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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-ev and cb-max-ev) are written out in one file.

!-------------------------------------------------------------!
$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-from-k-point              double_array  optional !
 k-direction-to-k-point                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 !
 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 !
 intraband-matrix-elements-operator    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/

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    ! It would have been better to call this amplitudes rather than complex-wave-functions.
 

Flag whether to print out the wave functions psi (amplitudes) including real and imaginary parts in addition to the output of the probability densities Psi².
If set to yes, then kp-spinors is also set to yes.

 

kp-spinors = yes / no

Flag whether to print out the n = 6 (6-band k.p) or n = 8 (8-band k.p) k.p spinors Psii2 in addition to the output of Psi² which is the sum of all six or eight spinors: Psi² = SUMi=1n Psii2

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.
This is just for visualization purposes in order to fit wavefunctions more nicely into band structure plots. This scaling has no physical meaning.

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

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.

(It holds for unscaled psi², i.e. scale = 1d0: A good check to see if psi² is normalized 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.

 

 

There are two different types of E(k) dispersion, i.e. energy vs. k = (kx,ky,kz) vector plot:

  • Bulk k.p dispersion (bulk-kp-dispersion)
    The bulk k.p is the dispersion E(kx,ky,kz) of the material at a specific grid point (grid-position) which can be a binary or ternary semiconductor.
    These are the energy levels Ei of the 8-band k.p Hamiltonian H(kx,ky,kz) which has the dimension 8.
     
  • k|| dispersion, i.e. kparallel (k-par-dispersion)
    In contrast to the bulk k.p dispersion which refers to a grid point, the k|| dispersion refers to the whole quantum cluster.
    1D simulation: The k|| dispersion is the dispersion E(kx,ky) where z is the quantization direction, e.g. for a quantum well grown along the z direction.
    2D simulation: The k|| dispersion is the dispersion E(kz) where x and y are the quantization directions, e.g. for a quantum wire oriented along the z direction.
    3D simulation: A k|| dispersion does not make sense.
    These are the energy levels Ei of the 8-band k.p Hamiltonian H(kx,ky,kz) which has the dimension of 8 times the number of grid points in the structure.

    Special case for superlattices
    k|| dispersion, i.e. kparallel (k-par-dispersion)
    1D simulation: The k dispersion is the dispersion E(kx,ky,kSL) where z is the quantization direction and the direction where periodic boundary conditions are employed.
    2D simulation: The k dispersion is the dispersion E(kSL,x,kSL,ykz) where x and y are the quantization directions and the directions where periodic boundary conditions are employed.
    3D simulation: The k dispersion is the dispersion E(kSL,xL,kSL,yLkSL,z) where x and y are the quantization directions and the directions where periodic boundary conditions are employed.
    These are the energy levels Ei of the 8-band k.p Hamiltonian H(kx,ky,kz) which has the dimension of 8 times the number of grid points in the structure.

 

Bulk k.p dispersion

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

bulk-kp-dispersion = yes                          ! Calculates energy dispersion E(k) where k is real.
                  
= real-and-imaginary-k-vector  !
Calculates energy dispersion E(k) where k is real and energy dispersion E(k) where k is imaginary.
                  
= no

Flag that signals if you want to output the pure bulk k.p dispersion E(k) = E(kx,ky,kz) along a line from k = 0 to k = (kx,ky,kz).
This is especially of interest in strained regions where the bands are shifted due to strain. The dispersion is taken at the material grid point grid-position which has to be located inside a quantum region.
The parabolic, isotropic dispersion of the effective-mass approximation (single-band) is included for comparison.

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(kx,ky,kz) over the three-dimensional k space.
This can be of interest in strained regions where the bands are shifted due to strain. The dispersion is taken at the material grid point grid-position which has to be located inside a quantum region.
The maximum value kmax is determined by MAX(|k-direction-from-k-point|,|k-direction-to-k-point|).
k space grid resolution: The number of k points from the Gamma point along each direction is number-of-k-points/4.

- 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.dat
Note: 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 2
which 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 = 20d0 ! 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).

 

The bulk k.p dispersion can be calculated along an arbitrary line from the k point 'k-direction-from-k-point' to the Gamma point and then to the k point 'k-direction-to-k-point'.
Either
k-direction-from-k-point or k-direction-to-k-point or both can be zero. If both are zero, then only the Gamma point is calculated.
If
k-direction-from-k-point is omitted, then this specifier takes the negative value of k-direction-to-k-point.
You can use this flag to specify a customized plot for the E(k) dispersion, e.g. along a line from [110] to the Gamma point and then to the [001] point.

k-direction-from-k-point = 0.0d0  1.0d0  1.0d0        ! [1/nm] along [011] direction with respect to simulation coordinate system
                         = 0.0d0  0.0d0  1.0d0        ! [1/nm]
along [001] direction with respect to simulation coordinate system
                         = - [k-direction-to-k-point] ! [1/nm] (default, i.e. if k-direction-from-k-point is omitted)

Determines k-direction and range for dispersion plot [1/nm] to the Gamma point.

k-direction-to-k-point   = 0.0d0  1.0d0  1.0d0        ! [1/nm] along [011] direction with respect to simulation coordinate system
                         = 0.0d0  0.0d0  1.0d0        ! [1/nm]
along [001] direction with respect to simulation coordinate system
                         = 0d0    0d0    1d0          ! [1/nm] (default, i.e. if k-direction-to-k-point is omitted))

Determines k-direction and range for dispersion plot [1/nm] from the Gamma point.

 

 

number-of-k-points = 50  ! If omitted, the default value of 50 is used.

Number of k points to be calculated (resolution) in addition to the Gamma point at (kx,kykz) = (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.

Inside the code, the k.p Hamiltonian is calculated with respect to the "calculation coordinate system" while it holds for all the output files that the [kx,ky,kz] directions are related to the "simulation coordinate system". So the software takes care of the rotations automatically. Care has to be taken when analyzing the eigenvectors as the quantization axis of spin is also relevant.

 

k|| dispersion

k-par-dispersion = no       ! no E(k||) dispersion
               
 = yes      !
E(k||) = E(kx,ky) dispersion for all values of kx, ky
               
 = 01-00    !
E(k||) = E(0,ky)  dispersion for ky = [ky,max,...0]
               
 = 01-00-11 !
E(k||) = E(0,ky)  dispersion for ky = [ky,max,...,0] (where kx=0) and for
                             
E(k||) = E(kx,ky) dispersion for kx = ky
                 = 01-00-10 ! E(k||) = E(0,ky)  dispersion for ky = [ky,max,...,0] (where kx=0) and for
                             
E(k||) = E(kx,0)  dispersion for kx = [kx,max,...,0] (where ky=0)

Flag that signals if you want to have the E(k||) = E(kx,ky) dispersion to be put out. This is for the quantum mechanically calculated eigenvalues.
This also works for 2D (but then only no and yes make sense), i.e. E(kz) 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(kx,ky) dispersion.
- 01-00:       A plot of E(kx,ky) along the line from E(0,ky,max) to E(0,0) .
- 01-00-11: A plot of E(kx,ky) along the line from E(0,ky,max) to E(0,0) (where kx=0) and along the line from E(0,0) to E(kx,max,ky,max=kx,max) for kx = ky
- 01-00-10: A plot of E(kx,ky) along the line from E(0,ky,max) to E(0,0) (where kx=0) and along the line from E(0,0) to E(kx,max,0) (where ky=0)

Note: If you use k-par-dispersion = 01-00, 01-00-11 or 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(kx,ky) (i.e. the 2D plot) or the density of states is not correct in this case.

 

Specify the eigenvalue numbers (i.e. subband numbers) for which the energy dispersion should be written out.

The following two specifiers do not distinguish between electrons and holes.

k-par-disp-ev-min =  1
k-par-disp-ev-max = 10

Lower (min) and upper (max) boundary for range of eigenvalues (ev) for which the k|| dispersion E(kx,ky) should be written out.
This also works for 2D, i.e. E(kz) but does not make sense for 3D.
This specifier affects the energy dispersion plots E(k||), the k||-space-resolved density plots n
(kx,ky) and the effective masses calculated from the dispersion.
This specifier can be used to output only the dispersion of the 1st or 2nd subband and not the dispersion all of calculated subbands.

 

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(kx,ky)

 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-Emin and DOS-Emax (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 kz 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 kz points in 2D for which eigenvalues and eigenfunctions are put out.
(Does not make sense in 3D.)

cb-k-SL-min = 1

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

cb-k-SL-max = 1

Only for superlattices (should be greater than or equal to 1).
Upper bound for range of superlattice vectors Kz (1D), Kx,Ky (2D), Kx,Ky,Kz (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 kz 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 kz points in 2D for which eigenvalues and eigenfunctions are put out.
(Does not make sense in 3D.)

vb-k-SL-min  = 1

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

vb-k-SL-max  = 1

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

 

interband-matrix-elements = yes  ! calculates interband matrix elements
                          
= no
==> square of spatial overlap matrix element | <psi_i* | psi_j> |^2
The 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         2.065139
which 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) / distance
with 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-operator = "z^2"                     ! (needed for standard deviation)
                                   = '0.0002 * x * ( x - 10)'  !
(useful to study for perturbation theory)
see single-band documentation $output-1-band-schroedinger
(not implemented yet)

intraband-matrix-elements = p          ! < psif* | p   | psii >
                          
= z          ! < psif* |  | psii >
                          
= o          ! < psif* |       psii
(spatial overlap)
                          
= yes        !
(prints out both matrix elements 'p' and  'z')
                          
= everything !
(prints out all matrix elements, i.e. 'p',  'z',  'o', and the one specified in intraband-matrix-elements-operator)
                          
= no         !
Calculates intersubband dipole moment and oscillator strength where the subscript i means initial and f means final state.

The output files
  intraband_p_cb1_qc1_6x6kp.txt
            z                   !
kind of matrix element ('p' / 'z' / 'o')
            o                   !
kind of matrix element ('p' / 'z' / 'o')
                vb1             !
kind of band (Gamma conduction band 'cb001' / heavy, light and split-off hole bands 'vb001')
                      8x8       !
kind of k.p ('6x6' / '8x8')

 contain data like

Note that the two-fold spin-degeneracy in the single-band approximation ('effective-mass') is counted explicitely in k.p.

(Note: | < psif* | p | psii > | 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_f* | z | psi_i > |  [Angstrom]
           
       Intersubband dipole moment  | < psi_f* | p | psi_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|psi0011.8126842E-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

    | Mfi | = | integral  (psif* (z) z psii (z) dz) |

and the oscillator strengths

    ffi = 2m* / hbar² (Ef - Ei)  | Mfi

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


Unfortunately, the commonly used
                  Intersubband dipole moment
  | < psi_f* | z | psi_i > |  [Angstrom]

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

which are the intersubband dipole moments

    | Nfi | = | integral  (psif* (z) pz psii (z) dz) | = | - i hbar integral  (psif* (z) d/dz psii (z) dz) |

and the oscillator strengths

    ffi
= 2m* / hbar² (Ef - Ei)  | Mfi  = 2 / ( m* (Ef - Ei) )  | Nfi

 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 (eigenvalues and square of the wave functions)

  • kp_8x8eigenvalues_qc001_el_kpar0005_1D_dir.dat
    kp_8x8eigenvalues_qc001_el_kpar0005_1D_dir_info.dat
    contains all electron eigenvalues for k = 0 and quantum cluster 1 for k|| vector 5.
    Similar for holes and 6-band k.p.
    The file with the suffix _info contains information about the character of the eigenvalues, i.e. their S-like, heavy-hole, light-hole and split-off hole character.
    num_ev energy[eV] P_CB   P_HH   P_LH   P_SO    sum       P_CB1  P_CB2  |3/2,3/2> |3/2,-3/2> |3/2,1/2> |3/2,-1/2> |1/2,1/2> |1/2,-1/2>
    1      0.1678     0.7543 0.0000 0.2208 0.02482 1.0000    0.7543 0.0000 0.0000    0.0000     0.2208    0.0000     0.02482   0.0000
  • kp_8x8psi_squared_qc001_el_kpar0005_1D_dir.dat
    kp_8x8psi_squared_qc001_el_kpar0005_1D_dir_shift.dat -
    Note that in this file 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.
    contains the square of all electron wave functions (psi2) (for each grid point) for k = 0 and quantum cluster 1 for k|| vector 5.

    Similar for holes and 6-band 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. Note that this file is written out only if detailed-output = yes.

 

k.p eigenfunctions

Filename

kp_8x8_el1D_wv_qc001_ev003_kpar001_Kz001_dir.dat
8x8             Kind of kp solved (8-band or 6-band)
  _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 Kz 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 8-band k.p / six for 6-band 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 (8-band or 6-band)
    _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 Kz point (superlattice vector)
              _dir Boundary condition:  Neumann/Dirichlet (_neu/_dir)

Structure:        

position[nm] PSI_real001 PSI_imag001 ...
position Real part of 1st component Imaginary part of 1st component ...
  • schroedinger-kp-discretization = box-integration

The eight spinor components of the 8-band k.p wave function are (for zinc blende): | cb+ >,  | cb- >,  | hh1+ >,  | lh1 >,  | lh2 >,  | hh2- >,  | so1 >,  | so2 >
 cb =
conduction band
 hh =
heavy hole
 lh =
light hole
 so =
split-off hole
 +  = spin up
 -  = spin up
 + and - correspond to the spin projection along the z axis of the crystal system ( + = up, - = down).
Note that the states | lh1 >,  | lh2 >,  | so1 >,  | so2 > are a mixture of spin up and spin down.

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  >
 

  • schroedinger-kp-discretization = box-integration-XYZ
                                  
    = finite-differences

The eight spinor components of the 8-band k.p wave function are (for zinc blende): | cb+ >,  | cb- >,  | x+ >,  | y+ >,  | z+ >,  | x- >,  | y- >,  | z- >

| x >, | y >, | z > correspond to x, y, z of the calculation system.
 

Note: 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. kx, ky are in calculation system.

For more details, see eq. (3.85) in the PhD thesis of S. Birner.


 

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-to-k-point [1/nm] 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 (6-band 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. [kx,ky] = [0,0] = [00]) to [10] (i.e. [kx,ky] = [kx,0] where kx > 0) of the 2D plot of the E(k||)=E(kx,ky) 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 (6-band or 8x8)
    _ev001 Number of eigenvalue of subband

Structure:

abscissa eigenvalue kx ky num_kpar
kx Eigenvalue of k|| point kx position [1/Angstrom] ky position [1/Angstrom] Number of k|| point

 

The file kpar1D_disp_00_10hl_8x8kp_ev_min001kp_ev_max010.dat contains 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(kx,ky) 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 (6-band 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 kx position [1/Angstrom] ky 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 file kpar1D_disp_01_00_11hl_8x8kp_ev_min001kp_ev_max010.dat contains 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 (6-band or 8-band)
    _ev001 Number of eigenvalue of subband

Structure:

num_kpar kx ky eigenvalue
Number of k|| point kx position [1/Angstrom] ky position [1/Angstrom] Eigenvalue of k|| point

This file cannot be plotted. It just contains for each k|| point the corresponding kx and ky 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(kx,ky).
 

  • '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-Emin and DOS-Emax:
DOS-points    = 1000
==>  Energy_grid = (DOS_Emax - DOS_Emin) / DOS_Points = 2.5 eV / 1000 = 0.0025 eV

 

The files Density_at_energy_el.dat and Density_at_energy_hl.dat contain the energy resolved density n(E).
n(E) = DOS(E) * f(E) where f(E) is the Fermi function and DOS(E) is the density of states at energy E.
  energy[eV]    n(E)[eV^-1cm^-2]

 


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: ni(kx,ky) for each eigenvalue i

The units are dimensionless, i.e. if one integrates over kx and ky (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(kx, ky) in units of [eV].
In addition, a 2D plot for the optical matrix element: P(kx, ky) in units of SQRT([2eV/m0]) is generated if optical-matrix-element-output = complex.
Note that the intensity and the matrix element are three-dimensional vectors with components Ix, Iy, Iz or Px, Py, Pz, respectively, and kx, ky is in units of [1/Angstrom].
The optical matrix element will be calculated between band-min and band-max eigenvalues.
The k|| points are the same that are chosen for the energy dispersion output.

Output units:
a) Intensity: I = 2/m0 * |< e.P >|2:  Ix, Iy, Iz, Ixy, Ixz, Iyz in units of [eV] where Ii = 2/m0 |Pi|2,  Iij = 2/m0 |Pij|2  (i=x,y,z; j=1,2,3).
b) complex value of the momentum matrix element Px, Py, Pz, Pxy, Pxz, Pyz in units of SQRT([2eV/m0]).
e is the light polarization vector
P is the momentum operator (vector)
Px(k)  = <psi1(k)| e.P |psi2(k)>, where |e|=1, e || x      (i.e. e || [1,0,0])
Pxy(k) = <psi1(k)| e.P |psi2(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       Ix              Iy             Iz              Ixy               Ixz              Iyz

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

   optical-matrix-element-output = complex  ! Units: SQRT([2eV/m0])
     kx    ky    kz    Re(Px) Im(Px)    Re(Py) Im(Py)    Re(Pz) Im(Pz)    Re(Pxy) Im(Pxy)    Re(Pxz) Im(Pxz)    Re(Pyz) Im(Pyz)

Optical oscillator strength

 

Example: Optical oscillator strength of the fundamental transition in a 2D quantum wire calculated by 8-band 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                          6-band k.p Dresselhaus parameters L,M,N. Note: N = N+ + N-

 

This file 6x6kp_Luttinger_parameters1D_used.dat contains the following columns:

  material_grid                            position in [nm] on material grid
  gamma1 gamma2 gamma3 (kappa)            
6-band k.p Luttinger parameters (if kappa is 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 [hbar2/2m0], 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 6-band k.p, but for 8-band 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 holds gamma2 = gamma3 (spherical approximation) then the valence band energy dispersion is isotropic. (This is equivalent to: L - M = N.)