Relaxation of optically excited carriers in graphene: Anomalous diffusion and Lévy flights

Physical Review B - Condensed Matter and Materials Physics

Volumn 89, Issue 7, 2014, Pages

  Briskot, U ^a   Dmitriev, I A ^a,b,c   Mirlin, A D ^a,d  

^a KARLSRUHE INSTITUTE OF TECHNOLOGY   (Germany)

^b MAX PLANCK INSTITUT FÜR FESTKÖRPERFORSCHUNG   (Germany)

^c IOFFE INSTITUTE   (Russian Federation)

^d PETERSBURG NUCLEAR PHYSICS INSTITUTE   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84897653330     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.89.075414     Document Type: Article

References (63)

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). SCIEAS 0036-8075 10.1126/science.1102896 (Pubitemid 39440910)
2. Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature (London) 438, 201 (2005). NATUAS 0028-0836 10.1038/nature04235 (Pubitemid 41599868)
3. K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal'ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim, Nature Physics 2, 177 (2006). 1745-2473 10.1038/nphys245 (Pubitemid 43553498)
4. K. Bolotin, K. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. Stormer, Solid State Commun. 146, 351 (2008). SSCOA4 0038-1098 10.1016/j.ssc.2008.02.024
5. X. Du, I. Skachko, A. Barker, and E. Y. Andrei, Nature Nanotechnology 3, 491 (2008). 1748-3387 10.1038/nnano.2008.199
6. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). RMPHAT 0034-6861 10.1103/RevModPhys.81.109
7. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nature Photonics 4, 611 (2010). 1749-4885 10.1038/nphoton.2010.186
8. P. Avouris, Nano Lett. 10, 4285 (2010). NALEFD 1530-6984 10.1021/nl102824h
9. M. Engel, M. Steiner, A. Lombardo, A. C. Ferrari, H. v. Löhneysen, P. Avouris, and R. Krupke, Nat. Commun. 3, 906 (2012). 2041-1723 10.1038/ncomms1911
10. T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, Phys. Rev. Lett. 108, 167401 (2012). PRLTAO 0031-9007 10.1103/PhysRevLett.108.167401
11. N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, Science 334, 648 (2011). SCIEAS 0036-8075 10.1126/science.1211384
12. J. C. W. Song, M. S. Rudner, C. M. Marcus, and L. S. Levitov, Nano Lett. 11, 4688 (2011). NALEFD 1530-6984 10.1021/nl202318u
13. D. E. Sheehy and J. Schmalian, Phys. Rev. Lett. 99, 226803 (2007). PRLTAO 0031-9007 10.1103/PhysRevLett.99.226803 (Pubitemid 350198425)
14. J. González, F. Guinea, and M. A. H. Vozmediano, Phys. Rev. Lett. 77, 3589 (1996). PRLTAO 0031-9007 10.1103/PhysRevLett.77.3589
15. L. Fritz, J. Schmalian, M. Müller, and S. Sachdev, Phys. Rev. B 78, 085416 (2008). PRBMDO 1098-0121 10.1103/PhysRevB.78.085416
16. S. Das Sarma, E. H. Hwang, and W.-K. Tse, Phys. Rev. B 75, 121406 (2007). PRBMDO 1098-0121 10.1103/PhysRevB.75.121406
17. E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007). PRBMDO 1098-0121 10.1103/PhysRevB.75.205418
18. M. Schütt, P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. B 83, 155441 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.83.155441
19. A. B. Kashuba, Phys. Rev. B 78, 085415 (2008). PRBMDO 1098-0121 10.1103/PhysRevB.78.085415
20. S. Kim, I. Jo, J. Nah, Z. Yao, S. K. Banerjee, and E. Tutuc, Phys. Rev. B 83, 161401 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.83.161401
21. R. V. Gorbachev, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. Tudorovskiy, I. V. Grigorieva, A. H. MacDonald, S. V. Morozov, K. Watanabe, T. Taniguchi, and L. A. Ponomarenko, Nature Physics 8, 896 (2012). 1745-2473 10.1038/nphys2441
22. M. Titov, R. V. Gorbachev, B. N. Narozhny, T. Tudorovskiy, M. Schütt, P. M. Ostrovsky, I. V. Gornyi, A. D. Mirlin, M. I. Katsnelson, K. S. Novoselov, A. K. Geim, and L. A. Ponomarenko, Phys. Rev. Lett. 111, 166601 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111.166601
23. W.-K. Tse, BenYu-Kuang Hu, and S. Das Sarma, Phys. Rev. B 76, 081401 (R) (2007). PRBMDO 1098-0121 10.1103/PhysRevB.76.081401
24. M. I. Katsnelson, Phys. Rev. B 84, 041407 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.84.041407
25. B. N. Narozhny, M. Titov, I. V. Gornyi, and P. M. Ostrovsky, Phys. Rev. B 85, 195421 (2012). PRBMDO 1098-0121 10.1103/PhysRevB.85.195421
26. J. Lux and L. Fritz, Phys. Rev. B 86, 165446 (2012). PRBMDO 1098-0121 10.1103/PhysRevB.86.165446
27. J. C. W. Song and L. S. Levitov, Phys. Rev. Lett. 111, 126601 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111.126601
28. M. Schütt, P. M. Ostrovsky, M. Titov, I. V. Gornyi, B. N. Narozhny, and A. D. Mirlin, Phys. Rev. Lett. 110, 026601 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.110.026601
29. D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. de Heer, P. N. First, and T. B. Norris, Phys. Rev. Lett. 101, 157402 (2008). PRLTAO 0031-9007 10.1103/PhysRevLett.101.157402
30. J. M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M. G. Spencer, Appl. Phys. Lett. 92, 042116 (2008). APPLAB 0003-6951 10.1063/1.2837539 (Pubitemid 351198838)
31. S. Winnerl, F. Göttfert, M. Mittendorff, H. Schneider, M. Helm, T. Winzer, E. Malic, A. Knorr, M. Orlita, M. Potemski, M. Sprinkle, C. Berger, and W. A. de Heer, J. Phys.: Condens. Matter 25, 054202 (2013). JCOMEL 0953-8984 10.1088/0953-8984/25/5/054202
32. M. Breusing, C. Ropers, and T. Elsaesser, Phys. Rev. Lett. 102, 086809 (2009). PRLTAO 0031-9007 10.1103/PhysRevLett.102.086809
33. J. Shang, Z. Luo, C. Cong, J. Lin, T. Yu, and G. G. Gurzadyan, Appl. Phys. Lett. 97, 163103 (2010). APPLAB 0003-6951 10.1063/1.3504704
34. K. J. Tielrooij, J. C. W. Song, S. A. Jensen, A. Centeno, A. Pesquera, A. Zurutuza Elorza, M. Bonn, L. S. Levitov, and F. H. L. Koppens, Nature Physics 9, 248 (2013). 1745-2473 10.1038/nphys2564
35. E. H. Hwang, BenYu-Kuang Hu, and S. Das Sarma, Phys. Rev. B 76, 115434 (2007). PRBMDO 1098-0121 10.1103/PhysRevB.76.115434
36. M. Polini, R. Asgari, Y. Barlas, T. Pereg-Barnea, and A. MacDonald, Solid State Commun. 143, 58 (2007). SSCOA4 0038-1098 10.1016/j.ssc.2007.04.035 (Pubitemid 46874509)
37. M. Polini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea, and A. H. MacDonald, Phys. Rev. B 77, 081411 (2008). PRBMDO 1098-0121 10.1103/PhysRevB.77. 081411
38. M. R. Ramezanali, M. M. Vazifeh, R. Asgari, M. Polini, and A. H. MacDonald, J. Phys. A: Math. Theor. 42, 214015 (2009). 1751-8113 10.1088/1751-8113/42/21/214015
39. M. Breusing, S. Kuehn, T. Winzer, E. Malić, F. Milde, N. Severin, J. P. Rabe, C. Ropers, A. Knorr, and T. Elsaesser, Phys. Rev. B 83, 153410 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.83.153410
40. R. Kim, V. Perebeinos, and P. Avouris, Phys. Rev. B 84, 075449 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.84.075449
41. A. Tomadin, D. Brida, G. Cerullo, A. C. Ferrari, and M. Polini, Phys. Rev. B 88, 035430 (2013). PRBMDO 1098-0121 10.1103/PhysRevB.88.035430
42. E. Mariani and F. von Oppen, Phys. Rev. Lett. 100, 076801 (2008); PRLTAO 0031-9007 10.1103/PhysRevLett.100.076801 (Pubitemid 351393575)
43. F. von Oppen, F. Guinea, and E. Mariani, Phys. Rev. B 80, 075420 (2009). 1098-0121 PRBMDO 10.1103/PhysRevB.80.075420
44. I. V. Gornyi, V. Y. Kachorovskii, and A. D. Mirlin, Phys. Rev. B 86, 165413 (2012). PRBMDO 1098-0121 10.1103/PhysRevB.86.165413
45. J. C. W. Song, M. Y. Reizer, and L. S. Levitov, Phys. Rev. Lett. 109, 106602 (2012). PRLTAO 0031-9007 10.1103/PhysRevLett.109.106602
46. D. Brida, A. Tomadin, C. Manzoni, Y. J. Kim, A. Lombardo, S. Milana, R. R. Nair, K. S. Novoselov, A. C. Ferrari, G. Cerullo, and M. Polini, Nat. Commun. 4, 1987 (2013). 2041-1723 10.1038/ncomms2987
47. J. C. W. Song, K. J. Tielrooij, F. H. L. Koppens, and L. S. Levitov, Phys. Rev. B 87, 155429 (2013). PRBMDO 1098-0121 10.1103/PhysRevB.87.155429
48. W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed., Vol. 2 (Wiley, New York, 1971).
49. M. S. Foster and I. L. Aleiner, Phys. Rev. B 79, 085415 (2009). PRBMDO 1098-0121 10.1103/PhysRevB.79.085415
50. E. M. Lifshitz and L. P. Pitaevskii, in Physical Kinetics, Course of Theoretical Physics Vol. 10, translated from the Russian by J. B. Sykes and R. N. Franklin (Butterworth Heinemann, Linacre House, Jordan Hill, Oxford, 1999).
51. S. Sachdev, Phys. Rev. B 57, 7157 (1998). PRBMDO 0163-1829 10.1103/PhysRevB.57.7157
52. A. Schmid, Z. Phys. 271, 251 (1974). EPJAFV 1434-6001 10.1007/BF01677931
53. B. L. Al'tshuler and G. Aronov, JETP Lett. 30, 482 (1979).
54. E. Abrahams, P. W. Anderson, P. A. Lee, and T. V. Ramakrishnan, Phys. Rev. B 24, 6783 (1981). PRBMDO 0163-1829 10.1103/PhysRevB.24.6783
55. In fact secondary electrons generated during the cascade will also relax. The account for this relaxation requires the full solution of the kinetic equation which is beyond the scope of this work.
56. H. E. Hurst, Trans. Am. Soc. Civ. Eng. 116, 770 (1951).
57. H. C. Fogedby, T. Bohr, and H. J. Jensen, J. Stat. Phys. 66, 583 (1992). JSTPBS 0022-4715 10.1007/BF01060082
58. J. Feder, Fractals, Physics of solids and liquids (Plenum Press, New York, 1988).
59. S. Jespersen, R. Metzler, and H. C. Fogedby, Phys. Rev. E 59, 2736 (1999). 1063-651X 10.1103/PhysRevE.59.2736
60. F. Mainardi, G. Pagnini, and R. Saxena, J. Comput. Appl. Math. 178, 321 (2005). JCAMDI 0377-0427 10.1016/j.cam.2004.08.006 (Pubitemid 40356094)
61. For a Gaussian initial density f (É) = (n 0 / δ 2 π) exp [(É-ω pump / 2) 2 / 2 δ 2] with width δ, our results remain valid for large times t ≠δ α / D, when the initial condition is washed out and the form of the probability density is determined by diffusion.
62. The propagator can be written for arbitrary α and β in terms of the Fox H function.
63. V. Ryzhii, M. Ryzhii, and T. Otsuji, J. Appl. Phys. 101, 083114 (2007). JAPIAU 0021-8979 10.1063/1.2717566 (Pubitemid 46691139)