 
Overview
Note: This documentation is nearly 10 years old and
might not be related any more to the actual version of the nextnano³
code.
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
website.
 nextnano^{3} is a simulator for calculating, in a
consistent manner, the realistic electronic structure of threedimensional
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 selfconsistently.
 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
and holes: 

The current relations for electrons
and holes: 


 A realistic simulator of threedimensional 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 selfconsistently 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
oneband [kumar1],
or severalband k.p models [grundmann1,
pryor1,
pryor2,
cusack1],
tight binding methods [dicarlo1]
or pseudopotential techniques [wang1,
dicarlo1].
Some of them include free and bound charge redistributions selfconsistently [kumar1,
grundmann1].
Most quantum transport methods that include the electronic structure beyond a
oneband description are still limited to one spatial dimension [lake1,
fischetti1,
ferry1].
Thus, a simultaneous realistic treatment of the electronic structure and the
quantum transport problem for 3D structures still poses a challenging task.
 Stateof theart 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 threedimensional
device geometries, and invoke oneband, or severalband k.p models,
tight binding methods.
The second class of models focus on currentvoltage characteristics and
attempts to solve quantum transport, using nonequilibrium Green functions [lake1,
klimeck1],
Wigner functions [bordone1,
grubin1,
ferry1] or
the Pauli master equation [fischetti1,
fischetti2].
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
IIIV 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 threedimensional
structures: we solve the electronic structure problem accurately but restrict
the current evaluation to situations close to equilibrium where the concept of
local quasiFermi levels is still justifiable. This approach may be viewed as
a lowfield approximation to the Pauli master equation [jones1].
 The nanodevice simulator that we have developed so far solves the 8bandk.pSchrödingerPoisson
equation for arbitrarily shaped 3D heterostructure device geometries, and for
any (IIIV and Si/Ge) combination of materials and alloys. It includes band
offsets of the minimal and higher band edges, absolute deformation potentials
[vandewalle1],
local density exchange and correlations (i.e. the KohnSham equations),
total elastic strain energy [pryor3,
grundmann2]
that is minimized for the whole device, the longrange Hartree potential
induced by charged impurity distributions, voltage induced charge
redistribution, piezo and pyroelectric charges, as well as surface charges,
in a fully selfconsistent 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 energyband dependent local quasiFermi
levels E_{Fc}(x) for charge carriers of type c, (i.e.
in the simplest case, one for holes and one for electrons)
(1)
 These local quasiFermi levels are determined by global current
conservation,
where the current is assumed to be proportional to the density and to the
gradient of the quasiFermi level (associated with each band) exactly as in
the semiclassical limit (see e.g. [selberherr1]).
Recombination and generation processes are included additionally. The carrier
wave functions Psi_{ic} and energies E_{ic} are
calculated by solving the multiband SchrödingerPoisson equation.
The open
system is mimicked by using mixed Dirichlet and Neumann boundary
conditions [fischetti1,
lent1,
frensley1]
at ohmic contacts. The charge density at these contacts is assumed to be equal
to the bulk equilibrium density. Thus, the quasiFermi levels and the
potential in the contact region are fixed according to the applied voltage.
Our method leads to globally orthogonal eigenstates including valence (splitoff,
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
Psi_{ic}(x) may get occupied differently at different positions
according to the spatial dependence of E_{Fc}(x). This
is a consequence of invoking the welldefined 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 quasiFermi
level is nearly constant.
 The computational methods [see
computational
scheme for more details] solve the KohnShamSchrödinger,
Poisson and
current continuity equations iteratively using
conjugate gradient, inverseiteration and predictorcorrector methods [trellakis1]
in an inhomogeneous
finite difference
framework.

 For a given nanostructure, the computations start by globally minimizing
the total elastic energy [pryor3,
grundmann2]
using a
conjugate gradient method. This determines the piezoinduced charge
distributions, the deformation potentials and band offsets. Subsequently, the
8bandSchrödinger,
Poisson, and
current continuity equations are solved iteratively. All equations are
discretized according to the
finite difference
method invoking the box integration scheme [kumar1,
selberherr1]. The irregular rectilinear mesh is kept fixed during the
calculations. As a preparatory step, the builtin potential is calculated for
zero applied bias by solving the Schrödinger and Poisson equation
selfconsistently employing a predictorcorrector 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
quasiFermi levels are calculated selfconsistently from the
current continuity equations, employing a
conjugate gradient method and a simple relaxation scheme.
In the second part, the quasiFermi levels are kept constant, and the density
and the potential are calculated selfconsistently from the Schrödinger and
Poisson equation. The
discrete 8band Schrödinger equation represents a huge
sparse matrix
(typically of dimension 10^{5} for 3D structures) and is diagonalized
using the
iterative methods
that yields the required inner eigenvalues and eigenfunctions close to the
energy gap. We very slightly shift the spinup and spindown 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
predictorcorrector 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
iteration [trellakis1].
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 electronhole localization in quantum dots
We have applied our simulator to study theoretically single quantumdot
photodiodes [findeis1,
frey1,
itskevich1,
kapteyn1,
hackenbuchner1] consisting of selfassembled InGaAs quantum dots with a
diameter of 3040 nm and heights of 48 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,
P. Vogl
submitted to Journal of Computational Electronics (2001)
Nonequilibrium
band structure of nanodevices
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
 strain
 piezo and pyroelectric charges
 electron/hole densities (space charge)
 electrostatic potential
 current
 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
vandeWalle 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
builtin potential.
 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...
(pdf/dvi/doc files)
