# 2D - 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.

## 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

• $$m_e = 0.026 m_0$$

• $$V(r) = 0$$ for $$r < a$$

• $$V(r) = \infty$$ for $$r > a$$

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

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

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 $$=|l|$$

## 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 $$|l|\ge1$$. 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)