nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Applying the NEGF method to a quantum well
Author:
Stefan Birner,
Tillmann
Kubis
If you are interested in the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 1DNEGF_InGaAs_QW_ballistic_CBR.in  nextnano^{3}
input file using the CBR method
> 1DNEGF_InGaAs_QW_ballistic.in 
nextnano^{3} input file using the NEGF method
> 1DNEGF_InGaAs_QW_scattering.in 
nextnano^{3} input file using the NEGF method
> 1DNEGF_InGaAs_QW_scattering_bias.in  nextnano^{3}
input file using the NEGF method
Applying the NEGF method to a quantum well
This tutorial is based on the following paper:
[Kubis_HCIS14_2005]
Selfconsistent quantum transport theory of carrier capture in
heterostructures
T. Kubis, A. Trellakis, P. Vogl
Proceedings of the 14^{th} International Conference on
Nonequilibrium Carrier Dynamics in Semiconductors, M. Saraniti and U.
Ravaioli, eds., Chicago, USA, July 2519, 2005, Springer Proceedings in
Physics, vol. 110, pp. 369372
Quantum well structure  InGaAs/GaAs nin heterostructure
 12 nm In_{0.14}Ga_{0.86}As QW in the center surrounded
on each side by GaAs barriers (of width 19 nm each)
zcoordinates = 19.0d0 31.0d0
! 19  31 [nm]
 Doping: nin
GaAs nin structure with an InGaAs quantum well within the intrinsic
region.
 0 nm  17 nm: 17 nm ndoped region with n = 1 * 10^{18}
cm^{3}
 17 nm  33 nm: 16 nm intrinsic region, i.e. 12 nm InGaAs QW + 2
nm in GaAs barriers on each side
 33 nm  50 nm: 17 nm ndoped region with n = 1 * 10^{18}
cm^{3}
 T = 300 K (room temperature)
 device length = 50 nm
zcoordinates = 0.0d0 50.0d0
! 0  50 [nm]
 constant effective electron mass throughout the device (parabolic
dispersion): m* = 0.067 m_{0} (GaAs)
conductionbandmasses = 0.067d0
0.067d0 0.067d0 ! [m0] electron mass of GaAs
Note: This NEGF code is so general that it can take into account a
spatially varying, material dependent, nonparabolic effective electron mass.
To keep this tutorial as simple as possible, we use here the GaAs mass also
for the InGaAs QW.
 Dielectric constants
staticdielectricconstants = 12.93d0
12.93d0 12.93d0 ! epsilon(0)
opticaldielectricconstants = 10.89d0
! epsilon(infinity)
Note: This NEGF code is so general that it can take into account
spatially varying, material dependent dielectric constants.
To keep this tutorial as simple as possible, we use here the GaAs values
also for the InGaAs QW.
 constant polar optical phonon energy throughout the device: 0.035 eV
(GaAs)
LOphononenergy = 0.035d0 ! [eV]
 GaAs/In_{0.14}Ga_{0.86}As conduction band offset: CBO =
0.150 eV
(For simplicity, we do not take into account strain.)
 Grid resolution: 1 nm, i.e. 49 grid points in total (plus the two
boundary points).
This corresponds to 50 equidistant grid points within the NEGF part.
 Ohmic contacts: The two reservoirs at the left and right boundaries are
in thermal equilibrium.
Ballistic NEGF calculation
> 1DNEGF_InGaAs_QW_ballistic.in
 V_{SD} = 0 eV
(sourcedrain), see Fig. 1 (a) in
[Kubis_HCIS14_2005]
 ballistic device (absence of any scattering)
 electronelectron scattering in Hartree approximation, see Fig. 1 (a) in
[Kubis_HCIS14_2005]
Necessary feature for ballistic calculation
$scatteringmechanisms
...
ballistic
= yes ! Fig.
1(a): switch off scattering (ballistic calculation)  to make calculation faster
Doping profile
The following figure shows the doping profile of the structure.
The structure is ntype doped (n = 1 * 10^{18} cm^{3}) with a
16 nm intrinsic region in the center.
The conduction band edge is also indicated to indicate the position of the 12 nm
wide InGaAs quantum well.
The output for the doping profile can be found in the following files:
 doping_concentration1D.dat  doping profile as specified in input
file for each doping function
 doping_profile1D.dat  total doping
profile as specified in input file (ntype + ptype)
 NEGF/dopingV_new.dat  doping profile used by NEGF code
(should by ntype)
Conduction band edge
The following figure shows the conduction band edge energy of the GaAs/InGaAs
heterostructure.
The black line is the band edge that is input for the calculation. It
can be found in this file:
NEGF/ex_potential_new.dat (in units of [eV])
The red line is the selfconsistently
calculated conduction band profile (calculated by NEGF method) and can be found
in this file:
NEGF/pot_new.dat (in units of
[eV])
Note: The conduction band edge of nextnano³ is usually contained in this
file:
band_structure/cb1D_001.dat (in units of [eV])
(or)
Energyresolved electron density
The following figure on the left shows the contour plot of the local,
energyresolved electron density n as a function of position z and
energy E at zero bias and in the absence of any scattering (ballistic
device) for the nin GaAs structure with an InGaAs quantum well in the center.
n(z,E) is in units of [10^{18} eV^{1}cm^{3}].
Integrating n(z,E) over the energy E yields the electron
density n(z).
The black line shows the selfconsistently calculated conduction band
edge profile.
The figure on the right shows a contour plot of the energyresolved local
density of states LDOS(z,E) in units of [eV^{1} Angstrom^{1}].
The LDOS(z,E) is known as the spectral function.
The zero of energy is put at the Fermi level of the left contact.
In the calculation, the Hartree Coulomb interaction was taken into account
(Poisson equation).
Due to the absence of inelastic scattering, the carriers cannot fill the
quantum well but occupy a resonance that is derived from the third quantum well
state. This explains the interference pattern in the well region at room
temperature.
The energyresolved electron density n(z,E) can be found
in this file:
NEGF/density_energy_resolved.fld  in units of
[10^{18} cm^{3 }eV^{1}]
(AVS format)
The energyresolved electron density n(z,E_{z}) can be found
in this file:
NEGF/density_Ez_energy_resolved.fld  in units of
[10^{18 }cm^{3 }eV^{1}]
(AVS format)
The energyresolved local density of states (spectral function) LDOS(z,E)
can be found in this file:
NEGF/spectral_real_avs.fld
 in units of [eV^{1} Angstrom^{1}] (AVS
format)
The conduction band profile can be found in this file:
NEGF/pot_avs.fld
 in units of [eV]
(AVS format)
NEGF/pot_new.dat
 in units of [eV]
NEGF calculation including scattering  zero bias (equilibrium)
> 1DNEGF_InGaAs_QW_scattering.in
Acoustic (both, elastic and inelastic) and polaroptical phonon scattering, electronelectron scattering in
Hartree approximation, see Fig. 1 (b) and 2 (a) in [Kubis_HCIS14_2005] .
 V_{SD} = 0 eV
(sourcedrain), see Fig. 1 (b) in
[Kubis_HCIS14_2005] .
The following scattering mechanisms were used:
$scatteringmechanisms
!
! Note: lattice_constant a and sound_velocity v
determine the
! dispersion relation of acoustic phonons: E_LA = h_bar v q
where q is from 0 to pi/a .
!
acoustic_phonons =
inelastic ! inelastic acoustic phonon scattering
! acoustic_phonons =
elastic
! elastic acoustic phonon scattering  as in paper (Fig.
1(b))
lattice_constant =
5.6534d0 ! GaAs [Angstrom]  Note: [Angstrom]
not [nm].
sound_velocity =
5.2d13 ! GaAs [Angstrom/s]
optical_phonons =
yes ! Fig. 1(b)
longitudinal polaroptical phonon scattering (polar LO phonon scattering)
charged_impurity = no
! charged impurity scattering not included in
Fig. 1(b)
interface_roughness = no
! interface roughness scattering not included in Fig. 1(b)
ballistic
= no !
Fig. 1(b): include scattering mechanisms
! maximum number of scattering events in the contacts
contact_scat =
7 !
contact scattering (number of scattering iterations in contact)
contact_sc_pot =
0d0 ! only for
periodic contacts ! scattering potential height in the contacts (only for
periodic contacts) in units of [eV]
Note: For longitudinal polaroptical phonon scattering (polar LO phonon
scattering, optical_phonons =
yes ), the LO phonon energy has to be
specified in the material section:
LOphononenergy =
0.035d0 ! [eV] LO phonon energy, GaAs LO phonon energy:
0.035 eV
Note: In contrast to the paper, where only elastic acoustic
phonon scattering was used, we consider here in this tutorial inelastic
scattering.
(Note: Elastic scattering is thus included automatically.)
The effect of scattering can be seen in the following figures.
The figure on the left shows a contour plot of the energy resolved electron
density in the same situation as above, but this time including scattering fully
selfconsistently. All phonon scattering mechanisms are fully included.
The lowest well state is a true bound state whereas the second one is a
resonance state an electron can tunnel out from into the leads.
Integrating n(z,E) over the energy E yields the electron
density n(z) which is shown in the
following figure, including the selfconsistently calculated conduction band
edge. The Fermi level E_{F} is also
shown.
NEGF calculation including scattering  150 mV bias
> 1DNEGF_InGaAs_QW_scattering_bias.in
Acoustic (both, elastic and inelastic) and polaroptical phonon scattering, electronelectron scattering in
Hartree approximation, see Fig. 1 (b) and 2 (a) in [Kubis_HCIS14_2005] .
 V_{SD} = 0.150 eV (sourcedrain), see Fig. 2 (a) in
[Kubis_HCIS14_2005] .
Results
The following two figures show the energy resolved electron density and the
energy resolved local density of states for the QW under an applied bias of
0.150 V.
The following figure shows the energy resolved current density j(z,E)
in units of [Ampere/(cm^2 eV)] .
The relevant data is contained in these files:
NEGF/EnergyResolvedCurrent_avs.v / *.dat / *.coord / *.fld .
The following figure shows the selfconsistently calculated conduction
band profile and the electron density n(z).
Boundary conditions for the Poisson equation
$globalparametersNEGF
...
!
! for Poisson equation: flat band boundary condition
! ==> Neumann boundary condition: d phi / d z = 0
!
given_slope = yes ! The slope of the poisson potential at the contacts is fixed: flat
band
poisson_slope = 0d0 ! [V/Angstrom]
chemical_potential = 0d0 ! [eV]
chemical potential (Fermi level) at the right contact
zero_drift =
yes ! yes =
equilibrium contacts
Grid points  Resolution
$globalparametersNEGF
grid_points_in_z = 50
! number of grid points in real space along z direction.
! It must hold: nextnano3 grid points  1 = grid_points_in_z
grid_points_in_E = 80
! E = total energy
grid_points_in_Ez = 120 ! k_{}
resolution, E_{z} = E  h_{bar}^{2} * k_{}^{2}
/ [2m(1)], 1 = 1^{st} grid point
Applied bias (a voltage sweep is optional)
$poissonboundaryconditions
!
! Note: For NEGF, the voltage sweep applies to the left contact.
!
poissonclusternumber = 1
! Source (left contact)
regionclusternumber = 1
! an integer number which refers to an existent regioncluster
!appliedvoltage =
0.0d0 ! 0
V ! [V] apply
voltage to poissoncluster
appliedvoltage = 0.150d0
! 0.155 V = 150 mV ! [V] apply voltage to poissoncluster
boundaryconditiontype = ohmic
contactcontrol =
voltage
poissonclusternumber = 2
! Drain (right contact)
regionclusternumber = 4
! an integer number which refers to an existent regioncluster
appliedvoltage =
0.0d0 ! 0 V
! [V] apply voltage to poissoncluster
boundaryconditiontype = ohmic
contactcontrol =
voltage
This keyword is necessary in order to loop over different voltages.
$voltagesweep
sweepnumber
= 1
!
sweepactive
= no ! 'yes'/'no'
poissonclusternumber = 1
! Note: For NEGF, the voltage sweep applies to the left contact.
stepsize
= 0.010d0 ! [V] 0.010
V = 10 mV
numberofsteps =
5 !
Chemical potentials (often called Fermi levels) at the contacts:
==> right contact = chemical potential
! appliedvoltage = 0.0d0
==> left contact = right contact + e * applied voltage
! appliedvoltage = 0.150d0
(preliminary, has to be generalized!)
 Please help us to improve our tutorial. Send comments to
support
[at] nextnano.com .
