How are dipole moment matrix elements calculated in practice for a semiconductor?

Question

I have been trying to understand paper [1] and replicate their plot in figure 1. They apparently explain their calculation method, but I cannot seem to understand how to get started.

In short, the paper aims to study the optical absorption that stems from the transitions of a heavy-hole valence band to the split-off valence band. They follow the work from the density matrix theory to get the absorption equation, given by:

$$ \alpha = \omega \sqrt{\frac{\mu}{\epsilon}}\int_0^\infty \langle R_{sh}^2\rangle \frac{f_v(\hbar/\tau_{in})k^2}{(E_{sh}-\hbar\omega)^2+(\hbar/\tau_{in})^2}\frac{dk}{\pi^2} $$

I can accept most of it. However, when it comes to $\langle R_{sh}^2\rangle$, which is the dipole moment formed between a heavy hole and a spin split-off hole. As far as I understand, the$\langle R_{sh}^2\rangle $ should be given by:

$$ \langle R_{sh}^2\rangle = \left| \langle u_{ss,k}| \mathbf{e}\cdot\mathbf{\hat{p}}|u_{hh,k}\rangle \right|^2 $$

where $|u_{hh,k}\rangle$ represents the envelope of the bloch wavefunction as $\phi_{hh,k}(\mathbf{r},t) = u_{hh,k}(\mathbf{r}) e^{i\mathbf{k}\cdot \mathbf{r}}$ and likewise for the spit off band. If this were to be true, how would we estimate the envelopes $u_{i,k}(\mathbf{r})$? This same dillema appears if I try to follow the optical absorption stemming from Fermi's golden rule, which also requires the dipole moment matrix element. In the cited paper, they say that:

$$ \langle R_{sh}^2\rangle =\frac{1}{3}\left(\frac{e}{E_{sh}}\right)^2\mathbf{p}^2a_s^2 \delta\mathbf{k_{hh}}\cdot\mathbf{k_s} $$

where $\frac{jm_0}{\hbar}\mathbf{p}$ is the momentum matrix element formed between a hole in the heavy hole band and one in the split off band, $m_0$ is the free electron mass, $a_s^2$ is the component ratio of the s-like function for a spin split-off valence band (as is expressed by $a_{v3}$ in $\mathbf{k}\cdot\mathbf{p}$ perturbation theory). But again the problem arises in evaluating the moment matrix element...

Question

How are dipole moment matrix elements $\left| \langle u_{ss,k}| \mathbf{e}\cdot\mathbf{\hat{p}}|u_{hh,k}\rangle \right|^2$ calculated in practice when dealing with semiconductor bandstructures?

PS: on the last equation cited I also don't know what that $\delta$ stands for.

[1] - https://doi.org/10.1063/1.1521260

Answer 1

I I can offer this point of view:

If we were to look at just a conduction and valence band dipole transition element, i.e : $$ d_{cv}=\langle c|\vec{e} \cdot \vec{p}|v\rangle $$ where $\langle c|$ and $|v\rangle$ describes the valence and conduction band wave functions. Let's consider a simple case of an out-of-plane dipole moment, this would give $$ d^z_{cv}=e\langle c|\vec{z}|v\rangle $$ lets say $$ |c\rangle = u_{c,k}(\vec{r})e^{i\vec{k}\cdot \vec{r}} $$ $$ |v\rangle = u_{v,k}(\vec{r})e^{i\vec{k}\cdot \vec{r}} $$ such that $$ d^z_{cv}=e\int d\vec{r}\hspace{2pt} u^*_{c,k}(\vec{r})\hspace{2pt} \vec{z} \hspace{2pt}u_{v,k}(\vec{r}) $$ where $u_{c/v,k}(\vec{r}) $ describes the wavefunction/eigenvectors of the valence and conduction band. These can be obtained by many means, e.g from DFT and tight-binding models - where in the tight binding model you obtain the eigenvectors and you can change the above integration to a summation over the orbitals of the system