 
nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Quantum Confined Stark Effect (QCSE)
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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> 1DQuantumConfinedStarkEffect.in
Quantum Confined Stark Effect (QCSE)
This tutorial aims to reproduce figure 3.22 (p. 96) of
Paul Harrison's
excellent book "Quantum
Wells, Wires and Dots" (Section 3.12 "Quantum Confined Stark
Effect"), thus the following description is based on the
explanations made therein.
We are grateful that the book comes along with a CD so that we were able to
look up the relevant material parameters and to check the results for
consistency.
Single quantum well: 20 nm AlGaAs / 6 nm GaAs / 20 nm AlGaAs
 Our structure consists of a 6 nm GaAs quantum well that is surrounded by 20 nm Al_{0.2}Ga_{0.8}As barriers on each side.
We thus have the following layer sequence: 20 nm Al_{0.2}Ga_{0.8}As
/ 6 nm GaAs / 20 nm Al_{0.2}Ga_{0.8}As.
The barriers are printed in bold.
 This figure shows the conduction band edge and the square of the
ground state electron
wave function (psi²) that is confined inside the well for two cases:
 no applied electric field (0 kV/cm)
 applied electric field (70 kV/cm)
In the case of an applied electric field, the wave function moves to the right
and its ground state energy decreases slightly. The reason is that a charged
particle prefers to move to areas of lower potential in order to lower the
total energy.
(Note that the energies were shifted so that the conduction band edge of GaAs
equals 0 eV.)
The origin of the electric field is chosen automatically to be the center of
the well. This makes it possible to compare
energies by varying the applied electric field as shown in this tutorial.
For holes, the wave function would move to the left (not shown here) thus
making it possible to produce a space charge or a polarization of the charge
carriers.

Technical details:
We use flowscheme = 21
$simulationflowcontrol
flowscheme = 21 !
apply constant electric field
...
and
$numericcontrol
...
zeropotential = yes
This flowscheme includes the following:
1. We calculate the strain (if any).
2. We calculate the piezo and pyroelectric charges (if any).
3. We do not solve Poisson's equation. (This is the difference to
flowscheme = 20 ).
4. We apply the electric field.
5. We calculate the eigenstates and wave functions by solving Schrödinger's
equation (either singleband or k.p).
Note that in this case, this is not a selfconsistent calculation of the
PoissonSchrödinger equation.
 Output
a) The conduction band edge of the Gamma conduction band can be
found here:
band_structure / cb1D_Gamma.dat
b)
This file contains the eigenenergies of the ground state. The units are [eV] .
Schroedinger_1band / ev1D_cb001_qc001_sg001_deg001_dir.dat
For example, the values for zero applied electric field
read:
nextnano³:
num_ev: eigenvalue [eV]:
1 0.053287
Paul Harrison's book: 1
0.05326045
(or 0.053310 )
c) This file contains the eigenenergies and the squared wave functions (Psi²):
Schroedinger_1band / cb001_qc001_sg001deg001_dir_psi_squared.dat
This file contains the eigenenergies and the wave functions (Psi):
Schroedinger_1band / cb001_qc001_sg001deg001_dir_psi.dat
a) and b) can be used to plot the data as shown in the figure above.
Varying the electric field...
 Now we vary the applied electric field from 0 kV/cm to 70 kV/cm.
There
are two possibilities to do so:Possibility 1:
We perform several individual calculations and vary the strength of the
electric field by specifying its value for each individual calculation.
$electricfield electricfieldon =
yes ! 'yes'
/ 'no' electricfieldstrength =
0d0
! in units of [V/m]  Here: 0 kV/cm ! electricfieldstrength =
5d5
! in units of [V/m]  Here: 5 kV/cm ! electricfieldstrength =
10d5 !
in units of [V/m]  Here: 10 kV/cm ! electricfieldstrength =
15d5 !
in units of [V/m]  Here: 15 kV/cm ! electricfieldstrength =
20d5 !
in units of [V/m]  Here: 20 kV/cm ! electricfieldstrength =
25d5 !
in units of [V/m]  Here: 25 kV/cm ! electricfieldstrength =
30d5 !
in units of [V/m]  Here: 30 kV/cm ! electricfieldstrength =
40d5 !
in units of [V/m]  Here: 40 kV/cm ! electricfieldstrength =
50d5 !
in units of [V/m]  Here: 50 kV/cm ! electricfieldstrength =
60d5 !
in units of [V/m]  Here: 60 kV/cm ! electricfieldstrength =
70d5 !
in units of [V/m]  Here: 70 kV/cm electricfielddirection =
0 0 1 ! [001]
direction, i.e. along z axis. $end_electricfield
Possibility 2:
An alternative (and much more userfriendly approach)
would be the usage of an "electric field sweep".
Here, only one calculation is necessary. The variation of the electric field
strength in done automatically.
$electricfield electricfieldon =
yes ! 'yes'
/ 'no' electricfieldstrength =
0d0
! in units of [V/m]  Here: 0 kV/cm electricfielddirection =
0 0 1 ! [001]
direction, i.e. along z axis.
!
electricfieldsweepactive
= yes ! 'yes'
/ 'no'
electricfieldsweepstepsize
= 5d5 ! in units of [V/m] 
Here: 5 * 10^{5} V/m = 5 kV/cm
electricfieldsweepnumberofsteps = 14
! 14 steps, starting from 0 kV/cm
to 70 kV/cm
$end_electricfield
Here, the electric field is varied from 0 kV/cm to
70 kV/cm , in steps of 5 kV/cm .
The output of the eigenvalues are then contained in
Schroedinger_1band /
electric_ev1D_cb001_qc001_sg001_deg001_dir_Kx001_Ky001_Kz001.dat .
The first column contains the strength of the electric field in units of
[kV/cm] .
The second column contains the 1^{st} eigenvalue for the
specified electric field in units of [eV] ,
the third column contains the 2^{nd}
eigenvalue for the specified electric field in units of [eV] , ...
 The following figure shows the ground state energy of the 6 nm quantum
well as a function of the applied electric field strength F.
The calculated energies can be represented by a parabolic fit. Over the range of electric fields investigated, the ground state energy can be
represented by the parabola:
E_{1}(F) = E_{1}(0)  0.000365 F²
where E_{1}(0) refers to the ground state energy at zero electric
field (in units of meV). Here, the electric field strength F is given
in units of kV/cm. (Note: Paul Harrison's value is 0.00036.)
This suppression of the confined energy level by an electric field is called
"Quantum Confined Stark Effect (QCSE)".
The data that were plotted here are contained in this file.
Schroedinger_1band\electric[kV/cm]_ev1D_cb1_qc1_sg1_deg1_dir.dat
The energy levels are contained as a function of the sweep variable
"electric field".
 The plot is (almost) in agreement with Fig. 3.22 (p. 96) of
Paul Harrison's
book "Quantum
Wells, Wires and Dots".
The energies differ slightly although we used  identical effective masses  identical conduction band offsets  identical grid resolution (0.1 nm) The only difference is that we use multiple points at the heterointerfaces but
this should not explain the difference. The answer lies probably in a tiny inconsistency in the book. On page 96 it says: "where E_{1}(0) refers to the groundstate energy
(53.310 meV) at zero field" However, the value that was used in Fig. 3.22 is E_{1}(0) = 53.26045
meV. (This value can be found on the CD that accompanies the book.) The nextnano³ value is 53.287 meV which
lies in the middle between these two values.
 Output: The energy values were taken from this file:
Schroedinger_1band / ev1D_cb001_qc001_sg001_deg001_dir_Kx001_Ky001_Kz001.dat
For example, the values for zero applied electric field read:
nextnano³:
num_ev: eigenvalue [eV]:
1 0.053287
Paul Harrison's book: 1
0.05326045 (or 0.053310 )
The values for an applied electric field of 70 kV/cm read:
nextnano³:
num_ev: eigenvalue [eV]:
1 0.051497
Paul Harrison's book: 1
0.051472
