5.5.2. Infinite quantum well

Changed in version 2.4.12.

Files for the tutorial located in nextnano++\examples\quantum_mechanics

quantum_confinement_1D_infinite_well.nnp

Scope:

Electron in vacuum, wave functions, energies, momentum matrix elements, dipole matrix elements, and oscillator strengths

Eigenstates and wave functions

Wave functions of an electron \(\psi_\mathrm{n}\left(x\right)\) confined in an infinite quantum well potential with thickness \(L\) are given by

\[\psi_\mathrm{n}\left(x\right) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}\right),\quad\text{where}\;n=1,2,3,...\]

with eigenenergies

\[E_\mathrm{n} = \frac{n^2\hbar^2}{2 m_0 L^2},\]

where \(m_0\) is the mass of a stationary electron.

Let us choose \(L\) such that \(E_1 = 1 eV\). Then the allowed energies are expressed simply as \(E_n = n^2\) and \(L\approx 0.6132\;\text{nm}\).

Table 5.5.2.1 Comparison of eigenenergies obtained in numerical simulation and calculated analytically.
n	computed values (eV)	analytical solutions (eV)	relative error
1	0.999778988	1	\(2.2\cdot10^{-4}\)
2	3.996464719	4	\(8.8\cdot10^{-4}\)
3	8.982110520	9	\(2.0\cdot10^{-3}\)
4	15.94349536	16	\(3.5\cdot10^{-3}\)
5	24.86215888	25	\(5.5\cdot10^{-3}\)

The computed eigenvalues can be found in bias_00000\Quantum\quantum_well\Gamma\energy_spectrum_k00000.dat. Eigenfunctions and are stored in bias_00000\Quantum\quantum_well\Gamma\amplitudes_k00000.dat.

Note

Better solutions can be obtained by reducing the grid spacing.

Attention

The grid spacing can be reduced only down to 1e-3 nm, as it is more that sufficient for simulations of electrons in semiconductors.

Momentum matrix elements

To evaluate probabilities of optical due to absorption of a photon flying in the plane of the quantum well with electric field oscillating perpendicular to it (the x direction), one should calculate momentum matrix elements.

(5.5.2.1)\[\MatMomentum_\mathrm{ij} = \bra{\psi_{\mathrm{i}}}\hat{p}_x\ket{\psi_{\mathrm{j}}} = \int_{-\infty}^{\infty} \psi_\mathrm{i}^* (x) \hat{p}_x \psi_\mathrm{j} (x) dx = - i \hbar \int_{-\infty}^{\infty} \psi_\mathrm{i}^* (x) \frac{\partial}{\partial x} \psi_\mathrm{j} (x) dx.\]

Note

Detailed definitions covering for crystals can be found in envelope momentum matrix elements.

The wave functions in this case are either even or odd. As the derivatives change the parity of any function they act upon, the integral (5.5.2.1) is non zero only for wave functions of different parities, e.g., \(\MatMomentum_{13}=\MatMomentum_{15}=\MatMomentum_{25}=0\) but \(\MatMomentum_{12}=\MatMomentum_{23}=\MatMomentum_{14}\neq 0\).

In the case of an infinite quantum well, the momentum matrix elements can be calculated analytically as

\[\begin{split}\begin{aligned} |\MatMomentum_{12}| &= |\MatMomentum_{21}| = \frac{8}{3}\frac{\hbar}{L} \\ |\MatMomentum_{14}| &= |\MatMomentum_{41}| = \frac{16}{15}\frac{\hbar}{L} \\ |\MatMomentum_{23}| &= |\MatMomentum_{32}| = \frac{24}{5}\frac{\hbar}{L} \\ |\MatMomentum_{25}| &= |\MatMomentum_{52}| = \frac{40}{21}\frac{\hbar}{L} \\ |\MatMomentum_{34}| &= |\MatMomentum_{43}| = \frac{48}{7}\frac{\hbar}{L} \\ |\MatMomentum_{45}| &= |\MatMomentum_{54}| = \frac{80}{9}\frac{\hbar}{L} \\ \end{aligned}\end{split}\]

Table 5.5.2.2 Comparison of selected non-zero momentum matrix elements obtained in numerical simulation and calculated analytically.
\(\psi_i\)	\(\psi_j\)	computed \(\|\MatMomentum_{ij}\|\) (\(\hbar/\text{nm}\))	analytical solution (\(\hbar/\text{nm}\))	relative error
1	2	4.34387	4.348679405	\(1.1\cdot10^{-3}\)
1	4	1.73293	1.739471762	\(3.8\cdot10^{-3}\)
2	3	7.80515	7.827622928	\(2.9\cdot10^{-3}\)
2	5	3.08629	3.106199575	\(6.4\cdot10^{-3}\)
3	4	11.12060	11.18231847	\(5.5\cdot10^{-3}\)
4	5	14.36460	14.49559802	\(9.0\cdot10^{-3}\)

The computed momentum matrix elements can be found in bias_00000\Quantumquantum_well\Gamma_Gamma\momentum_matrix_elements_k00000.*.

Oscillator Strengths

Momentum matrix elements can be further used to evaluate dimensionless oscillator strengths, measuring probability of absorption and emission of a photon.

\[f_{ij} = \frac{2}{m_0}\cdot \frac{| \MatMomentum_{ij} |^2}{E_\mathrm{j} - E_\mathrm{i}}\]

In the case of an infinite quantum well one can get \(f_{21} = 256 m_{eff} /27\mathrm{\pi}^2 \approx 0.063885\) which is independent of the well width.

In the case of an infinite quantum well, the matrix elements can be calculated analytically as

\[\begin{split}\begin{aligned} f_{12} &= -f_{21} = \frac{256}{27\pi^2} \\ f_{14} &= -f_{41} = \frac{1024}{3375\pi^2} \\ f_{23} &= -f_{32} = \frac{2304}{125\pi^2} \\ f_{25} &= -f_{52} = \frac{6400}{9261\pi^2} \\ f_{34} &= -f_{43} = \frac{9216}{343\pi^2} \\ f_{45} &= -f_{54} = \frac{25600}{729\pi^2} \\ \end{aligned}\end{split}\]

Table 5.5.2.3 Comparison of selected non-zero oscillator strengths obtained in numerical simulation and calculated analytically.
\(\psi_i\)	\(\psi_j\)	computed \(f_{ij}\)	analytical solution	relative error
1	2	0.9596130	0.960674926	\(1.1\cdot10^{-3}\)
1	4	0.0306258	0.030741598	\(3.8\cdot10^{-3}\)
2	3	1.8621900	1.867552057	\(2.9\cdot10^{-3}\)
2	5	0.0695704	0.070020038	\(6.4\cdot10^{-3}\)
3	4	2.7073500	2.722379092	\(5.5\cdot10^{-3}\)
4	5	3.5258800	3.558055283	\(9.0\cdot10^{-3}\)

The computed momentum matrix elements can be found in bias_00000\Quantum\quantum_well\Gamma_Gamma\oscillator_strengths_k00000.*.

Dipole moment

In some cases dipole moment is preferred to be used to analyze optical transitions.

(5.5.2.2)\[\MatDipoleMoment_\mathrm{ij} = \bra{\psi_{\mathrm{i}}}\hat{d}\ket{\psi_{\mathrm{j}}} = \int_{-\infty}^{\infty} \psi_\mathrm{i}^* (x) \ElementaryCharge x \psi_\mathrm{j} (x) dx,\]

where \(\ElementaryCharge\) is the elementary charge.

In the case of an infinite quantum well, the dipole matrix elements can be calculated analytically as

\[\begin{split}\begin{aligned} |\MatDipoleMoment_{12}| &= |\MatDipoleMoment_{21}| = \frac{16}{9}\frac{L}{\pi^2} \\ |\MatDipoleMoment_{14}| &= |\MatDipoleMoment_{41}| = \frac{32}{225}\frac{L}{\pi^2} \\ |\MatDipoleMoment_{23}| &= |\MatDipoleMoment_{32}| = \frac{48}{25}\frac{L}{\pi^2} \\ |\MatDipoleMoment_{25}| &= |\MatDipoleMoment_{52}| = \frac{80}{441}\frac{L}{\pi^2} \\ |\MatDipoleMoment_{34}| &= |\MatDipoleMoment_{43}| = \frac{96}{49}\frac{L}{\pi^2} \\ |\MatDipoleMoment_{45}| &= |\MatDipoleMoment_{54}| = \frac{160}{81}\frac{L}{\pi^2} \\ \end{aligned}\end{split}\]

Table 5.5.2.4 Comparison of selected non-zero electric dipole moment matrix elements obtained in numerical simulation and calculated analytically.
\(\psi_i\)	\(\psi_j\)	computed \(\|\MatMomentum_{ij}\|\) (\(\hbar/\text{nm}\))	analytical solution (\(\hbar/\text{nm}\))	relative error
1	2	0.11045600	0.110455938	\(-5.6\cdot10^{-7}\)
1	4	0.00883642	0.008836475	\(+6.2\cdot10^{-6}\)
2	3	0.11929200	0.119292413	\(+3.5\cdot10^{-6}\)
2	5	0.01127090	0.011271014	\(+1.0\cdot10^{-5}\)
3	4	0.12172700	0.121726952	\(-3.9\cdot10^{-7}\)
4	5	0.12272900	0.122728820	\(-1.5\cdot10^{-6}\)

The computed momentum matrix elements can be found in bias_00000\Quantum\quantum_well\Gamma_Gamma\dipole_moment_matrix_elements_k00000.*.

Last update: 2025-10-27