Width and Crystal Orientation Dependent Band Gap Renormalization

Letter pubs.acs.org/JPCL

Width and Crystal Orientation Dependent Band Gap Renormalization in Substrate-Supported Graphene Nanoribbons Neerav Kharche*,† and Vincent Meunier*,† †

Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, United States S Supporting Information *

ABSTRACT: The excitation energy levels of two-dimensional (2D) materials and their one-dimensional (1D) nanostructures, such as graphene nanoribbons (GNRs), are strongly affected by the presence of a substrate due to the long-range screening effects. We develop a first-principles approach combining density functional theory (DFT), the GW approximation, and a semiclassical image-charge model to compute the electronic band gaps in planar 1D systems in weak interaction with the surrounding environment. Application of our method to the specific case of GNRs yields good agreement with the range of available experimental data and shows that the band gap of substrate-supported GNRs are reduced by several tenths of an electronvolt compared to their isolated counterparts, with a width and orientation-dependent renormalization. Our results indicate that the band gaps in GNRs can be tuned by controlling screening at the interface by changing the surrounding dielectric materials.

O

computed from a Slater determinant of Kohn−Sham orbitals, has enabled a much better description of the electronic band gap. Further, when excited states are of central interest, DFT becomes inadequate, and methods “beyond DFT”, such as many-body perturbation theory and in particular the GW approximation, can provide a rigorous framework for accurate band gap calculations.13,14 The GW method has been applied to a wide variety of 2D materials and to their prototypical 1D nanostructures, nanoribbons.10,15−20 However, these calculations are performed either in the isolated configuration or using simplified model substrates because the GW method is computationally intensive and challenging to converge numerically. Specifically, for GNRs, the presently available GW calculations performed on isolated systems significantly overestimate the band gaps compared to experiments, where GNRs are supported on either metallic or insulating substrates. Given their promising technological applications and the recent developments in atomically precise fabrication and characterization techniques for GNRs,6,21,22 it is imperative for the effects of realistic dielectric environment surrounding GNRs to be included in the theoretical predictions, especially in the quest to develop GNRbased devices, such as field effect transistors.23 In this Letter, we develop a new first-principles framework for calculating the quasiparticle band gap of GNRs in weak (i.e., noncovalent) contact with a substrate. The objective is to develop a predictive tool to yield a quantitative evaluation of the electronic band gap that is both material (composition, structure, and size) and substrate dependent. In this approach,

wing to their unique physical properties, atomically thin two-dimensional (2D) materials and their one-dimensional (1D) nanostructures are expected to enable revolutionary applications in electronics, photonics, and energy harvesting.1−4 Of particular interest are graphene nanoribbons (GNRs), the 1D nanostructures of the prototypical 2D material, graphene. Quantum confinement in GNRs is responsible for opening of sizable band gaps in the otherwise semimetallic graphene, making GNRs promising candidates for semiconductor digital electronics, logic gates, and optoelectronics.5 Due to the absence of surface dangling bonds, the pristine surfaces of GNRs tend to interact with the surrounding materials predominately via the weak van der Waals (vdW) interaction. In these vdW heterostructures composed of a GNR and substrate, GNRs largely retain their intrinsic properties making them highly relevant for novel technological applications.6−9 However, even in this weak interaction regime, the long-range electron correlation effects induced by screening from the surrounding dielectric media significantly modifies their excited-state properties such as the quasiparticle band gaps and exciton binding energies.10 Quantitatively accurate computation of band gaps from firstprinciples has been a major challenge for modern electronicstructure methods. This difficulty is particularly acute in systems governed by fully nonlocal interactions, since this type of interactions involves the understanding of many-body effects. Density functional theory (DFT) is a computationally efficient first-principles method that is known to underestimate the band gaps,11 due in large part to the inability of local and semilocal approximations to properly account for the discontinuity in the derivative of the exchange-correlation energy functional.12 The development of nonlocal functionals, especially hybrid functionals built from the exact exchange © XXXX American Chemical Society

Received: February 23, 2016 Accepted: April 8, 2016

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DOI: 10.1021/acs.jpclett.6b00422 J. Phys. Chem. Lett. 2016, 7, 1526−1533

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screening response of the adsorbate, which is crucial for extended systems such as GNRs. Our new model yields band gaps in significantly improved agreement with available experiments, as we will demonstrate at the end of the paper. Our results are expected to have important implications for GNR-based electronics, as screening from the surrounding dielectrics would modulate the energy levels and transport in GNRs thereby influencing device performance. The quasiparticle band gaps of isolated GNRs have been previously studied using a first-principles many-body Green’s function approach within the GW approximation.17,26 In this method, the quasiparticle energies are determined by perturbatively evaluating the self-energy corrections to the DFT Kohn−Sham eigenvalues.11,14 Here, we use the GW quasiparticle band gaps of isolated GNRs reported by Yang et al.17 These calculations were performed at the “one-shot” G0W0 level of theory, where the local density approximation (LDA) is used for DFT calculations while the quasiparticle corrections are obtained using the single-shot G0W0 approach. Representative atomic structures of two common types of GNRs, armchair GNRs (AGNRs) and zigzag GNRs (ZGNRs), are shown in Figure 1b. Their GW quasiparticle band gaps in the isolated configuration are shown as a function of the width in Figure 2. For reference, the LDA band gaps are also provided in Figure S1 in the Supporting Information. All AGNRs exhibit direct band gaps at the Brillouin zone center. They are classified into three groups N = 3p, 3p + 1, and 3p + 2, where p is an integer and N denotes the number of carbon chains along the

the energy levels of the substrate-supported GNRs are determined by correcting the GW quasiparticle energies of isolated GNRs with the energy shifts arising from screening of quasiparticle excitations by the substrate (Figure 1a). The

Figure 1. (a) Schematic illustration of the substrate screening induced renormalization of the valence band maximum (VBM) and conduction band minimum (CBM) as a function of the distance between a GNR and the substrate. (b) Atomic structure of AGNR-7 with seven C−C dimer lines and ZGNR-6 with six zigzag C chains along their length.

energy shifts are determined using an image-charge model with input parameters calculated using DFT. In departure from our previously employed simple perturbative image-charge model,19,20,24 the proposed framework builds on the imagecharge model rigorously derived from the GW self-energy by Neaton et al.25 and we extend it here to include the internal

Figure 2. Quasiparticle band gap of AGNR-N, where is N the number of C−C dimer lines, supported on (a) Au (111) substrate and (b) composite Au-NaCl substrate consisting of a NaCl (001) monolayer deposited on the Au (111) surface. AGNRs are classified into three groups N = 3p, 3p + 1, and 3p + 2, where p is an integer. Band gap of ZGNR-N, where is N the number of zigzag C chains, supported on (c) Au (111) substrate and (d) composite Au-NaCl substrate. The GW quasiparticle band gap of isolated GNRs, taken from ref 17, are shown by the dashed lines. The experimental data for AGNR-7 and AGNR-13 are taken from refs 7−9 1527

DOI: 10.1021/acs.jpclett.6b00422 J. Phys. Chem. Lett. 2016, 7, 1526−1533

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The Journal of Physical Chemistry Letters GNR width (Figure 1b).5 For the AGNRs considered here, N varies from 4 to 14. We study two energy gaps for the ZGNRs: the direct band gap (Δ0) located approximately three-quarters of the way from the Brillouin zone edge and the energy gap (Δ1) at the Brillouin zone edge. As discussed in ref 17, the quasiparticle band gaps of both AGNRs and ZGNRs are significantly larger than the DFT Kohn−Sham band gaps (Figure S1). This is a well-known shortcoming of DFT within the local or semilocal approximations, which is particularly pronounced in GNRs due to the strong 1D quantum confinement and associated enhanced Coulomb interaction. Owing to the challenges involved in the fabrication of atomically precise GNRs, only very limited experimental data on atomically well-defined structures are presently available.7−9 Existing data are summarized in Table I where we also show a

tunneling spectroscopy (STS), photoemission and inverse photoemission experiments.7−9 In the interacting electron picture, the image potential is an exchange-correlation (XC) effect. Due to its long-range nature, the image potential cannot be accurately modeled using the conventional DFT XC functionals such as the local-density or generalized gradient approximation, or even the hybrid functionals.32 These DFT XC potentials decay exponentially away from the surface and fail to reproduce the image potential, since the image potential should decay much slower as −1/z.32 Therefore, the renormalization of energy levels in the weakly interacting adsorbates cannot be accurately modeled at the DFT level of theory.25,27,28 As shown by Neaton et al.,25 in the van der Waals bonding regime, where the hybridization between the adsorbate and substrate is negligible, the quasiparticle energy levels of an adsorbate can be obtained using a two-step approach schematically shown in Figure S2 in the Supporting Information. In the initial step, the quasiparticle energy levels in the isolated configuration are determined by computing the self-energy corrections to the Kohn−Sham eigenvalues using the GW method. Next, the change in the quasiparticle selfenergy (ΔEQP) upon adsorption is estimated using a classical image-charge model. This simplification is based on two main approximations: (i) the static Coulomb-hole and screenedexchange (COHSEX) approximation to the GW self-energy and (ii) the change in the screened Coulomb interaction (ΔW) evaluated as the classical image-potential (Vim). Under these approximations, ΔEQP model = ± 1/2∫ ∫ drdr′ρ(r)ρ(r′)Vim(r,r′), where the sign is positive (negative) for empty (occupied) states, ρ(r) = |ψ(r)|2 is the charge density of the one-body wave function, and the image-potential is given by

Table I. Comparison of Experimental and Theoretical Values of the Band Gaps, in Electronvolts, of Isolated and Substrate-Supported Armchair GNRs AGNR-7 theory isolated Aua Au-NaClb

3.79 2.44 2.72

AGNR-13

experiment

theory

experiment

2.30c, 2.50d 2.86e

2.25 1.29 1.44

1.40d

a

GNRs supported on the Au (111) substrate. bGNRs supported on a NaCl monolayer deposited on the Au (111) surface. cRef 8. dRef 7. e Ref 9.

comparison between theoretical and experimentally measured band gaps of AGNR-7 and AGNR-13. Both systems belong to the 3p + 1 family. The GW quasiparticle band gaps calculated for isolated AGNRs are significantly larger compared to the experimentally measured band gaps, where the AGNRs are supported on a Au (111) surface7,8 or on a NaCl (001) monolayer which is itself on top of a Au (111) surface.9 As discussed below, this overestimation is attributed to the lowering of Coulomb interaction in AGNRs by the screening from the underlying substrate. We note that there is no hybridization between the frontier orbitals of GNRs and the surface states of the Au and Au-NaCl substrates so that the interaction between them is of a weak van der Waals nature. When an adsorbate is deposited on a substrate, its quasiparticle (or single-particle) excitation energy levels are renormalized even in the absence of strong chemical bonding or charge transfer.25,27,28 This effect can be easily understood as follows: quasiparticles are charged excitations resulting from the addition or removal of one electron from the material. As illustrated schematically in Figure 1a, when an adsorbate approaches closer to the substrate surface, the energy levels of the charged excitations change as a result of the polarizationinduced screening effect.25,27,28 A classic example of this effect is that of a unit positive or negative point charge approaching a metal surface. The electrostatic energy due to the interaction of the charge with the induced surface charge amounts to (in atomic units) + 1/4z or −1/4z, respectively, where z is the distance relative to the position of the image plane.29,30 In addition to the rescaling of the quasiparticle energy levels, substrate screening also renormalizes the exciton (two-particle electron−hole excitations) binding energy that is relevant to the optical absorption.10,31 In this work, we will only consider the single-particle excitations as probed in the scanning

Vim(r , r′) = −1/ (x − x′)2 + (y − y′)2 + (z + z′)2 . Physically ΔEQP model represents the change in the electrostatic potential energy of the charge distribution ρ(r) as it approaches the substrate surface due to the interaction with its image charge ρ(r′) induced by the polarization of the substrate.25 This simple image-charge model is intrinsically nonlocal and captures the renormalization of quasiparticle energies with quantitative accuracy in small molecules weakly adsorbed on a substrate.25,33 However, it neglects the internal polarization response of the adsorbate to the added electron or hole, which plays an important role in extended systems such as 2D and 1D materials. To tackle this effect, we adopt the methodology presented by Spataru et al.34 for a closely related system, carbon nanotubes (CNTs) adsorbed on the metal substrate, where polarizability of the adsorbate is taken into account. Here, we adapt this methodology to the geometry corresponding to narrow ribbons, including GNRs. We adapt the formalism developed for cylindrical geometry34 to flat one-dimensional (i.e., ribbon) systems to determine ΔEQP using a charge density having the same spatial distribution as the given wave function added to a continuum ribbon having the same polarizability as the GNR. To reduce computational costs, we average out the charge distribution along the periodic direction of the GNR and in the direction normal to the GNR plane. This procedure yields the unscreened charge density ρ0(x) varying along the ribbon width, which incorporates effect of GNR’s confinement and crystal orientation, Figure 3. The presence of an external charge ρ0(x) is associated with an attractive force on screening charge of opposite sign, which in turn polarizes the substrate and the 1528

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approach that does not require any empirical parameters. The exponential decay of the integrand in eq 1 ensures that it is sufficient to consider only the small q behavior of the of P(q). We calculate P(q) using the 1D Penn model P(q) = αq2, which is valid in the long-wavelength limit (q → 0).34,35 The static polarizability α is obtained from the 1D Clausius−Mossotti expression α = (A/4π)(ε − 1), where A is the cross sectional area of the supercell and ε is the longitudinal dielectric constant, which is obtained using density functional perturbation theory (DFPT).36 The image plane position z0 is obtained from DFT calculations of the self-consistent response of the substrate to the external electric field normal to the surface.37 The technical details of the DFT calculations are given in the Computational Methods section. Figure 4 shows the image plane position z0 or the effective edge of the substrate for the clean Au (111) surface and the Au

Figure 3. Exemplary GNR wave functions averaged out along the periodic direction of the GNR and in the direction normal to the GNR plane. (a) Nonspin polarized wave functions at the valence and conduction band edges of AGNR-13. (b) Spin-up component of the spin-polarized wave functions at the valence and conduction band edges of ZGNR-8.

ribbon along its length. The resulting screened charge is given by ρ(x,q,h) = ρ0(x)/εads(q,h), in the Fourier representation where εads(q,h) is the dielectric function of the substratesupported ribbon located at a distance h from the image plane (z0) of the surface of the substrate and q is along the length of the GNR. The change in the electrostatic potential energy of this charge distribution as it moves from ∞ to a distance h0 from the surface gives34 QP ΔEmodel (h 0 ) = ±

1 2

dq

Vim(q , h0) iso(q)εads(q , h0)

∫ 2π ε

Figure 4. Image plane positions (dashed black vertical lines) and the classical image-charge potential (solid red lines) felt by a unit positive point charge as it approaches the substrate surface. The results are shown for the Au (111) surface and the surface of a composite substrate consisting of a NaCl (001) monolayer deposited on the Au (111) surface. The inset shows the atomistic schematic of the composite Au-NaCl slab used in the DFT calculations.

(1)

where εiso(q) = 1 − P(q)Viso(q) and εads(q,h0) = 1 − P(q)(Viso(q) + Vim(q,h0)) are the 1D dielectric functions of the isolated and substrate-supported ribbons, respectively. P(q) is the irreducible intrinsic polarizability of the ribbon. Viso(q) is the Coulomb potential in the isolated ribbon and Vim(q,h) is the Coulomb interaction with the induced image charge given by w /2

Viso(q) =

(111) surface covered with a NaCl (001) monolayer superimposed on top of the atomistic schematic of the Au-NaCl surface supercell used in the DFT calculations. The NaCl monolayer is weakly bound to the Au surface with a separation of 3.0 Å as determined using the optB86b-vdW functional that includes long-range van der Waals interactions.38,39 The image plane for the clean Au surface is found to be located 1.42 Å above the outer Au layer, whereas for the composite Au-NaCl surface, the image plane is located 0.40 Å above the NaCl layer. Thus, the image plane is pulled away from the Au surface by the presence of the NaCl monolayer. Similar behavior is determined for a closely related system, Ag(111) surface covered with a NaCl bilayer.40 The image potential for a unit positive point charge Vim = −1/2|z − z0| is also shown in Figure 4. We calculate the distance between the GNR and the substrate for two illustrative systems: AGNR-7 adsorbed on the Au (111) surface and AGNR-7 adsorbed on the monolayer NaCl (001) supported on the Au (111) surface. As mentioned above, these calculations are carried out using the optB86bvdW functional to account for the long-range van der Waals interactions.38,39 The AGNR-7 adsorbed on the Au (111) surface is found to be located at 3.11 Å above the surface, while the AGNR-7 adsorbed on the NaCl (001) monolayer is found to be located 3.45 Å above the surface. Thus, the relative distance of the AGNR-7 from the image plane is h0 = 3.11−

w /2

∫−w/2 dx1 ∫−w/2 dx2ρ0 (x1)ρ0 (x2)v(q , |x1 − x2|, 0) (2)

and Vim(q , h) w /2

=−

w /2

∫−w/2 dx1 ∫−w/2 dx2ρ0 (x1)ρ0 (x2)v(q , |x1 − x2|, h) (3) 2

2

respectively, where v(q , x , h) = 2K 0(q x + h ) is the Coulomb kernel and K0 is the modified Bessel function of second kind. The detailed derivation of eq 1 is provided in the Supporting Information. These equations describe the following phenomena: when a nanoribbon approaches the substrate, it is affected by the presence of an image charge in the substrate, which modifies both the ribbon band gap and its dielectric function. The image-charge correction in eq 1 requires knowledge of the irreducible polarizability P(q) of the GNR, the position of the image plane z0, and the distance between the GNR and the image plane h0. All three quantities can be obtained using DFT. It follows that the proposed framework is a fully first-principles 1529

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Figure 5. Variation of the static polarizability with the width for (a) AGNRs and (b) ZGNRs.

1.42 = 1.69 Å for the Au substrate and h0 = 3.45−0.40 = 3.05 Å for the composite Au-NaCl substrate. Given the weak interaction and the chemical similarities, we use the same values of h0 for all the other GNRs. As shown in Figure 5, the static polarizability of both the AGNRs and ZGNRs increases as a function of their width. All the three classes (3p, 3p + 1, and 3p + 2) of AGNRs show a different behavior as expected from their band gaps. The 3p + 2 (3p + 1) AGNRs that feature the smallest (largest) band gaps have the highest (lowest) polarizability. Using the polarizability of GNRs and their relative distance from the image plane, we now compute the substrate screening induced renormalization of the quasiparticle energy levels, according to eq 1. We consider quasiparticle states at the valence band maximum (VBM) and the conduction band minimum (CBM). The image-charge corrections are applied on the GW energy levels evaluated for isolated GNRs using QP GW − QP QP Eads = E iso + ΔEmodel

lower static polarizability results in a weaker screening of the added charge entering in eq 1 through the dielectric functions εiso(q) and εads(q,h0). We note that the dielectric function of the substrate supported ribbon εads(q,h0) is smaller than the dielectric function of the isolated ribbon εiso(q). This is because the image-charge interaction Vim(q,h0) reduces the Coulomb interaction from Viso(q) in the isolated ribbon to Vads(q) = Viso(q) + Vim(q,h0) in the substrate-supported ribbon. In addition to the Au and Au-NaCl substrates considered above that exhibit strong metallic screening, we have also estimated the polarization induced band gap reduction in two GNRs, AGNR-7 and AGNR-13, deposited on two popular and representative semiconductor/insulator substrates, Si and SiO2. In this case, the image charge is scaled down by a factor (ε − 1)/(ε + 1), where ε = ε∞ is the high frequency dielectric constant of the substrate material.41,42 Our objective here is to illustrate the effect of screening properties of the substrate rather than providing quantitatively accurate band gaps. Here, we use experimentally measured dielectric constants (ε∞) 11.7 and 2.4 for Si and SiO2 (α-quartz) substrates, respectively.43,44 The distance between the substrate surface and the GNRs is taken as the same as that for the Au-NaCl (3.45 Å), whereas the image plane is assumed to be located 1 Å above the substrate surface. Using these parameters the renormalized band gaps of AGNR-7 and AGNR-13 come out to be 2.80 and 1.52 eV, when they are deposited on the model Si substrate and 3.30 and 1.89 eV, when they are deposited on the model SiO2 substrate. Due to reduced screening, GNRs deposited on both the model Si and SiO2 substrates show smaller band gaps reduction and hence larger band gaps compared to the GNRs deposited on the metallic substrates (Table I). Finally, we provide a brief discussion on the shortcomings of simplified image-charge model employed in our previous work19,20,24 and how the more rigorous model used in the present work addresses those shortcomings. In the earlier model, the renormalization of the quasiparticle energy was estimated by the first order correction over the perturbation, ΔU = Uim − VDFT XC (z), where Uim = −1/4(z − z0) is the potential energy of a point charge near the metal surface and VDFT XC (z) is the planar averaged DFT exchange-correlation QP potential. The correction is given by ΔE model = ± ∫ z>z′drΔU(z)ρ(r); z′ is where the transition from the shortrange to long-range limit takes place and Uim(z) starts to deviate from VDFT XC (z). In contrast to the earlier model, the new model used in the present work is an extension of the imagecharge model rigorously derived from the GW self-energy as shown in Neaton et al.25 Additionally, the internal screening response of the adsorbate to the added charge was not taken

(4)

where EQP ads is the renormalized quasiparticle energy of the given electronic state in the GNR adsorbed on the substrate, EGW−QP iso is the quasiparticle energy of that state in the isolated GNR within the GW approximation, and ΔEQP model is the image-charge correction. The corrections are opposite in sign with the VBM shifting upward and the CBM shifting downward relative to the vacuum level, thus reducing the band gap in the substratesupported GNRs, as illustrated schematically in Figure 1a.25,27,28 This correction follows from the intuition that the polarization response of the substrate to a charge excitation in the adsorbate gives rise to induced image charge, which acts back on the added charge lowering the energy of that charged state. The magnitude of the corrections for the VBM and CBM are found to be almost identical, differing by less than 0.02 eV. Figure 2 shows the band gaps of the AGNRs and ZGNRs supported on the Au (111) substrate and on the composite substrate consisting of a NaCl (001) monolayer on the Au (111) surface. As seen from Figure 2a and b and Table I, the calculated band gaps are in good quantitative agreement with the available experimental data for AGNR-7 and AGNR-13. A closer inspection of Figure 2 shows that the image-charge correction ΔEQP model for both AGNRs and ZGNRs increases with decreasing width. ΔEQP model as a function of the GNR width is shown separately in Figure S1 in the Supporting Information. The width dependence of ΔEQP model arises from two effects: (i) the increasingly more confined charge density for reduced width and (ii) the decreasing static polarizability of GNRs with decreasing width, the latter being the dominating effect. The 1530

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The Journal of Physical Chemistry Letters into account in our earlier work.19,20,24 In the present work, we account for the internal screening of the added charge through the dielectric function ε(q,h0) in eq 1. Comparing the imagecharge corrections for the specific case of AGNR-7 supported on the Au (111) surface, our earlier model yields band gap renormalization of 0.68 eV20 significantly smaller compared to 1.35 eV as obtained using the new model. The renormalized band gaps using the earlier and new models are 2.44 and 3.11 eV, the latter being in a better agreement with the experimental result (Table I). Thus, in addition to be based on stronger theoretical principles, our new image-charge model also yields a significant improvement over the earlier model. The computational approach presented here provides a fully first-principles framework for predicting quasiparticle electronic structure of GNRs in the presence of external screening environment ubiquitous in realistic experiments and functional devices. In addition to GNRs, our computational approach can also be applied to other 1D nanostructures45 such as phosphorene and transition metal dichalcogenides (TMDCs), which are emerging as highly promising materials for electronics and optoelectronics. Directly including the external screening environment in GW calculations is not feasible using the presently available state-of-the-art computational tools.46 To circumvent these limitations, we included the effects of screening environment by means of a classical image-charge model. We note that our approach is valid in the weak van der Waals bonding or the physisorption regime (i.e., when no significant hybridization takes place between substrate and adsorbate). This situation is common to many 2D materials due to the absence of surface dangling bonds.47 The proposed approach is therefore applicable to a wide variety of systems. For the chemisorbed systems, the strong bonding will significantly alter the character of excited states and the local screening response of the substrate, making it necessary to treat the whole system at the first-principles GW level of theory. In addition, we have demonstrated that the band gap in GNRs can be tuned by changing the screening properties of the dielectric environment, for example, using atomically thin spacer layers or using dielectric materials such as Si/SiO2. This suggests the interesting possibility of using spatially varying dielectric environment to engineer the screening effects and induce band offsets into 1D materials without any chemical modification. In summary, we developed an integrated first-principles approach to calculate the quasiparticle energies of GNRs weakly interacting with the underlying substrate. In this approach, the quasiparticle energies of isolated GNRs are calculated using the computationally intensive GW method. The renormalizations of GW quasiparticle energies due to the long-range screening effects at the GNR-substrate interface are then determined using an image-charge model that includes the internal screening response of GNR. All input parameters to the image-charge model are calculated at the DFT level of theory, thus providing a computationally tractable fully firstprinciples framework. We demonstrate our method for AGNRs and ZGNRs supported on two chemically distinct substrates, the Au (111) substrate and the composite substrate consisting of NaCl (001) monolayer deposited on the Au (111) surface. The band gaps in the substrate-supported GNRs are reduced by several tenths of an electronvolt compared to the isolated GNRs. For both substrates, we find good agreement of the calculated band gaps with the presently available experimental data. More broadly, the integrated first-principles approach

presented here should support improved understanding of quasiparticle electronic structure of 1D nanostructures weakly interacting with the surrounding dielectric materials in realistic experiments and functional devices.

■

COMPUTATIONAL METHODS DFT calculations are performed with the Vienna Ab initio Simulation Package (VASP)48,49 using the projector augmented wave (PAW) method.50 All atomic coordinates are relaxed using a conjugate-gradient algorithm until all forces are smaller in magnitude than 0.05 eV/Å. The image plane positions are determined from the selfconsistent response of a Au (111) slab covered on one side by a NaCl (001) monolayer. The commensurate supercell contains a 5 × 5 surface cell of NaCl monolayer and a six-layer-thick orthogonal 7 × 8 surface cell of Au (111). The resulting strain (