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Non-linear photoconductivity of strongly driven graphene
by Lukas Broers, Ludwig Mathey
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Lukas Broers |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2312.13217v1 (pdf) |
| Date submitted: | 2023-12-21 10:05 |
| Submitted by: | Broers, Lukas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We present the non-linear DC photoconductivity of graphene under strong infra-red (IR) radiation. The photoconductivity is obtained as the response to a strong DC electric field, with field strengths outside of the linear-response regime, while the IR radiation is described by a strong AC electric field. The conductivity displays two distinct regimes in which either the DC or the AC field dominates. We explore these regimes and associate them with the dynamics of driven Landau-Zener quenches in the case of a large DC field. In the limit of large AC field, we describe the conductivity in a Floquet picture and compare the results to the closely related Tien-Gordon effect. We present analytical calculations for the non-linear differential photoconductivity, for both regimes based on the corresponding mechanisms. As part of this discussion of the non-equilibrium state of graphene, we present analytical estimates of the conductivity of undriven graphene as a function of temperature and DC bias field strength that show very good agreement with our simulations.
Current status:
Reports on this Submission
Report
This is a single-blind review of SciPost Physics manuscript 2312.13217v1 titled "Non-linear photoconductivity of strongly driven graphene" by Lukas Broers and Ludwig Mathey. The manuscript is a 27-pages long PDF file, including 7 figures, 117 equations, a 62-entry bibliography and 3 appendices with mathematical calculations (equations 29-117). The same manuscript is also on arxiv, https://arxiv.org/abs/2312.13217.
The work presented in the manuscript consists of the theoretical analysis of the longitudinal photoconductivity produced when a graphene monolayer is subjected to an E-field with a strong DC and/or strong AC (harmonic THz) component; the E-field is parallel to graphene. The quantum system studied is open, i.e., it assumes various dissipation and dephasing mechanisms, introduced by corresponding rates. The Authors identify two distinct regimes (AC>>DC or AC<<DC), for which they extract analytical formulas identifying the relevant boundaries and evaluating the response, i.e., the differential photo/conductivity (change in electric current due to a strong AC/DC E-field presence), which is highly nonlinear. The analytical results match the numerical ones, nothing that some of the formulas contain heuristically introduced parameters which can be tuned to fit the numerically computed response.
The paper is in overall very well written and presented, scientifically correct (theory matches numerical computation) and is positioned in the area of 2D material high-frequency nonlinearities which is highly relevant to the analysis and design of non-equilibrium quantum-electronic/photonic devices. For these reasons, I recommend publication of this paper after minor improvements. Below follows a list of comments that the Authors can use to improve their work by extending its impact on areas beyond mathematical physics:
A/ Even though the introduction is rather clear about the contents of the paper, it should further highlight the novelty of its core findings/results, with respect to the relevant body of the work in the literature. For instance, one wonders if these findings are consistent with this work https://doi.org/10.1103/PhysRevB.103.245406
B/ The experimental audience of the journal would appreciate a linking of these theoretical results to measured ones, at the FIR frequency chosen (~10 THz) or elsewhere (e.g., at NIR/optical or low-THz bands). As the paper mentions, such a comparison can be potentially exploited in the characterization of graphene quality, e.g., in the extraction of the effective dissipation-rate parameters from measured spectra.
C/ The Authors could consider to compare their results to quantities relevant to nonlinear optics, such third-order/Kerr-like effect [:https://doi.org/10.1103/PhysRevB.91.235320] or saturable absorption regimes [https://doi.org/10.1103/PhysRevB.100.115416]. Such cases could arise by their model in the limiting case of zero DC component (only AC drive) and by including energy-dependent relaxation (dissipation) rates. Coherent nonlinearities, in the phase and/or the magnitude of AC photoconductivity, are highly relevant to graphene-comprising electro-optic applications such as lasers or photodiodes.
D/ In several places in the paper, the dissipation is termed "small" (and has been further reduced, in some figures, for visual clarity); this refers to all the dissipation rates employed or to some of them? Do the Authors expect a different picture (for the two regimes identified) for more highly dissipative systems? The latter could be a parametric study using the derived formulas. As with the previous comment, the study of complex valued (i.e., in both magnitude and phase) photoconductivity in such high-loss nonlinear systems is relevant to applications in coherent and non-Hermitian photonic systems and devices/components.
E/ Some minor issues: (i) The paper should be swiped for consistency in the terminology used, e.g., use "bias" only for the DC component and "drive" only the AC one; presently, the word "probe" is sometimes used for DC and the word "drive" is sometimes used for either AC or DC; there is a typo (AC-->DC) in the first line of section 3.1. (ii) The reference list should be checked as some titles contain inappropriate lowercase characters (e.g., in names like zener or in acronyms like thz).
Report
The authors report the results of a theoretical study of graphene conductivity under the influence of strong ac and/or dc electric fields. The topic of this manuscript lies in the broad field of research on the nonlinear electromagnetic response of graphene, which began in 2007 after it was predicted that graphene should have highly nonlinear electrodynamic properties. The paper is potentially interesting and important but it is unclear whether the results of this work are new. There is an extensive literature on the relevant theoretical studies, first of all, the papers by Mikhailov and the group of Sipe, but the authors do not cite any of them and do not compare their results with those already available in the literature. Among the theoretical results obtained both within the framework of perturbation theory and non-perturbatively, both for the general third-order response and for the photoconductivity, the following works can be mentioned:
Mikhailov, Non-linear electromagnetic response of graphene, Europhys. Lett. 79, 27002 (2007)
Cheng et al., DC current induced second order optical nonlinearity in graphene, Optics Express 22 (13), 15868-15876 (2014)
Cheng et al., Third order optical nonlinearity of graphene, New Journal of Physics 16 (5), 053014 (2014)
Cheng et al., Third-order nonlinearity of graphene: Effects of phenomenological relaxation and finite temperature, Phys. Rev. B 91 (23), 235320 (2015)
Mikhailov, Quantum theory of the third-order nonlinear electrodynamic effects of graphene, Phys. Rev. B 93, 085403 (2016)
Mikhailov et al., Negative dynamic conductivity of a current-driven array of graphene nanoribbons, Phys. Rev. B 94, 035439 (2016)
Cheng et al., Numerical study of the optical nonlinearity of doped and gapped graphene: From weak to strong field excitation, Phys. Rev. B 92 (23), 235307
Al-Naib et al., Nonperturbative model of harmonic generation in undoped graphene in the terahertz regime, New Journal of Physics 17 (11), 113018 (2015)
Mikhailov, Nonperturbative quasiclassical theory of the nonlinear electrodynamic response of graphene, Phys. Rev. B 95, 085432 (2017)
Mikhailov, Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model, Phys. Rev. B 100, 115416 (2019)
Mikhailov, Nonperturbative quasiclassical theory of graphene photoconductivity”, Phys. Rev. B 103, 245406 (2021)
Authors should check whether their results are new compared to those already available in the literature, compare their and published results in the manuscript, and cite relevant articles accordingly. If there is anything new in the submitted manuscript, it may be published after the proposed changes have been made.
Author: Lukas Broers on 2024-09-30 [id 4808]
(in reply to Report 1 on 2024-02-15)
We thank the referee for their positive assessment of our work. We appreciate the referee drawing our attention to this body of work that we had not referenced. After studying the provided literature, we agree that the works are largely related and relevant. These studies cover different aspects of the large research field of electromagnetic responses in graphene. They all differ from our work in that they describe optical rather than direct responses, that the driving frequency is small compared to the chemical potential, that they utilize a semi-classical Boltzmann approach, or that they focus on third-order susceptibilities, to name the main differences. In most cases all or most of these points apply. We therefore did not find any instances of overlap in the results that would take away from the novelty of our results, nor stand in contradiction to them.
Regardless, we agree that these works should be mentioned and put into context in the introduction of our manuscript. We have therefore cited the provided references in the revised manuscript. We note that some of these references were also brought up by the other referee.
Author: Lukas Broers on 2024-09-30 [id 4809]
(in reply to Report 2 on 2024-02-19)We thank the referee for this very detailed reply, and for their positive assessment, as well as recommending publication of our work.
A/
We thank the referee for drawing our attention to this reference. It describes a non-perturbative Boltzmann approach for describing the photoconductivity of extrinsic graphene for small driving frequencies. In particular, they consider the driving photon energy to be below the Fermi energy, and therefore interband transitions become negligible. The work is consistent with our work in the sense, that it is complementary to our approach as it describes a different regime of photoconductivity and treats it with a distinct method. We have included this reference in the introduction of our revision. We note that this reference is also included in the list of references provided by the other referee, who raises similar concerns. We have compared these references to the results of our work, and find that they do not take away from our results.
B/
We thank the referee for this suggestion. A direct and in-depth comparison to experimental results would be valuable. We understand that the proper way of doing so is in a separate collaborative study, which we find to be out of the scope of our presented manuscript.
C/
We thank the referee for this suggestion. Third-order susceptibilities and saturation effects capture very interesting optical responses that we might explore in detail in future work. In this manuscript, we set on focusing on the non-perturbative DC responses in the presence of a DC field. While it is true that in the edge-case of a vanishing DC field, it is possible to study the third-order susceptibilities of the AC photoconductivity, we find this to be a topic that warrants a separate analysis and publication. We note that the two references the referee provides are also included in the list of references provided by the other referee. We include these references and contextualize them in the introduction of the revised manuscript.
D/
Yes, the reduction is applied to all dissipation coefficients equally, such that it keeps the ratios between coefficients fixed. In the revision we are more precise in how we phrase this. Indeed, for very strong dissipation we expect the characteristic features of the photoconductivity to be gradually lost. At that point we expect the dynamics to be dominated by semi-classical carrier dynamics. We believe that this comparison would be a very interesting analysis for future work.
E/
We thank the referee for these remarks, as they improve the manuscript. As suggested, we have gone over the manuscript to make the language more consistent, and we have fixed the typo. We have also gone over the reference list and have corrected the improper use of lowercase characters.