nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Singleband ('effectivemass') is a special case of 8band k.p ('8x8kp') if the
electrons are decoupled from the holes
Author:
Stefan Birner
If you want to obtain the input file that is used within this tutorial, please contact stefan.birner@nextnano.de.
> effective_mass_vs_8x8kp_sg_1D.in
> effective_mass_vs_8x8kp_kp_1D.in
> effective_mass_vs_8x8kp_density_sg_1D.in
> effective_mass_vs_8x8kp_density_kp_1D_gendos.in
> effective_mass_vs_8x8kp_density_kp_1D_simple.in
> effective_mass_vs_8x8kp_density_kp_1D_special.in
> effective_mass_vs_8x8kp_same_mass_sg_1D.in
> effective_mass_vs_8x8kp_same_mass_kp_1D_gendos.in
> effective_mass_vs_8x8kp_same_mass_kp_1D_simple.in
> effective_mass_vs_8x8kp_same_mass_kp_1D_special.in
This tutorial is a bit complicated and contains probably too much details.
Nevertheless, an experienced user might want to understand in detail how certain
properties are calculated in order to convince himself about the correctness of
the nextnano³ results.
1) Singleband ('effectivemass') is a special case of 8band k.p ('8x8kp') if the
electrons are decoupled from the holes
> 1Deffective_mass_vs_8x8kp_sg.in
> 1Deffective_mass_vs_8x8kp_kp.in
The 8band k.p parameters are specified as follows:
8x8kpparameters = L' M
N' ! L',M,N'
[hbar^2/2m] (> divide by hbar^2/2m)
old version: B P S
! B
[hbar^2/2m], P [eV * Angstrom],
S []
B E_P S
! B
[hbar^2/2m], E_P [eV] ,
S []
To decouple the electrons from the holes one has to perform the following
modifications on the 8band k.p material parameters
S and P :
S = 1 /
m_{e} :
m_{e} = electron mass at the the
Gamma point
conductionbandmasses = m_{e}
m_{e} m_{e}
... ... ...
... ... ...
E_P = 0
: (equivalent to P = 0; E_{p }= Kane momentum matrix
element; E_{p} = 2m_{0}/h_{bar}² * P²)
 The hole parameters
L' ,
M, N'
can be arbitrary. However, they should be chosen so that spurious solutions do
not arise.
With these modifications, both eigenvalues and wave functions must be
identical for the cases
$quantummodelelectrons
...
modelname
= effectivemass ! singleband
Schrödinger equation
modelname
= 8x8kp
! multiband (8x8) k.p Schrödinger equation
Note that for effectivemass the
eigenvalues are twofold degenerate due to spin. So we take twice as much in the
case of k.p:
numberofeigenvaluesperband = 4
! singleband Schrödinger equation
numberofeigenvaluesperband = 8
! multiband (8x8) k.p Schrödinger equation
As an example we will take a simple unstrained AlAs/GaAs/AlAs quantum well
structure where AlAs acts as the barrier material.
The conduction band offset is calculated as follows: 4.049  2.979 = 1.07 eV
conductionbandenergies = 4.049d0
... ... !
(AlAs)
conductionbandenergies = 2.979d0 ...
... ! (GaAs)
The conduction band effective masses at the Gamma point are:
conductionbandmasses =
0.15d0 0.15d0 0.15d0 ! (AlAs)
... ... ...
... ... ...
conductionbandmasses =
0.067d0 0.067d0 0.067d0 ! (GaAs)
... ... ...
... ... ...
The relevant S = 1 /
m_{e } parameters are:
8x8kpparameters =
... ...
...
... ...
6.66666667d0 ! (AlAs)
8x8kpparameters =
... ...
...
... ...
14.92537313d0 ! (GaAs)
Results
 The following figure shows the conduction band profile and the three
lowest (confined) eigenstates for k=0 including their charge densities (wave function
Psi²) for a 5 nm AlAs/GaAs quantum well.
The calculated eigenvalues can be found in the file
Schroedinger_1band/ev1D_cb001_qc001_sg001_deg001_dir_Kx001_Ky001_Kz001.dat :
e1 = 3.09100 eV
e2 = 3.43473 eV
e3 = 3.97526 eV
Both, effectivemass and
8x8kp lead to exactly the same eigenvalues. Also the squared wave functions
(Psi²) are identical. The squared wave functions are normalized in units of
[1/nm] so that they integrate to 1.0 if the integration interval is between 10
nm and 35 nm. (The quantum cluster extends from 10 nm to 35 nm.) Note that the
probabilities Psi² are shifted by the corresponding eigenvalues so that they fit
nicely into the graph. To integrate Psi² one has to substract the appropriate
eigenvalues in the data files.
New discretization routines:
! 3) => effectivemass, boxintegration,
LAPACKZHBGVX
schroedinger1bandevsolv =
LAPACKZHBGVX
schroedingermassesanisotropic = box
! 4) => 8band k.p, boxintegration, LAPACKZHBGVX
schroedingerkpevsolv
= LAPACKZHBGVX
schroedingerkpdiscretization = boxintegration
kpcvtermsymmetrization = yes
kpvvtermsymmetrization = yes
e1 = 3.09100 eV
e2 = 3.43473 eV
e3 = 3.97526 eV
! Note: Instead of 'LAPACKZHBGVX', one can also use 'arpack' or 'chearn' to
get identical results.
! For '8x8kp' also 'LAPACK' works.
! 'chearn' only works for 'effectivemass'.
! 'arpack': "Error calculate_kp_eigenvalues:
Eigenfunction zero." (eigenvalues are okay)
2) Calculating the quantum mechanical density within the singleband
approximation and 8band k.p selfconsistently
> 1Deffective_mass_vs_8x8kp_density_sg.in
1Deffective_mass_vs_8x8kp_density_kp_gendos.in
1Deffective_mass_vs_8x8kp_density_kp_simple.in
1Deffective_mass_vs_8x8kp_density_kp_special.in
Our input file contains a similar 5 nm quantum well but instead of AlAs we
take Al_{0.2}Ga_{0.8}As as the barrier material. However, this
time we include some ntype doping with a concentration of 1*10^{17} cm^{3}.
Only the AlGaAs region between 35 and 40 nm is doped. The donor atom has a donor
level of 0.007 eV below the conduction band.
We use flowscheme = 2 , i.e.
 we solve the Poisson equation including doping to obtain the electrostatic
potential
 we use this electrostatic potential to solve Schrödinger's equation
effectivemass (singleband) approximation
 we calculate the electron density (including quantum mechanical density) that
enters the Poisson equation
 we solve the Poisson and Schrödinger equation selfconsistently
The following figure shows the conduction band profile and the lowest
eigestate as well as its probability amplitude. The chemical potential (i.e. the
Fermi level) is not shown in the figure. It is equal to 0 eV. There are no
further eigenstates that are confined inside the well. (The figure also shows
the k.p results which are identical.)
Note: In our example we take into account only the first quantized
eigenstate. Usually one has to sum over all (occupied) eigenstates to calculate
the quantum mechanical density.
The threedimensional electron density of a 1D quantum structure
(homogeneous along the y and z directions) can be calculated
in the singleband effectivemass approximation (i.e. parabolic and isotropic bands) as
follows:
n(x) = g m_{,av} k_{B}T / (2 pi h_{bar}²) Psi_{1}(x)²
ln(1 + exp( ( E_{1} + E_{F}(x))/ (k_{B}T) ) )
Note the squared wave function Psi_{1}² must be normalized over the
length of the quantum cluster, i.e. the units are 1/m.
(Confer also the corresponding equation in the overview section: "Purely
quantum mechanical calculation of the density".)
 g is the degeneracy factor for both spin and valley degeneracy. In our
case, the Gamma conduction band valley is not degenerate, so we have g = 2 to
account for spin degeneracy.
 Usually m_{} depends on the distance x, so we should have
written m_{}(x) instead of m_{,av}. However, this leads to
discontinuities in the quantum charge density as m_{}(x) is
discontinuous at material interfaces. A possible solution is to take an
average of all m_{} values in the quantum cluster weighted by the
quantum charge density (Psi²). This leads to m_{,av} = 0.0698240.
 k_{B}T = 0.025852 eV (at 300 K)
 Psi_{1}(x)²: Probability amplitude of the wave function Psi_{1}
of the first eigenstate.
 E_{F}(x): Chemical potential (i.e. Fermi level). In our example it
is equal to zero: E_{F}(x) = 0
 E_{1} = 0.07117 eV: energy of the first quantized state
Thus the logarithm at position x can be simplified to: ln(1 + exp( 0.07117 / 0.025852)) =
ln 1.0637 = 0.06178
Evaluating the expression g m_{0} k_{B}T / (2 pi h_{bar}²)
leads to:
g m_{0} k_{B}T / (2 pi h_{bar}²) = 6.7403355*10^{35}
[kg eV / (J²s²)] = 1.07992*10^{17} 1/m²
Note that: [kg VAs / (J²s²)] = [Ns²/m / (Js²)] = [N/m / J] = [1 / m²]
Psi_{1}²(x = 22.5 nm) = 0.212373 1/nm
(Note that one has to subtract the eigenvalue of 0.07117 from the Psi² values
that are printed out in the file:
Schroedinger_1band/cb001_qc001_sg001deg001_dir_Kz001.dat : 0.28354  0.07117 =
0.21237)
We calculate the electron density for the middle of the quantum well where
the maximum of the probability amplitude occurs:
=> n(x = 22.5 nm) = 0.069824 * 1.07992*10^{17} 1/m² * 0.212373 1/nm *
0.06178 = 9.893 * 10^{22 }1/m³
This number divided by the threedimensional number density (1*10^{24})
leads to 0.09893, i.e. the output is in units of 1 * 10^{18} cm^{3},
thus the density equals 0.09893 * 10^{18} cm^{3}.
This value is in agreement with the plot of the electron density. The density
is dominated by the quantum mechanical contribution to the density. The
classical contribution is negligible (and is not plotted here).
A calculation with 8band k.p
rather than effectivemass
where we integrate over different k_{} will lead to the same
results._{
} methodofbrillouinzoneintegration =
specialaxis
numkpparallel = 361
! 361 = (2 * N_{kx} + 1)
* (2 * + N_{ky} + 1) in the whole 2D Brillouin
zone corresponds to 10 (= N_{kx} + 1) k_{} vectors that have to be calculated.
Again we assume a "decoupled 8band k.p Hamiltonian" with P = 0 and S = 1/m_{e} in order
to compare to the singleband case.
Here, we use different effective masses for the two materials so the results
of k.p and singleband cannot be exactly identical as the methods differ
slightly. Nevertheless, both selfconsistent calculations leed to almost
identical eigenvalues, wave functions and densities.
The folder densities/ contains also the density of the subbands,
for singleband and for 8band k.p, as well as the integrated
subband density.
 subband1D_el_qc001_sg001_deg001.dat / *_integrated.dat
(singleband)
 subband1D_el8x8kp_qc001.dat /
*_integrated.dat (8band k.p)
Again, the values agree. Note that the singleband eigenstates are twofold
spindegenerate. Thus the subband density is twice as high as in the case of
k.p.
3) Calculation of the quantum mechanical density from the k.p dispersion (no
selfconsistency)
> 1Deffective_mass_vs_8x8kp_same_mass_sg.in
1Deffective_mass_vs_8x8kp_same_mass_kp_gendos.in
1Deffective_mass_vs_8x8kp_same_mass_kp_simple.in
1Deffective_mass_vs_8x8kp_same_mass_kp_special.in
To compare the k.p results with the effectivemass results it is easier
to use for both materials the same conduction band effective mass. So even for
Al_{0.2}Ga_{0.8}As we take the GaAs mass of 0.067 this time. The
relevant 8band k.p parameter S is then given by S = 1/m_{e} =
1/0.067 = 14.925.
We use the same doping as in 2) but this time we use flowscheme =
3 , i.e.
 we solve the Poisson equation including doping to obtain the electrostatic
potential
 we use this electrostatic potential to solve Schrödinger's equation with both
effectivemass (singleband) approximation and 8band k.p. As expected the
eigenvalues and wave functions coincide for these two cases. The energy of the
lowest eigenstate is E_{1} = 0.1071 eV.
In order to calculate the k.p dispersion E(k_{y},k_{z}),
we have to solve the 8band k.p Hamiltonian for different combinations of k_{y}
and k_{z} (inplane wavevector). As our electrons in the Hamiltonian are
decoupled from the holes, we have the special case where the dispersion is
isotropic and parabolic, thus E(k_{y},k_{z}) = E(k_{})
where k_{}² = k_{y}² + k_{z}². Thus it is possible to
plot (and calculate) the E(k) dispersion along a line only.
numkpparallel = 361
! 361 = (2 * N_{kx} + 1)
* (2 * + N_{ky} + 1) in the whole 2D Brillouin
zone corresponds to 10 (= N_{kx} + 1) k_{} vectors that have to be calculated.
Our k space, i.e. k=(k_{y},k_{z}), contains
361 k points in the 2D Brillouin zone. However, due to the
crystal symmetry and its orientation with respect to the simulation system we
only need to calculate the k points lying in a triangle (~1/8^{th}
of the total k points, the exact number needed is 55). Due to the isotropy of the dispersion, it would
be sufficient to take even less points, i.e. to integrate over the k
points that are lying along any line of the 2D Brillouin zone.
The relevant data for the E(k_{y},k_{z}) dispersion along the
[010] and [011] direction is stored
in the file: Schroedinger_kp/kpar1D_disp_100_0_110_el_ev001.dat
It contains the dispersion into the [010] direction in k space, i.e. for k_{z}=0 and k_{} = k_{y }= [0,...]),
and into the [011] direction, i.e. for k_{} = SQRT(k_{y}² + k_{z}²).
The parabolic dispersion is given by E(k_{}) = E_{1} + h_{bar}²/2m_{0}
k_{}²/m_{c
}where h_{bar}²/2m_{0} is given by 3.81 [eV Angstrom²] and
the conduction band effective mass m_{c} is a dimensionless quantity
here in this equation. k_{} is given in [1/Angstrom]. E_{1} is
the energy of the lowest eigenstate E_{1} = 0.10707 eV. As expected, the
dispersion for the effectivemass approximation agrees perfectly to the 8band
k.p case.
The files for the bulk singleband (isotropic and parabolic dispersion) and the
bulk k.p energy dispersions E(kx,ky) are contained in these files:
 Schroedinger_kp/bulk_sg_dispersion000_kxkykz.dat
 Schroedinger_kp/bulk_6x6kp_dispersion000_kxkykz.dat
In this example, the energy dispersions for the conduction bands are
identical.
(In this example, the k.p bulk dispersion for the valence bands is not
correct because we set E_{p }= 0.)
To calculate the k.p densities one has three options to choose from for
the integration over the 2D Brillouin zone:
methodofbrillouinzoneintegration = specialaxis
= gendos
= simpleintegration
specialaxis is only
applicable to this artificial tutorial example and not to the general
zinc blende case where E_{P} /= 0 because it works only for the special
case of isotropic dispersion and thus the 2D integration can be reduced to a
1D integration. The number of k_{} points is calculated by N_{kx}
+ 1 = 10. They are lying
along a line where k_{x }>= 0 and k_{y}= 0.
gendos calculates the density
of states (DOS) and integrates over the DOS for obtaining the k.p
density. This is the method of choice.
simpleintegration integrates
over the k points in the 2D Brillouin zone.
More details can be found in the
glossary.
The following figure compares the quantum mechanical electron charge
densities calculated with these three different methods with the effectivemass
approximation. Note that the number of calculated k points corresponds to
a total of 361 k points of the whole 2D Brillouin zone. Note: In the
figure, the results for simpleintegration
look very bad. This is due to a bug that has been fixed meanwhile.
(For this tutorial example, we used krangedeterminationmethod =
bulkdispersionanalysis as here the automatic determination of krange works well because
the inplane dispersion is the same as the bulk k.p dispersion.)
 Please help us to improve our tutorial. Send comments to
support
[at] nextnano.com .
