# nextnano.NEGF - Software for Quantum Transport

### nextnano.MSB

Summary and introduction

### nextnano++

Software documentation

Examples

### nextnano³

Software documentation

### nextnanomat

Software documentation

### nextnano GmbH

qcl:electronic_band_structure

# Electronic Band Structure

(page in construction)

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

### 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:

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

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.

If the non-parabolicty flag is specified

<NonParabolicity>yes</NonParabolicity>

the effective mass is evaluated at the energy of the first 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 of the first miniband with respect to the conduction band edge.

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.

### 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 $$H(z) = \left( {\begin{array}{cc} E_c(z) + \frac{\hbar^2 k_z^2}{2S(z) m_0} & i P(z) k_z \\ - i P(z) k_z & E_v(z)\\ \end{array} } \right)$$

where $P$ is the interband momentum matrix element, which is related to the Kane energy $E_p$ through: $$P(z) = \sqrt{\frac{m_0 E_p(z)}{2}}$$

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 has to specified by using:

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

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