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qcl:electronic_band_structure [2020/02/27 18:55]
thomas.grange
qcl:electronic_band_structure [2020/04/15 15:40] (current)
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 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 50: 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 64: 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.1582829710.txt.gz · Last modified: 2020/02/27 18:55 by thomas.grange