| # Ellipticity-dependent high-harmonic emission from graphene |
|
|
| ## Background |
|
|
| Elliptically polarized light can drive electrons through a two-dimensional |
| crystal in ways that have no direct gas-phase analogue. In graphene, the |
| emitted high-harmonic signal may therefore contain information about the |
| driven band dynamics. |
|
|
| Consider undoped monolayer graphene in a nearest-neighbour, two-band |
| tight-binding model. Let |
|
|
| $$ |
| \mathbf a_1=\frac{a}{2}(\sqrt{3},1),\qquad |
| \mathbf a_2=\frac{a}{2}(\sqrt{3},-1), |
| $$ |
|
|
| with $a=2.46\ \mathring{\mathrm A}$. Use the standard nearest-neighbour |
| two-band graphene Hamiltonian with hopping energy $\gamma=3.03\ \mathrm{eV}$ |
| and work in the Houston basis, with $\mathbf q=\mathbf k+\mathbf A(t)$ and |
| $\mathbf F(t)=-\partial_t\mathbf A(t)$. Work in atomic units after conversion |
| of the supplied dimensional quantities. Use an undoped 300 K Fermi-Dirac |
| initial state, include an interband dephasing time $T_2=35\ \mathrm{fs}$, and |
| neglect longitudinal population relaxation. |
|
|
| The driving field is |
|
|
| $$ |
| F_x(t)=\frac{F_0}{\sqrt{1+\epsilon^2}} |
| e^{-2\ln2\,(t/t_d)^2}\cos(\omega_0t), |
| $$ |
|
|
| $$ |
| F_y(t)=\frac{\epsilon F_0}{\sqrt{1+\epsilon^2}} |
| e^{-2\ln2\,(t/t_d)^2}\sin(\omega_0t), |
| $$ |
|
|
| with $0\leq\epsilon\leq1$, central wavelength $4.7\ \mu\mathrm m$, |
| $t_d=160\ \mathrm{fs}$, $F_0=8.2348\times10^6\ \mathrm{V\,cm^{-1}}$, |
| and zero carrier-envelope phase. Fix the major axis along $k_x$. |
| |
| ## Research problem |
| |
| Determine how the fifth-, seventh-, and ninth-harmonic responses of this model |
| depend on the driver ellipticity. Establish, using a physically discriminating |
| evidence package of your own design, which microscopic part or parts of the |
| model account for any nontrivial ellipticity dependence you find. Explain the |
| physical and numerical evidence on which your conclusion rests and delimit the |
| scope supported by that evidence. |
| |
| Your evidence design, number of plots, intermediate diagnostics, and order of |
| argument are for you to decide. No particular current decomposition, |
| momentum-space intervention, or source-style panel sequence is required. |
| |
| ## Method boundary |
| |
| Use the stated two-band Houston-basis density-matrix model over a complete |
| reciprocal primitive cell. All band, optical-coupling, and emission quantities |
| must be mutually consistent with the same eigenstates and gauge. |
| |
| Construct the physical total current coherently in the time domain. For a |
| two-component current, define the total spectral strength by |
| |
| $$ |
| I(\omega)\propto |
| |\omega\widetilde J_x(\omega)|^2+ |
| |\omega\widetilde J_y(\omega)|^2. |
| $$ |
| |
| Obtain a harmonic yield by integrating a declared neighbourhood of the |
| relevant harmonic peak. For the principal ellipticity comparison, normalize |
| each harmonic order to its own value at $\epsilon=0$. If you compare resolved |
| current components, keep them on one common declared scale; do not normalize |
| each component independently in a way that erases their relative magnitude. |
| Absolute spectral normalization is not a grading target. |
| |
| The reciprocal-space grid, time grid, propagator, finite-record interval, FFT |
| window, zero padding, harmonic integration bandwidth, convergence strategy, |
| and display convention are student choices. State them and show that the |
| physical claims are supported at the resolution used. Physically equivalent |
| choices are accepted; no reference-program setting is prescribed. |
| |
| Do not add doping, additional bands, electron-hole interactions, macroscopic |
| propagation, or a fitted relaxation model to the required calculation. Such |
| extensions may be discussed only as clearly separated exploratory work. |
| |
| ## Submission and resources |
| |
| Submit one self-contained scientific report and an auditable offline |
| calculation package. Include the derivation, code, numerical data underlying |
| all material claims, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise run/environment |
| information. No fixed file tree, filename scheme, figure count, or complete |
| exploration history is required. |
| |
| Resource limits are supplied exclusively by the accompanying |
| `resource_limits.json`. Its fixed CPU allocation, candidate-active |
| wall time, network boundary are binding. No separate aggregate CPU-hour limit applies. Greater computational expense |
| receives no additional credit. |
|
|
| ## Reference and integrity policy |
|
|
| This examination is conducted without network access or live literature |
| retrieval. You may rely on your existing general scientific knowledge and, if any are |
| explicitly supplied with the examination, on those offline materials or |
| software documentation. Record every such source actually consulted. For formulas, |
| methods, constants, or conventions used from general knowledge rather than a |
| consulted source, state enough of the method basis, definition, value, units, |
| and convention to make the calculation auditable; do not invent a citation. |
| You are not required to identify or prove a publication date or to retrieve |
| literature during the examination. |
|
|
| The hidden target article and its identifying metadata, figures or captions, |
| supplementary material, author code or data, target-specific reproduction |
| code, solution or scoring material, online solutions, and worked |
| reproductions of this case must not be used. All submitted scientific figures |
| and numerical data must be generated by your own implementation of the stated |
| problem. |
|
|
| See [submission requirements](../../submission_requirements.md) for the common submission entry points. |
|
|
