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nextnano3 - Tutorial

next generation 3D nano device simulator

1D Tutorial

Density in n-doped GaAs - Comparison of classical, quantum, k.p and full-band density k.p approach

Author: Stefan Birner
Please send comments to support (-at-) nextnano.com.

If you want to obtain the input file that is used within this tutorial, please contact stefan.birner@nextnano.de.
==> DensityTest_GaAs_n_doped1D.in                          - classical density of a 1D structure
==> DensityTest_GaAs_n_doped2D.in                          -
classical density of a 2D structure
==> DensityTest_GaAs_n_doped3D.in                          -
classical density of a 3D structure
==> DensityTest_GaAs_n_doped1Dqm.in                        -
quantum density (single-band Schrödinger of a 1D structure)
==> DensityTest_GaAs_n_doped2Dqm_box.in                    -
quantum density (single-band Schrödinger of a 2D structure)
==> DensityTest_GaAs_n_doped1Dqm_kp_simple.in              -
quantum density (8-band k.p Schrödinger with k|| integration method simple-integration)
==> DensityTest_GaAs_n_doped1Dqm_kp_special.in             -
quantum density (8-band k.p Schrödinger with k|| integration method special-axis)
==> DensityTest_GaAs_n_doped1Dqm_kp_gendos.in              -
quantum density (8-band k.p Schrödinger with k|| integration method gen-dos)
==> DensityTest_GaAs_n_doped1Dqm_kp_simple_fullband.in     -
quantum density (8-band k.p Schrödinger with k|| integration method simple-integration and full-band-density for electrons)
==> DensityTest_GaAs_n_doped1Dqm_kp_simple_fullband_hl.in  -
quantum density (8-band k.p Schrödinger with k|| integration method simple-integration and full-band-density for holes)
==> DensityTest_GaAs_n_doped1Dqm_kp_special_fullband.in    -
quantum density (8-band k.p Schrödinger with k|| integration method special-axis         and full-band-density for electrons)
==> DensityTest_GaAs_n_doped1Dqm_kp_special_fullband_hl.in -
quantum density (8-band k.p Schrödinger with k|| integration method special-axis         and full-band-density for holes)
 


Density in n-doped GaAs - Comparison of classical, quantum, k.p and full-band density k.p approach

The aim of this tutorial is to compare the density calculation of different methods that are implemented in the nextnano³ software.

As an example, we use an n-doped bulk GaAs sample of width 20 nm with periodic boundary condition for the Schrödinger equation.

  • The temperature is set to 300 K.
  • The donor concentration is 1 x 1020 cm-3.
  • The donor level is Si with 5.8 meV below the conduction band edge.
  • In order to compare the 8-band k.p results to the simpler models for the density, we assume for all k.p calculations a parabolic and isotropic energy dispersion E(k) of electrons and holes where electrons and holes are decoupled.
  • The number of grid points is 41, leading to a grid spacing of 0.5 nm.

First we solve the Poisson equation without quantum mechanics. For the obtained potential, we calculate the density using different approaches.

 

In the following, we compare the results of the calculations:

The electron density is contained in this file: densities/density1Del.dat
However, as this file contains the contribution of all bands, i.e. Gamma, L and X bands,
we have to look at the electron density at the conduction band edge at Gamma only, in order to compare the results to the full-band density approach.
This information is contained in the second column of this file: densities/density1DGamma_L_X.dat

 

input file electron density (Gamma only) 1018 cm-3 electron density (Gamma, L, X) 1018 cm-3  
clasical density (1D structure)  1.9511  1.9546  
clasical density (2D structure) (not part of output yet)  1.9546  
clasical density (3D structure) (not part of output yet)  1.9546  
       
quantum density (single-band effective-mass, 1D structure)  1.9731  1.9766  
quantum density (single-band effective-mass, 2D structure, box) (not part of output yet)  1.9792  
       
quantum density (8-band k.p, simple-integration)  1.9726  1.9761  
quantum density (8-band k.p, special-axis)  1.9624  1.9659  
quantum density (8-band k.p, gen-dos)  1.9788  1.9823  
       
quantum density (8-band k.p, simple-integration, full-band-density electrons)  1.9726  (classical electron density at L and X set to zero)  
quantum density (8-band k.p, special-axis,         full-band-density electrons)  1.9624  (classical electron density at L and X set to zero)  
quantum density (8-band k.p, gen-dos,               full-band-density electrons)  (not implemented yet)  (classical electron density at L and X set to zero)  
  hole density (Gamma only) 1018 cm-3    
quantum density (8-band k.p, simple-integration, full-band-density holes) -1.9726  (classical electrons density at Gamma, L and X conduction bands set to zero)  
quantum density (8-band k.p, special-axis,         full-band-density holes) -1.9624  (classical electrons density at Gamma, L and X conduction bands set to zero)  
quantum density (8-band k.p, gen-dos,               full-band-density holes)  (not implemented yet)  (classical electrons density at Gamma, L and X conduction bands set to zero)  
       

As one can see, all results are in reasonable agreement.
In particular, one can see the equivalence of the full-band-density electron and full-band-density hole method.
The k|| integration method simple-integration seems to have the best agreement to the single-band results.

Note: The k|| integration method special-axis is only applicable for materials with isotropic energy dispersion (e.g. for 1D simulations of wurtzite along the hexagonal c axis) or for 2D simulations.

 

Full-band-density (8-band k.p)

In order to understand the full-band density k.p approach, it is necessary to read at least one of these papers.

  • Full-band envelope function approach for type-II broken-gap superlattices
    T. Andlauer, P. Vogl
    Physical Review B 80, 035304 (2009)
  • Self-consistent electronic structure method for broken-gap superlattices
    T. Andlauer, T. Zibold, P. Vogl
    Proc. SPIE 7222, 722211 (2009)

The following switch is required to turn on "full-band density".

$numeric-control
 ...
 broken-gap = full-band-density

As the structure consists of 41 grid points, we get 8 x 41 = 328 eigenstates for 8-band k.p in total.
The lowest 6 x 41 = 246 eigenstates belong to the hole states with their energies below the valence band edge.
There are 2 x 41 = 82 electron states above the conduction band edge.

  • full-band density for electrons: $quantum-model-electrons
    Here, we calculate all hole states, and the relevant electron states (number-of-eigenvalues-per-band = 40),
    i.e. we need the eigenstate numbers 1 - 286, where 286 = 246 + 40. For the output, we plot only 241 - 278, i.e. the highest 6 holes states are included in the output of the wave functions.

    cb-num-ev-min = 241 ! lower boundary for range of conduction band eigenvalues
    cb-num-ev-max = 278 !
    upper boundary

    For full-band density electrons, the eigenvalues are numbered from the bottom of the spectrum, with eigenvalue number 1 having the lowest energy, and being a hole eigenstate.
    All eigenstates are treated as electrons, and occupied as electrons, and contribute to the (negative) electron charge density.
    We then subtract a positive background charge density to obtain the final net charge density.
    The file densities/density1Del.dat contains the electron charge carrier density which is positive in this example because a net electron density is present.
     
  • full-band density for holes $quantum-model-holes
    Here, we calculate all electron states, and the relevant hole states (number-of-eigenvalues-per-band = 40  40  40),
    i.e. we need the eigenstate numbers 1 - 122, where 122 = 82 + 40. For the output, we plot only 77 - 100, i.e. the 6 lowest electron states are included in the output of the wave functions.

    vb-num-ev-min =  77 ! lower boundary for range of valence band eigenvalues
    vb-num-ev-max = 100 !
    upper boundary

    For full-band density holes, the eigenvalues are numbered from the top of the spectrum, with eigenvalue number 1 having the highest energy, and being an electron eigenstate.
    All eigenstates are treated as holes, and occupied as holes, and contribute to the (positive) hole charge density.
    We then subtract a negative background charge density to obtain the final net charge density.
    The file densities/density1Dhl.dat contains the hole charge carrier density which is negative in this example because a net electron density is present.

Full-band-density holes ($quantum-model-holes) is recommended, as one has less eigenvalues to calculate. This will make the numerical effort smaller.

The background charge density is contained in this file:
density1DFullBandBackground.dat

If using $quantum-model-electrons, this number contains the positive background charge density.
If using $quantum-model-holes    , this number contains the negative background charge density.

 

  • Please help us to improve our tutorial! Send comments to support [at] nextnano.com.