 
nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Density in ndoped GaAs  Comparison of classical, quantum, k.p and
fullband 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 (singleband Schrödinger of a 1D structure)
==> DensityTest_GaAs_n_doped2Dqm_box.in

quantum density (singleband Schrödinger of a 2D structure)
==> DensityTest_GaAs_n_doped1Dqm_kp_simple.in

quantum density (8band k.p Schrödinger with k_{}
integration method simpleintegration)
==> DensityTest_GaAs_n_doped1Dqm_kp_special.in

quantum density (8band k.p Schrödinger with k_{}
integration method specialaxis)
==> DensityTest_GaAs_n_doped1Dqm_kp_gendos.in

quantum density (8band k.p Schrödinger with k_{}
integration method gendos)
==> DensityTest_GaAs_n_doped1Dqm_kp_simple_fullband.in

quantum density (8band k.p Schrödinger with k_{}
integration method simpleintegration and fullbanddensity for electrons)
==> DensityTest_GaAs_n_doped1Dqm_kp_simple_fullband_hl.in 
quantum density (8band k.p Schrödinger with k_{}
integration method simpleintegration and fullbanddensity for holes)
==> DensityTest_GaAs_n_doped1Dqm_kp_special_fullband.in 
quantum density (8band k.p Schrödinger with k_{}
integration method specialaxis
and fullbanddensity for electrons)
==> DensityTest_GaAs_n_doped1Dqm_kp_special_fullband_hl.in 
quantum density (8band k.p Schrödinger with k_{}
integration method specialaxis
and fullbanddensity for holes)
Density in ndoped GaAs  Comparison of classical, quantum, k.p and
fullband 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 ndoped 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 10^{20} cm^{3}.
 The donor level is Si with 5.8 meV below the conduction band edge.
 In order to compare the 8band 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 fullband density approach.
This information is contained in the second column of this file:
densities/density1DGamma_L_X.dat
input file 
electron density (Gamma only) 10^{18}
cm^{3} 
electron density (Gamma, L, X) 10^{18}
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 (singleband effectivemass, 1D
structure) 
1.9731 
1.9766 

quantum density (singleband effectivemass, 2D
structure, box) 
(not part of output yet) 
1.9792 





quantum density (8band k.p, simpleintegration) 
1.9726 
1.9761 

quantum density (8band k.p, specialaxis) 
1.9624 
1.9659 

quantum density (8band k.p, gendos) 
1.9788 
1.9823 





quantum density (8band k.p, simpleintegration,
fullbanddensity electrons) 
1.9726 
(classical electron density at L
and X set to zero) 

quantum density (8band k.p, specialaxis,
fullbanddensity electrons) 
1.9624 
(classical electron density at L
and X set to zero) 

quantum density (8band k.p, gendos,
fullbanddensity electrons) 
(not implemented yet) 
(classical electron density at L
and X set to zero) 


hole density (Gamma only) 10^{18}
cm^{3} 


quantum density (8band k.p, simpleintegration,
fullbanddensity holes) 
1.9726 
(classical electrons density at
Gamma, L and X conduction bands set to zero) 

quantum density (8band k.p, specialaxis,
fullbanddensity holes) 
1.9624 
(classical electrons density at
Gamma, L and X conduction bands set to zero) 

quantum density (8band k.p, gendos,
fullbanddensity 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 fullbanddensity electron and
fullbanddensity hole method.
The k_{} integration method simpleintegration seems to
have the best agreement to the singleband results.
Note: The k_{} integration method specialaxis 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.
Fullbanddensity (8band k.p)
In order to understand the fullband density k.p approach, it is
necessary to read at least one of these papers.
 Fullband envelope function approach for typeII brokengap
superlattices
T. Andlauer, P. Vogl
Physical Review B 80, 035304 (2009)
 Selfconsistent electronic structure method for brokengap superlattices
T. Andlauer, T. Zibold, P. Vogl
Proc. SPIE 7222, 722211 (2009)
The following switch is required to turn on "fullband density".
$numericcontrol
...
brokengap = fullbanddensity
As the structure consists of 41 grid points, we get 8 x 41 = 328 eigenstates
for 8band 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.
 fullband density for electrons:
$quantummodelelectrons
Here, we calculate all hole states, and the relevant electron states (numberofeigenvaluesperband
= 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.
cbnumevmin = 241 ! lower
boundary for range of conduction band eigenvalues
cbnumevmax = 278 ! upper boundary
For fullband 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.
 fullband density for holes
$quantummodelholes
Here, we calculate all electron states, and the relevant hole states (numberofeigenvaluesperband
= 40 40 40 ),
i.e. we need the eigenstate numbers 1  122, where 122 = 82 + 40 .
For the output, we plot only 77 
100 , i.e. the 6 lowest electron states are included in the
output of the wave functions.
vbnumevmin = 77 ! lower
boundary for range of valence band eigenvalues
vbnumevmax = 100 ! upper boundary
For fullband density holes, the eigenvalues are numbered from the
top of the spectrum, with eigenvalue number 1 having the highest energy, and
being an electron eigenstate.
All eigenstates are treated as holes, and occupied as holes,
and contribute to the (positive) hole charge density.
We then subtract a negative background charge density to obtain the
final net charge density.
The file
densities/density1Dhl.dat
contains the hole charge carrier density which is negative in this
example because a net electron density is present.
Fullbanddensity holes ($quantummodelholes )
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
$quantummodelelectrons , this number contains the positive
background charge density.
If using
$quantummodelholes , this number contains the
negative background charge density.
 Please help us to improve our tutorial! Send comments to
support
[at] nextnano.com .
