5. Multivalley effectivemass theory solver
Differential equation
We consider a material with conductionband minima (valleys) located at \(\mathbf k_\nu\), with \(\nu\) being the valley index. Within multivalley effectivemass theory, we approximate the conductionelectron wavefunctions as [GJN+15, SN76]
where the sum is over all degenerate valleys \(\nu\). In Eq. (5.1), we have introduced \(F_\nu(\mathbf r)\), the envelope function for valley \(\nu\), and \(\xi_\nu(\mathbf r)\equiv u_\nu(\mathbf r)\mathrm e^{i\mathbf k_\nu\cdot\mathbf r}\), the Bloch function evaluated at the \(\nu\)th conduction band minimum, with \(u_\nu (\mathbf r)\) being the latticeperiodic Bloch amplitude of this state.
Assuming that the lengthscale for variations of the envelope functions is much larger than the lengthscale for variations of the Bloch functions (which is commensurate with the lattice constant), a set of coupled Schrödinger equations may be written as
where \(\mathbf M_\nu^{1}\) is the effective mass tensor for valley \(\nu\). Remark that we have one equation for each valley, and that these equations are coupled by the second term on the lefthand side of Eq. (5.2), which contains
where \(\Omega_{\mathbf r}\) is the lattice unit cell located at position \(\mathbf r\), and \(\Omega_{\mathbf r}=\Omega\;\forall\;\mathbf r\) is the unit cell volume (which is the same for all \(\mathbf r\) for a homogeneous semiconductor).
The quantity \(V^\mathrm{VO}_{\nu\nu'}(\mathbf r)\) introduced in Eq. (5.3) describes the impact of the confinement potential on the electronic structure of the device. If the confinement potential varies significantly over the lattice spacing, the term \(V^\mathrm{VO}_{\nu\nu'}(\mathbf r)\) in Eq. (5.3) gives rise to significant coupling between the valleys \(\nu\) and \(\nu'\) (valley–orbit coupling). In contrast, if the confinement potential varies negligibly over the lattice spacing, no significant coupling between the valleys arises due to the orthonormality of the Bloch functions, resulting in \(V^\mathrm{VO}_{\nu\nu'}(\mathbf r)\approx V_\mathrm{conf}(\mathbf r)\delta_{\nu\nu'}\).
To calculate the valley–orbit coupling term defined in Eq. (5.3), it is useful to write the Bloch amplitudes as a Fourier decomposition:
where \(A_\mathbf G^\nu\) is the Fourier component of the Bloch amplitude
associated with the reciprocal lattice vector \(\mathbf G\). These
Fourier components may be calculated using densityfunctional theory (e.g.
with RESCU+).
In QTCAD, it is possible
to import the Fourier components of Bloch amplitudes from text files.
In particular, it is possible to import
Fourier components of the silicon Bloch amplitudes calculated in
[SCalderonC+11]; these Fourier components are stored in
qtcad/device/valleycoupling/bloch/silicon
.
More practical information on how to use the multivalley effective mass theory
solver and the bloch_tools
module may be found
in the Valley coupling (MVEMT) tutorial.
Relevant device attributes
Parameter 
Symbol 
QTCAD name 
Unit 
Default 
Setter 
Confinement potential 
\(V\) 

J 
0 

External conf. potential 
\(V_\mathrm{ext}\) 

J 
0 
Note
The total confinement potential \(V_\mathrm{conf} (\mathbf r)\) employed by the multivalley effective mass theory solver is given by \(V_\mathrm{conf} (\mathbf r) = V(\mathbf r)+V_\mathrm{ext}(\mathbf r)\).
Boundary conditions
The boundary conditions for the multivalley effective mass theory solver are the same as for the regular Schrödinger solvers. These boundary conditions are described in Boundary conditions.