Optical third harmonic generation in black phosphorus

We present a calculation of Third Harmonic Generation (THG) for two-band systems using the length gauge that avoids unphysical divergences otherwise present in the evaluation of the third order current density response. The calculation is applied to bulk and monolayer black Phosphorus (bP) using a non-orthogonal tight-binding model. Results show that the low energy response is dominated by mixed inter-intraband processes and estimates of the magnitude of THG susceptibility are comparable to recent experimental reports for bulk bP samples.Comment: 9 pages, 5 figure

Harmonic analysis of fractal measures induced by representations of a certain C∗^*-algebra

We describe a class of measurable subsets $\Omega$ in \br^d such that $L^2(\Omega)$ has an orthogonal basis of frequencies $e_\lambda(x)=e^{i2\pi\lambda\cdot x}(x\in\Omega)$ indexed by \lambda\in\Lambda\subset\br^d. We show that such spectral pairs $(\Omega ,\Lambda)$ have a self-similarity which may be used to generate associated fractal measures $\mu$ with Cantor set support. The Hilbert space $L^2(\mu)$ does not have a total set of orthogonal frequencies, but a harmonic analysis of $\mu$ may be built instead from a natural representation of the Cuntz C$^*$- algebra which is constructed from a pair of lattices supporting the given spectral pair $(\Omega ,\Lambda)$. We show conversely that such a pair may be reconstructed from a certain Cuntz-representation given to act on $L^2(\mu)$.Comment: 7 page

Iterative approach to arbitrary nonlinear optical response functions of graphene

Two-dimensional materials constitute an exciting platform for nonlinear optics with large nonlinearities that are tunable by gating. Hence, gate-tunable harmonic generation and intensity-dependent refraction have been observed in e.g. graphene and transition-metal dichalcogenides, whose electronic structures are accurately modelled by the (massive) Dirac equation. We exploit on the simplicity of this model and demonstrate here that arbitrary nonlinear response functions follow from a simple iterative approach. The power of this approach is illustrated by analytical expressions for harmonic generation and intensity-dependent refraction, both computed up to ninth order in the pump field. Moreover, the results allow for arbitrary band gaps and gating potentials. As illustrative applications, we consider (i) gate-dependence of third- and fifth-harmonic generation in gapped and gapless graphene, (ii) intensity-dependent refractive index of graphene up to ninth order, and (iii) intensity-dependence of high-harmonic generation.Comment: 6 pages, 5 figures. Supplemental material: 6 pages, 2 figure

Linear and nonlinear optical response of crystals using length and velocity gauges: Effect of basis truncation

We study the effects of a truncated band structure on the linear and nonlinear optical response of crystals using four methods. These are constructed by (i) choosing either length or velocity gauge for the perturbation and (ii) computing the current density either directly or via the time-derivative of the polarization density. In the infinite band limit, the results of all four methods are identical, but basis truncation breaks their equivalence. In particular, certain response functions vanish identically and unphysical low-frequency divergences are observed for few-band models in the velocity gauge. Using hexagonal boron nitride (hBN) monolayer as a case study, we analyze the problems associated with all methods and identify the optimal one. Our results show that the length gauge calculations provide the fastest convergence rates as well as the most accurate spectra for any basis size and, moreover, that low-frequency divergences are eliminated.Comment: 11 pages, 7 figure