nextnano3 - Tutorial
next generation 3D nano device simulator
3D Tutorial
Transmission through a 3D nanowire (3D example)
Note: This tutorial's copyright is owned by Stefan Birner,
www.nextnano.de.
Author:
Stefan Birner
Please send comments to nextnano3 (-at-) wsi.tum.de.
If you want to obtain the input files that are used within this tutorial, please contact:
support [at] nextnano.de.
-> 3D_CBR_nanowire_10x10x20.in
(-> 3D_CBR_nanowire_10x10x20_holes.in - same but for holes
instead of electrons)
Transmission through a 3D nanowire (3D example)
-> 3D_CBR_nanowire_10x10x20.in
(-> 3D_CBR_nanowire_10x10x20_holes.in - same but for holes
instead of electrons)
In this tutorial we apply the Contact Block Reduction (CBR) method to a
simple GaAs nanowire of cuboidal shape.
|
schematic sketch of the 3D device showing the
GaAs region that is placed between two contacts
(red and
green leads)
|
|
nextnanomat screenshot
of the wavefunction of the 2nd lead mode
(nextnanomat has been developed by Jörg
Ehehalt) |
The device consists of two leads of 10 nm x 10 nm each. Each lead has a total
of 121 grid points (11 x 11 grid points).
The leads are described as quantum regions, i.e. in each lead (quantum region) a
two-dimensional Schrödinger equation has to be solved to obtain the
eigenenergies and wavefunctions of the lead modes.
The device dimensions are 10 nm x 10 nm x 20 nm.
The grid spacing is 1 nm in all directions.
The effective electron mass is assumed to be constant throughout the device
and equal to 0.067 m0.
The device region consists of 11 x 11 x 21 = 2541 grid points, which is equivalent to
a grid spacing of 1.0 nm.
This means that the device Hamiltonian is a matrix of size 2541 x 2541.
The conduction band profile is constant and set to Ec = 0 eV.
The following figure shows the calculated transmission coefficient as a
function of energy between the leads 1 and 2.
For the blue lines 23.6 % (600 of 2541) of all eigenvectors were used whereas for
the red lines only 15.7 % (400 of 2541) had to be calculated, i.e. one does not
have to calculate all eigenvalues of the device Hamiltonian which grossly
reduces CPU time.
For the black lines 7.9 % (200 of 2541) of
all eigenvectors were used.
A small percentage of eigenvalues suffices for T(E) in relevant energy range of
interest.
Note that the transmission drops significantly once the cutoff energy of the
highest eigenvector taken into account is reached.
The transmission coefficient can be found in this file:
CBR_data1/transmission3D_cb_sg001_CBR.dat
The same data as above is shown again as a zoom into the energy range 0 eV
- 0.5 eV.
The colored figures show the wavefunction amplitude of the lowest energy lead
modes.
They can be found in these files:
- CBR_data1/2Dmodes_lead001_sg001_psi_ev001.dat
- ...
- CBR_data1/2Dmodes_lead001_sg001_psi_ev080.dat
- CBR_data1/2Dmodes_lead002_sg001_psi_ev001.dat
- ...
- CBR_data1/2Dmodes_lead002_sg001_psi_ev080.dat
Once the energy reaches 78 meV, the first lead mode energy is reached and then
this mode transmits perfectly, giving a transmission of 1.
The second and third lead mode states are degenerate due to the symmetry of the
lead cross-section, thus they have the same energy (191 meV). Consequently, once
the energy of 191 meV is reached, the transmission increases by 2.
The total transmission is now equal to 3 as all lead modes transmit perfectly.
The energy of the 4th lead mode is at 305 meV.
The degeneracy of the 5th and 6th mode is accidental. They
have the same energy.
As once can clearly see, in this low energy limit, it is sufficient to
calculate only a few percent of all eigenfunctions of the device Hamiltonian.
For the leads, in all cases 80 eigenstates have been calculated.
Density of states
Using
calculate-CBR-DOS = yes
! flag: yes / no
the CPU time increases but information like the density of states (DOS) can be
obtained.
This is shown in the next figure where the same transmission data as above is
plotted but this time including the density of states.
Again, the colors indicate taking into account 200,
400 or 600
eigenvectors of the decoupled system (closed system).
Technical details
Definition of contacts
For each contact (lead), a quantum cluster ("lead quantum cluster") has to be
defined because in each lead,
a two-dimensional Schrödinger equation has to be solved which gives us the lead
modes (i.e. energies and eigenvectors of the leads).
In addition, a quantum cluster is required for the device itself ("main quantum
cluster").
!---------------------------------------------------!
$quantum-regions
!
region-number = 1
! 'device'
base-geometry = cuboid
!
region-priority = 1
!
x-coordinates = -5d0 5d0
! [nm] width of 'device' along x = 10 nm
y-coordinates = -5d0 5d0
! [nm] width of 'device' along y = 10 nm
z-coordinates = 0d0 20d0
! [nm] device length along z = 20 nm
region-number = 2
! 'source' (lead 1)
base-geometry = cuboid
!
region-priority = 2
!
x-coordinates = -5d0 5d0
! [nm] width of 'lead 1' along x = 10 nm
y-coordinates = -5d0 5d0
! [nm] width of 'lead 1' along y = 10 nm
z-coordinates = -2d0 -1d0
! [nm] (including 2 grid points along this direction)
region-number = 3
! 'gate' (lead 2)
base-geometry = cuboid
!
region-priority = 2
!
x-coordinates = -5d0 5d0
! [nm] width of 'lead 1' along x = 10 nm
y-coordinates = -5d0 5d0
! [nm] width of 'lead 1' along y = 10 nm
z-coordinates = 21d0 22d0
! [nm]
!---------------------------------------------------!
For each quantum cluster, the number of eigenstates to be calculated.
!-------------------------------------------------!
$quantum-model-electrons
!
...
cluster-numbers
= 1
...
number-of-eigenvalues-per-band = 200
! corresponds to 7.87 % of 2541
!number-of-eigenvalues-per-band = 400
!
!number-of-eigenvalues-per-band = 600
!
!-------------------------------------------------!
! lead 1 = source: number of modes per lead = 80 maximum number of relevant quantum grid points in lead 1
= 121
!-------------------------------------------------!
...
cluster-numbers
= 2
!
number-of-eigenvalues-per-band = 80
!
!-------------------------------------------------!
! lead 2 = source: number of modes per lead = 80
!-------------------------------------------------!
...
cluster-numbers
= 3
!
number-of-eigenvalues-per-band = 80
!
$end_quantum-model-electrons
!
!-------------------------------------------------!
!-------------------------------------------------!
$CBR-current
!
!
destination-directory = CBR_data1/
! directory for output and data files
calculate-CBR =
yes
! "yes"/"no"
!
main-qr-num =
1
! number of main quantum cluster for which transport is calculated:
'device'
num-leads
= 2
! total number of leads attached to main region
lead-qr-numbers =
2 3
!
propagation-direction = 3 3
!
'3'=z,
! 1,2,3)
for each lead
num-modes-per-lead = 80 80
!
! must be <= number of eigenvalues specified in
corresponding quantum model
!num-eigenvectors-used = 200
!
!num-eigenvectors-used = 400
!
num-eigenvectors-used = 600
!
!num-eigenvectors-used = 2541 !
2541=11*11*21=N_x*N_y*N_y!
E-min
= 0.0d0
! [eV]
E-max
= 2.5d0
! [eV]
num-energy-steps =
250
!
!
$end_CBR-current
!
!-------------------------------------------------!
Loop over energy (parallelization)
For each energy E (num-energy-steps = 250)
where the transmission coefficient T(E) has to be calculated,
a matrix of size 160 x 160 has to be inverted.
The size of 160 is determined by the sum over the number of lead modes taken
into account for each lead.
The upper limit would be the number of grid points in each lead that are in
contact to the device, i.e. in this example where each lead has 11 x 11 = 121
grid points, the maximum size of the matrix to be inverted could be 242 = 121 +
121.
- Lead 1 (Source): 121 grid points (80 lead modes taken into account)
- Lead 2 (Drain): 121 grid points (80 lead modes taken
into account)
==> in
total: 242 grid points (but only 80 modes are taken into account for each
lead: 160 = 80 + 80)
==> The total CPU time for calculation of the
transmission T(E) in this example was less than a minute for 200 eigenstates,
80 lead modes in each lead, and 250 energy steps.
This loop can be executed in parallel to improve computational performance.
$global-settings
...
!number-of-parallel-threads = 2
! (for dual-core CPU)
number-of-parallel-threads = 4
!
For further information, please study this section:
$CBR-current
- Please help us to improve our tutorial. Send comments to
nextnano3 (-at-) wsi.tum.de.
| 
