Funct. Mater. 2016; 23 (4): 599-611.
Modified participation ratio approach: application to edge-localized states in carbon nanoclusters
SSI Institute of Single Crystals, National Academy of Sciences of Ukraine, 60 Nauky Ave., 61078 Kharkiv, Ukraine
For nanoclusters and solids, the localization analysis of one-electron states, or MOs (molecular orbitals), is frequently provided by using the so-called participation ratio (PR) index. To this conventional PR approach, we add for each MO the new index σIPR which we define as an average fluctuation of the inverse PR (IPR) value. Typically, the σIPR index displays a significant sensitivity to any spatial irregularity in the MO distribution over molecule. We apply the thus extended PR analysis to the graphene nanoflakes of different types, and small nanodiamond structures including NV color centers as well. The proposed scheme has the virtue of being quite simple, and in case of huge clusters it allows one to rapidly detect orbitals with unusual non-uniform distribution. In particular, the localization of edge states in graphene molecules is examined in this way.
1. J.T.Edwards, D.J.Thouless, J. Phys. C, 5, 807 (1972); D.J.Thouless, Phys. Rep., 13, 93 (1974). https://doi.org/10.1016/0370-1573(74)90029-5
2. F.Evers, A.D.Mirlin, Rev. Mod. Phys., 80, 1355 (2008). https://doi.org/10.1103/RevModPhys.80.1355
3. R.J.Bell, P.Dean, D.C.Hibbins-Butler, J. Phys. C, 3, 63 (1970).
4. A.V.Luzanov, V.E.Umanski, Teor. Eksperim. Chem., 13, 162 (1977).
5. A.V.Luzanov, O.A.Zhikol, Int. J. Quantum Chem., 104, 167 (2005). https://doi.org/10.1002/qua.20511
6. V.May, O.Kuhn, Charge and Energy Transfer Dynamics in Molecular Systems, Wiley-VCH, Weinheim (2011). https://doi.org/10.1002/9783527633791
7. J.Pipek, G.Mezey, J. Chem. Phys., 90, 4916 (1989). https://doi.org/10.1063/1.456588
8. I.Tamm, Fiz. Zh. Soviet Union, 1, 733 (1932).
9. J.Koutecky, Adv. Chem. Phys., 9, 85 (1965). https://doi.org/10.1002/9780470143551.ch2
10. S.G.Davison, J.D.Levine, Solid State Phys., 25, 32 (1970). https://doi.org/10.1016/S0081-1947(08)60008-9
11. S.G.Davison, M.Steslicka, Basic Theory of Surface States, Clarendon Press, Oxford (1996).
12. K.Sattler, in: Handbook of Thin Film Materials, v.5, ed. by H.S.Nalwa, Academic, New York (2002), p.61.
13. Y.-W.Son, M.L.Cohen, S.G.Louie, Nature, 444, 347 (2006). https://doi.org/10.1038/nature05180
14. M.Nishida, J. Appl. Phys., 104, 086101 (2008). https://doi.org/10.1063/1.3000656
15. G.W.Bryant, J. Comput. Theor. Nanosci., 6, 1262 (2009). https://doi.org/10.1166/jctn.2009.1174
16. L.Jiang, Y.Zheng, C.Yi et al., Phys. Rev. B, 80, 155454 (2009) https://doi.org/10.1103/PhysRevB.80.155454
17. H.M.Luhavaya, M.V.Pavlov, A.Yu.Ermilov, N.F.Stepanov, Zh. Fiz. Khim. A, 86, 1261 (2012).
18. I.A.Denisov, A.A.Zimin, L.A.Bursill, P.I.Belobrov, J. Sib. Fed. Univ. Math. Phys., 7, 3 (2014).
19. O.-A.Dobrescu, M.Apostol, Can. J. Phys., 93, 580 (2015). https://doi.org/10.1139/cjp-2014-0313
20. A.V.Luzanov, J. Struct. Chem., 55, 799 (2014); Functional Materials, 22, 514 (2015). https://doi.org/10.15407/fm22.04.514
21. A.V.Luzanov, Functional Materials, 21, 437 (2014). https://doi.org/10.15407/fm21.04.437
22. N.C.Murphy, R.Wortis, W.A.Atkinson, Phys. Rev. B, 83, 184206 (2011). https://doi.org/10.1103/PhysRevB.83.184206
23. J.E.Lennard-Jones, Proc. Roy. Soc. A, 158, 280 (1937). https://doi.org/10.1098/rspa.1937.0020
24. I.S.Gradshteyn, I.M.Ryzhik, Table of Integrals, Series, and Products, Academic, San Diego, CA (2007).
25. G.F.Kventsel, Teor. Eksper. Khim., 5, 287 (1972).
26. S.E.Stein, R.L.Brown, Carbon, 23, 105 (1985); J. Am. Chem. Soc., 109, 3721 (1987). https://doi.org/10.1021/ja00246a033
27. K.Nakada, M.Fujita, G.Dresselhaus, M.S.Dresselhaus, Phys. Rev. B, 54, 17954 (1996). https://doi.org/10.1103/PhysRevB.54.17954
28. M.Fujita, K.Wakabayashi, K.Nakada, K.Kushakabe, J. Phys. Soc. Jpn., 65, 1920 (1996). https://doi.org/10.1143/JPSJ.65.1920
29. T.Enoki, Y.Kobayashi, K.Fukui, Int. Rev. Phys. Chem., 26, 609 (2007). https://doi.org/10.1080/01442350701611991
30. M.Vanevic, V.M.Stojanovi, M.Kindermann, Phys. Rev. B, 80, 045410 (2009). https://doi.org/10.1103/PhysRevB.80.045410
31. Y.Kobayashi, K.Fukui, T.Enoki et al., Phys. Rev. B, 71, 193406 (2005). https://doi.org/10.1103/PhysRevB.71.193406
32. Y.Niimi, T.Matsui, H.Kambara et al., Phys. Rev. B, 73, 085421 (2006). https://doi.org/10.1103/PhysRevB.73.085421
33. K.Sugawara, T.Sato, S.Souma et al., Phys. Rev. B, 73, 045124 (2006). https://doi.org/10.1103/PhysRevB.73.045124
34. A.V.Luzanov, in: Practical Aspects of Computational Chemistry IV, ed. by J.Leszczynski, M.K.Shukla, Springer, New York (2016).
35. Physics and Applications of CVD Diamond, ed. by C.Nebel, M.Nesladek, Wiley, Weinheim, p. 151 (2008).
36. A.Kruger, Carbon Materials and Nanotechnology, Wiley-VCH, Weinheim (2010). https://doi.org/10.1002/9783527629602
37. V.Georgakilas, J.A.Perman, J.Tucek, R.Zboril, Chem. Rev., 115, 4744 (2015). https://doi.org/10.1021/cr500304f
38. A.V.Luzanov, O.A.Zhikol, I.V.Omelchenko et al., Functional Materials, in press.
39. R.Hoffmann, J. Chem. Phys., 39, 1397 (1963). https://doi.org/10.1063/1.1734456
40. A.V.Luzanov, O.A.Zhikol, Functional Materials, 23, 63 (2016). https://doi.org/10.15407/fm23.01.063
41. A.Konishi, T.Kubo, in: Chemical Science of Electron Systems, ed. by T.Akasaka, A.Osuka, S.Fukuzumi et al., Springer, Japan (2015), p.337. https://doi.org/10.1007/978-4-431-55357-1_20
42. O.Ivanciuc, D.J.Klein, L.Bytautas, Carbon, 40, 2063 (2002). https://doi.org/10.1016/S0008-6223(02)00065-9
43. W.Jaskolski, A.Ayuela, M.Pelc et al., Phys. Rev. B, 83, 235424 (2011). https://doi.org/10.1103/PhysRevB.83.235424
44. E K.Fujisawa, R.Cruz-Silva, K.-S.Yang et al., J. Mater. Chem. A, 2, 9532 (2014). https://doi.org/10.1039/C4TA00936C
45. F.London, J. Phys. Radium, 8, 397 (1937); https://doi.org/10.1051/jphysrad:01937008010039700 T.E.Peacock, Electronic Properties of Aromatic and Heterocyclic Molecules, Academic, London (1965).
46. T.E.Stacey, D.C.Fredrickson, Dalton Trans., 41, 7801 (2012). https://doi.org/10.1039/c2dt30298e
47. N.A.Popov, J. Struct. Chem., 11, 670 (1971). https://doi.org/10.1007/BF00743441
48. P.W.Fowler, E.Stainer, Chem. Phys. Lett., 364, 259 (2002). https://doi.org/10.1016/S0009-2614(02)01244-7
49. K.Wakabayashi, M.Fujita, H.Ajiki, M.Sigrist, Phys. Rev. B, 59, 8271 (1999). https://doi.org/10.1103/PhysRevB.59.8271
50. K.Kusakabe, M.Maruyama, Phys. Rev. B, 67, 092406 (2003). https://doi.org/10.1103/PhysRevB.67.092406
51. J.Fernandez-Rossier, J.J.Palacios, Phys. Rev. Lett., 99, 177204 (2007). https://doi.org/10.1103/PhysRevLett.99.177204
52. O.V.Yazyev, Rep. Prog. Phys., 73, 056501 (2010). https://doi.org/10.1088/0034-4885/73/5/056501
53. A.V.Luzanov, O.V.Prezhdo, J. Chem. Phys., 125, 154106 (2006). https://doi.org/10.1063/1.2360262
54. M.Head-Gordon, Chem. Phys. Lett., 372, 508 (2003). https://doi.org/10.1016/S0009-2614(03)00422-6
55. T.Pedersen, C.Flindt, J.Pedersen et al., Phys. Rev. Lett., 100, 136804 (2008). https://doi.org/10.1103/PhysRevLett.100.136804
56. LP.Biro, P.Nemes-Incze, P.Lambin, Nanoscale, 4, 1824 (2012). https://doi.org/10.1039/C1NR11067E
57. M.Wierzbicki, R.Swirkowicz, J.Barnas, Phys. Rev., 88, 235434 (2013). https://doi.org/10.1103/PhysRevB.88.235434