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2D QWR magnetic

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 check if you can find them in the installation directory.
If you cannot find them, please submit a Support Ticket.
-> 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. m_hh = 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 m_hh is assumed to be negative):

E_n1,_n2 = h_bar² pi² / 2m_hh              ( n₁²/ L_x² + n₂²/ L_y²) =
          = -9.1714667 * 10^-19 eVm² ( n₁²/ L_x² + n₂²/ L_y²) =
          = -0.0091714667 eV           ( n₁² + n₂²      )           (if L_x = L_y = 10 nm)

E_n1,_n2 is the heavy hole energy in the two transverse directions, or the total heavy hole energy for k_z = 0.
Note that the total heavy hole energy is given by E_hh = E_n1,_n2 + h_bar²k_z²/ 2m_hh where k_z is the wavevector along the z direction leading to a one-dimensional E(k_z) dispersion.
n₁, n₂² are two discrete quantum numbers (because we have two directions of quantization).
L_x and L_y are the lengths along the x and y directions. In our case, L_x = L_y = 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 (E_n1,_n2 = E_n2,_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


1^st eigenstate: `-0.0183 eV`	2^nd eigenstate: `-0.0458 eV`

3^rd eigenstate: `-0.0458 eV`	4^th 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 2^nd and 3^rd eigenstate is slightly lifted in comparison to the case where no magnetic field is applied.


1^st eigenstate: `-0.0151 eV`	2^nd eigenstate:`-0.0375 eV`

3^rd eigenstate:`-0.0378 eV`	4^th 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 optionschroedinger-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]


1^st /2^nd eigenstate:`-0.0342 eV`	3^rd/4^theigenstate:`-0.0367 eV`

5^th/6^th eigenstate:`-0.0557 eV`	7^th/8^th 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 = chearnleads 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:


1^st /2^nd eigenstate:`-0.0383 eV`	3^rd/4^theigenstate:`-0.0419 eV`

5^th/6^th eigenstate:`-0.0679 eV`	7^th/8^th 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 featurevalence-band-masses-from-kpdoes 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:
m_xx =   0.333
m_yy =   0.333
m_zz =   0.333
m_xy = -0.943
m_xz = -0.943
m_yz = -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.


1^st eigenstate:`-0.0182 eV (-0.0182)`	2^nd eigenstate:`-0.0407 eV (-0.0406)`

3^rd eigenstate:`-0.0500 eV (-0.0500)`	4^th eigenstate:`-0.0665 eV (-0.0662)`	5^th eigenstate:`-0.0896 eV (-0.0895)`

In the brackets, the eigenvalues forschroedinger-masses-anisotropic = boxare 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.
m_xy = -0.943 := 0 <- assumed to be zero!!!
m_xz = -0.943 := 0 <- assumed to be zero!!!
m_yz = -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:


1^st eigenstate: `-0.0186 eV`	2^nd eigenstate: `-0.0463 eV`

3^rd eigenstate: `-0.0463 eV`	4^th eigenstate: `-0.0741 eV`

nextnano3 - Tutorial

next generation 3D nano device simulator

2D Tutorial

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

Quantum wire

Quantum cluster

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

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

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

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

nextnano³ - Tutorial