Collaborator: M. Baro,
A quantum transmitting Schrödinger-Poisson system
A. Jüngel (Johannes Gutenberg-Universität Mainz),
P. Degond, N. Ben Abdallah (Université Paul Sabatier,
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
This project is part of a long-term investigation of quantum
mechanical models for semiconductor nanostructures,
[2, 9, 10, 11,
12, 13, 14],
and their embedding into macroscopic models, like drift-diffusion and
energy models, cf.
and p. ,
for semiconductor devices,
in particular optoelectronic ones, cf.
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., .
We look for stationary states of a quasi-two-dimensional
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
is the single-particle effective-mass Hamiltonian in Ben-Daniel-Duke
form, is the reduced Planck constant, m = m(x) > 0 is the
spatially varying effective mass of the particle species under
= (k) is a dispersion relation, e.g.,
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
weighted by values of a distribution function f:
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
u(x) = cdk f ((k) - )|(x)|2 + cdk f ((k) - )|(x)|2, x (a, b).
where T is the temperature and kB Boltzmann's constant.
(2) can be written in the following way:
be the multiplication operator on
2() induced by the function
v : 2()2()be the Fourier transform
which diagonalizes the operator Kv on
vKvv * = ,where
is the maximal multiplication operator induced by
the dispersion relation
Then the operator
is a steady-state, that means a self-adjoint, positive operator
on the Hilbert space
2() which commutes
Moreover, any steady state can be expressed in the form
(4) by means of a function
The particle density u, defined by (2), is the
Radon-Nikodým derivative of the (Lebesgue) absolutely continuous
(a, b) tr (v)M()(
M() denotes the multiplication operator induced by the
of the set ) that means
for all Lebesgue measurable subsets 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 ,
cf. [4, Section 5.2].
In the asymptotic regions x < a and x > b the generalized
eigenfunctions 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
= (k), by means of
the quantum transmitting boundary method,
The corresponding homogeneous boundary conditions are
k , is the group velocity defined by
The differential expression (1), together with the boundary
conditions (6), sets up a family of maximal dissipative
operators on the Hilbert space
2(a, b). We call this
family, in the style of ,
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
, 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,
references cited there:
where q denotes the elementary charge, C is the density of ionized
dopants in the semiconductor device,
> 0 is the dielectric
permittivity function, and is the electrostatic potential,
v = wq 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
+ defined as the map of a
potential v to the density (5) with steady states
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
Ben Abdallah, Degond, and Markowich have investigated a special case
of this model in 
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.
- (x)(x) = qC(x) + +(v+)(x) - -(v-)(x), x (a, b),
The quantum transmitting Schrödinger-Poisson system is
closely related to the dissipative Schrödinger-Poisson
which we have investigated in , 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.
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