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A quantum transmitting Schrödinger-Poisson system

Collaborator: M. Baro, H.-Chr. Kaiser, H. Neidhardt, J. Rehberg

Cooperation with: A. Jüngel (Johannes Gutenberg-Universität Mainz), P. Degond, N. Ben Abdallah (Université Paul Sabatier, Toulouse, France), V.A. Zagrebnov (Université de la Méditerranée, Aix-Marseille II and Centre de Physique Théorique, France), P. Exner (Academy of Sciences of the Czech Republic, Prague)

Supported by: DFG: DFG-Forschungszentrum ``Mathematik für Schlüsseltechnologien'' (Research Center ``Mathematics for Key Technologies''), project D4; ``Kopplung von van Roosbroeck- und Schrödinger-Poisson-Systemen mit Ladungsträgeraustausch'' (Coupling between van Roosbroeck and Schrödinger-Poisson systems with carrier exchange); DAAD (PROCOPE): ``Numerics of hybrid models for quantum semiconductors''

Description:

This project is part of a long-term investigation of quantum mechanical models for semiconductor nanostructures, cf. [2, 9, 10, 11, 12, 13, 14], and their embedding into macroscopic models, like drift-diffusion and energy models, cf. [15] and p. [*], for semiconductor devices, in particular optoelectronic ones, cf. [1, 3] and pp. [*], [*].

We investigate, from a mathematical point of view, a basic quantum mechanical model for the transport of electrons and holes in a semiconductor device. More precisely, our subject is the distribution of electrons and holes in a device between two reservoirs within a self-consistent electrical field, thereby taking into account quantum phenomena such as tunneling and the quantization of energy levels in a quantum well. These very quantum effects are the active principle of many nanoelectronic devices: quantum well lasers, resonant tunneling diodes et cetera, cf., e.g., [17]. We look for stationary states of a quasi-two-dimensional electron-hole gas in a semiconductor heterostructure which is translationally invariant in these two dimensions. Thus, neglecting any magnetic field induced by the carrier currents, we are dealing with an essentially one-dimensional physical system. The transport model for a single band, electrons or holes, in a given spatially varying potential v is as follows: The potential v as well as the material parameters of the physical system are constant outside a fixed interval (a, b), cf. [7, 16]. The possible wave functions are given by the generalized solutions of

Kv$\displaystyle \psi_{k}^{}$ = $\displaystyle \lambda$(k)$\displaystyle \psi_{k}^{}$

where

Kv = - $\displaystyle {\frac{{\hbar^2}}{{2}}}$$\displaystyle {\frac{{d}}{{dx}}}$$\displaystyle {\frac{{1}}{{m}}}$$\displaystyle {\frac{{d}}{{dx}}}$ + v (1)
is the single-particle effective-mass Hamiltonian in Ben-Daniel-Duke form, $ \hbar$ is the reduced Planck constant, m = m(x) > 0 is the spatially varying effective mass of the particle species under consideration, and $ \lambda$ = $ \lambda$(k) is a dispersion relation, e.g.,

$\displaystyle \lambda$(k) = $\displaystyle \begin{cases} \frac{\hbar^2k^2}{2m_a}+v_a \quad&\text{for $k>0$,} [1ex] \frac{\hbar^2k^2}{2m_b}+v_b \quad&\text{for $k<0$;} \end{cases}$

ma, mb are the effective masses, and va, vb are the potentials in the asymptotic regions x < a and x > b, respectively. If there are no bounded states, then the particle density u is a composition of the wave functions $ \psi_{k}^{}$ weighted by values of a distribution function f:

u(x) = c$\displaystyle \int_{0}^{{\infty}}$dk f ($\displaystyle \lambda$(k) - $\displaystyle \epsilon_{a}^{}$)|$\displaystyle \psi_{k}^{}$(x)|2 + c$\displaystyle \int_{{-\infty}}^{0}$dk f ($\displaystyle \lambda$(k) - $\displaystyle \epsilon_{b}^{}$)|$\displaystyle \psi_{k}^{}$(x)|2, x $\displaystyle \in$ (a, b). (2)
$ \epsilon_{a}^{}$ and $ \epsilon_{b}^{}$ is the quasi-Fermi potential of the reservoir in the asymptotic region x < a and x > b, respectively, and c is the two-dimensional density of states. The distribution function is

f ($\displaystyle \xi$) = $\displaystyle \begin{cases} \exp\left(-\frac{\xi}{k_B T}\right) \quad&\text{... ...frac{\xi}{k_B T})\right) \quad&\text{for Fermi-Dirac statistics,} \end{cases}$

where T is the temperature and kB Boltzmann's constant. (2) can be written in the following way: Let $ \widehat{{\rho}}$ be the multiplication operator on $ \mathsf {L}$2($ \mathbb {R}$) induced by the function

$\displaystyle \rho$(k) = $\displaystyle \begin{cases} c f(\lambda(k)-\epsilon_a) \quad & \text{for $k>0$,} \\ c f(\lambda(k)-\epsilon_b) \quad & \text{for $k<0$,} \end{cases}$ (3)
and let $ \mathcal {F}$v : $ \mathsf {L}$2($ \mathbb {R}$)$ \to$$ \mathsf {L}$2($ \mathbb {R}$)be the Fourier transform which diagonalizes the operator Kv on $ \mathsf {L}$2($ \mathbb {R}$), that means $ \mathcal {F}$vKv$ \mathcal {F}$v * = $ \widehat{{\lambda}}$,where $ \widehat{{\lambda}}$ is the maximal multiplication operator induced by the dispersion relation $ \lambda$ = $ \lambda$(k). Then the operator

$\displaystyle \varrho$(v) = $\displaystyle \mathcal {F}$v * $\displaystyle \widehat{{\rho}}$$\displaystyle \mathcal {F}$v (4)
is a steady-state, that means a self-adjoint, positive operator on the Hilbert space $ \mathsf {L}$2($ \mathbb {R}$) which commutes with Kv. Moreover, any steady state can be expressed in the form (4) by means of a function $ \rho$ = $ \rho$(k). The particle density u, defined by (2), is the Radon-Nikodým derivative of the (Lebesgue) absolutely continuous measure (a, b) $ \supset$ $ \omega$ $ \mapsto$ tr $ \big($$ \varrho$(v)M($ \chi_{\omega}^{}$)$ \big)$( M($ \chi_{\omega}^{}$) denotes the multiplication operator induced by the characteristic function $ \chi_{\omega}^{}$ of the set $ \omega$) that means

$\displaystyle \int_{\omega}^{}$ u(x)dx = tr$\displaystyle \big($$\displaystyle \varrho$(v)M($\displaystyle \chi_{\omega}^{}$)$\displaystyle \big)$, (5)
for all Lebesgue measurable subsets $ \omega$ of (a, b). By replacing the real-valued distribution function (3) by a generalized distribution function with 2x2-matrix values, this concept of particle density carries over to the setup we investigate in this project, cf. [4, Section 5.1]. It should be noted that the species current density between the reservoirs also can be expressed in terms of the $ \psi_{k}^{}$, cf. [4, Section 5.2].

In the asymptotic regions x < a and x > b the generalized eigenfunctions $ \psi_{k}^{}$ can be written as a superposition of plane waves. This allows to define boundary conditions at a and b, with respect to the dispersion relation $ \lambda$ = $ \lambda$(k), by means of the quantum transmitting boundary method, cf. [7, 16]. The corresponding homogeneous boundary conditions are

$\displaystyle {\frac{{\hbar}}{{m(a)}}}$$\displaystyle \psi^{\prime}_{}$(a) = - i $\displaystyle \mathfrak{v} (k) %\flow(k) \psi(a) ,\quad \frac{\hbar}{m(b)}\p... ...me(b) = i \mathfrak{v} (-k) %\flow(-k) \psi(b) ,\quad k\in\mathbb{R} , $ (6)
where $ \mathfrak{v} (k)$, k $ \in$ $ \mathbb {R}$, is the group velocity defined by $ \mathfrak{v} = \frac{1}{\hbar}\frac{d\lambda}{dk} .$ The differential expression (1), together with the boundary conditions (6), sets up a family of maximal dissipative operators on the Hilbert space $ \mathsf {L}$2(a, b). We call this family, in the style of [16], the quantum transmitting boundary operator family (QTB operator family), cf. [4, Section 2]. The QTB operator family already contains all the information needed to define, in conjunction with a generalized distribution function $ \rho$, physical quantities such as the particle density, the current density, and the scattering matrix.

The interaction between an electric field and carriers of charge within a semiconductor device can be modeled by Poisson's equation, cf. [8] and the references cited there:

- $\displaystyle {\frac{{d}}{{dx}}}$$\displaystyle \epsilon$(x)$\displaystyle {\frac{{d}}{{dx}}}$$\displaystyle \varphi$(x) = q$\displaystyle \left(\vphantom{ C(x) + \mathcal{N}^+(v^+)(x) -\mathcal{N}^-(v^-)(x) }\right.$C(x) + $\displaystyle \mathcal {N}$+(v+)(x) - $\displaystyle \mathcal {N}$-(v-)(x)$\displaystyle \left.\vphantom{ C(x) + \mathcal{N}^+(v^+)(x) -\mathcal{N}^-(v^-)(x) }\right)$, x $\displaystyle \in$ (a, b), (7)
where q denotes the elementary charge, C is the density of ionized dopants in the semiconductor device, $ \epsilon$ > 0 is the dielectric permittivity function, and $ \varphi$ is the electrostatic potential, v$\scriptstyle \pm$ = $ \mp$w$\scriptstyle \pm$$ \pm$q$ \varphi$ are the potential energies of electrons (``-'') and holes (``+''), and w-, w+ are the conduction and valence band offset, respectively. The quantum transmitting Schrödinger-Poisson system is a Poisson equation (7) with nonlinear electron and hole density operators $ \mathcal {N}$- and $ \mathcal {N}$+ defined as the map of a potential v to the density (5) with steady states $ \varrho^{-}_{}$(v) and $ \varrho^{+}_{}$(v), respectively. In [4, Section 6] we have demonstrated that the thus defined carrier density operators are continuous; the corresponding currents are uniformly bounded for all potentials v. We have proved that the quantum transmitting Schrödinger-Poisson system comprising electrons and holes always admits a solution provided the function inducing the steady states has reasonable decay properties with increasing energy. Furthermore, we give a priori estimates for the solutions. The a priori bounds for the electrostatic potential and the electron and hole density of solutions are explicit expressions in the data of the problem. Ben Abdallah, Degond, and Markowich have investigated a special case of this model in [6] and prove the existence of solutions for the unipolar case. Unfortunately, the mathematical techniques used in their proof do not apply to the bipolar case, which we treat in this project.

The quantum transmitting Schrödinger-Poisson system is closely related to the dissipative Schrödinger-Poisson system, which we have investigated in [5], cf. Annual Research Report 2002, pp. 26-28. In particular, the dissipative Schrödinger-Poisson system and the quantum transmitting Schrödinger-Poisson system coincide for fixed energy, modulo a unitary transformation.

References:

  1. U. BANDELOW, H. GAJEWSKI, H.-CHR. KAISER, Modeling combined effects of carrier injection, photon dynamics and heating in strained multi-quantum well lasers, in: Physics and Simulation of Optoelectronic Devices VIII, R.H. Binder, P. Blood, M. Osinski, eds., vol. 3944, Proceedings of SPIE, SPIE, Bellingham, WA, August 2000, pp. 301-310.

  2. U. BANDELOW, H.-CHR. KAISER, TH. KOPRUCKI, J. REHBERG, Spectral properties of k . p Schrödinger operators in one space dimension, Numer. Funct. Anal. Optimization, 21 (2000), pp. 379-409.

  3. H.-CHR. KAISER, U. BANDELOW, TH. KOPRUCKI, J. REHBERG, Modeling and simulation of strained quantum wells in semiconductor lasers, in: Mathematics--Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.J. Krebs, eds., Springer-Verlag, Berlin Heidelberg, 2003, pp. 377-390.

  4. M. BARO, H.-CHR. KAISER, H. NEIDHARDT, J. REHBERG, A quantum transmitting Schrödinger-Poisson system, Rev. Math. Phys., 16 (2004), pp. 281-330.

  5.          , Dissipative Schrödinger-Poisson systems, J. Math. Phys., 45 (2004), pp. 21-43.

  6. N. BEN ABDALLAH, P. DEGOND, P.A. MARKOWICH, On a one-dimensional Schrödinger-Poisson scattering model, Z. Angew. Math. Phys., 48 (1997), pp. 135-155.

  7. W. FRENSLEY, Boundary conditions for open quantum systems driven far from equilibrium, Rev. Mod. Phys., 62 (1990), pp. 745-791.

  8. H. GAJEWSKI, Analysis und Numerik von Ladungstransport in Halbleitern (Analysis and numerics of carrier transport in semiconductors), Mitt. Ges. Angew. Math. Mech., 16 (1993), pp. 35-57.

  9. H.-CHR. KAISER, H. NEIDHARDT, J. REHBERG, Density and current of a dissipative Schrödinger operator, J. Math. Phys., 43 (2002), pp. 5325-5350.

  10.          , Convexity of trace functionals and Schrödinger operators, WIAS Preprint no. 835, 2003, submitted.

  11.          , Macroscopic current induced boundary conditions for Schrödinger-type operators, Integral Equations Oper. Theory, 45 (2003), pp. 39-63.

  12.          , On one-dimensional dissipative Schrödinger-type operators, their dilations and eigenfunction expansions, Math. Nachr., 252 (2003), pp. 51-69.

  13. H.-CHR. KAISER, J. REHBERG, About a one-dimensional stationary Schrödinger-Poisson system with Kohn-Sham potential, Z. Angew. Math. Phys., 50 (1999), pp. 423-458.

  14.          , About a stationary Schrödinger-Poisson system with Kohn-Sham potential in a bounded two- or three-dimensional domain, Nonlinear Anal., Theory Methods Appl., 41 (2000), pp. 33-72.

  15.          , About some mathematical questions concerning the embedding of Schrödinger-Poisson systems into the drift-diffusion model of semiconductor devices, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., vol. 2, World Scientific, Singapore [u. a.], 2000, pp. 1328-1333.

  16. C.S. LENT, D.J. KIRKNER, The quantum transmitting boundary method, J. Appl. Phys., 67 (1990), pp. 6353-6359.

  17. C. WEISBUCH, B. VINTER, Quantum Semiconductor Structures: Fundamentals and Applications, Academic Press, Boston, 1991.


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2004-08-13