| |
nextnano3 - Tutorial
next generation 3D nano device simulator
1D Tutorial
Optical intersubband transitions in a quantum well - Intraband matrix elements and selection rules
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.
If you cannot find them, please submit a
Support Ticket.
-> 1DQW_intraband_matrixelements_infinite_nn3.in / *_nnp.in - input files for the nextnano3 and nextnano++
software
-> 1DQW_intraband_matrixelements_infinite_kp_nn3.in
Optical intersubband transitions in a 10 nm AlAs / GaAs / AlAs quantum well -
Intraband matrix elements and selection rules
Eigenstates and wave functions in the quantum well
- We consider a 10 nm GaAs quantum well embedded between AlAs barriers. The structure
is assumed to be unstrained.
We assume "infinite" AlAs barriers. (This can be achieved by chosing a
band offset of 100 eV.)
This way we can compare our results to analytical text books results.
-> 1DQW_intraband_matrixelements_infinite_nn3.in
$output-1-band-schroedinger
...
scale
= 0.3d0 !
scale the wave function psi and psi² so that it is easier to visualize
them
intraband-matrix-elements = z
!
calculate intersubband dipole moment | < psif* | z | psii
> | and oscillator strength ffi
!intraband-matrix-elements = p
!
calculate intersubband dipole moment | < psif* | pz | psii
> | and oscillator strength ffi
complex-wave-functions =
yes !
to print out psi in addition to psi²
The figure shows the six lowest eigenfunctions of the 1D GaAs quantum
well. The conduction band edge of GaAs is assumed to be located at 0 eV.
For "infinite" barriers we obtain using single-band Schrödinger
effective-mass approximation (i.e. isotropic and parabolic effective masses)
the following eigenvalues:
E1 = 0.05652 eV
(0.05655)
E2 = 0.22601 eV
(0.22618 = 2² E1)
E3 = 0.50831 eV
(0.50891 = 3² E1)
E4 = 0.90314 eV
(0.90473 = 4² E1)
E5 = 1.41011 eV
(1.41365 = 5² E1)
E6 = 2.02872 eV (2.03565
= 6² E1)
The analytic formula in the infinite barrier QW model reads: En = hbar²/2m*
(pi n / L )² = 0.056546 n²
where L is the width of the quantum well (L = 10 nm). The analytically
calculated values are given in brackets and are in excellent agreement.
We used an electron effective mass of 0.0665 m0 and a 0.1 nm grid
resolution.
We used:
schroedinger-1band-ev-solv =
LAPACK-ZHBGVX ! 'LAPACK', 'LAPACK-ZHBGVX',
'arpack',
'it_jam', 'chearn'
schroedinger-masses-anisotropic = box
! 'yes', 'no',
'box'
Intersubband matrix elements
- Light that propagates normal to the quantum well layers cannot be
absorbed by intraband transitions.
However, if the light propagates in the plane of the well (i.e. the
electric field is oriented normal to the quantum well layers), intersubband
absorption occurs.
To understand optical intersubband (or intraband) transitions for light that
travels in the plane of the QW, we have to examine the intersubband dipole
moment
| Mfi | = | integral (psif* (z) z psii
(z) dz) |
where psi is the envelope function of the relevant state (within the same
band).
In our case, we have a symmetric quantum well with infinite barriers, thus our envelope functions are
either symmetric or antisymmetric. The intersubband matrix elements
will vanish if the envelope functions have the same parity, e.g. | M13
| = | M31 | = 0.
In this simple example, the matrix elements can be calculated analytically,
e.g. | M12 | = 16 / (9 pi²) L = 1.8013
[nm].
nextnano³ result: | M12 | = | M21 | = 1.8017
[nm]
| M13 | = | M31 | = 0.00000006
[nm]
For the "infinite" QW barrier model, this matrix element is independent
of the effective mass, thus the matrix elements in the conduction band are the
same as in the valence bands (single-band approximation).
A useful quantity is the oscillator strength ffi which is
defined as follows:
ffi = 2m* / hbar² (Ef
- Ei) | Mfi |²
ffi = - fif
f21 for our simple infinite barrier example is given by
f21 = 256 / (27 pi²) = 0.9607
and is independent of the well width.
The nextnano³ result is: f21 = 0.9603
= - f12
We can also see that this is a strong transition because all transitions from
state '1' to state 'f' must add up to unity (so-called "f-sum rule"):
sumf (ff1) = 1.0
(Thomas-Kuhn sum rule for constant effective mass m*.)
Thus all other transitions are much weaker.
It is interesting to look at the transitions starting from the second level i
= 2. The lowest oscillator strength f12 = - 0.96 is
negative, but the sum over all ff2 must still give unity, thus
oscillator strengths larger than 1 are possible, e.g. f32 = 1.87.
- The intersubband dipole moments and the oscillator strenghts are contained in this
file:
Schroedinger_1band/intraband_z1D_cb001_qc001_sg001_deg001_dir.txt
- Gamma conduction band
For each transition, the transition energy is given.
The effective masses that have been used for the calculation of the oscillator
strengths are also indicated. They are calculated by building an average of
the parallel effective masses for each grid point, weighted by the
square of the wave function on each grid point. In this particular example, the
effective masses are constant and do not vary with position (m||
= 0.0665 m0 ).
(Assuming that the masses are isotropic, it is fine to use the parallel
effective masses.)
-------------------------------------------------------------------------------
Intersubband transitions
=> Gamma conduction band
-------------------------------------------------------------------------------
Electric field in z-direction [kV/cm]: 0.0000000E+00
-------------------------------------------------------------------------------
-------------------------------------------------------------------------------
Intersubband dipole moment | < psi_f* | z | psi_i >
| [Angstrom]
------------------|------------------------------------------------------------
Oscillator strength []
------------------|--------------|---------------------------------------------
Energy of transition [eV]
------------------|--------------|--------------|------------------------------
m* [m_0]
------------------|--------------|--------------|----------|-------------------
<psi001*|z|psi001> 249.0000
<psi002*|z|psi001> 18.01673
0.9602799
0.1694912 6.6500001E-02
<psi003*|z|psi001> 6.1430171E-07
2.9757722E-15 0.4517909 6.6500001E-02
(same parity: symmetric)
<psi004*|z|psi001> 1.441336
3.0698571E-02 0.8466209 6.6500001E-02
<psi005*|z|psi001> 1.6007220E-07
6.0536645E-16 1.353592 6.6500001E-02
(same parity: symmetric)
<psi006*|z|psi001> 0.3971010
5.4281605E-03 1.972205 6.6500001E-02
<psi007*|z|psi001> 5.1874160E-08
1.2690011E-16 2.701849 6.6500001E-02
(same parity: symmetric)
<psi008*|z|psi001> 0.1634139
1.6508275E-03 3.541806 6.6500001E-02
...
<psi020*|z|psi001> 1.0178176E-02 3.9451432E-05
21.81846 6.6500001E-02
Sum rule of oscillator strength: f_psi001 =
0.9994023
<psi001*|z|psi002> 18.01673
-0.9602799
-0.1694912 6.6500001E-02
<psi002*|z|psi002> 249.0000
<psi003*|z|psi002> 19.45806
1.865556 0.2822997
6.6500001E-02
<psi004*|z|psi002> 2.0636767E-06
5.0333130E-14 0.6771297 6.6500001E-02
(same parity: antisymmetric)
<psi005*|z|psi002> 1.838436
6.9852911E-02 1.184101 6.6500001E-02
<psi006*|z|psi002> 1.4976163E-08
7.0571038E-18 1.802713 6.6500001E-02
(same parity: antisymmetric)
<psi007*|z|psi002> 0.5605143
1.3886644E-02 2.532358 6.6500001E-02
<psi008*|z|psi002> 8.7380023E-08
4.4941879E-16 3.372315 6.6500001E-02
(same parity: antisymmetric)
<psi009*|z|psi002> 0.2461317
4.5697703E-03 4.321757 6.6500001E-02
<psi010*|z|psi002> 8.3240280E-07
6.5062044E-14 5.379748 6.6500001E-02
(same parity: antisymmetric)
<psi011*|z|psi002> 0.1302904
1.9393204E-03 6.545245 6.6500001E-02
...
<psi020*|z|psi002> 2.7233656E-07
2.8025147E-14 21.64897 6.6500001E-02
Sum rule of oscillator strength: f_psi002 = 0.9975320
<psi001*|z|psi003> 6.1430171E-07
-2.9757722E-15 -0.4517909 6.6500001E-02
(same parity: symmetric)
<psi002*|z|psi003> 19.45806
-1.865556 -0.2822997 6.6500001E-02
<psi003*|z|psi003> 249.0000
<psi004*|z|psi003> 19.85515
2.716784 0.3948300
6.6500001E-02
<psi005*|z|psi003> 6.4708888E-07
6.5907892E-15 0.9018011 6.6500001E-02
(same parity: symmetric)
<psi006*|z|psi003> 2.001849
0.1063465 1.520414 6.6500001E-02
<psi007*|z|psi003> 3.9201248E-07
6.0352080E-15 2.250058 6.6500001E-02
(same parity: symmetric)
<psi008*|z|psi003> 0.6432316
2.2314854E-02 3.090015 6.6500001E-02
<psi009*|z|psi003> 2.6240454E-07
4.8547223E-15 4.039457 6.6500001E-02
(same parity: symmetric)
...
<psi020*|z|psi003> 3.1797737E-02 3.7707522E-04
21.36667 6.6500001E-02
Sum rule of oscillator strength: f_psi003 = 0.9945912
...
- The commonly used
Intersubband dipole moment | < psi_f* | z
| psi_i > | [nm]
depends on the choice of origin for the matrix elements when f = i , thus the user might prefer to output the
Intersubband dipole moment | < psi_f* | p
| psi_i > | [h_bar / nm]
which are the intersubband dipole moments
| Nfi | = | integral (psif* (z) pz
psii
(z) dz) | = | - i hbar integral
(psif* (z) d/dz psii
(z) dz) |
and the oscillator strengths
ffi = 2m* / hbar²
(Ef
- Ei) | Mfi |² = 2 / ( m* (Ef
- Ei) ) | Nfi |²
between all calculated states in each band from eigenvalues 'min-ev '
to 'max-ev '.
In the simple QW of this tutorial, the matrix elements can be
calculated analytically, e.g. | N21 | = 8 hbar / (3 L) =
26.66 hbar /nm.
nextnano³ result: | N21 | = | N12 | = 2.665
hbar /Angstrom
| N31 | = | N13 | = 0
Here, the definition of the oscillator strength ffi has to
be adjusted slightly:
ffi = 2m* / hbar² (Ef
- Ei) | Mfi |² = 2 / ( m* (Ef
- Ei) ) | Nfi |²
ffi = - fif
f21 for our simple infinire barrier example is given by
f21 = 256 / (27 pi²) = 0.9607
and is independent of the well width.
The nextnano³ result is: f21 = 0.9603
= - f12
The intersubband dipole moments and the oscillator strenghts are contained in this
file:
Schroedinger_1band/intraband_p1D_cb001_qc001_sg001_deg001_dir.txt
- Gamma conduction band
The numbers show a comparison betwenn the
z and the pz
matrix elements (in green).
-------------------------------------------------------------------------------
Intersubband dipole moment
| < psi_f*
| z | psi_i > | [Angstrom]
Intersubband
dipole moment | < psi_f* | p | psi_i > | [h_bar / Angstrom]
------------------|------------------------------------------------------------
Oscillator strength []
------------------|--------------|---------------------------------------------
Energy of transition [eV]
------------------|--------------|--------------|------------------------------
m* [m_0]
------------------|--------------|--------------|-----------|------------------
<psi001*|z|psi001> 249.0000
(matrix element <1|1> depends on choice of origin!)
<psi001*|p|psi001> 4.3405972E-19
(matrix element <1|1> independent of origin)
<psi002*|z|psi001> 18.01673
0.9602799
0.1694912 6.6500001E-02
<psi002*|p|psi001> 2.6649671E-02
0.9602799 0.1694912 6.6500001E-02
<psi003*|z|psi001> 6.1430171E-07
2.9757722E-15 0.4517909 6.6500001E-02
(same parity: symmetric)
<psi003*|p|psi001> 2.7325134E-18
<psi004*|z|psi001> 1.441336
3.0698571E-02 0.8466209 6.6500001E-02
<psi004*|p|psi001> 1.0649348E-02
3.0698579E-02 0.8466209 6.6500001E-02
<psi005*|z|psi001> 1.6007220E-07
6.0536645E-16 1.353592 6.6500001E-02
(same parity: symmetric)
<psi005*|p|psi001> 6.9518724E-18
<psi006*|z|psi001> 0.3971010
5.4281605E-03 1.972205 6.6500001E-02
<psi006*|p|psi001> 6.8347314E-03
5.4281540E-03 1.972205 6.6500001E-02
<psi007*|z|psi001> 5.1874160E-08
1.2690011E-16 2.701849 6.6500001E-02
(same parity: symmetric)
<psi007*|p|psi001> 2.8686024E-19
<psi008*|z|psi001> 0.1634139
1.6508275E-03 3.541806 6.6500001E-02
<psi008*|p|psi001> 5.0510615E-03
1.6508278E-03 3.541806 6.6500001E-02
...
<psi020*|z|psi001> 1.0178176E-02 3.9451432E-05
21.81846 6.6500001E-02
<psi020*|p|psi001> 1.9380626E-03
3.9452334E-05 21.81846 6.6500001E-02
Sum rule of oscillator strength: f_psi001 =
0.9994023
Sum rule of oscillator strength: f_psi001 = 0.9994023
...
8-band k.p calculation for k|| = (kx,ky)
= 0
- The following input file performs the same calculations as above but this
time using the 8-band k.p model.
-> 1DQW_intraband_matrixelements_infinite_kp_nn3.in
We modified the 8-band k.p parameters and decoupled (!) the
electrons from the holes (EP = 0 eV, S = 1/me). This way
we have an effective single-band model and thus we are able to compare the
k.p results to the single-band results in order to check for consistency.
- The numbering of the k.p eigenstates differs slightly from the
single-band eigenstates because the k.p eigenstates are two-fold
spin-degenerate. The actual values for the matrix elements are identical
(assuming a decoupled k.p Hamiltonian, i.e. a single-band
Hamiltonian).
- Note that the single-band definition of the oscillator strength does not
really make sense for a k.p calculation where the masses usually are
anisotropic, nonparabolic and are different on each grid point (due to
different materials and different strain tensors).
For the calculation of the oscillator strength in a k.p calculation,
the user can specify suitable masses by overwriting the default entries:
conduction-band-masses = 0.0665d0
0.0665d0 0.0665d0 ! Gamma band (only used for oscillator
strength in k.p)
1.32d0 0.15d0 0.15d0 !
L band (ignored in k.p)
0.97d0 0.22d0 0.22d0 ! X band
(ignored in k.p)
valence-band-masses =
0.500d0 0.500d0 0.500d0 ! heavy hole (only
used for oscillator strength in k.p)
0.068d0 0.068d0 0.068d0 !
light hole (ignored in k.p)
0.172d0 0.172d0 0.172d0 ! split-off hole (ignored in
k.p)
Of course, the masses that are used to calculate the k.p
eigenstates have to be specified via the 6-band and 8-band k.p parameters.
- The intersubband dipole moments and the oscillator strenghts are contained in this
file:
Schroedinger_kp/intraband_p1D_cb001_qc001_8x8kp_dir.txt
- Gamma conduction band
intraband_z1D_cb001_qc001_8x8kp_dir.txt
-
Gamma conduction band
Note that the two-fold spin-degeneracy in single-band is counted
explicitely in k.p.
-------------------------------------------------------------------------------
Intersubband dipole moment
| < psi_f* |
z | psi_i > | [Angstrom]
Intersubband dipole moment
|
< psi_f* | p | psi_i > | [h_bar / Angstrom]
------------------|------------------------------------------------------------
Oscillator strength []
------------------|--------------|---------------------------------------------
Energy of transition [eV]
------------------|--------------|--------------|------------------------------
m* [m_0]
------------------|--------------|--------------|-----------|------------------
<psi001*|z|psi001> 249.0000
(matrix element <1|1> depends on choice of origin!)
<psi002*|z|psi001>
249.0000
(matrix element <2|1> depends on choice of origin!)
<psi001*|p|psi001>
1.8126842E-18 (matrix element
<1|1> independent of
origin)
<psi002*|p|psi001> 1.8126842E-18
(matrix element <2|1> independent of origin)
<psi003*|z|psi001> 18.01673
0.9602799 0.1694912 6.6500001E-02
<psi004*|z|psi001> 18.01673
0.9602799 0.1694912 6.6500001E-02
<psi003*|p|psi001> 2.6649671E-02 0.9602798
0.1694912 6.6500001E-02
<psi004*|p|psi001> 2.6649671E-02 0.9602798
0.1694912 6.6500001E-02
<psi005*|z|psi001> 3.5382732E-13
<psi006*|z|psi001> 3.5382732E-13
<psi005*|p|psi001>
2.1414240E-15
<psi006*|p|psi001> 2.1414240E-15
<psi007*|z|psi001> 1.441336
3.0698583E-02 0.8466209 6.6500001E-02
<psi008*|z|psi001> 1.441336
3.0698583E-02 0.8466209 6.6500001E-02
<psi007*|p|psi001> 1.0649348E-02 3.0698583E-02
0.8466209 6.6500001E-02
<psi008*|p|psi001> 1.0649348E-02 3.0698583E-02
0.8466209 6.6500001E-02
<psi009*|z|psi001>
7.2598817E-13
<psi010*|z|psi001> 7.2598817E-13
<psi009*|p|psi001>
1.0445775E-14
<psi010*|p|psi001> 1.0445775E-14
<psi011*|z|psi001> 0.3971008
5.4281550E-03 1.972205 6.6500001E-02
<psi012*|z|psi001> 0.3971008
5.4281550E-03 1.972205 6.6500001E-02
<psi011*|p|psi001> 6.8347319E-03
5.4281550E-03 1.972205
6.6500001E-02
<psi012*|p|psi001> 6.8347319E-03
5.4281550E-03 1.972205
6.6500001E-02
...
<psi039*|z|psi001> 1.0178294E-02
3.9452352E-05 21.81846 6.6500001E-02
<psi040*|z|psi001> 1.0178294E-02
3.9452352E-05 21.81846 6.6500001E-02
<psi039*|p|psi001> 1.9380630E-03
3.9452349E-05 21.81846
6.6500001E-02
<psi040*|p|psi001> 1.9380630E-03
3.9452349E-05 21.81846
6.6500001E-02
Sum rule of oscillator strength: f_psi001 = 0.9994023
Sum rule of oscillator strength: f_psi001 = 0.9994023
(The deviations from the single-band calculation are indicated
in red.)
We used:
schroedinger-kp-ev-solv
= LAPACK-ZHBGVX
! 'LAPACK', 'LAPACK-ZHBGVX',
'arpack',
'it_jam', 'chearn'
schroedinger-kp-discretization = box-integration
! 'finite-differences', 'box-integration'
|