Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model

S. A. Mikhailov

REVIEW 3 major objections 4 minor 13 references

Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that graphene's strongly nonlinear absorption is governed by a hot electron model whose intensity dependence changes sign at the interband threshold, and that the model matches recent pump-probe data.

desk verdict A transparent, useful hot-electron framework for strong-field graphene optics, with a genuine approximation worry in the absorption coefficient that the authors should benchmark before publication. read the letter →

arxiv 1908.04631 v1 pith:BFE4AOXG submitted 2019-08-13 cond-mat.mes-hall cond-mat.mtrl-sci

S. A. Mikhailov This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-sci

keywords graphenehotelectronmodelnonlinearopticalresponsesaturableabsorptioninducedinterbandrecombinationterahertzspectroscopyquasi-equilibriumFermidistribution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a hot electron model for graphene under intense radiation, in which photoexcited electrons and holes keep quasi-equilibrium Fermi distributions with their own chemical potentials and a common elevated temperature. It claims that coupled equations for energy relaxation and recombination let one compute the hot-electron temperature, the non-equilibrium carrier densities, and the absorption coefficient as functions of frequency, intensity, doping, and relaxation times. The central result is that the absorption coefficient's intensity dependence changes sign near $\hbar\omega \approx 2|\mu_0|$: absorption grows with intensity below this threshold and falls above it. The paper reports that this behavior, and the accompanying spectra, agree with recent experiments.

What carries the argument

The machinery is the quasi-equilibrium hot-electron ansatz: after fast electron-electron thermalization, the electron and hole gases are described by Fermi functions with separate chemical potentials and a common temperature $T$. The absorption coefficient is then evaluated by inserting these nonequilibrium parameters into the linear-response conductivity, and the unknowns are fixed by two balance equations — energy relaxation of the total electron-hole energy and generation-recombination balance of the carrier densities — together with inversion relations linking densities to chemical potentials. The steady-state problem reduces to one dimensionless equation depending only on the ratio $\tau_\epsilon/\tau_{\rm rec}$, which is the paper's single essential fitting parameter.

What would settle it

Measure the absorption coefficient versus intensity at several photon energies on both sides of $\hbar\omega = 2|\mu_0|$ in a single gated graphene sample at fixed lattice temperature; the model predicts absorption curves that cross at one point near the threshold, with $dA/dI>0$ below and $dA/dI<0$ above. Observing the same sign of $dA/dI$ on both sides, or a crossing point that moves strongly with pulse duration, would refute the quasi-equilibrium absorption formula.

Watch

Extended reading notes

Core claim

The central claim is that the strongly nonlinear electrodynamic response of graphene is controlled by three nonequilibrium quantities — the electron chemical potential $\mu_e$, the hole chemical potential $\mu_h$, and the common hot-carrier temperature $T$ — which obey a closed set of steady-state equations. Given the frequency $\omega$, intensity $I$, equilibrium chemical potential $\mu_0$, temperature $T_0$, and the relaxation-time ratio $\tau_\epsilon/\tau_{\rm rec}$, the model predicts the density change $\delta n$, the temperature change $\delta T$, and the absorption coefficient $A$. Its quantitative signature is the opposite sign of $dA/dI$ on the two sides of $2|\mu_0|$: at low frequencies the heating of carriers fills initial valence-band states and absorption is induced, while at high frequencies the occupation of final conduction-band states saturates the absorption. The paper identifies a crossing point where absorption curves at different intensities intersect and shows that this feature, as well as the magnitude of the absorption changes, is consistent with measured pump-probe data in epitaxial graphene.

Load-bearing premise

The load-bearing premise is that the nonlinear absorption can be computed by taking the equilibrium linear-response conductivity formula and only replacing the chemical potentials and temperature by their nonequilibrium values, while neglecting explicit higher-order current terms; if those frequency-dependent higher-order conductivities matter under strong fields, the predicted absorption and its sign change could be altered.

Editorial extensions

If this is right

Below the interband threshold $\hbar\omega \lesssim 2|\mu_0|$, intense radiation heats carriers and increases absorption; above it, absorption saturates and drops by factors of order ten at $I/I_0 \sim 10^6$.
The full model predicts that the hot-electron temperature $T$ drops steeply as $\hbar\omega$ approaches $2|\mu_0|$, so the effective temperature cannot be treated as frequency-independent when interpreting experiments.
The model explains the apparent two-time-scale decay of photoexcited carrier densities as a single nonlinear recombination process, with time constant $\tau_{\rm rec}/(\Delta_0+1)$ at strong excitation and $\tau_{\rm rec}$ at weak excitation.
For weakly doped graphene at low temperature, absorption curves at different intensities intersect near $\hbar\omega \approx 28$ meV, slightly below $2|\mu_0| = 32$ meV, matching the sign-change location observed in pump-probe experiments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension would be to solve the time-dependent equations (29) and (35) for pulsed excitation instead of the steady-state limit, which would predict how the crossing point and the magnitude of $dA/dI$ evolve with pump-probe delay without invoking two independent relaxation times.
Because the model makes $T(\omega)$ and $\mu_{e,h}(\omega)$ explicit, one could use the predicted collapse of $T$ near $\hbar\omega \approx 2|\mu_0|$ as a spectroscopic calibration of the local chemical potential in graphene devices; this is not proposed in the paper.
The central approximation suggests its own stress test: compute or measure the first neglected higher-order conductivity $\sigma^{(3)}$ away from equilibrium and check whether inserting it shifts the predicted crossing frequency; the paper leaves this to future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The useful thing here is a closed, self-consistent hot-electron model for strongly driven graphene: one set of equations for mu_e, mu_h, T and delta_n, with a full frequency-intensity study and a transparent explanation of why dA/dI changes sign around hbar*omega ~ 2|mu_0|. That is genuinely new relative to earlier simplified HEM treatments, and it is a natural framework for interpreting saturation and induced absorption experiments.

The model has real structure. The carrier-density/temperature equations are clean, the steady-state reduction leaves only tau_epsilon/tau_rec in the central equation, and the nonlinear recombination decay in Eq. (32) is a nice corrective to papers that force a biexponential fit. I also like the explicit disclaimer that Eqs. (23)-(25) are approximations. The reference list is appropriate; the author's own earlier nonlinear-response papers are the relevant background, not padding.

The main soft spot is exactly where the stress-test note points. The paper computes nonlinear absorption by injecting hot mu_e, mu_h, T into the linear-response conductivity, dropping explicit sigma^(3), sigma^(5), etc. It acknowledges the approximation, but never checks the low-intensity limit against the known exact sigma^(3) results from Refs. 28-30. If the thermalized linear response disagrees with the perturbative sigma^(3) absorption, the sign-change prediction could be an artifact of the ansatz, not hot-electron physics. I would not reject over this, because the phenomenological picture is well motivated at high excitation, but the authors should be asked to add that comparison before publication.

Two secondary issues. The "one fitting parameter" claim understates the actual parameter freedom: |mu_0|=16 meV is chosen in Figure 11 to equalize the 20/30 meV changes, tau_p is imported from Ref. [48], and zeta, E_i come from a separate conductivity fit. A short sensitivity/uncertainty discussion would make the experimental comparison honest. Also, the Ref. [48] sample is multilayer epitaxial graphene while the theory is monolayer; the authors mention this only implicitly, and a monolayer theory applied to a multilayer sample can at most be semi-quantitative.

Verdict: conditionally solid. The paper deserves a serious referee and, if the weak-field benchmark and sensitivity analysis are added, it is publishable. I would bring it to a reading group for the graphene photonics crowd, and I would cite it as the standard hot-electron framework when interpreting this class of experiments.

Referee Report

3 major / 4 minor

Summary. The paper develops a hot-electron model (HEM) for the strongly nonlinear electrodynamic response of graphene. The carriers are described by quasi-equilibrium Fermi distributions with electron and hole chemical potentials μe, μh and a common temperature T, Eq. (6), and the coupled equations (29), (35)–(39) determine these quantities as functions of frequency ω, intensity I, equilibrium chemical potential μ0, temperature T0, and scattering parameters. The authors solve the steady-state equations, present density-temperature diagrams, power and frequency dependences of the absorption coefficient, and compare their results with pump-probe experiments on multilayer epitaxial graphene (Ref. 48). The central claim is that the model quantitatively describes the observed sign change of dA/dI across ℏω ≈ 2|μ0| and that the results agree well with experiment.

Significance. If the central claim holds, the paper would provide a tractable closed framework for strongly nonlinear graphene optics beyond perturbation theory, with explicit predictions for the hot-carrier temperature, chemical potentials, densities, and absorption. The formulation is transparent, the dimensionless steady-state equations (42)–(46) are elegant, and the paper gives a useful scattering-rate model, Eq. (19), fitted to static conductivity data in the Appendix. However, the energy-balance inconsistency identified below affects all quantitative results in their current form, and the comparison with Ref. 48 is partly fitted rather than predictive. With the energy balance corrected and the weak-field limit checked, the model could be a valuable contribution.

major comments (3)

[II.G.1, Eq. (29)] The energy balance equation (29) omits the inter-band absorption power Ainter·I from the right-hand side. The recombination-generation equation (31) explicitly includes Ainter·I as the rate of electron-hole pair generation, and each generated pair carries the photon energy ℏω. Therefore the total carrier energy E must increase at the rate (Aintra + Ainter)·I, not Aintra·I alone. As written, the steady-state equation (41) forces I = [E − E0]/(τϵ·Aintra), so for ℏω > 2|μ0| the carrier temperature is controlled only by the small intra-band absorption even though the dominant absorbed power is inter-band. This likely explains why T ≈ T0 throughout the inter-band-dominated regime in Figs. 3(a) and 6(a). The authors should add the missing Ainter·I term to Eq. (29) and redo the numerical analysis; many reported results may change quantitatively.
[II.F, Eqs. (23)–(25)] The nonlinear absorption coefficient is computed by substituting the hot-carrier chemical potentials and temperature into the linear-response conductivity formulas, while explicit higher-order conductivities σ^(3), σ^(5), etc. are neglected. The paper acknowledges this in the text below Eq. (25), but the procedure is load-bearing for the central claim. A necessary consistency check is to expand the HEM equations in the weak-field limit and compare the linear-in-I correction to A with the exact third-order nonlinear absorption computed from Refs. 28–30. If the linearized HEM disagrees with the known σ^(3), then the predicted sign change of dA/dI and the agreement in Fig. 11 could be artifacts of the incomplete conductivity model rather than consequences of hot-electron redistribution. This test should be performed before the quantitative agreement claim is accepted.
[III.C, Fig. 11] The comparison with Ref. 48 is not a stringent quantitative test. The text states that |μ0| = 16 meV was chosen specifically to make the absorption changes at 20 and 30 meV approximately equal, τp is taken from Ref. 48, τϵ/τrec is fixed, and the calculation is for a monolayer while the experimental sample is multilayer epitaxial graphene. No direct overlay of the calculated spectra with the measured data is shown. The abstract's claim of 'good agreement' should be supported either by a fit with a fixed, pre-specified parameter set or by a quantitative goodness-of-fit measure; otherwise the agreement is partly a product of parameter choice.

minor comments (4)

[II.A, Eq. (3)] The equilibrium densities for doped graphene at T0 = 300 K are given to three significant figures, but no uncertainty or experimental source is cited; a brief reference would help readers assess the representative parameters.
[II.E.1, Eq. (19)] The parameters ζ and Ei are introduced as fitting parameters, but the main text then fixes ζ = 4 and Ei = 30 meV. It would be useful to state how sensitive the main conclusions (e.g., the sign-change frequency) are to the range of realistic ζ and Ei values.
[III.B.1, Fig. 4] The right panel of Fig. 4(b) shows that at very high intensities the hole chemical potential becomes positive while the electron chemical potential becomes negative; the text explains this, but a short physical explanation of why the band occupations invert would improve readability.
[III.C, p. 16] The phrase 'results were found to be weakly dependent on τϵ/τrec as long as this parameter is small compared to unity' should be quantified: specify the range over which the calculated absorption changes by less than, say, a few percent.

Circularity Check

1 steps flagged · score 4.0 of 10

The hot-electron model derivation is self-consistent, but the quantitative comparison with Ref. [48] is partly calibrated by choosing the chemical potential to match the measured absorption changes.

fitted input called prediction [Section III C, comparison with Ref. [48] (Fig. 11); see also Section IV Summary]
"Right and left from 28 meV the absorption decreases and increases respectively, and we have chosen the chemical potential |µ0| = 16 meV to get approximately equal (in absolute values) changes of the absorption coefficient at ℏω = 20 and 30 meV."

The abstract claims the results 'are shown to be in good agreement with recent experimental data', and the Section IV summary says the theory 'reasonably describes available experimental data'. For the quantitative comparison with Ref. [48], the chemical potential is not derived from the model's inputs but is selected precisely to make the computed absorption changes at the two measured frequencies (20 and 30 meV) equal in magnitude; τ_eps/τ_rec = 0.01 and τp = 300 fs are also set to the experimental situation. The agreement at those two frequencies is therefore partly enforced by parameter choice rather than predicted.

full rationale

The core hot-electron-model derivation is self-contained: Eqs. (9)-(13) define densities and energies from the quasi-equilibrium ansatz (6); Eqs. (15)-(22) give the linear-response conductivity with non-equilibrium parameters; Eqs. (23)-(25) approximate the nonlinear absorption by those conductivities; and Eqs. (29), (35)-(39) close the system. The approximation in Eqs. (23)-(25) is explicitly acknowledged by the paper and is a modeling assumption, not a circular identification: the absorption coefficient is not defined as the energy-relaxation term divided by intensity, but is computed from sigma^(1) and then required to satisfy the energy and number balance equations. The sign change of dA/dI follows from the Fermi-function form of Eq. (21), so it has independent physical content. No load-bearing self-citation or imported uniqueness theorem appears; self-citations to Refs. 28-30 and Ref. 1 are contextual and do not force the present result. The main circularity concern is the experimental validation: for the Ref. [48] comparison the chemical potential is chosen as 16 meV to make the absorption changes at 20 and 30 meV approximately equal, and tau_eps/tau_rec and tau_p are set to the experimental values. The quantitative agreement at those frequencies is therefore partly fitted, though the model still predicts the spectral shape, the crossing point, and the intensity dependence. A score of 4 reflects this partial calibration while recognizing that the central theoretical derivation is not circular.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The model relies on phenomenological relaxation equations and an interpolation formula for the scattering rate. It has five adjustable inputs: zeta, E_i, tau_epsilon/tau_rec, tau_p (in the simplified model), and the chemical potential used for the experimental comparison. The quasi-equilibrium ansatz and the linear-response substitution are the main physical axioms.

free parameters (5)

zeta = 4 in main text; also fitted zeta=5.13 and zeta=4.32 in Appendix
Minimal static conductivity parameter in the scattering rate model (19), fitted to experimental gate-voltage dependence in Fig. 13.
E_i = 30 meV in main text; fitted Ni values correspond to Ei 26.5-83.9 meV
Coulomb impurity energy in the scattering rate model, fitted to experimental data in Fig. 13.
tau_epsilon/tau_rec = 0.1 or 0.01 (assumed)
Ratio of intra-band energy relaxation time to recombination time; the paper calls it the essentially one fitting parameter of the model.
tau_p = 300 fs or 100 fs
Momentum relaxation time used in the simplified constant-gamma model (20) when comparing to Ref. 48; value taken from experiment.
mu_0 for Figure 11 = |mu_0| = 16 meV
Chosen so that the calculated absorption changes at 20 and 30 meV approximately equal the observed changes in Ref. 48; not independently determined.

assumptions (5)

domain assumption The electron system in graphene under intense radiation quickly reaches quasi-equilibrium Fermi-Dirac distributions with a common temperature T and separate chemical potentials mu_e and mu_h for electrons and holes (Eq. 6).
Justified by assumed hierarchy tau_ee << tau_p (Eq. 5), citing Refs. 76-81; if electron-electron scattering is not fast enough, the model fails.
domain assumption The hierarchy of relaxation times tau_ee << tau_p << tau_epsilon << tau_rec (Eq. 5).
Needed to separate the quasi-equilibrium and slow-recombination stages; not directly verified for all experimental conditions.
ad hoc to paper The nonlinear absorption coefficient is computed by inserting non-equilibrium mu_e, mu_h, T into the linear-response conductivity formulas (23)-(25), neglecting explicit higher-order conductivities.
The paper states this is approximate because sigma^(5), sigma^(7) are unknown and sigma^(3) is only known in equilibrium; this is a modeling assumption, not derived.
domain assumption The energy relaxation of electrons and holes follows simple relaxation-time equations (26)-(27) and recombination is a bi-particle process with rate alpha_rec (31).
Phenomenological semiconductor physics model; the paper uses it without microscopic derivation.
ad hoc to paper The intra-band scattering rate model (19) with constants zeta and E_i captures the momentum relaxation for all energies.
An interpolation formula (A5) matched to high-energy impurity scattering; low-energy behavior is assumed.

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Cite this review

Pith. "Pith review of Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model." pith.science (2026). https://pith.science/paper/BFE4AOXG

@misc{pith2026190804631,
  author       = {Pith},
  title        = {Pith review of: Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFE4AOXG}},
  note         = {Machine review of arXiv:1908.04631}
}

abstract

An electrodynamic response of graphene to a strong electromagnetic radiation is considered. A hot electron model (HEM) is introduced and a corresponding system of nonlinear equations is formulated. Solutions of this system are found and discussed in detail for intrinsic and doped graphene: the hot electron temperature, non-equilibrium electron and holes densities, absorption coefficient and other physical quantities are calculated as functions of the incident wave frequency $\omega$ and intensity $I$, of the equilibrium chemical potential $\mu_0$ and temperature $T_0$, scattering parameters, as well as of the ratio $\tau_\epsilon/\tau_{\rm rec}$ of the intra-band energy relaxation time $\tau_\epsilon$ to the recombination time $\tau_{\rm rec}$. The influence of the radiation intensity on the absorption coefficient $A$ at low ($\hbar\omega\lesssim 2|\mu_0|$, $dA/dI>0$) and high ($\hbar\omega\gtrsim 2|\mu_0|$, $dA/dI<0$) frequencies is studied. The results are shown to be in good agreement with recent experimental data.

Figures

Figures reproduced from arXiv: 1908.04631 by the authors.

**Figure 2.** FIG. 2. The relative change of the density ∆ = [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) The relative change of the density ∆ = [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: FIG. 4. (a) The relative change of the density ∆ = [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. The electron distribution function (6) in the valenc [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) The relative change of the density ∆ = [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7. The electron distribution function (6) in the valenc [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8. The absorption coefficient vs frequency at different va [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: FIG. 9. Inter-band transitions in graphene (a) in equilibri [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 10.** Figure 10: FIG. 10. The hot electron temperature [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

**Figure 11.** Figure 11: FIG. 11. (a) The absorption coefficient [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

**Figure 12.** Figure 12: FIG. 12. The Coulomb impurity energy [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

**Figure 13.** Figure 13: FIG. 13. Experimental data from Ref [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Intra-band conductivity 5

work page
[2]

Absorption coeﬃcient 6 G

Inter-band dynamic conductivity 6 F. Absorption coeﬃcient 6 G. Dynamics equations of the hot electron model 6

work page
[3]

Steady-state equations 9 III

Preliminary summary and discussion 8 H. Steady-state equations 9 III. Results 10 A. Density-temperature diagrams 10 B. Power dependencies of diﬀerent physical quantities 11

work page
[4]

Frequency dependencies of diﬀerent physical quantities 14 IV

Intrinsic graphene 13 C. Frequency dependencies of diﬀerent physical quantities 14 IV. Summary 18 Acknowledgments 18 A. A model for the intra-band scattering rate 18 References 20 I. INTRODUCTION The nonlinear electrodynamic response of graphene attracted gr eat attention in recent years. After the pioneer- ing prediction 1 of the strongly nonlinear elect...

work page
[5]

Intra-band conductivity In order to calculate the intra-band conductivity (16)–(17) we ne ed a model for the scattering rate γintra(E) ≡ 1/τ p(E), where τp(E), see Eq. (5), is the energy-dependent momentum relaxation time due to the scattering of electrons and holes with impurities, phonons and other lattice imperf ections (but not with each other). As a ...

work page
[7]

We assume that γinter → 0 since a ﬁnite γinter does not inﬂuence the ﬁnal result under the condition ℏγinter ≪ T which is typically satisﬁed in experiments

Inter-band dynamic conductivity The inter-band conductivity (18) depends on the inter-band scat tering rate γinter. We assume that γinter → 0 since a ﬁnite γinter does not inﬂuence the ﬁnal result under the condition ℏγinter ≪ T which is typically satisﬁed in experiments. Then Eq. (18) can be simpliﬁed so that the real part a ssumes the form Re σ (1) inte...

work page
[8]

Energy relaxation Assume that the system is excited by a powerful incident radiation w ith the intensity I. Since the recombination is a slow process, the electron and hole Fermi gases are independen t from each other in that sense that they are characterized (at the time t /greaterorsimilarτee after the excitation is switched on) by their own chemical po...

work page
[9]

Recombination At a longer time scale ∼ τrec electrons and holes recombine. Taking into account that they are g enerated and recombine by pairs, ne(t) = n0 e + δn(t), n h(t) = n0 h + δn(t), (30) we write the generation-recombination rate equation in the form ∂n e ∂t = ∂n h ∂t = G − R = Ainter(ω, µ e, µ h, T ) ℏω I − α rec(nenh − n0 en0 h). (31) The ﬁrst (g...

work page

Show all 13 references

[10]

Quantitatively, ∆( t) decreases from its initial value ∆ 0 by a factor of two during the time τrec/ (∆ 0 + 1) ≪ τrec and by a factor of ∆ 0 during the time ∼ τrec. The generation-recombination rate equation (31) can be also rewr itten in the form explicitly containing τrec, ∂(...
[11]

48,84,85 and other

Preliminary summary and discussion The recombination of charge carriers in graphene characterized b y a more complicated than e− t/τ -decay was com- monly observed in time-resolved pump-probe experiments, see, e.g ., Refs. 48,84,85 and other. It was often interpreted by introd...
[12]

2 eV, under the condition ℏω > 2|µ 0| when the inter-band transitions dominate

Doped graphene Figure 3 exhibits the power dependencies of the chemical potentials µ e and µ h, temperature T , photo-excited charge carrier density δn and the absorption coeﬃcient A, in the doped graphene sample with µ 0 = −0. 2 eV, under the condition ℏω > 2|µ 0| when the in...
[13]

In the rest of the paper we u se ζ = 4 and Ei = 30 meV, which corresponds to the mobility of about 7260 cm 2/Vs

Thus found parameters ζ and Ei and the model expression (19) are then used in formulas (16)–(17 ) for the high-frequency nonlinear conductivities of graphene. In the rest of the paper we u se ζ = 4 and Ei = 30 meV, which corresponds to the mobility of about 7260 cm 2/Vs. For t...
[14]

Non-linear electromagnetic response of g raphene,

Intrinsic graphene Figure 6 shows the power dependencies of diﬀerent physical quant ities in intrinsic graphene with µ 0 = 0 and ℏω = 0 . 4 eV. Other parameters ( T0, Ei and τǫ/τ rec) are the same as in Figures 3, 4 and 5. Since at µ 0 = 0 the condition ℏω > 2|µ 0| is always s...

2007

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Intra-band conductivity 5

work page

[2] [2]

Absorption coeﬃcient 6 G

Inter-band dynamic conductivity 6 F. Absorption coeﬃcient 6 G. Dynamics equations of the hot electron model 6

work page

[3] [3]

Steady-state equations 9 III

Preliminary summary and discussion 8 H. Steady-state equations 9 III. Results 10 A. Density-temperature diagrams 10 B. Power dependencies of diﬀerent physical quantities 11

work page

[4] [4]

Frequency dependencies of diﬀerent physical quantities 14 IV

Intrinsic graphene 13 C. Frequency dependencies of diﬀerent physical quantities 14 IV. Summary 18 Acknowledgments 18 A. A model for the intra-band scattering rate 18 References 20 I. INTRODUCTION The nonlinear electrodynamic response of graphene attracted gr eat attention in recent years. After the pioneer- ing prediction 1 of the strongly nonlinear elect...

work page

[5] [5]

Intra-band conductivity In order to calculate the intra-band conductivity (16)–(17) we ne ed a model for the scattering rate γintra(E) ≡ 1/τ p(E), where τp(E), see Eq. (5), is the energy-dependent momentum relaxation time due to the scattering of electrons and holes with impurities, phonons and other lattice imperf ections (but not with each other). As a ...

work page

[6] [7]

We assume that γinter → 0 since a ﬁnite γinter does not inﬂuence the ﬁnal result under the condition ℏγinter ≪ T which is typically satisﬁed in experiments

Inter-band dynamic conductivity The inter-band conductivity (18) depends on the inter-band scat tering rate γinter. We assume that γinter → 0 since a ﬁnite γinter does not inﬂuence the ﬁnal result under the condition ℏγinter ≪ T which is typically satisﬁed in experiments. Then Eq. (18) can be simpliﬁed so that the real part a ssumes the form Re σ (1) inte...

work page

[7] [8]

Energy relaxation Assume that the system is excited by a powerful incident radiation w ith the intensity I. Since the recombination is a slow process, the electron and hole Fermi gases are independen t from each other in that sense that they are characterized (at the time t /greaterorsimilarτee after the excitation is switched on) by their own chemical po...

work page

[8] [9]

Recombination At a longer time scale ∼ τrec electrons and holes recombine. Taking into account that they are g enerated and recombine by pairs, ne(t) = n0 e + δn(t), n h(t) = n0 h + δn(t), (30) we write the generation-recombination rate equation in the form ∂n e ∂t = ∂n h ∂t = G − R = Ainter(ω, µ e, µ h, T ) ℏω I − α rec(nenh − n0 en0 h). (31) The ﬁrst (g...

work page