Semiconductor Bloch equations conserving momentum

We sketch a procedure to theoretically study the light-matter interaction without neglecting the orbital angular momentum of light.

There are several ways to treat the problem of light-matter interaction, for example the simple evaluation of matrix elements with Fermi's Golden Rule in mind [1], the use of a single-particle picture with the Schrödinger's equation [2], and the use equations of motion.

A common practice has been to neglect the momentum of the photon in the excitation of solids, arguing that the momentum of light is small compare to the quasi-momentum of electrons in the crystal. The neglect of the linear momentum is an approximation that proved very useful, however, not in all cases, as we discussed in a previous post. Actually, it is imperative to consider the conservation of momentum in the twisted light - matter interaction.

We sketch now the procedure to derive Semiconductor Bloch Equations of motion [3]:

1. Write the Hamiltonian in second quantization form, using electron's creation $a^\dagger_{bi}$ and annihilation $a_{bi}$ operators of band $b$ and momentum $i$ of a bare Hamiltonian $H_0$. For example, the interaction with light is

\begin{eqnarray}\Large h_I=\sum_{b,i',b',i} \langle b',i' | -\frac{q}{m} \mathbf p \cdot \mathbf A | b,i \rangle a^\dagger_{b'i'} a_{bi} \end{eqnarray}

and we do this for each Hamiltonian term, including if we want electron-electron interactions [5].

The matrix elements are explicitly calculated. It is here where we depart from the common derivation and allow for conservation of momentum: the sum will be over $i'\neq i$. For example, in the case of a one-dimensional quantum ring with orbital quantum number $m$, the conservation of orbital angular momentum becomes $m' = m+\ell$.

2. Use the Heisenberg's equations of motion

to set up equations for pairs of creation and annihilation operators: $a^\dagger_{b'i'}a_{bj}$. At this stage all quantities are operators. See Ref. 4 for an example in bulk.

3. We convert the operator equations into c-number equations for the so-called coherences and populations (same quantum numbers)  density matrix elements $\rho_{b'i,bj}=\langle a^\dagger_{b'i}a_{bj} \rangle$, by taking expectation values over the initial state of the system, for example the ground state of all electrons in the valence band.

We are careful with terms that have products of 4 operators: one has to choose a way to separate the average of 4 into products of averages of 2 operators.

The system of equations may be simplified to a smaller set of equations using the conservation of momentum. See for example the case of quantum rings, where one ends up with a set of three equations of motion.

We are all set to study the kinetics/dynamics of electrons!

In the next post I will continue with examples and deducing complex quantities such as the electric current.


REFERENCES

[1] G. F. Quinteiro and P. I. Tamborenea, Electronic transitions in disk-shaped quantum dots induced by twisted light, Phys. Rev. B 79, 155450 (2009).
[2] G. F. Quinteiro and P. I. Tamborenea, Theory of the optical absorption of light carrying orbital angular momentum by semiconductors, EPL 85, 47001 (2009).
[3] Hartmut Haug, Stephan W Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th Edition.
[4] G. F. Quinteiro and P. I. Tamborenea, Twisted-light-induced optical transitions in semiconductors: Free-carrier quantum kinetics, Phys. Rev. B 82, 125207 (2010).
[5] G. F. Quinteiro, Below-bandgap excitation of bulk semiconductors by twisted light, EPL 91 27002 (2010).