nextnano³ - Tutorial

2D Tutorial

Fock-Darwin states of a 2D parabolic potential in a magnetic field

Author: Stefan Birner.

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

 -> 2DGaAs_BiParabolicQW_4meV_GovernalePRB1998.in
 -> 2DGaAs_BiParabolicQW_3meV_FockDarwin.in
 

Fock-Darwin states of a 2D parabolic potential in a magnetic field

In this tutorial we study the electron energy levels of a two-dimensional parabolic 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,y) plane.

• The motion in the z direction is not influenced by the magnetic field and is thus that of a free particle with energies and wave functions given by:
     E_z = h_bar² k_z² / (2 m*)
     psi(z) = exp (+- i k_z z)
  For that reason, we do not include the z direction into our simulation domain, and thus only simulate in the (x,y) plane (two-dimensional simulation).

2D parabolic confinement with h_barw₀ = 4 meV

 -> 2DGaAs_BiParabolicQW_4meV_GovernalePRB1998.in

First, we want to benchmark the nextnano³ code to some other numerical calculation:
This input file aims to reproduce the figures 1, 2, 3 and 4 of

    M. Governale, C. Ungarelli
    Gauge-invariant grid discretization of the Schrödinger equation
    Phys. Rev. B 58 (12), 7816 (1998).
 

• The GaAs sample extends in the x and y directions (i.e. this is a two-dimensional simulation) and has the size of 240 nm x 240 nm.
  At the domain boundaries we employ Dirichlet boundary conditions to the Schrödinger equation, i.e. infinite barriers.
  The grid is chosen to be rectangular with a grid spacing of 2.4 nm, in agreement with Governale's paper.
   
• The magnetic field is oriented along the z direction, i.e. it it perpendicular to the simulation plane which is oriented in the (x,y) plane).
  We calculate the eigenstates for different magnetic field strengths (1 T, 2 T, ..., 20 T), i.e. we make use of the magnetic field sweep.
  
    $magnetic-field
   magnetic-field-on                    = yes
   magnetic-field-strength              = 0.0d0   ! 1 Tesla = 1 Vs/m²
   magnetic-field-direction             = 0 0 1   ! [001] direction
   magnetic-field-sweep-active          = yes     !
   magnetic-field-sweep-step-size       = 0.5d0   ! 0.5 Tesla = 0.5 Vs/m²
   magnetic-field-sweep-number-of-steps = 40      ! 40 magnetic field sweep steps
  $end_magnetic-field
   
• A useful quantitiy is the magnetic length (or Landau magnetic length) which is defined as:
      l_B = [h_bar / (m_e* w_c)]^1/2 = [h_bar / (|e| B)]^1/2
  It is independent of the mass of the particle and depends only on the magnetic field strength:
  -  1 T: 25.6556 nm
  -  2 T: 18.1413 nm
  -  3 T: 14.8123 nm
  - ...
- 20 T:   5.7368 nm
   
• 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).
  Another useful quantity is the cyclotron frequency:
      w_c = |e| B / m_e*
  Thus for the electrons in GaAs, it holds for the different magnetic field strengths:
  -  1 T: h_barw_c =   1.7279 meV
  -  2 T: h_barw_c =   3.4558 meV
  -  3 T: h_barw_c =   5.1836 meV
  -  ...
- 20 T: h_barw_c = 34.5575 meV
• The two-dimensional parabolic confinement (conduction band edge confinement) is chosen so that the electron ground state has the following energy: E₁ = h_barw₀ = 4 meV   (without magnetic field)
• The following figure shows the lowest fifteen eigenvalues for a magnetic field magnitude of B = 10 T. It is in perfect agreement with Fig. 1 of Governale's paper. The ground state has the energy E₁ = 9.44 meV (at B = 10 T).
  
  
   
• The following figure shows the ground state probability density (psi²) for a magnetic field magnitude of B = 10 T. It is in perfect agreement with Fig. 2(a) of Governale's paper.
  The ground state has the energy E₁ = 9.44 meV (at B = 10 T).
  The left, vertical axis shows psi² in units of nm^-2 (the peak value is 0.00267 nm^-2).
  
  The parabolic conduction band edge confinement potential is also shown.
  The horizontal axis shows the colormap of the conduction band edge values. In the middle of the sample the conduction band edge is 0 eV, and at the boundary region, the conduction band edge has the value 1.0092 eV.
  
  
   
• The following figure shows the probability density (psi²) of the 14th excited state (i.e. E₁₅) for a magnetic field magnitude of B = 10 T. It is in perfect agreement with Fig. 3(a) of Governale's paper.
  Its energy is E₁₅ = 21.72 meV (at B = 10 T).
  The left, vertical axis shows psi² in units of nm^-2 (the peak value is 0.000284 nm^-2).
  
  The parabolic conduction band edge confinement potential is also shown.
  The horizontal axis shows the colormap of the conduction band edge values. In the middle of the sample the conduction band edge is 0 eV, and at the boundary region, the conduction band edge has the value 1.0092 eV.
  
  
   
• The following figure shows the ground state energy as a function of magnetic field magnitude. It is in perfect agreement with Fig. 4 of Governale's paper. The ground state has the energy E₁ = 4.01 meV at B = 0 T.
  
  

2D parabolic confinement with h_barw₀ = 3 meV - Fock-Darwin spectrum

 -> 2DGaAs_BiParabolicQW_3meV_FockDarwin.in

Here, we calculate the single-particle states of a two-dimensional harmonic oscillator.
The eigenvalues for such a system are given by
    E_n,l = (2n + |l| - 1) h_barw₀          for n = 1,2,3,... and l = 0,+-1,+-2,...
(n = radial quantum number, l = angular momentum quantum number, w₀ = oscillator frequency)

The degeneracy of the eigenvalues is as follows (neglecting spin, for zero magnetic field):
- the ground state is not          degenerate
- the second state is two-fold   degenerate
- the third     state is three-fold degenerate
- the forth     state is four-fold   degenerate
- ...

Magnetic field

This input file aims to reproduce the Figs. 5(a) and 6(a) (which are analytical results) of
    L.P. Kouwenhoven, D.G. Austing, S. Tarucha
    Few-electron quantum dots
    Rep. Prog. Phys. 64, 701 (2001).

• We chose the parabolic confinement such that h_barw₀ = 3 meV in agreement to this paper.
  The electron effective mass is taken to be m_e* = 0.067 m₀ (GaAs).
   
• The eigenvalues of a two-dimensional parabolic potential that is subject to a magnetic field can be solved analytically. The spectrum of the resulting eigenstates is known as the Fock-Darwin states (1928):
  
      E_n,l = (2n + |l| + 1) h_bar [w₀² + 1/4 w_c²]^1/2 - 1/2 l h_barw_c     for n = 0,1,2,3,... and l = 0,+-1,+-2,...
  
                                                                            (Note that the last term is w_c and not w₀ as in Kouwenhoven's paper.)
      (w_c = |e| B / m_e* = cyclotron frequency, for GaAs: h_barw_c = 1.728 meV at 1 T)
  
  Each of these states is two-fold spin-degenerate. A magnetic field lifts this degeneracy (Zeeman splitting). However, this effect is not taking into account in this tutorial.
   
• The following figure shows the calculated Fock-Darwin spectrum, i.e. the eigenstates as a function of magnetic field magnitude:
  
  
  
  The figure is in excellent agreement with Fig. 5(a) of Kouwenhoven's paper.
   
• The following figure show the probability densities (psi²) of some of these eigenstates for a magnetic field of B = 0.05 T.
  The figures are in excellent agreement with Fig. 6(a) of Kouwenhoven's paper.
  The parabolic conduction band edges are also shown.
   

  (n,l) = (0,0)    (1st)	(n,l) = (0,1)     (2nd)	(n,l) = (0,2)     (4th)
  (n,l) = (1,0)     (5th)	(n,l) = (2,0)     (13th)	(n,l) = (2,2)     (18th)

  
  '

• For very high magnetic fields, eventually all states become degenerate Landau levels as can be seen in this figure. The reason is that the electrons are confined only by the magnetic field and not any longer by the parabolic conduction band edge.
  
  
  
  The red line shows the fan of the lowest Landau level at 1/2 h_barw_c. The higher lying states (not shown) will collect in the second, third, ..., and higher Landau fans (not shown).
  
  The left part of the figure (black region) contains exactly the same Fock-Darwin spectrum that has been shown in the figure further above (from 0 T to 3.5 T).