Electronic band structure#

The band structure is modeled in the envelope function approximation, using either the single-band effective mass approximation or a multiband model.

The number of bands is specified in the input file by the keyword NumberOfBands.

Single-band effective mass model can be sufficient for intraband devices where the transitions energies are small compared to the band gap. However, to describe nonparabolicity in intraband devices, 2- or 3-band models are needed. The 3-band modeled is strongly recommended for structures based on electrons in III-V heterostructures.

For interband devices, or devices involving only the valence band, the 8-band model is recommended.

Definition of band offsets#

The definition is identical to nextnano++. The database contains the values \(E_\mathrm{c}^\Gamma, E_\mathrm{v,av}, \Delta_\mathrm{so}\) and the band gap parameters \(E_\mathrm{g}^\Gamma(T=0), \alpha, \beta\). The choice UseConductionBandOffset results in different temperature dependence of the heterostructure band offsets via the Varshni formula described in nextnano++.

Note

The band offsets get an additional shift if strain is present (see nextnano++)

Single-band model#

In this case a 1-dimensional Schrödinger equation is solved:

(2)#\[-\frac{\hbar^2}{2m_{\perp}^*(z)} \frac{\partial^2}{\partial z^2} \psi(z) + V(z) \psi(z) = E \psi(z)\]

where \(m_{\perp}^*(z)\) is a position-dependent effective mass along the growth direction.

Effective mass#

Effective mass from \(\mathbf{k} \cdot \mathbf{p}\) parameters#

If Material{ EffectiveMassFromKpParameters } is set to yes, the effective mass is calculated from the \(\mathbf{k} \cdot \mathbf{p}\) parameters using the following equation:

(3)#\[\frac{m_0}{m_{\perp}^*} = S + \frac{E_P(E_g+2\Delta_{\text{SO}}/3)}{E_g(E_g+\Delta_{\text{SO}})}\]

where \(E_P\) is the Kane energy, \(E_g\) the band gap and \(\Delta_{\text{SO}}\) is the spin-orbit splitting [Vurgaftman2001]. The effective mass of the database is ignored in this case.

Effective mass without \(\mathbf{k} \cdot \mathbf{p}\) parameters#

On the other hand, if Material{ EffectiveMassFromKpParameters } is set to no, the effective mass is taken directly from the nextnano.NEGF database. Note that the material database can be overwritten in the input file using the Material{ Overwrite{ } } option.

Anisotropic effective mass#

In the case of an anisotropic effective masse, the axial \(m_{\perp}^*\) and in-plane \(m_{\parallel}^*\) effective masses can be individually overwritten using ElectronMass and ElectronMassInPlane respectively.

2-band model#

When NumberOfBands is set to 2, the considered bands are:

  • the conduction band, with bandedge energy \(E_{c}(z)\)

  • an averaged valence band, defined as \(E_{v} = (E_{\text{hh}} + E_{\text{lh}} + E_{\text{so}})/3\)

The 2-band Hamiltonian describing the 1-D Schrödinger equation along the z-axis reads

\[\begin{split}H(z) = \left( {\begin{array}{cc} E_c(z) + S(z)\frac{\hbar^2 k_z^2}{2 m_0} & i P(z) k_z \\ - i P(z) k_z & E_v(z) + (1+L'(z))\frac{\hbar^2 k_z^2}{2 m_0}\\ \end{array} } \right)\end{split}\]

where \(P\) is the interband momentum matrix element, which is related to the Kane energy \(E_p\) through:

\[P(z) = \sqrt{\frac{m_0 E_p(z)}{2}}\]

\(L'\) corresponds to the Dresselhaus parameter. By default, this value is set to -1. Note that when \(S<0\) and \(L'\neq-1\), spurious solutions are likely to occur. Hence it is recommended either (i) to let \(L=-1\) (default value), or (ii) to rescale the \(\mathbf{k} \cdot \mathbf{p}\) parameters \(E_p\), \(S\) and \(L'\) with \(S=1\) (see Material{ RescaleSTo }).

3-band model#

Its aim is to account more accurately for the nonparabolicity of the conduction band. It accounts for the 3 following bands:

  • conduction band

  • light-hole (lh) band

  • split-off (so) band

In this basis, the Hamiltonian reads:

\[\begin{split}H = \left( {\begin{array}{ccc} E_c(z) + S(z)\frac{\hbar^2 k_z^2}{2 m_0} & i \sqrt{\frac{2}{3}} P(z) k_z & -i \sqrt{\frac{1}{3}} P(z) k_z\\ - i \sqrt{\frac{2}{3}} P(z) k_z & E_{\text{lh}}(z) + (1+L'(z)) \frac{\hbar^2 k_z^2}{2 m_0} & 0\\ i \sqrt{\frac{1}{3}} P(z) k_z & 0 & E_{\text{so}}(z) + (1+L'(z))\frac{\hbar^2 k_z^2}{2 m_0}(z) \end{array} } \right)\end{split}\]

Electron effective mass as input#

When Material{ EffectiveMassFromKpParameters } is set to no, the effective mass \(m^*_0\) of the database (or overwritten in the input file) is used as an input to calculate the \(\mathbf{k} \cdot \mathbf{p}\) parameters. In this case, the parameter \(S\) is set to \(S=0\), and the interband energy is overwritten using:

\[E_p = \frac{m_0}{m^*} \frac{E_g(Eg+\Delta_{SO})}{Eg+\frac{2}{3}\Delta_{SO}}\]

Output of axial effective masses (2- and 3-band cases)#

In the multiband case (2 or 3 bands), a position and state-dependent effective mass along the growth axis is output in the file EffectiveMasses.dat. It allows comparison with other models and to analyze the multiband calculation.

In the 2-band case, its value is given for the level \(i\) by:

\[\frac{m_0}{m_{\perp}^*(z,i)} = S(z) + \frac{E_P(z)}{\epsilon_i-E_{v,av}(z)} = S(z) + \frac{E_P(z)}{\epsilon_i-\left(E_{g}(z)+\frac{1}{3}\Delta_{SO}(z)\right)}\]

In the 3-band case, it reads

\[\frac{m_0}{m_{\perp}^*(z,i)} = S(z) + \frac{2}{3} \frac{E_P(z)}{\epsilon_i-E_{lh}(z)} + \frac{1}{3} \frac{E_P(z)}{\epsilon_i-E_{SO}(z)}\]

where \(\epsilon_i\) is the energy of level \(i\). Note that this formula accounts for nonparabolicity as the effective mass depends on the energy difference between the energy level and the valence bandedges.

From this position-dependent effective masses, an averaged effective mass can de defined for each level, accounting for nonparabolicity. The state-dependent effective mass \(m_{\perp}^*(i)\) is output in the file EffectiveMasses.dat and is defined by the following averaging:

\[\frac{m_0}{m_{\perp}^*(i)} = \int dz \frac{m_0}{m_{\perp}^*(z,i)} |\Psi_i(z)|^2\]

In-plane nonparabolicity in the 2- and 3-band cases#

In the 2- and 3-band multiband models, the in-plane nonparabolicity of the conduction band is modeled by considering an energy-dependent in-plane effective mass \(m_{\parallel}^*(i,E)\).

\[\frac{m_0}{m_{\parallel}^*(i,E)} = \int dz \left( S(z) + \frac{E_P(z)}{E-E_{v,av}(z)} \right) \vert \psi_i(z) \vert ^2\]
\[\frac{m_0}{m_{\parallel}^*(i,E)} = \int dz \left( \S(z) + \frac{2}{3} \frac{E_P(z)}{E-E_{lh}(z)} + \frac{1}{3} \frac{E_P(z)}{E-E_{SO}(z)} \right) \vert \psi_i(z) \vert ^2\]

in the 2 and 3-band cases respectively, with

\[E = \epsilon_i + \frac{\hbar^2}{2m_{\parallel}^*(i,E)}\]

where \(\epsilon_i\) is the energy of level \(i\) along the growth axis. The above equations are solved self-consistently at zero bias

Note

This simplified treatment of the in-plane nonparabolicity is used only in the 2- and 3-band cases when the focus is the conduction band. A full treatment is made in the 8-band model.

8-band model#

See Hamiltonian: 8-band model for zincblende on nextnano++ documentation.

Rescaling of \(\mathbf{k} \cdot \mathbf{p}\) parameters (for multiband)#

When diagonalizing the \(\mathbf{k} \cdot \mathbf{p}\) Hamiltonian for a given wave vector, if the coefficient \(S(L+1)\) of \(k^4\) in the secular equation is positive, two different \(k\) may correspond to the same eigenenergy. One is the expected correct solution, but the other is an oscillatory solution with a large \(k\), and a smooth wave function may not be obtained. To prevent this, the Material{ RescaleS } option rescales \(S\) to 0 (per default) while maintaining the effective mass of the conduction band.

The effect of rescaling on \(S\) and \(E_P\) is the following:

(4)#\[ \begin{align}\begin{aligned}S \to S'\\E_P \to E_P'\end{aligned}\end{align} \]

while the effective mass at bandedge is conserved.

(5)#\[S + \frac{E_P}{E_g} = S' + \frac{E_P'}{E_g}\]

while for 3 bands it corresponds to:

(6)#\[S + \frac{E_P(E_g + 2\Delta_\mathrm{SO}/3)}{E_g(E_g + \Delta_\mathrm{SO})} = S' + \frac{E_P'(E_g + 2\Delta_\mathrm{SO}/3)}{E_g(E_g + \Delta_\mathrm{SO})}\]

Smoothing of the \(\mathbf{k} \cdot \mathbf{p}\) parameters#

In the multiband case, spurious solutions can occur at the interface in the case of a discontinuity of the \(\mathbf{k} \cdot \mathbf{p}\) parameters \(E_p\) and \(S\) To avoid this, a smoothing of the \(\mathbf{k} \cdot \mathbf{p}\) parameters \(E_p(z)\) and \(S(z)\) is done.

The smoothing is performed by performing a convolution of the \(E_p(z)\) and \(S(z)\) spatial functions by a Gaussian \(e^{-z^2/L_s^2}\) where \(Ls\) is the smoothing length.

The smoothing length can be set manually using the command SmoothingLengthKP. Otherwise it is set automatically. In any case, the used value is displayed in the log file.

Oscillator strength#

The oscillator strength is calculated from the formula

(7)#\[f_{\alpha\beta} = \frac{2|p_{\alpha\beta}|^2}{m_0 (E_\beta - E_\alpha)}\]

The electron mass \(m_0\) entering the above formula is the bare electron mass.

This oscillator strength (sometimes referred to as the unnormalized one) differs from the usual definition in the single band case by the ratio \(m^*/m_0\), i.e. \(\frac{m^*}{m_0}f_{\alpha\beta}\) is called the normalized oscillator strength.

The advantage of this unnormalized definition is that it is general enough to be applied to the multiband case.

Note

In the parabolic single-band model, the usual sum-rule is retrieved by using the normalized definition

(8)#\[\sum_{\beta \neq \alpha} \frac{m^*}{m_0} f_{\alpha\beta} = 1\]

Last update: 29/09/2026