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Cassiano, Tiago de Sousa Araújo; Júnior, Luiz Antônio Ribeiro; Silva, Geraldo Magela e; Neto, Pedro Henrique de Oliveira

Advanced Theory and Simulations
, Volume Early View – Mar 16, 2023

/lp/wiley/regulating-polaron-transport-regime-via-heterojunction-engineering-in-FoBvpIHLkD

- Publisher
- Wiley
- Copyright
- © 2023 Wiley‐VCH GmbH
- eISSN
- 2513-0390
- DOI
- 10.1002/adts.202200877
- Publisher site
- See Article on Publisher Site

IntroductionGraphene holds great promise due to its outstanding electronic proprieties,[1] being central to several theoretical and experimental breakthroughs[2–5] since its isolation two decades ago.[6] However, the gap‐less linear electronic band structure restricts its use in semiconducting applications, motivating the development of techniques to open the gap. An elegant way of doing it consists of cutting laterally the graphene sheet up to sub nm scale. The new material, known as graphene nanoribbon (GNR),[7] may display semiconducting behavior due to quantum confinement effects while inheriting many of the appealing features of graphene, such as high mechanical resilience,[8] excellent charge transport properties,[9,10] and high surface area.[11] Moreover, structural changes in edge shape and width allow the tailoring of electronic attributes through minimal and elegant strategies.[12] All these features are triggering great interest in GNRs, motivating studies to insert them in the next‐generation nanoelectronics and optoelectronics.[7] Tremendous effort was made toward this goal, culminating on the current developments of biomedical devices,[13] organic field‐effect transitors (OFETS),[14,15] and organic photovoltaic[16,17] GNR‐based applications.The edge shape in GNRs dictates most of their intrinsic properties. Besides the most common ones, the armchair (AGNR) and zigzag (ZGNR),[12] nowadays there are reports of an extensive collection of novel GNR types[7,18–20] being synthesized. Among the many approaches, bottom‐up‐based synthesis is especially promising since they deliver high‐quality nanoribbons with atomic precision. In this context, a recent report of a liquid‐phase bottom–up synthesis describes the fabrication of a new semiconducting GNR with a cove‐shaped edge symmetry.[9] The so‐called cove‐type graphene nanoribbons (CGNR) have an encouraging future in nanoelectronics since they display excellent intramolecular mobility and are structurally well‐defined.[9,21] Recently, we reported a smooth gap strategy that takes advantage of the unique edge shape of CGNRs.[22] Considering the prospects, it is imperative to carry out further investigations on CGNR's attributes to fulfill its potential in future applications.In this context, GNR's versatility is improved through heterojunction (HJ), using two nanoribbons as building blocks to form a new one that mix the attributes of the predecessor systems. The geometric mismatch at the junction induces an abrupt change in the local electronic structure,[23,24] giving rise to tunable exotic electronic properties.[25–30] As a result, an entirely new pathway to fabricate GNRs for innovative devices is emerging through HJ. Currently, several applications are exploring this route, rendering promising results in photovoltaic,[31,32] spintronic,[33] OFET,[34] and flexible devices[24] applications.All things considered, HJ remains an unexplored subject in some cases. Due to the diversity of GNR types available, several configurations involving recently discovered GNRs still require individual investigation. The scenario aggravates if we consider the recent advances in HJ and GNR synthesis since they enormously expand the set of possibilities to explore. One of the key steps in bottom–up synthesis is the precursor choice, directly impacting the GNR's edge shape and width length.[7] However, with recent advances, there is a variety of precursors available spanning GNRs with unconventional shapes, suggesting that arbitrary edge design will become a feasible feature in the future. In addition, it was demonstrated that scanning tunneling microscopy can manufacture, with atomic precision, armchair edges in armchair/polyanthrylene HJ,[35] representing an advance toward the design of arbitrary combinations of junctions. Therefore, to fully explore the potential of GNRs, one must consider the combinations beyond the previously synthesized nanoribbons, digging into the hidden geometries that arise from simple modifications of standard nanoribbons. In this context, an HJ composed from the proposed CGNR with elongated zigzag extensions[22] was not considered up to now.An adequate characterization of GNR requires deep analysis of the inner workings of charge transport, especially when integration into OFET applications is desired. That ultimately falls on studying the properties of the charge carriers present. In this context, it is well established that self‐trapped charged states known as quasiparticles[12,36,37] mediate the transport in GNR systems. Polarons, half‐spin polarized structures with charge ∓e$\mp e$, have a distinct relevance, being the main carriers in several organic materials.[38] Besides spin and charge, polaron's transport characteristics such as mobility and effective mass depend on the host material. For this reason, a thorough investigation of polaron's properties in new nanoribbons is needed to gauge their potential for transport‐related applications. The use of similar approaches yielded insightful results for other edge symmetries and polymeric systems.[39–44]Several modelings were developed to simulate intramolecular charge transport. For instance, DFT‐based studies use the deformation potential theory combined with the semiclassical Boltzmann transport equation to estimate the carrier's effective mass and mobility in a free carrier picture that only considers the acoustic phonons.[45–47] Although widely diffused, the approach does not explicitly address inner transport mechanisms such as polaron‐phonon collisions[21,42,44,48] and quasiparticle formation[40,43,49] when there is a tight coupling between lattice and electronic phenomena. An alternative lies in non‐adiabatic model Hamiltonians like the Su–Schrieffer–Heeger (SSH) model, which can represent the intrinsic bound nature of the quasiparticles. Under this formalism, charge transport is modeled as the drifting of a localized collective excitation along the lattice. The use of similar approaches yielded insightful results for other edge symmetries,[39,41–43,50] polymeric systems[40,44,51] and crystalline arrangements under the Holstein‐polaron modeling for intermolecular transport.[52–54]In this work, we assessed the effects on transport and electronic properties due to the heterojunction engineering of cove‐type graphene nanoribbons with different zigzag segments as building blocks. The heterojunctions will be simulated through the extended two‐dimensional Su–Schrieffer–Heeger (SSH) model. Stationary states reveal that controlling the HJ's junction provides multiple pathways for smooth and monotonic bandgap and ground state energy modulation. Investigations of the charged states reveal that HJ formation regulates polaron morphology, hopping mechanism, and transport properties such as mobility and effective mass. Our findings can serve as theoretical background to produce CGNR‐based devices with tunable charge carriers, expanding even more the potential of CGNRs in high‐end nanoelectronics.Experimental SectionModel HamiltonianHeterojunctions formed by combining two CGNRs with distinct zigzag/armchair edge ratios aCGNR and bCGNR as illustrated in Figure 1a were proposed here. The parameters a and b indicated the number of lacking hexagons at the border. Their corresponding edges are highlighted in red and blue, respectively. For instance, a=2$a=2$ and b=3$b=3$ in Figure 1a. The HJ resulting from their combination is called the 2‐3CGNR heterojunction.1FigureIllustration of the methodology used. (a) The proposed junction formation, combining two cove‐type with different zigzag edge lengths aCGNR and bCGNR to form the a‐bCGNR heterojunction. The colors highlight their borders, which we measure through the number of lost hexagons. In the figure, a=2$a=2$ and b=3$b=3$. (b) The index labeling convention used for the sites i, j, and their corresponding neighboring sites marked with prime signs in the SSH formalism. (c) Illustrates our algorithm to measure morphological spreading through the distribution of bond lengths.The Su–Schrieffer–Heeger (SSH) model Hamiltonian was adopted[55,56] to simulate the nanoribbons. Past works made comprehensive discussion toward this modeling.[21,40,44,57–59] In the presence of an external electric field E$\mathbf {E}$, the hybrid Hamiltonian consisted of the sum of an electronic (Hele$H_{\text{ele}}$) and a lattice (Hlatt$H_{\text{latt}}$) components. The first one is a tight‐binding term that reads, in the second quantization formalism, as1Htb=−∑<i,j>,s(ti,jCi,s†Cj,s+h.c.)$$\begin{equation} H_{\text{tb}} = -\sum _{<i,j>,s}(t_{i,j}C^{\dagger }_{i,s}C_{j,s} + h.c.) \end{equation}$$in which Ci,s†$C^{\dagger }_{i,s}$ and Cj,s$C_{j,s}$ are, respectively, the creation and annihilation operators of a π‐electron with spin s at sites i and j. The brackets in the lower index indicated a pair‐wise sum over the sites. ti,j$t_{i,j}$ denotes the hopping integral from the ith site to the jth site. The suitable form of ti,j$t_{i,j}$ for organic systems arises by expanding the term up to the first order around the symmetric hopping integral t0. In addition, the Peierls substitution allows the inclusion of an external field in the modeling. If E(t)=−(1/c)dA(t)dt$\mathbf{E}(t)=-(1/c)\frac{d\bm{A}(t)}{\textit{dt}}$ was chosen, then these two modifications leads to[49,57,58]2ti,j=e−iγAi,j(t0−αηi,j)$$\begin{equation} t_{i,j} = \text{e}^{-i\gamma A_{i,j}}(t_{0} - \alpha \eta _{i,j}) \end{equation}$$Here, γ=el0/(ℏc)$\gamma = el_{0}/(\hbar c)$, where e is the elementary charge, l0 is the lattice parameter, and c is the speed of light. In addition, Ai,j$A_{i,j}$ is the projection of the potential vector in the direction connecting the sites i and j, while ηi,j$\eta _{i,j}$ is the relative displacement of the σ‐bond that links them. Finally, α is the electron–phonon coupling constant, a parameter that connects the interaction strength between electronic and lattice phenomena.Under the harmonic approximation, the lattice was modeled as a mass‐spring system of harmonic oscillators. That is3Hlatt=K2∑<i,j>ηi,j2+12M∑iPi2$$\begin{equation} H_{\text{latt}} = \frac{K}{2}\sum _{<i,j>} \eta _{i,j}^2 + \frac{1}{2M}\sum _{i}P_{i}^{2} \end{equation}$$where K is the Hook's constant, M is the site's mass, and Pi$P_{i}$ is the conjugated momentum of the ith site.The stationary algorithm is as follows. First Pi,Ai,j$P_{i}, A_{i,j}$ was set to zero. Here, It should be stressed that the hybrid Hamiltonian explicitly depended on ηi,j$\eta _{i,j}$, requiring the use of a self‐consistent procedure that simultaneously solves the lattice and the electronic parts. In that sense, Htb${H}_{\textit{tb}}$ is built using an initial guess for ηi,j${\eta}_{i,j}$. Then, the Hamiltonian was diagonalized, which returned the corresponding set of eigenvectors {ψk,s}$\lbrace \psi _{k,s}\rbrace$. Next, the lattice part was to be solved. Let ⟨L⟩=⟨ψ|L|ψ⟩$\mathinner {\langle {L}\rangle }=\mathinner {\langle {\psi }|}L\mathinner {|{\psi }\rangle }$ be the expected value of the Lagrangian calculated with a Slater determinant state |ψ⟩$\mathinner {|{\psi }\rangle }$ of the occupied π‐electrons. Then,4ddt∂⟨L⟩∂q̇i=∂⟨L⟩∂qi$$\begin{equation} \frac{\text{d}}{\text{d}t}{\left(\frac{\partial \mathinner {\langle {L}\rangle }}{\partial \dot{q}_{i}}\right)} = \frac{\partial \mathinner {\langle {L}\rangle } }{\partial q_{i}} \end{equation}$$Explicit manipulation of the equations returns an expression that allows to calculate ηi,j$\eta _{i,j}$ in terms of {ψk,s}$\lbrace \psi _{k,s}\rbrace$. Then, Htb$H_{\text{tb}}$ was re‐evaluated by using this new set of spatial displacements. Re‐diagonalization of the electronic Hamiltonian yielded a new {ψk,s}$\lbrace \psi _{k,s}\rbrace$. If the set was equal to the one that preceded it according to a convergence criterion, the stationary state was found and the algorithm stopped. On the other hand, if the condition was not met, the steps were repeated using the newly found ηi,j$\eta _{i,j}$ as the initial guess. Under this context, one can obtain charged states by unbalancing the number of electrons. In the case of polarons, this meant removing an electron from the highest occupied molecular orbital (HOMO) or adding one to the lowest unoccupied molecular orbital (LUMO).To evolve the states in time, the restrictions made in the stationary case for momentum and potential vector were relaxed. Time progression will unravel through the simultaneous evolution of the lattice and electronic states using the Euler–Lagrange and Schrodinger equations. Starting with the lattice part, the explicit evaluation of Equation (4) returned the equations of motions that express the force experienced by the sites5Fi,j(t)=Mη̈i,j=K2(ηi,i′+ηi,i′′+ηj,j′′+ηj,j′−4ηi,j)+α2(Bi,i′+Bi,i′′+Bj,j′+Bj,j′′−4Bi,j+h.c.)$$\begin{equation} \begin{aligned} F_{i,j}(t) &= M \ddot{\eta }_{i,j} = \frac{K}{2}(\eta _{i,i^{\prime }} + \eta _{i,i^{\prime \prime }} + \eta _{j,j^{\prime \prime }} + \eta _{j,j^{\prime }} -4\eta _{i,j})\\ &\quad\, + \frac{\alpha }{2}(B_{i,i^{\prime }} + B_{i,i^{\prime \prime }} + B_{j,j^{\prime }} + B_{j,j^{\prime \prime }} -4B_{i,j} + h.c.) \end{aligned} \end{equation}$$Here, the electronic force field was written in terms of Bs, that have the following form6Bi,j≡e−iγAi,j∑′k,sψk,s∗(i,t)ψk,s(j,t).$$\begin{equation} B_{i,j} \equiv e^{-i\gamma A_{i,j}}\mathop {{\sum\nolimits ^{\prime }}}\limits _{k,s} \psi _{k,s}^{*}(i,t)\psi _{k,s}(j,t). \end{equation}$$The prime indexes in Equation (5) denote neighboring sites, as illustrated in Figure 1b, while the primed sum in Equation (6) indicates a sum over the occupied orbitals. Finally, having built Equation (5), one can find the sites' position and velocity after a discrete time integration.The electronic part evolved according to the time‐dependent Schrodinger equation, which for a time progression from t to t+dt$t + \text{d}t$ becomes7|ψk,s(t+dt)⟩=e−iH(t)dt/ℏ|ψk,s(t)⟩$$\begin{equation} \mathinner {|{\psi _{k,s}(t+\text{d}t)}\rangle } = \text{e}^{-i H(t)\text{d}t/\hbar }\mathinner {|{\psi _{k,s}(t)}\rangle } \end{equation}$$Expanding the right side over the energy basis {|ϕl(t)⟩}$\lbrace \mathinner {|{\phi _{l}(t)}\rangle }\rbrace$ returns8|ψk(t+dt)⟩=∑l⟨ϕl(t)|ψk(t)⟩e−iεldt/ℏ|ϕl(t)⟩$$\begin{equation} \mathinner {|{\psi _{k}(t+\text{d}t)}\rangle } = \sum _{l} \mathinner {\langle {\phi _{l}(t)|\psi _{k}(t)}\rangle }\text{e}^{-i\epsilon _{l}\text{d}t/\hbar }\mathinner {|{\phi _{l}(t)}\rangle } \end{equation}$$where {εl(t)}$\lbrace \epsilon _{l}(t)\rbrace$ are the corresponding eigenvalues of H at the instant t. The expression in Equation (8) allows to obtain the electronic states after a time increment dt$\text{d}t$.The algorithm is as follows. First, the stationary solution was obtained to compose {ηi,j}$\lbrace \eta _{i,j}\rbrace$ and {ψk,s}$\lbrace \psi _{k,s}\rbrace$ for the time = 0. Then, Equations (8) and (5) were evaluated using these sets, returning, respectively, the electronic states and lattice spatial distribution after a time dt$\text{d}t$. Because ηi,j$\eta _{i,j}$ changed at each time step, further time progression required re‐diagonalization of H using the updated lattice, followed by the reapplication of Equations (8) and (5) on the new state.The parameters of the hybrid Hamiltonian were in agreement with previous studies of GNRs. That is t0 = 2.7 eV,[60,61] K$K\nobreakspace $= 21 eV Å−2[55,62] and l0=1.41 Å.[63] In a previous report by the authors', they had estimated 2‐2CGNR's α as 4.6 eV Å−1[22] through an empirical tuning method[43,64] based on the experimentally measured bandgap of 1.88 eV.[9] The value of the electron–phonon coupling constant used here was within the expected range of 4–14.1 eV Å−1 obtained in early estimates for graphene[65–67] which is also applicable for GNRs[41–43] and π‐conjugated polymers,[40,49,55] where quasiparticle formation is vastly observed.[38,68–71] Since the conformations in the present study were structurally similar, the 2‐2CGNR's α was extended for them. Finally, the external electric field was included adiabatically[21] in dynamics simulations of about 700 fs. As a final remark, the sites directly interact only with their first neighbors, as indicated in Figure 1b. Thus, different HJ conformations were simulated by adequately considering the neighbors and their corresponding positions in the lattice for each site when numerically evaluating the Hamiltonian.Polaron Transport ModelingStable polarized quasiparticles maintain their characteristic charge profile during the drift. Because of that, one can track the motion by calculating the time evolution of the center of the charge density ρ. Under a constant electric field regime, the resulting trajectory X(t)$X(t)$ could be modeled through projectile dynamics of a single particle of effective mass Meff$M_{\text{eff}}$ subjected to a linear Stokes‐type dissipation. The solution of the corresponding equation of motion is [21,50]9X(t)=X0+vtt+v0−vtk(1−e−kt)$$\begin{equation} X(t) = X_{0} + v_{\text{t}}t +\frac{v_{0} - v_{\text{t}}}{k}(1-\text{e}^{-kt}) \end{equation}$$where X0 and v0 are, respectively, the carrier's position and velocity at the instant that the field becomes constant (t=0$t=0$). In addition, vt$v_{t}$ is the terminal velocity and k=B/Meff$k=B/M_{\text{eff}}$, where B is the Stokes dissipation coefficient. Fitting Equation (9) provided estimates of coefficients k and vt$v_{t}$, which in turn could be rearranged to determine B and Meff$M_{\text{eff}}$.[21] Under this regime, the mobility μ$\umu$ could be estimated using vt$v_{\text{t}}$ and the electric field strength E0 as10μ=vt/E0$$\begin{equation} \mu = v_{\text{t}}/E_{0} \end{equation}$$ABS FactorThere is a tight coupling between lattice and electronic phenomena in GNRs. Because of that, the material's morphological profile might provide insight into electronic mechanisms and vice‐versa. For instance, previous analysis of the morphological spreading, that is how different are double bond length distributions, revealed the impact on the lattice after the increment of zigzag edges in CGNRs.[22] However, that investigation relied on visual confirmation, which might lead to limited conclusions. Therefore, a more reliable and reproducible way to measure morphological spreading was proposed here. The idea consisted of calculating the difference between the sets of single bonds. Figure 1c illustrates the procedure. In the upper region, four straight red lines were placed in the length axis, representing the set of unique bond lengths of a reference nanoribbon. Right below it, four blue lines characterized the morphological profile of another GNR. The red dashed lines symbolized the unique bonds of the reference system. Let Di$D_{i}$ be the length difference between the ith blue line to the nearest red one. Averaging over all blue bond lengths returned what was called the average bond spreading (ABS) factor, an estimate of the spreading degree that a system of interest had in comparison with a reference. The higher the ABS, the greater will be the morphological spreading. Throughout this work, the original cove‐type nanoribbon 2‐2CGNR was chosen as a reference system.[9]ResultsOur investigation begins through the stationary solutions of CGNR heterojunctions in the neutral state. Figure 2 displays three heatmaps of energy bandgap (a), energy per site (b), and ABS factor (c) for all a‐bCGNR HJ in which a and b are between 2 and 10. In the first heatmap, hot colors represent geometries with high gap magnitudes, while cold tones indicate lower values. The first geometry considered is the 2‐2CGNR,[9] owning the highest gap of 1.88 eV. Any other combination of a‐b parameters leads to narrower bandgaps. More interestingly, the drop occurs monotonically as the number of zigzag segments increases. For instance, if parameter a is fixed while b increases, the gap decays monotonically, nearing 0 eV when b=10$b=10$.2FigureHeatmaps of (a) energy band gap, (b) energy per site, and the average bond spreading (ABS) factor for the a‐bCGNR heterojunctions with a and b varying from 2 to 10. The colors in each graph have a specific meaning. In (a), hot tones indicate geometries with high bandgap values, while the opposite is symbolized in cold colors. On the other hand, the geometries in hot at heatmap (b) are less stable than those in cold tones. Finally, hot‐colored specimens in (c) display a high degree of morphological spreading, as the opposite is indicated by the cold colors.In general, beginning from the 2‐2CGNR, the heatmap's origin, a negative gradient shape forms throughout the graph. In that way, HJs with an equal sum a2+b2$a^2 + b^2$ display approximately the same gap. As a result, the heatmap exhibits a radial‐like symmetry composed of these “equipotential” domains. This response is not unexpected. Gap decays because zigzag segments are being infused into the CGNR.[22] The closer the edge is to a pure ZGNR, a gapless GNR, the narrower the gap should be. Since there is no external agent such as an electric or a strain field that uniformly distorts the nanoribbon's energy landscape, there will be no preferential location to place the zigzag edges. Therefore, as long as the amount of zigzag segments maintains a2+b2$a^2 + b^2$ constant, one can arrange them in an arbitrary order without affecting the gap substantially. Because the equipotential regions contain more than one HJ, multiple routes to smoothly tune the gap are available by jumping through these domains. Therefore, we found that the monotonic decay behavior is not an exclusive feature of the a‐aCGNRs previously studied.[22] The result exemplifies the potential behind HJ design since it provides alternative options to regulate the band structure of CGNRs. Future applications could explore this feature because some routes may be easier to access than others. In addition, we emphasize that the bandgap engineering displayed in Figure 2 closely relates to experimental reports on other synthesized HJ shapes, namely the 7/9‐AGNR,[26] 7/13‐AGNR,[25] pristine‐doped chevron GNR[31] heterojunctions, where the band structure was tailored using the precursor blocks. This mechanism is also reported in other theoretical investigations.[43,72]The energy heatmap in Figure 2b provides some insight into the HJ's stability. Geometries in hot tones have energies closer to −4.02 eV, while the cold colors address the ones near −4.06 eV. Interestingly, here we notice a radial‐like shape that mirrors the gap heatmap. Starting from the origin, the 2‐2CGNR, the energy moves closer to −4.02 eV as the a and b parameters increase, revealing a clear correlation between gap and energy: as the gap decays, energy lowers in magnitude. Moreover, within the SSH modeling, the semiconducting CGNRs are more stable than the ones with big zigzag chains. This reveals a potential feature in HJ design to directly tune the optimized GNRs energy, which we believe that extends to excited states. If this turns out to be the case, HJ engineering could be enforced to regulate mechanisms that depend on energy differences between states such as exciton dynamics[73,74] and intermolecular charge transfer.[75–77] We highlight that although the energy per site changes slightly between the conformations, the difference increases considerably the total energy since such molecules may have 1000–3000 sites.The last heatmap shows the morphological spreading due to the HJ formations via the absolute bond spreading factor. In contrast with the band gap and energy, the graph does not have a radial behavior. ABS shows a growth trend at the diagonal as parameters a and b increase. However, this trend does not hold indefinitely. When they reach about 7, ABS begins to drop slowly. Then, a plateau regime arises, meaning that morphological spreading remains unchanged regardless of further increments of zigzag edges. ABS shows this profile because moving through the diagonal makes the CGNRs progressively resemble pure ZGNRs. Eventually, the nanoribbon becomes so structurally similar to a zigzag nanoribbon that further inclusion of edge segments does not cause significant changes. Alternatively, the ABS changes through non‐diagonal transformations occur at a different pace. For instance, fixing b$b\nobreakspace $= 4 while increasing parameter a also leads to the profile of the diagonal case. However, ABS decreases considerably less here, leading to a higher plateau value. That is because parameter b was fixed, forcing a still relevant proportion of non‐zigzag segments. Consequently, the nanoribbon is not as close to a pure ZGNR as the ones in the diagonal. This result reveals that structural attributes may not respect the radial profile displayed in the gap and energy heatmaps. Because of that, mechanisms that involve the lattice, such as charge transport, may exhibit more complex behavior.Probed the characteristics of the HJs in the neutral state, we now investigate the polaron's properties on them. Figure 3 displays the charge density of the 2‐3, 2‐5, 2‐7, and 2‐9CGNR of positive polaron states. Hot and cold colors represent high and low local charge accumulation, respectively. One can observe that the 2‐3CGNR's charge density concentrates in a region that we recognize as the polaron quasiparticle. Here, ρ is non‐vanishing throughout the entire structure. However, its magnitude is not uniform such that the zigzag edges have a significantly higher charge density than the armchair edges, as can be seen by the red dots. Contrastingly, 2‐5CGNR's ρ is visibly fading at the armchair regions, while the red dots at the zigzag edges grow in number. In fact, these high‐density domains mainly locate at the longest zigzag segments, derived from 5‐5CGNR. Additional increments in parameter b enhance this trend to a point where charge accumulates essentially only at those regions, as in 2‐9CGNR's heatmap.3FigureCharge density heatmaps of 2‐bCGNR. In descending order, the profiles of 2‐3CGNR, 2‐5CGNR, 2‐7CGNR, and 2‐9CGNR polaron are presented. The colors indicate the charge accumulation degree. Places in hot tones have locally a high charge density, while the opposite occurs for those in cold colors.Computational simulations of the 7‐9AGNR in ref. [43] show a similar phenomenon, in which the quasiparticle's charge density concentrates in the 9AGNR's segment. It was conjectured that entropic effects triggered by the increase of aromatic bonds on the wider nanoribbon were the cause of it. We do not believe that it is the case since entropy is usually a source of stability loss in these highly symmetrical systems.[78] Alternatively, we propose a simpler view. Under the SSH model optic, the sites are a spring‐mass system. Accumulating the charge in narrow spaces would lead to deeper local lattice deformations because the springs would be more distorted, which is energetically costly. In opposition, the largest zigzag chain has a “bigger” room to accommodate the polaron, attenuating the lattice stress. We can make a parallel to polaron formation in GNRs with defects.[57] In this situation, the polarized region forms away from the defect's localized lattice distortion. Alternatively, the places far from it have a smoother lattice landscape. Because of that, they are more energetically favored to bear a polaron. Evidently, the quasiparticle can collide with the defects, provided that an external agent like an electric field helps it overcome the energy barrier. However, in the absence of such influence, the defect‐less zone is favored. The findings reveal that HJ engineering can regulate polaron morphology, which can become a tool to tailor charge carriers' properties. Toward this goal, we now address the drifting of these structures.Figure 4 displays the time evolution of charge densities of HJ in the presence of an external field of E0=1.21$E_{0}=1.21$ kV cm−1. Figure 4a–c exhibits five snapshots in different time frames of polaron's drifting in the 2‐5, 2‐7, and 2‐9CGNR, respectively. The heatmaps' colors share the same meaning as Figure 3. Qualitatively, the three structures drift identically. Initially, the polarized region gains momentum from the electric field, going from an inert to a mobilized carrier through a continuous velocity gain regime. Finally, the velocity ceases to increase, reaching a terminal state that remains throughout the rest of the simulation. However, a closer look shows that they cover different distances in the same period. During the time interval of 520‐693 fs, 2‐5CGNR's polaron covers a distance of about 67 Å. At the same time, 2‐7, and 2‐9CGNR's carriers had advanced 55 and 34 Å, respectively. Thus, the polaron of the 2‐5CGNR, the HJ with the smallest zigzag segments, is the most mobile carrier among them, suggesting that increasing parameter b depletes transport efficiency. That is an unexpected result since 2‐5CGNR polaron's charge density is the most localized among the four cases. Usually, a higher degree of charge localization translates into higher transport inertia, contrasting with our observation. In addition, 2‐7CGNR and 2‐9CGNR are closer to a pure ZGNR than 2‐5CGNR. As discussed earlier, nanoribbons of zigzag type are semi‐metal with massless charge carriers. Therefore, it would not be surprising that they outperform 2‐5CGNR's polaron. The reason behind that behavior lies in the asymmetry of the HJ edges. Although the nanoribbons have a considerable amount of zigzag chains, the mismatch between the building blocks does not let the properties of a pure ZGNR take over smoothly. We recall that Figure 2c revealed a similar phenomenon, that the morphological spreading of the non‐diagonal HJ does not transition to ZGNR's as easily as the diagonal elements.4FigureDrifting polarons through the 2‐bCGNR heterojunctions. The colors carry the same meaning as Figure 3. (a)–(c) Selected snapshots of charge density heatmaps of 2‐5CGNR, 2‐7CGNR, and 2‐9CGNR with a single polaron state are displayed, respectively. (d) Zoomed view of the drifting of the last carrier, showing in detail the semi‐hopping transport mechanism.In addition, we report that changes in the parameter b lead to modifications in the polaron's transport mechanism. Close examination of the charge density snapshots reveals this. Figure 4d shows a zoomed look of the drifting inside the 2‐9CGNR. Here we notice that as the polarized structure advances, the charge density progressively hops through the lengthy ZGNR blocks, jumping the immediate region between them. That occurs to preserve the characteristic morphology seen in Figure 3, indicating its stability and robustness. More importantly, the transport mechanism is different from 2‐2CGNR's polaron,[21] in which charge density fills out the entire region where the quasiparticle is confined. Here, the structure has a hopping‐like behavior internally, although the motion is continuous in the bigger picture. That characterizes a semi‐hopping transport, which is regulated by the parameter b in the 2‐bCGNR. Therefore, one concludes that heterojunction design can tailor the charge transport regime of the polarons. The result reveals a potential path to smoothly control the polaron transport mechanism in CGNR HJ.Our findings qualitatively agree with recent reports of charge transport on other HJs. Experimental probing of the mechanism on AGNRs heterojunctions reveals that the structural engineering can control the transport, making one of the two building blocks behave as a tunneling barrier.[34,79,80] Our findings display a similar outcome. Along the a‐bCGNR extension, the polaron's charge density is confined at the longest zigzag extensions. When the carrier drifts, the shortest blocks remain bearing no charge, suggesting that these regions act as potential barriers for the quasiparticle. This result indicates a promising path to include CGNR‐based HJs in customizable nanoelectronics with tailored barriers due to heterojunction engineering.Although it is visible that HJ manipulates polaron's morphology, the practical effects on transport properties is convoluted. Figure 5a,b displays the polaron's position as a function of time to 2‐bCGNR and 3‐bCGNR, respectively. Their colors label the number of parameter b for that given geometry, while the shape indicates which family they belong to: the 2‐bCGNR are depicted in triangles, and the 3‐bCGNR are in diamonds. The lines are the fitted curves, calculated according to Equation (9). Here we recognize the same profile discussed during the analysis of Figure 4. All curves begin with a timid rise, followed by a rapid velocity gain regime. However, numerically, the polarons develop different dynamics. In the case of 2‐bCGNR, during the same time interval, the polarons become monotonically slower as b increases. That corroborates the visual observation made in Figure 4 and suggests that the trend will hold if the parameter continues to increase. The mismatch of zigzag chains may be the reason behind it, favoring the semi‐hopping mechanism. If this turns out to be the case, we do not expect similar observations on the diagonal CGNRs.5FigureTransport properties of polarons in 2‐bCGNR and 3‐bCGNR heterojunctions. The carriers' trajectories in those sets are displayed, respectively, in (a) and (b) for an electric field strength of 1.21 kV cm−1. (c) and (d) Mobility as a function of E0 for the corresponding nanoribbons.Contrastingly, the 3‐bCGNRs' trajectories exhibit a more complex behavior. Their response to the electric field qualitatively agrees with the 2‐bCGNR case. However, increasing parameter b does not lead to a monotonic drop in the covered distance. The fastest carrier among them is the 3‐4CGNR's polaron, followed by 3‐2CGNR, 3‐6CGNR, and 3‐8CGNR, showing no apparent correlation between the number of zigzag edges and transport efficiency. Even so, we note that the member with the highest number of zigzag segments still hosts the slowest carrier. Therefore, we conjuncture that it is the result of two competing effects. The first one is in the direction toward pure ZGNR properties that boosts the motion. The other one is the rising of the semi‐hopping transport provoked by the mismatch in the HJ, depleting the motion efficiency. Because the trends have opposite effects on the polaron, the overall mechanism shows no clear correlation. That said, one may ponder why there was an explicit correlation in the 2‐bCGNR case. The reason is that the first zigzag block, derived from the 2‐2CGNR, is smaller, favoring the mismatch effect and, consequently, the semi‐hopping mechanism.Estimates of mobility for the 2‐bCGNR and 3‐bCGNR specimens as a function of the electric field strength are shown in Figure 5c,d, respectively. Their insets display the corresponding effective masses. The points represent the calculated points, while the lines are eye guides. Similar to the trajectory graph, the qualitative response of the two sets is the same. Mobility shows its maximum value in the low‐field regime. As the field progressively increases, μ$\umu$ decays monotonically. The 2‐2CGNR's polaron shares the same behavior,[21] which we had attributed to the collision with phonons. Here we extend the explanation for the HJs. During the polaron's motion, breathers are excited in the lattice. As the field rises, the quasiparticle can cover greater distances, turning the collision with phonons more frequently. Since polarons have a modest lattice distortion, the impact disrupts it, depleting the transport.A closer look at the 2‐bCGNR graph reveals an interesting trend. Increasing the parameter b makes mobility drop monotonically. That corroborates our previous observation regarding the polarons' trajectories. Equivalent analysis can be made for Meff$M_{\text{eff}}$. First, we note that all values are independent of field strength. 2‐3, 2‐5, 2‐7, and 2‐9CGNR's effective masses are, respectively, 0.33, 0.35, 0.47, and 0.71 in units of electron's mass (me$m_{\text{e}}$). Therefore, greater zigzag extensions lead to heavier carriers in the 2‐bCGNR. The trend holds for 2‐2CGNR's polaron Meff$M_{\text{eff}}$, as it was estimated as 0.285 me$m_{\text{e}}$. The responses from mobility and effective mass reveal that enforcing the asymmetric confining of charge in HJ depletes the transport efficiency in 2‐bCGNR. More importantly, the findings show that HJ engineering can control polaron's transport regime. The overall mobility range accessible through junction formation has high upper limits, with smooth transition between the values, potentially outperforming non‐cove‐shaped state‐of‐art GNRs. Therefore, future investigations might use these features to produce highly specific materials in nanoelectronics.On the other hand, varying parameter b in 3‐bCGNR does not lead to monotonic behavior on either mobility or effective mass. In fact, the carrier with the highest mobility is hosted in the 3‐4CGNR, followed by the 3‐2CGNR, 3‐6CGNR, and 3‐8CGNR. Therefore, there is no clear relation between b and mobility response. Interestingly, effective masses do not share the ordering of mobility. In units of me$m_{\text{e}}$ the Meff$M_{\text{eff}}$ of 3‐2CGNR, 3‐4CGNR, 3‐6CGNR, and 3‐8CGNR are, respectively, 0.33, 0.24, 0.32, and 0.49. As a result, some carriers display a non‐trivial relationship between mobility and effective mass. For instance, 3‐2CGNR's polaron has the second greatest mobility, while having the second heaviest Meff$M_{\text{eff}}$. That is surprising since more mobile carriers tend to have lower transport inertia. Indeed, some heterojunctions of the 3‐bCGNR set follow this expectation. 3‐4CGNR's polaron has the highest mobility and the lowest effective mass. 3‐8CGNR shows opposite properties, hosting the heaviest carrier with the lowest mobility.ConclusionIn conclusion, we explored the electronic and lattice effects of heterojunctions with CGNRs. Results show that controlling the mismatch of the junction provides a smooth tuning of energy bandgap through multiple routes. Moreover, both total energy and bandgap display a curious symmetry toward variations of the number of zigzag extensions on the HJ, such that structurally different nanoribbons exhibit equivalent responses. Regarding the lattice effects, we report a non‐trivial morphological response due to the engineering of the edges. All these findings clearly show that HJ formation can modulate several properties of the nanoribbons. Thus, revealing a potential tool to realize the precise design of cove‐type GNR‐based applications.Besides the general properties of HJ, we also investigated the charge transport in the heterojunctions. Stationary solutions reveal that polaron morphology depends on the proportion of zigzag edges of the building blocks of HJ. The more asymmetric the mismatch is, the more concentrated the charge density will be on the long zigzag chains. As a result, the inner drifting of the carrier switches to a hopping mechanism if the asymmetry between two adjacent blocks is big enough. Thus, the junction formation dictates the carriers' transport regimes.A closer examination of the finding shows that transport properties such as mobility and effective mass are controllable too. In fact, there is a monotonic dependence of these properties for HJ from the set 2‐bCGNR, allowing access to a wide range of such attributes through simple engineering of the junctions. For instance, we report changes up to 10 000 cm2 V−1$^{-1\nobreakspace }$s−1 in the mobility between the geometries, a notable flexibility that could be explored in nanoelectronic devices.An equivalent analysis was extended to the 3‐bCGNR, where we found some qualitative agreement with the results of the 2‐bCGNR. However, mobility and effective mass do not display a monotonic behavior due to changes in parameter b. Because of that, some polarons displayed interesting sets of transport properties such as the 3‐2CGNR's carrier that simultaneously has the second greatest mobility and the second heavier effective mass. The reason for this unexpected result comes from the interplay between localization and scattering. Our work demonstrates the potential of carrying out heterojunction engineering to smoothly modulate lattice, transport, and electronic properties of CGNRs. We hope our findings will aid future theoretical and experimental studies regarding these systems and inspire similar approaches to other GNR types.AcknowledgementsThe authors gratefully acknowledge the financial support from Brazilian Research Councils DC, (grant number 304637/2018–1) CAPES, and FAPDF. P.H.O.N. and G.M.S. gratefully acknowledge, respectively, the financial support from FAPDF grants 00193.00001217/2021‐13 and 0193.001766/2017. G.M.S. and P.H.O.N. gratefully acknowledges the financial support from CNPq grants 304637/2018‐1 and 310473/2019‐5, respectively. L.A.R.J. acknowledges the financial support from FAP‐DF grants 00193−00000857/2021−14$00193-00000857/2021-14$, 00193−00000853/2021−28$00193-00000853/2021-28$, 00193.00001808/2022−71$00193.00001808/2022-71$, and 00193−00000811/2021−97$00193-00000811/2021-97$, and CNPq grants 302922/2021−0$302922/2021-0$ and 350176/2022−1$350176/2022-1$.Conflict of InterestThe authors declare no conflict of interest.Data Availability StatementThe data that support the findings of this study are available from the corresponding author upon reasonable request.K. Novoselov, S. Morozov, T. Mohinddin, L. Ponomarenko, D. Elias, R. Yang, I. Barbolina, P. Blake, T. Booth, D. Jiang, J. Giesbers, E. W. Hill, A. K. Geim, Phys. Status Solidi B 2007, 244, 4106.C. Chung, Y.‐K. Kim, D. Shin, S.‐R. Ryoo, B. H. Hong, D.‐H. Min, Acc. Chem. Res. 2013, 46, 2211.P. Avouris, F. Xia, MRS Bull. 2012, 37, 1225.Y. He, C. Yi, X. Zhang, W. Zhao, D. Yu, TrAC, Trends Anal. Chem. 2021, 136, 116191.A. K. Geim, K. S. Novoselov, Nat. 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Advanced Theory and Simulations – Wiley

**Published: ** Mar 16, 2023

**Keywords: **graphene nanoribbon; heterojunction; polaron dynamics

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