5.5.1. Harmonic Oscillator
- Files for the tutorial located in nextnano++\examples\quantum_mechanics
quantum_confinement_1D_harmonic_oscillator.nnp
Introduction
1D quantum harmonic oscillator is described by a Hamiltonian
The second term corresponds to a potential energy of the particle \(V(x)\). Let us assume that we are describing an electron and this second term originates form an electrostatic potential \(\phi(x)\). Then \(V(x)=-\ElementaryCharge\phi(x)\), where \(-\ElementaryCharge\) is the charge of the electron, and
This allow us to easily define \(V(x)\) in the simulation by specifying \(\phi(x)\) in the section analytic_function{ } if the input file.
import{
analytic_function{
name = "potential"
function = "-x*x"
label = potential_label
}
}
Eigenenergies of the quantum harmonic oscillator state are given by
where \(n\) = 0, 1, 2, …
Note that the smallest possible energy is not zero, but \(\frac{1}{2}\hbar\omega\).
In the simulation we chose \(\omega = 1/\hbar\) such that allowed energies are simply \(E_n = n + 1/2\).
Warning
Further content is going to be replaced soon with the version consistent as described above.
Simulation Results
Figure 5.5.1.1 Potential energy profile of the harmonic oscillator with shifted (a) wave functions and (b) probability densities of 5 confined stated with the lowest energies.
The conduction band edge of the Gamma conduction band can be found in bias_00000\bandedge_Gamma.dat. In this simulation it directly corresponds to the potential energy of the electron. The file amplitudes_shift_quantum_region_Gamma.dat contains the eigenenergies and the wave functions (\(\Psi_n\)), while biass_00000\Quantum\quantum_harmonic_oscillator\Gamma\probabilities_shift_k00000.dat contains the eigenenergies and the squared wave functions (\(Psi^2_n\)) of all calculated states. Note that both \(\Psi_n\) and \(\Psi^2_n\) are shifted to their eigenenergies \(E_n\) so that they can be comfortably visualized together with the potential profile.
state number |
computed value (meV) |
analytical solution (meV) |
|---|---|---|
1 |
0.20 |
|
2 |
0.59 |
|
3 |
0.98 |
|
4 |
1.37 |
|
5 |
1.76 |
Making use of equation (5.5.1.1) with \(\omega_0 = \sqrt{C/m^*}\) (\(m^*\) = effective mass, \(C\) = constant which is related to the parabolic potential \(V(x) = 1/2 Kx^2\)), one can calculate \(E_n\) = \(\hbar \omega_0\):
\(\hbar \omega_0\) = 2\(E_1\) - 0 eV = 0.276 eV
\(\hbar \omega_0\) = \(E_2\) - \(E_1\) = 0.274 eV
\(\hbar \omega_0\) = \(E_3\) - \(E_2\) = 0.264 eV
Obviously, due to the finite \(\rm AlAs\) barrier that we have employed, the higher lying states deviate slightly from the analytical results where infinite barriers have been assumed.
Figure 5.5.1.2 shows the eigenenergies for the confined states \(E_1\), \(E_2\) and \(E_3\). As expected they are lying on a straight line because they are separated by \(\hbar\omega_0\). The figure is in perfect agreement with Fig. 3.12 (p. 84) of [HarrisonQWWD2005].
Figure 5.5.1.2 Eigenenergies for the three lowest states.
Matrix elements
The following matrix elements have been calculated:
overlap_integrals{}: This spatial overlap matrix elements \(\left\langle\psi_f|\psi_i\right\rangle\) simply returns the Kronecker delta as expected, because the wave functions are orthogonal.
momentum_matrix_elements{}: \(\left\langle\psi_f|p_x|\psi_i\right\rangle\) (see Tutorial on intraband transition)
dipole_moment_matrix_elements{}: \(\left\langle\psi_f|x|\psi_i\right\rangle\) (see Tutorial on intraband transition)
“Infinite” (30 eV) parabolic QW confinement for GaAs
Inputfile: 1DGaAs_ParabolicQW_infinite.nnp
Figure 5.5.1.3 shows the eigenstates of a parabolic quantum well (\(\rm GaAs\)) where the confinement is assumed to be 30 eV. Now up to 37 eigenstates are confined in the quantum well (grid resolution: 0.025 nm inside the well, 0.05 nm inside the barrier).
Figure 5.5.1.3 Calculated conduction band profile and probability densities (\(\Psi^2\)) for eigenstate \(n\) (\(n\) = 1, 2, …, 37).
Figure 5.5.1.4 shows the energies of the 37 confined electron states as a function of eigenstate \(n\). As expected, the curve shows a linear dependence because the eigenstates are equally spaced by \(\hbar\omega_0\) = 0.826 eV (where we used \(E_n\) = ( \(n\) - 1/2 ) \(\hbar\omega_0\)).
\(\hbar\omega_0\) = 2 \(E_1\) - 0 eV = 0.8261 eV |
\(E_1/(2 E_1)\) = 0.5000 |
\(\hbar\omega_0\) = \(E_2\) - \(E_1\) = 0.8260 eV |
\(E_2/(2 E_1)\) = 1.4999 |
\(\hbar\omega_0\) = \(E_3\) - \(E_2\) = 0.8260 eV |
\(E_2/(2 E_1)\) = 2.4997 |
\(\hbar\omega_0\) = \(E_4\) - \(E_3\) = 0.8259 eV |
\(E_4/(2 E_1)\) = 3.4994 |
\(\hbar\omega_0\) = \(E_5\) - \(E_4\) = 0.8259 eV |
\(E_5/(2 E_1)\) = 4.4991 |
\(\hbar\omega_0\) = \(E_6\) - \(E_5\) = 0.8258 eV |
\(E_6/(2 E_1)\) = 5.4987 |
\(\hbar\omega_0\) = \(E_7\) - \(E_6\) = 0.8258 eV |
\(E_7/(2 E_1)\) = 6.4982 |
\(\hbar\omega_0\) = \(E_8\) - \(E_7\) = 0.8257 eV |
\(E_8/(2 E_1)\) = 7.4978 |
Still, due to the “infinite” barrier of 30 eV (which is still a finite barrier) that we have employed, the higher lying states deviate slightly from the analytical results where infinite barriers have been assumed.
Figure 5.5.1.4 Eigenenergies of 37 eigenstates for an infinite parabolic QW.
One should bear in mind that the energy level spacing of such parabolic quantum wells is inversely proportional to both the well width and the square root of the effective mass.
It is also interesting to look at the momentum matrix elements, i.e. to investigate the probability for intersubband transitions. The relevant output is contained in these two files:
\bias_00000\Quantum\dipole_moment_matrix_elements_quantum_region_Gamma_100.txt - \(p_x\)
\bias_00000\Quantum\momentum_matrix_elements_quantum_region_Gamma_100.txt - \(x\)
From the calculated oscillator strengths it can be seen that only transitions from one level to the neighboring levels (+1 and -1) are allowed. Because in the case of a harmonic oscillator the momentum operator is proportional to the sum of the creation and the annihilation operators, thus only states can couple that have different occupation numbers with the difference equal to 1.
“Infinite” (30 eV) half-parabolic QW confinement for GaAs
Input file: 1DGaAs_ParabolicQW_infinite_half_nnpp.nnp
Figure 5.5.1.5 shows the eigenstates when taking only the right half of the parabolic quantum well (\(\rm GaAs\)) that has been calculated above. The confinement is 30 eV on the right and infinite confinement on the left (Dirichlet boundary conditions). Now only 18 eigenstates are confined in the quantum well, i.e. half the number of the eigenvalues compared with the full parabolic QW (grid resolution: 0.025 nm inside the well, 0.05 nm inside the barrier). The figure shows the conduction band profile and the square of the wave functions (\(\Psi_n^2\)) for eigenstate \(n\) (\(n\) = 1, 2, …, 18).
Figure 5.5.1.5 Calculated conduction band profile and probability densities (\(\Psi^2\)) for all confined eigenstates in an half-parabolic potential.
Again, the eigenstates are equally spaced. However, the separation energy is now twice as large as before, i.e. \(\hbar\omega_0\) = 2 \(\cdot\) 0.826 eV = 1.65 eV.
The ground state energy this time is given by: \(E_1\) = 3/2 \(\hbar\omega_0\)/2.
\(\hbar\omega_0\) = 4/3 \(E_1\) - 0 eV = 1.639 eV
\(\hbar\omega_0\) = \(E_2\) - \(E_1\) = 1.647 eV
\(\hbar\omega_0\) = \(E_3\) - \(E_2\) = 1.648 eV
\(\hbar\omega_0\) = \(E_4\) - \(E_3\) = 1.648 eV
It is also interesting to look at the momentum matrix elements, i.e. to investigate the probability for intersubband transitions. The relevant output is contained in these two files:
\bias_00000\Quantum\dipole_moment_matrix_elements_quantum_region_Gamma_100.txt - \(p_x\)
\bias_00000\Quantum\momentum_matrix_elements_quantum_region_Gamma_100.txt - \(x\)
Conclusion
We note that also more realistic parabolic quantum wells can be calculated with nextnano++. Assuming that the alloy profile is parabolic,
strain can be included (the strain tensor depends on the alloy profile),
as well as effective masses that depend on the alloy profile,
an 8-band k.p model (necessary to get correct intersubband transition energies)
and bowing parameters (especially important for \(AlGaAs\)).
All these features are automatically included in the nextnano++ code.
Last update: 2025-10-16