Electronic band structure¶

Attention

This page is under construction.

Definition of band offsets¶

The definition is identical to nextnano++, see Figure 2.3.1. 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 temperature_dependent_bandgap.

Note

The band offsets get an additional shift if strain is present.

Single-band model¶

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2-band model¶

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3-band model¶

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8-band model¶

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Output of effective mass in the multiband case¶

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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:

(4.2.1)¶\[S \to S' E_P \to E_P'\]

while the effective mass at bandedge is conserved.

(4.2.2)¶\[S + \frac{E_P}{E_g} = S' + \frac{E_P'}{E_g}\]

while for 3 bands it corresponds to:

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

Smoothing of the \(\mathbf{k} \cdot \mathbf{p}\) parameters (for multiband)¶

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Oscillator strength¶

The oscillator strength is calculated from the formula

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

Note

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

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

Last update: 30/10/2024