nextnano3 - Tutorial
next generation 3D nano device simulator
3D Tutorial
Energy levels in idealistic 3D cubic and cuboidal shaped quantum dots
Author:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please
check if you can find them in the installation directory.
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-> 3DcubicQD.in -
10 nm x 10 nm x 10 nm QD
-> 3DcuboidQD.in -
10 nm x 10 nm x 5 nm QD
Energy levels in an idealistic 3D cubic quantum dot
-> 3DcubicQD.in -
10 nm x 10 nm x 10 nm QD
Here, we want to calculate the energy levels and the wave functions of a cubic
quantum dot of length 10 nm.
We assume that the barriers at the QD boundaries are infinite. This way we
can compare our numerical calculations to analytical results.
The potential inside the QD is assumed to be 0 eV.
As effective mass we take the electron effective mass of InAs, i.e. me
= 0.026 m0.
conduction-band-masses = 0.026 0.026 0.026 ! electron
effective mass at Gamma conduction band
...
A discussion of the analytical solution of the 3D Schrödinger equation of a
particle in a box (i.e. quantum dot) with infinite barriers can be found in e.g.
Quantum Heterostructures (Microelectronics and Optoelectronics) by V.V.
Mitin, V.A. Kochelap and M.A. Stroscio.
The solution of the Schrödinger equation leads to the following eigenvalues:
En1,n2,n3 = hbar2 pi2
/ 2me
( n12 / Lx2 + n22
/ Ly2 + n32 / Lz2
) =
=
1.4462697 * 10-17 eVm2 ( n12
/ Lx2 + n22 / Ly2
+ n32 / Lz2 ) =
=
0.1446269 eV
( n12
+ n22
+ n32
) (if Lx
= Ly = Lz = 10 nm)
- En1,n2,n3 is the total electron energy.
- n1, n2, n3
are three discrete quantum numbers (because we have three
directions of quantization).
- Lx, Ly, Lz are the lengths along the x, y
and z directions. In our case, Lx = Ly = Lz =
10 nm.
Generally, the energy levels are not degenerate, i.e. all energies are
different.
However, some energy levels with different quantum numbers coincide, if
the lengths along two or three directions are identical or
if their ratios are integers. In our cubic QD case, all three lengths are
identical.
Consequently, we expect the following degeneracies:
- E111 = 0.43388 eV (ground state)
- E112 = E121 = E211 = 0.86776 eV = 2 E111
- E122 = E212 = E221 = 1.30164 eV = 3 E111
- E113 = E131 = E311 = 1.59090 eV = 11/3 E111
- E222 = 1.73552 eV = 4 E111
- E123 = E132 = E213 = E231 = E312
= E321 = 2.02478 eV = 14/3 E111
- E333 = 3.90493 eV = 27/3 E111
nextnano³ numerical results for a 10 nm cubic quantum dot with 0.50 nm
grid spacing:
(The grid spacing is rather coarse but has the advantage that the calculation
takes only a minute.)
Output file name:
Schroedinger_1band/ev_cb1_sg1_deg1.dat
num_ev: eigenvalue [eV]:
(0.50 nm grid)
1 0.432989
= E111
2 0.862425
E112/E121/E211
3 0.862425
E112/E121/E211
4 0.862425
E112/E121/E211
5 1.291860
E122/E212/E221
6 1.291860
E122/E212/E221
7 1.291860
E122/E212/E221
8 1.566392
E113/E131/E311
9 1.566392
E113/E131/E311
10 1.566392
E113/E131/E311
11 1.721296
12 1.995828
E123/E132/E213/E231/E312/E321
13 1.995828
E123/E132/E213/E231/E312/E321
14 1.995828
E123/E132/E213/E231/E312/E321
15 1.995828
E123/E132/E213/E231/E312/E321
16 1.995828
E123/E132/E213/E231/E312/E321
17 1.995828
E123/E132/E213/E231/E312/E321
18 2.425263
E223/E232/E322
19 2.425263
E223/E232/E322
20 2.425263
E223/E232/E322
21 2.527557
E114/E141/E411
22 2.527557
E114/E141/E411
23 2.527557
E114/E141/E411
24 2.699795
E233/E323/E332
25 2.699795
E233/E323/E332
26 2.699795
E233/E323/E332
27 2.956993
E124/E142/E214/E241/E412/E421
28 2.956993
E124/E142/E214/E241/E412/E421
29 2.956993
E124/E142/E214/E241/E412/E421
30 2.956993
E124/E142/E214/E241/E412/E421
31 2.956993
E124/E142/E214/E241/E412/E421
32 2.956993
E124/E142/E214/E241/E412/E421
...
48 3.833198
...
The following figures show the isosurfaces of the electron wave function
(psi²) of a 10 nm cubic quantum dot with infinite barriers for
- the ground state E111
- the 11th eigenstate E222.
Both states are nondegenerate.
 
Intraband (=intersublevel) transitions
$output-1-band-schroedinger
...
intraband-matrix-elements = p
!
calculate intersublevel dipole moment < psif* | pz | psii
> and oscillator strength ffi
In this cubic QD with infinite barriers, optical intersublevel
transitions are only allowed between states with odd difference
quantum numbers along the same axes:
E111 <==> E112/E121/E211
1 <==> 2 / 3 / 4
E111 <==> E114/E141/E411
1 <==> 21 / 22 / 23
E211 <==> E311
2 <==> 8
E121 <==> E131
3 <==> 9
E112 <==> E113
4 <==> 10
...
The following transitions are forbidden:
E111 <==> E113/E131/E311
1 <==> 8 / 9 / 10
E211 <==> E112/E121
2 <==> 3 / 4
E121 <==> E211/E112
3 <==> 2 / 4
E112 <==> E211/E121
4 <==> 2 / 3
...
Energy levels in an idealistic 3D cubodial shaped quantum dot with Lx
= Ly /= Lz
-> 3DcuboidQD.in -
10 nm x 10 nm x 5 nm QD
This time we use a similiar quantum dot as above but the lengths are Lx
= Ly = 10 nm and Lz = 5 nm.
Therefore, the degeneracies of the eigenenergies are different. We expect the
following:
En1,n2,n3 = hbar2 pi2
/ 2me
( n12 / Lx2 + n22
/ Ly2 + n32 / Lz2
) =
=
1.4462697 * 10-17 eVm2 ( n12
/ Lx2 + n22 / Ly2
+ n32 / Lz2 ) =
=
0.1446269 eV
( n12
+ n22 ) + 0.5785079 eV
n32 )
(if Lx = Ly = 10 nm and Lz = 5 nm)
Generally, the energy levels are not degenerate, i.e. all energies are
different.
However, some energy levels with different quantum numbers coincide, if
the lengths along two or three directions are identical or
if their ratios are integers. In our cubic QD case, all three lengths are
identical.
Consequently, we expect the following degeneracies:
- E111 = 0.86776 eV (ground state)
- E121 = E211 = 1.301642 eV
- E221 = 1.73552 eV = 2 E111 (This
is a coincidence because Lx,y / Lz are integers and have
the value 2.)
- E131 = E311 = 2.02478 eV
- E231 = E321 = 2.45866 eV
- E112 = 2.60329 eV = 2 E121 (This
is a coincidence because Lx,y / Lz are integers and have
the value 2.)
- E122 = E212 =
E141 = E411 = 3.03717 eV
(This is a coincidence because Lx,y / Lz are integers
and have the value 2.)
- E331 = 3.18180 eV
- E222 = 2 E221 =
(This is a coincidence because Lx,y / Lz are integers
and have the value 2.)
E241 = E421 = 3.47105 eV
(This is a coincidence because Lx,y / Lz are integers
and have the value 2.)
- E132 = E312 = 3.76030 eV
- E341 = E431 =
E232 = E322 = 4.19418 eV
(This is a coincidence because Lx,y / Lz are integers
and have the value 2.)
- E151 = E511 = 4.33881 eV
- E142 = E412 =
E251 = E521 = 4.77269 eV
(This is a coincidence because Lx,y / Lz are integers
and have the value 2.)
- E332 = 4.91731 eV
- E441 =
E242 = E422 = 5.20657 eV
(This is a coincidence because Lx,y / Lz are integers
and have the value 2.)
- E113 = 5.49582 eV
- E123 = E213 = 5.92971 eV
nextnano³ numerical results for a 10 nm cubic quantum dot with
- 0.50 nm grid spacing (left column) and
- 0.25 nm grid spacing (right column):
(The grid spacing is rather coarse (for 0.50 nm) but has the advantage that the
calculation takes only a minute.)
Output file name:
Schroedinger_1band/ev_cb1_qc1_sg1_deg1.dat
num_ev: eigenvalue [eV]:
(0.50 nm grid)
(0.25 nm grid)
1 0.862425
0.866424
= E111
2 1.291860
1.299191
3 1.291860
1.299191
4 1.721296
1.731958
5 1.995828
2.017504
6 1.995828
2.017504
7 2.425263
2.450270
8 2.425263
2.450270
9 2.527557
2.584167
10 2.956993
3.016933
11 2.956993
3.016933
12 2.956993
3.016933
13 2.956993
3.016933
14 3.129231
3.168583
15 3.386428
3.449700
16 3.386428
3.449700
17 3.386428
3.449700
18 3.660960
3.735246
19 3.660960
3.735246
20 4.090396
4.168013
21 4.090396
4.168013
22 4.090396
4.168013
23 4.090396
4.168013
24 4.151688
4.291319
25 4.151688
4.291319
26 4.581124
4.724086
27 4.581124
4.724086
28 4.622125
4.734676
29 4.622125
4.734676
30 4.794363
4.886326
...
34 5.121061
5.400036
...
The following figures show the isosurfaces of the electron wave function (psi²)
of a 10 nm x 10 nm x 5 nm cuboidal shaped quantum dot with infinite barriers for
- the ground state E111
- the 4th eigenstate E221
-
- the 14th eigenstate E331.
All these states are nondegenerate.
 
 
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