nextnano^{3}  Tutorial
next generation 3D nano device simulator
2D Tutorial
Schrödinger equation of a twodimensional coreshell structure
Hexagonal 2DEG  Twodimensional electron gas in a deltadoped hexagonal shaped
GaAs/AlGaAs nanowire heterostructure
Author:
Stefan Birner
Note: This tutorial's copyright is owned by Stefan Birner,
www.nextnano.com.
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
==> 2DGaAs_AlGaAs_circle.in
/ *_nnp.in  input file for the nextnano^{3} and nextnano++
software
==> 2DGaAs_AlGaAs_hexagon.in
/ *_nnp.in  input file for the nextnano^{3} and nextnano++
software
==> 2DGaAs_AlGaAs_hexagon_AlloySweep.in
 input file for the nextnano^{3} software
==> 2D_Hexagonal_Nanowire_2DEG.in / *_nnp.in  input file
for the nextnano^{3} and nextnano++ software
Schrödinger equation of a twodimensional coreshell structure
In this tutorial we solve the twodimensional Schrödinger equation of a
circular and a hexagonal GaAs/Al_{0.33}Ga_{0.67}As coreshell
structure.
Circular coreshell structure
==> 2DGaAs_AlGaAs_circle.in
The following figure shows the probability density of the 6^{th}
eigenstate of the circular GaAs/AlGaAs structure.
Schroedinger_1band/2Dcb001_qc001_sg001_deg001_dir_ev006.fld
Its energy level is higher than the AlGaAs barrier energy, i.e. this
state is not confined in the circular shaped GaAs quantum well.
The horizontal and vertical slices are through the center and show the square of
the probability amplitude of this eigenstate.
The GaAs core has a radius of 5 nm. (It cannot be recognized on this plot.)
The AlGaAs shell has a radius of 15 nm. It is surrounded by an infinite barrrier
which is defined by the circular quantum region of radius 15 nm (nextnano^{3}
input file).
(Note: The nextnano++ input file cannot use a circular quantum region.
Therefore a 15 nm x 15 nm square is used in this case as the quantum region. The
circular confinement in this case comes from the "band offset" due to the
surrounding material "air".)
Hexagonal coreshell structure
==> 2DGaAs_AlGaAs_hexagon.in
The following figure shows the conduction band edge of the hexagonal
GaAs/AlGaAs structure.
The GaAs region is indicated in black, the AlGaAs region in
blue.
Horizontal and vertical slices through the center show the energy of the
conduction band edge profile.
band_structure/cb2D_001_sub001.fld
The diameter of the hexagonal shaped GaAs core is ~8.66 nm (corresponding to
an outer radius of the core of 5 nm).
The diameter of the hexagonal shaped AlGaAs shell is ~26 nm (corresponding to an
outer radius of the shell of 15 nm).
The following figure shows the probability density of the 10^{th}
eigenstate of the circular GaAs/AlGaAs structure.
Schroedinger_1band/2Dcb001_qc001_sg001_deg001_dir_ev010.fld
Its energy level is higher than the AlGaAs barrier energy, i.e. this
state is not confined in the hexagonal shaped GaAs quantum well.
The horizontal and vertical slices are through the center and show the square of
the probability amplitude of this eigenstate.
The hexagonal GaAs core has an outer radius of 5 nm. It cannot be seen on
this plot.
The AlGaAs shell has a diameter of 26 nm. It is surrounded by an infinite
barrrier which is defined by the hexagonal shaped quantum region of diameter 26
nm (nextnano^{3} input file).
(Note: The nextnano++ input file cannot use a hexagonal quantum region.
Therefore a 16 nm x 16 nm square is used in this case as the quantum region. The
hexagonal confinement in this case comes from the "band offset" due to the
surrounding material "air".)
Alloy sweep
==> 2DGaAs_AlGaAs_hexagon_AlloySweep.in
(There is no nextnano++ input file
available that performs an alloy sweep as this feature is not supported in the
input file. But you can use nextnanomat's
Template feature to perform an alloy
sweep with nextnano++.)
In the following, we vary the alloy content x of the ternary Al_{x}Ga_{1x}As
from 0 to 0.33 in 11 steps.
$alloyfunction
materialnumber
= 2
functionname
= constant ! spatially constant alloy
content
xalloy
= 0.0d0 !
alloy content x
alloysweepactive
= yes ! to
switch on alloy sweep
alloysweepstepsize =
0.03d0 ! increase alloy
concentration in each step by 0.03
alloysweepnumberofsteps = 11
! number of alloy sweep steps
For x = 0, we have pure GaAs.
For x = 0.33 we have an AlGaAs/GaAs conduction band offset of 0.285 eV, and a
valence band offset of 0.168 eV.
In the latter case, the quantum confinement is stronger.
Even for x = 0 we have "quantum confinement" due to the Dirichlet boundary
conditions (corresponding to infinite barriers) at the shell surface that we use
for the Schrödinger equation.
Consequently, even for x = 0, we get an e_{1}h_{1} transition
energy from the lowest electron state (e_{1}) to the highest heavy hole
state (h_{1}) that is larger than the band gap as shown in the following
figure.
The transition energies (e_{1}h_{1}), as well as the spatial
overlap intergral of the electron and hole ground state wave functions, are
contained in this file:
sweep_interband2D_qc001_vb001_sg001_cb001_sg001_deg001_dir.dat
It contains:
alloy type
elhl[eV] el[eV]
hl[eV] matrix_element
0.330000 <psi_vb001psi_cb001_>^2 1.522777615
2.965889676 1.443112062 0.936344908
0.300000 <psi_vb001psi_cb001_>^2 1.520316794
2.963669699 1.443352905 0.931291593
...
The spatial overlap of electron and hole wave functions is always very high.
When there is only confinement due to the shell boundary, the matrix element is
very high (99.8 %).
The matrix element must be smaller than 1 for x = 0 because the electron and
hole masses are different.
The matrix element must be even smaller (94 %) for x = 0.33 (strong confinement)
because in addition to the mass difference, the conduction and valence band
offsets are not equivalent.
The matrix element has a minimum at around x = 0.06 because in this case the
electron wave function penetrates into the barrier much stronger that the hole
wave function does.
Thus the differences in well and barrier masses (as well as band offsets) play a
high role for the spatial extension of the wave functions.
Sweep over further variables
It is also possible to sweep over variables, like e.g. the outer radius of
the GaAs core ("CoreOuterRadius") using nextnanomat's
"Template" feature.
Just select "Range of values", choose the variable
to be swept over, and specify a "from" "to"
range, and the "step".
The click on "Create input files" and switch to the
"Run" pane to start your batch list.
You can define your own variables using macros.
Macros are described here.
Hexagonal 2DEG  Twodimensional electron gas in a deltadoped hexagonal
shaped GaAs/AlGaAs nanowire heterostructure
==> 2D_Hexagonal_Nanowire_2DEG.in
The following example deals with a deltadoped GaAs/AlGaAs 2DEG
(twodimensional electron gas) structure.
In this case, the heterostructure consists of a hexagonal GaAs/AlGaAs nanowire.
The material layers are shown here in the following figure which shows the
content of this file: material_grid2D.fld
 black: GaAs core
 blue: AlGaAs spacer
 green:
Sidoped AlGaAs
 yellow:
AlGaAs
 red: GaAs capping layer

white: Schottky barrier contact: The white layer itself is not
included in the calculation. It only serves as a boundary condition.

Artist's view of a hexagonal nanowire.
(The artist is 4 years old. ;) 
The selfconsistently calculated conduction band edge (band_structure/cb2D_Gamma_sub001.fld )
is shown next.
The horizontal and vertical slices through the center indicate the triangular
potential well (conduction band minimum) where the 2DEG is located.
The resulting 2DEG electron density (densities/density2Del.fld )
is shown in the following figure.
At the corners, the electron density is significantly higher, thus
onedimensional conducting channels are formed.
Although the structure itself has a hexagonal symmetry, our rectangular grid
breakes this symmetry.
Therefore the density in the upper/lower corner are different from the density
at the left/right corners.
The 2D Poisson equation and the 2D Schrödinger equation have been solved
selfconsistently.
The dimension of the Schroedinger matrix is 28,625.
The CPU time for this calculation was about 18 minutes.
 Please help us to improve our tutorial. Send comments to
support
[at] nextnano.com .
