All-carbon approach to inducing electrical and optical anisotropy in graphene Cite as: AIP Advances 11, 115007 (2021); https://doi.org/10.1063/5.0062521 Submitted: 14 July 2021 • Accepted: 28 September 2021 • Published Online: 02 November 2021 Aleandro Antidormi and Aron W. Cummings COLLECTIONS This paper was selected as Featured This paper was selected as Scilight ARTICLES YOU MAY BE INTERESTED IN A review on low-dimensional novel optoelectronic devices based on carbon nanotubes AIP Advances 11, 110701 (2021); https://doi.org/10.1063/5.0063774 A simple structure laser three-focus scheme for thick glass separation AIP Advances 11, 115001 (2021); https://doi.org/10.1063/7.0000533 Optical rectification by ratchet transport in an asymmetric grating AIP Advances 11, 115006 (2021); https://doi.org/10.1063/5.0062816 AIP Advances 11, 115007 (2021); https://doi.org/10.1063/5.0062521 11, 115007 © 2021 Author(s). AIP Advances ARTICLE scitation.org/journal/adv All-carbon approach to inducing electrical and optical anisotropy in graphene Cite as: AIP Advances 11, 115007 (2021); doi: 10.1063/5.0062521 Submitted: 14 July 2021 • Accepted: 28 September 2021 • Published Online: 2 November 2021 Aleandro Antidormia) and Aron W. Cummingsb) AFFILIATIONS Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra 08193, Barcelona, Spain a) Electronic mail: aleandro.antidormi@icn2.cat b) Author to whom correspondence should be addressed: aron.cummings@icn2.cat ABSTRACT Owing to its array of unique properties, graphene is a promising material for a wide variety of applications. Being two-dimensional, the properties of graphene are also easily tuned via proximity to other materials. In this work, we investigate the possibility of inducing electrical and optical anisotropy in graphene by interfacing it with other anisotropic carbon systems, including nanoporous graphene and arrays of graphene nanoribbons. We find that such materials do indeed induce such anisotropy in graphene while also preserving the unique prop- erties offered by graphene’s Dirac band structure, namely, its superior charge transport and long-wavelength optical absorption. The optical anisotropy makes such heterostructures interesting for their use in applications related to long-wavelength polarimetry, while the electrical anisotropy may be valuable for enhancing the performance of graphene photothermoelectric detectors. © 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0062521 Graphene possesses unique properties with potential for a structure. This can be carried out with top-down fabrication meth- variety of applications.1 With respect to its optical properties, the ods, such as lithography of the dielectric substrate,13,14 with fea- absence of a bandgap and its linear Dirac band structure are ture sizes on the order of tens of nanometers. Superlattices can responsible for its uniform light absorption over a broad range also be induced by layering graphene with hexagonal boron nitride of frequencies, from the optical to the THz regime.2 Moreover, or another layer of graphene, leading to exotic electronic prop- graphene is fully compatible with standard silicon photonics and erties.15,16 In these systems, the periodicity of the Moiré super- exhibits tunable electro-absorption and electro-refraction with fast lattice is tunable by varying the twist angle between the two electron dynamics and a small Fermi surface. These factors moti- layers. vate the development of graphene-based photonics, with applica- Recently, superlattices with periodicity ≈1 nm have been real- tions in optical data communications,3–6 THz technologies,7,8 and ized in graphene. By using directed reactions among self-organized plasmonics.9,10 molecular precursors, it is possible to fabricate graphene containing Another fundamental feature of graphene is the tunability of its a periodic array of nanoscale holes.17 Called nanoporous graphene physical properties, which may be achieved electrostatically, chem- (NPG), this material has a large bandgap and is expected to exhibit ically, or via proximity to other materials. As an example, strong anisotropic optical and electrical properties, making it intriguing spin–orbit coupling can be induced in graphene by interfacing it for devices. It also represents a new approach for customizing with transition metal dichalcogenides, leading to phenomena such graphene, by using chemistry to design nanoscale superlattices from as large spin relaxation anisotropy and spin-charge conversion.11,12 the bottom-up. Similar techniques can also grow arrays of graphene Meanwhile, graphene’s Dirac cone and superior charge transport are nanoribbons (GNRs).18 maintained in these systems, allowing the transfer of spin over long In this paper, we explore the possibility of tuning the proper- distances. ties of graphene by interfacing it with NPG or arrays of GNRs. The In the ongoing search for new functionalities of graphene, goal is to combine one of the interesting properties of these mate- one strategy is to alter its features by imposing long range rials, namely, their anisotropy, with the unique properties offered periodicity, called a superlattice, on top of its underlying crystal by graphene’s Dirac band structure, namely, its superior charge AIP Advances 11, 115007 (2021); doi: 10.1063/5.0062521 11, 115007-1 © Author(s) 2021 AIP Advances ARTICLE scitation.org/journal/adv transport and long-wavelength optical absorption, into mid-IR and Ĥ = − γ0 ∑ (â†m,i b̂m,j + h.c.) beyond. Using numerical simulations, we show that NPG and GNRs ⟨ij⟩,m can indeed induce anisotropy in graphene, both in its optical absorp- − γ1 ∑(â†1,j â2,j + h.c.) tion and in its electrical transport. j With respect to optical absorption, this opens up graphene for its potential use in IR polarimetry. Polarimetry is the use of the − γ3 ∑(b̂†1,j b̂2,j + h.c.) polarization of detected light to analyze images better than that done j only with intensity and wavelength. In the IR region, analysis of the − γ4 ∑(â†1,j b̂2,j + â†2,j b̂1,j + h.c.), (1) polarity of detected light can be used, e.g., for the detection of land j mines,19,20 to conduct tissue analysis,21–23 or to study astrophysical phenomena.24,25 where â†m,j and b̂†m,j (âm,j and b̂m,j ) are the creation (annihilation) Meanwhile, anisotropic charge transport may be valuable for operators of the pz orbital on sublattice A or B, respectively, at enhancing the efficiency of graphene photodetectors based on the lattice site j of layer m = 1, 2. The parameter γ0 = 2.9 eV is the in- photothermoelectric effect.4–8 A recent extensive analysis indicates plane hopping energy between nearest neighbors, while γ1 = 0.4 eV, that suppressing the electronic thermal conductivity in the trans- γ3 = 0.3 eV, and γ4 = 0.04 eV are the interlayer hoppings, as depicted verse direction of such photodetectors, while maintaining it in the in Fig. 1(c). The values of these parameters are chosen to be those of longitudinal direction, would enhance their performance.26 bilayer graphene.27 The system we primarily focus on in this work is shown From this Hamiltonian, we can compute the electronic band in Fig. 1. Panel (a) shows two unit cells (along the y axis) of a structure, the optical absorption, and the transport properties of the graphene/NPG heterostructure in the lowest-energy AB stacking NPG/graphene heterostructure. We begin with the band structure, configuration. The NPG is the same as that synthesized in Ref. 17 shown in Fig. 2. The thinner, fainter lines show the band struc- and may be viewed as an array of GNRs oriented along the y axis and ture of the two layers when they are uncoupled (γ1 = γ3 = γ4 = 0), linked together via a single carbon–carbon bond. Panel (c) shows the i.e., they are the superposition of the band structures of the indi- hopping parameters that are used in our tight-binding model of this vidual graphene and NPG layers. Owing to the larger supercell, the system, with the Hamiltonian given by graphene Dirac cones have been folded from K/K′ to positions along the Γ–X line. Meanwhile, the minimum (maximum) of the conduc- tion (valence) band of the NPG remains at Γ, with a bandgap of 0.56 eV. The thicker, bolder lines show the band structure when we turn on the interlayer coupling. Owing to hybridization between the lay- ers, the bands no longer strictly belong to states in the graphene or NPG layer. Henceforth, we refer to the bands around Γ as “NPG- like” and those that belong to the graphene Dirac cone along Γ–X as “graphene-like” ones. In the presence of interlayer coupling, the bandgap of the NPG-like bands shrinks to 0.34 eV. The graphene- like bands remain within this gap but also exhibit a small gap open- ing of 32 meV. More striking, however, is a strong renormalization of the Fermi velocity of the graphene-like bands. This can be seen in the middle inset, where we plot the Fermi contour at a few energies near the charge neutrality point. The elliptical shape of the Fermi surface indicates the anisotropy induced in the graphene-like bands due to hybridization with the NPG. The Fermi velocity along the x-direction is suppressed with respect to that along the y direc- tion, with the ratio v F,x /v F,y ≈ 1.2 at a Fermi energy of 0.1 eV, and v F,x /v F,y ≈ 1.4 near the bottom of the Dirac cone. The anisotropy induced in the band structure of the graphene by the NPG is expected to affect its optical and electronic properties. We first focus on the optical absorption by computing the optical conductivity of the graphene/NPG heterostructure. In the electric dipole approximation, the optical conductivity is given by28 2πe2 σ(ω) = ∑∣⟨ j∣e ⋅ p̂∣i⟩∣ 2 m20 ωΩ i,j FIG. 1. Two unit cells of (a) the graphene/NPG heterostructure and (b) a ̵ × [ f (Ei ) − f (Ej )]δ(Ej − Ei − hω), (2) graphene/9-armchair graphene nanoribbon (aGNR) heterostructure, both with AB stacking. Light blue atoms belong to graphene, and dark red atoms belong to where ω is the optical frequency, m0 is the free electron mass, p̂ is the momentum operator, f (E) is the Fermi distribution function, the NPG or GNR. (c) The hopping parameters used in the tight-binding model of Eq. (1). and Ω is the volume of the unit cell. The unit vector e denotes the AIP Advances 11, 115007 (2021); doi: 10.1063/5.0062521 11, 115007-2 © Author(s) 2021 AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. Band structure of the graphene/NPG heterostructure of Fig. 1(a). The thin faint lines show the band structure when the two layers are uncoupled, while the thicker darker lines are when the interlayer coupling is turned on. The middle inset shows the Fermi surface of the graphene-like Dirac bands at a few energies, high- lighting the anisotropy induced by hybridization with the NPG. The lower right inset shows the Brillouin zone of the isolated graphene layer and that of the graphene/NPG heterostructure. polarization of the electric field, which is real for linear polariza- indicating that NPG is indeed effective at inducing optical tion. Equation (2) expresses the conductivity as the sum of all vertical anisotropy in graphene. transitions between states ∣i⟩ and ∣ j⟩, with the δ term imposing con- The optical anisotropy observed for small energies is associ- servation of energy. The translational invariance of the system allows ated with anisotropic modification of the graphene Fermi velocity us to use the Bloch eigenstates ∣k⟩ of H, and to use the relation induced by the NPG. We expect a similar impact on charge transport at these energies. To examine this, we compute the electrical conduc- tivity using a real-space order-N wave packet propagation method.33 im0 ∂Ĥ ⟨j∣e ⋅ p̂∣i⟩ = ̵ ⟨j∣e ⋅ ∣i⟩, The key quantity of this method is the energy- and time-dependent (3) h ∂k mean-square displacement of the wave packet, Tr[δ(E − Ĥ)∣X̂(t) − X̂(0)∣ ] 2 which involves the derivative of the Hamiltonian with respect to the ΔX (E, t) = 2 , (4) wavevector k. The sum in Eq. (2) is then performed via numerical ρ(E) integration over the Brillouin zone with 701 × 701 points, while the δ-function is approximated as a Lorentzian with a broadening of 10 meV. Here, we ignore excitonic effects as they are not present in the linear portion of the graphene spectrum,29 and the first excitonic absorption peak of NPG is predicted to be above 1 eV,30 beyond our range of interest. The optical conductivity of the NPG/graphene heterostructure is shown in Fig. 3 as a function of photon energy/wavelength at T = 300 K. Its value is normalized with respect to the “universal” optical conductivity of graphene, σ 0 = πe2 /(2h).31 The curves labeled Ex and Ey correspond to linearly polarized light aligned along the x and y axes, respectively. The optical conductivity is clearly anisotropic over the entire energy range, with larger absorp- tion for optical fields aligned along y. This anisotropy is largest for hω > 0.34 eV when optical absorption begins to occur in non- graphene-like bands. For photon energies smaller than 0.34 eV, highlighted by the vertical dashed line, only graphene-like states are present, and the resulting conductivity involves only optical transitions within these bands. In this case, we also observe that σ y (ω) is larger than σ x (ω) as a consequence of the anisotropy of the graphene-like bands induced FIG. 3. Optical conductivity of the graphene/NPG heterostructure for light linearly by NPG. As shown in the inset, the magnitude of this anisotropy polarized along the x and y axes. The vertical dashed line indicates the onset of is σ y (ω)/σ x (ω) ≈ 1.2–1.4. This anisotropy is similar to that in absorption within the NPG-like bands at Γ. The inset shows the optical anisotropy, given by the ratio σ y (ω)/σ x (ω). photodetectors based on aligned arrays of carbon nanotubes,32 AIP Advances 11, 115007 (2021); doi: 10.1063/5.0062521 11, 115007-3 © Author(s) 2021 AIP Advances ARTICLE scitation.org/journal/adv where ρ(E) = Tr[δ(E − Ĥ)] is the density of states and X̂ is the result of the anisotropic renormalization of the graphene Dirac position operator along the x axis. From this, we calculate the cones. We now examine this behavior for graphene interfaced with time-dependent diffusion coefficient Dxx (E, t) = 12 ∂t ∂ ΔX 2 (E, t) and other carbon-based anisotropic 2D systems. Specifically, we consider ̃ its long-time limit Dxx (E). The latter enters in the resulting elec- two other configurations of NPG that were studied in Ref. 36. These tronic conductivity, σxx (E) = e2 ρ(E)D ̃ xx (E). By separately comput- structures are the same as that in Fig. 1 except that, instead of a ing the evolution of the mean-square displacement along x and y single carbon–carbon bond connecting the GNRs, a benzene ring directions, we can evaluate the conductivities σ xx (E) and σ yy (E). serves as the bridge. It was found that the connection of this benzene We assume transport is dominated by charged impurity scat- bridge (meta vs para) had a notable impact on the charge transport tering, which is modeled as a random distribution of Gaussian elec- anisotropy in the NPG. Here, we see if this difference carries over to trostatic impurities.34 The electrostatic potential at each atomic site the anisotropy induced in graphene. To test the impact of the NPG i is then given by ϵi = ∑j Vj exp(−∣⃗ ri − r⃗j ∣2 /2ξ 2 ), where r⃗i is the posi- bandgap on the optical anisotropy induced in graphene, we have also tion of each carbon atom, r⃗j is the position of each impurity, ξ created two new NPG samples that have the same pore structure as is the width of each impurity, and the height V j of each impu- that in Fig. 1(a) but with wider regions between the pores. We call rity √is randomly distributed in [−V, V]. Here, we use V = 2.8 eV, these structures NPG-8 and NPG-10, as they are 8 and 10 carbon ξ = 3a, and an impurity density of 0.1%. This choice of parame- rings wide, respectively, at the widest point. ters leads to a charge mobility of around 2000 cm2 /Vs, similar to We also investigate heterostructures of graphene with periodic experiments.35 arrays of armchair graphene nanoribbons (aGNRs). We consider the All calculations have been performed on a graphene/NPG het- collection of m-aGNRs, with m = {3p, 3p + 1}, where m is the num- erostructure with 22 × 106 atoms, consisting of 672 × 168 replicas of ber of atoms across the width of the ribbon and p > 0 is an integer. the unit cell shown in Fig. 1(a). The conductivity has been averaged These aGNRs are semiconducting, with the bandgap inversely pro- over ten randomly chosen initial wave packets, and the calculation portional their width.37 A unit cell of graphene interfaced with a of the mean-square displacement employed an efficient Chebyshev 9-aGNR is shown in Fig. 1(b). polynomial expansion with 5000 moments, corresponding to an In Fig. 5, we plot a summary of our simulation results. Panel (a) energy resolution of ∼10 meV. shows the optical anisotropy of the heterostructures as a function The electronic conductivities of the NPG/graphene het- of the GNR or NPG bandgap. Open circles correspond to graphene erostructure along the x and y directions are shown in Fig. 4. At interfaced with (3p + 1)-aGNRs, solid circles to 3p-aGNRs, and the low energies, transport is only within the graphene-like bands, with other red symbols to the NPGs. In this set of results, we have varied the vertical dashed lines indicating the onset of transport in the the ribbon width up to m = 57, corresponding to a width of 7 nm. NPG-like bands at Γ (see Fig. 2). As with the optical conductiv- For many values of m, we have also varied the inter-GNR distance. ity, the electrical conductivity along y is larger than that along x. The optical anisotropy has been averaged over the energy range for The resulting anisotropy is shown in the inset and takes on val- which only graphene-like bands are involved. ues of σ yy /σ xx ≈ 1.2–1.4, the same as the optical conductivity. Upon Here, we see a few interesting trends. First, the 3p-aGNRs the onset of charge transport in the NPG-like bands, the anisotropy show an almost constant and small anisotropy (≈1), independent grows dramatically as a result of saturation of σ xx . of the bandgap. Meanwhile, an optimal ribbon width exists for the We have shown that interfacing graphene with NPG induces (3p + 1)-aGNRs, reaching an anisotropy of nearly 1.6 for m = 37, anisotropy in its optical and electrical conductivities, which is a or a width of 4.6 nm. For the NPGs, all fall within the trend of the GNRs. We note that the meta- and para-NPGs show little differ- ence in the anisotropy they induce in graphene, despite their sig- nificantly different transport properties in isolation.36 Finally, the NPG-8 and NPG-10 structures fall on the other side of the opti- mal width/bandgap. These results suggest that there may exist an as-yet undiscovered NPG structure that can optimize the anisotropy induced in graphene. Importantly, while reducing the bandgap of the nanoribbon increases the anisotropy, it also reduces the energy range in which the linearity of the graphene bands is preserved. This denotes the maximum photon energy, which could excite graphene-like states, hence giving the upper limit for absorption frequency of “anisotropic” graphene. We present this quantity as a function of the NPG or GNR bandgap in Fig. 5(b). As would be expected, a smaller bandgap results in a smaller energy window for optical absorption only in the graphene layer. To summarize, we have tested the hypothesis that anisotropic semiconducting carbon materials—such as nanoporous graphene FIG. 4. Longitudinal electrical conductivity of the graphene/NPG heterostructure or arrays of graphene nanoribbons—can induce anisotropy in along the x and y directions. The vertical dashed lines indicate the onset of graphene while also maintaining its Dirac-like band structure. We the NPG-like bands at Γ (see Fig. 2). The inset shows the anisotropy of charge transport, given by σ yy /σ xx . have shown that this is indeed the case with such systems induc- ing an anisotropy of 20%–50% in graphene’s Fermi velocity, optical AIP Advances 11, 115007 (2021); doi: 10.1063/5.0062521 11, 115007-4 © Author(s) 2021 AIP Advances ARTICLE scitation.org/journal/adv anisotropic charge transport may also prove to be useful in some graphene-based sensor applications. For example, sensors based on the photothermoelectric effect in graphene may benefit from anisotropic thermal conductivity by allowing electronic heat to spread only along one direction.4–8,26 We hope that our initial results presented here will inspire further research in these directions. Finally, we would like to note that our discussion has cen- tered on how NPG or GNRs may induce anisotropy in graphene. However, an examination of Fig. 3 indicates that such a heterostruc- ture may be viewed in a different light, with graphene serving to extend the anisotropic optical properties of NPG to wavelengths into mid-IR and beyond. ICN2 is supported by the Severo Ochoa Centers of Excel- lence Program, funded by the Spanish State Research Agency (AEI, Grant No. SEV-2017-0706). ICN2 is also supported by the CERCA Program of the Generalitat de Catalunya. This work received funding from the European Union’s Horizon 2020 Research and Innovation Program under Grant Agreement No. 825272 (ULISSES) and from the project “Waveguide-Integrated Mid-Infrared Graphene Detectors for Optical Gas Sensor Systems,” Reference No. PCI2018-093128, funded by the Spanish Ministry of Science and Innovation—AEI. AUTHOR DECLARATIONS Conflict of Interest FIG. 5. Summary of the properties of graphene interfaced with NPG or with GNR The authors have no conflicts to disclose. arrays. (a) The optical anisotropy induced in graphene by a wide range of GNRs and NPGs, as a function of their bandgap. (b) The energy window over which only DATA AVAILABILITY graphene-like bands are present. The data that support the findings of this study are available from the corresponding author upon reasonable request. absorption, and electrical transport. This anisotropy depends on the REFERENCES bandgap of the NPG or GNR but can also depend strongly on the 1 type of GNR. This opens the possibility that other types of semi- A. C. Ferrari et al., “Science and technology roadmap for graphene, related two- conducting GNRs or NPG structures, which may be fabricated via dimensional crystals, and hybrid systems,” Nanoscale 7, 4598–4810 (2015). 2 bottom-up synthesis,38,39 may exhibit stronger effects than what we J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashekhar, F. Rana, have presented here. M. G. Spencer, D. Veksler, and Y. 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