Note: This documentation is nearly 10 years old and
might not be related any more to the actual version of the nextnano³
It is still displayed here for historical reasons until a decent manual for nextnano³
will be available.
Interested readers should consult the diploma and PhD theses of S.
Hackenbuchner, M. Sabathil, T. Zibold (nextnano++) and T. Andlauer (nextnano++)
available from the publications
- nextnano3 is a simulator for calculating, in a
consistent manner, the realistic electronic structure of three-dimensional
heterostructure quantum devices under bias and its current density close to
equilibrium. The electronic structure is calculated fully quantum mechanically,
whereas the current is determined by employing a semiclassical concept of
local Fermi levels that are calculated self-consistently.
- The solving of the Schrödinger, Poisson, current continuity equations for
electrons and holes and the current relations for electrons and holes.
- The basic semiconductor equations:
|The Poisson equation:
|The current continuity equation for electrons
|The current relations for electrons
- A realistic simulator of three-dimensional semiconductor nano structures
and optoelectronic nano devices should meet two requirements.
Firstly it should model the electronic structure of any combination of quantum
wells, wires, and dots accurately on a length scale from nm to µm.
Secondly, a device simulator should self-consistently account for the charge
redistribution under applied voltage and for the resulting current [compelectron1].
Recently, several methods have been published that can realistically predict
the equilibrium electronic structure of 3D nano structures. they are based on
or several-band k.p models [grundmann1,
tight binding methods [dicarlo1]
or pseudopotential techniques [wang1,
Some of them include free and bound charge redistributions self-consistently [kumar1,
Most quantum transport methods that include the electronic structure beyond a
one-band description are still limited to one spatial dimension [lake1,
Thus, a simultaneous realistic treatment of the electronic structure and the
quantum transport problem for 3D structures still poses a challenging task.
- State-of the-art simulators for semiconductor nano structures and
optoelectronic nano devices roughly fall into two classes [compelectron1]:
some models focus on the equilibrium electronic structure. They attempt
to predict as accurately as possible, the free and bound charge density as
well as optical properties of quantum wells, wires, and dots on a lengths
scale that ranges from nm to µm. Several models of this kind have been
developed in the last few years that can deal with fully three-dimensional
device geometries, and invoke one-band, or several-band k.p models,
tight binding methods.
The second class of models focus on current-voltage characteristics and
attempts to solve quantum transport, using nonequilibrium Green functions [lake1,
Wigner functions [bordone1,
the Pauli master equation [fischetti1,
Presently, they are still limited to one spatial dimension and/or put less
emphasis on details of the electronic structure and the quantum transport
problem for 3D structures still poses a challenging task.
- We are currently developing a simulator for a wide class of 3D Si and
III-V nano structures [hackenbuchner1].
It attempts to bridge the two types of approaches described above, albeit with
a stringent limitation that makes it feasible to simulate three-dimensional
structures: we solve the electronic structure problem accurately but restrict
the current evaluation to situations close to equilibrium where the concept of
local quasi-Fermi levels is still justifiable. This approach may be viewed as
a low-field approximation to the Pauli master equation [jones1].
- The nano-device simulator that we have developed so far solves the 8-band-k.p-Schrödinger-Poisson
equation for arbitrarily shaped 3D heterostructure device geometries, and for
any (III-V and Si/Ge) combination of materials and alloys. It includes band
offsets of the minimal and higher band edges, absolute deformation potentials
local density exchange and correlations (i.e. the Kohn-Sham equations),
total elastic strain energy [pryor3,
that is minimized for the whole device, the long-range Hartree potential
induced by charged impurity distributions, voltage induced charge
redistribution, piezo- and pyroelectric charges, as well as surface charges,
in a fully self-consistent manner. The charge density is calculated for a
given applied voltage by assuming the carriers to be in a local
equilibrium that is characterized by energy-band dependent local quasi-Fermi
levels EFc(x) for charge carriers of type c, (i.e.
in the simplest case, one for holes and one for electrons)
- These local quasi-Fermi levels are determined by global current
where the current is assumed to be proportional to the density and to the
gradient of the quasi-Fermi level (associated with each band) exactly as in
the semi-classical limit (see e.g. [selberherr1]).
Recombination and generation processes are included additionally. The carrier
wave functions Psiic and energies Eic are
calculated by solving the multi-band Schrödinger-Poisson equation.
system is mimicked by using mixed Dirichlet and Neumann boundary
at ohmic contacts. The charge density at these contacts is assumed to be equal
to the bulk equilibrium density. Thus, the quasi-Fermi levels and the
potential in the contact region are fixed according to the applied voltage.
Our method leads to globally orthogonal eigenstates including valence (split-off,
light and heavy hole) and conduction band states. Further, it automatically
includes tunneling, and yields optical transition energies and as well as
optical matrix elements.
- According to equation (1), one and the same bound state
Psiic(x) may get occupied differently at different positions
according to the spatial dependence of EFc(x). This
is a consequence of invoking the well-defined but semiclassical concept of
local Fermi levels together with nonlocal quantum mechanics. Fortunately, no
conflict arises for situations close to equilibrium since the spatial
variation of the occupancy of any given eigenstate turns out to be negligible
for three reasons:
(i) Deeply bound states do not contribute to the current and thus do not lead
to a gradient of the Fermi level.
(ii) The Fermi level has the largest variation in regions where the density is
very low (within barriers, for example).
(iii) Very extended states that are treated formally as bound states in our
method are either not occupied because of their high energy, or occur in
regions of high density (near contacts, for example) where the quasi-Fermi
level is nearly constant.
- The computational methods [see
scheme for more details] solve the Kohn-Sham-Schrödinger,
current continuity equations iteratively using
conjugate gradient, inverse-iteration and predictor-corrector methods [trellakis1]
in an inhomogeneous
- For a given nano-structure, the computations start by globally minimizing
the total elastic energy [pryor3,
conjugate gradient method. This determines the piezo-induced charge
distributions, the deformation potentials and band offsets. Subsequently, the
current continuity equations are solved iteratively. All equations are
discretized according to the
method invoking the box integration scheme [kumar1,
selberherr1]. The irregular rectilinear mesh is kept fixed during the
calculations. As a preparatory step, the built-in potential is calculated for
zero applied bias by solving the Schrödinger and Poisson equation
self-consistently employing a predictor-corrector approach [trellakis1]
and setting to zero the electric field at ohmic contacts. For applied bias,
the Fermi level and the potential at the contacts are then shifted according
to the applied potential which fixes the boundary conditions. The main
iteration scheme itself consists of two parts.
In the fist part, the wave functions and potential are kept fixed and the
quasi-Fermi levels are calculated self-consistently from the
current continuity equations, employing a
conjugate gradient method and a simple relaxation scheme.
In the second part, the quasi-Fermi levels are kept constant, and the density
and the potential are calculated self-consistently from the Schrödinger and
Poisson equation. The
discrete 8-band Schrödinger equation represents a huge
(typically of dimension 105 for 3D structures) and is diagonalized
that yields the required inner eigenvalues and eigenfunctions close to the
energy gap. We very slightly shift the spin-up and spin-down diagonal
Hamiltonian matrix elements with respect to each other in order to avoid
degeneracies and guarantee orthogonal eigenstates automatically. To reduce the
number of necessary diagonalizations, we employ an efficient
predictor-corrector approach [trellakis1]
to calculate the potential from the nonlinear Poisson equation. In this
approach, the wave functions are kept fixed within one iteration and the
density is calculated perturbatively from the wave functions of the previous
The nonlinear Poisson equation is solved using a modified
Newton method, employing a
conjugate gradient method and line minimizations. The code is written in
Fortran 2003 and consists of some 200.000 lines by now.
- Piezoelectric fields and electron-hole localization in quantum dots
We have applied our simulator to study theoretically single quantum-dot
hackenbuchner1] consisting of self-assembled InGaAs quantum dots with a
diameter of 30-40 nm and heights of 4-8 nm that are embedded in the intrinsic
region of a Schottky diode.
Towards fully quantum
mechanical 3D device simulation
M. Sabathil, S. Hackenbuchner, J.A. Majweski, G. Zandler,
submitted to Journal of Computational Electronics (2001)
band structure of nano-devices
S. Hackenbuchner, M. Sabathil, J.A. Majweski, G. Zandler, P. Vogl, E.
Beham, A. Zrenner, P. Lugli
to appear in Physica B (2001)
- dimension of sample (1D, 2D or 3D)
- materials and shape of the heterostructure
- applied bias if any
- doping if any
- Where are the quantum regions, where should be calculated classically?
- specification of desired output
- band structure
- piezo- and pyroelectric charges
- electron/hole densities (space charge)
- electrostatic potential
- wave functions
- The input file is processed, i.e. the material data will be read in
from the data base and the geometry will be mapped on the grid.
- The strain is calculated.
- The band edges will be calculated by taking account of the
van-de-Walle model and the strain. This can possibly lead to a splitting of
degenerate energy states.
- The piezoelectric and pyroelectric polarization charges
will be determined.
- The program sets up quantum regions and allocates quantum states,
and all other relevant variables that contain the physical solutions.
- The main program starts.
- A starting value for the potential is determined.
- The nonlinear Poisson
equation in thermodynamic equilibrium will be determined leading to the
- Then the program can continue differently which will be discussed later.
- Eventually the results will be written into the specified files.
- Finally, here is a more
detailed explanation of the structure
of the program ("Setup").
Go to Basics and you will get a more deeper
understanding about how all fits together...