2.2. Electron wave functions in a cylindrical well (2D Quantum Corral)

In this tutorial we demonstrate 2D simulation of a cilindrical quantum well. We will see the electron eigenstates and their degeneracy.

Input files used in this tutorial are the followings:

  • 2DQuantumCorral_nn3.in / *_nnp.in


Structure

  • A cylindrical InAs quantum well (diameter 80 nm) is surrounded by a cylindrical GaAs barrier (20 nm) which is surrounded by air. The whole sample is 160 nm x 160 nm.

  • We assume infinite GaAs barriers. This can be achieved by a circular quantum cluster with Dirichlet boundary conditions, i.e. the wave function is forced to be zero in the GaAs barrier.

  • The electron mass of InAs is assumed to be isotropic and parabolic (\(m_e = 0.026 m_0\)).

  • Strain is not taken into account.

../../_images/2DQuantumCorral_material_grid.png

Simulation outcome

Electron wave functions

The size of the quantum cluster is a circle of diameter \(2a=80\) nm.

The following figures shows the square of the electron wave functions (i.e. \(\psi^2\)) of the corresponding eigenstates. They were calculated within the effective-mass approximation (single-band) on a rectangular finite-differences grid.

  • 1st eigenstate, \((n,\ l)=(1,\ 0)\)

    ../../_images/wave1.png
  • 2nd eigenstate, \((n,\ l)=(1,\ 1)\)

    ../../_images/wave2.png
  • 3rd eigenstate, \((n,\ l)=(1,-1)\)

    ../../_images/wave3.png
  • 4th eigenstate, \((n,\ l)=(1,\ 2)\)

    ../../_images/wave4.png
  • 5th eigenstate, \((n,\ l)=(1,-2)\)

    ../../_images/wave5.png
  • 6th eigenstate, \((n,\ l)=(2,\ 0)\)

    ../../_images/wave6.png
  • 15th eigenstate, \((n,\ l)=(3,\ 0)\)

    ../../_images/wave15.png
  • 20th eigenstate, \((n,\ l)=(1,\ 6)\)

    ../../_images/wave20.png
  • 22th eigenstate, \((n,\ l)=(3,\ 1)\)

    ../../_images/wave22.png

The parameters of the quantum corral are the followings:

  • radius: \(a = 40\) nm

  • \(m_e = 0.026 m_0\)

  • \(V(r) = 0\) for \(r < a\)

  • \(V(r) = \infty\) for \(r > a\)

The analytical solution of the eigenstates of this quantum well is:

(2.2.1)\[\psi_{n,l}(r,\theta) \propto J_{l}\left(\frac{j_{l,n}r}{a}\right)\left[A\cos(l\theta)+B\sin(l\theta)\right]\]

where

  • \(J_{l}(x)\) is the Bessel function of the first kind (We cite them for \(l=0,1,2\) below.)

  • \(j_{l,n}\) is its zero point i.e. \(J_l(j_{l,n})=0\) and \(n=1,2,...\)

  • \(A,B\) are constant

  • \(l=0, \pm 1, \pm 2, ...\)

The corresponding eigenenergies are: \(E_{nl} = \frac{\hbar^2j_{l,n}^2}{2m_e a^2}\)

The Quantum number \(n\) comes from the boundary condition \(\psi(a, \theta)=0\). The requirement that \(\psi\) has the same value at \(\theta=0\) and \(2\pi\) leads to the quantum number \(l\). In the above figures of the eigenstates, we can know them through the following relations:

  • (the number of zero points in the radial direction) \(=n\)

  • (the number of zero points in the circumferential direction)/2 \(=|l|\)

../../_images/bessels.png

Figure 2.2.6 Bessel functions of the first kind for \(l=0,1,2\) generated by scipy.


Energy spectrum

The following figure shows the energy spectrum of the quantum corral. (The zero of energy corresponds to the InAs conduction band edge.)

../../_images/energy_levels.jpeg

The two-fold degeneracies of the states

  • (2, 3), (4, 5), (7, 8), (9, 10), (11, 12), (13, 14), (16, 17), (18, 19), (20, 21), (22, 23), (24, 25), (26, 27), (28, 29), (31, 32), (33, 34), (35, 36), (37, 38), (39, 40)

correponds to \(|l|\ge1\). On the other hand, the non-degenerate energy eigenvalues corresponds to \(l=0\)

The analytical energy values are: \(E_{nl} = \frac{\hbar^2j_{l,n}^2}{2m_e a^2}\).

There is a formula to approximate \(j_{l,n}\): \(j_{l,n} = ( n + \frac{1}{2} | l | - \frac{1}{4} ) \pi\) which is accurate as \(n \rightarrow \infty\).

Here we describe the comparison between the analytical values, approximate values, nextnano++ results and nextnano³ results.

\([n,l]\)

\(j_{l,n}\)

\(j_{l,n}\) (approx.)

\(E_{n,l}\) [eV]

\(E_{n,l}\) [eV] (approx.)

\(E_{n,l}\) [eV] (nextnano++)

\(E_{n,l}\) [eV] (nextnano³)

1st

[1, 0]

2.405

0.75\(\pi\simeq\)2.356

0.00530

0.00508

0.00510

0.00511

2nd

[1, 1]

3.832

1.25\(\pi\simeq\)3.926

0.01345

0.01412

0.01294

0.01298

3rd

[1,-1]

3.832

1.25\(\pi\simeq\)3.926

0.01345

0.01412

0.01294

0.01298

4th

[1, 2]

5.136

1.75\(\pi\simeq\)5.497

0.02416

0.02768

0.02320

0.02325

5th

[1,-2]

5.136

1.75\(\pi\simeq\)5.497

0.02416

0.02768

0.02329

0.02325

6th

[2, 0]

5.520

1.75\(\pi\simeq\)5.497

0.02791

0.02767

0.02685

0.02693

7th

[2, 1]

7.016

2.25\(\pi\simeq\)7.067

0.04508

0.04574

0.03584

0.03597


Further details about the analytical solution of the cylindrical quantum well with infinite barriers can be found in:

The Physics of Low-Dimensional Semiconductors - An Introduction
John H. Davies
Cambridge University Press (1998)

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