Flexural phonons in supported graphene: from pinning to localization

Zhao, Wei L.  Z.; Tikhonov, Konstantin S.; Finkel’stein, Alexander M.

doi:10.1038/s41598-018-34426-3

Title: Flexural phonons in supported graphene: from pinning to localization

Journal Article · 01 November 2018 · Scientific Reports

DOI: https://doi.org/10.1038/s41598-018-34426-3 · OSTI ID:1612099

^[1];

^[2]; Finkel’stein, Alexander M. ^[3]

Texas A & M Univ., College Station, TX (United States); DOE/OSTI
L. D. Landau Inst. for Theoretical Physics, Moscow (Russian Federation); Skolkovo Institute of Science and Technology (Russian Federation)
Texas A & M Univ., College Station, TX (United States); Weizmann Inst. of Science, Rehovot (Israel)

We identify graphene layer on a disordered substrate as a system where localization of phonons can be observed. Generally, observation of localization for scattering waves is not simple, because the Rayleigh scattering is inversely proportional to a high power of wavelength. The situation is radically different for the out of plane vibrations, so-called flexural phonons, scattered by pinning centers induced by a substrate. In this case, the scattering time for vanishing wave vector tends to a finite limit. One may, therefore, expect that physics of the flexural phonons exhibits features characteristic for electron localization in two dimensions, albeit without complications caused by the electron-electron interactions. We confirm this idea by calculating statistical properties of the Anderson localization of flexural phonons for a model of elastic sheet in the presence of the pinning centers. Finally, we discuss possible manifestations of the flexural phonons, including the localized ones, in the electronic thermal conductance.

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Research Organization:: Texas A & M Univ., College Station, TX (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: SC0014154

OSTI ID:: 1612099

Journal Information:: Scientific Reports, Journal Name: Scientific Reports Journal Issue: 1 Vol. 8; ISSN 2045-2322

Publisher:: Nature Publishing GroupCopyright Statement

Country of Publication:: United States

Language:: English

References (107)

Scattering of flexural waves on thin plates Norris, A. N.; Vemula, C. Journal of Sound and Vibration, Vol. 181, Issue 1 https://doi.org/10.1006/jsvi.1995.0129	journal	March 1995
Anomalous elasticity, fluctuations and disorder in elastic membranes Le Doussal, Pierre; Radzihovsky, Leo Annals of Physics, Vol. 392 https://doi.org/10.1016/j.aop.2017.08.033	journal	May 2018
Graphene Bimetallic-like Cantilevers: Probing Graphene/Substrate Interactions Conley, Hiram; Lavrik, Nickolay V.; Prasai, Dhiraj Nano Letters, Vol. 11, Issue 11 https://doi.org/10.1021/nl202562u	journal	November 2011
Direct Nanoscale Imaging of Ballistic and Diffusive Thermal Transport in Graphene Nanostructures Pumarol, Manuel E.; Rosamond, Mark C.; Tovee, Peter Nano Letters, Vol. 12, Issue 6 https://doi.org/10.1021/nl3004946	journal	May 2012
The structure of suspended graphene sheets Meyer, Jannik C.; Geim, A. K.; Katsnelson, M. I. Nature, Vol. 446, Issue 7131, p. 60-63 https://doi.org/10.1038/nature05545	journal	March 2007
Graphene kirigami Blees, Melina K.; Barnard, Arthur W.; Rose, Peter A. Nature, Vol. 524, Issue 7564 https://doi.org/10.1038/nature14588	journal	July 2015
Increasing the elastic modulus of graphene by controlled defect creation López-Polín, Guillermo; Gómez-Navarro, Cristina; Parente, Vincenzo Nature Physics, Vol. 11, Issue 1 https://doi.org/10.1038/nphys3183	journal	December 2014
Fluctuations in membranes with crystalline and hexatic order Nelson, D. R.; Peliti, L. Journal de Physique, Vol. 48, Issue 7 https://doi.org/10.1051/jphys:019870048070108500	journal	January 1987
High-resolution scanning tunneling microscopy imaging of mesoscopic graphene sheets on an insulating surface Stolyarova, E.; Rim, K. T.; Ryu, S. Proceedings of the National Academy of Sciences, Vol. 104, Issue 22 https://doi.org/10.1073/pnas.0703337104	journal	May 2007
Statistical mechanics of randomly polymerized membranes Radzihovsky, Leo; Nelson, David R. Physical Review A, Vol. 44, Issue 6 https://doi.org/10.1103/physreva.44.3525	journal	September 1991
Curvature disorder in tethered membranes: A new flat phase at T =0 Morse, David C.; Lubensky, T. C. Physical Review A, Vol. 46, Issue 4 https://doi.org/10.1103/physreva.46.1751	journal	August 1992
Localization in a disordered elastic medium near two dimensions John, Sajeev; Sompolinsky, H.; Stephen, Michael J. Physical Review B, Vol. 27, Issue 9 https://doi.org/10.1103/physrevb.27.5592	journal	May 1983
Weak localization and anharmonicity of phonons Akkermans, E.; Maynard, R. Physical Review B, Vol. 32, Issue 12 https://doi.org/10.1103/physrevb.32.7850	journal	December 1985
Flat glassy phases and wrinkling of polymerized membranes with long-range disorder Le Doussal, Pierre; Radzihovsky, Leo Physical Review B, Vol. 48, Issue 5 https://doi.org/10.1103/physrevb.48.3548	journal	August 1993
Mesoscopic fluctuations of eigenfunctions and level-velocity distribution in disordered metals Fyodorov, Yan V.; Mirlin, Alexander D. Physical Review B, Vol. 51, Issue 19 https://doi.org/10.1103/physrevb.51.13403	journal	May 1995
Statistics of prelocalized states in disordered conductors Fal’ko, Vladimir I.; Efetov, K. B. Physical Review B, Vol. 52, Issue 24 https://doi.org/10.1103/physrevb.52.17413	journal	December 1995
Scaling properties of flexible membranes from atomistic simulations: Application to graphene Los, J. H.; Katsnelson, M. I.; Yazyev, O. V. Physical Review B, Vol. 80, Issue 12 https://doi.org/10.1103/physrevb.80.121405	journal	September 2009
Pinning of a two-dimensional membrane on top of a patterned substrate: The case of graphene Viola Kusminskiy, S.; Campbell, D. K.; Castro Neto, A. H. Physical Review B, Vol. 83, Issue 16 https://doi.org/10.1103/physrevb.83.165405	journal	April 2011
Mechanical properties of warped membranes Košmrlj, Andrej; Nelson, David R. Physical Review E, Vol. 88, Issue 1 https://doi.org/10.1103/physreve.88.012136	journal	July 2013
Giant Intrinsic Carrier Mobilities in Graphene and Its Bilayer Morozov, S. V.; Novoselov, K. S.; Katsnelson, M. I. Physical Review Letters, Vol. 100, Issue 1, Article No. 016602 https://doi.org/10.1103/physrevlett.100.016602	journal	January 2008
Intrinsic and extrinsic corrugation of monolayer graphene deposited on SiO 2 Geringer, V.; Liebmann, M.; Echtermeyer, T. Physical Review Letters, Vol. 102, Issue 7 https://doi.org/10.1103/physrevlett.102.076102	journal	February 2009
Electronic Cooling in Graphene Bistritzer, R.; MacDonald, A. H. Physical Review Letters, Vol. 102, Issue 20 https://doi.org/10.1103/physrevlett.102.206410	journal	May 2009
Dephasing Time in Graphene Due to Interaction with Flexural Phonons Tikhonov, Konstantin S.; Zhao, Wei L. Z.; Finkel’stein, Alexander M. Physical Review Letters, Vol. 113, Issue 7 https://doi.org/10.1103/physrevlett.113.076601	journal	August 2014
Level Correlations Driven by Weak Localization in 2D Systems Kravtsov, Vladimir E.; Lerner, Igor V. Physical Review Letters, Vol. 74, Issue 13 https://doi.org/10.1103/physrevlett.74.2563	journal	March 1995
Two-Dimensional Phonon Transport in Supported Graphene Seol, J. H.; Jo, I.; Moore, A. L. Science, Vol. 328, Issue 5975 https://doi.org/10.1126/science.1184014	journal	April 2010
Crumpling Transition in Elastic Membranes: Renormalization Group Treatment David, F.; Guitter, E. Europhysics Letters (EPL), Vol. 5, Issue 8 https://doi.org/10.1209/0295-5075/5/8/008	journal	April 1988
Atomic Structure of Graphene on SiO2 Ishigami, Masa; Chen, J. H.; Cullen, W. G. arXiv https://doi.org/10.48550/arxiv.0811.0587	text	January 2008
Conductivity of suspended graphene at the Dirac point Gornyi, I. V.; Kachorovskii, V. Yu.; Mirlin, A. D. arXiv https://doi.org/10.48550/arxiv.1208.3240	text	January 2012
Van der Waals heterostructures Geim, A. K.; Grigorieva, I. V. arXiv https://doi.org/10.48550/arxiv.1307.6718	text	January 2013
Rippling and crumpling in disordered free-standing graphene Gornyi, I. V.; Kachorovskii, V. Yu.; Mirlin, A. D. arXiv https://doi.org/10.48550/arxiv.1505.04483	text	January 2015
Electron-phonon cooling in large monolayer graphene devices McKitterick, Christopher B.; Rooks, Michael J.; Prober, Daniel E. arXiv https://doi.org/10.48550/arxiv.1505.07034	text	January 2015
Electron-phonon coupling in suspended graphene: supercollisions by ripples Laitinen, Antti; Oksanen, Mika; Fay, Aurélien arXiv https://doi.org/10.48550/arxiv.1510.09176	text	January 2015
The Effect of Intrinsic Crumpling on the Mechanics of Free-Standing Graphene Nicholl, Ryan J. T.; Conley, Hiram J.; Lavrik, Nickolay V. arXiv https://doi.org/10.48550/arxiv.1511.02049	text	January 2015
Scattering of flexural waves on thin plates Norris, A. N.; Vemula, C. Journal of Sound and Vibration, Vol. 181, Issue 1 https://doi.org/10.1006/jsvi.1995.0129	journal	March 1995
Statistics of energy levels and eigenfunctions in disordered systems Mirlin, A. Physics Reports, Vol. 326, Issue 5-6 https://doi.org/10.1016/S0370-1573(99)00091-5	journal	March 2000
Anomalous elasticity, fluctuations and disorder in elastic membranes Le Doussal, Pierre; Radzihovsky, Leo Annals of Physics, Vol. 392 https://doi.org/10.1016/j.aop.2017.08.033	journal	May 2018
Atomic Structure of Graphene on SiO₂ Ishigami, Masa; Chen, J. H.; Cullen, W. G. Nano Letters, Vol. 7, Issue 6, p. 1643-1648 https://doi.org/10.1021/nl070613a	journal	June 2007
Graphene Bimetallic-like Cantilevers: Probing Graphene/Substrate Interactions Conley, Hiram; Lavrik, Nickolay V.; Prasai, Dhiraj Nano Letters, Vol. 11, Issue 11 https://doi.org/10.1021/nl202562u	journal	November 2011
Direct Nanoscale Imaging of Ballistic and Diffusive Thermal Transport in Graphene Nanostructures Pumarol, Manuel E.; Rosamond, Mark C.; Tovee, Peter Nano Letters, Vol. 12, Issue 6 https://doi.org/10.1021/nl3004946	journal	May 2012
Electron–Phonon Coupling in Suspended Graphene: Supercollisions by Ripples Laitinen, Antti; Oksanen, Mika; Fay, Aurélien Nano Letters, Vol. 14, Issue 6 https://doi.org/10.1021/nl404258a	journal	May 2014
The structure of suspended graphene sheets Meyer, Jannik C.; Geim, A. K.; Katsnelson, M. I. Nature, Vol. 446, Issue 7131, p. 60-63 https://doi.org/10.1038/nature05545	journal	March 2007
Van der Waals heterostructures Geim, A. K.; Grigorieva, I. V. Nature, Vol. 499, Issue 7459, p. 419-425 https://doi.org/10.1038/nature12385	journal	July 2013
Graphene kirigami Blees, Melina K.; Barnard, Arthur W.; Rose, Peter A. Nature, Vol. 524, Issue 7564 https://doi.org/10.1038/nature14588	journal	July 2015
The effect of intrinsic crumpling on the mechanics of free-standing graphene Nicholl, Ryan J. T.; Conley, Hiram J.; Lavrik, Nickolay V. Nature Communications, Vol. 6, Issue 1 https://doi.org/10.1038/ncomms9789	journal	November 2015
Increasing the elastic modulus of graphene by controlled defect creation López-Polín, Guillermo; Gómez-Navarro, Cristina; Parente, Vincenzo Nature Physics, Vol. 11, Issue 1 https://doi.org/10.1038/nphys3183	journal	December 2014
Anomalous interface adhesion of graphene membranes He, Y.; Chen, W. F.; Yu, W. B. Scientific Reports, Vol. 3, Issue 1 https://doi.org/10.1038/srep02660	journal	September 2013
Fluctuations in membranes with crystalline and hexatic order Nelson, D. R.; Peliti, L. Journal de Physique, Vol. 48, Issue 7 https://doi.org/10.1051/jphys:019870048070108500	journal	January 1987
High-resolution scanning tunneling microscopy imaging of mesoscopic graphene sheets on an insulating surface Stolyarova, E.; Rim, K. T.; Ryu, S. Proceedings of the National Academy of Sciences, Vol. 104, Issue 22 https://doi.org/10.1073/pnas.0703337104	journal	May 2007
Statistical mechanics of randomly polymerized membranes Radzihovsky, Leo; Nelson, David R. Physical Review A, Vol. 44, Issue 6 https://doi.org/10.1103/PhysRevA.44.3525	journal	September 1991
Curvature disorder in tethered membranes: A new flat phase at T =0 Morse, David C.; Lubensky, T. C. Physical Review A, Vol. 46, Issue 4 https://doi.org/10.1103/PhysRevA.46.1751	journal	August 1992
Localization in a disordered elastic medium near two dimensions John, Sajeev; Sompolinsky, H.; Stephen, Michael J. Physical Review B, Vol. 27, Issue 9 https://doi.org/10.1103/PhysRevB.27.5592	journal	May 1983
Localization of acoustic waves Kirkpatrick, T. R. Physical Review B, Vol. 31, Issue 9 https://doi.org/10.1103/PhysRevB.31.5746	journal	May 1985
Weak localization and anharmonicity of phonons Akkermans, E.; Maynard, R. Physical Review B, Vol. 32, Issue 12 https://doi.org/10.1103/PhysRevB.32.7850	journal	December 1985
Flat glassy phases and wrinkling of polymerized membranes with long-range disorder Le Doussal, Pierre; Radzihovsky, Leo Physical Review B, Vol. 48, Issue 5 https://doi.org/10.1103/PhysRevB.48.3548	journal	August 1993
Mesoscopic fluctuations of eigenfunctions and level-velocity distribution in disordered metals Fyodorov, Yan V.; Mirlin, Alexander D. Physical Review B, Vol. 51, Issue 19 https://doi.org/10.1103/PhysRevB.51.13403	journal	May 1995
Spectral correlations from the metal to the mobility edge Braun, D.; Montambaux, G. Physical Review B, Vol. 52, Issue 19 https://doi.org/10.1103/PhysRevB.52.13903	journal	November 1995
Statistics of prelocalized states in disordered conductors Fal’ko, Vladimir I.; Efetov, K. B. Physical Review B, Vol. 52, Issue 24 https://doi.org/10.1103/PhysRevB.52.17413	journal	December 1995
Electrostatic interactions between graphene layers and their environment Sabio, J.; Seoánez, C.; Fratini, S. Physical Review B, Vol. 77, Issue 19 https://doi.org/10.1103/PhysRevB.77.195409	journal	May 2008
Numerical simulation of the localization of elastic waves in two- and three-dimensional heterogeneous media Sepehrinia, Reza; Rahimi Tabar, M. Reza; Sahimi, Muhammad Physical Review B, Vol. 78, Issue 2 https://doi.org/10.1103/PhysRevB.78.024207	journal	July 2008
Spatially resolved spectroscopy of monolayer graphene on SiO 2 Deshpande, A.; Bao, W.; Miao, F. Physical Review B, Vol. 79, Issue 20 https://doi.org/10.1103/PhysRevB.79.205411	journal	May 2009
Scaling properties of flexible membranes from atomistic simulations: Application to graphene Los, J. H.; Katsnelson, M. I.; Yazyev, O. V. Physical Review B, Vol. 80, Issue 12 https://doi.org/10.1103/PhysRevB.80.121405	journal	September 2009
Anderson localization of phonons in dimension d = 1 , 2 , 3 : Finite-size properties of the inverse participation ratios of eigenstates Monthus, Cécile; Garel, Thomas Physical Review B, Vol. 81, Issue 22 https://doi.org/10.1103/PhysRevB.81.224208	journal	June 2010
Pinning of a two-dimensional membrane on top of a patterned substrate: The case of graphene Viola Kusminskiy, S.; Campbell, D. K.; Castro Neto, A. H. Physical Review B, Vol. 83, Issue 16 https://doi.org/10.1103/PhysRevB.83.165405	journal	April 2011
Electron-phonon mediated heat flow in disordered graphene Chen, Wei; Clerk, Aashish A. Physical Review B, Vol. 86, Issue 12 https://doi.org/10.1103/PhysRevB.86.125443	journal	September 2012
Conductivity of suspended graphene at the Dirac point Gornyi, I. V.; Kachorovskii, V. Yu.; Mirlin, A. D. Physical Review B, Vol. 86, Issue 16 https://doi.org/10.1103/PhysRevB.86.165413	journal	October 2012
Asymptotic freedom at zero temperature in free-standing crystalline membranes Kats, E. I.; Lebedev, V. V. Physical Review B, Vol. 89, Issue 12 https://doi.org/10.1103/PhysRevB.89.125433	journal	March 2014
Thermal transport and Wiedemann-Franz law in the disordered Fermi liquid Schwiete, G.; Finkel'stein, A. M. Physical Review B, Vol. 90, Issue 6 https://doi.org/10.1103/PhysRevB.90.060201	journal	August 2014
Rippling and crumpling in disordered free-standing graphene Gornyi, I. V.; Kachorovskii, V. Yu.; Mirlin, A. D. Physical Review B, Vol. 92, Issue 15 https://doi.org/10.1103/PhysRevB.92.155428	journal	October 2015
Electron-phonon cooling in large monolayer graphene devices McKitterick, Christopher B.; Prober, Daniel E.; Rooks, Michael J. Physical Review B, Vol. 93, Issue 7 https://doi.org/10.1103/PhysRevB.93.075410	journal	February 2016
Heat diffusion in the disordered electron gas Schwiete, G.; Finkel'stein, A. M. Physical Review B, Vol. 93, Issue 11 https://doi.org/10.1103/PhysRevB.93.115121	journal	March 2016
Response of thermalized ribbons to pulling and bending Košmrlj, Andrej; Nelson, David R. Physical Review B, Vol. 93, Issue 12 https://doi.org/10.1103/PhysRevB.93.125431	journal	March 2016
Non-Hookean statistical mechanics of clamped graphene ribbons Bowick, Mark J.; Košmrlj, Andrej; Nelson, David R. Physical Review B, Vol. 95, Issue 10 https://doi.org/10.1103/PhysRevB.95.104109	journal	March 2017
Mechanical properties of warped membranes Košmrlj, Andrej; Nelson, David R. Physical Review E, Vol. 88, Issue 1 https://doi.org/10.1103/PhysRevE.88.012136	journal	July 2013
Giant Intrinsic Carrier Mobilities in Graphene and Its Bilayer Morozov, S. V.; Novoselov, K. S.; Katsnelson, M. I. Physical Review Letters, Vol. 100, Issue 1, Article No. 016602 https://doi.org/10.1103/PhysRevLett.100.016602	journal	January 2008
Intrinsic and extrinsic corrugation of monolayer graphene deposited on SiO 2 Geringer, V.; Liebmann, M.; Echtermeyer, T. Physical Review Letters, Vol. 102, Issue 7 https://doi.org/10.1103/PhysRevLett.102.076102	journal	February 2009
Electronic Cooling in Graphene Bistritzer, R.; MacDonald, A. H. Physical Review Letters, Vol. 102, Issue 20 https://doi.org/10.1103/PhysRevLett.102.206410	journal	May 2009
Disorder-Assisted Electron-Phonon Scattering and Cooling Pathways in Graphene Song, Justin C. W.; Reizer, Michael Y.; Levitov, Leonid S. Physical Review Letters, Vol. 109, Issue 10 https://doi.org/10.1103/PhysRevLett.109.106602	journal	September 2012
Phonon Bottleneck in Graphene-Based Josephson Junctions at Millikelvin Temperatures Borzenets, I. V.; Coskun, U. C.; Mebrahtu, H. T. Physical Review Letters, Vol. 111, Issue 2 https://doi.org/10.1103/PhysRevLett.111.027001	journal	July 2013
Dephasing Time in Graphene Due to Interaction with Flexural Phonons Tikhonov, Konstantin S.; Zhao, Wei L. Z.; Finkel’stein, Alexander M. Physical Review Letters, Vol. 113, Issue 7 https://doi.org/10.1103/PhysRevLett.113.076601	journal	August 2014
Scaling Behavior and Strain Dependence of In-Plane Elastic Properties of Graphene Los, J. H.; Fasolino, A.; Katsnelson, M. I. Physical Review Letters, Vol. 116, Issue 1 https://doi.org/10.1103/PhysRevLett.116.015901	journal	January 2016
Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions Abrahams, E.; Anderson, P. W.; Licciardello, D. C. Physical Review Letters, Vol. 42, Issue 10 https://doi.org/10.1103/PhysRevLett.42.673	journal	March 1979
Self-consistent theory of polymerized membranes Le Doussal, Pierre; Radzihovsky, Leo Physical Review Letters, Vol. 69, Issue 8 https://doi.org/10.1103/PhysRevLett.69.1209	journal	August 1992
Level Correlations Driven by Weak Localization in 2D Systems Kravtsov, Vladimir E.; Lerner, Igor V. Physical Review Letters, Vol. 74, Issue 13 https://doi.org/10.1103/PhysRevLett.74.2563	journal	March 1995
Measurement of the Electronic Thermal Conductance Channels and Heat Capacity of Graphene at Low Temperature Fong, Kin Chung; Wollman, Emma E.; Ravi, Harish Physical Review X, Vol. 3, Issue 4 https://doi.org/10.1103/PhysRevX.3.041008	journal	October 2013
Anderson transitions Evers, Ferdinand; Mirlin, Alexander D. Reviews of Modern Physics, Vol. 80, Issue 4 https://doi.org/10.1103/RevModPhys.80.1355	journal	October 2008
Two-Dimensional Phonon Transport in Supported Graphene Seol, J. H.; Jo, I.; Moore, A. L. Science, Vol. 328, Issue 5975 https://doi.org/10.1126/science.1184014	journal	April 2010
Theory of thermal conductivity in the disordered electron liquid Schwiete, G.; Finkel’stein, A. M. Journal of Experimental and Theoretical Physics, Vol. 122, Issue 3 https://doi.org/10.1134/S1063776116030195	journal	March 2016
Statistical Mechanics of Membranes and Surfaces Nelson, D.; Piran, T.; Weinberg, S. https://doi.org/10.1142/5473	book	June 2004
Statistical Mechanics of Membranes and Surfaces: Proceedings of the Jerusalem Winter School for Theoretical Physics Nelson, D.; Piran, T.; Weinberg, S. 5th Jerusalem Winter School for Theoretical Physics, Statistical Mechanics of Membranes and Surfaces https://doi.org/10.1142/9789814541602	conference	June 1989
Crumpling Transition in Elastic Membranes: Renormalization Group Treatment David, F.; Guitter, E. Europhysics Letters (EPL), Vol. 5, Issue 8 https://doi.org/10.1209/0295-5075/5/8/008	journal	April 1988
Anderson Transitions Evers, F.; Mirlin, A. D. arXiv https://doi.org/10.48550/arxiv.0707.4378	text	January 2007
Electrostatic interactions between graphene layers and their environment Sabio, J.; Seoanez, C.; Fratini, S. arXiv https://doi.org/10.48550/arxiv.0712.2232	text	January 2007
Atomic Structure of Graphene on SiO2 Ishigami, Masa; Chen, J. H.; Cullen, W. G. arXiv https://doi.org/10.48550/arxiv.0811.0587	text	January 2008
Spatially resolved spectroscopy of monolayer graphene on SiO2 Deshpande, A.; Bao, W.; Miao, F. arXiv https://doi.org/10.48550/arxiv.0812.1073	text	January 2008
Electronic Cooling in Graphene Bistritzer, R.; MacDonald, A. H. arXiv https://doi.org/10.48550/arxiv.0901.4159	text	January 2009
Pinning of a two-dimensional membrane on top of a patterned substrate: the case of graphene Kusminskiy, S. Viola; Campbell, D. K.; Neto, A. H. Castro arXiv https://doi.org/10.48550/arxiv.1007.1017	text	January 2010
Disorder-Assisted Electron-Phonon Scattering and Cooling Pathways in Graphene Song, Justin C. W.; Reizer, Michael Y.; Levitov, Leonid S. arXiv https://doi.org/10.48550/arxiv.1111.4678	text	January 2011
Electron-phonon mediated heat flow in disordered graphene Chen, Wei; Clerk, Aashish A. arXiv https://doi.org/10.48550/arxiv.1207.2730	text	January 2012
Mechanical Properties of Warped Membranes Kosmrlj, Andrej; Nelson, David R. arXiv https://doi.org/10.48550/arxiv.1306.5941	text	January 2013
Numerical simulation of the localization of elastic waves in two- and three-dimensional heterogeneous media Sepehrinia, Reza; Tabar, M. Reza Rahimi; Sahimi, Muhammad arXiv https://doi.org/10.48550/arxiv.1308.1775	text	January 2013
Thermal Transport and Wiedemann-Franz Law in the Disordered Fermi Liquid Schwiete, G.; Finkel'stein, A. M. arXiv https://doi.org/10.48550/arxiv.1312.7810	text	January 2013
Electron-phonon cooling in large monolayer graphene devices McKitterick, Christopher B.; Rooks, Michael J.; Prober, Daniel E. arXiv https://doi.org/10.48550/arxiv.1505.07034	text	January 2015
Response of thermalized ribbons to pulling and bending Kosmrlj, Andrej; Nelson, David R. arXiv https://doi.org/10.48550/arxiv.1508.01528	text	January 2015
The Effect of Intrinsic Crumpling on the Mechanics of Free-Standing Graphene Nicholl, Ryan J. T.; Conley, Hiram J.; Lavrik, Nickolay V. arXiv https://doi.org/10.48550/arxiv.1511.02049	text	January 2015
The structure of suspended graphene sheets Meyer, Jannik C.; Geim, A. K.; Katsnelson, M. I. arXiv https://doi.org/10.48550/arxiv.cond-mat/0701379	text	January 2007
Statistics of energy levels and eigenfunctions in disordered systems Mirlin, Alexander D. arXiv https://doi.org/10.48550/arxiv.cond-mat/9907126	text	January 1999
Anderson transitions Evers, Ferdinand; Mirlin, Alexander D. Universität Regensburg https://doi.org/10.5283/epub.46205	text	January 2008

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