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 ($m_e=0.026m_0$).

• Strain is not taken into account.

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, $(n,l)=(1,0)$

  

• 2nd eigenstate, $(n,l)=(1,1)$

  

• 3rd eigenstate, $(n,l)=(1,-1)$

  

• 4th eigenstate, $(n,l)=(1,2)$

  

• 5th eigenstate, $(n,l)=(1,-2)$

  

• 6th eigenstate, $(n,l)=(2,0)$

  

• 15th eigenstate, $(n,l)=(3,0)$

  

• 20th eigenstate, $(n,l)=(1,6)$

  

• 22th eigenstate, $(n,l)=(3,1)$

  

The parameters of the quantum corral are the followings:

• radius: $a=40$ nm

• $m_e=0.026m_0$

• $V(r)=0$ for $r<a$

• $V(r)=\mathrm{\infty }$ for $r>a$

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

(2.4.45)$${\psi }_{n,l}(r,\theta )\propto J_l\left(\frac{j_{l,n}r}{a}\right)[A\cos (l\theta )+B\sin (l\theta )]$$

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: $E_{nl}=\frac{{\hslash }^2{j}_{l,n}^2}{2m_e{a}^2}$

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 $=|l|$

Figure 2.4.164 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.)

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 $|l|\ge 1$. On the other hand, the non-degenerate energy eigenvalues corresponds to $l=0$

The analytical energy values are: $E_{nl}=\frac{{\hslash }^2{j}_{l,n}^2}{2m_e{a}^2}$.

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 $n\to \mathrm{\infty }$.

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)

---

Last update: nn/nn/nnnn