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qcl:electronic_band_structure
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| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
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qcl:electronic_band_structure [2020/02/27 18:47] thomas.grange |
qcl:electronic_band_structure [2020/04/15 15:40] thomas.grange [2-band model] |
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| - | ====== Electronic Band Structure (in construction) ====== | + | ====== 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. | The band structure is modelled in the enveloppe function approximation, using either the single-band effective mass approximation or a multiband model. | ||
| Line 21: | Line 22: | ||
| where $m_{\perp}^*(z)$ is a position-dependent effective mass along the growth direction. | 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 | If the following option is used, the effective mass is calculated from the k.p parameters | ||
| <code> | <code> | ||
| Line 33: | Line 35: | ||
| $$ \frac{m_0}{m^*} = S + \frac{E_P(E_g+2\Delta_{\text{SO}}/3)}{E_g(E_g+\Delta_{\text{SO}})} $$ | $$ \frac{m_0}{m^*} = S + \frac{E_P(E_g+2\Delta_{\text{SO}}/3)}{E_g(E_g+\Delta_{\text{SO}})} $$ | ||
| + | |||
| + | where $E_P$ is the Kane energy, $E_g$ the band gap and $\Delta_{\text{SO}}$ is the spin-orbit splitting [Vurgaftman2001]. The effective mass of the database is ignored in this case. | ||
| + | |||
| + | 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> | ||
| + | <Effective_mass_from_kp_parameters>no</Effective_mass_from_kp_parameters> | ||
| + | </code> | ||
| + | is specified, the effective mass is taken directly from the nextnano.QCL database. | ||
| + | Note that the database can be overwritten in the input file. | ||
| ==== 2-band model ==== | ==== 2-band model ==== | ||
| Line 46: | 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 60: | Line 128: | ||
| </code> | </code> | ||
| + | $$ H = | ||
| + | \left( {\begin{array}{ccc} | ||
| + | E_c(z) + \frac{\hbar^2 k_z^2}{2S(z) m_0} & i \sqrt{\frac{2}{3}} P(z) k_z & -i \sqrt{\frac{1}{3}} P(z) k_z\\ | ||
| + | - i \sqrt{\frac{2}{3}} P(z) k_z & E_{lh}(z) & 0\\ | ||
| + | i \sqrt{\frac{1}{3}} P(z) k_z & 0 & E_{so}(z) | ||
| + | \end{array} } \right) | ||
| + | $$ | ||
qcl/electronic_band_structure.txt · Last modified: 2022/07/23 17:22 by thomas.grange
