Semiconductor Bloch equations conserving momentum, examples

We provide examples to the procedure used to study the light-matter interaction with transfer of momentum.

In our previous post we sketched a procedure to derive the kinetics and dynamics of electrons in a solid, taking into account the conservation of momentum. Here, we provide examples and how to deduce complex quantities.

Quantum Ring + TL

1. Write the Hamiltonian in second quantization form:

with the light-matter matrix element between valence and conduction band: $\langle vm'|h_I|cm \rangle = const. \times \delta_{m-m',\pm \ell}$, where $\ell$ is the orbital angular momentum of the light beam. $\epsilon_{bm}$ is the bear energy of an electron in the ring.

2. Use the Heisenberg's equations of motion, to set up equations + 3. Convert the operator equations into c-number equations for the so-called coherences and populations:
$$\begin{eqnarray}     i\hbar \frac{d}{dt}{\rho}_{v, m' \,m} &=& (\varepsilon_{v m } - \varepsilon_{v m' })     \, {\rho}_{v, m' \,m} +     \xi^*\,     \widetilde{\rho}_{v m', c m+l} -     \xi\,     \widetilde{\rho}_{c m'+l, v m},     \label{eq:rho_v_TL} \\     i\hbar \frac{d}{dt} {\rho}_{c, m' \,m} &=& (\varepsilon_{c m } - \varepsilon_{c m' })     \, {\rho}_{c, m'\, m} +     \xi\,     \widetilde{\rho}_{c m', v m-l} -     \xi^*\,     \widetilde{\rho}_{v m'-l, c m}, \label{eq:rho_c_TL} \\     i\hbar \frac{d}{dt}\widetilde{\rho}_{v m', c m} &=& (\varepsilon_{c m } - \varepsilon_{v m' } - \hbar \omega)     \, \widetilde{\rho}_{v m', c m} +     \xi\,     \left(     {\rho}_{v, m' \,m-l} - {\rho}_{c, m'+l\, m}     \right)     \,, \label{eq:rho_vc_TL} \end{eqnarray}$$
4. Derive complex quantities: The orbital angular momentum of the whole system is divided into a coherence and population contributions
$$\begin{eqnarray*} {L}_z^{(coh)} &=&     \sum_{m m^\prime }     2\Re\left[     {\cal L}^{(p)}_{vm' cm}     \, \rho_{v m^\prime, c m}     \right]   \\ {L}_z^{(pop)} &=& \sum_{m  }     \hbar\,m     \, \rho_{c, m m} +     \sum_{m }     \left(         l_{z}  +  \hbar \,m     \right)     \, \rho_{v, m m} \,, \end{eqnarray*}$$
where one uses the solutions of the EOM [steps 2 and 3] to obtain explicit expressions for the OAM. [1]

Bulk + TL

1. Write the Hamiltonian in second quantization form: the Hamiltonian looks the same as before, but with matrix elements
$$\begin{eqnarray} \label{eq:Matrix_Element}     \langle{c \mathbf k' m'} |h_I^{(+)} |{v \mathbf k m}\rangle &=& \xi_{c k_r' m', v k_r m} \, e^{-i\omega t}\, \delta_{l, m' - m } \delta_{q_z, k_z' - k_z } \nonumber \\     \langle{v \mathbf k' m'} |h_I^{(-)} |{c \mathbf k m}\rangle &=& \xi_{c k_r m, v k_r' m'}^* \, e^{i\omega t}\, \delta_{-l, m' - m } \delta_{-q_z, k_z' - k_z } \,, \end{eqnarray}$$
we note that we used cylindrical electron's states, and this is why there is a $\delta_{\pm l, m' - m }$ on angular quantum numbers $m$.

2. Use the Heisenberg's equations of motion, to set up equations + 3. Convert the operator equations into c-number equations for the so-called coherences and populations using $$\begin{eqnarray} \rho_{c, \alpha' \alpha} = \langle \hat{\rho}_{c \alpha', c \alpha} \rangle, \nonumber \\ \rho_{v, \alpha' \alpha} = \langle \hat{\rho}_{v \alpha', v \alpha} \rangle, \nonumber \\ \rho_{v \alpha', c \alpha} = \langle \hat{\rho}_{v \alpha', c \alpha} \rangle \,. \end{eqnarray}$$
For example:
$$\begin{eqnarray} i\hbar \frac{d}{dt} \rho_{v k_z k_r' m, \, k_z k_r m} &=&     \Delta_{v k_z k_r m, \, k_z k_r' m}     \, {\rho}_{v k_z k_r' m, \, k_z k_r m} + \nonumber \\ &&  \hspace{-25mm}     e^{i\omega t} \sum_{k_r''} \,     \xi_{c k_r'' m+l, \, v k_r m}^* \,     {\rho}_{v k_z k_r' m, \, c k_z+q_z k_r'' m+l} -     e^{-i\omega t} \sum_{k_r''}     \xi_{c k_r'' m+l, \, v k_r m}\,     {\rho}_{c k_z+q_z k_r'' m+l, \, v k_z k_r m} \, . \label{eq:TL-SBE_v} \end{eqnarray}$$
We note that, because in these examples there are no 4 operators' terms, it is trivial to go from step 2 to 3, and so we have put together both steps. For a situation with 4 operators see Ref. [2].

4. Derive complex quantities: The paramagnetic electric current is $$\begin{eqnarray} \hat{\mathbf j}^{(p)}(\mathbf r,t) &=& -i\,\frac{q\,\hbar}{2\,m_e} \sum_{\scriptsize \begin{array}{c} b' \mathbf k' m' \\ b \mathbf k m \end{array} } \left[ \psi_{b' \mathbf k' m'}^*(\mathbf r) \, \nabla \psi_{b \mathbf k m}(\mathbf r) - \psi_{b \mathbf k m}(\mathbf r) \, \nabla \psi_{b' \mathbf k' m'}^*(\mathbf r) \right] \, {\rho}_{b' \mathbf k' m',b \mathbf k m}^\dagger (t)\,, \label{eq:jparamag} \end{eqnarray}$$
See Ref. [3]

REFERENCES

[1] Quinteiro, G. F., and J. Berakdar. Electric currents induced by twisted light in Quantum Rings, Optics express 17, no. 22 (2009): 20465-20475.
[2] G. F. Quinteiro, Below-bandgap excitation of bulk semiconductors by twisted light, EPL 91 27002 (2010).
[3] G. F. Quinteiro and P. I. Tamborenea, Twisted-light-induced optical transitions in semiconductors: Free-carrier quantum kinetics, Phys. Rev. B 82, 125207 (2010).