DFG Research Center Matheon

D4: Quantum mechanical and macroscopic models for optoelectronic devices


K. Hoke (supported by DFG),   H.-Chr. Kaiser,   J. Rehberg (project head)

Former members

H. Gajewski (project head until July 2004),   R. Hünlich (project head until October 2005),   M. Baro (until 2006; supported by DFG),



The ongoing progress of industrial semiconductor device technologies permits to fabricate devices which inherently employ quantum phenomena in their operation, e.g. nano-transistors, tunneling diodes, quantum-well lasers, etc. The drift-diffusion models, which served as a backbone for semiconductor device simulations in the past decades, are not adequate to these semiconductor devices, since they do not take into account the quantum phenomena. On the other hand, a concise quantum mechanical simulation of the whole semiconductor device is not feasible from the numerical point of view. In many semiconductor devices quantum effects take place in a localized region (microstructure), e.g. around the double barrier in a resonant tunneling device, or the active zone of a quantum well laser, whereas the rest of the device (macrostructure) can be well described by approved classical models. Thus, it makes sense to follow a hybrid strategy, i.e. to couple quantum mechanical and macroscopic models. This enforces a detailed mathematical analysis (existence, uniqueness, regularity) and numerical analysis (convergence of discretized approximative problems) of each individual model and the hybrid model itself.

The set-up of stand-alone drift-diffusion models for the macroscopic description of charge transport in general semiconductor devices is by now well established, see [Gaj93]. Open questions are the regularity of solutions and how to deal with wider classes of reactions, vital issues for coupling with external systems, see [KNR05].

The most prominent quantum mechanical model for the description of semiconductor devices are Schrödinger-Poisson systems, see [Kap99]. These models are non-linear systems comprising Schrödinger operators and the Poisson equation, which accounts for the Coulomb interaction, one of the most important effects in semiconductor devices. In contrast to the drift-diffusion models, the mathematical analysis of Schrödinger-Poisson systems is less advanced. In the past years the analysis and numerical analysis of Schrödinger-Poisson systems focused mainly on the situation of thermal equilibrium and scalar Schrödinger operators, [Nie90, Nie93, KR00]. In order to describe the transport of particles in a nanoscaled semiconductor device one has to consider non-equilibrium situations, which leads to open quantum systems, see [Fre94]. The mathematical analysis of open Schrödinger-Poisson systems is a completely open problem, though first steps have been done recently, see e.g. [KNR03a, KNR03b]. Contrary to the equilibrium case, one cannot expect that the Schrödinger-Poisson system admits a unique solution away from its equilibrium. Thus, monotonicity techniques cannot be applied in this case and more general techniques have to be developed. Additionally, it is indispensable to derive regularity properties of the macroscopic physical quantities in order to set up a coupling of quantum mechanical and drift-diffusion models.

The inclusion of multi-band and additional many-particle effects into the Schrödinger-Poisson model is desirable and possible by introducing k.p-Schrödinger operators and exchange-correlation potentials, see [Ker94, Ker96], but the rigorous analysis and numerical analysis of these models are up to now an open challenge.

Current status of the project

The set-up of stand-alone drift-diffusion models for the macroscopic description of charge transport in general semiconductor devices is well established. This is not the case, however, for microscopic transport. We have investigated a model which describes the transport on the quantum mechanical level. This model essentially is a one-dimensional open quantum system. Thus, we can deal with non-equilibrium situations occurring when transport is involved. Moreover, open quantum systems inherently interact with their environment. The Hamiltonians of these open quantum systems are intrinsically non-selfadjoint. We have analyzed the spectral properties of non-selfadjoint Schrödinger operators which especially comprise non-selfadjoint boundary conditions. These boundary conditions allow for the transmission of scattering states through the device boundary, see [4]. For the open quantum systems under consideration we have performed a rigorous derivation of the physical quantities, such as density, current, reflection and transmission coefficients. Moreover, we have proved properties of these quantities which allow matching them with the corresponding quantities from macroscopic models. The electrostatic potential of the open quantum system is determined self-consistently by a Poisson equation, which leads to open Schrödinger-Poisson systems. We proof the existence of solutions for open Schrödinger-Poisson systems comprising non-selfadjoint Hamiltonians [3] and families of non-selfadjoint Hamiltonians [2]. Moreover, we can give conditions which guarantee the uniqueness of solutions of the Schrödinger-Poisson system involving non-selfadjoint Schrödinger operators [8, 9]

A one-dimensional hybrid model, which is capable to describe the transport of particles in a quantum semiconductor device has been developed. The model consists of a drift-diffusion model and an open quantum model as described before. The two models are coupled such that the continuity of the current density over the whole device domain is ensured. This is a consistent condition for a coupling and leads to a nonlinearly coupled drift-diffusion and quantum model. Here, the transport in the microstructure is assumed to be ballistical. In order to take into account overall Coulomb interaction we solve the Poisson equation for the electrostatic potential selfconsistently on the whole device domain. To illustrate the validity of the model, it has been implemented and simulations of real-world quantum semiconductor devices have been carried out. In particular, resonant tunneling diodes (RTD) are considered. For a wide class of RTDs the numerical results compare well with the measurements of the corresponding devices, if the microstructure is chosen properly, see [1]. After this successful number-crunching validation of our concept we carried out a rigorous numerical analysis of an energy discretized version of the hybrid model, see [5].

The assumption of a ballistic transport in the microstructure is not adequate for some RTDs and consequently collisions in the microstructure have to be taken into account, too. In joint work with our colleagues from the Institut Math�matiques pour l'Industrie et la Physique in Toulouse, we extended the hybrid model described above by a Pauli master equation, which takes into account collisions in the microstructure and results in an additional nonlinearity in the hybrid model. The model has been implemented in our software and we have analyzed the influence of the collisions on the results, see [1]. Depending on the collision operator in the Pauli master equation, we obtain good results for quantum semiconductor devices in which collisions are strong.

Indium-Arsenide-Antimonide (InAsSb) heterostructures, having the lowest band gap of all III--V semiconductors, are of great interest in the recent development of infrared optoelectronic devices. The multi quantum-well (MQW) structures we investigate are grown at the Walter Schottky Institute, characterized at the Max Born Institute (MBI) and are potential candidates for the optical active region of infrared light emitting diodes (LED). The structures consist of a stack of strained InAsxSb1-x quantum wells sandwiched between Al0.15In0.85As0.78Sb0.22 barriers. The Arsenic content of the well material varies between the samples, which leads to different strain for each sample. The investigation of the influence of the strain on the bandgap is a central point for the design of InAsSb based optoelectronic devices. In joint work with the WIAS-QW group, we were able to model mathematically and simulate the electronic structure of these devices by means of pure k.p-calculations. Our simulation results are in good agreement with the measured values obtained by the MBI, see [6].

Publications originated in the project

[1] H. Cornean, K. Hoke, H. Neidhardt, P.N. Racec and J. Rehberg: A Kohn-Sham system at zero temperature. J. Phys. A, 41:85304/1--385304/21, 2008. (preprint version)
[2] J. Griepentrog, W. Höppner, H.-Chr. Kaiser and J. Rehberg: A bi-Lipschitz, volume preserving map from the unit ball onto a cube. Note di Matematica, 28:185--201, 2008. (preprint version)
[3] P. Merino, J.C. de Los Reyes, J. Rehberg and F. Tröltzsch: Optimality Conditions for State-Constrained PDE Control Problems with Finite-Dimensional Control Space. Control and Cybernetics, 37(1):5-38, 2008.
[4] M. Hieber and J. Rehberg: Quasilinear systems with mixed boundary conditions on nonsmooth domains. SIAM J. Math, Anal., 40(1):292-305, 2008 (preprint version)
[5] R. Haller-Dintelmann, H.-Chr. Kaiser and J. Rehberg: Elliptic model problems including mixed boundary conditions and material heterogeneities. Journal de Mathématiques Pures et Appliques, 89(1):25-48, 2008. (preprint version)
[6] R. Haller-Dintelmann, M. Hieber and J. Rehberg: Irreducibility and Mixed Boundary Conditions. Positivity, 12(1):83-91, 2008. (publicated version)
[7] J. Elschner, J. Rehberg and G. Schmidt: Optimal regularity for elliptic transmission problems including $C^1$ interfaces. Interfaces Free Bound., 9:233--252, 2007. (preprint version)
[8] J. Elschner, H.-Chr. Kaiser, J. Rehberg and G. Schmidt: ^{1,q}$ regularity results for elliptic transmission problems on heterogeneous polyhedra. Math. Models Methods Appl. Sci., 17(4):593-615, 2007. (preprint version)
[9] J. Behrndt, H. Neidhardt, and J. Rehberg: Block operator matrices, optical potentials, trace class perturbations and scattering. Operator Theory: Advances and Applications, 175:33-49, 2007. (preprint version)
[10] H. Neidhardt and J. Rehberg: Scattering matrix, phase shift, spectral shift and trace formula for one-dimensional Schrödinger-type operators. Integral Equations Operator Theory, 58(3):407-431, 2007. (preprint version)
[11] T. Koprucki, H.-Chr. Kaiser and J. Fuhrmann: Electronic states in semiconductor nanostructures and upscaling to semi-classical models. In Alexander Mielke,editor, Analysis, modeling and Simulation of Multiscale Problems, pages 367-396.Srinr, Berlin Heidelberg New York, 2006.
[12] H.-Chr. Kaiser, H. Neidhardt and J. Rehberg: Convexity of trace functionals and Schrödinger operators. J. Funct. Anal., 234:45-69, 2006. (preprint version)
[13] H.-Chr. Kaiser, H. Neidhardt and J. Rehberg: Classical solutions of quasilinear parabolic systems on two-dimensional domains. NoDEA Nonlinear Differ. Equ. Appl., 13(3):287-310, 2006. (preprint version)
[14] M. Baro, N. Ben Abdallah, P. Degond, and A. El Ayyadi: A 1D coupled Schrödinger drift-diffusion model including collisions. J. Comput. Phys., 203(1):129-153, 2005. (preprint version)
[15] M. Baro, H. Neidhardt, and J. Rehberg: Current coupling of drift-diffusion models and Schrödinger-Poisson systems: dissipative hybrid models. SIAM J. Math. Anal., 37(3):941--981, 2005. (preprint version)
[16] T. Koprucki, M. Baro, U. Bandelow, T. Tien, F. Weik, J. Tomm, M. Grau, and M.-C. Amann: Electronic structure and optoelectronic properties of strained InAsSb/GaSb multi quantum-wells. Appl. Phys. Lett., 87, 181911, 2005. (preprint version)
[17] M. Baro One-dimensional open Schrödinger-Poisson systems. PhD thesis, Humboldt-Universität zu Berlin, 2005.
[18] H. Neidhardt and J. Rehberg: Uniqueness for dissipative Schrödinger-Poisson systems. J. Math. Phys., 46:113513/1--113513/28, 2005. (preprint version)
[19] V. Maz'ya, J. Elschner, J.Rehberg and G. Schmidt: Solutions for quasilinear nonsmooth evolution systems in $L^p$. Arch. Rational Mech. Anal., 171:219-262, 2004. (preprint version)
[20] M. Baro, H.-Chr. Kaiser, H. Neidhardt and J. Rehberg: A quantum transmitting Schrödinger-Poisson system. Rev. Math. Phys., 16(3):281-330, 2004. (preprint version)
[21] M. Baro, H.-Chr. Kaiser, H. Neidhardt and J. Rehberg: Dissipative Schrödinger-Poisson systems. J. Math. Phys., 45(1):21-43, 2004. (preprint version)
[22] M. Baro and H. Neidhardt: Dissipative Schrödinger-type operator as a model for generation and recombination. J. Math. Phys., 44(6):2373-2401, 2003. (preprint version)
[23] H.-Chr. Kaiser, H. Neidhardt and J. Rehberg: On one dimensional dissipative Schrödinger-type operators, their dilations and eigenfunction expansions. Mathematische Nachrichten, 252:51-69, 2003. (preprint version)
[24] H.-Chr. Kaiser, H. Neidhardt and J. Rehberg: Macroscopic current induced boundary conditions for Schrödinger-type operators. Integral Equations and Operator Theory, 45:39-63, 2003. (preprint version)
[25] H.-Chr. Kaiser, H. Neidhardt and J. Rehberg: Density and current of a dissipative Schrödinger operator. Journal of Mathematical Physics, 43(11):5325-5350, 2002. (preprint version)
[26] H.-Chr. Kaiser, H. Neidhardt and J. Rehberg: Van Roosbroeck's equations admit classical solutions in $L^p$: The 2D case. to appear in Nonlinear Analysis. (preprint version)
[27] H.-Chr. Kaiser, H. Neidhardt and J. Rehberg: Monotonicity properties of the quantum mechanical particle density. to appear in Monatshefte für Mathematik. (preprint version)
[28] R. Haller-Dintelmann, Chr. Meyer, J. Rehberg and A. Schiela: Hölder continuity and optimal control for nonsmooth elliptic problems. to appear in Applied Mathematics and Optimization. (preprint version)
[29] D. Hömberg, CHr. Meyer, J. Rehberg and W. Ring: Optimal control for the thermistor problem. to appear in SIAM J. Control and Optimization.
[30] M. Hieber and J. Rehberg: Quasilinear parabolic systems with mixed boundary conditions. to appear. (preprint version)

Further references

[KNR05] H.-Chr. Kaiser, H. Neidhardt, and J. Rehberg: Classical solutions of quasilinear parabolic systems on two dimensional domains. NoDEA Nonlinear Differential Equations Appl., 2005. In press. (preprint version)
[KNR03b] H.-Chr. Kaiser, H. Neidhardt, and J. Rehberg: On one dimensional dissipative Schrödinger-type operators their dilations and eigenfunction expansions. Math. Nachr., 252:51-69, 2003. (preprint version)
[KNR03a] H.-Chr. Kaiser, H. Neidhardt, and J. Rehberg: Macroscopic current induced boundary conditions for Schrödinger-type operators. Integral Equations Operator Theory, 45:39-63, 2003. (preprint version)
[KR00] H.-Chr. Kaiser and J. Rehberg: About a stationary Schrödinger-Poisson system with Kohn-Sham potential in a bounded two- or three-dimensional domain. Nonlinear Anal., 41:33-72, 2000. (preprint version)
[Kap99] E. Kapon: Semiconductor Lasers: Fundamentals. Optics and Photonics. Academic Press, 1999.
[Ker96] T. Kerkhoven: Numerical nanostructure modeling. Z. Angew. Math. Mech., 76(Suppl. 2):297-300,1996.
[Fre94] W. R. Frensley: Quantum transport. In N. G. Einspruch and W. R. Frensley, editors, Heterostructures and Quantum Devices, volume 24 of VLSI Electronics: Microstructure Science. Academic Press, San Diego, 1994.
[Ker94] T. Kerkhoven: Mathematical modelling of quantum wires in periodic hetero junction structures. In Semiconductors Part II, volume 59 of The IMA Volumes in Mathematics and its Applications, pages 237-253. Springer Verlag, New York, 1994.
[Gaj93] H. Gajewski: Analysis und Numerik von Ladungstransport in Halbleitern. Mitt. Ges. Angew. Math. Mech., 16(1):35-57, 1993.
[Nie93] F. Nier: A variational formulation of Schrödinger-Poisson systems in dimensions d ≤ 3. Commun. in Partial Differential Equations, 18:1125-1147, 1993.
[Nie90] F. Nier: A stationary Schrödinger-Poisson system arising from the modelling of electronic devices. Forum Math., 2:489-510, 1990.

Last reviewed on October 20, 2009 by K. Hoke.