# quantum{}¶

Specifications that define quantum models, i.e. how the Schrödinger equation should be solved.

quantum{
debuglevel      = 1
allow_overlapping_regions = no

#----------------
# Quantum regions
#----------------
region{
name = "qr1"

quantize_x{}
quantize_y{}
quantize_z{}

no_density = yes
x = [10.0, 20.0]
y = [10.0, 20.0]
z = [10.0, 20.0]

# Boundary conditions
#--------------------
boundary{
x = dirichlet
y = dirichlet
z = neumann
classical_boundary_x = no
classical_boundary_y = no
classical_boundary_z = no
num_classical_x  = [1,1]
num_classical_y  = [1,1]
num_classical_z  = [1,1]
}

# Output definitions
#-------------------
output_wavefunctions{
max_num        = 10
all_k_points   = yes/no
structured     = no
amplitudes = "S_X_Y_Z CB_HH_LH_SO"
probabilities = "yes CB_HH_LH_SO"
scale          = 0.7
in_one_file    = yes
energy_shift   = both
include_energies_in_shifted_files = yes
}
output_subband_densities{
max_num     = 10
in_one_file = yes
}
output_sparse_matrix{
type = all
structured = no
}
output_rotated_inverse_mass_tensor{
boxes      = yes
structured = no
}

# Quantum models and solver definitions
#--------------------------------------
Gamma{
num_ev           = 10
# Eigensolvers (choose one)
lapack{}
arpack{}
accuracy         = 1e-6
iterations       = 200
preconditioner   = chebyshev
cutoff           = 0.3
abs_cutoff       = 2.5
order_chebyshev  = 20

# Dispersion
#-----------
dispersion{
path{
name = "100"
point{
k = [1.0, 0.0, 0.0]
k = [1.0, 1.0, 0.0]
}
spacing    = 0.5
num_points = 10
}
lines{
name = "lines"
spacing  = 0.5
k_max    = 1.0
}
full{
name = "3D"
kxgrid{
line{
pos = -1
spacing = 0.02
}
}
kygrid{
line{
pos = -1
spacing = 0.02
}
}
kzgrid{
line{
pos = -1
spacing = 0.02
}
}
}
superlattice{
name = "superlattice"
num_points_x   = 10
num_points_y   = 15
num_points_z   = 20
num_points     = 20
}
}
}

L{
... (same as Gamma)
}

X{
... (same as Gamma)
}

Delta{
... (same as Gamma)
}

HH{
... (same as Gamma)
}

LH{
... (same as Gamma)
}

SO{
... (same as Gamma)
}

kp_6band{
... (same as Gamma)

kp_parameters{
use_Luttinger_parameters = no
approximate_kappa        = no
}

lapack{}
#arpack{}

k_integration{
relative_size = 0.2
num_points    = 5
num_subpoints = 2
max_symmetry  = no
force_k0_subspace = yes
}
}

kp_8band{
num_electrons     = 6
num_holes         = 12
accuracy          = 1e-8
iterations        = 200

kp_parameters{
use_Luttinger_parameters = no
from_6band_parameters    = no
approximate_kappa        = no
evaluate_S               = no
rescale_S_to             = 1.0
}

k_integration{
... (same as kp_6band)
}

lapack{}
#arpack_inv{}
shift_window      = 0
shift             = 0.2
abs_shift         = 2.5

linear_solver{
iterations     = 500
abs_accuracy   = 1e-9
rel_accuracy   = 1e-9
use_cscg       = no
force_diagonal_preconditioner = no
}

#advanced settings for 8-band k.p quantum density
shift_min_CB      = 0.0
shift_max_VB      = 0.0
tunneling         = yes

classify_kspace   = 0
threshold_classification = 0.5

full_band_density = no
}

#Matrix elements definitions
#---------------------------
interband_matrix_elements{
KP6_Gamma{
direction = [1,1,0]
}
HH_Gamma{ ... }                # < HH_i | Gamma_j >
LH_Gamma{ ... }                # < LH_i | Gamma_j >
SO_Gamma{ ... }                # < SO_i | Gamma_j >
HH_Delta{ ... }                # < HH_i | Delta_j >
LH_Delta{ ... }                # < LH_i | Delta_j >
SO_Delta{ ... }                # < SO_i | Delta_j >
HH_X{ ... }                    # < HH_i | X_j >
LH_X{ ... }                    # < LH_i | X_j >
SO_X{ ... }                    # < SO_i | X_j >
HH_L{ ... }                    # < HH_i | L_j >
LH_L{ ... }                    # < LH_i | L_j >
SO_L{ ... }                    # < SO_i | L_j >

output_matrix_elements      = yes
output_transition_energies  = yes/no #
}

intraband_matrix_elements{
Gamma{
direction = [1,1,0]
}
Delta{ ... }
X{ ... }
L{ ... }
HH{ ... }
LH{ ... }
SO{ ... }
KP6{ ... }
KP8{ ... }

output_matrix_elements      = yes/no                output_transition_energies  = yes/no
output_oscillator_strengths = yes/no
}

dipole_moment_matrix_elements{
Gamma{
direction = [1,1,0]
}
Delta{ ... }
X{ ... }
L{ ... }
HH{ ... }
LH{ ... }
SO{ ... }
KP6{ ... }
KP8{ ... }

output_matrix_elements      = yes
output_transition_energies  = yes
output_oscillator_strengths = yes
}

transition_energies{
Gamma{}
KP6_Gamma{}
HH_Gamma{}
LH_Gamma{}
SO_Gamma{}
Delta{}
HH_Delta{}
LH_Delta{}
SO_Delta{}
X{}
HH_X{}
LH_X{}
SO_X{}
L{}
HH_L{}
LH_L{}
SO_L{}
HH{}
LH{}
SO{}
KP6{}
KP8{}
}

phonon_energy = 0.036
}

bulk_dispersion{
path{
name = "from_Gamma_to_L"
position{
x = 5.5
y = 10.0
z = -1.1
}
shift_holes_to_zero = yes
point{
k = [1.0, 0.0, 0.0]
}
...

spacing             = 0.5
num_points          = 10
}

lines{
name = "lines"
position{
x = 5.5
y = 10.0
z = -1.1
}
shift_holes_to_zero = yes
spacing             = 0.5
k_max               = 1.0
}

full{
name = "3D"
position{
x = 5.5
y = 10.0
z = -1.1
}
shift_holes_to_zero = yes
kxgrid{
line{
pos     = -1
spacing = 0.02
}
line{
pos     = 1
spacing = 0.02
}
...
}
kygrid{
...
}
kzgrid{
...
}
}

output_bulk_dispersions{}
output_masses{}

} # end: bulk_dispersion{}

} # end: region{}

#Many body effects
#-----------------
exchange_correlation{
type             = lda
initial_spin_pol = 1.0
output_spin_polarization{}
output_exchange_correlation{}
}

}


## debuglevel¶

value

any integer between -1 and 3

default

1

The higher this integer number, the more information on the numerical solver is printed to the screen output. Increasing the respective debuglevel to 2 or more significantly increases the volume of the diagnostic output displayed in nextnanomat (or a shell window). As result of the additional I/O load, particularly 1D simulations will slow down correspondingly (especially for current{ } and poisson{ }).

## allow_overlapping_regions¶

value

yes or no

default

no

Overlapping quantum regions computing the same band(s) are not allowed. Note that, in case such overlap is allowed, the quantum densities of the respective regions are added in the overlap region and a too high density will be computed. Thus, please only allow such overlap when the quantum densities are known to be extremely small in the overlap region.

## region{ }¶

Inside the quantum region, the Schrödinger equation is solved.

name
value

“string”

Provides the name of the quantum region.

quantize_x{}

In 2D or 3D simulation, the Schrödinger equation is solved within the slices perpendicular to x-direction. This results in the reduction of the calculation time.

For example, if a 2D simulatin has 100 grids in x-direction and 50 grids in y-direction, the normal calculation solves the eigenvalue problem of a (100x50) x (100x50) matrix. When quantize_x{} is specified, on the other hand, nn++ solves the 1D Schroedinger equations along y-direction at each grid point in x-direction so 100 eigenvalue problems of 50x50 matrixes are solved. Thus the runtime of the eigenvalue solver could be roughly estimated as (number of x-grids)$$^{-1}$$ times, but we should note that the runtime also depends on the number of eigenvalues to be calculated.

Currently, only one-band (Gamma, X, Delta, LH, HH, etc.) without k-integration and without magnetic field is supported, and QM output is limited to local spectra and occupations. If strain is enabled, deformation potentials are ignored. Similarly, quantum boundary conditions are always Neumann or periodic, irrespective of what is specified in the input file. And quantum decomposition regions cannot be used for CBR or optics.

Only one quantization direction (x, y, z) can be simultaneously specified when quantum decomposition is used. Typically, the quantization direction is the growth direction.

Note that a similar number of states should be requested as for a corresponding 1D simulation (i.e. much less than normally needed in 2D or 3D), and that lateral (i.e. orthogonal to the quantization direction) grid spacings can be much larger than for “normal” quantum simulation, as the density from quantum decomposition is NOT affected by wide lateral grid spacings.

quantize_y{}

The same as quantize_x{}, but the slices are in y-direction.

quantize_z{}

The same as quantize_x{}, but the slices are in z-direction.

no_density
value

yes or no

default

no

Tells if to not calculate quantum mechanical charge density.

x
value

2D float vector

Provides the extension of quantum region in x direction in nanometers (nm)

y
value

2D float vector

Provides the extension of quantum region in y direction in nanometers (nm). To be used for 2D or 3D calculations only.

z
value

2D float vector

Provides the extension of quantum region in x direction in nanometers (nm). To be used for 3D calculation only.

• boundary{ }

Specifies the boundary condition for Schrödinger equation along various axis dimensions. In general, Dirichlet boundary conditions correspond to $$f = \mathrm{constant}$$ and Neumann boundary conditions correspond to $$df/ dx = \mathrm{constant}$$. Quantum densities may exhibit pathological density values on the boundary (e.g. 0 in the case of Dirichlet boundary conditions). Using classical_boundary_x, classical_boundary_y, classical_boundary_z, the computation of a classical density can be enforced on the respective boundary points for the respective band(s). (The quantum calculation itself and respective results such as wavefunctions are not affected by this setting). Using num_classical_x, num_classical_y, num_classical_z you can explicitly specify the number of points to be cut at each side.

x
value

dirichlet/neumann/cbr

default

neumann

y
value

dirichlet/neumann/cbr

default

neumann

z
value

dirichlet/neumann/cbr

default

neumann

classical_boundary_x
value

yes or no

default

no

classical_boundary_y
value

yes or no

default

no

classical_boundary_z
value

yes or no

default

no

num_classical_x
value

2D integer vector

default

[1 , 1]

num_classical_y
value

2D integer vector

default

[1 , 1]

num_classical_z
value

2D integer vector

default

[1 , 1]

Note

Periodic boundary conditions along the appropriate direction(s) are taken automatically if global { ... periodic{ x/y/z = yes} } is specified and if the quantum region extends over the whole simulation region along the appropriate direction. In this case, the dirichlet or neumann specifications under quantum{ ... {region{ ... boundary{...} } } are ignored along the appropriate direction(s).

• output_wavefunctions{ }

Provides options for output of wavefunction data

max_num
value

any integer between 1 and 9999

default

1.0

all_k_points
value

yes or no

default

false

Prints out the wavefunctions for all $$k_{||}$$ points (1D: $$k_{||} = (k_y,k_z)$$, 2D: $$k_{||} = k_z$$) that are used in the k_integration{ } or dispersion{ }. Enabling this option can produce a large number of output files.

structured
value

yes or no

default

no

The whole output for quantum{ } is written in subdirectory Quantum/. If enabled, additional subdirectories are created in subdirectory Quantum/ to organize the structure of the output files in a meaningful way. It is recommended to set this parameter to yes if a lot of output files are created, e.g. in case all_k_points = yes, and both amplitudes and probabilities are printed out.

amplitudes
value

string

default

” no “

Prints out the wavefunctions $$\psi$$ in units of 1D: $$\mathrm{nm}^{-1/2}$$, 2D: $$\mathrm{nm}^{-1}$$, 3D: $$\mathrm{nm}^{-3/2}$$.

options

” yes “ : for k.p it is equivalent to S_X_Y_Z

” no “ : no output is done for amplitudes.

” S_X_Y_Z “ : prints out the wavefunctions (psi) with respect to the basis (k.p only) $$| S+ \rangle | S- \rangle | X+ \rangle | Y+ \rangle | Z+ \rangle | X- \rangle | Y- \rangle | Z- \rangle$$. $$| X+ \rangle | Y+ \rangle | Z+ \rangle$$ correspond to the x, y, z of the simulation coordinate system (and not crystal coordinate system) and + and - correspond to the spin projection along the z axis of the crystal system.

” CB_HH_LH_SO “ : prints out the wavefunctions (psi) with respect to the basis (k.p only) $$| cb+ \rangle | cb- \rangle | hh+ \rangle | lh+ \rangle | lh- \rangle | hh- \rangle | so+ \rangle | so- \rangle$$. This basis is the same as used in L. C. Lew Yan Voon, M. Willatzen, The k.p method(2009) (Table 3.4); G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (1988) and B. A. Foreman, PRB 48, 4964 (1993).

If multiple choices are required type them together inside a string like

amplitudes = "
S_X_Y_Z
CB_HH_LH_SO
"

probabilities
value

string

default

yes

Prints out the wavefunctions $$|\psi|^2$$ in units of 1D: $$\mathrm{nm}^{-1}$$, 2D: $$\mathrm{nm}^{-2}$$, 3D: $$\mathrm{nm}^{-3}$$.

options

yes : for k.p it is the sum of the squares of all components of a spinor no : no output

S_X_Y_Z : same as for the amplitudes (k.p only)

CB_HH_LH_SO : same as for the amplitudes (k.p only)

If multiple choices are required type them together inside a string like

probabilities = "
yes
CB_HH_LH_SO
"

scale
value

float

default

1.0

scale factor for output of amplitudes and probabilities

in_one_file
value

yes or no

default

yes

Prints out the amplitudes into one file and the probabilities into one file. If no is chosen, for each eigenvalue a separate file is written out.

energy_shift
value

string

default

both

options

shifted : prints out the amplitudes and the probabilities shifted by the energy.

not_shifted : prints out the amplitudes and the probabilities as they are (an integral over volume is equal to 1).

both : prints out the amplitudes and the probabilities with and without energy shift.

include_energies_in_shifted_files
value

yes or no

default

yes

Selects if the energy levels are added in output of shifted amplitudes and probabilities or not. If no is selected a separate file with energy levels is written out.

Note

The energy spectrum (i.e. the eigenvalues) are always written into the files energy_spectrum_*.dat. The projections of the eigenfunctions on the basis states of the bulk Hamiltonian are written into the files spinor_composition_*.dat.

• output_subband_densities{ }

Provides options for output of subband densities.

max_num
value

any integer between 0 and 9999

default

1

number of subband densities to be printed out. If max_num is not present, the subband density is written out for each eigenvalue.

in_one_file
value

yes or no

default

yes

Prints out the subband densities into one file. If no is chosen, for each subband density a separate file is written out. This feature only makes sense for 1D simulations.

• output_sparse_matrix{ }

Provides options for output of sparse matrix, used in Schroedinger equation solver, in .mtx format

type
value

string

default

values

options

values : output sparse matrix as it is (also imaginary part, if sparse matrix is complex valued).

zero_nonzero : output matrix containing ‘0’ and ‘1’ for zero and non-zero entries of sparse matrix (same for imaginary part, if sparse matrix is complex valued)

zero_nonzero_absolute : output matrix containing ‘0’ and ‘1’ for zero and non-zero absolute values of entries of sparse matrix

all : output all types listed above

structured
value

yes or no

default

no

Whole output is written in subdirectory Quantum/. IF yes is selected additional subdirectories are created in subdirectory Quantum/ to organize the structure of the output files in a meaningful way.

• Gamma{ }

Solves single-band effective mass Schrödinger equation for the Gamma conduction band.

num_ev
value

integer >= 0

default

0

Provides the number of eigenvalues to be calculated.

force_complex_solver
value

yes or no

default

no

Set flag to yes, when optics{} is complaining about real-valued quantum solvers.

lapack{ }

LAPACK eigensolver is used to solve dense matrix problem (should be used for 1D and small 2D systems). For 1D simulations without periodic boundary conditions a tridiagonal LAPACK solver is used for the single-band Hamiltonian as default.

arpack{ }

ARPACK eigensolver is used to solve eigenvalue problem using sparse matrix routines. It ARPACK should be faster for large matrices (N > 1000) where only a few eigenvalues are sought (~5-30). Memory usage of arpack (and also arpack_inv) only depends on the number of eigenvectors requested, and is not influenced by the type of preconditioner used. Essentially, for each requested eigenvector (i.e. wave function), additional temporary space corresponding to 2.5 eigenvectors is needed during runtime. Among the preconditioners, chebyshev preconditioning and legendre preconditioning are comparably fast, but require both the specification of a cutoff energy under (above) which all eigenvalues of interest are assumed to be located. If this assumption is violated, only spurious parts of the energy spectrum will be computed. On the other hand, setting the cutoff energy too generous will slow down convergence. Since the energy spectrum often shifts during the Quantum-Poisson iteration, a more generous initial cutoff energy is also needed for the first Quantum-Poisson iteration step. If this initial cutoff energy is not provided, much slower but more predictable polynomial preconditioning will be used for the first Quantum-Poisson iteration step instead of the specified chebyshev / legendre preconditioner. Alternatively, this slower polynomial preconditioning can also be used for the entire Quantum-Poisson iteration. In this case, no cutoff energies need to be specified at all. Generally, it is advisable to use polynomial preconditioning when simulating a new structure until the distribution of the eigenvalues, the location of the Fermi level(s), and the required numbers of eigenvalues are better known. Performance of all preconditioners can be further tuned by changing the order of the respective polynomial used, with optimal values typically lying between 10 and 30. arpack will terminate once the desired accuracy has been reached or the specified number of iterations has been exceeded. In the latter case, not all requested eigenvectors may have been calculated, or convergence may be incomplete.

accuracy
value

any float > 0

default

1e-10 for LAPACK 1e-7 for ARPACK

accuracy of eigenvalue

iterations
value

any integer > 1

default

500

number of iterations for eigenvalue solver

preconditioner
value

0 or polynomial, 1 or chebyshev, 2 or legendre

default

1 or chebyshev

Polynomial preconditioner is the slowest but does not require to specify cutoff energy whereas chebyshev or legendre preconditioner requires you to specifiy cutoff energy.

order_polynomial
value

any integer > 1

default

15

order of the polynomial used for polynomial preconditioning

order_chebychev
value

any integer > 1

default

20

order of the polynomial used for Chebyshev preconditioning

order_legendre
value

any integer > 1

default

20

order of the polynomial used for Legendre preconditioning

cutoff
value

any float >= 0.001

default

0.3 # [eV]

abs_cutoff
value

any float >= 0.001

default

0.0 # [eV]

Note

The default behaviour of ARPACK eigensolver is the following: When the Schrödinger equation is solved for the first time, the polynomial preconditioner is used, because there is no suitable cutoff energy known. In all later Quantum-Poisson iterations the chebyshev preconditioner will be used (up to two times faster) with a cutoff energy slightly above the highest eigenvalue, which was calculated in the last iteration.

• dispersion{ … }

calculate the $$\mathbf{k_{||}}$$ and $$\mathbf{k_{\tiny{superlattice}}}$$ (if applicable) dispersion. The energy dispersion E(k) along the specified paths and for the specified k space resolutions are completely independent from the k space resolution that was used within the self-consistent cycle where the k.p density has been calculated. The latter is specified in k_integration{ }. For more details, see dispersion{}.

• L{ }

solves single-band Schrödinger equation for the L conduction band. The options are the same as Gamma{ }.

• X{ }

solves single-band Schrödinger equation for the X conduction band. The options are the same as Gamma{ }.

• Delta{ }

solves single-band Schrödinger equation for the Delta conduction band. The options are the same as Gamma{ }.

• HH{ }

solves single-band Schrödinger equation for the heavy hole valence band. The options are the same as Gamma{ }.

• LH{ }

solves single-band Schrödinger equation for the light hole valence band. The options are the same as Gamma{ }.

• SO{ }

solves single-band Schrödinger equation for the split-off hole valence band. The options are the same as Gamma{ }.

• kp_6band{ }

solves 6-band k.p Schrödinger equation for the ** heavy, light and split-off hole** valence band. The options are the same as Gamma{ } with some additional options, which are

• kp_parameters{ }

advanced manipulation of k.p parameters from the database.

use_Luttinger_parameters
value

yes or no

default

no

By default the solver uses the DKK (Dresselhaus-Kip-Kittel) parameters (L, M, N). If enabled then it uses Luttinger parameters ($$\gamma_1$$, $$\gamma_2$$, $$\gamma_3$$) instead.

approximate_kappa
value

yes or no

default

no

By default the $$\kappa$$ for zinc blende crystal structure is taken from the database or input file. If this is enabled then the solver is forced to approximate kappa through others 6-band k.p parameters, even though kappa is given in database or input file.

lapack{}

LAPACK eigensolver: solves dense matrix problem (for 1D and small 2D systems only)

arpack{}

ARPACK eigensolver (default) ARPACK should be faster for large matrices (N > 1000) where only a few eigenvalues are sought (~5-30).

• k_integration{ }

Provides options for integration over $$\mathbf{k_{||}}$$ space for k.p density calculations (for 1D and 2D only). By default the quantum mechanical charge density is calculated (no_density = no). Therefore, k_integration{ } is required. If you do not need a quantum mechanical density, e.g. because you are not interested in a self-consistent simulation, the calculation is much faster if you use (no_density = yes). Then you can omit k_integration{ } and only the eigenstates for $$\mathbf{k_{||}} = (k_y,k_z) = (0,0) = 0$$ are calculated.

relative_size
value

float between 0.0 and 1.0

default

1.0

Range of $$\mathbf{k_{||}}$$ integration relative to size of Brillouin zone. Often a value between 0.1-0.2 is sufficient.

num_points
value

integer > 1

default

10

number of $$\mathbf{k_{||}}$$ points, where Schrödinger equation has to be solved (in one direction). In 1D, the number of Schrödinger equations that have to be solved depends quadratically on num_points. In 2D, the number of Schrödinger equations that have to be solved depends linearly on num_points.

num_subpoints
value

integer > 1

default

5

number of points between two $$\mathbf{k_{||}}$$ points, where wavefunctions and eigenvalues will be interpolated.

max_symmetry
value

1 or no 2 or C2 3 or full

default

full

no does not use symmetry of Brillouin zone to reduce number of $$\mathbf{k_{||}}$$ points.

C2 uses up to $$C_2$$ symmetry of Brillouin zone to reduce number of $$\mathbf{k_{||}}$$ points.

full uses full symmetry of Brillouin zone to reduce number of $$\mathbf{k_{||}}$$ points. For example for a cubic k space the 1/8th of the zone.

force_k0_subspace
value

yes or no

default

no

If set to yes, $$k_\parallel$$ integration in quantum{} is modified in that only states for point $$k=0$$ are computed exactly, whereas all other k points are computed in the subspace of the $$k=0$$ wavefunctions. As a result of this approximation, computational speed is much improved (you may even be able to also enlarge the number of eigenvalues). In case you are planning to use this approximation for final results, please make sure to check whether the resulting loss of accuracy in density is acceptable.

• kp_8band{ }

It solves 8-band k.p Schrödinger equation for the Gamma conduction band and the heavy, light and split-off hole valence bands.

num_electrons
value

integer >= 0

default

0

number of electron eigenvalues

num_holes
value

integer >= 0

default

0

number of hole eigenvalues

accuracy
value

any float > 0

default

1e-7

accuracy of eigenvalue

iterations
value

any integer > 1

default

500

number of iterations for eigenvalue solver

• kp_parameters{ }

Provides options for advanced manipulation of k.p parameters from database.

use_Luttinger_parameters
value

yes or no

default

no

By default the solver uses the DKK (Dresselhaus-Kip-Kittel) parameters (L, M, N). If enabled then it uses Luttinger parameters ($$\gamma_1$$, $$\gamma_2$$, $$\gamma_3$$) instead.

from_6band_parameters
value

yes or no

default

no

By default the 8-band k.p parameters are taken from database or input file. If enabled then it evaluates the 8-band k.p parameters from 6-band k.p parameters, Kane parameter $$E_P$$ and temperature dependent band gap $$E_g$$.

approximate_kappa
value

yes or no

default

no

By default the $$\kappa$$ for zinc blende crystal structure is taken from the database or input file. If this is enabled then the solver is forced to approximate kappa through others 8-band k.p parameters, even though kappa is given in database or input file.

evaluate_S
value

yes or no

default

no

By default $$S$$ ($$S_1$$, $$S_2$$ for wurtzite) k.p parameter(s) is (are) taken from database or input file. If enabled it evaluates $$S$$ ($$S_1$$, $$S_2$$ for wurtzite) k.p parameter(s) from effective mass $$m_e$$ ($$m_{e,par}$$, $$m_{e,perp}$$ for wurtzite), Kane parameter(s), spin-orbit coupling(s) and temperature dependent band gap.

rescale_S_to
value

float for zinc blende crystal structure

2D float vector for wurtzite crystal structure

set $$S$$ for zinc blende crystal structure to specified value and rescale $$E_P$$, $$L'$$, $$N^{+}$$ in order to preserve electron’s effective mass.

set $$S_1$$, $$S_2$$ for wurtzite crystal structure to specified values respectively and rescale $$E_{P1}$$, $$E_{P2}$$, $$L_{1}'$$, $$L_{2}'$$, $$N^+_1$$, $$N^+_2$$ in order to preserve electron’s effective masses.

• k_integration{ }

Provides options for integration over $$\mathbf{k_{||}}$$ space for k.p density calculations (for 1D and 2D only) same as kp_6band{ k_integration{ }}

lapack{ }

LAPACK eigensolver: solves dense matrix problem (for 1D and small 2D systems only)

arpack_inv{ }

ARPACK shift invert eigensolver. ARPACK should be faster for large matrices (N > 1000) where only a few eigenvalues are sought (~5-30).

shift_window
value

integer

default

0

When LAPACK is used, shifts the window of computed states by the specified number of states up (for positive integers) or down (for negative integers). Adjust when the computed states are not centered around the band gap.

shift
value

float >=0

default

0.1 # [eV]

energy shift relative to band edges in arpack_inv.

abs_shift
value

float >=0

default

0.0 # [eV]

energy shift on an absolute energy scale in arpack_inv.

• linear_solver{ }

Provides parameters for linear equation solver in arpack_inv shift invert preconditioner

iterations
value

integer > 1

default

10000

number of iterations in arpack_inv. Occasionally, using even larger values than 10000 may be necessary to avoid diagonalization failure.

abs_accuracy
value

float between 0.0 and 0.01

default

1e-8

absolute accuracy in arpack_inv.

rel_accuracy
value

float between 0.0 and 0.01

default

1e-8

relative accuracy in arpack_inv.

use_cscg
value

yes or no

default

no

When arpack_inv is used, forces the slower but occasionally more robust CSCG (Composite Step Conjugate Gradient ) linear solver to be used rather than the cg (Conjugate Gradient) linear solver. May occasionally prevent a diagonalization failure.

force_diagonal_preconditioner
value

yes or no

default

no

When arpack_inv is used, forces the use of a slower but more robust diagonal preconditioner. As result, total runtime and stability of the arpack_inv solver may actually become much better and diagonalization failures may be avoided.

shift_min_CB
value

float

default

0.0

(relevant only if classify_kspace = 0) Shifts the minimum of the conduction band to manipulate cutoff energy and thereby the quantum density classification.

shift_max_VB
value

float

default

0.0

(relevant only if classify_kspace = 0) Shifts the maximum of the valence band to manipulate cutoff energy and thereby the quantum density classification.

tunneling
value

yes or no

default

yes

(relevant only if classify_kspace = 0) Choice of the (position-dependent) cutoff energy. yes defines the cutoff energy at max((minimum of the conduction band in the structure), (position-dependent valence band edge)), while no sets it to min((maximum of the valence band in the structure), (position-dependent conduction band edge)).

classify_kspace
value

0, 1, 2, or 3

default

0

Choice of the classification method in the 8-band k.p quantum density calculation.

• classify_kspace = 0: Eigenstates are classified by comparing the zone-center eigenvalues with the (possibly position-dependent) cutoff energies. For the definition of cutoff energies, see shift_min_CB, shift_max_VB, and tunneling.

• classify_kspace = 1: Eigenstates are classified by comparing the zone-center spinor composition with threshold_classification.

• classify_kspace = 2: Eigenstates are classified at each in-plane k vector (1D simulation) and at each k value (2D simulation) using spinor composition averaged with the neighbouring k points.

• classify_kspace = 3: Eigenstates are classified at each in-plane k vector (1D simulation) and at each k value (2D simulation) using spinor composition averaged with the neighbouring k points, but skipping the average if any of the neighbouring k points has the opposite sign of charge. The resulting quantum density will be different from the case classify_kspace = 2 if electron-hole hybridization occurs (e.g. type-II broken-gap superlattices).

threshold_classification
value

0.0 <= float <= 1.0

default

0.5

(relevant only if classify_kspace >= 1) Classify states to electrons if the electron spinor composition is greater than this threshold and otherwise to holes.

full_band_density
value

yes or no

default

no

Calculate density by filling all states above Fermi level with holes and subtracting a negative background charge (lapack only). This ignores classify_kspace.

spurious_handling
value

six dimensional double vector

default

[0.0, 1.0, -1.0, 1.0, 0.0, 0.0]

• first component: If value > 0, forward-/backward differences are used for the first derivative discretization of the P material parameter (Kane parameter) in the 8-band k.p Hamiltonian. Default is 0 (= FALSE), i.e. centered differences are used instead. This parameter might affect spurious solutions of the wavefunctions. See eq. (1.50) and eq. (1.51) of PhD thesis T. Andlauer.

• second component: farband contribution to electrons = value - 1.0 (conduction band g factor, should be a material parameter but it is not) (default is: 1.0) S = 1 + farband contribution, by default farband contribution = 0. This corresponds to setting S=1. It can be useful to set this value to 0.0 (farband contribution = -1). Then it corresponds to setting S=0. Otherwise the default is rescaling to that S=1.

• third component: correction for electron g factor [eV] (default is: -1.0)

• fourth component: If value > 0, rescale everywhere (default is: 1 = TRUE)

• fifth component: If value > 0, upwinding is TRUE (default is: 0 = FALSE) ==> It seems that upwinding is not used at all.

• sixth component: If value > 0, avoid spurious solutions. (default is: 0 = FALSE)

To avoid spurious solutions, an example configuration could be given by spurious_handling = [0.0, 1.0, -1.0, 1.0, 0.0, 1.0].

• interband_matrix_elements{ }

Provides the option to calculate interband matrix elements between wave functions of two different bands.

KP6_Gamma{ }

$$\sum_k \langle kp6_{k,i} | \Gamma_j \rangle$$ , with k = 1 .. 6 indexing the component of the six-component k.p wave function and $$i$$, $$j$$ indexing the wave function numbers. kp_6band{ } and Gamma{ } calculation must be present.

HH_Gamma{ }

Matrix element of the transition between the heavy hole valence band and the gamma conduction band $$\langle HH_{i} | \Gamma_j \rangle$$

LH_Gamma{ }

Matrix element of the transition between the light hole valence band and the gamma conduction band $$\langle LH_{i} | \Gamma_j \rangle$$

SO_Gamma{ }

Matrix element of the transition between the split-off hole valence band and the gamma conduction band $$\langle SO_{i} | \Gamma_j \rangle$$

HH_Delta{ }

Matrix element of the transition between the heavy hole valence band and the Delta conduction band $$\langle LH_{i} | \Delta_j \rangle$$

LH_Delta{ }

Matrix element of the transition between the light hole valence band and the Delta conduction band $$\langle LH_{i} | \Delta_j \rangle$$

SO_Delta{ }

Matrix element of the transition between the split-off hole valence band and the Delta conduction band $$\langle SO_{i} | \Delta_j \rangle$$

HH_X{ }

Matrix element of the transition between the heavy hole valence band and the X conduction band $$\langle HH_{i} | X_j \rangle$$

LH_X{ }

Matrix element of the transition between the light hole valence band and the X conduction band $$\langle LH_{i} | X_j \rangle$$

SO_X{ }

Matrix element of the transition between the split-off valence band and the X conduction band $$\langle SO_{i} | X_j \rangle$$

HH_L{ }

Matrix element of the transition between the heavy hole valence band and the L conduction band $$\langle HH_{i} | L_j \rangle$$

LH_L{ }

Matrix element of the transition between the light hole valence band and the L conduction band $$\langle LH_{i} | L_j \rangle$$

SO_L{ }

Matrix element of the transition between the split-off valence band and the L conduction band $$\langle SO_{i} | L_j \rangle$$

output_matrix_elements = yes/no

Output matrix elements.

output_transition_energies = yes/no

Output transition energies.

• intraband_matrix_elements{ }

Calculate intraband matrix elements $$\langle i | \epsilon\cdot\hat{\mathbf{p}} | j \rangle$$ for wave functions within one band. The light polarization direction $$\epsilon$$ is automatically normalized in the program. $$\hat{\mathbf{p}} = i\hbar\nabla$$ is the momentum vector.

For further reading: J. H. Davies, The Physics of Low-Dimensional Semiconductors. An Introduction, 2006, Chapters 10 and 8.

Gamma{ }

Calculates the matrix element $$\langle \Gamma_i | \epsilon\cdot\hat{\mathbf{p}} | \Gamma_j \rangle$$.

direction
value

3D integer vector

default

[1 , 0 , 0]

It defines the polarization direction $$\epsilon$$. From it a vector of unit length is calculated, which enters the calculation. In 1D simulation it can be omitted and [1,0,0] is then assumed.

Delta{ }

Calculates the matrix element $$\langle \Delta_i | \epsilon\cdot\hat{\mathbf{p}} | \Delta_j \rangle$$. See Gamma{ ... } for direction option.

X{ }

Calculates the matrix element $$\langle X_i | \epsilon\cdot\hat{\mathbf{p}} | X_j \rangle$$. See Gamma{ ... } for direction option.

L{ }

Calculates the matrix element $$\langle L_i | \epsilon\cdot\hat{\mathbf{p}} | L_j \rangle$$. See Gamma{ ... } for direction option.

HH{ }

Calculates the matrix element $$\langle HH_i | \epsilon\cdot\hat{\mathbf{p}} | HH_j \rangle$$. See Gamma{ ... } for direction option.

LH{ }

Calculates the matrix element $$\langle LH_i | \epsilon\cdot\hat{\mathbf{p}} | LH_j \rangle$$. See Gamma{ ... } for direction option.

SO{ }

Calculates the matrix element $$\langle SO_i | \epsilon\cdot\hat{\mathbf{p}} | SO_j \rangle$$. See Gamma{ ... } for direction option.

KP6{ }

Calculates the matrix element $$\sum_k \langle kp6_{k,i} | \epsilon\cdot\hat{\mathbf{p}} | kp6_{k,j} \rangle$$, $$k$$ = 1,…,6. See Gamma{ ... } for direction option.

KP8{ }

Calculates the matrix element $$\sum_k \langle kp8_{k,i} | \epsilon\cdot\hat{\mathbf{p}} | kp8_{k,j} \rangle$$, $$k$$ = 1,…,8. See Gamma{ ... } for direction option.

output_matrix_elements = yes/no

Output matrix elements.

output_transition_energies = yes/no

Output transition energies.

output_oscillator_strengths = yes/no

Output oscillator strengths. Currently, only a simple formula is used, i.e. the free electron mass is used and not the real effective mass one.

• dipole_moment_matrix_elements{ }

Calculate dipole moment matrix elements $$\langle i | \epsilon\cdot\hat{\mathbf{d}} | j \rangle$$ for wave functions within one band. The light polarization direction $$\epsilon$$ is automatically normalized in the program. $$\hat{\mathbf{d}} = e\hat{\mathbf{r}}$$ is the dipole moment vector.

For further reading: J. H. Davies, The Physics of Low-Dimensional Semiconductors. An Introduction, 2006, Chapters 10 and 8.

Gamma{ }

Calculates the matrix element $$\langle \Gamma_i | \epsilon\cdot\hat{\mathbf{d}} | \Gamma_j \rangle$$.

direction
value

3D integer vector

default

[1 , 0 , 0]

It defines the polarization direction $$\epsilon$$. From it a vector of unit length is calculated, which enters the calculation. In 1D simulation it can be omitted and [1,0,0] is then assumed.

Delta{ }

Calculates the matrix element $$\langle \Delta_i | \epsilon\cdot\hat{\mathbf{d}} | \Delta_j \rangle$$. See Gamma{ ... } for direction option.

X{ }

Calculates the matrix element $$\langle X_i | \epsilon\cdot\hat{\mathbf{d}} | X_j \rangle$$. See Gamma{ ... } for direction option.

L{ }

Calculates the matrix element $$\langle L_i | \epsilon\cdot\hat{\mathbf{d}} | L_j \rangle$$. See Gamma{ ... } for direction option.

HH{ }

Calculates the matrix element $$\langle HH_i | \epsilon\cdot\hat{\mathbf{d}} | HH_j \rangle$$. See Gamma{ ... } for direction option.

LH{ }

Calculates the matrix element $$\langle LH_i | \epsilon\cdot\hat{\mathbf{d}} | LH_j \rangle$$. See Gamma{ ... } for direction option.

SO{ }

Calculates the matrix element $$\langle SO_i | \epsilon\cdot\hat{\mathbf{d}} | SO_j \rangle$$. See Gamma{ ... } for direction option.

KP6{ }

Calculates the matrix element $$\sum_k \langle kp6_{k,i} | \epsilon\cdot\hat{\mathbf{d}} | kp6_{k,j} \rangle$$, $$k$$ = 1,…,6. See Gamma{ ... } for direction option.

KP8{ }

Calculates the matrix element $$\sum_k \langle kp8_{k,i} | \epsilon\cdot\hat{\mathbf{d}} | kp8_{k,j} \rangle$$, $$k$$ = 1,…,8. See Gamma{ ... } for direction option.

output_matrix_elements = yes/no

Output matrix elements.

output_transition_energies = yes/no

Output transition energies.

output_oscillator_strengths = yes/no

Output oscillator strengths. Currently, only a simple formula is used, i.e. the free electron mass is used and not the real effective mass one.

• transition_energies{ }

Calculate transition energies (energy difference) between two states in certain bands. Use this if you want to calculate transition energies but but do not want to calculate the matrix elements. Note that the matrix elements defined above also include specifiers for transition energies: output_transition_energies  = yes.

• Gamma{ }

• KP6_Gamma{ }

• HH_Gamma{ }

• LH_Gamma{ }

• SO_Gamma{ }

• Delta{ }

• HH_Delta{ }

• LH_Delta{ }

• SO_Delta{ }

• X{ }

• HH_X{ }

• LH_X{ }

• SO_X{ }

• L{ }

• HH_L{ }

• LH_L{ }

• SO_L{ }

• HH{ }

• LH{ }

• SO{ }

• KP6{ }

• KP8{ }

Calculate the lifetimes of the state due to LO phonon scattering. For more information check R. Ferreira, G. Bastard, PRB 40, 1074 (1989) and Section 2.1.3 of the PhD thesis of G. Scarpa, Technische Universität München.

phonon_energy
value

any float > 0.0

default

0.01

LO phonon energy

• bulk_dispersion{ … }

Calculate bulk k.p dispersion of the material at a specific position in the simulation domain. For more details, see bulk_dispersion{}.

## exchange_correlation{ }¶

Provides options to calculate exchange-correlation effects. This is not calculated by default.

type
value

string

options

lda : Include exchange-correlation effects in the LDA approximation (Local Density Approximation)

lsda : Include exchange-correlation effects in the LSDA approximation (Local Spin Density Approximation)

initial_spin_pol
value

float between 0.0 and 1.0

default

0.0

Breaks spin up/down symmetry if no magnetic field is present.

output_spin_polarization{}

output spin polarization [dimensionless]

output_exchange_correlation{}

output exchange correlation potentials in [eV]`.