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(C) 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE 684 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 5, MAY 2001 Dynamical Simulation of Quantum-Well Structures Marco Wiedenhaus, Andreas Ahland, Dirk Schulz, and Edgar Voges Abstract—For the design and development of optical semiconductor devices based on quantum-well structures, the investigation of saturation phenomena is necessary for high optical power operation. By applying stationary physical models, nonlinear effects cannot be described adequately; hence, transient models are important for an accurate analysis. By utilizing transient models, saturation phenomena, signal delays, and distortions can be investigated. For the analysis of integrated optoelectronic devices, such as lasers and modulators, transient transport or density matrix equations for carriers and photons and the Poisson equation have to be solved self-consistently. A transient model which is useful for the investigation of a wide range of optoelectronic applications is presented. Quantum optical phenomena are included by applying the interband density matrix formalism in real-space representation, where the Coulomb singularity is treated exactly in the limits of the discretization. As we focus on electroabsorption modulators, a drift-diffusion model adequately approximates the transport properties. Here, quantum effects are considered by a quantum correction, the Bohm potential. The model is applied to investigate transport effects in InPbased waveguide electroabsorption modulators including strained lattices. Index Terms—Dynamical solutions, excitons, modulators, quantum wells, semiconductor device modeling. I. INTRODUCTION O PTOELECTRONIC devices are based on both the electrical properties of a semiconductor and the optical characteristics. It is important, therefore, to couple the simulations of these two systems to describe important optoelectronic effects like the Franz–Keldysh effect (FKE), the quantum-confined Stark effect (QCSE), and the Wannier–Stark effect (WSE). As these devices are operated at high frequencies, special attention is directed toward the simulation of dynamical processes like saturation effects. This is the basis for the determination of, e.g., cutoff frequencies and phase shifts. Simulation tools allowing an adequate coupling of electrical, optical, and transient properties of applicable structures have not been presented up to now. The dynamical simulation of transport phenomena in optoelectronic components demands a treatment that considers computational effort, stability of the numerical algorithms and realistic results. We present an approach in this paper that meets these requirements. For the first time, to our knowledge, dynamical simulations are performed, which describe a whole optoelectronic system. In particular, the optical system and the coupling of optics and electronics are a new feature of transient device simulations. It becomes possible, therefore, to realize novel time-varying sim- Manuscript received October 3, 2000; revised February 1, 2001. This work was supported by the Deutsche Forschungsgemeinschaft (DFG). The authors are with the Universität Dortmund, D-44221 Dortmund, Germany. Publisher Item Identifier S 0018-9197(01)03489-3. ulations of electroabsorption modulators, as described in this paper. Electrical properties are obtained from the solution of the fundamental semiconductor equations. These are the Poisson equation and an adequate transport model. Classical transport can be described by a drift diffusion model (DDM) [1] or a hydrodynamic model (HDM) [2], which accounts for temperature dependence. For electroabsorption modulators (EAMs) the DDM is adequate. A HDM should be used for the simulation of lasers. A quantum correction given by Bohm [3] specifies the quantum mechanical transport in the semiconductor. These fundamental equations can be implemented in a coupled scheme [1]. This leads to the introduction of an artificial time derivative into Poisson’s equation, which actually does not depend on time [4], [5]. Several approaches solved a decoupled scheme by a stable explicit algorithm, including a numerical damping term [6]–[8]. But this resulted in rapid damping or unphysical oscillations. In this paper, the hydrodynamic model and the quantum correction are implemented. An implicit decoupled algorithm is preferred here as it provides wider time-steps. Optical properties, on the other hand, are determined by the interband density matrix equation describing the excitonic effect [9]. The singularity of the Coulomb interaction term in this equation makes a decisive contribution to the optical spectrum. It can be treated exactly in the limits of the discretization [10], which is realized here for the first time. The density matrix formalism allows one to account for nanostructures, which, due to the QCSE, yield a higher bandwidth and an improved modulation depth. In contrast to [11], where a “leap-frog” algorithm is used to handle the time dependence, we prefer an implicit scheme allowing wider time-steps due to the stability of implicit methods. As outlined above, the coupling of the optical and electrical systems is essential for an analysis of optoelectronic effects. The interband density matrix equation includes the impact of slanted band edges and, thus, describes an influence of the electrical properties on the optical system. Saturation terms in this equation cause an optical generation of carriers regarded in the recombination-generation-rate of the continuity equations. The influence of the optical on the electrical system is thereby as well described. Coupling is therefore an important feature of our calculations. We implemented different algorithms to investigate derivatives in time. This enables a flexible choice concerning stability and oscillations. Explicit algorithms provide a simple calculation of each time-step [1]. Its main drawback is the restrictive stability condition implying very short steps. Hence, we perform implicit algorithms, which allow much longer time-steps. Results are computed faster this way although a system of linear equations has to be solved. 0018–9197/01$10.00 © 2001 IEEE (C) 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE WIEDENHAUS et al.: DYNAMICAL SIMULATION OF QW STRUCTURES 685 First, we introduce the fundamental semiconductor equations, which describe the electrical properties (Section II). Briefly, the interband density matrix equation is presented in Section III. Sections IV and V show the implementation of the equations. In our opinion, these should be presented in detail, as important features of device simulations are outlined. For a more superficial reading, these sections may be skipped. Finally, examples are given in Section VI to demonstrate the usefulness of dynamical solutions in the simulation of modulators. The argument reads (10) and is determined by the Fermi energy and the potential energy . contains the intrinsic Fermi energy and the , the quasi-Fermi potential , includes the band energy self-consistent potential calculated in Poisson’s equation and the quantum potential in the form given by Bohm [3] II. SEMICONDUCTOR EQUATIONS (11) A system containing Poisson’s equation (1), the continuity equations (2) and (3) and carrier transport equations of the DDM with an additional quantum correction (4) and (5) is solved in a decoupled manner to determine the electrical properties of the structure [1] (12) (1) (2) (3) (4) (5) These equations contain the permittivity , the self-consistent (electrons) and potential , the carrier concentrations (holes) and the concentrations of singly ionized accepters and donors . Further, the difference of recombination- and generation-rate , the carrier current densities and for electrons ( ) and holes ( ), respectively, and the mobilities and are used, as well as the quasi-Fermi potentials and . is the elementary charge. In addition to these fundamental semiconductor equations, carrier densities are calculated for the Fermi distribution and parabolic bands from (6) (7) is the effective density of states for parabolic bands The optical properties of a semiconductor are described by the interband density matrix equation in the real-space representation linearized in the electric field , i.e., saturation terms for carrier densities comparable to the saturation density are neglected [9]. These neglected terms depend on the carrier densities. They account for carrier density based effects such as the Burnstein–Moss effect described in Section VI. Beyond this, the source terms can be used to describe the optical generation of carriers [see (21)]. The effective mass approximation and an approach corresponding to the slowly varying envelope (SVE) yield function (13) The solution is the “pair wave function” describing a coherent wave function due to the well defined phase relation of the linearization on the right-hand side (13). Its envelope function leads to the observable , the macroscopic polarization by (16). An incoming optical wave of photon energy causes this polarization damped by a phenomenological relaxation term according to the inverse relaxation time. This accounts for a dephasing of the transition amplitude under influence of random is set to 7 meV for transitions implying heavy local fields. holes and to 8 meV for light holes. represents the undisturbed Hamiltonian (14) (8) where effective mass; temperature; Boltzmann constant; reduced Planck constant In general, the Fermi integral of order III. OPTICAL PROPERTIES . is defined as (9) It cannot be solved in closed form and is treated by applying asymptotic approximations [1] or, for arguments close to zero, by a polynomial expression that yields an error below 0.5% [12]. where the second term, the Coulomb term, includes the absolute distance between electron and hole. The potential energy is ; and are the potentreated as [10]. tials at the respective positions of electron and hole Consequently, the main mechanisms in electroabsorption modulators, the effect of electroabsorption (the FKE), the QCSE, and the WSE, can be described. The FKE leads to a flattened absorption edge, which can be shifted by the QCSE in quantum confined structures. Thus, in a quantum well (QW) it is possible to control the absorption of the material by an external voltage and, therefore, to modulate an incoming wave. A distinction between the radial and -directions (growth and confinement direction) becomes necessary to describe this effect. Beyond that, the density matrix formalism accounts for free states as well (C) 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE 686 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 5, MAY 2001 as for bound and quasibound states. Accordingly, it is possible to describe field dependent effects like, as mentioned above, QCSE, WSE, and FKE. The real-space representation in relative coordinates simplifies the quantum mechanical calculation because the termination of the calculation domain by applying a perfectly matched layer (PML) [13] becomes possible. As already mentioned, the source term on the right-hand side of (13) is linear in the electric field (15) weighted with the delta funcis the dipole matrix element . It is assumed to be of rotational symmetry. The polartion ization is given by (16) and the complex linear interband susceptibility is defined as (17) which determines the optical absorption coefficient and the refractive index. As we merely consider -excitons, a radial symmetry is assumed. Higher excitations can be neglected [14] having only an indirect impact on the optical absorption. Since the complex susceptibility is calculated, there is no need for using the Kramers–Kronig relation. From this, we derive an effective dielectric constant [10] Fig. 1. Flowchart of the nonlinear calculation of the electrical and the optical properties. IV. IMPLEMENTATION OF THE SEMICONDUCTOR EQUATIONS Poisson’s equation (1) can be efficiently solved by an algorithm introduced by Gummel et al. [15]–[17]. There are no time derivatives in this equation. Solving a decoupled system of equations there is no need for introducing an artificial time derivative as described in [1], [4], [5]. At each time-step, Poisson’s equation is solved using the carrier density values of the last time-step. The calculated potential in turn is used for the subsequent solution of the continuity equations (2) and (3). A general time-dependent problem (22) is discretized according to (18) (23) (integration over the center-of-mass coordinate ). Assuming the absorption and the effective refractive index are including a parameter ( ), where is the time-step size. Our programs are implemented using this parameter, which for an explicit algorithm, can be chosen to be for a Crank–Nicolson scheme, or for a fully implicit algorithm. Selberherr [1] remarks that for a coupled system, the best results are obtained by using a fully implicit scheme. For the decoupled system, we compared this fully implicit method to a Crank–Nicolson scheme. Since the latter unfortunately tended to oscillations [18], the fully implicit scheme also gives best results for the decoupled system. , no restrictive stability condition has to be reFor garded. Hence, the step size can be chosen much larger than for an explicit algorithm. Its choice may be aligned to the feature that the main variations of the electrical properties take place within less than 1 ps. Though no stability problem occurs, numerical dispersion must be taken into account because step sizes on the order of 0.1 ps are not suitable for description of the given problem. The semiconductor equations (1)–(5) are expressed by the not ). But two other sets of varimandatory set of variables ( ) based on the ables are feasible as well, e.g., the set ( quasi-Fermi potentials. The latter leads to nonlinear continuity equations and, therefore, is not advantageous as the solution of the decoupled scheme converges quite slowly. These variables and (19) (20) The vacuum wave number is given by the wavelength of the incident light. The electrical and the optical system can be treated separately near the local equilibrium. The density matrix equation even holds for processes out of equilibrium, whereas the continuity equations (2) and (3) have to be modified. The saturation terms in the source terms of the density matrix equation neglected in (13) lead to an optical generation of carriers. This generation can be accounted for in the continuity equations (2) and (3)—a coupling of the electrical (Section II) and the optical system just mentioned can be realized this way. Consequently, an additive generation-recombination rate can be derived (21) leading to a nonlinear calculation, as depicted in Fig. 1. (C) 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE WIEDENHAUS et al.: DYNAMICAL SIMULATION OF QW STRUCTURES 687 should be considered in a coupled solution because they are of nearly the same order of magnitude and provide an improved behavior concerning convergence. ), with and defined by We prefer the set ( (24) (25) Though these variables reveal an enormous range of 32 orders of magnitude for a difference of voltage from 1 to 1 V at a temperature of 300 K [1], this set supplied stable results in our case because no noteworthy differences between two succeeding results occurred in an iterative method. The spatial derivatives are implemented by a simple finite differences approach. V. IMPLEMENTATION OF THE DENSITY MATRIX EQUATION Different from the implementation of the semiconductor equations, a polynomial finite element scheme is used in the spatial domain as mentioned in [10]. As the Coulomb in the second term on the right-hand side singularity for of (14) is constitutive for the solution of the density matrix equation, it is of interest to include this contribution as exactly as possible. Our finite-elements approach allows an exact treatment of this Coulomb singularity within the limits of the discretization since the integral over this singularity is finite. In particular, the application of spherical coordinates yields a simple solution of this integral because angular dependence can be neglected. In bulk material and ideal 2-D collocations, a volume element containing emerges from the integral trans-dependence of the Coulomb formation. This cancels the term. The second derivative in the first term on the right-hand side of the Hamiltonian is subject to an integration by parts as the resulting first derivative is applicable to the finite elements scheme. A nonequidistant discretization near the Coulomb singularity can be avoided in contrast to [19], as well as it is not necessary to calculate the ground state for the effective potential as described in [11]. In order to reduce computational effort, it is advisable to shorten the calculation domain. Reflections at the boundaries are eliminated by a PML absorber [13], a nonphysical absorber which is matched independent of the angle of incidence. Here, we assume a small Coulomb term and a plane wave in cartesian coordinates. Again, we implemented different algorithms to treat the time dependence. The “leap-frog” scheme uses central differences for the time derivatives. Thus, the problem (22) mentioned above, which is comparable to beam propagation methods [20], [21], can be rewritten as (26) leading to a directly solvable scheme in the general form (27) Fig. 2. Structure of a GaInAsP QW system. [A matrix in front of the sought emerging from the finite elements method can be constricted to a lumped stiffness matrix. This diagonal matrix arises by taking the sum of each row. It can easily be inverted, which again leads to a directly solvable scheme (27).] This explicit algorithm underlies very restricted stability conditions. We found step widths in the order of 0.1 fs, a really restrictive condition, which does not allow efficient computations. An implicit algorithm, on the other hand, leads to tedious calculations counterbalanced by a possible choice of wider time-steps. In opposition to the solution of the semiconductor equations, a fully implicit scheme should not be applied here, since energy conservation is not guaranteed this way. Wave functions are preferably solved by a Crank–Nicolson scheme, which does not tend to oscillations as much as in the electrical system. Expressed in terms of (23), the parameter has to be chosen to . If the impact of oscillations is not acceptable, can be set to a higher value, e.g., 0.7 or 0.8. The main aspect to be considered, is that dephasing is sufficiently sampled. The step size should, therefore, be about an order below the relaxation time, which is typically of order 500 fs in our calculations. Compared to the stability condition for the “leap-frog” method, this enables wider time-steps up to about an order of magnitude below the relaxation time. Equation (13) becomes, discretized in time, the following: (28) VI. EXAMPLES First, we show the results for a simple GaInAsP system. The QW structure has a special behavior because of a field-induced absorption edge merging (see [14] for details). Its structure is shown in Fig. 2. The structure consists of the following material. Barriers I and V are of lattice-matched Ga In As P , layer II of compressively strained InAs P and layer IV of tensile strained Ga In As. A well layer of InAs P (layer III) provides an increased overlap of the electron and hole wavefunctions; but there is no bound state within this narrow layer. (C) 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE 688 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 5, MAY 2001 Fig. 3. Potential energies inside the QW for a voltage step of 0.5 V. Calculation 1), without optics. Fig. 5. Band edges at two different times. At 0.75 ps, the outer region of the device near the contact is affected, while the inner parts are nearly unaffected. Fig. 4. Potential energies inside the QW for a voltage step of 0.5 V. Calculation 2), nonlinear with optical system. Fig. 6. Actuated voltage and imaginary part of the optical susceptibility for an incoming wave of 0.675 eV. Here, a voltage step of 0.5 V is actuated to show how the material follows the external voltage and approaches the stationary state again. Two calculations were performed: 1) a linear calculation, where the optical system has no influence on the electrical properties; 2) a nonlinear simulation, where the optical system affects the potential energies. Figs. 3 and 4 exhibit the evolution of the potential energies in the inner QW for an incoming wave of 0.81 eV (1530 nm) and V/m. From the initial state, the thermodynamic equilibrium, it takes around 3–4 ps in calculation 1) to reach the stationary state. Calculation 2) results in a much slower behavior of the device, which can be explained by a generation of carriers as described in (21). The transport of these carriers is the reason for the retarded reaction. Fig. 5 shows the temporal behavior of the inner parts of the structure. First of all, the actuated voltage affects the outer region of the device near the active contact. At 0.75 ps, no remarkable change can be seen in the QW region. A retarded reaction of the active region is the consequence. The presented method achieves stable results for a discretization of 1 fs. The electrical system is calculated by a fully implicit scheme, while we chose for the optical system. No further attention is devoted to a detailed description of the absorption edge merging and strained material parameters, for these are given in [14]. For an applied sinusoidal voltage of 40 GHz, the result of the optical calculations is shown in Fig. 6. The voltage alternates with an amplitude of 0.5 V around 0.5 V, the energy of the wave is set to 0.675 eV and the electric field to 300 V/m. A short delay can be seen due to the evolution of the material. Again, the outer parts are affected first. After around 1 ps, the active region responds to the external voltage, which leads to the delay of the device resulting in a phase shift of the modulated signal. Here, the electrical and the optical system are coupled. It is interesting to see how this algorithm copes with more complicated structures controlled by a sinusoidal voltage. We give a demonstration of a wave-guide electroabsorption modulator that accounts for the QCSE in a SPQW structure. This property, in the present case, results in a blue-shift of the absorption edge [22], [23]. Normally, the Stark effect results in a red-shift of the edge, but in structures with an asymmetrically stepped-potential, a blueshift can be obtained for small electric fields directed from the narrow to the wide bandgap. The SPQW has a small QW at one side of the main QW, as shown in Fig. 7. In principle, a tilting of the bands leads to a slightly raised electron resonance level followed by a blue-shift, as mentioned. But a further increase of the field yields the Stark effect, a redshift again. In this case, a barrier is included in the QW separating the small trap from the rest of the well (Fig. 8), which may lead to a blue-shift of more than 13 meV. All information concerning these structures can be obtained from [23]–[25]. Fig. 8 shows the layered structure of the modulator applied. The periodically arranged active region is depicted in detail for the shaded region. The doping of the n- and p-contact regions is cm . (C) 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE WIEDENHAUS et al.: DYNAMICAL SIMULATION OF QW STRUCTURES 689 Fig. 7. Principle band diagram of a stepped-potential QW. Fig. 9. Optical susceptibility for an incoming wave of 0.81 eV, stationary calculations. Fig. 8. Structure of the applied blue-shift modulator: band energies and Fermi level. The shaded region is depicted in detail. Fig. 10. Actuated voltage and imaginary part of the optical susceptibility for an incoming wave of 0.81 eV, transient calculations. The calculations presented are done with a discretization in space of 1 nm and a discretization in time of 1 fs. In the beginning, the left contact is biased by 1 V while the right contact is fixed at 0 V. Thus, we first calculate the stationary state. A sinusoidal voltage with an amplitude of 1 V and a frequency of 40 GHz is actuated in addition to the stationary bias. The material is exposed to a wave of energy 0.81 eV and an electric field of 300 V/m. The electrical system is again coupled to the optical system. Saturation effects have to be considered [23]. A small bandgap and a small density of states cause a fully occupied band near the band edge, the so-called Burnstein–Moss effect,which appears in terms of a screened Coulomb term. This is another reason for a blue-shift since the generation of electron-hole pairs needs higher energies. As a counterpart to the Burnstein–Moss effect, a reduction of the bandgap has to be regarded. Concerning high carrier concentrations, this follows from exchange terms in the interband density matrix equations. For small carrier densities, the two effects compensate one another. The reduction of the Coulomb potential, even a rejection for higher concentrations, results in a decrease of the absorption. The time derivatives of the electrical system are handled by a fully implicit algorithm, whereas the optical properties are treated by an implicit scheme with the parameter in (28). It is chosen this way because a parameter tended to oscillations. A compromise had to be found that accounts for ), as well as avoids oscillations energy conservation ( ). ( Fig. 9 shows the optical susceptibility depending on the actuated voltage in the stationary case. This graph is quite useful for comparison. Fig. 11. Actuated voltage (dashed line), band energies and Fermi levels in one of the QWs, transient solutions. Starting from the stationary case at 1 V the dynamical calculations are done. Its result is shown in Fig. 10. The graph reveals that the device again has a retarded reaction. It takes about 0.5 ps before the inner parts of the device are subject to a significant variation of the band energies and the Fermi levels. The band energies and the Fermi levels inside the left QW are depicted in Fig. 11. Variations of the actuated voltage have an immediate influence on the outer parts of the device. Gradually, this effects the inner parts of the structure and, therefore, causes a delayed variation of the optical behavior in the active region leading to the phase shift. The calculations for such a complicated structure are a question of days, and therefore, only the first few picoseconds are shown. Fortunately, it is possible to choose a simpler, but more improper treatment, e.g., a noncoupled simulation. The simula- (C) 2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE 690 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 5, MAY 2001 tions can be adjusted to individual requirements. It should, at any rate, be mentioned that transient calculations demand an increased effort. The last example shows that the implemented algorithms provide a stable calculation for even a quite complex device. This is a remarkable confirmation of our approach. VII. CONCLUSION We have demonstrated an efficient method to apply dynamical solutions of the optical and electrical properties on EAMs, which account for computational time and numerical stability. The response of a blue-shift waveguide EAM on field-induced effects was calculated concerning delay and nonlinearities. The methods presented allow the investigation of transient effects of QW semiconductor devices at high frequencies, which are constitutive for the study of cutoff frequencies, time constants and phase shifts. Implicit schemes were preferred to explicit algorithms in order to treat time dependence in our calculations. This is due to wider time-steps, which are not feasible in an explicit method because of stability considerations. In particular, the transient solutions of the density matrix approach are a new feature of simulating QW structures providing such a wide field of applications. The tools developed are a successful approach to close the gap between existing investigations of the physical basics and an applicable and fast simulation of optoelectronic devices. [15] H. K. Gummel, “A self-consistent iterative scheme for one-dimensional steady state transistor calculations,” IEEE Trans. Electron Devices, vol. 11, pp. 455–465, Oct. 1964. [16] D. L. Scharfetter and H. K. Gummel, “Large-signal analysis of a silicon read diode oscillator,” Solid-State Electron., vol. 16, no. 1, pp. 64–77, 1969. [17] R. Andrew, “Improved formulation of Gummel’s algorithm,” Solid-State Electron., vol. 15, no. 1, pp. 1–4, 1972. [18] G. D. Smith, Numerical solution of partial differential equations: Finite difference method. Oxford, U.K.: Clarendon, 1985. [19] N. Grün, A. Mühlhans, and W. Scheid, “Numerical treatment of the time-dependent Schrödinger-equation in rotating coordinates,” J. Phys. B, vol. 15, p. 4042, 1982. [20] Y. Chung and N. Dagli, “Explicite finite difference beam propagation method: Application to semiconductor rib waveguide Y-junction analysis,” Electron. Lett., vol. 26, no. 11, pp. 711–713, 1990. [21] Y. Chung, N. Dagli, and L. Thylen, “Explicite finite difference vectorial beam propagation method,” Electron. Lett., vol. 27, no. 23, pp. 2119–2121, 1991. [22] M. Morita, K. Goto, and T. Suzuki, “Quantum-confined stark effect in stepped-potential quantum wells,” Jpn. J. Appl. Phys., vol. 29, no. 9, pp. 1663–1665, 1990. [23] A. Ahland, D. Schulz, and E. Voges, “Nonlinear effects in blue-shift waveguide EAM’s,” Opt. Quantum Electron., vol. 32, pp. 769–780, 2000. [24] M. Krijn, “Heterojunction band offset and effective masses in III-V quaternary alloys,” Semicond. Sci. Technol., vol. 6, pp. 27–31, 1991. [25] Landolt- Boernstein, Semiconductors, ser. New Series III. New York: Springer-Verlag, 1985/1987, vol. 17a, 22a, New Series III. Marco Wiedenhaus was born in Lippstadt, Germany, in 1974. He received the Dipl.-Ing. degree in electrical engineering from the Universität Dortmund, Dortmund, Germany, in 2000. In March 2000, he joined the Department for High Frequency, Universität Dortmund, and is currently working in the field of modeling and simulation of semiconductor devices and optical communication systems. REFERENCES [1] S. Selberherr, Analysis and Simulation of Semiconductor Devices. New York: Springer-Verlag, 1984. [2] K. Bløtekjær, “Transport equations for electrons in two-valley semiconductors,” IEEE Trans. Electron Devices, vol. ED-17, pp. 38–47, 1970. [3] D. Bohm, “A suggested interpretation of the quantum theory in terms of “Hidden” Variables,” Phys. Rev., vol. 85, no. 2, pp. 166–193, 1952. [4] J. P. Kreskowsky and H. L. Grubin, “Numerical solution of the transient, multidimensional semiconductor equations using the LBI techniques,” in Proc. NASECODE III Conf., Dublin, Ireland, 1983, pp. 155–160. [5] M. S. Mock, “Time discretization of a nonlinear initial value problem,” J. Comp. Phys., vol. 21, pp. 20–37, 1976. , Analysis of Mathematical Models of Semiconductor De[6] vices. Dublin, Ireland: Boole, 1983. [7] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic, 1970. [8] E. H. Zarantonello, “Solving functional equations by contractive averaging,” Univ. of Wisconsin, Rep. 160, MRC, 1960. [9] A. Stahl and I. Balslev, Electrodynamics of the Semiconductor Band Edge. New York: Springer-Verlag, 1987. [10] A. Ahland, M. Wiedenhaus, D. Schulz, and E. Voges, “Calculation of exciton absorption in arbitrary layered semiconductor nanostructures with exact treatment of the Coulomb Singularity,” IEEE J. Quantum Electron., vol. 36, pp. 842–848, July 2000. [11] S. Glutsch and D. S. Chemla, “Numerical calculation of the optical absorption in semiconductor quantum structures,” Phys. Rev. B, vol. 54, no. 16, pp. 11 592–11 600, 1996. [12] J. S. Blakemore, “Approximations for Fermi-Dirac integrals, especially the function F (x), used to describe electron density in a semiconductor,” Solid-State Electron., vol. 25, no. 11, pp. 1067–1076, 1982. [13] A. Ahland, D. Schulz, and E. Voges, “Accurate mesh truncation for Schrödinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra,” Phys. Rev. B, vol. 60, no. 8, pp. 5109–5112, 1999. , “Efficient modeling the optical properties of MQW modulators on [14] In GaAsP with absorption edge merging,” IEEE J. Quantum Electron., vol. 34, pp. 1597–1603, Sept. 1998. Andreas Ahland was born in Münster, Germany, in 1971. He received the Dipl.-Ing. degree in electrical engineering from the Universität Dortmund, Dortmund, Germany, in 1996. In April 1996 he joined the Department for High Frequency, Universität Dortmund, and is currently working in the field of modeling and simulation of semiconductor devices and optical communication systems. Dirk Schulz was born in Düsseldorf, Germany, in 1965. He received the Dipl.-Ing. degree in electrical engineering from the RWTH Aachen, Aachen, Germany, and the Dr.-Ing. degree from the Universität Dortmund, Dortmund, Germany, in 1990 and 1994, respectively, and is now pursuing his venia legendi. Since August 1990, he has been a Research Assistant at the Institute for High Frequency Techniques, Universität Dortmund, where he worked on beam propagation techniques for guided wave devices in integrated optics and optoelectronics. His current interests are in optical communication systems, modeling, and simulation techniques for integrated optical, optoelectronic, and high-frequency devices, and high-frequency circuits. Edgar Voges was born in Braunschweig, Germany, in 1941. He received the Dipl.-Phys. and Dr.-Ing. degrees from the Technische Universität Braunschweig, Braunschweig, Germany, where he was engaged in investigations on microwave semiconductor devices and acoustic surface waves. From 1974 to 1977, he was a Professor with the Department of Electrical Engineering, Universität Dortmund, Dortmund, Germany. From 1977 to 1982, he was a Professor of Communication Engineering at the Fern Universität, Hagen, Germany. Since 1982, he has again been with the Department of Electrical Engineering, Universität Dortmund, as Head of the Institute for High Frequency Techniques. His current interests are in optical communication, integrated optics, and high-frequency integrated circuits.