Charge Localization and Hopping in a Topologically Engineered GNR

Graphene nanoribbons (GNRs) are promising two-dimensional materials with various technological applications, in particular for the armchair GNR families that have a semiconductor character. Recently, methods that allowed for the control of GNR's topology have been developed, resulting in the production of nanoribbons composed of alternating segments of two distinct armchair GNR families (7 and 9-AGNRs) connected in heterojunctions. This GNR displays two topological bands that lie between the valence and conduction bands that effectively modulates the nanoribbon bandgap. Here, we employ a two-dimensional extension of the Su-Schrieffer-Heeger model to study morphological and electronic properties of this new material in both neutral and charged states. Results demonstrate that charge injection in this system results in the formation of polarons that localize strictly in the 9-AGNRs segments of the system and whose mobility is highly impaired by the system's topology. We further show polaron displacement by means of hopping between 9-AGNR portions of the system, suggesting this mechanism for charge transport in this material.


Introduction
Optoelectronic devices are responsible for the capture, control, and emission of light and are widely used nowadays [1][2][3]. The most common materials used in the production of these devices are inorganic [4,5]. However, ever since the discovery of conductive organic crystals in the 1960s [6], the search for the production and improvement of devices manufactured from organic materials has been drawing much attention from the scientific and industrial communities, as these devices offer lower manufacturing costs due to the abundance of raw material, their mechanical flexibility, [34][35][36][37][38][39][40][41]. To account for such effects, the electron-phonon coupling term should be considered.
This feature, absent in the study conducted in reference [33]. In this sense, other approaches are required to describe the presence of polarons and bipolarons in organic-based lattices.
In semiconductor AGNRs, the electronic properties of the system are substantially altered by the deformations of the lattice sites, as the displacements of atoms and electrons is coupled and responds to external factors in a connected manner. As a result, charge injection to these systems produces quasi-particles such as the polaron, which corresponds to an electron or hole coupled to the lattice deformations that appear due to polarization of the system in the region of excess charge and play the role of charge carriers in these materials. In this work, the electronic and morphological properties of this recently synthesized 7,9-AGNR are studied in both neutral and charged states by means of a two-dimensional extension of the Su-Schrieffer-Heeger (SSH) model. Lattice dynamics is investigated within an Ehrenfest molecular dynamics approach. Results demonstrate that charge injection in this system results in polaron formation. This quasi-particle is seen to localize strictly in the 9-AGNR segments of the system and, for a range of electron-phonon coupling, it moves under the influence of an external electric field by means of a hopping mechanism between the 9-AGNRs portions of the system.

Methodology
To study the transport of quasi-particles in hybrid structures formed by the heterojunction of AGNRs with widths of 7 (3 + 1 family) and 9 (3 family), we used a SSH Hamiltonian model, in which the electronic part of the system is described quantum mechanically while the lattice part is treated classically. The two parts of the Hamiltonian are connected by an electron-phonon coupling term that is used to include lattice relaxation to a two-dimensional tight-binding model.
Since the position of atoms in graphene nanoribbons is not substantially altered, the electronic transfer integrals for electrons can be expanded in first order [30]. As such, the hopping term is given by where 0 is the hopping integral of the system with all atoms equally spaced, represents the electron-phonon coupling that is responsible for the interaction between the electronic and lattice degrees of freedom, and , are the variations in the bond-lengths of two neighboring sites and .
The Hamiltonian model used here is given by the expression where ⟨ , ⟩ represents the indexes of neighboring sites (see Figure 1), , is the -electron annihilation operator on site with spin and † , represents the corresponding creation operator. The second term is the effective potential associated with sigma bonds between carbon atoms, modeled according to the harmonic approximation with being the elastic constant. The last term describes the kinetic energy of the sites in terms of their momenta and mass .
Starting the iteration from an initial set of coordinates { , }, a self-consistent stationary solution (with = 0) of the system is determined [47]. The ground state is obtained with the diagonalization of the electronic Hamiltonian, according to the expression where are the eigenenergies of the electronic system. To do this procedure, it is necessary to From these considerations, the electronic Hamiltonian becomes which is diagonalized and becomes Equation 3 as long as the condition (Equation 6) is satisfied is satisfied for neighboring sites , ; , ′ ; and , ′′ (see Figure 1). The result of the procedure of diagonalization is the energies of the electronic states and the wave functions for the ground state.
The concomitant self-consistent lattice solution is obtained from the Euler-Lagrange equations: wherein the static case is Thus, to take into account lattice effects, it is necessary to obtain the expectation value of the thus, with, where the sum is realized only for the occupied states. Note also that the last equation is responsible for the connection between the electronic and lattice parts of the system.
Thus, an initial set of coordinates { , } is used to start an auto-consistent calculation, where a corresponding electronic set { , , } is obtained, which when solved for the lattice returns a new set of coordinates { , }. The process is repeated until a given convergence criterion is satisfied.  [48,49]. Thus, the electronic time evolution is given by Expanding the ket | ( )⟩ in a basis of eigenstates of the electronic Hamiltonian at a given time , we obtain and finally, we obtain the temporal evolution of the electronic part of the system, according to the final expression Or, in terms of wave functions where { } and { } are the eigenfunction and eigenvalues of the electronic Hamiltonian, respectively. The numerical integration of the last equation is performed as usual and has already been reported in our previous work [47].
For the classical treatment governing the lattice part of the system, the complete Euler-Lagrange equations are required. Its solution can be written as a Newtonian equation able to describe the movements of the sites in the system and is given by To perform the quasiparticle dynamics in the system, an external electric field, E( ), was included in our model. Here, this is done by inserting a time-dependent vector potential, A( ), through a Peierls Substitution for the electronic transfer integrals of the system, making the hopping term where ≡ ∕(ℏ ), with being the lattice parameter ( = 1.42 Å in graphene nanoribbons), being the absolute value of the electronic charge, and the speed of light. The relationship between the time-dependent electric field and the potential vector is given by E( ) = −(1∕ )Ȧ( ). In our model, the electric field is activated adiabatically to avoid numerical oscillations that appear when the electric field is turned on abruptly [30].

Results and Discuss
The structure of the graphene nanoribbon employed in the simulations is shown in the inset of Figure 2. This nanoribbon is composed of alternating segments of 7-AGNRs and 9-AGNRs   in the 9-AGNR portion, as evidenced by their homogeneity [50]. The overall distribution of bond lengths in the 7,9-AGNR can be seen in Figure 3 The more interesting aspects of the 7,9-AGNR heterojunction can be seen when a hole is injected in the system. The behavior of the excess charge is dependent on the intensity of the electronphonon coupling. This can be seen in Figure 4(a), that presents a charge density plot for different . For < 4.75 eV/Å, the excess charge is delocalized over the entire nanoribbon. As grows larger, the excess charge becomes progressively more localized. The excess charge polarizes the structure and distorts the lattice, as it can be seen in Figure 4  observed to be symmetric with respect to both nanoribbon axes and localized within the 9-AGNR segment. This combination of charge accumulation and localized bond length distortion is a feature of the electron-phonon coupling and characterizes, in this case, a polaron. From an electronic standpoint, the presence of a polaron is confirmed by the appearance of two intragap energy levels.
In the case of the 7,9-AGNR, these intragap levels lies between the two topological bands produced by the heterojuntion.
A remarkable feature of polaron formation in 7,9-AGRNs is the fact that regardless of the set of initial coordinates employed in the simulation, charge accumulation always takes place in a 9-AGNR portion of the system. This is so even though both 7 and 9-AGNR, as members of the 3 +1 and 3 AGNR families, are known to be prone to polaron formation [51]. We conjecture that this behavior is due to the fact that the distortion of aromatic bonds found only in the 9-AGNR segments constitute significantly contribution to entropy increase in the nanoribbon, making the localization of polarons in these regions a process that minimizes the free energy of the system. This curious preference for polaron formation in the 9-AGNR segments of the system raises the question of whether polarons are able to move in the 7-AGNR regions. In general, for systems in which charge transport is mediated by polarons, the application of an electric field results in charge drift with the center of the quasi-particle moving continuously through the system. In the simulations, the position of the polaron center can be calculated as a function of time by considering the center of the charge distribution as the polaron position [52,53].
With this assumption at hand, we investigate the motion of a polaron in the 7,9-AGNR under the influence of an electric field. Figure 5(a) shows the behavior of such motion in the case of a 0.3 mV/Å electric field for = 5.0 eV/Å. The same qualitative behavior is observed for polarons as long as ≥ 4.75 eV/Å. However, as a larger electron-phonon coupling increases the polaron inertia, longer simulations are necessary to observe polaron movement. It can be seen in Figure   5(a) that in the first 80 fs of simulation the polaron is mainly localized in a 9-AGNR segment, with an increase in charge density being observed in the adjacent 9-AGNR portion. This gradual charge transfer is mostly concluded within 100 fs, when the polaron becomes localized in the following 9-AGNR segment. This process is repeated as time increases, but the residence time of the quasiparticle within each 9-AGNR portion is reduced as the polaron gains more momentum. The main aspect of this charge transport process is the discrete nature of the polaron movement between 9-AGNRs as opposed to the expected continuous motion through each alternating AGNR type in the system. This is further corroborated by the behavior of the polaron center, which is shown in Figure 5(b) for electric fields ranging from 0.2 mV/Å to 0.5 mV/Å. In all cases, the polaron center is seen to remain for a given time in the same position inside a 9-AGNR segment before hopping to the next one. Polaron residence times at each site decrease with time, indicating that the corresponding hopping rate increases. Hopping distances, on the other hand, are kept constant around 15 Å, which is the distance between neighboring 9-AGNR portions. Finally, compiling results for different intensities of the electric field allows us to estimate charge mobility in the 7,9-AGNR to be 0.144 cm 2 /(Vs), which is three orders of magnitude lower than the 350 cm 2 /(Vs) charge carrier mobility measured for 9-AGNRs [54]. On the other hand, this constitutes a typical mobility value for organic semiconductors. As such, it is clear that the even though the 7,9-AGNR heterojunction allows for the engineering of bandgpaps in GNRs, it also severely hinders charge mobility in comparison to regular GRNs, restricting polaron motion to a hopping process.

Conclusions
A semiclassical model with tight-binding approximation was used to describe the structural and electronic properties as well as the charge transport mechanism of an AGNR heterojunction composed of alternating segments of 7-AGNR and 9-AGNR. A sweep of electron-phonon coupling values was conducted, from which its relationship to the energy bandgaps was established.
Bond length distortions were presented, indicating that an important structural difference between 7 and 9-AGNR segments was the presence in the latter of aromatic rings. These rings were then shown to suffer strong distortion when a charge was injected to the system, which could be the reason why polarons become localized in the 9-AGNR portions of the system. Furthermore, it was demonstrated that even under the influence of an electric field, the charge carries never localize in the 7-AGNR regions, rather moving by means of a hopping process between 9-AGNRs segments.
Hopping rates are seen to increase with time in the initial moments of the simulation but charge mobility reaches only 0.1 cm 2 /(Vs), which is significantly lower than the experimentally obtained charge mobilities in pure 9-AGNRs but similar to what is found in typical organic semiconductors.
We conclude that the engineering of such sequence of heterojunctions in GNRs may allow for gap tuning but simultaneously hinder charge transport in this class of material.