Bands groups in database{ …_zb{} } and database{ …_wz{} }¶
There are about 23 identical groups available directly under all zincblende- and wurtzite-related groups. In this section we describe four of them, specifically all groups related to band paramters:
conduction_bands{}
valence_bands{}
kp_6_bands{}
kp_8_bands{}
Bands for zincblende in database{ }¶
database{ …{ conduction_bands{} } } for zincblende¶
- Gamma{}
material parameters for the conduction band valley at the Gamma point of the Brillouin zone:
- mass
electron effective mass (isotropic, parabolic)
- value:
double
- unit:
m0
This mass is used for the single-band Schrödinger equation and for the calculation of the densities.
- bandgap
band gap energy at 0 K
- value:
double
- unit:
eV
- bandgap_alpha
Varshni parameter \(\alpha\) for temperature dependent band gap
- value:
double
- unit:
eV/K
- bandgap_beta
Varshni parameter \(\beta\) for temperature dependent band gap
- value:
double
- unit:
K
- defpot_absolute
absolute deformation potential of the Gamma conduction band: \(a_{c, \Gamma} = a_v + a_{\Gamma}\)
- value:
double
- unit:
eV
- g
g-factor (for Zeeman splitting in magnetic fields)
- value:
double
- L{}
Material parameters for the conduction band valley at the L point of the Brillouin zone
- mass_l
longitudinal electron effective mass (parabolic)
- value:
double
- unit:
m0
- mass_t
transversal electron effective mass (parabolic)
- value:
double
- unit:
m0
These masses are used for the single-band Schrödinger equation and for the calculation of the densities.
- bandgap
band gap energy at 0 K
- value:
double
- unit:
eV
- bandgab_alpha
Varshni parameter \(\alpha\) for temperature dependent band gap
- value:
double
- unit:
eV/K
- bandgab_beta
Varshni parameter \(\beta\) for temperature dependent band gap
- value:
double
- unit:
K
- defpot_absolute
absolute deformation potential of the L conduction band: ac, L = av + agap, L
- value:
double
- unit:
eV
- defpot_uniaxial
uniaxial deformation potential of the L conduction band
- value:
double
- unit:
eV
- g_l
longitudinal g factor (for Zeeman splitting in magnetic fields)
- value:
double
- g_t
transversal g factor (for Zeeman splitting in magnetic fields)
- value:
double
- X{}
material parameters for the conduction band valley at the X point of the Brillouin zone. The options are the same as for
L{}
Note
In Si, Ge and GaP we have a Delta valley instead of the X conduction band valley.
- Delta{}
material parameters for the conduction band valley at the X point of the Brillouin zone. The options are the same as
L{}
, howeverDelta{}
has an extra paramterposition
:
- position
- value:
double
Note
At present, the value for
position
does not enter into any of the equations.
database{ …{ valence_bands{} } } for zincblende¶
material parameters for the valence band valley at the Gamma point of the Brillouin zone
- bandoffset
average valence band energy \(E_{v,av} = (E_{hh} + E_{lh} + E_{so}) / 3\)
- value:
double
- unit:
eV
- HH{}
- mass
heavy hole effective mass (isotropic, parabolic!)
- value:
double
- unit:
m0
- g
g factor (for Zeeman splitting in magnetic fields)
- value:
double
- LH{}
- mass
light hole effective mass (isotropic, parabolic!)
- value:
double
- unit:
m0
- g
g factor (for Zeeman splitting in magnetic fields)
- value:
double
- SO{}
- mass
split-off hole effective mass (isotropic, parabolic!)
- value:
double
- unit:
m0
- g
g factor (for Zeeman splitting in magnetic fields)
- value:
double
- defpot_absolute
absolute deformation potential of the valence bands (average of the three valence bands: \(a_v\))
- value:
double
- unit:
eV
- defpot_uniaxial_b
uniaxial shear deformation potential b of the valence bands
- value:
double
- unit:
eV
- defpot_uniaxial_d
uniaxial shear deformation potential d of the valence bands
- value:
double
- unit:
eV
- delta_SO
spin-orbit split-off energy \(\Delta_{so}\)
- value:
double
- unit:
eV
database{ …{ kp_6_bands{} } } for zincblende¶
- gamma1
Luttinger parameter \(\gamma\)1
- value:
double
- gamma2
Luttinger parameter \(\gamma\)2
- value:
double
- gamma3
Luttinger parameter \(\gamma\)3
- value:
double
Note
The user can either specify the Luttinger parameters (\(\gamma\)1, \(\gamma\)2, \(\gamma\)3) or the Dresselhaus parameters (L, M, N) parameters
- L
Dresselhaus parameter L
- value:
double
- unit:
\(\hbar^2/(2m_0)\)
- M
Dresselhaus parameter M
- value:
double
- unit:
\(\hbar^2/(2m_0)\)
- N
Dresselhaus parameter N
- value:
double
- unit:
\(\hbar^2/(2m_0)\)
Warning
There are different definitions of the L and M parameters available in the literature. Definition used in nextnano++:
\[\mathrm{L = (-\gamma_1 - 4 \gamma_2 - 1) \cdot \left[\frac{\hbar^2}{2m_0}\right]}\]\[\mathrm{M = (2 \gamma_2 - \gamma_1 - 1 ) \cdot \left[\frac{\hbar^2}{2m_0}\right]}\]
database{ …{ kp_8_bands{} } } for zincblende¶
- S
electron effective mass parameter S for 8-band k.p. The S parameter (S = 1 + 2F) is also defined in the literature as F, where F = (S - 1)/2, e.g. I. Vurgaftman et al., JAP 89, 5815 (2001).
- value:
double
Note
The S parameter (S = 1 + 2F) is also defined in the literature as F where F = (S - 1)/2, e.g. I. Vurgaftman et al., JAP 89, 5815 (2001).
- E_p
Kane’s momentum matrix element. The momentum matrix element parameter P is related to Ep: \(P^2 = \hbar^2/(2m_0) \cdot E_p\)
- value:
double
- unit:
eV
- B
bulk inversion symmetry parameter (B=0 for diamond-type materials)
- value:
double
- unit:
\(\hbar^2/(2m_0)\)
- gamma1
Luttinger parameter \(\gamma\)1’
- value:
double
- gamma2
Luttinger parameter \(\gamma\)2’
- value:
double
- gamma3
Luttinger parameter \(\gamma\)3’
- value:
double
Note
The user can either specify the modified Luttinger parameters (\(\gamma\)1’, \(\gamma\)2’, \(\gamma\)3’) or the L’, M’ = M, N’ parameters.
- L
Dresselhaus parameter L’
- value:
double
- unit:
\(\hbar^2/(2m_0)\)
- M
Dresselhaus parameter M’
- value:
double
- unit:
\(\hbar^2/(2m_0)\)
- N
Dresselhaus parameter N’
- value:
double
- unit:
\(\hbar^2/(2m_0)\)
Bands for Wurtzite in database{ }¶
database{ …{ conduction_bands{} } } for wurtzite¶
- Gamma{}
material parameters for the conduction band valley at the Gamma point of the Brillouin zone:
- mass_t
electron effective mass perpendicular to hexagonal c axis (parabolic)
- value:
double
- unit:
m0
- mass_l
electron effective mass along hexagonal c axis (parabolic)
- value:
double
- unit:
m0
This mass is used for the single-band Schrödinger equation and for the calculation of the densities.
- bandgap
band gap energy at 0 K
- value:
double
- unit:
eV
- bandgap_alpha
Varshni parameter \(\alpha\) for temperature dependent band gap
- value:
double
- unit:
eV/K
- bandgap_beta
Varshni parameter \(\beta\) for temperature dependent band gap
- value:
double
- unit:
K
- defpot_absolute_t
absolute deformation potential of the Gamma conduction band perpendicular to hexagonal c axis ac,a = a2
- value:
double
- unit:
eV
- defpot_absolute_l
absolute deformation potential of the Gamma conduction band perpendicular along hexagonal c axis ac,c = a1
- value:
double
- unit:
eV
Note
Note that I. Vurgaftman et al., JAP 94, 3675 (2003) lists a1 and a2 parameters. They refer to the interband deformation potentials, i.e. to the deformation of the band gaps. Thus, we have to add the deformation potentials of the valence bands to get the deformation potentials for the conduction band edge.
\[\mathrm{a_{c,a} = a_{2} = a_{2, Vurgaftman} + D2}\]\[\mathrm{a_{c,c} = a_{1} = a_{1, Vurgaftman} + D1}\]
- g_t (optional)
g factor perpendicular to hexagonal c axis (for Zeeman splitting in magnetic fields)
- value:
double
- g_l (optical)
g factor along hexagonal c axis (for Zeeman splitting in magnetic fields)
- value:
double
database{ …{ valence_bands{} } } for wurtzite¶
material parameters for the valence band valley at the Gamma point of the Brillouin zone
- bandoffset
- value:
double
- unit:
eV
average energy of the three valence band edges (S.L. Chuang, C.S. Chang, “\(\mathbf{k} \cdot \mathbf{p}\) method for strained wurtzite semiconductors”, Phys. Rev. B 54 (4), 2491 (1996)):
\[\mathrm{E_{v,av} = (E_{hh} + E_{lh} + E_{ch}) / 3 - 2/3 \cdot \mathrm{Delta_{cr}}}\]The valence band energies for heavy hole (HH), light hole (LH) and crystal-field split-hole (CH) are calculated by defining an “average” valence band energy Ev (=Ev,av) for all three bands and adding the spin-orbit-splitting and crystal-field splitting energies afterwards. The “average” valence band energy Ev (=Ev,av) is defined on an absolute energy scale and must take into accout the valence band offsets which are “averaged” over the three holes.
Note
This energy determines the valence band offset (VBO) between two materials:
\[\mathrm{VBO_{v,av} = bandoffset_{material1} - bandoffset_{material2}}\]
- HH{}
- mass_t
heavy hole effective mass perpendicular to hexagonal c axis (parabolic !)
- value:
double
- unit:
m0
- mass_l
heavy hole effective mass along hexagonal c axis (parabolic !)
- value:
double
- unit:
m0
- g_t (optional)
g factor perpendicular to hexagonal c axis (for Zeeman splitting in magnetic fields)
- value:
double
- g_l (optional)
g factor along hexagonal c axis (for Zeeman splitting in magnetic fields)
- value:
double
- LH{}
- mass_t
light hole effective mass perpendicular to hexagonal c axis (parabolic !)
- value:
double
- unit:
m0
- mass_l
light hole effective mass along hexagonal c axis (parabolic !)
- value:
double
- unit:
m0
- g_t (optional)
g factor perpendicular to hexagonal c axis (for Zeeman splitting in magnetic fields)
- value:
double
- g_l (optional)
g factor along hexagonal c axis (for Zeeman splitting in magnetic fields)
- value:
double
- SO{}
- mass_t
crystal-field split-off hole effective mass perpendicular to hexagonal c axis (parabolic !)
- value:
double
- unit:
m0
This mass is used for the single-band Schrödinger equation and for the calculation of the densities.
- mass_l
crystal-field split-off hole effective mass along hexagonal c axis (parabolic !)
- value:
double
- unit:
m0
This mass is used for the single-band Schrödinger equation and for the calculation of the densities.
- g_t (optional)
g factor perpendicular to hexagonal c axis (for Zeeman splitting in magnetic fields)
- value:
double
- g_l (optional)
g factor along hexagonal c axis (for Zeeman splitting in magnetic fields)
- value:
double
- defpotentials
deformation potential of the valence bands: [D1, D2, D3, D4, D5, D6]
- value:
vector of 6 real numbers
- units:
eV
- example:
[-3.7, 4.5, 8.2, -4.1, -4.0, -5.5]
(for GaN)- delta
crystal-field splitting energy Deltacr = Delta1, spin-orbit splitting energy parameter Delta2, spin-orbit splitting energy parameter Delta3: [Delta1, Delta2, Delta3]
- value:
vector of 3 real numbers
- units:
eV
- example:
[0.010, 0.00567, 0.00567]
(for GaN)Very often one assumes Delta2 = Delta3 = 1/3 Deltaso.
database{ …{ kp_6_bands{} } } for wurtzite¶
- A1
6-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A1 (Rashba-Sheka-Pikus parameter)
- value:
double
- A2
6-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A2 (Rashba-Sheka-Pikus parameter)
- value:
double
- A3
6-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A3 (Rashba-Sheka-Pikus parameter)
- value:
double
- A4
6-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A4 (Rashba-Sheka-Pikus parameter)
- value:
double
- A5
6-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A5 (Rashba-Sheka-Pikus parameter)
- value:
double
- A6
6-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A6 (Rashba-Sheka-Pikus parameter)
- value:
double
database{ …{ kp_8_bands{} } } for wurtzite¶
- S1
electron effective mass parameter S1 = Sparallel for 8-band \(\mathbf{k} \cdot \mathbf{p}\)
- value:
double
- S2
electron effective mass parameter S2 = Sperpendicular for 8-band \(\mathbf{k} \cdot \mathbf{p}\)
- value:
double
- E_P1
Kane’s momentum matrix elements Ep1 = Ep, parallel
- value:
double
- E_P2
Kane’s momentum matrix elements Ep2 = Ep,perpendicular
- value:
double
Note
The momentum matrix element parameter P is related to Ep : P2 = \(\frac{\hbar^2}{2m_0}\) Ep
- B1
bulk inversion symmetry parameter B1
- value:
double
- B2
bulk inversion symmetry parameters B2
- value:
double
- B3
bulk inversion symmetry parameters B3
- value:
double
- A1
8-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A1’ (Rashba-Sheka-Pikus parameter)
- value:
double
- A2
8-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A2’ (Rashba-Sheka-Pikus parameter)
- value:
double
- A3
8-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A3’ (Rashba-Sheka-Pikus parameter)
- value:
double
- A4
8-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A4’ (Rashba-Sheka-Pikus parameter)
- value:
double
- A5
8-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A5’ (Rashba-Sheka-Pikus parameter)
- value:
double
- A6
8-band \(\mathbf{k} \cdot \mathbf{p}\) hole effective mass parameter A6’ (Rashba-Sheka-Pikus parameter)
- value:
double