[ Home ] [ News ] [ Contact ] [ Search ] [ nextnano.com ] 2D Quantum corral

 Download Software
 Copyright notice
 Publications

 GUI: nextnanomat
 Tool: nextnano
 Tool: nextnano++
 Tool: nextnano.QCL
 * password protected

 

 
Up
 

nextnano3 - Tutorial

next generation 3D nano device simulator

2D Tutorial

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

Author: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
 -> 2DQuantumCorral_nn3.in / *_nnp.in  - input file for the nextnano3 and nextnano++ software


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

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 (me =  0.026 m0). Strain is not taken into account.

 

Electron wave functions

The size of the quantum cluster is a circle of diameter 80 nm.

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

1st
2nd
3rd
4th
5th
6th
15th
20th
22th

 

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



One can clearly see 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

The parameters of the quantum corral are the following:
- radius: a = 40 nm
- me =  0.026 m0
- V(r) = 0 for r < a
- V(r) = infinity for r > a

The energies are: Enl = hbar2 / (2me a2) * jl,n2

which can be simplified to this formula: Enl = 0.0381 / (me a2) eV * jl,n2
In the latter formula, me is dimensionless and a in units of [nm]. (In the formula above, me is in units of [m0] and a in [m].)

There is a formula to approximate jl,n:  jl,n = ( n + 1/2  | l |  - 1/4 ) pi
which is accurate as n -> infinity.
However, it seems that this formula is not accurate enough to compare the results to the nextnano≥ results..
 

   Enl jl,n jl,n (approx.) [eV] (approx.) [eV] nextnano≥ [eV]
E1 =  E10 2.405 = 0.765 pi 0.75 pi 0.00508 0.00528 0.00511
E2 =  E11   1.25 pi 0.01412   0.01298
E3 =  E1-1   1.25 pi 0.01412   0.01298
E4 =  E12   1.75 pi 0.02768   0.02325
E5 =  E1-2   1.75 pi 0.02768   0.02325
E6 =  E20   1.75 pi 0.02768   0.02693
E7 =  E21   2.25 pi 0.04576   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)

  • Please help us to improve our tutorial. Send comments to support [at] nextnano.com.