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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.

To describe non-parabolicity, 2- or 3-band models are needed. The 3-band modeled is strongly recommended for structures based on electrons in III-V heterostructures.

Single-band model

The single-band model is the default case, or can also be specified explicitly by using:


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

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

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

Effective mass

If the following option is used, the effective mass is calculated from the k.p parameters


using the following equation:

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

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.

On the other hand, if


is specified, the effective mass is taken directly from the nextnano.NEGF database. Note that the database can be overwritten in the input file.

If the non-parabolicty flag is specified


the effective mass is evaluated at the energy of the lowest miniband: $$ \frac{m_0}{m^*} = S + \frac{E_P((E_g+E_0)+2\Delta_{\text{SO}}/3)}{(E_g+E_ 0)((E_g+E_0)+\Delta_{\text{SO}})} $$ where $E_0$ is the energy difference between the lowest miniband and the conduction band edge.

Note, however, that in this case the effective mass is not state-dependent, i.e. higher energy states have the same effective mass than the lowest miniband. A full nonparabolic description is provided by 2 or 3 band models described below.

2-band model

The two-band model has to specified by using:


In this case, a conduction band is coupled to an effective valence band. The 2-band Hamiltonian describing the 1-D Schrödinger equation along the z-axis reads $$ H(z) = \left( {\begin{array}{cc} E_c(z) + S(z)\frac{\hbar^2 k_z^2}{2 m_0} & i P(z) k_z \\ - i P(z) k_z & E_v(z) + (1+L(z))\frac{\hbar^2 k_z^2}{2 m_0}\\ \end{array} } \right) $$

where $P$ is the interband momentum matrix element, which is related to the Kane energy $E_p$ through: $$ P(z) = \sqrt{\frac{m_0 E_p(z)}{2}} $$ $L$ correponds to the Dresselhaus parameter $L'$ used in 8-band k.p parameters of nextnano3/nextnano++. 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 use renormalized k.p parameters $E_p$, $S$ and $L$ given by nextnano++ with $S=1$.



is specified, the band structure is calculated from the parameters $E_p$, $S$, $E_g$, and band offsets. The effective mass of the database is not used.

Input from effective mass and non-parabolicity



is specified, the effective mass $m^*_0$ of the database (or overwritten in the input file) is used. $S=0$ is considered, and the interband energy is overwritten using: $$ E_p = \frac{m_0}{m^*_0} E_g $$

In this case, non-parabolicity is controlled by the value of the bandgap.

To further modify non-parabolicity, a non-parabolic coefficient $a_{np}$ can be further specified in the database or in the input file:


The parameters $E_g$ and $E_p$ are then modified according to: $$ E'_g = E_g (1- a_{np}) $$ and $$ E'_p = E_p (1- a_{np}) $$

In this case, the dispersion relation is given by: $$ E(k) = \frac{\hbar^2 k^2}{2m^*_0} \frac{1}{1 + \frac{(1-a_{np}) E}{E_g}} \simeq \frac{\hbar^2 k^2}{2m^*_0} \left[ 1 - \frac{(1-a_{np}) E}{E_g} \right] $$

3-band model

The 3-band model consider has to specified by using:


It accounts for: - conduction band - light-hole (lh) band - split-off (so) band

The considered Hamiltonian reads:

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

Note that in this case, the k.p parameters are considered, and the input of a nonparabolic coefficient is not supported.

Smoothing of the k.p parameters (2 or 3 band case)

In the multiband case, spurious solutions can occur at the interface in the case of a discontinuity of the k.p parameters $E_p$ and/or $S$. To avoid this, a smoothing of the k.p parameters is done. The smoothing length can be set manually using the following command (otherwise it is set automatically);

       <Smoothing_Length_kp unit="nm">0.2</Smoothing_Length_kp> 

In any case, the used value is displayed in the log file:

Calculating axial electronic structure...
00:00:07   46.2  Smoothing of k.p parameters: auto mode
00:00:07   46.1  Smoothing length for Ep (Kane) parameter = 0.274702188030413 nm
00:00:07   46.1  Smoothing length for S (remote bands) parameter = 0.260335418344245 nm

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 $L_s$ is either the $E_p$ or $S$ smoothing length.

qcl/electronic_band_structure.txt · Last modified: 2020/10/30 11:15 by thomas.grange