nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

Valence band masses from k.p

Authors: Stefan Birner, Michael Povolotskyi

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.
-> 1DGaAs_vb_masses_from_kp.in
 

Valence band masses from k.p

-> 1DGaAs_vb_masses_from_kp.in

The energy dispersion of the valence band masses in a cubic crystal has cubic symmetry.
Thus it is usually not a good approximation to take a spherical effective mass tensor for the single-band Schrödinger equation for the holes.
For that reason, usually a 6-band or 8-band k.p Hamiltonian should be used.

However, there might be situations where the user wants to use the single-band Schrödinger equation (e.g. to minimize CPU time) but wants to use a local (i.e. grid point dependent) effective mass tensor that takes into account the local strain tensor of each grid point.
This means that for each grid point, the bulk E(k) band structure (which takes into account the local strain tensor) is used to calculate the (strain dependent) components of the symmetric (3x3) effective mass tensors of the heavy hole, light hole and split-off hole.

The relation of effective mass tensor to energy dispersion is as follows:

(1/m)_ij = 1/h_bar²   d² E / (dk_i dk_j)

 $numeric-control !
  ...
  valence-band-masses-from-kp = yes   ! makes only sense for use with single-band Schrödinger equation

 

 $output-1-band-schroedinger
  ...
  effective-mass-tensor       = yes   ! to output the effective mass tensor compontents (1/m)ij

  ==> Schroedinger_1band/mass_tensor1D_vb001_qc001_sg001_deg001.dat - heavy hole
  ==> Schroedinger_1band/mass_tensor1D_vb002_qc001_sg001_deg001.dat - light hole
  ==> Schroedinger_1band/mass_tensor1D_vb003_qc001_sg001_deg001.dat - split-off hole

Approximation scheme

From the energy dispersion function, E(k_x,k_y,k_z), the components of the symmetric (3x3) effective mass tensor are calculated as follows:

mxx = h_bar² * dk2 / (2*[-E(dk,0,0)             + E(0,0,0)])    where 'E(0,0,0)' is the Gamma point.
mxy = h_bar² * dk2 /    [ E(dk,0,0) + E(0,dk,0) - E(0,0,0) - E(dk,dk,0)]
...

dk is chosen to be equal to: dk = 1d-4

GaAs

 !Luttinger-parameters =  6.98d0   2.06d0   2.93d0  ! gamma1, gamma2, gamma3
                          1.72d0   0.04d0           ! kappa, q
  6x6kp-parameters     = -16.22d0 -3.86d0 -17.58d0  ! L,M,N [hbar^2/2m] (--> divide by hbar^2/2m)
                          0.341d0                   ! deltasplit-off in [eV]
 

Heavy hole: components of the effective mass tensor

(1/m)xx =  2.86000   ==>  mxx =  0.34965 [m0]
(1/m)yy =  2.86000   ==>  myy =  0.34965 [m0]
(1/m)zz =  2.86000   ==>  mzz =  0.34965 [m0]
(1/m)xy = -1.35707   ==>  mxy = -0.73688 [m0]
(1/m)xz = -1.35707   ==>  mxz = -0.73688 [m0]
(1/m)yz = -1.35707   ==>  myz = -0.73688 [m0]

Light hole: components of the effective mass tensor

(1/m)xx =  11.1000   ==>  mxx =  0.09009 [m0]
(1/m)yy =  11.1000   ==>  myy =  0.09009 [m0]
(1/m)zz =  11.1000   ==>  mzz =  0.09009 [m0]
(1/m)xy =  1.35704   ==>  mxy =  0.73690 [m0]
(1/m)xz =  1.35704   ==>  mxz =  0.73690 [m0]
(1/m)yz =  1.35704   ==>  myz =  0.73690 [m0]

Split-off hole: components of the effective mass tensor

(1/m)xx =  6.98001   ==>  mxx =  0.14327 [m0]
(1/m)yy =  6.98001   ==>  myy =  0.14327 [m0]
(1/m)zz =  6.98001   ==>  mzz =  0.14327 [m0]
(1/m)xy =  0.000034  ==>  mxy =  29128.7 [m0]
(1/m)xz =  0.000034  ==>  mxz =  29128.7 [m0]
(1/m)yz =  0.000034  ==>  myz =  29128.7 [m0]

These results can be compared with analytical formulas (e.g. I. Vurgaftman et al., J. Appl. Phys. 89, 5815 (2001)):

• m_hh[100] = 1 / ( gamma₁ - 2 gamma₂) = 1 / 2.86 = 0.34965 [m₀]
• m_lh [100] = 1 / ( gamma₁ + 2 gamma₂) = 1 / 11.1 = 0.09009 [m₀]
• m_hh[110] = 2 / (2 gamma₁ - gamma₂ - 3 gamma₃) = 1 / 1.555  = 0.64309 [m₀]
• m_lh [110] = 2 / (2 gamma₁ + gamma₂ + 3 gamma₃) = 1 / 12.405 = 0.08061 [m₀]
• m_hh[111] = 1 / (gamma₁ - 2 gamma₃) = 1 / 1.12   = 0.89286 [m₀]
• m_lh [111] = 1 / (gamma₁ + 2 gamma₃) = 1 / 12.84  = 0.07788 [m₀]

Principal axes system

!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++!
! ==> Leading to a diagonal effective mass tensor: m_xx = 0.145, m_yy = m_zz = 4.21
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++!
 hkl-x-direction-zb =  1 1  1  ! [111]
 hkl-y-direction-zb =  0 1 -1  ! [01-1]
!hkl-z-direction-zb =  . .  .  ! [...] (calculated internally)