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qcl:electronic_band_structure

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:

  <Materials>
     ...
     <Number_of_bands>1</Number_of_bands>
    ...
  </Materials>

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

  <Materials>
    <Material>
      ...
      <Effective_mass_from_kp_parameters>yes</Effective_mass_from_kp_parameters>
    </Material>
   </Materials>

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

      <Effective_mass_from_kp_parameters>no</Effective_mass_from_kp_parameters>

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

<NonParabolicity>yes</NonParabolicity>

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:

  <Materials>
     ...
     <Number_of_bands>2</Number_of_bands>
    ...
  </Materials>

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 $S=1$.

If

<Effective_mass_from_kp_parameters>yes</Effective_mass_from_kp_parameters>

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

If

<Effective_mass_from_kp_parameters>no</Effective_mass_from_kp_parameters>

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:

<NonParabolicityRelative>0.02</NonParabolicityRelative>

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:

  <Materials>
     ...
     <Number_of_bands>3</Number_of_bands>
    ...
  </Materials>

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);

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

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