Absorption of a GaAs spherical quantum dot
This tutorial calculates the optical absorption spectrum of a GaAs spherical quantum dot with infinite barriers. We will see which output file we should refer to in order to understand the absorption spectrum.
Also, the formula used for the absorption calculation is presented. For the detailed scheme of the calculation of the optical matrix elements and absorption spectrum, please see our 1D optics tutorial: Optical absorption for interband and intersubband transitions
Input file:
3Dspherical_infinite_dot_GaAs_intra_nnp.in
3Dspherical_infinite_dot_GaAs_inter_nnp.in
Structure

Figure 2.4.381 Left: GaAs region as a spherical quantum dot. Right: Slice of the Gamma band edge along
The above figures show the Gamma band edge of the spherical GaAs region and the barrier region.
We model the infinite barrier by assigning 100 eV for the band edge of AlAs barrier region from database{ }
section.
Please see the input file for the details.
The parameters used in this simulation are as follows.
Property |
Symbol |
Value [unit] |
---|---|---|
quantum dot radius |
5 [nm] |
|
barrier height |
92 [eV] |
|
effective electron mass |
0.0665 |
|
refractive index |
3.3 |
|
doping concentation (n-type) |
8 |
|
linewidth (FWHM) |
0.01 [eV] |
|
temperature |
300 [K] |
Scheme
The run{ }
section is specified as follows:
run{
poisson{ }
quantum{ }
quantum optics{ }
}
Then the simulation follows these steps:
Poisson equation is solved with the setting specified in the poisson{ } section.
“Schrödinger” equation is solved with the setting specified in the quantum{ } section.
“Schrödinger” equation is solved again with the setting specified in the optics{ } section and optical properties are calculated.
Note
If
quantum_poisson{ }
is specified instead ofquantum{ }
, Poisson and Schrödinger equations are solved self-consistently.optics{ }
requires that kp8 model is used in the quantum region specified inquantum{ }
.In this tutorial the kp parameters are adjusted so that the conduction and valence bands are decoupled from each other. Thus the single-band Schrödinger equations are solved effectively by the kp solver.
The optical absorption accompanied by the excitation of charge carriers (state
where
is the energy of eigenstate . The first sum runs over the pair of states where . is the occupation of eigestate . is the optical polarization vector defined in optics{ quantum_spectra{ polarization{ } } }. where is the canonical momentum operator and is the contribution of spin-orbit interaction. .we call
as the optical matrix elements. is the energy broadening function:When
energy_broadening_lorentzian
is specified in optics{ quantum_spectra{ energy_broadening_lorentzian } },where
is the FWHM defined byenergy_broadening_lorentzian
.When
energy_broadening_gaussian
is specified in optics{ quantum_spectra{ energy_broadening_gaussian } },where
energy_broadening_lorentzian
defines the FWMHWhen neither
energy_broadening_lorentzian
norenergy_broadening_gaussian
is specified in optics{ quantum_spectra{ } }, is replace by the delta function .It is also possible to include both Lorentzian and Gaussian broadening (Voigt profile).
The detailed calculation scheme of the optical matrix elements
Results
Absorption

Figure 2.4.382 Calculated absorption spectrum
Figure 2.4.382 shows the calculated
Note
When we use the realistic k.p paramters,
They are identical in this tutorial since the single-band model is emulated.
Eigenvalues, transition energies, and occupations

Figure 2.4.383 Calculated energy spectrum and Fermi energy (=0 eV).
Figure 2.4.383 shows the calculated energy eigenvalues specified in \Quantum\energy_spectrum_~.dat.
Please note that the output in Quantum\ counts the eigenstates with different spins individually when k.p model is used, while they are counted jointly in Optics\.
Comparing the excitation energy of other upper states to
We can see the peak energy of P1 in Figure 2.4.382 corresponds to the transition energy from the ground states (no. 1 and 2) to the 1st excited states (no. 3,4,5,6,7 and 8).
Note
The eigenstates with different spins are counted individually in Quantum\ when k.p model is used, while they are counted jointly in Optics\.
For example, the two ground states counted as no.1 and 2 in Figure 2.4.383 due to spin are put together as one eigenstate in Optics\.
From the above data of eigenvalues, we could see which pair of states contributes to the peak in the absorption spectrum Figure 2.4.382.
In order to understand why some pairs of states do not appear as peaks, we will see the output data for
Transition intensity (Momentum matrix element)
One of the key element for the calculation of optical absorption is the transition intensity
which has the dimension of energy [eV].
The intensity
Energy[eV] From To Intensity_k0[eV] 1/Radiative_Rate[s]
0.233098 1 2 2.02882 4.43013e-08
0.233098 1 3 2.42777 3.70214e-08
0.233098 1 4 2.30413 3.90079e-08
The transtions from 1 to 5~10 are zero and these pairs of states do not contribute to the absorption (They are omitted here since Intensity_k0
are too small).
Eigenstates
The probability distribution of eigenfunctions
Here the probability distribution of eigenfunctions calculated by single-band model are shown.

Figure 2.4.384 |wave function|

Figure 2.4.385 |wave function|

Figure 2.4.386 |wave function|

Figure 2.4.387 |wave function|
|wave function|
Last update: nn/nn/nnnn