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Output-kp-data

The output of the k.p eigenvalues and eigenfunctions is controlled by this keyword. All eigenfunctions and eigenvalues between vb-min-ev and vb-max-ev (for conduction band:cb-min-evand cb-max-ev) are put out in one file for each band.

k.p dispersion without density integration

The program chooses the maximum k vector up to which the states are calculated automatically (see above). However, if one wants to know the dispersion up to a given k vector, one has to specify the value in the input file under the keyword $output-kp-data and its specifier 'k-par-disp-range' [1/Angstrom]. Then the k.p states will be calculated again in the end and put out.

!-------------------------------------------------------------! $output-kp-data optional ! destination-directory character required ! complex-wave-functions character optional ! kp-spinors character optional ! detailed-output character optional ! scale double optional ! shift-wavefunction-by-eigenvalue character optional ! ! bulk-kp-dispersion character optional ! bulk-kp-dispersion-3D character optional ! grid-position double_array optional ! k-direction double_array optional ! number-of-k-points integer optional ! shift-holes-to-zero character optional ! 'yes' / 'no' ! k-par-dispersion character optional ! k-par-disp-ev-min integer optional ! k-par-disp-ev-max integer optional ! k-par-disp-k-range double optional ! k-par-disp-num-k-par integer optional ! DOS-density-of-states character optional ! DOS-Emin-Emax double_array optional ! [eV] DOS-points integer optional ! ! !-------- range for eigenvalues and eigenfunctions ----------! ! cb-num-ev-min integer optional ! cb-num-ev-max integer optional ! cb-k-par-min integer optional ! 1D/2D cb-k-par-max integer optional ! 1D/2D cb-k-SL-min integer optional !for superlattice cb-k-SL-max integer optional !for superlattice
! vb-num-ev-min integer optional ! vb-num-ev-max integer optional ! vb-k-par-min integer optional ! 1D/2D vb-k-par-max integer optional ! 1D/2D vb-k-SL-min integer optional !for superlattice vb-k-SL-max integer optional !for superlattice ! interband-matrix-elements character optional ! intraband-matrix-elements character optional ! intraband-lifetime character optional ! ! optical-matrix-element-output character optional ! dipole-transition-type character optional ! initial-band-min integer optional ! initial-band-max integer optional ! final-band-min integer optional ! final-band-max integer optional ! ! $end_output-kp-data ! !-------------------------------------------------------------!

Syntax:

destination-directory = kp_data/

Name of directory to which the files should be written. Must exist and directory name has to include the slash (\ for DOS and / for UNIX).

complex-wave-functions = yes / no

Flag whether to print out the wave functions psi including real and imaginary parts in addition to the output of Psi².

kp-spinors = yes / no

Flag whether to print out the n = 6 (6x6 k.p) or n = 8 (8x8 k.p) k.p spinors Psi_i² in addition to the output of Psi² which is the sum of all six or eight spinors: Psi² = SUM_i=1ⁿ Psi_i²

Filename:3Dkp8x8_spinori_el_qc001_kpar001_ev001.fld / *.coord / *.dat 3Dkp8x8_spinori_complx_el_qc001_kpar001_ev001.fld / *.coord / *.dat (if complex-wave-functions = yes)

detailed-output = yes / no

Flag whether to print out additional output for k.p.
In particular more information about the eigenvalues for each k|| vector.

scale = 0.1d0

The scale parameter can be used to scale the size of the wave functions in the output file.

1D: So far, it scales the wave function psi and psi² (i.e. the probability amplitude or charge density) in the same way:
psi²' = scale * psi² psi' = scale * psi

2D/3D: No scaling via input file implemented so far.
However, internally in 3D psi² is scaled in any case by 1 * 10¹⁰.
??? In 3D, if specified in input file: scale * 0.001 / MAXVAL(spinorV) * 1 * 10¹⁰ = 1 * 10¹⁷ / MAXVAL(spinorV)

1D: The units of psi² are [1/nm], the units of psi are SQRT([1/nm]).
   This way the integrated psi² over the whole device (which is in units of [nm]) equals 1.
2D: The units of psi² are [1/nm²], the units of psi are SQRT([1/nm²]).
   This way the integrated psi² over the whole device (which is in units of [nm²]) equals 1.
3D: The units of psi² are [1/nm³], the units of psi are SQRT([1/nm³]).
   This way the integrated psi² over the whole device (which is in units of [nm³]) equals 1.

A good check to see if psi² is scaled correctly is to apply Neumann boundary conditions at all boundaries of the quantum cluster. The ground state probability density then is constant over the whole device. This value in units of [1/nm] (1D), [1/nm²] (2D) or [1/nm³] (3D) multiplied by the length (1D), area (2D) or volume (3D) of the quantum cluster must equal 1.

shift-wavefunction-by-eigenvalue = yes !(1D: yes =default) = no !(2D/3D:no =default)

If yes, in addition to default output, the wave function psi and the probability density psi² are shifted with respect to their eigenvalue.
This is sometimes useful when plotting the wave functions together with the band edge profile.
The relevant output files have the label _shift in their file names.

Bulk dispersion:

See tutorial: k.p dispersion in bulk GaAs (strained / unstrained)

bulk-kp-dispersion = yes / no

Flag that signals if you want to put out the pure bulk k.p dispersion E(k) = E(k_x,k_y,k_z) along a line from k = 0 to k = (k_x,k_y,k_z).
This can be of interest in strained regions where the bands are shifted due to strain. The dispersion is taken at one point in space grid-position which has to be located inside a quantum region.
It is also possible to output the parabolic, isotropic dispersion of the effective-mass approximation.

bulk-kp-dispersion-3D = yes / no = graphene !(special option for bulk graphene tight-binding band structure)

Flag that signals if you want to put out the pure bulk k.p dispersion E(k) = E(k_x,k_y,k_z) over the three-dimensional k space in the AVS/Express format.
This can be of interest in strained regions where the bands are shifted due to strain. The dispersion is taken at one point in space grid-position which has to be located inside a quantum region.
- bulk_8x8kp_dispersion3D_cb1.fld / *.dat / *.coord / *_ijk.dat - bulk_8x8kp_dispersion3D_cb2.fld / *.dat / *.coord / *_ijk.dat - bulk_8x8kp_dispersion3D_hh1.fld / *.dat / *.coord / *_ijk.dat - bulk_8x8kp_dispersion3D_hh2.fld / *.dat / *.coord / *_ijk.dat - bulk_8x8kp_dispersion3D_lh1.fld / *.dat / *.coord / *_ijk.dat - bulk_8x8kp_dispersion3D_lh2.fld / *.dat / *.coord / *_ijk.dat - bulk_8x8kp_dispersion3D_soh1.fld / *.dat / *.coord / *_ijk.dat - bulk_8x8kp_dispersion3D_soh2.fld / *.dat / *.coord / *_ijk.datNote: The individual components are called
cb1 = electron 1 cb2 = electron 2 hh1 = heavy hole 1 hh2 = heavy hole 2 lh1 = light hole 1 lh2 = light hole 2 soh1 = split-off hole 1 soh2 = split-off hole 2which is correct ONLY for the unstrained case. In the strained case the character is no longer "purely" heavy, light and split-off hole-like if strain is present because all the states are mixed! Note that the '1' and '2' states might be degenerate.To see an example, please refer to the Tutorial "Strained Silicon".

grid-position = 100d0 ! in units of[nm]

grid-position = 10d0 ! x = 10 [nm] (1D) = 10d0 20d0 ! x = 10 [nm], y = 20 [nm] (2D) = 10d0 20d0 20d0 ! x = 10 [nm], y = 10 [nm], z = 20 [nm] (3D)

Determines position of point in structure for bulk dispersion (must be within a k.p quantum cluster).

k-direction = 0d0 1.0d0 1.0d0 ! [1/Angstrom]along [011] direction with respect to simulation coordinate system = 0d0 0d0 1.0d0 ! [1/Angstrom]along [001] direction with respect to simulation coordinate system

Determines k-direction and range for dispersion plot [1/Angstrom].

number-of-k-points = 50

Number of k points to be calculated (resolution) in addition to the Gamma point at (k_x,k_yk_z) = (0,0,0) for the bulk k.p dispersion along a particular direction.

shift-holes-to-zero = yes = no

This is to shift the bulk dispersion curve, i.e. to set the zero point of the energy axis to the topmost hole energy:
E(k=0) = 0 eV for the heavy and light holes dispersion (unstrained case) or to the topmost hole energy (strained case or wurtzite).
The conduction band dispersion then starts at the band gap energy and the split-off energy dispersion at delta_split-off.
The strain can lift the degeneracy of heavy and light holes. Thus we define for the zero of energy the topmost valence band energy if
shift-holes-to-zero = yes.

Output files
- bulk_sg_dispersion000_kxkykz.dat     isotropic and parabolic dispersion of single-band effective masses from [000] to [kx ky kz] bulk_8x8kp_dispersion000_kxkykz.dat  dispersion of bulk k.p Hamiltonian                                              from [000] to [kx ky kz]
- bulk_sg_dispersion100_000_001.dat    isotropic and parabolic dispersion of single-band effective masses from [100] to [000] to [001] bulk_8x8kp_dispersion100_000_001.dat dispersion of bulk k.p Hamiltonian                                              from [100] to [000] to [001]
bulk_8x8kp_dispersion110_000_001.dat dispersion of bulk k.p Hamiltonian                                              from [110] to [000] to [001]
bulk_8x8kp_dispersion111_000_001.dat dispersion of bulk k.p Hamiltonian                                              from [111] to [000] to [001]
These directions might not be general enough for wurtzite or strained structures but one can rotate the crystal coordinate system with respect to the simulations coordinate system accordingly in the input file.

k_|| dispersion:

k-par-dispersion = no #no E(k_||) dispersion = yes # E(k_||) = E(k_x,k_y) dispersion for all values of k_x, k_y = 01-00 #E(k_||) = E(0,k_y) dispersion for k_y = [k_y,max,...0] = 01-00-11 #E(k_||) = E(0,k_y) dispersion for k_y = [k_y,max,...,0] (where k_x=0) and for E(k_||) = E(k_x,k_y) dispersion for k_x = k_y = 01-00-10 #E(k_||) = E(0,k_y) dispersion for k_y = [k_y,max,...,0] (where k_x=0) and for E(k_||) = E(k_x,0) dispersion for k_x = [k_x,max,...,0] (where k_y=0)

Flag that signals if you want to have the E(k_||) = E(k_x,k_y) dispersion to be put out. This is for the quantum mechanically calculated eigenvalues.
This also works for 2D, i.e. E(k_z) but does not make sense for 3D.
'01-00', '01-00-11' and'01-00-10'are special options for slices along special directions through the two-dimensional E(k_||) = E(k_x,k_y) dispersion.
- 01-00: A plot of E(k_x,k_y) along the line from E(0,k_y,max) to E(0,0) .
- 01-00-11: A plot of E(k_x,k_y) along the line from E(0,k_y,max) to E(0,0) (where k_x=0) and along the line from E(0,0) to E(k_x,max,k_y,max=k_x,max) for k_x = k_y - 01-00-10: A plot of E(k_x,k_y) along the line from E(0,k_y,max) to E(0,0) (where k_x=0) and along the line from E(0,0) to E(k_x,max,0) (where k_y=0)

Note: If you use k-par-dispersion = 01-00, 01-00-11or 01-00-10, then the calculation is much faster because only at the k_|| points at these symmetry lines the eigenvalues are calculated.
Obviously in this case you cannot do a self-consistent calculation. Also plotting the 2D dispersion of E(k_x,k_y) (i.e. the 2D plot) or the density of states is not correct in this case.

k-par-disp-ev-min = 1

Lower boundary for range of eigenvalues for which the k_|| dispersion should be written out.
This also works for 2D, i.e. E(k_z) but does not make sense for 3D.
This specifier affects the energy dispersion plots E(k_||).

k-par-disp-ev-max = 10

Upper boundary for range of eigenvalues for which the k_|| dispersion should be written out.
This also works for 2D, i.e. E(k_z) but does not make sense for 3D.
This specifier affects the energy dispersion plots E(k_||).

k-par-disp-k-range = 0.1d0 ! [1/Angstrom]

Radius of k_|| points [1/Angstrom].
Absolute value of k radius [1/Angstrom], if not present or zero, an internally used value is taken.
This also works for 2D, i.e. E(k_z) but does not make sense for 3D.

k-par-disp-num-k-par = 20

Number of k_|| points to be calculated. At present this specifier is basically ignored as the number of k_|| vectors is specified in$quantum-model-electronsand$quantum-model-holes.
Therefore you can and should omit it.
If no values for range and number are given, default values are used in self-consistent calculation.
However, it is useful to make use of this specifier if the self-consistent calculation is on a coarse k space grid, and you want for the output a fine k space resolution, then you should use this specifier with a higher number of k_|| points.
This also works for 2D, i.e. E(k_z) but does not make sense for 3D.

The conception of the k.p output is the following:

We define the k_|| grid within the keywords:
- $quantum-model-electrons - $quantum-model-holes
It cannot be a very fine grid because of a big amount of time necessary for the self-consistent calculation.
But the user may wish to have a k.p output on a finer grid after the self-consistent potential has been found.

There are two possibilities:

- k-par-disp-k-range - k-par-disp-num-k-par
If none of these entries is present within the keyword$output-kp-data(or not different to the default ones as specified in $quantum-model-electrons/$quantum-model-holes) then the k.p eigenfunctions/eigenenergies that were used for the self-consistent calculations will be written out.
If any of these entries is present then another set of k_|| points will be considered, a new set of eigenfunctions will be calculated and then written out.
At present, k-par-disp-k-range must be present unless specified within $quantum-model-electrons/$quantum-model-holes: k-range-determination-method = k-max-input ! 1D/2D k-range = 1.0d0

Density of states (DOS) for k_|| dispersion

DOS-density-of-states = yes ! 'yes' / 'no' flag to output the density of states (DOS) (default: 'no')
! only implemented for 1D so far, i.e. E(k_x,k_y)

DOS-Emin-Emax         = 0d0 2.5d0 !minimum and maximum energy for density of states (DOS) [eV]
                                    ! used for DOS calculation
                                    ! for Monte Carlo part: used for DOS calculation and scattering tables

DOS-points = 1000 ! number of points between DOS-E_min and DOS-E_max (default: 1000).
! This number determines the grid resolution of the energy grid used for the density of states (DOS) output. ==> Energy_grid = (DOS_Emax - DOS_Emin) / DOS_Points = 2.5 eV / 1000 = 0.0025 eV

For more information on the output (e.g. units), see below.

For more information on the density of states, have a look at the tutorial: Density of states (DOS) in a GaAs quantum well with infinite barriers

Conduction bands:

cb-num-ev-min = 1

Lower boundary for range of conduction band eigenvalues for which those and the eigenfunctions are put out.

cb-num-ev-max = 1

Upper boundary for range of conduction band eigenvalues for which those and the eigenfunctions are put out.

cb-k-par-min = 1

Lower boundary for range of conduction band k_|| points in 1D for which eigenvalues and eigenfunctions are put out.
Lower boundary for range of conduction band k_z points in 2D for which eigenvalues and eigenfunctions are put out.
(Does not make sense in 3D.)

cb-k-par-max = 1

Upper boundary for range of conduction band k_|| points for which eigenvalues and eigenfunctions are put out.
Upper boundary for range of conduction band k_z points in 2D for which eigenvalues and eigenfunctions are put out.
(Does not make sense in 3D.)

cb-k-z-min = 1 cb-k-SL-min = 1

Only for superlattices (should be greater than or equal to 1).
Lower bound for range of superlattice vectors K_z(1D),K_x,K_y(2D),K_x,K_y,K_z(3D).

cb-k-z-max = 1 cb-k-SL-max = 1

Only for superlattices (should be greater than or equal to 1).
Upper bound for range of superlattice vectorsK_z(1D), K_x,K_y(2D),K_x,K_y,K_z(3D).

Valence bands:

vb-num-ev-min = 1

Lower boundary for range of valence band eigenvalues for which those and the eigenfunctions are put out.

vb-num-ev-max = 1

Upper boundary for range of valence band eigenvalues for which those and the eigenfunctions are put out.

vb-k-par-min = 1

Lower boundary for range of valence band k_|| points in 1D for which eigenvalues and eigenfunctions are put out.
Lower boundary for range of valence band k_z points in 2D for which eigenvalues and eigenfunctions are put out.
(Does not make sense in 3D.)

vb-k-par-max = 1

Upper boundary for range of valence band k_|| points in 1D for which eigenvalues and eigenfunctions are put out.
Upper boundary for range of valence band k_z points in 2D for which eigenvalues and eigenfunctions are put out.
(Does not make sense in 3D.)

vb-k-z-min = 1 vb-k-SL-min = 1

Only for superlattices (should be greater than or equal to 1).
Lower bound for range of superlattice vectors K_z(1D),K_x,K_y(2D),K_x,K_y,K_z(3D).

vb-k-z-max = 1 vb-k-SL-max = 1

Only for superlattices (should be greater than or equal to 1).
Upper bound for range of superlattice vectors K_z(1D),K_x,K_y(2D),K_x,K_y,K_z(3D).

interband-matrix-elements = yes ! calculates interband matrix elements =no==> square of spatial overlap matrix element | <psi_i* | psi_j> |^2The output files
interband1D_vb001_cb001_qc001_hlsg001_deg001_dir.dat(heavy hole <-> Gamma conduction band) interband1D_vb002_cb001_qc001_hlsg002_deg001_dir.dat(light hole <-> Gamma conduction band) interband1D_vb003_cb001_qc001_hlsg003_deg001_dir.dat(split-off hole <-> Gamma conduction band) contain data like Spatial overlap matrix elements | < psi_hl_i | psi_el_j > |^2 and energy of transition in [eV] heavy hole <-> Gamma conduction band ------------------------------------------------------------------------ |<psi_vb001|psi_cb001>|^2 0.987507995852382 1.654103 |<psi_vb001|psi_cb002>|^2 1.336279027563441E-030 |<psi_vb001|psi_cb003>|^2 -0.145559411422541 2.538366 |<psi_vb002|psi_cb001>|^2 1.133344425625580E-030 |<psi_vb002|psi_cb002>|^2 -0.964789984970279 2.065139which are the spatial overlap matrix elements between all calculated states in bands 'cband' and 'vband' from eigenvalues 'min-ev' to 'max-ev'.

To plot matrix elements vs. electric field.The electric field is calculated by('voltage-offset' + vbias) / distancewith v_bias = sweep_index * sweep_voltage.The E_field is scaled to [kV/cm].The 'voltage-offset' is the built-in potential in [V].distance:The length of the region with constant field [in scaled units: nm] is specified via 'lever-arm-length'.This only works for intrinsic regions with quasi constant electric field, but imitates the way experimental physicists approximate the electric field.
In order to use only bound states the MODULE bound_states_1D must be initialized.
=> $quantum-bound-states

-> momentum matrix <psi*|p|psi> (not implemented yet)-> Coulomb element <psi*|V|psi> (not implemented yet)with V = Int( 1/4pi (r1-r2) * |psi(r2)|‹dr2 )

intraband-matrix-elements = p    ! < psi_j* | p | psi_i >                           = z    ! < psi_j* | z | psi_i >                           = o    ! < psi_j*   |   psi_i > (spatial overlap)                           = yes !(prints out all matrix elements, i.e. 'p', 'z' and 'o')                          = no   !Calculates intersubband dipole moment and oscillator strength.

The output files
intraband_p1D_cb001_qc001_6x6kp_dir.dat             z                         !kind of matrix element ('p' / 'z' / 'o')            o                         !kind of matrix element ('p' / 'z' / 'o')             2D                       !'1D', '2D', '3D'             3D                       !'1D', '2D', '3D'                vb001                 !kind of band (Gamma conduction band 'cb001' / heavy, light and split-off hole bands 'vb001')                            8x8       !kind of k.p ('4x4' / '6x6' / '8x8')                                  neu !kind of boundary condition (Dirichlet 'dir' / Neumann 'neu') contain data likeNote that the two-fold spin-degeneracy in the single-band approximation ('effective-mass') is counted explicitely in k.p.(Note:< psi_j* | p | psi_i >matrix element given in green color.)
------------------------------------------------------------------------------- Intersubband transitions => Gamma conduction band ------------------------------------------------------------------------------- Electric field in z-direction [kV/cm]: 0.0000000E+00 ------------------------------------------------------------------------------- -------------------------------------------------------------------------------        Intersubband dipole moment < psi_j* | z | psi_i > [Angstrom]        Intersubband dipole moment < psi_j* | p | psi_i > [- i h_bar / Angstrom] ------------------|------------------------------------------------------------                                   Oscillator strength [] ------------------|--------------|---------------------------------------------                                                  Energy of transition [eV] ------------------|--------------|--------------|------------------------------                                                              m* [m_0] ------------------|--------------|--------------|-----------|------------------ <psi001*|z|psi001> 249.0000      (matrix element <1|1> depends on choice of origin!)
<psi002*|z|psi001> 249.0000      (matrix element <2|1> depends on choice of origin!)<psi001*|p|psi001> 1.8126842E-18 (matrix element <1|1> independent of origin)<psi002*|p|psi001> 1.8126842E-18 (matrix element <2|1> independent of origin)<psi003*|z|psi001> 18.01673       0.9602799      0.1694912 6.6500001E-02 <psi004*|z|psi001> 18.01673       0.9602799      0.1694912 6.6500001E-02 <psi003*|p|psi001> 2.6649671E-02 0.9602798      0.1694912 6.6500001E-02 <psi004*|p|psi001> 2.6649671E-02 0.9602798      0.1694912 6.6500001E-02 <psi005*|z|psi001> 3.5382732E-13 <psi006*|z|psi001> 3.5382732E-13 <psi005*|p|psi001> 2.1414240E-15 <psi006*|p|psi001> 2.1414240E-15 <psi007*|z|psi001> 1.441336       3.0698583E-02 0.8466209 6.6500001E-02 <psi008*|z|psi001> 1.441336       3.0698583E-02 0.8466209 6.6500001E-02 <psi007*|p|psi001> 1.0649348E-02 3.0698583E-02 0.8466209 6.6500001E-02 <psi008*|p|psi001> 1.0649348E-02 3.0698583E-02 0.8466209 6.6500001E-02 <psi009*|z|psi001> 7.2598817E-13 <psi010*|z|psi001> 7.2598817E-13 <psi009*|p|psi001> 1.0445775E-14 <psi010*|p|psi001> 1.0445775E-14 <psi011*|z|psi001> 0.3971008      5.4281550E-03 1.972205   6.6500001E-02 <psi012*|z|psi001> 0.3971008      5.4281550E-03 1.972205   6.6500001E-02 <psi011*|p|psi001> 6.8347319E-03 5.4281550E-03 1.972205   6.6500001E-02 <psi012*|p|psi001> 6.8347319E-03 5.4281550E-03 1.972205   6.6500001E-02 ... <psi039*|z|psi001> 1.0178294E-02 3.9452352E-05 21.81846   6.6500001E-02 <psi040*|z|psi001> 1.0178294E-02 3.9452352E-05 21.81846   6.6500001E-02 <psi039*|p|psi001> 1.9380630E-03 3.9452349E-05 21.81846   6.6500001E-02 <psi040*|p|psi001> 1.9380630E-03 3.9452349E-05 21.81846   6.6500001E-02 Sum rule of oscillator strength: f_j,001 = 0.9994023 Sum rule of oscillator strength: f_j,001 = 0.9994023 ...which are the intersubband dipole moments

    | M_ji | = | integral (psi_j* (z) z psi_i (z) dz) |

and the oscillator strengths

    f_ji = 2m* / h_bar² (E_j - E_i) | M_ji |²

between all calculated states in each band from eigenvalues 'min-ev' to 'max-ev'.

Unfortunately, the commonly used                  Intersubband dipole moment < psi_j* | z | psi_i >   [Angstrom]
depends on the choice of origin for the matrix elements where j = i (also j+1 = i in some cases because of spin degeneracy, see above), thus the user might prefer to output the
                  Intersubband dipole moment < psi_j* | p | psi_i >   [- i h_bar / Angstrom]

which are the intersubband dipole moments

    | N_ji | = | integral (psi_j* (z) p_z psi_i (z) dz) | = | - i h_bar integral (psi_j* (z) d/dz psi_i (z) dz) |

and the oscillator strengths

    f_ji = 2m* / h_bar² (E_j - E_i) | M_ji |² = 2 / ( m* (E_j - E_i) ) | N_ji |²

between all calculated states in each band from eigenvalues 'min-ev' to 'max-ev'.

For more details, have a look at the tutorial: Optical intersubband transitions in a quantum well - Intraband matrix elements and selection rules

intraband-lifetime = yes ! calculates the lifetime of intersubband transitions = no ! does nothing (default)

This feature is useful for e.g. quantum cascade lasers.
Note: intraband-matrix-elements must not be set to no in order to print out the lifetimes (scattering rates).
See tutorial "Scattering times for electrons in unbiased and biased single and multiple quantum wells" for more details.

Output:

Simple files

kp_8x8eigenvalues_qc001_el1D_dir.dat
contains all electron eigenvalues for k = 0 and quantum cluster 1 (for each grid point), so that they can be plotted into the conduction/valence band edge diagram.
Similar for holes and 6x6 k.p.
kp_8x8psi_squared_qc001_el1D_dir.datcontains the square of all electron wave functions (psi²) for k = 0 and quantum cluster 1 (for each grid point). Note that psi² is shifted by the corresponding eigenvalues so that the square of the wave functions can be plotted into the conduction/valence band edge diagram.
Similar for holes and 6x6 k.p.

dir = Dirichlet boundary conditions
neu = Neumann boundary conditions
per = periodic boundary conditions

k.p eigenvalues

e.g. kp_8x8eigenvalues_pos_qc001_el_kpar0001_1D_dir.dat

This file contains all eigenvalues. It can be plotted into a conduction or valence band profile diagram.

k.p eigenfunctions

Filename:

`kp_8x8_el1D_wv_qc001_ev003_kpar001_Kz001_dir.dat`
`8x8`							Kind of kp solved (8x8 or 6x6)
	`_el1D_``wv`						Electron/hole (`_el/_hl`) eigenvectors (`_``wv`)
		`_qc001`					Number of quantum cluster
			`_ev001`				Number of eigenvalue
				`kpar001`			Number of k_\|\| point
					`Kz001`		Number of K_z point (superlattice vector)
						`_dir`	Boundary condition: Neumann/Dirichlet (`_neu`/`_dir`)

Structure:

position[nm]	PSI^2_SUM	PSI^2(1)	PSI^2(2)	PSI^2(3)	PSI^2(4)	PSI^2(5)	PSI^2(6)	PSI^2(7)	PSI^2(8)
Position	Sum of components^2	Squares of components of k.p-vector. Eight in case of 8x8 k.p / six for 6x6 k.p / four for 4x4 k.p.

k.p complex eigenfunctions:

Filename:

`kp_comp_8x8_el1D_wv_qc001_ev002_kpar001_Kz001_dir.dat`
`_comp`								Complex wave functions
	`_8x8`							Kind of k.p solved (8x8 or 6x6)
		`_el1D_``wv`						Electron/hole (`_el/_hl`) wavevectors (`_``wv`)
			`_qc001`					Number of quantum cluster
				`_ev001`				Number of eigenvalue
					`kpar001`			Number of k_\|\| point
						`Kz001`		Number of K_z point (superlattice vector)
							`_dir`	Boundary condition: Neumann/Dirichlet (`_neu`/`_dir`)

Structure:

position[nm]	`PSI_real001`	`PSI_imag001`	...
position	Real part of 1^st component	Imaginary part of 1^st component	...

schroedinger-kp-discretization = finite-differences

The components are: | x+ >, | y+ >, | z+ >, | x- >, | y- >, | z- >

| x >, | y >, | z > correspond to x, y, z of the calculation system.
+ and - correspond to the spin projection along the z axis of the crystal system.

In nextnano³ k.p subroutine we introduced an unnecessary rotation. When we get rid of it in the future, we will have| x >, | y >, | z >in crystal system.

k_x, k_y are in calculation system.

schroedinger-kp-discretization = box-integration

The components are: | hh1 >, | lh1 >, | lh2 >, | hh2 >, | so1 >, | so2 > hh =heavy hole lh =light hole so =split-off hole

| J , m > | 3/2, 3/2 > = | hh1 > | 3/2, 1/2 > = | lh1 > | 3/2, -1/2 > = | lh2 > | 3/2, -3/2 > = | hh2 > | 1/2, 1/2 > = | so1 > | 1/2, -1/2 > = | so2 >

k.p bulk dispersion:

Bulk k.p dispersion E(k).

Filename:

bulk_8x8kp_dispersion000_kxkykz.dat (k.p dispersion)bulk_sg_dispersion000_kxkykz.dat (effective-mass dispersion)

Structure:

k	hi ( i = 1-8, or 1-6)
Position in k space in direction of `k-direction [1/Angstrom]`	Eigenvalues of bulk k.p Hamiltonian

See tutorial: k.p dispersion in bulk GaAs (strained / unstrained)

k_|| dispersion:

Remark:

1D: The output is two-dimensional because the dispersion in the whole plane perpendicular to the quantization direction is calculated. The output therefore consists of three files (AVS/Express format):

AVS/Express
Field file:      kpar1D_disp_hl_6x6kp_qc001_ev001_2Dplot.fld
                     AVS/Express field file
Coordinates:   kpar1D_disp_hl_6x6kp_qc001_ev001_2Dplot.coord           AVS/Express coordinate files, x/y coordinates
Data file:    kpar1D_disp_hl_6x6kp_qc001_ev001_2Dplot.dat                    AVS/Express data file, dispersion in nonquantized directions

Filename:

`kpar1D_disp_hl_6x6kp_qc001_ev001_2Dplot.dat`
`_hl/_el`			File containing data for conduction band (`_el`) or valence band (`_hl`) eigenstates
	`_6x6kp/_8x8kp`		kind of k.p (6x6 or 8x8)
	`_qc001`		Number of quantum cluster
		`_ev001`	Number of eigenvalue of subband

Structure:

E_val_k_par:

Eigenvalues of k_|| points

Example: See Tutorial "Energy dispersion of holes in quantum wells".

k_|| dispersion from [00] to [10]:

1D: This file contains data of a cut along the line from zero (i.e. [k_x,k_y] = [0,0] = [00]) to [10] (i.e. [k_x,k_y] = [k_x,0] where k_x > 0) of the 2D plot of the E(k_||)=E(k_x,k_y) dispersion.

Filename:

kpar1D_disp_00_10_hl_6x6kp_ev001.dat kpar1D_disp_00_10_el_8x8kp_ev001.dat

`kpar1D_disp_00_10_hl_6x6kp_ev001.dat`
`_hl/_el`			File containing data for conduction band (`_el`) or valence band (`_hl`) eigenstates
	`_6x6kp/_8x8kp`		kind of k.p (6x6 or 8x8)
		`_ev001`	Number of eigenvalue of subband

Structure:

`abscissa`	`eigenvalue`	`kx`	`ky`	`num_kpar`
`kx`	Eigenvalue of k_\|\| point	k_x position `[1/Angstrom]`	k_y position `[1/Angstrom]`	Number of k_\|\| point

The filekpar1D_disp_00_10hl_8x8kp_ev_min001kp_ev_max010.datcontains essentially the same data but here all subband dispersions from eigenvalue min001 to eigenvalue max011 are contained in one single file.

k_|| dispersion from [01] to [00] and from [00] to [11]:

1D: This file contains data of cuts along the line from [01] to zero and from zero to [11] of the 2D plot of the E(k_||)=E(k_x,k_y) dispersion.

Filename:

kpar1D_disp_01_00_11_hl_6x6kp_ev001.dat kpar1D_disp_01_00_11_el_8x8kp_ev001.dat

`kpar1D_disp_01_00_11_hl_6x6kp_ev001.dat`
`_hl/_el`			File containing data for conduction band (`_el`) or valence band (`_hl`) eigenstates
	`_6x6kp/_8x8kp`		kind of k.p (6x6 or 8x8)
		`_ev001`	Number of eigenvalue of subband

Structure:

`abscissa`	`eigenvalue`	`kx`	`ky`	`num_kpar`
`-ky -- for [01] to [00] SQRT(kx²+ky²) -- for [00] to [11]`	Eigenvalue of k_\|\| point	k_x position `[1/Angstrom]`	k_y position `[1/Angstrom]`	Number of k_\|\| point

To plot the data file, only the first (abscissa) and the second column (eigenvalue) are important.

Example: See Tutorial "Energy dispersion of holes in quantum wells".

The filekpar1D_disp_01_00_11hl_8x8kp_ev_min001kp_ev_max010.datcontains essentially the same data but here all subband dispersions from eigenvalue min001 to eigenvalue max011 are contained in one single file.

Simple k_|| dispersion data:

Filename:

kpar1D_disp_simple_hl_6x6kp_ev001.dat kpar1D_disp_simple_el_8x8kp_ev001.dat

`kpar1D_disp_simple_hl_ev001.dat`
`_hl/_el`			File containing data for conduction band (`_el`) or valence band (`_hl`) eigenstates
	`_6x6kp/_8x8kp`		kind of k.p (6x6 or 8x8)
		`_ev001`	Number of eigenvalue of subband

Structure:

`num_kpar`	`kx`	`ky`	`eigenvalue`
Number of k_\|\| point	k_x position `[1/Angstrom]`	k_y position `[1/Angstrom]`	Eigenvalue of k_\|\| point

This file cannot be plotted. It just contains for each k_|| point the corresponding k_x and k_y values as well as its corresponding eigenvalue.

Density of states (DOS)

The two-dimensional density of states (DOS) is written to the files'DOS_hl_6x6kp.dat'and 'DOS_hl_6x6kp_sum.dat' (and similar for electrons).
The DOS has been calculated from the energy dispersion E(k) = E(k_x,k_y).

'DOS_hl_6x6kp_norm.dat'
=======================
This file contains the density of states (DOS) for each eigenvalue (i.e. subband).
The first column contains the energy in units of [eV], the other columns the DOS of each individual subband in units of [eV^-1 m^-2].
'DOS_hl_6x6kp_sum_norm.dat'
===========================This file contains the density of states (DOS) , summed over each eigenvalue (i.e. subband).
The first column contains the energy in units of [eV], the second column the DOS (sum over all subbands) in units of [eV^-1 m^-2].

For the above two files, the x axis is in units of [eV] and its boundaries can be adjusted by the following specifier:
DOS-Emin-Emax = 0d0 2.5d0 ! [eV]The energy grid of the x axis (energy grid resolution) can be adjusted by the following specifier which determines the number of points between DOS-E_min and DOS-E_max:
DOS-points = 1000 ==> Energy_grid = (DOS_Emax - DOS_Emin) / DOS_Points = 2.5 eV / 1000 = 0.0025 eV

Density n(kx,ky)

kpar1D_density_el_8x8kp_qc001_2Dplot.fld -contains density as a function of kx and ky: n(kx,ky)

kpar1D_density_el_8x8kp_qc001_ev001_2Dplot.fld - contains density as a function of kx and ky: n_i(kx,ky)for each eigenvalue i

The units are dimensionless, i.e. if one integrates over k_x and k_y (which are in units of [1/m]), one gets a sheet charge density in units of [m^-2].

Optics

For the option to calculate and output the optical matrix elements we need the following specifiers.

optical-matrix-element-output = no = yes ! output of intensity, oscillator strength and transition energies = complex !output of intensity, oscillator strength and transition energies, as well as complex matrix element dipole-transition-type = el-el = el-hl = hl-el = hl-hl initial-band-min = 1 initial-band-max = 2 final-band-min = 8 final-band-max = 9

Note: For 'el-hl' initial means 'el', final means 'hl'. For 'hl-el' initial means 'hl', final means 'el'.
Example: In this example, the following transitions are considered:
1 -> 8 1 -> 9 2 -> 8 2 -> 9

1D:
This generates a 2D plot (in the AVS/Express format) for the intensity I(k_x, k_y) in units of [eV].
In addition, a 2D plot for the optical matrix element: P(k_x, k_y) in units ofSQRT([2eV/m₀])is generated ifoptical-matrix-element-output = complex.
Note that the intensity and the matrix element are three-dimensional vectors with components I_x, I_y, I_z or P_x, P_y, P_z, respectively, and k_x, k_y is in units of [1/Angstrom].
The optical matrix element will be calculated between band-min andband-maxeigenvalues.
The k_|| points are the same that are chosen for the energy dispersion output.

Output units:
a) Intensity: I = 2/m₀ * < e.P >: I_x, I_y, I_z, I_xy, I_xz, I_yz: [eV]
b) complex value of the momentum matrix element P_x, P_y, P_z, P_xy, P_xz, P_yz:SQRT([2eV/m₀])
e is the light polarization vector
P is the momentum operator (vector)
P_x(k) = <psi_1(k)| e.P |psi_2(k)>, where |e|=1, e || x (i.e. e || [1,0,0])
P_xy(k) = <psi_1(k)| e.P |psi_2(k)>, where |e|=1, e || xy (i.e. e || [1,1,0])

The file 'opt_Intensity_info...txt' contains the following data:
optical-matrix-element-output = yes !Units:[eV] kx ky kz I_x I_y I_z I_xy I_xz I_yz

The file 'opt_MatrixElement_info...txt' contains the following data:

optical-matrix-element-output = complex !Units:SQRT([2eV/m₀]) kx ky kz Re(P_x) Im(P_x) Re(P_y) Im(P_y) Re(P_z) Im(P_z) Re(P_xy) Im(P_xy) Re(P_xz) Im(P_xz) Re(P_yz) Im(P_yz)

Optical oscillator strength

Example: Optical oscillator strength of the fundamental transition in a 2D quantum wire calculated by 8x8 k.p theory. The optical matrix elements (complex) were calculated. In the picture one can see the dependence on the polarization.

Note that the oscillator strength is a dimensionless quantity. It is useful in order to compare the probabilities of different dipole transitions.

(The calculations were performed by Michael Povolotskyi, University of Rome "Tor Vergata".)

Most attention is paid to the matrix element. It is not taken from the database (or input file), instead it is calculated from the k.p Hamiltonian. Due to this approach all kinds of dipole transitions are calculated in the same manner. All the symmetry is taken into account automatically.
Example: Optical intraband TE mode (transverse electric) transitions are treated correctly within k.p. (Note that they are not zero within k.p but they are zero for electrons in a single band approach.)

k.p material parameters

The k.p material parameters can be found in the folder material_parameters/.

The file 6x6kp_parameters1D_used.dat contains the following columns:
material_grid position in [nm] on material grid L M N+ N- N 6x6 k.p Dresselhaus parameters L,M,N. Note: N = N⁺ + N^-

This file6x6kp_Luttinger_parameters1D_used.datcontains the following columns:

material_grid                           position in [nm] on material grid gamma1 gamma2 gamma3 (kappa)            6x6 k.p Luttinger parameters (ifkappais not used, it is in brackets) (A) (B) (C^2)                           A,B,C k.p parameters of Dresselhaus, Kip, Kittel (DKK Hamiltonian) in units of [h_bar²/2m₀], Phys. Rev. 98, 368 (1955)                                          The warping (anisotropy) is caused by C. If C is zero, than the energy dispersion is isotropic (spherical).
                                          The dispersion is nonparabolic if both B and C are nonzero. (m_hh_iso) (m_lh_iso) (m_so_iso)        spherically averaged heavy and light hole masses according to Yu, Cardona, Fundamentals of semiconductors, p. 201 (Problem 4.4),
                                          A. Baldereschi, N.O. Lipari, PRB 8, 2697 (1973)                                          The split-off mass is currently 1/gamma₁ for 6x6 k.p, but for 8x8 k.p according to I. Vurgaftman et al., JAP 89, 5815, eq. (2.18). (m_hh_100) (m_lh_100)                   heavy and light hole masses along [100] according to I. Vurgaftman et al., JAP 89, 5815, eq. (2.16) and eq. (2.17) (m_hh_110) (m_lh_110)                   heavy and light hole masses along [110] according to I. Vurgaftman et al., JAP 89, 5815, eq. (2.16) and eq. (2.17) (m_hh_111) (m_lh_111)                   heavy and light hole masses along [111] according to I. Vurgaftman et al., JAP 89, 5815, eq. (2.16) and eq. (2.17)

The A, B, C parameters and the masses are currently not used inside the code. These values are just calculated and printed out to make the k.p parameters "more transparent".

If it holdsgamma2 = gamma3(spherical approximation) then the valence band energy dispersion is isotropic. (This is equivalent to: L - M = N.)


		Last modified: 27-Jan-2012