nextnano³ - Tutorial

2D Tutorial

Fock-Darwin states of a 2D parabolic, anisotropic (elliptical) potential in a magnetic field

Author: Stefan Birner

==> This is the old website: A new version of this tutorial can be found here.

 -> 1DGaAs_ParabolicQW_infinite_4_6meV.in
 -> 1DGaAs_ParabolicQW_infinite_6_1meV.in
 -> 2DGaAs_BiParabolicEllipticQD_Austing.in
 

Fock-Darwin states of a 2D parabolic, anisotropic (elliptical) potential in a magnetic field

In this tutorial we study the electron energy levels of a two-dimensional parabolic, anisotropic (elliptical) confinement potential that is subject to a magnetic field.
Such a potential can be constructed by surrounding GaAs with an Al_xGa_1-xAs alloy that has a parabolic alloy profile in the x and y directions.

First, it is necessary to study the energy states of a 1D parabolic confinement.

1D parabolic confinement along the x direction with h_barw₀ = 4.6 meV (1D simulation)

 -> 1DGaAs_ParabolicQW_infinite_4_6meV.in

For similar results and a discussion, we refer to this tutorial: Parabolic Quantum Well (GaAs / AlAs)

1D parabolic confinement along the y direction with h_barw₀ = 6.1 meV (1D simulation)

 -> 1DGaAs_ParabolicQW_infinite_6_1meV.in

For similar results and a discussion, we refer to this tutorial: Parabolic Quantum Well (GaAs / AlAs)

First, it is a good idea to get familiar with the results of a 2D parabolic and isotropic confinement: Fock-Darwin states of a 2D parabolic potential in a magnetic field

Now we turn to the anisotropic confinement...

2D parabolic, anisotropic (elliptical) confinement with h_barw_x = 4.6 meV and and h_barw_y = 6.1 meV -
Fock-Darwin-like spectrum  (2D simulation)

 -> 2DGaAs_BiParabolicEllipticQD_Austing.in
 

• The electron effective mass in GaAs is m_e* = 0.067 m₀. We assume this value for the effective mass in the whole region (i.e. also inside the AlGaAs alloy).
   
• The following figure shows the parabolic, anisotropic (elliptical) conduction band edge confinement potential, as well as the ground state wave function (psi²) at B = 0 T.
  In the middle of the sample the conduction band edge is at 0 eV and at the boundary region the conduction band edge has the value 0.84 eV. The radii of the ellipse are 300 nm along the x axis and 226 nm along the y axis.
  The parabolic confinement along the x direction is: h_barw_x = 4.6 meV
  The parabolic confinement along the y direction is: h_barw_y = 6.1 meV
  Thus the ellipticity is roughly 4/3.
  
  
   

At zero magnetic field, the eigenvalues for such a system are given by:
    E_nx,ny = (n_x + 1/2) h_barw_x + (n_y + 1/2) h_bar w_y
    n_x = n + 1/2 |l| - 1/2 |l|
    n_y = n + 1/2 |l| + 1/2 |l|
          for n = 0,1,2,3,... and l = 0,+-1,+-2,...
(n = radial quantum number, l = angular momentum quantum number, w_x and w_y = oscillator frequencies)
For more details, see A.V. Madhav, T. Chakraborty, Physical Review B 49, 8163 (1994).

The eigenvalue spectrum of a 2D parabolic and isotropic potential shows a shell-like structure: Energy levels of an "artificial atom" - 2D harmonic potential
For the anisotropic elliptical potential, this degeneracy at B = 0 T is lifted.

The following figure shows the calculated Fock-Darwin-like spectrum, i.e. the eigenstates as a function of magnetic field magnitude.
Here, each of these states is two-fold spin-degenerate. However, a magnetic field lifts this degeneracy (Zeeman splitting) but this effect is not taking into account in this tutorial.

• Such a spectrum can be related to experimental transport measurements which give insight into the single-particle energy spectrum of a quantum dot.
  
  The rectangles in the above figure are related to the figures of the following publications:
  
  cyan rectangle: Fig. 2 of
       Two-level anti-crossings high up in the single-particle energy spectrum of a quantum dot
       C. Payette, D.G. Austing, G. Yu, J.A. Gupta, S.V. Nair, B. Partoens, S. Amaha, S. Tarucha
       arXiv:0710.1035v1 [cond-mat.mes-hall] (2007)
  
  green rectangle: Fig. 2(b) of
  red rectangle: Fig. 3(a) of
       Probing by transport the single-particle energy spectrum up to high energy of
       one quantum dot with the ground state of an adjacent weakly coupled quantum dot
       D.G. Austing, G. Yu, C. Payette, J.A. Gupta, M. Korkusinski, G.C. Aers
       physica status solidi (a), 508 (2007)
  
  (Comments red rectangle: In Fig. 3(a) of the publication by Austing et al., the ground state energy has been subtracted from the excited states. Thus the slope of the energy spectrum look slightly different.)
  
  It is interesting to note that there are exact crossings in the calculated spectrum whereas the experiment reveals anti-crossings.