User Tools

Site Tools


qcl:electronic_band_structure

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
qcl:electronic_band_structure [2020/10/22 14:37]
thomas.grange [Smoothing of the k.p parameters (2 or 3 band case)]
qcl:electronic_band_structure [2022/07/23 17:22] (current)
thomas.grange [Single-band model]
Line 1: Line 1:
 ====== Electronic Band Structure ====== ​ ====== Electronic Band Structure ====== ​
-(page in construction) 
  
 The band structure is modeled in the envelope function approximation,​ using either the single-band effective mass approximation or a multiband model. ​ The band structure is modeled in the envelope function approximation,​ using either the single-band effective mass approximation or a multiband model. ​
Line 25: Line 24:
  
 === Effective mass === === Effective mass ===
 +== Effective mass from k.p parameters ==
 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 36: Line 36:
 using the following equation: using the following equation:
  
-$$ \frac{m_0}{m^*} = S + \frac{E_P(E_g+2\Delta_{\text{SO}}/​3)}{E_g(E_g+\Delta_{\text{SO}})} $$+$$ \frac{m_0}{m_{\perp}^*} = S + \frac{E_P(E_g+2\Delta_{\text{SO}}/​3)}{E_g(E_g+\Delta_{\text{SO}})} $$
  
 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. 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.
  
 +== Effective mass without k.p parameters ==
 On the other hand, if On the other hand, if
 <​code>​ <​code>​
Line 47: Line 48:
 Note that the database can be overwritten in the input file. Note that the database can be overwritten in the input file.
  
 +<​code>​
 +<​Materials>​
 +  <​Material>​
 +      ...
 +      <​Effective_mass_from_kp_parameters>​no</​Effective_mass_from_kp_parameters>​
 +      <​Overwrite>​
 +        <​ElectronMass>​0.07</​ElectronMass>​
 +      </​Overwrite> ​   ​
 +  </​Material>​
 +</​Materials>​
 +</​code>​
 +
 +== Overwitting in-plane effective mass ==
 +
 +For anisotropic effective masses, the axial $m_{\perp}^*$ and in-plane $m_{\parallel}^*$ effective masses can be individually overwritten in a separate way in the following way:
 +
 +<​code>​
 +<​Materials>​
 +  <​Material>​
 +      ...
 +      <​Effective_mass_from_kp_parameters>​no</​Effective_mass_from_kp_parameters>​
 +      <​Overwrite>​
 +        <​ElectronMass>​0.07</​ElectronMass>​
 +        <​ElectronMass_inPlane>​0.12</​ElectronMass_inPlane>​
 +      </​Overwrite> ​   ​
 +  </​Material>​
 +</​Materials>​
 +</​code>​
 +
 +
 +== Non-parabolicity within single-band approach (not recommended) ==
 If the non-parabolicty flag is specified If the non-parabolicty flag is specified
 <​code>​ <​code>​
Line 56: Line 88:
 where $E_0$ is the energy difference between the lowest miniband and the conduction band edge. where $E_0$ is the energy difference between the lowest miniband and the conduction band edge.
  
-Note, however, that in this case the effective mass is not state-dependent,​ i.e. higher energy states have the same effective mass than the lowest miniband. A full nonparabolic description is provided by 2 or 3 band models described below.+Note, however, that in this case the effective mass is not state-dependent,​ i.e. higher energy states have the same effective mass than the lowest miniband. A full nonparabolic description is provided by 2or 3-band models described below.
  
 ==== 2-band model ==== ==== 2-band model ====
  
-The two-band model has to specified ​by using:+The two-band model allows for a more accurate description of the nonparabolicity. It can be activated ​by using the following syntax:
  
 <​code>​ <​code>​
Line 74: Line 106:
 $$ H(z) =  $$ H(z) = 
 \left( {\begin{array}{cc} \left( {\begin{array}{cc}
-   ​E_c(z) + \frac{\hbar^2 k_z^2}{2S(z) m_0} &  i P(z) k_z \\ +   ​E_c(z) + S(z)\frac{\hbar^2 k_z^2}{m_0} &  i P(z) k_z \\ 
-   - i P(z) k_z & E_v(z)\\+   - i P(z) k_z & E_v(z) ​+ (1+L(z))\frac{\hbar^2 k_z^2}{2 m_0}\\
   \end{array} } \right)   \end{array} } \right)
 $$  $$ 
Line 83: Line 115:
 P(z) = \sqrt{\frac{m_0 E_p(z)}{2}} P(z) = \sqrt{\frac{m_0 E_p(z)}{2}}
 $$ $$
 +$L$ correponds to the Dresselhaus parameter $L'$ used in 8-band k.p parameters of nextnano3/​nextnano++. By default, this value is set to -1. Note that when $S<0$ **and** $L\neq-1$, spurious solutions are likely to occur. Hence it is recommended either (i) to let $L=-1$ (default value), or (ii) to use renormalized k.p parameters $E_p$, $S$ and $L$ given by nextnano++ with $S=1$.
  
 If If
Line 95: Line 128:
 <​Effective_mass_from_kp_parameters>​no</​Effective_mass_from_kp_parameters>​ <​Effective_mass_from_kp_parameters>​no</​Effective_mass_from_kp_parameters>​
 </​code>​ </​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:+is specified, the effective mass $m^*_0$ of the database (or overwritten in the input file) is used as an input to calculate the k.p parameters.  
 +In this case, the parameter $S$ is set to $S=0$, and the interband energy is overwritten using:
 $$ $$
-E_p = \frac{m_0}{m^*_0} E_g+E_p = \frac{m_0}{m^*} E_g
 $$ $$
  
 In this case, non-parabolicity is controlled by the value of the bandgap. 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:+To further modify non-parabolicity,​ a non-parabolic coefficient $a_{np}$ can be further specified in the database or in the input file (optional):
 <​code>​ <​code>​
 <​NonParabolicityRelative>​0.02</​NonParabolicityRelative>​ <​NonParabolicityRelative>​0.02</​NonParabolicityRelative>​
Line 122: Line 156:
 ==== 3-band model ==== ==== 3-band model ====
  
-The 3-band model consider has to specified by using:+The 3-band model can be activated ​using:
  
 <​code>​ <​code>​
Line 132: Line 166:
 </​code>​ </​code>​
  
-It accounts for: +Its aim is to account accurately for the nonparabolicity of the conduction band. It accounts for the 3 following bands
-conduction band +  ​* ​conduction band 
-light-hole (lh) band +  ​* ​light-hole (lh) band 
-split-off (so) band+  ​* ​split-off (so) band
  
-The considered Hamiltonian reads:+In this basis, the considered Hamiltonian reads:
  
 $$ H =  $$ H = 
 \left( {\begin{array}{ccc} \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\\ +   ​E_c(z) + S(z)\frac{\hbar^2 k_z^2}{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{2}{3}} P(z) k_z &  E_{lh}(z) ​+ (1+L(z)) \frac{\hbar^2 k_z^2}{2 m_0} & 0\\ 
-i \sqrt{\frac{1}{3}} P(z) k_z & 0 & E_{so}(z)+i \sqrt{\frac{1}{3}} P(z) k_z & 0 & E_{so} + (1+L(z))\frac{\hbar^2 k_z^2}{2 m_0}(z)
   \end{array} } \right)   \end{array} } \right)
 $$  $$ 
  
-Note that in this case, the k.p parameters ​are considered, and the input of a nonparabolic ​coefficient ​is not supported.+ 
 +=== Input from effective mass and non-parabolicity === 
 +When 
 +<​code>​ 
 +<​Effective_mass_from_kp_parameters>​no</​Effective_mass_from_kp_parameters>​ 
 +</​code>​ 
 +is specifiedthe effective mass $m^*_0$ of the database (or overwritten in the input file) is used as an input to calculate ​the k.p parameters.  
 +In this case, the parameter $S$ is set to $S=0$, and the interband energy is overwritten using: 
 +$$ 
 +E_p = \frac{m_0}{m^*} \frac{E_g(Eg+\Delta_{SO})}{Eg+\frac{2}{3}\Delta_{SO}} 
 +$$ 
 + 
 +==== Output ​of the effective mass in the multiband case==== 
 + 
 +In the multiband case (2 or 3 bands), ​position and state-dependent effective mass is output in the file "​EffectiveMassesPosition.dat"​. Note that this effective mass is only an output in order to allow comparison with other models and to analyze the multiband calculation. 
 + 
 +In the 2-band case, its value is given for the level $i$ by: 
 + 
 +$$ \frac{m_0}{m_{\perp}^*(z,​i)} = S(z) + \frac{E_P(z)}{\epsilon_i-E_{v,​av}(z)} = S(z) + \frac{E_P(z)}{\epsilon_i-\left(E_{g}(z)+\frac{1}{3}\Delta_{SO}(z)\right)} $$ 
 + 
 +In the 3-band case, it reads 
 +$$ \frac{m_0}{m_{\perp}^*(z,​i)} = S(z) + \frac{2}{3} \frac{E_P(z)}{\epsilon_i-E_{lh}(z)} +  \frac{1}{3} \frac{E_P(z)}{\epsilon_i-E_{SO}(z)}$$ 
 + 
 +where $\epsilon_i$ is the energy of level $i$. 
 +Note that this formula accounts for nonparabolicity as the effective mass depends on the energy difference between the energy level and the valence bandedges. 
 + 
 +From this position-dependent effective masses, an averaged effective mass can de defined for each level, accounting for nonparabolicity. The state-dependent effective mass $m_{\perp}^*(i)$ is  output in the file "​EffectiveMasses.dat"​ and is defined by the following averaging:​ 
 +$$ 
 +\frac{m_0}{m_{\perp}^*(i)} = \int dz \frac{m_0}{m_{\perp}^*(z,​i)} |\Psi_i(z)|^2  
 +$$ 
 + 
 +=== In-plane nonparabolicity === 
 +In the multiband models, to account that the in-plane effective mass depends on both the energies along the growth axis $z$ and the in-plane directions ($x$,$y$), the following command has to included in ''<​Materials>'':​ 
 +<​code>​ 
 +<​Materials>​ 
 +   ... 
 +   <​InPlaneNonParabolicity>​yes</​InPlaneNonParabolicity>​ 
 +   ... 
 +</​Materials>​ 
 +</​code>​ 
 +If ''​no''​ is specified, the dispersion will be nonparabolic ​along the growth axis, while parabolic in the plane. 
 + 
 +==== Rescaling of k.p parameters (for multiband) ==== 
 +It is possible to rescale the $S = 1+2F$ parameter (where $F$ is the remote-band contribution) by using the following command: 
 +<​code>​ 
 +<​Material>​ 
 +            <​Name>​In(x)Ga(1-x)As</​Name>​ 
 +            ... 
 +            <​Effective_mass_from_kp_parameters>​yes</​Effective_mass_from_kp_parameters>​ 
 +            <​Rescale_S>​yes</​Rescale_S>​ 
 +            <​Rescale_S_to>​1.0</​Rescale_S_to>​ 
 +        </​Material>​ 
 +</​code>​ 
 + 
 +The effect on $S$ and $Ep$ of rescaling is the following:​ 
 +$$S \rightarrow S'$$ 
 +$$E_p \rightarrow E_p' $$ 
 +while the effective mass at bandedge is conserved. 
 +For 2 bands this corresponds to: $$ S + \frac{E_P}{E_g} = S' + \frac{E'​_P}{E_g}$$ 
 +while for 3 bands it corresponds to: 
 +$$ S + \frac{E_P(E_g+2\Delta_{\text{SO}}/​3)}{E_g(E_g+\Delta_{\text{SO}})} = S' + \frac{E'​_P(E_g+2\Delta_{\text{SO}}/​3)}{E_g(E_g+\Delta_{\text{SO}})}$$ 
 + 
 +Note that the effective mass is conserved at the bandedge energy, but that the nonparabolicity will be affected by such rescaling. 
 + 
 +In the above example, rescaling $S$ to 1, means that only the free electron kinetic energy term will remain (i.e. no remote-band contribution,​ $F=0$).
  
 ==== Smoothing of the k.p parameters (2 or 3 band case) ==== ==== Smoothing of the k.p parameters (2 or 3 band case) ====
Line 171: Line 269:
  
 The smoothing is performed by performing a convolution of the $E_p(z)$ and $S(z)$ spatial functions by a Gaussian $e^{-z^2/​L_s^2}$ where $L_s$ is either the $E_p$ or $S$ smoothing length. The smoothing is performed by performing a convolution of the $E_p(z)$ and $S(z)$ spatial functions by a Gaussian $e^{-z^2/​L_s^2}$ where $L_s$ is either the $E_p$ or $S$ smoothing length.
 +
 +
 +==== Definition of band offsets ====
 +
 +There are two options to define band offsets:
 +  * Option a) Specify conduction band offset (CBO) directly, and then the valence band offset (VBO) is calculated.
 +  * Option b) Specify valence band offset (VBO), and then the conduction band offset (CBO) is calculated.
 +
 +{{ :​qcl:​valenceband_edges.gif?​direct |}}
 +
 +$E_{\rm v,av}$ is the average energy of heavy hole (hh), light hole (lh) and split-off hole (so).
 +$\Delta_{\rm so}$ is the spin-orbit split-off energy.
 +
 +  * Option a) Specify conduction band offset (CBO) $E_{\rm c}$\\ ''<​UseConductionBandOffset>​yes</​UseConductionBandOffset>''​
 +The valence bandedges are then defined according to
 +$$E_{\rm lh}(T) = E_{\rm c} - E_{\rm g}(T)$$
 +$$E_{\rm so}(T) = E_{\rm c} - E_{\rm g}(T) - \Delta_{\rm so} = E_{\rm lh}(T) - \Delta_{\rm so} $$
 +
 +  * Option b) Specify valence band offset (VBO) $E_{\rm v,av}$\\ The conduction band edge $E_{\rm c}$ is calculated and depends on temperature.\\ ''<​UseConductionBandOffset>​no</​UseConductionBandOffset>''​ (default)
 +\begin{align*}
 +  E_{\rm lh}   & = E_{\rm v,​av}+\frac{1}{3}\Delta_{\rm so}\\
 +  E_{\rm so}   & = E_{\rm v,​av}-\frac{2}{3}\Delta_{\rm so}\\
 +  E_{\rm c}(T) & = E_{\rm lh} + E_{\rm g}(T)
 +\end{align*}
 +
 +Note that the band gap $E_{\rm gap}$ is temperature dependent (Varshni formula),
 +$$E_{\rm gap}(T)=E_{\rm gap}(T=0 {\rm ~K})-\frac{\alpha T^2}{T+\beta},​$$
 +where $\alpha$ and $\beta$ are the Varshni parameters. On the other hand, the bandoffsets of the database don't have any tabulated temperature dependence.
 +Hence the two options lead to different temperature dependence, as the temperature dependence of the bandgap is  attributed to the conduction band if ''<​UseConductionBandOffset>​no</​UseConductionBandOffset>'' ​ is used, and to the valence band in the ''<​UseConductionBandOffset>​yes</​UseConductionBandOffset>''​ case.
 +----
 +
qcl/electronic_band_structure.1603377426.txt.gz · Last modified: 2020/10/22 14:37 by thomas.grange