Polaron Properties in Armchair Graphene Nanoribbons - The Journal

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Polaron Properties in Armchair Graphene Nanoribbons Wiliam Ferreira da Cunha, Paulo H Acioli, Pedro Henrique de Oliveira Neto, Ricardo Gargano, and Geraldo Magela e Silva J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b12491 • Publication Date (Web): 26 Feb 2016 Downloaded from http://pubs.acs.org on February 28, 2016

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The Journal of Physical Chemistry

Polaron Properties in Armchair Graphene Nanoribbons Wiliam F. da Cunha,∗,†,‡ Paulo H. Acioli,¶ Pedro H. de Oliveira Neto,†,§ Ricardo Gargano,† and Geraldo M. e Silva† Institute of Physics, University of Brasilia, Brasilia 70.919-970, Brazil, Quantum Theory Project, University of Florida, Gainesville FL 32611-2085, USA, Department of Physics, Northeastern Illinois University, Chicago IL 60625, USA, and Department of Chemistry, MIT, Cambridge MA 02139 E-mail: [email protected]

Phone: +55 (61)3107 7714. Fax: +55 (61)3107 2363

∗ To

whom correspondence should be addressed of Physics, University of Brasilia, Brasilia 70.919-970, Brazil ‡ Quantum Theory Project, University of Florida, Gainesville FL 32611-2085, USA ¶ Department of Physics, Northeastern Illinois University, Chicago IL 60625, USA § Department of Chemistry, MIT, Cambridge MA 02139 † Institute

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Abstract By means of a two dimensional tight-binding model with lattice relaxation in a first order expansion, we report different polaron properties depending on the armchair graphene nanoribbons width family as well as on its size. We obtain that representatives of 3p+2 family do not present a polaronic mediated charge transport. As for 3p and 3p+1 families, the polaron behavior was completely dependent on the system’s width. Particularly, we observed a greater degree of delocalization for broader nanoribbons; narrower nanoribbons of both families, on the other hand, typically presented a more localized polaronic-type transport. Energy levels and occupation numbers analysis are performed in order to rigorously assess the nature of the charge carrier. Time evolution in the scope of the Ehrenfest Molecular Dynamics was also carried out to confirm the collective behavior and stability of the carrier as a function of time. We were able to determine that polarons in nanoribbons of 3p family present higher mobility than those in 3p+1 nanoribbons. These results are identification of the transport process that takes place for each system and it allows the prediction of the mobility of the charge carriers as a function of the structural properties of the system. Thus, providing guidance on how to improve the efficiency of graphene nanoribbon based devices.

1 - Introduction Graphene based structures are celebrated as promising candidates for the development of an alternative nanoelectronic technology. Through the use of such systems, the hope is to create efficient and inexpensive devices whose fabrication process generates less environmental impact. 1 Among these materials, Armchair Graphene Nanoribbons (AGNRs) draw special attention from the scientific community due to their possibility of exhibiting a tunable sized bandgap which is dependent of the system’s width. 2 This feature, along with other known properties such as high thermal and electrical conductivities, excellent mechanical strength, and reasonable transparency, makes the understanding of the charge transport mechanism in AGNRs one of the last hurdles to be overcome towards the development of a new competitive carbon–based electronics. 3 In this context, a 2 ACS Paragon Plus Environment

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considerable amount of effort has been devoted to the study of the charge transport in GNRs as a function of electric field, 4,5 temperature 6 and doping, 7 properties that are obviously important to the development of an AGNR based electronics. These works, however, were conducted by considering the electronic transport picture. It was not until very recently that the quasi-particle–like behavior scenario and the possibility of these system to present alternative transport mechanism was taken into account. 8 After developing a model that predicted with great accuracy the experimental charge distribution through the AGNR lattice, this work suggested a collective behavior of the charge typical of the quasi-particle response to an applied external electric field. It has been known for decades that the structures responsible for charge transport in organic semiconductors arise from lattice distortions that yields self–trapped quantum states due to strong electron–lattice interactions. 9 The conduction scheme carried out by such structures is commonly referred as quasi-particle mediated, in contrast to the conventional electronic mediated transport observed in metals and inorganic semiconductors. Throughout the years the difference of performance among systems in which different kinds of quasi-particles take part has been extensively pointed out. 10 This has to do with the very nature of the quasi-particle, the distortion it creates on the lattice and the way charge is distributed in the chain. For instance, a polaron possess ± 1/2 spin and ±e charge, 11 thus responding to electric and magnetic field simultaneously. A bipolaron, on the other hand, which is a ±2e charged structure and no spin 12 can only be affected by the former. As different kinds of charge transport schemes are conducted to carriers with different properties, one can reason that the difference in performance of the organic electronic devices will strongly depend on the nature of the carrier. In this sense, the correct description of which quasi-particle is responsible to the charge transport in AGNRs under specific circumstances is of fundamental importance to the development of graphene based technology. Among the crucial properties that are known to strongly affect the charge transport mechanism in AGNRs those that have to do with geometric parameters, such as different crystallographic orientations and variable widths, 13 stand out. Particularly, the importance of the latter could not be



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stressed enough because depending on the width, AGNRs can acquire different band gap values ranging from zero (metallic) to several finite (semiconducting) values. This fact has been extensively reported by means of tight-binding and DFT 14,15 methodologies and has been attributed to edge states and quantum confinement. As the synthesis methods for graphene based nanostructures have evolved to the point of allowing increasingly precise controlled shape and roughness, 2 theoretical calculations aiming to investigate the underlying transport mechanism for different widths becomes mandatory. Specifically, since the charge transport efficiency is strongly dependent on the nature of the quasi-particle involved, a thorough investigation concerning the relationship between the nanoribbon width and the nature of the arising carrier and its dynamics is strongly desired. We have recently developed a series of publications concerning the quasi-particle transport scheme in AGNR under different conditions of electron-phonon coupling, 8 applied electric field 16 and impurities. 17 Due to the similarities between organic conductors and the graphene nanoribbons, we decided to refer to the quasi-particles present in the treated systems generically as “polarons”. However, in ref. 17 we explicitly emphasized that this was, in principle, a terminology adopted in favor of using terms that were familiar to the scientific community. This argument was raised because at that point no specific study on the nature of the quasi-particle was presented in order to allow for a conclusive denomination of that charge carrier as being a polaron. In order to rigorously access the nature of the charge transport in each case, one needs to specifically investigate properties such as the energy levels and occupation numbers as well as their time evolution during the process. We have just argued that the width of the nanoribbon affects its transport due to the quantum effects generated by the edges. It is only natural to conclude that these effects will directly affect the nature of the quasi-particle. Therefore, the width of the nanoribbon is expected to play a crucial role on determining which quasi-particle arises in a given situation. For instance, one can not rule out the possibility of a different quasi-particle, a different transport mechanism or even no transport at all to take place for a given situation. Even of more importance than the nature of the quasi-particle is how they might possibly behave differently — in terms of its distribution, dynamics and kinematics — as a function of the nanorribon width.


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Having this perspective in mind we dedicate the present article to investigate the AGNR width influence on the nature of the quasi-particle responsible for the charge transport, as well as on its transport properties. We aim to provide general rules describing the nature of the carrier depending on the width of the system, as well as its behavior. In order to do so, we divided the AGNRs in their three characteristic families, depending on the width formation rule for the number of carbon sites: 15 3p, 3p+1 and 3p+2, where p is a natural number . By doing so, we are able to obtain the influence of the family width on the kind of the charge carrier. Moreover, we included a quantitative analysis of the width value itself, by considering different values of p (p=1,2,3), in order to investigate the role played by the size as well as the parity of the number of carbons forming each nanoribbon. By means of an extensive numerical investigation in the scope of a two dimensional tightbinding hamiltonian with lattice relaxation in a first order expansion we were able to observe that both the width family as well as its value and parity plays a fundamental role on determining what kind of transport process is to take place. As the 3p+2 constitutes an AGNR family that does not present energy gap as a result no quasi-particle was formed and the charge just distributed among the system, regardless of the width considered. For the other families, the kind of transport mode is determined by its width. As a rule of thumb, we observed that the narrower the nanoribbon, the more polaronic localized the transport tends to be. These facts are consistent with the two limiting structural conditions and their typical transport mechanism: for a cis-symmetry organic conductor, we have localized polarons as the main charge carrier, whereas for the entire graphene sheet delocalization is observed. Besides rescuing intermediate responses between these limiting cases, we were able to obtain the correct trend of systematically increasing delocalization with size. As our main achievement, we provided the literature with the correct kind of charge transport mode that will take place for a given size of the AGRN device and also determine how the nanoribbon family affects the charge carrier transport. This is of fundamental importance for the correct prediction of the device’s performance as well as of its possible ways of improvement.



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2 - Model and Method We have been investigating AGNR in the scope of a hybrid hamiltonian, in which electrons are treated within a second quantization approach, whereas the lattice is treated classically. The two parts of the system are connected through an electron-phonon coupling term used to include lattice relaxation to an otherwise unrelaxed two dimensional tight-binding model. This is a key feature of our model, and it allows quasi-particle mechanism to arise. Considering the Hückel approximation, the π-electron integrals can be expanded up to first order, provided the bond distance variations are small. It is worthy to mention that this is certainly the case for the nanoribbon lattice, the typical oscillations observed are usually not much greater than 2%. In this sense we consider the hopping integral to be of the form:

ti, j = t0 − αyi, j .

(1)

This expression provides the probability amplitude of finding an electron originally from site i in site j. Here α is the electron-phonon coupling constant, which is responsible for including interdependency between electron and lattice degrees of freedom. yi, j represents the variation on the existing bond distance between sites i and j in Figure 1. In our work the values used for the constants t0 and α were, respectively, 2.7 eV and 4.1 eV/Å. 18 We chose these values after performing preliminary simulations considering values of α that lie in the range from 3.5 to 10 eV /Å. 19 According to the literature, these are the most suitable values for GNRs. A similar analysis was carried out in terms of the harmonic elastic constant K. 8,16,18 Considering the equation (1), the system Hamiltonian is given by

H = −

∑

† (ti, jCi,s C j,s + t ∗j,iC†j,sCi,s )

,s

+

1 Pi2 1 2 K(y ) + ∑ i, j 2 ∑ M . 2 i

(2)

Where < i, j > represents the indices of neighboring sites. Note that the first term is nothing but the 6 ACS Paragon Plus Environment

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Figure 1: Schematic labeling of an armchair GNR. two electron tight binding hamiltonian, except that the amplitude of probability is given by equation (1), so that electron and lattice parts of the system are coupled. In this term, Ci,s is the annihilation operator of a s spin π electron in the i-th site and C†j,s the corresponding creation operator in site j. The second term is the effective potential associated to the σ bonds, considered in an usual harmonic (Hooke-type) approximation; K is the elastic constant associated to this approximation. Finally, the third term describes the kinetic energy of the sites in terms of their momenta Pi , M being the mass. From an initial set of coordinates {yi, j } and given the values of t0 , K and α one can construct the stationary initial electronic Hamiltonian, according to equation (8), for Pi = 0 (static case). 20 In order to solve our problem, an initial self consistent state is obtained by diagonalizing the stationary



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Hamiltonian as H = − ∑ Ek a†k ak ,

(3)

k

where Ek are the eigenenergies of the electronic hamiltonian. The procedure is to find the set {ak } of operators that yields a diagonal hamiltonian. These operators are used in a LCAO fashion to obtain the annihilation and creation operators: ∗ ak = ∑ ψk,i Ci

(4)

i

Substitution of these expressions in the electronic Hamiltonian yields

H =−

∑

,s,k,k0

∗ (ti, j ψi,k ψk0 , j + t ∗j,i ψ ∗j,k ψk0 ,i )a†k ak0 .

(5)

Finally, the above expression is diagonalized as long as the following condition is satisfied

Ek ψi,k = −ti, j ψ j,k − ti, j0 ψ j0 ,k − ti, j00 ψ j00 ,k .

(6)

where i, j, j0 and j00 stand for neighboring sites. This diagonalization procedure results in the eigenvalues — energies — and eigenvectors — wave functions — for the initial instant. The lattice is solved classically by means of the EulerLagrange equations: d dt

∂ hLi ∂ hLi − =0 ∂ q˙i ∂ qi

=⇒

∂ hLi = 0, ∂ yi, j

(7)

as we are considering the static case. So, in order to take lattice effects into account, one needs to obtain the expectation value of the system’s Lagrangean hΨ|L|Ψi, where |Ψi is the Slater type wave function represented in the second quantization formalism as |Ψi = a†1 a†2 a†3 . . . a†n | i.


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Since,

L=

1 Pi2 1 † − ∑ K(yi, j )2 + ∑ (t0 − αyi j )(Ci,s C j,s +C†j,sCi,s ), 2∑ M 2 i ,s

thus,

hLi =

1 Pi2 1 − ∑ K(yi, j )2 + ∑ (t0 − αyi j )(Bi, j ) 2∑ M 2 i ,s

where, ∗ (i,t)ψk,s ( j,t). Bi, j ≡ ∑ 0 ψk,s

(8)

k,s

In the latter expression, the sum is carried out over the occupied states only. It is important to note that the electronic and the lattice parts of the system are coupled by the terms expressed by equation (8). Now, we return to Eq. (7) and solve the Euler-Lagrange equation for the stationary case. Summarizing, an initial state, which is self consistent to the degrees of freedom of both electrons and lattice, is constructed by considering a set {yi, j }, obtaining a corresponding set {ψk,i }, and solving to obtain new values {yi, j } until convergence is achieved. The next step is to proceed to the time evolution of the system. This is accomplished by solving the time dependent Schrödinger equation for the electrons together with the Euler-Lagrange equations for the lattice. Starting from the stationary solution {yi, j } and {ψk,i }, the electronic part is governed by

i¯h

∂ |ψk (t)i = H|ψk (t)i, ∂t

and the time evolution is accomplished as i

|ψk (t + dt)i = e− h¯ H(t)dt) |ψk (t)i.


(9)


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By expanding the wave function |ψk (t)i in a basis set of eigenstates of the electronic Hamiltonian at time t (|φl (t)i)

|ψk (t)i = ∑ Dk,l |φl (t)i,

(10)

l

we can obtain the electronic time evolution from of equation (9) as i

|ψk (t + dt)i = ∑hφl (t)|ψk (t)ie− h¯ εl dt |φl (t)i.

(11)

l

The numeric integration of such expression is carried out as usual. 20 The lattice dynamics time evolution is described through the solution of equation (11) coupled to the full Euler-Lagrange equations. Straightforward application of Euler-Lagrange Equations yields a Newton-type 8 expression that describes the lattice behavior (see Figure 1) 1 Fi j (t) = M y¨i, j = K[yi,i0 + yi,i00 + y j, j0 + y j, j00 − 4yi, j ] + 2 1 + α[Bi,i0 + Bi,i00 + B j, j0 + B j, j00 − 4Bi, j + h.c.]. 2

(12)

In this work we also included an electric field E(t) in our model. This is accomplished by considering the time–dependent vector potential A(t) through a Peierls substitution of the phase factor to the hopping integral. 20 Equation (1) is modified as ti j = e−iγA [t0 − αyi, j ],

(13)

where γ ≡ ea/(}c), with a being the lattice parameter, e the absolute value of the electronic charge, and c the speed of light. The relation between the time dependent electric field and the vector ˙ potential is given as usual by E(t) = −(1/c)A(t). In this work, we included the electric field adiabatically in the system by considering equation (14) as the definition of the potential vector A(t):


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A(t) =

                      

0 −1 τ 2 cE t − π sin −c t − τ2 −1 2 cE

t + t f − τ + πτ sin

if πt

0 ≤ t < τ,

if

τ

if τ π (t − t f

+ π)

τ ≤ t < tf ,

(14)

t f ≤ t < t f + τ,

if

−cEt f

t < 0,

if

t ≥ τ.

In this expression, τ is the period for which the field acts on the system and t f denotes the instant where electric field is at its maximum. This choice is important to avoid artificially included numerical oscillations that would appear in the lattice as a relaxation response to an otherwise abrupt implementation of the electric field.

3 - Results and Discussion In the present work we are concerned with the characterization of charge carriers in AGNR of different sizes. Particularly, we seek to obtain a qualitative relationship between the nanoribbon width and the behavior of the charge carrier in terms of its distribution as well as of its dynamics. We perform simulations of nanoribbons of 423 Åof length (in Figure 1) with periodic boundary conditions along the vertical direction, in order to mitigate length edge effects. For the number of columns of Figure 1, whose dependence is the main focus of this work, we consider n =3, 4, 5, 6, 7, 8, 9 and 10 sites width nanoribbons. These values are consistent with the well known three families of AGNR, namely 3p, 3p+1 and 3p+2, for integer p. This is a particularly interesting choice because these families are known to present different electronic properties. 15 We thus have, as representative of the 3p family, the nanoribbons 3x200, 6x200 and 9x200 where this notation stands for nxm. The 3p+1 family is represented by the nanoribbons 4x200, 7x200 and 10x200, and the 3p+2 is constituted of 5x200 and 8x200 cases. We discuss three different aspects of charge transport in AGNR for the aforementioned cases. We begin by studying static calculations to investigate how the charge is initially distributed over



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the chain depending on the width. This initial distribution is expected to have profound influence on the behavior of the time evolved system. An accurate analysis of energy levels allows us to determine the kind of quasi-particle that is present in each case. Following, we carry out a dynamic description of the charge carriers for an applied external electric field, so that the collective behavior expected for a quasi-particle is confirmed. Finally, we summarize the results of the dynamic calculations by performing a kinematic analysis concerning the dependency of carrier’s velocities on the AGNR width. We start by presenting static simulations results of charge distribution in AGNRs. The idea is to study how the charge density profile presents itself depending on different widths. We do this by considering the extraction of one electron from the lattice, so that all cases are equally positively charged. Depending on whether or not a high delocalization degree is present we can rule out or look further into the possibility of a polaron transport scheme to take place. Should the latter hypothesis be confirmed, additional investigation are carried for a complete characterization of the quasi-particle. The first set of simulations to be discussed concerns how an excess of (or lack of) electronic charge is distributed in AGNRs of family 3p, i.e., those with n= 3, 6 and 9. The charge density profile is shown in Figure 2. We can clearly observe that the charge density is symmetrically distributed around the region where the quasi-particle is formed. The length of the localization plays a fundamental role in the dynamical behavior of the quasi-particle. An interesting feature to be noted is that narrower nanoribbons present much more localized charge density. This fact has to do with the nature of the quasi-particle that takes place in this systems, a property that will be discussed shortly. If one turns the attention on how the charge density is spread along the width of the nanoribbon, it is possible to see that whereas in the n=3 and n=9 cases the charge is concentrated in the middle of the chain, for the n=6 nanoribbon the charge is mainly localized near its edges. This, again, arises from symmetry considerations: as no defects were simulated in the chain, the simplest equilibrium condition for lattices with even number of sites along its width is to have two evenly spaced centers


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of charge. When n is odd, on the other hand, an odd number of centers of charge naturally appears on the chain. Another important feature that stands out in Figure 2 is the expected fact that charge in broader nanoribbons tends to present a broader distribution. Since our goal is to characterize the kind of charge carrier that arises in each case, an investigation concerning the energy levels is required. For the systems in Figure 2 we obtained that the energy gap decreases as the width increases as follows: for n=3 we obtained a HOMO-LUMO energy of 2.5 eV; for n=6, 1.5 eV and for n=9, a 1.04 eV value was achieved. This decreasing trend is consistent with the known fact that a whole graphene sheet does not present energy gap. Moreover, we can see that these energies values lie in an intermediate range that characterizes the systems as semiconducting. Indeed, this is necessary condition for a polaronic transport to take place. It is worthy to mention that these values are in reasonable agreement to those obtained by Y-W Son et al., 15 even though their work disregarded lattice relaxation. The next step towards a complete characterization of the charge carrier is to investigate the nature of typical energy levels inside the gap. By means of this analysis we can unambiguously determine what kind of charge carrier is present in the chain. For the three cases of the 3p family we observed two symmetrically displaced energy levels: the upper level stands at the same distance below the conducting band than does the lower one above the valence band. We obtained as the absolute value of the difference between the level inside the gap and the nearest edge of the relative band as 0.075 eV for n=3; 0.025 eV for n=6 and 0.010 eV for n=9. These levels appear after an abrupt increase of density of states that characterizes the edge of the valence and conducting bands, therefore, we can be sure that they lie inside the gap. Due to the analyzed nature of these levels, one can be sure that, indeed, the charge density presents itself by means of a true quasi-particle of the polaron type. It is now clear why a higher degree of localization is obtained for the narrower nanoribbon: the broader the system becomes, the less polaronic character it presents in favor to a more delocalized scheme. This is confirmed by the continued closing of the energy gap that, in the limit, approaches zero for the completely delocalized case of an infinitely broad graphene sheet. The next family to be analyzed, the 3p+1, presents qualitatively similar results concerning



Charge Density (e) 255

0.010 225

0.008

230

0.006

0

Length (A)

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205

215 205 195

0.004

185

n=9

0.002

180

0.000 155

n=3

n=6

n=9

Figure 2: Charge density distribution for 3p family the static charge distribution pattern. We can see in Figure 3 that the same considerations about the parity dependence on charge distribution can be made. Symmetry arguments allows one to conclude that the even widths of n=4 and 10 should present even number of equally spaced charge centers (2 and 4, respectively) including one near each edge of the nanoribbon. Correspondingly, the odd width n=7 presents 3 well defined centers, including one at the middle of the lattice. Also, the pattern of increasing charge density distribution in broader nanoribbons is confirmed. Thus, we conclude that the static disposition of the charge carrier is qualitatively invariant for systems from family 3p to family 3p+1. The similarity between the 3p and 3p+1 families also presents itself in an energy levels analysis. The energy gap varies from 2.50 eV, for a n=4 nanoribbon to 1.48 eV for n=7 up to 1.04 eV for n=10. These values are similar to those obtained for 3p family representatives of n=3, 6 and 9, respectively. In other words, nanoribbons of similar widths from both families present energy gaps of almost identical values. Equally similar is the disposition of the energy levels inside the gap. 14 ACS Paragon Plus Environment

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For these levels we obtain the following displacements: 0.075 eV for n=4, 0.018 eV for n=7 and 0.010 eV for n=10. As these values are in strikingly accordance with the ones from 3p family, we conclude that for the 3p+1 family too the charge carriers shown in Figure 3 are typical polarons.

Charge Density (e) 225 225 230

215 205

0

Length (A)

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


205

195 185

180

155

n=4

n=7

n=10

n=10

Figure 3: Charge density distribution for 3p+1 family The last family that remains to be discussed is 3p+2, whose charge density profiles are shown in Figure 4 for n=5 and n=8. Unlike the results discussed so far, one can readily see that in both cases the charge is completely delocalized. Symmetry considerations similar to those applied for the other families allow us to reach the same conclusions regarding the charge being mostly dispersed in an odd number of centers, including the center of the nanoribbon width for n=5 and alternatively in an even number of centers including the edges for n=8. Other than that, Figure 4 is completely different from Figure 2 and Figure 3. As no charge localization is present we can readily conclude that no polaron is present in the system. Energy levels calculations confirmed this conclusion by resulting in zero energy gap for this system. This is in complete agreement to results of literature that predicts 3p+2 AGNRs to be metallic.



Charge Density (e) 255

225 230

215 0

Length (A)

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

205

205

195 180

185 155

n=5

n=5

n=8

n=8

Figure 4: Charge density distribution for 3p+2 family We have reached a point that allow us to conclude that 3p+2 AGNR does not show a polaron transport mechanism. Before moving on to a time evolution approach we discuss one more aspect of the static results by presenting, in Figure 5 the charge profile of polarons for the other families. Although this Figure still does not account for a clear distinction between 3p and 3p+1 family, it is useful to understand the charge localization dependency with the AGNR width size. One can see that the broader the nanoribbon, the broader the corresponding polaron. This means that delocalization of polarons is mainly a function of the AGNR size, being relatively insensitive to its family. This relative insensitivity is observed from the lack of pattern obtained through analysis of similar width of different families. It is important to remark that high delocalizations are indicative of higher mobility. This point will be further explored shortly. Although of crucial importance to the determination of the charge carrier nature, the static results presented so far do not allow us to observe a clear qualitative difference between the behavior of polarons from families 3p and 3p+1. As we are concerned with characterization of charge carriers as a function of the nanoribbon width (including its family), we will dedicate the remaining of this work on discussing the differences presented between the polaron transport in these two 16 ACS Paragon Plus Environment

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3x200 4x200 6x200 7x200 9x200 10x200

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Nanoribbon (Å) Figure 5: Charge density profile of polarons with n=3,4,6,7,9 and 10. AGNRs families. This is performed by means of a time evolution of the system. The dynamics of all our cases follows the pattern presented in Figure 6(a) and Figure 6(b). In these Figures, a polaron in a 6x200 AGNR (n=6, of 3p family) is subjected to an electric field of 1.5 mV/Å directed along the nanoribbon length. Note that the seemingly repeating pattern of the charge distribution on these figures is due to the consideration of periodic boundary conditions in the system. One can see that after a transient of time necessary for the polaron to respond to the electric field, the structure presents the collective behavior, following a trajectory expected for a quasi particle of its kind. 16 The delay in the response is attributed to the adiabatic application of the electric field according to equation (14). After about 20 fs the polaron reaches its terminal velocity, which can be confirmed by the straight path it describes in the nanoribbon length as a function of time. As the dynamical behavior of all the considered simulations were qualitatively the same. We found that an useful manner to study the differences between polarons of different lengths is to compute each terminal velocity and compare their values as a function of the family and the nanoribbon width. In a recent work, 21 the mobility of charge carriers was measured in Field Effect Transistors made from graphene nanoribbons. It was observed that those charge carriers could present mobilities



of the order of 120 cm2 /(Vs). Different methods of producing GNR’s yield different values for mobility, but at the same order of magnitude (see, for example, the works 22–24 ). Our work was able to reproduce the same order of magnitude for the charge carriers mobilities that was obtained in these experimental results. For instance, taking into account the 4x200 AGNR case, subjected to a 1.5mV/Åelectric field, we reach the value of 133cm2 /(Vs).

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Figure 6: Polaron dynamics for 6x200 AGNR subjected to 1.5 mV/Åelectric field. Table 1 provides a summary of such velocities for the polarons of both families in Å/fs units. One can see that polarons of family 3p systematically present much higher terminal velocities than 3p+1 representatives of similar size. Actually, we note that polarons in nanoribbon n=10 (of 3p+1 family) are even slower than those in n=6 (of 3p family), which is much narrower. The conclusion is clear: the family of the nanoribbon plays a more decisive role in the terminal velocity of polarons than does the width itself. This remark is of crucial importance for the design of electronic devices whose performance requires a pre-determined range of charge carrier mobility. Just as a mean of visual comparison we can plot the data of table 1 and perform a linear regression of the terminal velocity as a function of n, for the two families. This regression is expected to provide some qualitative insight on the behavior of terminal velocity of polarons for narrow nanoribbons, but it is important to emphasize that a saturation effect should occur for larger 18 ACS Paragon Plus Environment

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Table 1: Terminal velocities as a function of AGNR width and family 3p family 3p+1 family n=3 3.1Å/fs n=4 2.2Å/fs n=6 4.7Å/fs n=7 2.9Å/fs n=9 5.1Å/fs n=10 4.3Å/fs n, and the trend is expected to be no longer linear. This, however, is of limited importance to the present study since we have shown in Figure 5 that large nanoribbons present charge carriers way too delocalized. Still pertinent to this point is the previously discussed fact that energy levels of polarons in increasingly broader nanoribbons are increasingly closer to the valence and conduction bands so that in the limit, the quasi-particle is no longer present. The regression result are shown shown in Figure 7. We obtain a linear equation v(n) = 0.33n + 2.3 for the family 3p and v(n) = 0.35n + 0.68 for family 3p+1. Although crude approximations, these expressions allow a reasoning about the order of magnitude of charge carrier velocities in these systems. If we bear in mind that the velocity of a polaron in a 1-D polyacetylene chain is of about 0.5 Å/fs, we can see how faster does polaron moves in these 2D systems. Also, note that although the slope of the curves is the same for the two families, the 3p family has a positive offset of almost 2 Å/fs. Remember the results discussed in Figure 5 on the fact that the family was not a factor in the delocalization of the charge carrier, we conclude that this difference in velocities from 3p and 3p+1 families is a manifestation of two rather different transports modes, rather than just a delocalization effect. In other words, transport of polarons from different families are substantially different in nature. This is truly a phase transition of the velocity pattern due only to the nanoribbon family, i.e., a symmetry effect. These are important points to be addressed and taken into consideration when deciding what kind of material is to be used in a given AGNR technological application.



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3p family 3p+1 family

5 4 3 2 4

6 n

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Figure 7: Velocity regression for both families as a function of the width.

4 - Conclusions We were able to determine that polaron mediated charge transport indeed takes place in graphene nanoribbons in general, with the exception of those from the 3p+2 family. This is in accordance to experimental results that predict no energy gap for representatives of this family. In nanoribbons of 3p and 3p+1 families, the charge behavior was found to be completely dependent on geometric features of the lattice. For instance, the parity related to the number of sites in the nanoribbon width is found to play a role on the charge distribution: odd width presented polarons whose charge is preferentially localized on the center of the nanoribbons, whereas even widths show charge more localized in the edges. It was found that the width size is a crucial parameter to determine the delocalization of polarons. For 3p and 3p+1 families, the broader the nanoribbon, the greater terminal velocity the polarons tended to present. This is directly associated to a greater degree of charge delocalization for wider AGNR. Moreover, the behavior of terminal velocity was strongly dependent on the system’s family. As a general pattern, polarons in 3p nanoribbons present higher velocities when compared to those in 3p+1 of similar width. This was found to be an effect of the symmetry each family possess rather than being a consequence of polaron delocalization. By 20 ACS Paragon Plus Environment

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means of the characterization study conducted in this work, we provide the literature with valuable information for qualitatively predicting the behavior of charge transport in AGNR based on the geometric structure of the system.

Acknowledgement The gratefully acknowledge the financial support from the Brazilian Research Councils CAPES, CNPq, and FINATEC.

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(16) Ribeiro Jr, L.A.; da Cunha, W. F.; de Almeida Fonseca, A. L.; e Silva, G. M.; Stafström, S. Transport of Polarons in Graphene Nanoribbons. The Journal of Physical Chemistry Letters 2015, 6, 510–514. (17) da Cunha, W. F.; Junior, L. A. R.; de Almeida Fonseca, A. L.; Gargano, R.; e Silva, G. M. Impurity Effects on Polaron Dynamics in Graphene Nanoribbons. Carbon 2015, 91, 171 – 177. (18) Kotov, V. N.; Uchoa, B.; Pereira, V. M.; Guinea, F.; Castro Neto, A. H. Electron-Electron Interactions in Graphene: Current Status and Perspectives. Rev. Mod. Phys. 2012, 84, 1067– 1125. (19) Yan, J.; Zhang, Y.; Kim, P.; Pinczuk, A. Electric Field Effect Tuning of Electron-Phonon Coupling in Graphene. Phys. Rev. Lett. 2007, 98, 166802–4. (20) Lima, M. P.; e Silva, G. M. Dynamical Evolution of Polaron to Bipolaron in Conjugated Polymers. Phys. Rev. B. 2006, 74, 224303–6. (21) Ling, C.; Setzler, G; Lin, M. W.; Dhindsa, K. S.; Jin, J.; Yoon, H. J.; Kim, S. S.; Cheng, M. M. C.; Widjaja, N.; Zhou, Z. Electrical Transport Properties of Graphene Nanoribbons Produced From Sonicating Graphite in Solution. Nanotechnology. 2011, 22, 325201–325207. (22) Lin, Y. M.; Avouris, P. Strong Suppression of Electrical Noise in Bilayer Graphene Nanodevices. Nano Letters. 2008, 8, 2119–2125. (23) Wang, X.; Ouyang, Y.; Li, X.; Wang, H.; Guo, J.; Dai, H. Room-Temperature AllSemiconducting Sub-10-nm Graphene Nanoribbon Field-Effect Transistors. Phys. Rev. Lett. 2008, 100, 206803–6. (24) Jiao, L.; Wang, X.; Diankov, G.; Wang, H.; Dai, H. Facile Synthesis of High-Quality Graphene Nanoribbons. Nature nanotechnology 2010, 5, 321–325 . 23 ACS Paragon Plus Environment


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Polaron Properties in Armchair Graphene Nanoribbons - The Journal

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