 
nextnano^{3}  Tutorial
1D Tutorial
Simple quantum cascade structure
> 1DQCL_simple_nn3.in / *nnp.in
 input file for the nextnano^{3} and nextnano++ software
Simple quantum cascade structure  Results
This tutorial is based on the quantumcascade structure (Figures 12 (b) and
16 (b)) that has been presented in the following paper:
Resonant Tunneling Through Double Barriers, Perpendicular Quantum
Transport Phenomena in Superlattices, and Their Device Applications
F. Capasso, K. Mohammed, A.Y. Cho
IEEE Journal of Quantum Electronics QE22 (9), 1853 (1986)
The following picture is based on Fig. 3 of
Simulation of quantum cascade lasers  optimizing laser performance (in
English)
S. Birner, T. Kubis, P. Vogl
Photonik international 2, 60 (2008)
Simulation zur Optimierung von Quantenkaskadenlasern
(in German)
S. Birner, T. Kubis, P. Vogl
Photonik 1, 44 (2008)
It shows the conduction band edge profile of an Al_{0.48}In_{0.52}As/In_{0.53}Ga_{0.47}As
superlattice at an electric field of 89 kV/cm.
The singleband effectivemass Schrödinger equation is solved for this band
profile.
The wave functions (psi²) of this quantum cascade structure are shown.
The basic idea of such a structure is to depopulate the
lowest eigenstate of each quantum well
efficiently by bringing it into resonance with the
third eigenstate of the next quantum well (resonant tunneling).
The transition second eigenstate >
lowest eigenstate should be a nonradiative
intersubband transition whereas
the transition third eigenstate >
second eigenstate should be a radiative
intersubband transition, i.e. a photon is emitted.
Another important condition for a quantum cascade laser is population
inversion,
i.e. the occupation of the third eigenstate
must be much higher than the occupation of the second
eigenstate and lowest eigenstate.
 The input file "
1DQCL_simple.in " should be rather intuitive
and selfexplanatory.
Documentation for each keyword and each specifier can be found here:
keywords
 An example of a keyword (
$electricfield )
and a specifier (electricfieldstrength ) is the electric
field.
The electric field is set to 89 kV/cm.
$electricfield
...
electricfieldstrength = 89d5 ! in
units of [V/m]  Here: 89 kV/cm (d5
= 10^{5} , i.e. 89 * 10^{5} [V/m] )
Output
The output files are ASCII files and can be plotted with software like e.g.
Origin.
 The conduction band edge can be found in the following file:
band_profile/cb_Gamma.dat
1^{st} column: grid points in units of [nm]
2^{nd} column: Gamma conduction band edge in units of [eV]
If one plots the content of this file, one gets the following figure.
There are six Al_{0.48}In_{0.52}As barriers and five In_{0.53}Ga_{0.47}As
barriers.
The conduction band offset is 0.51 eV.
 The 40 eigenvalues that were calculated can be found in this file:
wavefunctions/ev_cb1_sg1_deg1.dat
The units are [eV].
The eigenvalues are also contained in this file, i.e. the eigenvalues
for each grid point
wavefunctions/cb1_qc1_sg1_deg1_psi_squared_shift.dat
1^{st} column: grid
points in units of [nm]
2^{nd} column: 1^{st}
eigenvalue in units of [eV]
3^{rd} column: 2^{nd}
eigenvalue in units of [eV]
...
41^{st} column: 40^{th}
eigenvalue in units of [eV]
If one plots these columns (together with the conduction band edge)
one obtains the following picture:
Note: The figure shows only the following energy levels:
1,2,3,4,5,9,10,12,16,18,20,26,27,30,37
 The square of the wave functions (psi²) of the 40 eigenstates can be
found in this file
wavefunctions/cb1_qc1_sg1_deg1_psi_squared_shift.dat
1^{st} column: grid
points in units of [nm]
...
42^{nd} column: psi² of 1^{st}
eigenstate
43^{rd} column: psi² of 2^{nd}
eigenstate
...
81^{st} column: psi² of 40^{th}
eigenstate
Note: In order to be able to plot the wave functions nicely into the
conduction band edge profile, we shift the square of the wave function by its
corresponding energy: psi²_{n}' = psi²_{n} + E_{n}
If one plots these columns (together with the conduction band edge)
one obtains the following picture:
Note: The figure shows only the following wave functions:
1,2,3,4,5,9,10,12,16,18,20,26,27,30,37
The basic idea of such a structure is to depopulate the lowest
eigenstate of each quantum well efficiently by bringing it into resonance
with the third eigenstate of the next quantum well (resonant tunneling).
The transition 2 >
1 should be a nonradiative intersubband
transition whereas
the transition 3 >
2 should be a radiative intersubband
transition, i.e. a photon is emitted.
Another important condition for a quantum cascade laser is population
inversion,
i.e. the occupation of the state 3 must
be much higher than the occupation of the states 2
and 1.
 The conduction band masses that were used for each grid point can be found in this
file:
conduction_band_masses1D.dat
1^{st} column: grid
points in units of [nm] other columns: effective mass tensor components of Gamma, L and X valleys in
units of [m_{0}]
m(Gamma)
m(Gamma) m(Gamma) m_{l}(L) m_{t}(L)
m_{t}(L) m_{l}(X) m_{t}(X) m_{t}(X)
These masses have been calculated from the binaries InAs, GaAs and
AlAs for the relevant ternaries, including bowing parameters.
 Experienced users might be interested in having a look at the
intersubband matrix elements:
The intersubband (or intraband) matrix elements p_{z} and the
oscillator strengths can be found in this file:
wavefunctions/intraband_pz1D_cb001_qc001_sg001_deg001_dir.txt
The content of this file should not be plotted! One has to open
this file with a text editor.
More information and documentation on these matrix elements is
available here: Intersubband matrixelements
