaxation via less energy level has better carrier confinement and higher 3 dB bandwidth. The improvement of bandwidth by tunnelling injection QD structure is investigated from the point of view of relaxation pathway. The different effects of tunnelling into ground state and excited state on the 3 dB bandwidth are analyzed. The enhanced carrier relaxation by p-type modulation doping and its effect on the bandwidth are investigated. It is found that there exists a tradeoff on the improvement of bandwidth by p-doping, which is explained as the competition between the bandwidth limitation of K -factor and relaxation dynamics. Increase in the bandwidth of QD lasers by improving both the carrier relaxation dynamics and K-factor limitation is discussed.

Island-size distribution and capture numbers in three-dimensional nucleation: Dependence on island morphology

The scaling of the monomer and island densities, island-size distribution (ISD), and capture-number distribution (CND) as a function of the fraction of occupied sites (coverage) and ratio D(h)/F of the monomer hopping rate D(h) to the (per site) monomer creation rate F are studied for the case of irreversible nucleation and growth of fractal islands in three dimensions (d=3) . We note that our model is a three-dimensional analog of submonolayer growth in the absence of island relaxation and may also be viewed as a simplified model of the early stages of vacancy cluster nucleation and growth under irradiation. In contrast to results previously obtained for point-islands in d=3 , for which mean-field behavior corresponding to a CND which is independent of island size was observed, our results indicate that for fractal islands the scaled CND increases approximately linearly with island size in the asymptotic limit of large D(h)/F . In addition, while the peak height of the scaled ISD for fractal islands appears to diverge with increasing D(h)/F , the dependence on D(h)/F is much weaker than for point-islands in d=3 . The results of a self-consistent rate-equation calculation for the coverage and D(h)/F dependence of the average island and monomer densities are also presented and good agreement with simulation results is obtained. For the case of point-islands, the value of the exponent chi describing the D(h)/F dependence of the island density at fixed coverage, e.g., N(sat) approximately (D(h)/F)-chi , is in good agreement with the value (chi=1/3) expected for irreversible growth. However, for both compact and fractal islands in d=3 , our results indicate that the value of chi (chi approximately 0.42) is significantly larger. In order to explain this behavior, an analytical expression [e.g., chi=d(f)/(3d(f)-2) ] for the dependence of chi on island fractal dimension d(f) in d=3 is derived and found to give reasonable agreement with our simulation and rate-equation results for the case of point-islands (d(f)=infinity) , compact islands (d(f)=3) , and fractal islands (d(f) approximately 2.5) . A general expression for the exponent chi , valid for d>or=2 , as a function of the critical island size i and d(f) is also derived.

Modelling the quasistatic and dynamical sensing response of Galfenol-based magnetostrictive devices

A computational model for the quasistatic and dynamical response of Galfenol based magnetostrictive devices in the sensing configuration is presented. The model calculates the fraction of magnetic moments oriented along the preferred orientations within the crystal as a function of time using a self-consistent rate equation technique. These magnetic moment fractions are then used to determine the total magnetization as a function of time. The model is compared to experiments for uniaxial, compressive, and quasistatic loading. Predictive calculations are presented for dynamical loading. Eddy currents and finite transition times lead to increasing hysteresis as the frequency increases.

Temperature Characteristics of 1.3-$\mu$m p-Doped InAs–GaAs Quantum-Dot Vertical-Cavity Surface-Emitting Lasers

In this paper, we present results from room-temperature continuous-wave operation of 1.3-mum p-doped InAs-GaAs quantum-dot (QD) vertical-cavity surface-emitting lasers (VCSELs) with high <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> of ~510 K and low threshold current density of ~65 A/cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> per QD layer. The highest output power from the device is over 0.74 mW. The temperature characteristics of the devices are investigated. It is demonstrated that deterioration in QD VCSEL performance due to self-heating results from the temperature sensitivity of QD emission, instead of mismatch between the gain wavelength and cavity modes. The real temperature at the QD VCSEL active region above threshold is estimated from the shift in lasing wavelength, which is in good agreement with calculations based on a self-consistent rate equation and thermal conduction model. The analysis shows that enhancing the carrier confinement in the QD wetting layer contributes to improving the saturated output power of the QD VCSEL.

Polarization properties of active semiconductor micro-ring structures

Polarization properties of active semiconductor micro-ring structures are studied theoretically emphasizing on the prospects for equalising the structure response for the two polarizations. The evaluation is performed with the help of a scheme for polarization insensitive operation of active MR that, unlike previous approaches, addresses both the waveguide and the heterostructure polarization properties. A self-consistent rate equation model is used to assess the spectral response of the polarization insensitive structures. Based on these findings the inherent spectral limitations for polarization insensitive performance imposed by the highly resonant nature of micro-ring are discussed and compared to that of conventional structures.

Electron transport in quantum cascade lasers in a magnetic field

A theoretical model of electron transport in quantum cascade lasers subjected to a magnetic field is developed. The Landau level electronic structure was calculated from the envelope-function Schr\"odinger equation within the effective-mass approximation. The electron transport in a magnetic field was modeled using the self-consistent rate-equation description for the full period of the cascade and its interaction with adjacent periods. The scattering processes included in the model are electron--longitudinal-optical-phonon, electron--longitudinal-acoustic-phonon, and electron-electron scattering. All these processes show oscillatory behavior with magnetic field, and their interplay determines the electron transport and the output characteristics of quantum cascade lasers in magnetic field. The model was applied to investigate the influence of magnetic field on the performance of a $\mathrm{GaAs}∕\mathrm{AlGaAs}$ quantum cascade laser emitting at $\ensuremath{\lambda}\ensuremath{\approx}11.4\phantom{\rule{0.3em}{0ex}}\mathrm{\ensuremath{\mu}}\mathrm{m}$ [P. Kruck et al., Appl. Phys. Lett. 76, 3340 (2000)]. The calculated results show good overall agreement with the available experimental data.

Design and simulation of InGaAs∕AlAsSb quantum-cascade lasers for short wavelength emission

The design and simulation of an In0.53Ga0.47As∕Al0.56As0.44Sb quantum-cascade laser emitting in the near infrared is presented. Designed using a self-consistent rate equation solver coupled with an energy balance rate equation, the proposed laser has a calculated population inversion of ∼20% at 77K and sufficient gain to achieve room-temperature laser emission at λ∼2.8μm. Threshold currents in the range 4–8kA∕cm2 are estimated as the temperature increases from 77Kto300K. The output characteristics of the proposed laser are compared to an existing λ∼3.1μm In0.53Ga0.47As∕Al0.56As0.44Sb quantum-cascade structure presented in the literature.

Relationship between carrier dynamics and temperature in terahertz quantum cascade structures: simulation of GaAs/AlGaAs, SiGe/Si and GaN/AlGaN devices

Analysis of carrier transport in cascaded GaAs/AlGaAs, p-Si/SiGe and GaN/AlGaN terahertz structures is performed using self-consistent rate equations with kinetic energy-balance simulations. The electron transport in n-type structures is simulated using standard envelope function and nonparabolic effective mass approximations with all relevant electron–electron and electron–optical phonon scattering. In the case of GaAs-based structures the model is extended to include ionized impurity and interface roughness scattering as well. The hole subband structure in the p-type device is calculated using the 6 × 6 k ⋅ p model, and then used to find carrier relaxation rates due to the alloy disorder, acoustic and optical phonon scattering, as well as hole–hole scattering. The simulation accounts for the in-plane k-space anisotropy of both the hole subband structure and the scattering rates. Results are presented for the original GaAs/AlGaAs THz quantum cascade laser and prototypes of GaN/AlGaN and Si/SiGe THz quantum cascade structures.

Optically pumped terahertz laser based on intersubband transitions in a GaN∕AlGaN double quantum well

A design for a GaN∕AlGaN optically pumped terahertz laser emitting at 34 μm (ΔE∼36meV) is presented. This laser uses a simple three-level scheme where the depopulation of the lower laser level is achieved via resonant longitudinal-optical-phonon emission. The quasibound energies and associated wave functions are calculated with the intrinsic electric field induced by the piezoelectric and the spontaneous polarizations. The structures based on a double quantum well were simulated and the output characteristics extracted using a fully self-consistent rate equation model with all relevant scattering processes included. Both electron-longitudinal-optical phonon and electron-acoustic-phonon interactions were taken into account. The carrier distribution in subbands was assumed to be Fermi–Dirac-like, with electron temperature equal to the lattice temperature, but with different Fermi levels for each subband. A population inversion of 12% for a pumping flux Φ=1027cm−2s−1 at room temperature was calculated for the optimized structure. By comparing the calculated modal gain and estimated waveguide and mirror losses the feasibility of laser action up to room temperature is predicted.

Towards automated design of quantum cascade lasers

We present an advanced technique for the design and optimization of GaAs∕AlGaAs quantum cascade laser structures. It is based on the implementation of the simulated annealing algorithm with the purpose of determining a set of design parameters that satisfy predefined conditions, leading to an enhancement of the device output characteristics. Two important design aspects have been addressed: improved thermal behavior, achieved by the use of higher conduction band offset materials, and a more efficient extraction mechanism, realized via a ladder of three lower laser states, with subsequent pairs separated by the optical phonon energy. A detailed analysis of performance of the obtained structures is carried out within a full self-consistent rate equations model of the carrier dynamics. The latter uses wave functions calculated by the transfer matrix method, and evaluates all relevant carrier–phonon and carrier–carrier scattering rates from each quantized state to all others within the same and neighboring periods of the cascade. These values are then used to form a set of rate equations for the carrier density in each state, enabling further calculation of the current density and gain as a function of the applied field and temperature. This paper addresses the application of the described procedure to the design of λ∼9μm GaAs-based mid-infrared quantum cascade lasers and presents the output characteristics of some of the designed optimized structures.