For citation:
Panferov A. D., Shcherbakov I. A. Tight-binding implementation of the quantum kinetic equation for graphene. Izvestiya of Saratov University. Physics , 2024, vol. 24, iss. 3, pp. 198-208. DOI: 10.18500/1817-3020-2024-24-3-198-208, EDN: EWJDQE
Tight-binding implementation of the quantum kinetic equation for graphene
Background and Objectives: Progress in the development of pulsed radiation sources with high energy density makes it possible to study the nonlinear response of condensed matter to the disturbing influence of high-intensity electromagnetic fields. To understand the processes occurring in this case, adequate models are needed that qualitatively and quantitatively reproduce the characteristics of the materials under study. In this area, graphene is considered one of the most promising materials due to the specificity of its band structure. The purpose of the work is to present and test a new model based on the quantum kinetic equation, free from restrictions on such parameters as the frequency and strength of the electric field of the disturbing influence. Materials and Method: The approach used in the work is based on the quantum kinetic equation for the distribution function of charge carriers in the state space. It makes it possible, in the one-electron approximation, to nonperturbatively reproduce the ultrafast dynamics of carriers in an external classical electric field. The system under consideration is specified by the electron dispersion law. The approach was developed and implemented for the pseudo-relativistic approximation of massless fermions, successfully used in describing the features of graphene. However, by its definition, this approximation quite accurately reproduces the real dispersion law only in the low-energy region in the vicinity of the Dirac points. Therefore, the direct use of this version of the model to describe processes in which electronic states with high excitation energies are known to participate raises questions about the accuracy of the results obtained. The problem can be resolved by moving to an exact definition of the dispersion law through the parameters of the tight-binding model of nearest neighbors in the crystal lattice of the graphene. The presented work proposes an implementation option for such a procedure and verifies the results obtained. A generalization of the formalism for a two-level system with a massless Hamiltonian of general form is used, which universally defines the explicit form of the quantum kinetic equation and expressions for macroscopic observable parameters. Results: A computational model based on the exact tight-binding model Hamiltonian has been determined, which strictly takes into account the real law of graphene dispersion in reciprocal space. The new model has been verified. For this purpose, the results of its use are compared with the results of a similar model based on the massless fermion approximation. Under conditions of limiting the parameters of the perturbing influence, ensuring the generation of excited states with only low energies in the immediate vicinity of the Dirac points, an exact coincidence has been demonstrated both at the stage of determining the values of the distribution function and for the observed parameters. It has been shown that going beyond the applicability limits of the massless fermion approximation is accompanied by the appearance of qualitative and quantitative differences in the results obtained. Conclusion: The results of the work provide new opportunities for studying the behavior of graphene under extreme conditions of strong high-frequency fields, modeling and searching for new nonlinear effects, and accurately reproducing the ultrafast quantum dynamics of its electrons for states with high energy values.
- Higuchi T., Heide Ch., Ullmann K., Weber H. B., Hommelhof P. Light-field-driven currents in grapheme. Nature, 2017, vol. 550, pp. 224–228. https://doi.org/10.1038/nature23900
- Aryasetiawan F., Gunnarsson O. The GW method. Rep. Prog. Phys., 1998, vol. 61, pp. 237. https://doi.org/10.1088/0034-4885/61/3/002
- Golze D., Dvorak M., Rinke P. The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy. Frontiers in Chemistry, 2019, vol. 7, article no. 377. https://doi.org/10.3389/fchem.2019.00377
- Provorse M. R., Isborn Ch. M. Electron Dynamics with Real-Time Time-Dependent Density Functional Theory. International Journal of Quantum Chemistry, 2016, vol. 116, pp. 739–749. https://doi.org/10.1002/qua.25096
- Mocci P., Malloci G., Bosin A., Cappellini G. Time-Dependent Density Functional Theory Investigation on the Electronic and Optical Properties of Poly-C, Si, Ge-acenes. ASC Omega, 2020, vol. 5, pp. 16654–16663. https://dx.doi.org/10.1021/acsomega.0c01516
- Wilhelm J., Grössing P., Seith A., Crewse J., Nitsch M., Weigl L., Schmid Chr., Evers F. Semiconductor Bloch-equations formalism: Derivation and application to high-harmonic generation from Dirac fermions. Phys. Rev. B, 2021, vol. 103, article no. 125419. https://dx.doi.org/10.1103/PhysRevB.103.125419
- Ornigotti M., Carvalho D. N., Biancalana F. Nonlinear optics in graphene: Theoretical background and recent advances. La Rivista del Nuovo Cimento, 2023, vol. 46, pp. 295–380. https://doi.org/10.1007/s40766-023-00043-8
- Novoselov K. S., Geim A. K., Morozov S. V., Jiang D., Katsnelson M. I., Grigorieva I. V., Dubonos S. V., Firsov A. A. Two-dimensional gas of massless Dirac fermions in grapheme. Nature, 2005, vol. 438, pp. 197–200. https://doi.org/10.1038/nature04233
- Castro Neto A. H., Guinea F., Peres N. M. R., Novoselov K. S., Geim A. K. The electronic properties of grapheme. Rev. Mod. Phys., 2009, vol. 81, pp. 109–162. https://doi.org/10.1103/RevModPhys.81.109
- Katsnelson M. I. The Physics of Graphene. Cambridge University Press, 2020. 425 p. https://doi.org/10.1017/9781108617567
- Greeb A. A., Mamaev S. G., Mostepanenko V. M. Vakuumnye kvantovye effekty v sil’nykh polyakh [Vacuum quantum effects in strong fields]. Мoscow, Energoatomizdat, 1988. 288 p. (in Russian).
- Bialynicky-Birula I., Gornicki P., Rafelski J. Phase space structure of the Dirac vacuum. Phys. Rev. D, 1991, vol. 44, pp. 1825–1835. https://doi.org/10.1103/PhysRevD.44.1825
- Schmidt S. M., Blaschke D., Röpke G., Smolyansky S. A., Prozorkevich A. V., Toneev V. D. A Quantum kinetic equation for particle production in the Schwinger mechanism. Int. J. Mod. Phys. E, 1998, vol. 7, pp. 709–718. https://doi.org/10.1142/S0218301398000403
- Blaschke D. B., Prozorkevich A. V., Röpke G., Roberts C. D., Schmidt S. M., Shkirmanov D. S., Smolyansky S. A. Dynamical Schwinger effect and high-intensity lasers. Realising nonperturbative QED. Eur. Phys. J. D, 2009, vol. 55, pp. 341–358. https://doi.org/10.1140/epjd/e2009-00156-y
- Panferov A., Smolyansky S., Blaschke D., Gevorgyan N. Comparing two different descriptions of the I–V characteristic of graphene: Theory and experiment. EPJ Web Conf., 2019, vol. 204, article no. 06008. https://doi.org/10.1051/epjconf/201920406008
- Smolyansky S. A., Panferov A. D., Blaschke D. B., Gevorgyan N. T. Nonperturbative kinetic description of electron-hole excitations in graphene in a time dependent electric field of arbitrary polarization. Particles, 2019, vol. 2, pp. 208–230. https://doi.org/10.3390/particles2020015
- Smolyansky S. A., Blaschke D. B., Dmitriev V. V., Panferov A. D., Gevorgyan N. T. Kinetic equation approach to graphene in strong external fields. Particles, 2020, vol. 3, pp. 456–476. https://doi.org/10.3390/particles3020032
- Panferov A. D., Novikov N. A. Characteristics of induced radiation under the action of short high-frequency pulses on graphene. Izvestiya of Saratov University. Physics, 2023, vol. 23, iss. 3, pp. 254–264. https://doi. org/10.18500/1817-3020-2023-23-3-254-264
- Tseryupa V. A., Churochkin D. V., Dmitriev V. V., Smolyansky S. A. Emission in graphene: A kinetic approach. Journal of Technical Physics, 2024, vol. 94, iss. 3, pp. 351–357 (in Russian). https://doi.org/10.61011/JTF.2024.03.57371.1-24
- Wallace P. R. The Band Theory of Graphite. Phys. Rev., 1947, vol. 71, pp. 622–634. https://doi.org/10.1103/PhysRev.71.622
- Martin P. C., Schwinger J. Theory of many-particle systems. I. Phys. Rev., 1959, vol. 115, pp. 1342–1373. https://doi.org/10.1103/PhysRev.115.1342
- Akhiezer A. I., Peletminskii S. V. Metody statisticheskoi fiziki [Methods of statistical physics]. Moscow, Nauka, 1977. 367 p. (in Russian).
- Sipe J. E., Shkrebtii A. I. Second-order optical response in semiconductors. Phys. Rev. B, 2000, vol. 61, pp. 5337–5352. https://dx.doi.org/10.1103/PhysRevB.61.5337
- Ishikawa K. L. Nonlinear optical response of graphene in time domain. Phys. Rev. B, 2010, vol. 82, article no. 201402. https://doi.org/10.1103/PhysRevB.82.201402
- Yoshikawa N., Tamaya T., Tanaka K. High-harmonic generation in graphene enhanced by elliptically polarized light excitation. Science, 2017, vol. 356, pp. 736–738. https://doi.org/10.1126/science.aam8861
- Sato Sh. A., Hirori H., Sanari Y., Kanemitsu Y., Rubio A. High-order harmonic generation in graphene: Nonlinear coupling of intraband and interband transitions. Phys. Rev. B, 2021, vol. 103, article no. L041408. https://dx. doi.org/10.1103/PhysRevB.103.L041408
- Reich S., Maultzsch J., Thomsen C. Tight-binding description of grapheme. Phys. Rev. B, 2002, vol. 669, article no. 035412. https://dx.doi.org/10.1103/PhysRevB.66.035412
- Panferov A. D., Posnova N. V., Ulyanova A. A. Simulation the behavior of a two-level quantum system using scalable regular grids. Program Systems: Theory and Applications, 2023, vol. 14, iss. 2, pp. 27–47 (in Russian). https://doi.org/10.25209/2079-3316-2023-14-2-27-47
- Gil-Villalba A., Meyer R., Giust R., Rapp L., Billet C., Courvoisier F. Single shot femtosecond laser nano-ablation of CVD monolayer grapheme. Scientific Reports, 2018, vol. 8, article no. 14601. https://dx.doi.org/10.1038/s41598-018-32957-3
- Weitz T., Heide Chr., Hommelhoff P. Strong-Field Bloch Electron interferometry for Band Structure Retrieval. arXiv:2309.16313v1. https://doi.org/10.48550/arXiv.2309.16313
- 185 reads