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Quantum transport - Green's functions (NEGF)

The method of calculating the carrier transport is defined as "fully self-consistent nonequilibrium Green's function (NEGF) approach for vertical quantum transport in open quantum devices with contacts". This part of the nextnano³ code is based on the original code of Tillmann Kubis which is described in this publication:

Self-consistent quantum transport theory: Applications and assessment of approximate models
T. Kubis, P. Vogl
Journal of Computational Electronics (2006)
DOI: 10.1007/s10825-006-0078-6

There are several possibilities:

to include several scattering mechanisms (e.g. inelastic scattering, elastic scattering)
no scattering at all ("ballistic transport")

The electrons are described within a one-band model with a variable effective mass, i.e. a spatially dependent (= material dependent) effective electron mass m_e(z). Alternatively, it is possible to use an energy dependent effective mass (nonparabolicity). This nonparabolicity parameters are grid point dependent. The static and optical dielectric constants are also grid point dependent, i.e. material dependent.

This method is well suited to study resonant tunneling diodes and quantum cascade lasers.

Restrictions for green:

homogeneous grid
not too much grid points (~50)
for nextnano³: quantum cluster must extend over the whole device
for nextnano³: two contacts at the boundaries having 2 grid points at the left and 2 grid points at the right contact,
material at the contacts should be the same as the semiconductor material

For an example of the Green's function functionality, have a look at the RTD tutorial.

Global parameters for Green's function code

!----------------------------------------------------------! $global-parameters-green optional ! grid_points_in_z integer required ! grid_points_in_Ez integer optional ! grid_points_in_E integer optional ! device_length double optional ! [Angstrom] non_diagonal_range double optional ! [Angstrom] contact_points integer optional ! chemical_potential double optional ! [eV] max_energy_factor double optional ! [eV] Ez_grid_power double optional ! E_grid_power double optional ! grid_exponent double optional ! zero_drift character optional ! given_slope character optional ! poisson_slope double optional ! [V/Angstrom] max_drift character optional ! drift_length double optional ! off_drift double optional ! grid_limit double optional ! long_conv_limit double optional ! ! gain character optional ! gain-output-every-nth-iteration integer optional ! gain-integrate-device-from-to double_array optional ! [nm] min_photon double optional ! [eV] max_photon double optional ! [eV] photon_number integer optional ! ! transmission character optional ! ! no_poisson character optional ! built_in_potential double optional ! [V] correlation character optional ! fermi character optional ! ! get-cb-from-nextnano character optional ! get-cb-masses-from-nextnano character optional ! get-nonparabolicity-from-nextnano character optional ! get-dielectric-from-nextnano character optional ! get-doping-from-nextnano character optional ! ! directory-green character optional ! directory-contact character optional ! directory-scattering-rates character optional ! directory-test-debug character optional ! directory-stop character optional ! !not really relevant (CHECK!): dielectric_eps0 double optional ! [] dielectric_epsinf double optional ! [] ! $end_global-parameters-green optional ! !----------------------------------------------------------!

!----------------------------------------------------------! $global-parameters-green ! !temperature = 300d0 ! [K], see$global-parameters grid_points_in_z = 40 !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 = 110 ! E =total energy grid_points_in_Ez = 110 !k_|| resolution,Ez = E - h_bar²* k_||²/ [2m(1)], 1 =1^st grid point device_length = 400d0 ! [Angstrom]length of device!non_diagonal_range = 10d0 ! [Angstrom]for ballistic and to make calculation faster non_diagonal_range = 30d0 ! [Angstrom]for ballistic and to make calculation faster!non_diagonal_range = 80d0 !correct physics ! [Angstrom] !has to be increased if the screening length is too large !scattering range relevant for polar optical phonon scattering and charged impurity scattering !must be larger thandevice_length / number of grid points contact_points = ! chemical_potential = 0d0 ! [eV](often called Fermi level) !chemical potential at the right contact !left contact = right contact + e * applied voltage zero_drift = yes ! yes =equilibrium contacts,no =nonequilibrium contacts given_slope = character ! poisson_slope = double ! [V/Angstrom]deleted: zero_slope = no ! yes = constant electrostatic potential at the boundary
!(derivate of electrostatic potential has zero slope, i.e. d phi / d z = 0) ! no =electrostatic potential is calculated self-consistently max_drift = no !if zero_drift = no, thenmax_driftcan beyes !(useful for extreme high current densities) off_drift = 0d0 !should be zero!Ez_grid_power = 2d0 !exponent n, i.e.xⁿ -if present, static Ez grid !if not present, a powerlaw grid for Ez is used (self-consistent multigrid)!E_grid_power = 1d0 !same as Ez_grid_power, but here for E grid grid_exponent = -3d-1 ! default:-3d-1 max_energy_factor = 12d0 ! []controls maximum of considered energy!grid_limit = 5d-2 ! []controls multigrid convergence long_conv_limit = 5d-5 ! []limit for long convergency, default is5d-5 ! ! gain = yes ! gain-output-every-nth-iteration = 10 !output gain every 10^th iteration (default: 10) gain-integrate-device-from-to = 5d0 65d0 ! [nm]Integrate alpha(z,E) fromz_min= 5 nmtoz_max= 65 nm. min_photon = 1d-3 ! [eV]minimum photon energy relevant for gain max_photon = 2d-2 ! [eV]maximum photon energy relevant for gain photon_number = 20 !number of energy grid steps between min_photon andmax_photon ! ! get-cb-from-nextnano = yes ! get-cb-masses-from-nextnano = yes ! get-nonparabolicity-from-nextnano = yes ! get-dielectric-from-nextnano = yes ! get-doping-from-nextnano = yes ! ! ! directory-green = NEGF_data/ ! directory-contact = contact/ ! ==> NEGF_data/contact/ directory-scattering-rates = sc_rates ! ==> NEGF_data/sc_rates/ directory-test-debug = test_debug/ ! directory-stop = stop/ ! ==> NEGF_data/stop/ ! ! dielectric_eps0 = 12.93d0 ! []static dielectric contstant (redundant, should be deleted) dielectric_epsinf = 10.89d0 ! []optical dielectric constant (redundant, should be deleted) ! $end_global-parameters-green ! !----------------------------------------------------------!

transmission = yes ! 'yes' / 'no' (default: no)
Flag to switch on/off calculation of transmission function T(E).

no_poisson = yes ! 'yes' / 'no' (default: no, i.e solve Poisson equation) Flag to switch off Poisson equation inside the NEGF algorithm.

built_in_potential = 0.5d0 !built-in potential in units of[V], default: 0 V This optional flag introduces an additional built-in potential, e.g. necessary for pn junctions.
First, the Poisson equation has to be solved in equilibrium to determine the built-in potential.
This value is then taken and specified in the input file.

correlation = yes ! 'yes' / 'no' (default: no) Flag to switch on/off plotting the correlation function(s).

fermi = yes ! 'yes' / 'no' (default: no) Flag to switch on/off the call to the subroutine that determines the quasi Fermi level by dividing greenL with the spectral function.
(correspond to Fig. 1 in IWCE-11 paper)

gain                              = yes      ! 'yes' / 'no' (default: no) Flag to switch on/off the calculation of the gain.
gain-output-every-nth-iteration   = 10       !output gain every10^th iteration (default: 10) When the whole calculation has converged, the gain will be printed out in any case. If one is interested in the gain, one also has to specify
gain-integrate-device-from-to     = 5d0 65d0 ! [nm]Integrate alpha(z,E) fromz_min= 5 nmto z_max= 65 nm.                                              !This is important in order to exclude the absorption of the contacts.                                              !The affected output files are:                                              ! gain_real_integrated_energy.dat     alpha(E)                                              ! gain_real_integrated_wavelength.dat alpha(lambda) min_photon                        = 1d-3     ! [eV]minimum photon energy relevant for gain max_photon                        = 2d-2     ! [eV]maximum photon energy relevant for gain photon_number                     = 20       !number of energy grid steps between min_photon and max_photon

get-cb-from-nextnano = yes ! 'yes' / 'no' Flag to read in conduction band edge profile (Gamma point) of nextnano³ calculation.

get-cb-masses-from-nextnano = yes ! 'yes' / 'no' Flag to read in conduction band effective masses profile (Gamma point) of nextnano³ calculation.

get-nonparabolicity-from-nextnano = yes ! 'yes' / 'no' Flag to read in nonparabolicity parameter of conduction band effective mass (Gamma point) of nextnano³ calculation.

get-dielectric-from-nextnano = yes ! 'yes' / 'no' Flag to read in the static and optical dielectric constants of nextnano³ calculation.

get-doping-from-nextnano = yes ! 'yes' / 'no' Flag to read in n-type doping profile of nextnano³ calculation.
Note: All donors are assumed to be ionized.

Specify directories for output files. If these specifiers are not present, the default values are taken.
Note that the directories must be present, as the nextnano³ code cannot create them.
Be sure to include the "/" (slash). On Windows systems, also the"\"(backslash) will work.

directory-green = NEGF_data/ ! directory-contact = contact/ ! ==> NEGF_data/contact/ directory-scattering-rates = sc_rates ! ==> NEGF_data/sc_rates/ directory-test-debug = test_debug/ ! directory-stop = stop/ ! ==> NEGF_data/stop/

Damping parameters (used to influence the convergence of the equations)

!----------------------------------------------------------! $damping-parameters optional ! poisson_damping1 double optional ! poisson_damping2 double optional ! poisson_damping3 double optional ! self_damping1 double optional ! self_damping2 double optional ! self_damping3 double optional ! drift_damping1 double optional ! drift_damping2 double optional ! drift_damping3 double optional ! slope_damping1 double optional ! slope_damping2 double optional ! slope_damping3 double optional ! $end_damping-parameters optional ! !----------------------------------------------------------!

All values for the damping parameter should be between zero and 1:0 <= x < 1

!----------------------------------------------------------! $damping-parameters ! poisson_damping1 = 0.8d0 !for Poisson equation poisson_damping2 = 0.8d0 !for Poisson equation poisson_damping3 = 0.8d0 !for Poisson equation self_damping1 = 0d0 !for scattering selfenergies self_damping2 = 0d0 !for scattering selfenergies self_damping3 = 0d0 !for scattering selfenergies drift_damping1 = 0d0 !for nonequilibrium contacts, not used ifzero_drift = yes drift_damping2 = 0d0 !for nonequilibrium contacts, not used ifzero_drift = yes drift_damping3 = 0d0 !for nonequilibrium contacts, not used ifzero_drift = yes slope_damping1 = ! slope_damping2 = ! slope_damping3 = ! $end_damping-parameters ! !----------------------------------------------------------!

Scattering mechanisms

!----------------------------------------------------------! $scattering-mechanisms optional ! acoustic_phonons character optional ! inel_acoustic_phonons character optional ! artificial_acoustic double optional !prefactor (for testing purposes) lattice_constant double optional ! sound_velocity double optional ! optical_phonons character optional ! LO_phonon_energy double optional ! charged_impurity character optional ! interface_roughness character optional ! roughness_width double optional ! correlation_length double optional ! ! ballistic character optional ! pauli_principle double optional ! !direct_contact character optional ! deleted!laser_contact character optional !deleted contact_scat integer optional !maximum number of scattering events in the contacts contact_sc_pot double optional ! scattering potential height in the contacts (only for periodic contacts) in units of [eV] scattering_boost character optional ! scattering_boost_factor integer optional ! scattering_boost_limit double optional !$end_scattering-mechanisms optional ! !----------------------------------------------------------!

!----------------------------------------------------------! $scattering-mechanisms                                     ! acoustic_phonons      = no                                !elastic acoustic phonon scattering                                                           !If yes, then inelastic (inel_acoustic_phonons) should be no. inel_acoustic_phonons = yes                               !inelastic acoustic phonon scattering                                                           !If yes, then elastic (acoustic_phonons) should be no
                                                           !because elastic is already automatically included. artificial_acoustic   = double        ! not relevant    !prefactor (for testing purposes) lattice_constant      = 5.6534d0                        ! [Angstrom] Note: [Angstrom] not [nm], default is GaAs lattice constant:5.6534 [Angstrom] sound_velocity        = 5.2d13                          ! [Angstrom/s], default is GaAs sound velocity:5.2d13 [Angstrom/s] optical_phonons       = yes                               !longitudinal polar-optical phonon scattering (polar LO phonon scattering) LO_phonon_energy      = 0.035d0                           ! [eV]LO phonon energy, default is GaAs LO phonon energy:0.035 [eV] charged_impurity      = no                                !charged impurity scattering                                                           !                                                            ! interface_roughness   = no                                !interface roughness scattering roughness_width       = 10d0                              !roughness width in [Angstrom] in growth direction (z direction) correlation_length    = 80d0                              !correlation length for interface roughness in x and y directions in [Angstrom]                                                            !see also$roughness-profile                                                            ! !pauli_principle       = 0.5d0                             !Pauli principle (should not be changed, default is0.5)                                                           ! !ballistic             = no                                !include scattering mechanisms ballistic             = yes                               !switch off scattering (ballistic calculation)
                                                           !to make calculation faster contact_scat          = 7                                 !contact scattering (number of scattering iterations in contact)                                                           !maximum number of scattering events in the contacts direct_contact        = no                                !direct contact (should not be changed, default isno)$end_scattering-mechanisms                                 ! !----------------------------------------------------------!

Note: For interface roughness scattering, the file BesselI.dat must be present.

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.

ballistic = yes ! 'yes' / 'no'Flag to switch between ballistic and nonballistic calculation.
Ballistic does not include any scattering (and is thus a rather fast calculation). Its results do not really correspond to physical reality but still might give a reasonable insight into a physical problem as it represents an extreme case where scattering is absent (i.e. it should yield an upper boundary for the expected current).
Nonballistic includes scattering (and is thus a very time-consuming calculation). Its results correspond (or are at least close) to physical reality.

Contacts

!----------------------------------------------------------! $contact-type optional !(for nextnano³)$contact-type required !(for T. Kubis' version) type character required ! contact_temperature double optional ! [K] contact_sc_limit double optional ! start_left integer optional ! end_left integer optional ! start_right integer optional ! end_right integer optional ! contact_occupation character optional ! contact_poisson character optional ! left_drift double optional ! right_drift double optional ! contact_den_diff double optional ! slope_limit double optional ! $end_contact-type required !(for T. Kubis' version)$end_contact-type optional !(for nextnano³)!----------------------------------------------------------!

type = direct ! direct contacts = indirect !indirect contacts = laser !laser contacts = periodic !periodic contacts = real_periodic !real periodic contacts

contact_occupation = no !A Fermi distribution in the contacts is used. = yes !using quasi periodic electron distribution in the contacts = periodic !using quasi periodic electron distribution in the contacts
!Note: yes and periodic is equivalent.
In all other cases, a Fermi distribution in the contacts is used.

contact_poisson = no !A flat conduction band in the contacts (except external potentials) is used. = yes !using quasi periodic Poisson potential in the contacts = periodic !using quasi periodic Poisson potential in the contacts
!Note: yes and periodic is equivalent.
In all other cases, a flat conduction band in the contacts (except external potentials) is used.

Instead of using bulk contacts, one can use quasi Stark ladder contacts.

!----------------------------------------------------------! $left-contact-potential-profile optional ! left_potential_height double optional ! [eV] left_start_point integer optional ! left_end_point integer optional ! $end_left-contact-potential-profile optional ! !----------------------------------------------------------! !----------------------------------------------------------! $right-contact-potential-profile optional ! right_potential_height double optional ! [eV] right_start_point integer optional ! right_end_point integer optional ! $end_right-contact-potential-profile optional ! !----------------------------------------------------------!

Interface roughness

Here, the use can specify
- the position dependent roughness width in growth direction and
- the position dependent correlation length in growth direction.

!----------------------------------------------------------! $roughness-profile optional ! roughness_width double required ! [Angstrom] correlation_length double optional ! [Angstrom] start_point integer optional ! end_point integer optional ! $end_roughness-profile optional ! !----------------------------------------------------------!

Output

All output files will be written to the folder "data/".

General files

ex_potential.dat: external potential = conduction band edge (without electrostatic potential) in units of[eV] ex_potential_new.dat: grid point in [nm] conduction band edge in [eV] (without electrostatic potential)
pot.dat:                                                conduction band edge (incl. electrostatic potential) in units of[eV] pot_new.dat:                              grid point in [nm]       conduction band edge in [eV] (incl. electrostatic potential)
pot_avs.dat/*.coord/*.fld:    AVS ooutput files that can be used to plot the
                                                               conduction band edge (incl. electrostatic potential) in units of[eV]                                                       with AVS/Express visualization software
dopingV_new.dat: grid point in [nm] doping concentration in [10¹⁸ e/cm^-3]
massV_new.dat: grid point in [nm] Gamma conduction band effective mass in [m₀]
nonparabolicity_new.dat: grid point in [nm] nonparabolicity parameter for Gamma conduction band effective mass in [1/eV]
eps_infinity_new.dat: grid point in [nm] optical dielectric constant epsilon_infinity in []
eps_static_new.dat: grid point in [nm] static dielectric constant epsilon₀ in []
roughness_width_new.dat grid point in [nm] roughness width in [Angstrom]
correlation_length_new.dat grid point in [nm] correlation length in [Angstrom]
Emapping.dat: energy resolution (energy grid) in units of[eV]
Ezmapping.dat: energy resolution in growth direction (z) (energy grid) in units of [eV]
poisson2.dat: electrostatic potential (will be updated for each Poisson iteration)
==> The feature "solving the Poisson equation" can be switched off:
no_poisson = yes ! 'yes' / 'no'
dissipated_power.dat !in units of[Watts/cm²]The dissipated power will be printed out for each grid point.
averaged_dissipated_power.dat ! in units of[Watts/cm²]The average dissipated power is the average of the dissipated power at each grid point.
This quantity is very interesting to study the heating of the device during operation.

AVS files

EnergyResolvedDensity.dat: energy resolved density "density(z,E)": z, energy, density in units of [eV^-1*10²⁴ cm^-3].

EnergyResolvedDensity_avs.fld, *.coord, *.datdensity(z,E): corresponds to Fig. 4 in IWC-11 paper
Note: For AVS output, we scale the density from [eV^-1*10²⁴ cm^-3] to [eV^-1*10¹⁸ cm^-3].
EnergyResolvedDensity_0.dat: energy resolved density: matrix z x E (contains density for each matrix element density(z,E))
spectral_real.dat: energy resolved local density of states (LDOS) (see Fig. in ICPS poster) (z,Ez,lDOS(z,Ez))
in units of [1 / (eVAngstrom)].
spectral_real_avs.fld , *.coord, *.dat (spectral_aimag_avs.fld, *.coord, *.dat)The imaginary part of the diagonal of the spectral function should be zero.
spectrum_ana.dat: matrix representation of 'spectral_real.dat'
Optical gain within linear response theory
These files contain the absorption alpha (and gain):
alpha(z,E) where z is the spatial coordinate and E is the photon energy.
Note that positive values correspond to absorption, negative values to gain.
- gain_real_avs.fld, *.coord, *.dat (gain_imag_avs.fld, *.coord, *.dat) -The x axis is the distance in units of [nm].
- The y axis is the photon energy in units of [eV].
    The y axis is from
      -'min_photon' (minimum photon energy relevant for gain) to
    -'max_photon' (maximum photon energy relevant for gain) as specified in the input file.
    -'photon_number' (e.g.= 20, = 100) is the number of energy grid steps between 'min_photon' and 'max_photon'.

- gain_real_integrated_energy.dat contains the integrated absorption over spatial coordinate: alpha(E)where E is in units of[eV] - gain_real_integrated_wavelength.dat contains the integrated absorption over spatial coordinate: alpha(lambda) = alpha(lambda)where lambda is in units of [µm]

- sigma_real_avs.fld, *.coord, *.dat (sigma_imag_avs.fld, *.coord, *.dat) -The x axis is the distance in units of [nm].
- The y axis is the photon energy in units of [eV].
    The y axis is from
      -'min_photon' (minimum photon energy relevant for gain) to
    -'max_photon' (maximum photon energy relevant for gain) as specified in the input file.
    -'photon_number' (e.g.= 20, = 100) is the number of energy grid steps between 'min_photon' and 'max_photon'.

Current

U_I_curve.dat: current-voltage characteristics (current (averaged value over all grid points (N-2)), achieved voltage, desired voltage)
U_I_curve.dat: current-voltage characteristics (current at each grid point (should be the same if converged), achieved voltage, desired
voltage)

Convergence files

convergency.dat: contains convergence parameter: relative change of density to previous iteration
(both Poisson (phi) self-consistency and scattering-self-consistency (sigma))
long_convergency.dat: contains convergence parameter: relative change of density to previous iteration
(only Poisson (phi) self-consistency)
see also specifier long_conv_limit
comp_current.dat: electron current density [A/Angstrom^2] - each line corresponds to an iteration
comp_density.dat: electron density [Angstrom^3] - each line corresponds to an iteration
density.dat: last line of comp_density.dat

For an example of the Green's function functionality, have a look at the RTD tutorial.


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