If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 2DTshapedQuantumWireCEO.in
Similar to the 1D confinement in a quantum well, it is possible to
confine electrons or holes in two dimensions, i.e. in a quantum wire.
The quantum wire is formed at the Tshaped intersection of two 10 nm GaAs
typeI quantum wells and is confined by Al_{0.35}Ga_{0.65}As barriers.
The electrons and holes are free to move along the z direction only, thus
the wire is oriented along the [011] direction. Such a heterostructure can
be manufactured by growing the layers along two different growth directions
with the CEO (cleaved egde overgrowth) technique. Due to the nearly
identical lattice constants of GaAs and AlAs it is possible to assume this
heterostructure as being unstrained.
It is sufficient to describe this heterostructure within a 2D simulation as
it is translationally invariant along the z direction.
The simulation coordinate system is oriented in the following way:
$domaincoordinates
domaintype =
1 1 0 !
(x,y) plane
hklxdirectionzb = 1 0 0
!
along the [100]
direction in the crystal coordinate
system
hklydirectionzb = 0 1 1
!
along the [011]
direction in the crystal coordinate
system
! hklzdirectionzb = 0 1 1 !
along the [011]
direction in the crystal
coordinate system
The hklzdirection
does not have to be specified. It
is calculated internally in the code.
The electron and hole wave functions can be calculated within the effective
mass theory (envelope function approximation) by using position dependent
effective masses. In our example, the effective masses are constant within each
material but have discontinuities at the material interfaces. For the electrons,
the effective mass was assumed to be isotropic (conduction band minimum at the
Gamma point) whereas for the heavy and light
holes we used anisotropic effective mass tensors that were derived from the 6band
k.p parameters (or Luttinger parameters).
Usually the database entries for the effective masses assume spherical
symmetry for the holes and are specified with respect to the crystal coordinate
system.
Their default values (isotropic) and the values which were derived from the
Luttinger parameters are given in this table:
In this tutorial, however, we calculated the effective masses for different
directions and therefore we do not have spherical symmetry anymore.
Thus we have to rotate the new eigenvalues of the effective mass tensors that
are given in the x=[100], y=[011], z=[011] simulation coordinate system into
the crystal coordinate system where x_{cr}=[100], y_{cr}=[010],
z_{cr}=[001].
First we have to overwrite the default entries in the database so that they
contain the eigenvalues of the effective mass tensors in the simulation system.
conductionbandmasses = 0.067d0
0.067d0 0.067d0 !
electron effective
mass (Gamma point) (default)
...
valencebandmasses =
0.350d0 0.643d0 0.643d0 !
eigenvalues of the heavy
hole effective mass tensor [100] [011] [011]
0.090d0 0.081d0 0.081d0 !
eigenvalues of the light hole effective mass tensor [100] [011] [011]
...
!
To project these eigenvalues onto the crystal coordinate system we
need to know the principal axes system which these eigenvalues refer to.
principalaxesvbmasses = 1d0 0d0
0d0 !
heavy hole [100]
0d0 1d0 1d0 !
[011]
0d0 1d0 1d0 !
[011]
1d0 0d0 0d0 !
light
hole [100]
0d0 1d0 1d0 !
[011]
0d0 1d0 1d0 !
[011]
...
(The normalization of these vectors will be done internally by the
program.)
Similar, the valence band masses of AlAs are chosen to be:
conductionbandmasses = 0.15d0
0.15d0 0.15d0 !
electron effective
mass (Gamma point) (default)
valencebandmasses =
0.472d0 0.820d0 0.820d0 !
eigenvalues of the heavy
hole effective mass tensor [100] [011] [011]
0.185d0 0.159d0 0.159d0 !
eigenvalues of the light hole effective mass tensor [100] [011] [011]
Further material parameters of relevance:
Conduction band offset Al_{0.35}Ga_{0.65}As / GaAs:
0.2847 eV
Valence band offset Al_{0.35}Ga_{0.65}As
/ GaAs: 0.1926 eV
E_{gap} Al_{0.35}Ga_{0.65}As: 2.2883 eV
E_{gap} GaAs:
1.5193 eV
As we do not have doping and no piezoelectric fields (the structure is
assumed to be unstrained) and as the temperature is assumed to be 4 K, we do not
have to deal with charge redistributions. Thus we can refrain from solving
Poisson's equation and we also do not have to take care about selfconsistency.
Both, the heavy hole and the light hole band edge energies are degenerate but
the effective mass tensors differ. Thus we have to solve three Schrödinger
equations, namely for the
 conduction band
 heavy hole band
 light hole band
The lowest hole state is the heavy hole state and the second hole state is the
light hole state. No further hole states are confined. Also, in the conduction
band only the ground state is confined.
The following figures show the charge densities (Psi²; Psi =
wave function) of the ground states of the confined electron, heavy and light hole eigenstates
of the quantum wire.

The probabiliy amplitudes of the electron (e),
the heavy hole (hh) and the light hole (lh) envelope functions at an
unstrained Tshaped intersection of two 10 nm wide GaAs quantum wells
embedded by Al_{0.35}Ga_{0.65}As barriers.
For the lower three pictures, the wave functions are normalized so that the
maximum of each equals one. 

Contour diagram of the probabiliy amplitudes
of the electron (e), heavy hole (hh) and light hole (lh) eigenfunctions
(same figures as the ones above but this time viewed from the top).
One can clearly see that each ground state wave function is localized at the
Tshaped intersection and shows the Tshaped symmetry.
Due to the anisotropy of the heavy hole effective mass, the heavy hole
wave function prefers to extend along the [100] direction and hardly
penetrates into the quantum well that is aligned along the [011] direction.
The heavy hole mass along the [100] direction is only half as heavy as along
the [011] direction.
The light hole anisotropy is only minor and thus its symmetry resembles the
one of the isotropic electron.
Again, the normalization is chosen so that the maximum of the wave function
equals one. 
The quantum cluster ($quantumregions
)
used in this calculation has the size 108 nm x 108 nm. (The whole simulation area
is 110 nm x 110 nm.) The figures show an extract
of 60 nm x 60 nm.
The calculated eigenvalues are:
Singleband Schrödinger equation (effectivemass
)
schroedingermassesanisotropic =

box 
yes 
no 
electron ground state energy (eV) 
3.00584185 
3.00584196 
3.00582875 
heavy hole ground state energy (eV) 
1.45511017 
1.45511001 
1.45511013 
light hole ground state energy (eV) 
1.43906098 
1.43906078 
1.43906813 
6band k.p Schrödinger equation (6x6kp
)
schroedingerkpdiscretization =

boxintegration 
boxintegration 
finitedifferences 
kpvvtermsymmetrization =

no 
yes 
yes 
1^{st} hole energy (eV) (6band k.p) 
1.4549 (2fold degenerate) 
 (2fold degenerate) 
 (2fold degenerate) 
2^{nd} hole energy (eV) (6band k.p) 
 (2fold degenerate) 
 (2fold degenerate) 
 (2fold degenerate) 
3^{rd} hole energy (eV) (6band k.p) 
 (2fold degenerate) 
 (2fold degenerate) 
 (2fold degenerate) 
4^{th} hole energy (eV) (6band k.p) 
 (2fold degenerate) 
 (2fold degenerate) 
 (2fold degenerate) 
5^{th} hole energy (eV) (6band k.p) 
 (2fold degenerate) 
 (2fold degenerate) 
 (2fold degenerate) 
Note:
 The grid lines have an equally spaced separation of 1 nm.
Choosing a denser gridding of 0.5 nm separation between 25 and 85 nm leads to
the following eigenvalues for box
:
3.00529
, 1.45525
,
1.43942
.
 Conduction band edge energy (GaAs) = 2.979 eV, hole band edge energy
(GaAs) = 1.459667 eV.
 Neumann boundary conditions were used. In case of confinement, the
wave function is zero at the quantum cluster boundaries.
(Here, schroedingermassesanisotropic = no
leads to the same results as the mixed derivatives d²/(dxdy) are zero
because our effective mass tensors are either isotropic (electrons) or oriented
with their principal axes parallel to the simulation system (holes).)
For these calculations, we neglected excitonic effects.
=>
electron  heavy hole transition energy = 1.551 eV
=>
electron  light hole transition energy =
1.567 eV
AVS/Express screenshots where the scale of the wave function (psi²) is given
in 1/nm². Integration of psi² over the simulation area of 110 * 110 nm² sums up
to one.
In addition to these ground states for k_{z}=0, excited states are possible as well.
Similar to the subbands of a 1D quantum well that show a E(k_{x},k_{y})
dispersion one can assign a subband with the energy dispersion E(k_{z})
to each quantum wire eigenvalue which describes the free motion along the
quantum wire axis (z axis). A more advanced treatment would be to use k.p
theory to calculate the eigenvalues for different k_{z} in order to
obtain the (nonparabolic) energy dispersion E(k_{z}).
Understanding the meaning of the 6band k.p parameters
For the same structure as above we perform the calculations again but this
time using 6band k.p instead of singleband.
The left figure shows the wave function (psi²) for the hole ground state where
we used for GaAs the same Luttinger parameters as above:
gamma_{1} = 6.98
, gamma_{2} = 2.06
,
gamma_{3} = 2.93
(This corresponds to: L = 16.220
,
M = 3.860
, N = 17.580
)
The right figure shows the wave function (psi²) for the hole ground state where
we used for GaAs the Luttinger parameters
gamma_{1} = 6.98
, gamma_{2} = 2.06 =
gamma_{3}
(This corresponds to: L = 16.220
, M
= 3.860
, N = 12.36
)
Choosing gamma_{2} = gamma_{3}
corresponds
to an isotropic effective mass.
These results are in very good qualitative agreement with the heavy hole and
light hole wave functions calculated within the singleband approach. The impact
of an isotropic (for electrons and light holes) or anisotropic (for heavy hole)
effective mass tensor should now be clear.
Within the 6band k.p calculations we get only one confined hole state,
in contrast to the singleband effectivemass approach where we obtained one
confined state for the heavy hole and another confined state for the light hole.
Note: Here we used Dirichlet boundary conditions.
Interband transitions
(Note: This part has to be updated: Now we output the square of this matrix
element.)
We evaluate the spatial overlap integral between the electron and hole
envelope wave functions of the ground states. These numbers are proportional to
the transition probability. Additionally, the polarization of light has
to be taken into account to get the correct probability.
$output1bandschroedinger
...
interbandmatrixelements = yes
The results are contained in these files:
Schroedinger_1band/interband2D_vb001_cb001_qc001_hlsg001_deg001_neu.dat
!
heavy hole <>
electron
Schroedinger_1band/interband2D_vb002_cb001_qc001_hlsg002_deg001_neu.dat
!
light hole <>
electron
vb001
: valence band 1
, i.e. heavy
hole
vb002
: valence band 1
, i.e. heavy hole
cb001
: conduction band 1
, i.e.
electron for Gamma point
heavy hole <> Gamma conduction band
matrix element transition energy e1hh1

<psi_vb001psi_cb001>
0.9048
1.5507 eV
light hole <> Gamma conduction band
matrix element transition energy e1lh1

<psi_vb001psi_cb001>
0.9946
1.5668 eV
vb001
: valence band hole state 1
,
i.e. (heavy or light) hole ground state
cb001
: conduction band state 1
, i.e.
electron ground state
By looking at the wave functions (psi²), one would expect a larger
overlap between the electron and the light hole rather than electron / heavy
hole.
As the calculatations show, this is exactly the case: The electron /
heavy hole overlap is only 0.9
whereas
the electron / light hole overlap is much larger:
0.99
.
Note: A comparison of these results with different eigenvalue solvers and
discretization routines:
heavy hole <> Gamma conduction band
matrix element transition energy e1hh1

<psi_vb001psi_cb001>
0.904760286602367 (box) 1.550732 eV
0.904842702923420 (yes) 1.550732 eV
0.904765324859528 (no) 1.550719 eV
(chearn,box)
0.556467704746565 (chearn,yes) 1.550732 eV > imaginary part of psi is NOT zero
0.904765324861447 (chearn,no) 1.550719 eV
0.574566655967155 (arpack,box) 1.550732 eV 0.574531849467085 (bug fix
diag_sgV)
0.556467704746344 (arpack,yes) 1.550732 eV > imaginary part of psi is NOT zero
> complex arpack
0.904765324861290 (arpack,no) 1.550719 eV
0.904765324859528 (no,magnetic) 1.550719 eV > imaginary part of psi is
zero
0.556442520299478 (chearn,no,magnetic) 1.550719 eV > imaginary part of psi is
NOT zero
0.556442520299654 (arpack,no,magnetic) 1.550719 eV > imaginary part of psi is
NOT zero > complex arpack
light hole <> Gamma conduction band
matrix element transition energy e1lh1

<psi_vb001psi_cb001>
0.994609977897094 (box) 1.566781 eV
0.994618620870019 (yes) 1.566781 eV
0.994618072678695 (no) 1.566761 eV
(chearn,box)
0.579198621913980 (chearn,yes) 1.566781 eV > imaginary part of psi is NOT zero
0.994618072678062 (chearn,no) 1.566761 eV
0.181672284480707 (arpack,box) 1.566781 eV 0.181668617064867 (bug fix diag_sgV)
0.665134365762336 (arpack,yes) 1.566781 eV > imaginary part of psi is NOT zero
> complex arpack
0.994618072678148 (arpack,no) 1.566761 eV
0.994618072678695 (no,magnetic) 1.56676 eV > imaginary part of psi
is zero
0.579560683962475 (chearn,no,magnetic) 1.566761 eV > imaginary part of psi is
NOT zero
0.579560683962440 (arpack,no,magnetic) 1.56676 eV > imaginary part of
psi is NOT zero > complex arpack
See $output1bandschroedinger
for an explanation of this.
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[at] nextnano.com
.