The semiconductor Bloch equations (SBEs) are an insightful and well-established formalism for studying light-matter interactions in solids. When Coulomb interactions between electrons are omitted, the SBEs are simplified to a single particle model. The SBEs in this single electron approximation have been used extensively to model strong-field interactions in condensed matter. The SBEs in the length gauge provide an intuitive and numerically efficient model of high harmonic generation (HHG) in solids. In this approach, the SBEs involve Berry connections and transition dipole moments, which are gauge dependent structural quantities. This thesis studies the role of gauge symmetry in the SBEs, and how it can be exploited to facilitate efficient numerical analysis of HHG in solids.
In the length gauge, the macroscopic current describing HHG can be decomposed into physically intuitive contributions. In particular, this leads to a contribution known as the "mixture" current, which has been overlooked by the HHG community until recently. We study the influence of this contribution using the analytic tight-binding model for gapped graphene. We derive an analytic gauge transformation that removes singular behaviour from the gapped graphene model, thus enabling efficient numerical integration of the SBEs.
We also present an alternative approach for simulating dynamics in tight-binding models. Instead of simulating the SBEs in the usual basis of Bloch functions, we transform to the basis in which the tight-binding Hamiltonian is represented. The dipole matrix elements necessarily vanish in this basis, and the SBEs can be integrated using only the Hamiltonian matrix elements. We first generalize the SBEs to accomodate a non-diagonal Hamiltonian matrix, and we demonstrate this formalism numerically using two different tight-binding models.
Finally, we derive a novel formulation of the SBEs which involve only gauge invariant matrix elements. Specifically, the Berry connections and transition dipole phases are replaced by a gauge invariant quantity known as the shift vector. This yields a fully gauge invariant description of HHG in solids, and the shift vector provides intuitive insight for HHG in systems with broken inversion symmetry. Further, the ability to describe HHG solely in terms of gauge invariant quantities raises new possibilities for tomographic reconstruction of crystal band structure, and this idea is discussed as a possible direction of future work.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/44307 |
| Date | 25 November 2022 |
| Creators | Parks, Andrew |
| Contributors | Brabec, Thomas |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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