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Graphene Bloch Equations

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Low-Dimensional Functional Materials

Abstract

The ultrafast carrier dynamics in graphene has been intensively investigated in recent years. From the theoretical side the graphene Bloch equations have been successfully applied to explain linear absorption and various features observed in optical pump-probe experiments. Here, we present a detailed derivation of the graphene Bloch equations and discuss different contributions resulting from the electron-electron and electron-phonon interaction.

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Notes

  1. 1.

    This work is part of the dissertation of Torben Winzer.

  2. 2.

    In the following the optical matrix element will be denoted only with a single electron wave vector. Nevertheless, it still contains the Kronecker \(\delta _{\mathbf{k},\mathbf{k}^\prime}\).

  3. 3.

    Due to the Kronecker \(\delta _{\mathbf{k}_{\mathbf{ 1}}+\mathbf{k}_{\mathbf{ 2}},\mathbf{k}_{\mathbf{ 3}}+\mathbf{k}_{\mathbf{ 4}}}\) in the Coulomb matrix element only the dynamics of momentum conserving correlations have to be determined. Nevertheless, for a more clear notation the equation of motion for a general \(\Lambda _{\mathbf{ 34}}^{\mathbf{ 12}c}\) is derived.

  4. 4.

    In this derivation the spatial homogeneity applied form the beginning.

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Acknowledgements

We acknowledge the financial support from the Deutsche Forschungsgemeinschaft through SPP 1459. Ermin Malić thanks the Einstein Foundation Berlin. We thank Faris Kadi (TU Berlin) for fruitful discussions.

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Authors and Affiliations

  1. Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universität Berlin, Hardenbergstr. 36, 10623, Berlin, Germany

    Torben Winzer, Ermin Malić & Andreas Knorr

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  1. Torben Winzer
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  2. Ermin Malić
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  3. Andreas Knorr
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Corresponding author

Correspondence to Andreas Knorr .

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Editors and Affiliations

  1. , Institut für Theoretische Physik IV, Heinrich-Heine-Universität, Universitätsstrasse 1, Düsseldorf, 40225, Germany

    Reinhold Egger

  2. Laboratory for Advanced Studies, Department of Energy, Turin Polytechnic University in Tashkent, Niyazov St. 17, Tashkent, 100174, Uzbekistan

    Davron Matrasulov

  3. , Department of Physics, National University of Uzbekistan, Niyazov St. 17, Tashkent, 100174, Uzbekistan

    Khamdam Rakhimov

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Winzer, T., Malić, E., Knorr, A. (2013). Graphene Bloch Equations. In: Egger, R., Matrasulov, D., Rakhimov, K. (eds) Low-Dimensional Functional Materials. NATO Science for Peace and Security Series B: Physics and Biophysics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6618-1_4

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