This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
qcl:electronic_band_structure [2020/04/13 16:16] thomas.grange [Single-band model] |
qcl:electronic_band_structure [2020/04/15 15:40] thomas.grange [2-band model] |
||
---|---|---|---|
Line 36: | Line 36: | ||
$$ \frac{m_0}{m^*} = S + \frac{E_P(E_g+2\Delta_{\text{SO}}/3)}{E_g(E_g+\Delta_{\text{SO}})} $$ | $$ \frac{m_0}{m^*} = S + \frac{E_P(E_g+2\Delta_{\text{SO}}/3)}{E_g(E_g+\Delta_{\text{SO}})} $$ | ||
- | where $E_P$ is the Kane energy, $E_g$ the band gap and $\Delta_{\text{SO}}$ is the spin-orbit splitting [Vurgaftman2001]. | + | 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 | + | If the non-parabolicty flag is specified |
+ | <code> | ||
+ | <NonParabolicity>yes</NonParabolicity> | ||
+ | </code> | ||
+ | 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 | ||
<code> | <code> | ||
<Effective_mass_from_kp_parameters>no</Effective_mass_from_kp_parameters> | <Effective_mass_from_kp_parameters>no</Effective_mass_from_kp_parameters> | ||
Line 57: | Line 66: | ||
</code> | </code> | ||
+ | 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 | ||
+ | <code> | ||
+ | <Effective_mass_from_kp_parameters>yes</Effective_mass_from_kp_parameters> | ||
+ | </code> | ||
+ | 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 | ||
+ | <code> | ||
+ | <Effective_mass_from_kp_parameters>no</Effective_mass_from_kp_parameters> | ||
+ | </code> | ||
+ | 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: | ||
+ | <code> | ||
+ | <NonParabolicityRelative>0.02</NonParabolicityRelative> | ||
+ | </code> | ||
+ | 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 ==== | ==== 3-band model ==== | ||
Line 71: | Line 128: | ||
</code> | </code> | ||
+ | $$ H = | ||
+ | \left( {\begin{array}{ccc} | ||
+ | E_c(z) + \frac{\hbar^2 k_z^2}{2S(z) m_0} & i \sqrt{\frac{2}{3}} P(z) k_z & -i \sqrt{\frac{1}{3}} P(z) k_z\\ | ||
+ | - i \sqrt{\frac{2}{3}} P(z) k_z & E_{lh}(z) & 0\\ | ||
+ | i \sqrt{\frac{1}{3}} P(z) k_z & 0 & E_{so}(z) | ||
+ | \end{array} } \right) | ||
+ | $$ |