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nextnano3 - Tutorial

next generation 3D nano device simulator

2D Tutorial

Hole wave functions in a quantum wire subjected to a magnetic field

Author: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
 -> 2Dwire.in                       (single-band approximation)
 -> 2Dwire_magnetic.in             
(single-band approximation including magnetic field)
 -> 2Dwire_6x6kp.in                
(6-band k.p approximation)
 -> 2Dwire_6x6kp_Burt.in           
(6-band k.p approximation)
 


Quantum wire

Similar to the 1D confinement in a quantum well, it is possible to confine electrons or holes in two dimensions, i.e. in a quantum wire.
The quantum wire consists of InAs (blue area) and is confined by GaAs barriers (red area).
Its size is 10 nm x 10 nm whereas the whole simulation dimension is 30 nm x 30 nm.

 


The blue area is the quantum wire which consists of InAs. The quantum wire rectangle has the size 10 nm x 10 nm.

 


This picture shows a possible configuration of rectangular grid lines.
Here, the grid spacing is 0.5 nm, thus the quantum wire (blue area) consists of 21 x 21 = 400 grid points.

 

Quantum cluster

Assumptions:
1) We apply Dirichlet boundary conditions to our quantum cluster (psi = 0 at the boundary).
2) The quantum cluster is defined only in the area of the quantum wire, i.e. from 10 nm to 20 nm.
These two conditions lead to an infinite GaAs barrier and thus the wave function is forced to be zero at the InAs quantum wire boundaries. Of course, this is not a realistic assumption but we simplify the sample to make the tutorial easier.

The energy levels and the wave functions of a rectangular quantum wire of length 10 nm with infinite barriers can be calculated analytically.

We assume that the barriers at the QD boundaries are infinite. This way we can compare our numerical calculations to analytical results.
The potential inside the quantum wire is assumed to be 0 eV.

As effective mass we take the isotropic heavy hole effective mass of InAs, i.e. mhh = 0.41 m0.
 valence-band-masses = 0.41d0   0.41d0   0.41d0  ! isotropic heavy hole effective
                       ...

 

A discussion of the analytical solution of the 2D Schrödinger equation of a particle in a rectangle (i.e. quantum wire) with infinite barriers can be found in e.g.
Quantum Heterostructures (Microelectronics and Optoelectronics) by V.V. Mitin, V.A. Kochelap and M.A. Stroscio.

The solution of the Schrödinger equation leads to the following eigenvalues (where mhh is assumed to be negative):

En1,n2 = hbar2 pi2 / 2mhh                ( n12 / Lx2 + n22 / Ly2) =
          = -9.1714667 * 10-19 eVm2 ( n12 / Lx2 + n22 / Ly2) =
          = -0.0091714667 eV           ( n12           + n22       )           (if Lx = Ly = 10 nm)

  • En1,n2 is the heavy hole energy in the two transverse directions, or the total heavy hole energy for kz = 0.
    Note that the total heavy hole energy is given by Ehh = En1,n2 + hbar2 kz2 / 2mhh where kz is the wavevector along the z direction leading to a one-dimensional E(kz) dispersion.
  • n1, n22 are two discrete quantum numbers (because we have two directions of quantization).
  • Lx and Ly are the lengths along the x and y directions. In our case, Lx = Ly = 10 nm.

Generally, the energy levels are not degenerate, i.e. all energies are different.
However, some energy levels with different quantum numbers coincide, if the lengths along two directions are identical (En1,n2 = En2,n1) or
if their ratios are integers. In our quadratic quantum wire, the two lengths are identical.
Consequently, we expect the following degeneracies:

  • E11 = -0.018343 eV (ground state)
  • E12 = E21 = -0.045857 eV
  • E22 = -0.073372 eV
  • E13 = E31 = -0.091715 eV
  • E23 = E32 = -0.119229 eV
  • E14 = E41 = -0.155915 eV
  • ...
  • E18 = E81 = E47 = E74 = -0.596145 eV (Here, the degeneracy is a coincidence.)

nextnano³ numerical results for a 10 nm quadratic quantum wire with 0.10 nm grid spacing:

Output file name: Schroedinger_1band/ev2D_vb001_qc001_sg001_deg001_dir_Kx001_Ky001_Kz001.dat

 num_ev: eigenvalue [eV]:
        (0.10 nm grid)

  1      0.018341
= E11
  2     -0.045845 
(two-fold degenerate) E12/E21
  3     -0.045845 
(two-fold degenerate) E12/E21
  4     -0.073348 = E22
  5     -0.091653  (two-fold degenerate) E13/E31
  6     -0.091653  (two-fold degenerate) E13/E31
  7     -0.119156  (two-fold degenerate) E23/E32
  8     -0.119156 
(two-fold degenerate) E23/E32
  9     -0.155721  (two-fold degenerate) E14/E41
 10     -0.155721  (two-fold degenerate) E14/E41
 ...
 42     -0.593061  (four-fold degenerate) E18/E81 or E47/E74
 43     -0.593061  (four-fold degenerate) E18/E81 or E47/E74
 44     -0.594144 
(four-fold degenerate) E18/E81 or E47/E74
 45     -0.594144  (four-fold degenerate) E18/E81 or E47/E74

 

 

Hole wave functions - Single-band effective-mass approximation (infinite GaAs barriers)

-> 2Dwire.in

The following figures show the charge densities (Psi²; Psi = hole wave function) of the four lowest energy confined hole eigenstates in an infinitely deep 10 nm x 10 nm InAs quantum wire. Due to the symmetry of the quantum wire, the second and the third eigenstate are degenerate.
 

1st eigenstate: -0.0183 eV

2nd eigenstate: -0.0458 eV

3rd eigenstate: -0.0458 eV

4th eigenstate: -0.0733 eV

The heavy hole valence band edge energy of InAs is set to be located at 0 eV. The hole eigenvalues are:

schroedinger-masses-anisotropic = yes  = no  = box
1st  eigenvalue  =>  confinement energy: -0.018341 eV -0.018341 eV -0.018341 eV
2nd  eigenvalue  =>  confinement energy: -0.045845 eV -0.045845 eV -0.045845 eV
3rd  eigenvalue  =>  confinement energy: -0.045845 eV -0.045845 eV -0.045845 eV
4th  eigenvalue  =>  confinement energy: -0.073348 eV -0.073348 eV -0.073348 eV

Note that these wave functions were obtained by using a single-band effective mass approximation for the holes. A more accurate and more realistic treatmeant would have been to use 6-band k.p. Note that the wire has been assumed to be unstrained (which is a rather unphysical situation) for the purpose to make this tutorial easier to understand.

 

 

Magnetic field - single-band approximation (infinite GaAs barriers)

-> 2Dwire_magnetic.in

Here we use: schroedinger-masses-anisotropic = yes ! 'yes', 'no', 'box'

We apply a magnetic field perpendicular to the (x,y) plane, i.e. along the [001] direction.

$magnetic-field
 magnetic-field-on        = yes
 magnetic-field-strength  = 1.0d0 ! 1 Tesla
 magnetic-field-direction = 0 0 1 ! [001] direction
$end_magnetic-field

This leads to an additional confinement in addition to the wire potential. However, for the first and forth eigenstate, the confinement does not play an important role whereas for the second and third it does. The effect is more dominant onto the wave functions but not so pronounced onto the values of the eigenenergies. Now the degeneracy of the 2nd and 3rd eigenstate is slightly lifted in comparison to the case where no magnetic field is applied.

 

1st eigenstate: -0.0151 eV

2nd eigenstate: -0.0375 eV

3rd eigenstate: -0.0378 eV

4th eigenstate: -0.0602 eV

Here, the probabilty density of the second eigenstate is plotted viewed from a different perspective.

Note: Currently magnetic field only works for option schroedinger-masses-anisotropic = yes.

 

 

 

Hole wave functions - 6-band k.p approximation (infinite GaAs barriers)

-> 2Dwire_6x6kp.in
-> 2Dwire_6x6kp_Burt.in

The following figures show the charge densities (Psi²; Psi = hole wave function) of the four lowest energy confined hole eigenstates in an infinitely deep 10 nm x 10 nm InAs quantum wire. This time we used 6-band k.p theory to describe the hole states. Here, the second and the third eigenstate are no longer degenerate. Only the ground state looks similar to the single-band effective-mass approximation results.

We used the following Luttinger parameters for InAs:
   gamma1 = 20.0, gamma2 = 8.5, gamma3 = 9.2
This corresponds to the Dresselhaus parameters L, M, N:
   L = -55, M = -4, N = -55.2

 6x6kp-parameters = -55.000d0 -4.000d0 -55.200d0 ! L, M, N [hbar^2/2m](--> divide by hbar^2/2m)
                     0.39d0                      ! delta_split-off [eV]
 

1st /2nd eigenstate: -0.0342 eV

3rd/4th eigenstate: -0.0367 eV

5th/6th eigenstate: -0.0557 eV

7th/8th eigenstate: -0.0581 eV

The energies are contained in this file: ev_hl_6x6kp_qc001_num_kpar001_2D.dat

 num_ev:  eigenvalue[eV]:    cb:       hh:       lh:       s/o:
 1        -0.34226968E-01    0.000E+00 0.894E-01 0.905E+00 0.519E-02
 2        -0.34226968E-01    0.000E+00 0.894E-01 0.905E+00 0.519E-02
 3        -0.36675499E-01    0.000E+00 0.285E+00 0.710E+00 0.528E-02
 4        -0.36675499E-01    0.000E+00 0.285E+00 0.710E+00 0.528E-02
 5        -0.55698284E-01    0.000E+00 0.263E+00 0.718E+00 0.192E-01
 6        -0.55698284E-01    0.000E+00 0.263E+00 0.718E+00 0.192E-01
 7        -0.58120024E-01    0.000E+00 0.384E+00 0.604E+00 0.113E-01
 8        -0.58120024E-01    0.000E+00 0.384E+00 0.604E+00 0.113E-01

 

We used:
 schroedinger-kp-ev-solv        = ARPACK          !
 schroedinger-kp-discretization = box-integration !
 kp-vv-term-symmetrization      = yes             ! 'yes' (= incorrect physics), 'no' (= correct physics)

Comment: schroedinger-kp-ev-solv = chearn leads to very similar (almost identical) results for 6-band k.p
for both options of kp-vv-term-symmetrization = yes/no.

 

However, using instead Burt's non-symmetrized discretization
 kp-vv-term-symmetrization      = no              ! 'yes', 'no'
leads to the following results:

1st /2nd eigenstate: -0.0383 eV

3rd/4th eigenstate: -0.0419 eV

5th/6th eigenstate: -0.0679 eV

7th/8th eigenstate: -0.0768 eV

 num_ev:  eigenvalue[eV]:    cb:       hh:       lh:       s/o:
 1        -0.38319736E-01    0.000E+00 0.531E-01 0.941E+00 0.606E-02
 2        -0.38319736E-01    0.000E+00 0.531E-01 0.941E+00 0.606E-02
 3        -0.41904230E-01    0.000E+00 0.265E+00 0.735E+00 0.860E-03
 4        -0.41904230E-01    0.000E+00 0.265E+00 0.735E+00 0.860E-03
 5        -0.67878674E-01    0.000E+00 0.414E+00 0.584E+00 0.180E-02
 6        -0.67878675E-01    0.000E+00 0.414E+00 0.584E+00 0.180E-02
 7        -0.76825362E-01    0.000E+00 0.111E+00 0.874E+00 0.145E-01
 8        -0.76825362E-01    0.000E+00 0.111E+00 0.874E+00 0.145E-01

 

 

 

The following is obsolete because the feature valence-band-masses-from-kp does not really make much sense.

Hole wave functions - Single-band effective-mass approximation (infinite GaAs barriers)

-> 2Dwire_vbmassesfrom6x6kp.in

Here we use a single-band effective-mass approximation but we use an effective-mass tensor that is obtained from the Luttinger parameters.
 valence-band-masses-from-kp     = yes ! 'yes', 'no'
 schroedinger-masses-anisotropic = yes ! 'yes', 'no', 'box'

We used the following Luttinger parameters for InAs:
   gamma1 = 20.0, gamma2 = 8.5, gamma3 = 9.2
This corresponds to the Dresselhaus parameters L, M, N:
   L = -55, M = -4, N = -55.2

 6x6kp-parameters = -55.000d0 -4.000d0 -55.200d0 ! L, M, N [hbar^2/2m](--> divide by hbar^2/2m)
                     0.39d0                      ! delta_split-off [eV]

This leads to the following effective-mass tensor components for the heavy hole:
  mxx =   0.333
  myy =   0.333
  mzz =   0.333
  mxy = -0.943
  mxz = -0.943
  myz = -0.943

The following figures show the charge densities (Psi²; Psi = hole wave function) of the five lowest energy confined hole eigenstates in an infinitely deep 10 nm x 10 nm InAs quantum wire. The degeneracy of the second and the third eigenstate is now lifted.

 

1st eigenstate: -0.0182 eV (-0.0182)

2nd eigenstate: -0.0407 eV (-0.0406)

 

3rd eigenstate: -0.0500 eV (-0.0500)

4th eigenstate: -0.0665 eV (-0.0662)

5th eigenstate: -0.0896 eV (-0.0895)

In the brackets, the eigenvalues for schroedinger-masses-anisotropic = box are given.

 

-> 2Dwire_vbmassesfrom6x6kp_iso.in

If we had chosen
 schroedinger-masses-anisotropic = no  ! 'yes', 'no', 'box'
then the off-diagonal elements of the effective-mass tensor are not taken into account.
  mxy = -0.943 := 0 <- assumed to be zero!!!
  mxz = -0.943 := 0 <- assumed to be zero!!!
  myz = -0.943 := 0 <- assumed to be zero!!!
Due to the symmetry of the quantum wire and the effective mass tensor, the second and the third eigenstate are degenerate.
The relevant wave functions in this case are:

1st eigenstate: -0.0186 eV

2nd eigenstate: -0.0463 eV

3rd eigenstate: -0.0463 eV

4th eigenstate: -0.0741 eV

 

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