nextnano^{3}  Tutorial
next generation 3D nano device simulator
3D Tutorial
Energy levels of an "artificial atom"  Spherical and ellipsoidal CdSe
Quantum Dot
Author:
Stefan Birner
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> 3DsphericCdSeQD5nm_ArtificialAtom.in 
spherical CdSe QD with radius 5 nm
> 3DsphericCdSeQD_ArtificialAtom_radius.in 
spherical CdSe QD with radius as a parameter
> 3DellipsoidalCdSeQD5nm5nm10nm_ArtificialAtom.in 
ellipsoidal, cigarshaped CdSe QD with radii 5 nm, 5nm and 10 nm
> 2DGaAs_BiParabolicQW_3meV.in 
2D harmonic potential
Energy levels of an "artificial atom"  Spherical CdSe Quantum Dot
> 3DsphericCdSeQD5nm_ArtificialAtom.in 
spherical CdSe QD with radius 5 nm
Here, we want to calculate the energy levels and the wave functions of a
spherical CdSe
quantum dot of radius 5 nm.
We assume that the barriers at the QD boundaries are infinite.
The potential inside the QD is assumed to be 0 eV.
We use a grid resolution of 0.5 nm.
We solve the singleband Schrödinger equation within the effectivemass
approximation.
The electron effective mass of CdSe is assumed to be m_{e}
= 0.112 m_{0}.
conductionbandmasses = 0.026d0 0.026d0 0.026d0 ! electron
effective mass at Gamma conduction band
...
A spherically symmetric potential leads to an energy spectrum where some
eigenvalues are degenerate.
We want to study the "shell structure" (degeneracy scheme) of a CdSe quantum dot
of radius 5 nm.
The following figure shows the calculated energy spectrum for the lowest 20
electron eigenvalues.
One can clearly identify the shell structure 1s, 2p, 3d, 2s, 4f and 3p which is
similar to the shell structure of the periodic table.
This is the reason why quantum dots are often called "artificial atoms".
Note that each eigenstate is twofold degenerate due to spin.
Thus the s states are twofold degenerate, the p
states are sixfold degenerate, the d states are tenfold
degenerate and the degeneracy of the f states is 14.
We have also solved the singleband Schrödinger equation for the holes
assuming an isotropic effective mass for simplicity.
Obviously, this is a crude approximation.
From the electron and hole wave functions, we calculate their spatial overlap
matrix elements (interband matrix elements).
In this simple model, due to symmetry arguments, only the following transitions
are allowed:
1s  1s, 2p  2p, 3d  3d, 2s  2s, 4f  4f, ...
The following figure shows the calculated interband matrix elements as a
function of energy.
(Note: The figure has to be updated: Now we output the square of this matrix
element.)
Both figures are in reasonable agreement with Fig. 1 and Fig. 2 (inset) of the
following paper:
Optical Properties of Ellipsoidal CdSe Quantum Dots
W.S. Ferreira, J.S. de Sousa, J.A.K. Freire, G.A. Farias, V.N. Freire
Brazilian Journal of Physics 36, 438 (2006)
Energy levels of an "artificial atom"  Ellipsoidal, cigarshaped CdSe
quantum dot
> 3DellipsoidalCdSeQD5nm5nm10nm_ArtificialAtom.in 
ellipsoidal, cigarshaped CdSe QD with radii 5 nm, 5nm and 10 nm
For an ellipsoidal, cigarshaped CdSe quantum dot (r_{x} = 5 nm, r_{y}
= 5 nm, r_{z} = 10 nm), we calculate the lowest 30 eigenvalues.
The energy spectrum (degeneracy spectrum) looks very different from the
spherical QD spectrum.
The interband matrix elements are shown in this figure:
(Note: The figure has to be updated: Now we output the square of this matrix
element.)
Energy levels of an "artificial atom"  2D harmonic potential
> 2DGaAs_BiParabolicQW_3meV.in 
2D harmonic potential
The following figure shows the energy spectrum of a "twodimensional disc" which
we approximate as a cylindrically symmetric parabolic (harmonic) potential.
We solve the 2D Schrödinger equation for this system.
The harmonic potential is assumed to be h_{bar}w = 3 meV. Each shell is
thus separated by 3 meV.
From the energy spectrum of this twodimensional shell structure, one can derive
"magic numbers". (They include spin
degeneracy.)
See also this tutorial for twodimensional parabolic confinement:
FockDarwin states
