Impact of the ElectronâPhonon Interactions on the Polaron Dynamics

Subscriber access provided by Purdue University Libraries

Article

Impact of the Electron-Phonon Interactions on the Polaron Dynamics in Graphene Nanoribbons Ana Virgínia Passos Abreu, Jonathan Fernando Teixeira, Antonio Luciano de Almeida Fonseca, Ricardo Gargano, Geraldo Magela e Silva, and Luiz Antonio Ribeiro J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b12482 • Publication Date (Web): 06 Apr 2016 Downloaded from http://pubs.acs.org on April 9, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Impact of the Electron-Phonon Interactions on the Polaron Dynamics in Graphene Nanoribbons Ana Virgínia Passos Abreu,† Jonathan Fernando Teixeira,† Antonio Luciano de Almeida Fonseca,† Ricardo Gargano,∗,† Geraldo Magela e Silva,† and Luiz Antonio Ribeiro Junior† Institute of Physics, University of Brasília, 70.919-970, Brasília, Brazil, University of Brasília, UnB Faculty of Planaltina, 73.345-010, Planaltina, Distrito Federal, Brazil, and Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83 Linköping, Sweden E-mail: [email protected]

Phone: +55 61 3107-3300.

∗

To whom correspondence should be addressed University of Brasilia ‡ University of Brasilia - FUP ¶ Linkoping University †

1

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 14

Abstract The influence of the electron-phonon (e-ph) interactions on the filed-included polaron dynamics in armchair graphene nanoribbons (GNRs) is theoretically investigated in the scope of a two-dimensional tight-binding model. The results show that the localization of the polaronic charge increases when the strength of e-ph coupling also increases. Consequently, the polaron saturation velocity decreases for higher e-ph coupling strengths. Interestingly, the interplay between the e-ph coupling strength and the GNR width changes substantially the polaron dynamics properties.

Introduction Carbon is recognized as one of the most interesting chemical elements for organic optoelectronic applications mainly by presenting several allotropes such as diamond, graphite, fullerenes, nanotubes, and specially the graphene. 1–3 The two-dimensional structure of graphene aggregates unique traits such as high mechanical strength, excellent electrical conductivity, and good transparency. 4,5 Graphene nanoribbons (GNRs), in turn, are structures obtained from specific cuts of a graphene sheet. 6 These structures preserve several interesting properties of the original system and, depending on its symmetry, may present a finite band gap. 7 One of the most important properties of GNRs for electronics applications is the semiconducting-like charge transport. 8 The description of this charge transport mechanism is the focus of this work and, with the approach used here, we describe a rather complex charge transport behavior which arises from changing parameters and the GNR width. Some relevant theoretical 9 and experimental 10 results have been performed in order to investigate the charge transport mechanism in GNRs. The metallic-like behavior for the electronic transport in these materials was investigated considering the effects of the electric field, 11 doping, 9 temperature, 12 several widths and crystallographic orientations. 13 Very recently, our previous study have shown that a family of GNR widths has no semiconducting bang gap and, consequently, the charge transport can not be mediated by polarons. 14 2


Page 3 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Nevertheless, studies that take into account the impact of the e-ph coupling on the polaron dynamics in GNRs are not available in the literature. Moreover, the role played by the interplay between the GNR width and e-ph coupling constant in the charge transport is not completely understood and requires more detailed investigations. In this work, the polaron dynamics in armchair GNRs is theoretically studied in the scope of a two-dimensional tight-binding model, which includes lattice relaxation. Ehrenfest-like simulations were performed in order to investigate the polaron transport considering different GNR widths and e-ph coupling strengths. The present work is intended to provide a physical picture of the polaron dynamics in GNRs contributing to the understanding of this important process, which may provide guidance for designing of optimal graphene-based optoelectronic materials.

Methodology In order to study the polarons dynamics in GNRs, we have developed a two-dimensional SSH-type Hamiltonian, 15 which includes an external electric field,

H=−

X

† ti,j Ci,s Cj,s

+ h.c. +

hi,ji,s

XK hi,ji

2

2 yi,j +

X p2 i . 2M i

(1)

In this model Hamiltonian, hi, ji represents the sum over nearest-neighbor sites i and j. † Considering the electronic part, the operator Ci,s (Ci,s ) creates (annihilates) a π–electron

with spin s at the ith site. ti,j = exp [−iγAi,j (t)] (t0 − αyi,j ) is the transfer integral, where t0 is the hopping of a π-electron between neighboring sites in an dimerized lattice, α is the e-ph coupling constant, and yi,j is the relative displacement coordinate between neighboring sites. The external electric field E(t) is turned on adiabatically and is included in the present model by means of the time-dependent vector potential A(t). 16 γ ≡ ea/(ℏc), where a is the lattice parameter, e the electronic charge, and c the speed of light. Ai,j is the vector potential ˙ component in the hi, ji direction. The equation E(t) = −(1/c)A(t) establishes the relation 3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 14

between the time-dependent vector potential and the electric field. pi denotes the conjugated momentum for the displacement coordinate of each site, K is the harmonic force constant, and M is the mass of a carbon atom. The lattice dynamics is described within a classical framework through the Euler-Lagrange equation with the expected value of the Lagrangian hLi, leading to a Newtonian equation M y¨i,j = Fi,j (t). The force expression is analogous to that originally developed by Silva and coworkers. 15 Furthermore, this equation is numerically integrated using the approach stated in reference. 17 The electron dynamics, in its turn, is described by solving the time-dependent Schrödinger equation (TDSE). The wave functions are obtained by means of a linear combination of instantaneous eigenstates, in order that the solutions of the TDSE can be expressed as

ψk,i (t + dt) =

" X X l

#

φ∗l,m (t)ψk,m (t) e(−iεl dt/ℏ) φl,i (t),

m

(2)

where {φl,i (t)} and {εl } are the eigenfunctions and the eigenvalues of the Hamiltonian at a given time t, respectively. 17 Here, we have adopted the following parameters: t0 = 2.7 eV and K = 21 eV/Å2 . 7,15 The e-ph coupling α ranging from 3.5 to 5.5 eV/Å, which corresponds to suitable values for GNRs. 18 The α value is incremented using a step of 0.1 eV/Å. However, for the sake of clarity, we present in the results section just some particular values which can represent better all the set of parameters. Moreover, this range for the α values is in agreement with experimental data for graphene. 18,19

Results The main focus of the present work is to investigate the interplay between the nanoribbon width and e-ph coupling strength on the polaron dynamics in GNRs. In order to do so, firstly, we discuss the polaron dynamics, for E = 0.5 mV/Å, using the time evolution of the mean charge density for the systems composed of a GNR-4, GNR-6, and GNR-10, where -N 4


Page 5 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


denotes the number of carbon atoms along the GNR width. For a more detailed analysis regarding the polaron saturation velocity, we present results for the wider GNRs. In this way, we begin our discussion by presenting the time evolution of the mean charge density in a GNR-4. Figure 1 depicts the polaron dynamics considering four different e-ph coupling strength (α) 3.0, 3.5, 4.0, and 4.5 eV/Å. From this figure, it is possible to verify that the charge localization increases when the strength of the e-ph coupling is increased. Moreover, it can be seen that the saturation velocity of the polaron decreases for higher e-ph coupling strengths. Naturally, these two behaviors are strictly connected in the sense that more localized charge carriers has lower degrees of mobility. In Figure 1(a), for α = 3.0 eV/Å, it is possible to note that the polaron delocalization is about 40 Å. This delocalization increases for higher α values (see Figures 1(b) and 1(c)) reaching 12 Å for 4.5 eV/Å, as depicted in Figure 1(d). Such increasing in the charge localization can be also noted observing the intensity of the red color through the lattice. Higher intensities indicate a increasing in the charge concentration. The saturation velocity mentioned above is the maximum velocity that the polaron attains in the presence of a constant external electric field. The polaron usually moves at an average drift speed proportional to the field strength that it experiences temporally. It is well known that the proportionality constant is the polaron mobility. For the polaron dynamics presented in our results, after a short acceleration period, the polaron velocity becomes constant and the excess energy, imposed by the field strength, is dissipated by creation of lattice vibrations. In this way, the polaron can keep its constant velocity (saturation velocity) by emitting phonons on its motion. However, as the action of the applied electric field is permanent in our simulations, this process is repeated systematically during the polaron motion. After a transient period, the polaron starts to release almost the same amount of excess energy at each time step, thus presenting a constant velocity. The connection between charge localization and strength of the e-ph coupling presented by the system in Figure 1 is consistent with the metalic-like behavior for the system in the

5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1: Time evolution of the mean charge density in a GNR-4 for α assuming the values of (a) 3.0, (b) 3.5, (c) 4.0, and (d) 4.5 eV/Å. The time scale is multiplied by 4, i. e., 4×fs. limit α → 0. Indeed, it is well known that the GNR band gap has an inverse dependence with the value of the e-ph coupling strength. 13,20 In metals, the charge carriers are distributed as an electron gas (Fermi gas). This physical picture resembles, therefore, a system with high degree of charge delocalization. In this way, it was already expected in our systems a decreasing in the polaron localization for small e-ph coupling values. Regarding the polaron dynamics, we can attribute the stronger interaction between the charge carrier and the lattice, for higher e-ph coupling values, as the main cause for the lowering in the polaron 6


Page 6 of 14

Page 7 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


saturation velocity. The higher the α value, the greater the interaction between π-electrons and the lattice. Figure 1(a) shows that a rapid response of the polaron to the application of an external electric field can be realized for α = 3.5 eV/Å. In Figures 1(b) and 1(c), with α assuming the values of 4.0 and 4.5 eV/Å, respectively, the polaron motion starts to be substantially compromised. Interestingly, it can be observed that for α = 4.5 eV/Å the saturation velocity is decreased for approximately 50% with respect to the case depicted in Figure 1(a). For the case shown in Figure 1(d), we obtained a critical value of α for the polaron motion in GNR-4. Considering this type of systems, practically, there is no polaron motion for α values higher than 5.0 eV/Å. Now we turn our attention to the polaron dynamics in systems composed of GNR-6. Figure 2 shows the polaron dynamics for α = 4.6, 4.9, 5.2, and 5.4 eV/Å. In these systems, the polaron dynamics occurs analogously to the cases depicted in Figure 1 for GNR-4. However, in Figure 2 the influence of the e-ph coupling strength can be noted more easily. Similarly to the cases discussed in Figure 1, the color scheme displayed in Figure 2 shows higher red and orange intensities for an increasing in the charge concentration. It worth to notice here that the polaron velocity in a GNR-6 decreases substantially increasing α from 4.6 to 4.9 eV/Å. In Figure 2(a) it is possible to note that, due to the periodic boundary conditions, the polaron runs over all the GNR length almost two times. For the case depicted in Figure 2(b), in its turn, it can be seen that the polaron reaches the beginning of the GNR and do not returns to its starting point, unlike the case shown in Figure 2(a). For α values higher than 4.9 eV/Å, the polaron saturation velocity decreases dramatically in a such way that the polaron is not able to reach the beginning of the GNR. Comparing the results depicted in Figures 2(a) and 2(b), we can note that the GNR-6 is more sensitive to variations of the α strength than systems composed by GNR-4 considering both charge localization and, consequently, the polaron saturation velocity. Moreover, one can see that the charge localization increases substantially when the α value is increased. Considering α = 4.6 eV/Å (Figure 2(a)) the charge is delocalized through almost all the nanoribbon. Increasing α for 4.9 eV/Å, as shown

7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

in Figure 2(b), the delocalization decreases considerably and the polaron still moves easily, which can be seen by the amount of charge that cross the periodic boundary condition. For α = 5.2 eV/Å, Figure 2(c), the charge density which was delocalized through basically all the nanoribbon is now localized in approximately 40 Å. In Figure 2(d) this charge localization is even more small, reaching about 30 Å.

Figure 2: Time evolution of the mean charge density in a GNR-6 for α assuming the values of (a) 4.6, (b) 4.9, (c) 5.2, and (d) 5.4 eV/Å. The time scale is multiplied by 4, i. e., 4×fs. We now consider the GNR-10 system. From Figure 3, it is easy to conclude that wider GNRs are more sensitive to variations of the α strengths. For α = 4.9 eV/Å, the polaronic charge is delocalized at about 60 Å, as shown in Figure 3(a) . Considering a small variation in the α value for 5.1 eV/Å, as depicted in Figure 3(b), the polaron starts to be localized essentially in 40 Å. We can not observe such difference degree in the polaron localization 8


Page 8 of 14

Page 9 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


for narrower GNRs when the α strength is increased. Increasing the α value even more (as shown in Figures 3(c) and 3(d)), the role played by the polaron localization and the α strength is similar to cases discussed for narrower GNRs, i. e., an increasing in the eph coupling strength leads, necessarily, to an increasing in the localization of the polaronic charge.

Figure 3: Time evolution of the mean charge density in a GNR-10 for α assuming the values of (a) 4.9, (b) 5.1, (c) 5.3, and (d) 5.5 eV/Å. The time scale is multiplied by 4, i. e., 4×fs. Finally, we summarize the discussion about the interplay between GNR width and e-ph coupling strength on the polaron saturation velocity for different GNRs. In order to do so, Figure 4(a) depicts the saturation velocity for GNR-4, GNR-6, GNR-9, and GNR-15. For the sake of clarity, Figures 4(b), 4(c), and 4(d) zoom in the regions that present the polaron velocity for GNR-4, GNR-9, and GNR-15, respectively. It is possible to conclude that the wider the GNR the higher is the polaron velocity. It means that the polaron moves faster in GNRs where its charge is more delocalized. As shown above, the polaronic charge is more delocalized for wider GNRs. Considering the same α strength for different widths, it can be seen that the polaron moves faster for wider GNRs, as shown in Figure 4. Now considering a particular width and different α values, it is possible to note that the polaron velocity is 9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment

Page 10 of 14

Page 11 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


different GNR families. Furthermore, in that work the set of parameters adopted for the simulations was based in a particular experimental reference, which is not commonly used, in order to discuss the generality of the model. Another point is that we do not discuss or present the particularities of the charge transport for different GNR families when the effects of the electron-phonon coupling are considered. Here, we depicted precisely the polaron dynamics for different GNR families, discussing substantially all of their particularities, an issue absent in the reference. 14 Another crucial difference is that here we have adopted the standard set of parameters, which may produces more trustable results.

Conclusions In summary, a two-dimensional tight-binding approach was employed to study numerically the impact of the electron-phonon interactions on the polaron dynamics in GNRs. The findings showed that the polaron localization increases when the strength of electron-phonon coupling is increased. Moreover, the interplay between GNR width and the coupling strength plays the role of modifying substantially the polaron dynamics. For a particular e-ph coupling strength, and different GNR widths, the polaron can moves faster for wider GNRs. On the other hand, considering a certain width and different e-ph coupling values, the polaron velocity decreases for higher values of coupling strengths. This detailed knowledge about the polaron dynamics in GNRs may open new channels for the developing of optimal graphenebased optoelectronic devices.

Acknowledgement The authors gratefully acknowledge the financial support from the Brazilian Research Councils CNPq, CAPES, FAPDF, and FINATEC and CENAPAD-SP for provide the computational facilities. The author L.A.R.J gratefully acknowledges the financial support from FAP-DF grant 0193.000942/2015. 11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

References (1) Stankovich, S.; Dikin, D. A.; Dommett, G. H. B.; Kohlhaas, K. M.; Zimney, E. J.; Stach, E. A.; Piner, R. D.; Nguyen, S. B. T.; Ruoff, R. S. Graphene-Based Composite Materials. Nat. Lett. 2006, 442, 282–286. (2) Yip, H.-L.; Jen, A. K. Y. Recent Advances in Solutions-Processed Interfacial Materials for Efficient and Stable Polymer Solar Cells. Energy Environ. Sci. 2012, 5, 5994–6011. (3) Pumera, M. Graphene-Based Nanomaterials for Energy Storage. Energy Environ. Sci. 2011, 4, 668–674. (4) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric Field Effect in Atomically Thin Carbon Films. Science 2004, 306, 666–669. (5) Geim, A. K.; Novoselov, K. S. The Rise of Graphene. Nat. Mat. 2007, 6, 183–191. (6) Li, X.; Wang, X.; Zhang, L.; Lee, S.; Dai, H. Chemically Derived, Ultrasmooth Graphene Nanoribbon Semiconductors. Science 2008, 319, 1229–1231. (7) Neto, A. H. C.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. The Electronic Properties of Graphene. Rev. Mod. Phys. 2009, 81, 109–162. (8) Mak, K. F.; Lui, C. H.; Shan, J.; Heinz, T. F. Observation of an Electric-Field-Induced Band Gap in Bilayer Graphene by Infrared Spectroscopy. Phys. Rev. Lett. 2009, 102, 256405–256409. (9) Martins, T. B.; Miwa, R. H.; da Silva, A. J. R.; Fazzio, A. Electronic and Transport Properties of Boron-Doped Graphene Nanoribbons. Phys. Rev. Lett. 2007, 98, 196803– 196807.

12


Page 12 of 14

Page 13 of 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(10) Baringhaus, J.; Ruan, M.; Edler, F.; Tejeda, A.; Sicot, M.; Taleb-Ibrahimi, A.; Li, A.-P.; Jiang, Z.; Conrad, E. H.; Berger, C. et al. Exceptional Ballistic Transport in Epitaxial Graphene Nanoribbons. Nat. Lett. 2014, 506, 349–354. (11) Son, Y. W.; Cohen, M. L.; Louie, S. G. Half–Metalic Graphene Nanoribbons. Nat. Lett. 2006, 444, 347–342. (12) Han, M. Y.; Brant, J. C.; Kim, P. Electron Transport in Disordered Graphene Nanoribbons. Phys. Rev. Lett. 2010, 104, 056801–056805. (13) Han, M. Y.; Özyilmaz, B.; Zhang, Y.; Kim, P. Energy Band-Gap Engineering of Graphene Nanorribbons. Phys. Rev. Lett. 2007, 98, 206805–206809. (14) Ribeiro, L. A.; da Cunha, W. F.; Fonseca, A. L. A.; e Silva, G. M.; Stafström, S. Transport of Polarons in Graphene Nanoribbons. J. Phys. Chem. Lett. 2015, 6, 510– 514. (15) de Oliveira Neto, P. H.; Teixeira, J. F.; da Cunha, W. F.; Gargano, R.; e Silva, G. M. Electron-Lattice Coupling in Armchair Graphene Nanoribbons. J. Phys. Chem. Lett 2012, 3, 3039–3042. (16) e Silva, G. M. Electric-Field Effects on the Competition Between Polarons and Bipolarons in Conjugated Polymers. Phys. Rev. B. 2000, 61, 10777–10781. (17) Lima, M. P.; e Silva, G. M. Dynamical Evolution of Polaron to Bipolaron in Conjugated Polymers. Phys. Rev. B. 2006, 74, 224303–224309. (18) Yana, J.; Zhang, Y.; Golerb, S.; Kim, P.; Pinczuk, A. Raman Scattering and Tunable Electron–Phonon Coupling in Single Layer Graphene. Sol. Sta. Commun. 2007, 143, 39–43. (19) Yan, J.; Zhang, Y.; Kim, P.; Pinczuk, A. Electric Field Tunin of Electron–Phonon Coupling in Graphene. Phys. Rev. Lett. 2007, 98, 166802–166806. 13



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(20) Son, Y.-W.; Cohen, M. L.; Louie, S. G. Energy Gaps in Graphene Nanoribbons. Phys. Rev. Lett. 2006, 97, 216803–216807.

TOC Graphic

Polaron dynamics in a graphene nanoribbon.

14


Page 14 of 14

Impact of the ElectronâPhonon Interactions on the Polaron Dynamics

Recommend Documents

Impact of the ElectronâPhonon Interactions on the Polaron Dynamics