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THE JOURNAL OF CHEMICAL PHYSICS 142, 184708 (2015)

Electronic, structural, and substrate effect properties of single-layer covalent organic frameworks Liangbo Liang,1 Pan Zhu,1 and Vincent Meunier1,2,a) 1

Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA 2 Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, USA

(Received 15 February 2015; accepted 20 April 2015; published online 14 May 2015) Recently synthesized two-dimensional covalent organic frameworks (COFs) exhibit high surface area, large pore size, and unique structural architectures, making them promising materials for various energy applications. Here, a total of nine COFs structures, including two deposited on a hexagonal boron nitride substrate, are investigated using density functional theory, quasi-particle many-body theory within the GW approximation, and an image charge model. The structures considered belong to two major families (thiophene-based COF-n (T-COF-n) and tetrakis (4-aminophenyl) porphyrin-x (TAPP-x)) differing from the presence of B—O or C==N linkers. While T-COF-n structures are shown to constitute planar networks, TAPP-x systems can display non-negligible corrugation due to the out-of-plane rotation of phenyl rings. We find that the electronic properties do not differ significantly when altering the chain molecules within each family. Many-body effects are shown to lead to large band-gap increase while the presence of the substrate yields appreciable reductions of the gaps, due to substrate polarization effects. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4919682]

I. INTRODUCTION

Two-dimensional covalent organic frameworks (2D COFs) organized into layers of periodic networks linked by strong covalent bonds have been attracting increasing interest since their first successful synthesis by Côté et al. in 2005.1 COFs feature high surface area, large pore size, and unique structural architectures, making them promising targets for a variety of energy related applications such as photovoltaics, gas storage, and fuel cells.2–6 Although the synthesis of 2D COF networks is dominated by boronic acid condensation with the formation of either boronate esters or boroxine rings,7 other reactions such as the Schiff-base/imine formation have also been successfully used as an efficient route to assemble 2D COF networks.8 These two condensation reactions used for the formation of boroxines or imines under reversible covalent-bond-forming conditions have been the most widely applied chemical strategies to synthesize 2D COFs. This type of synthesis offers an opportunity for rational material design since it enables the relative ease to tune the network structures in a predictive way by selecting appropriate chemical groups as starting reactants. A donor-acceptor scheme was recently introduced where the 2D COF network can be construed as a system where one node and chain molecule act as a donor unit while the other ones play the role of an acceptor.9 Using this scheme, a design strategy is adopted where the donor or acceptor molecules are varied to explore the impact on physical properties of the assembly using different electronic structure methods. The recently reported synthesis of two kinds of 2D COFs using the above two chemical approaches provides good a)[email protected] URL: http://homepages.rpi.edu/∼meuniv/

0021-9606/2015/142(18)/184708/9/$30.00

examples of this design strategy applied in a predictive and systematic way to tune the structures and properties of 2D COF networks. The first kind of systems we are concerned with is the family of thiophene-based COFs (T-COFs) obtained from the condensation of 2, 3, 6, 7, 10, 11-hexahydroxytriphenylene (HHTP) with thiophene-, bithiophene-, and thienothiophenediboronic acids,7 which are synthesized through boronateester formation as a result of B—O bonding (Figure 1(a), left panel). The other type of COFs considered here results from the condensation of 5, 10, 15, 20-tetrakis (4-aminophenyl) porphyrin (TAPP) with terephthaldicarboxaldehyde (TPA), 2, 2-bithiophene-5, 5-dicarboxaldehyde (BCA), and 2, 5thiophenedicarboxaldehyde (TCA),8 which are based on the Schiff-base/imine formation as a result of C==N bonding (right panel in Fig. 1(a)). We choose these two COF families since they share very similar structure-tuning strategy of varying the composition of chain molecules while at the same time keeping the same node molecule, as shown in Fig. 1. The understanding of these new 2D COFs constitutes an open and unexplored area, and thus first-principles density functional theory (DFT) is used here to systematically determine their electronic and structural properties. A first-principles manybody Green’s function approach within the GW approximation is also used to yield a more accurate description of the electronic properties. In addition to free-standing 2D COFs, we also consider how the presence of a substrate affects the electronic properties, since a substrate is usually required for practical applications.9–11 Here, we consider a single-layer of hexagonal boron nitride (h-BN) as the substrate to support 2D COFs (Figs. 1(e) and 1(h)). Monolayer h-BN is structurally similar to graphene but with a wide band gap and is promising as a 2D dielectric substrate for nanoscale electronics.12–14

142, 184708-1

© 2015 AIP Publishing LLC

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FIG. 1. (a) Chemical structures of the two COF families considered in this work. (b)–(e) Optimized atomic structures of free-standing T-COF-1, -2, -3, and T-COF-4 deposited on a h-BN substrate. (f)–(h) DFT-optimized atomic structures of free-standing TAPP-TPA, -BCA, and TAPP-TCA deposited on a h-BN substrate. The TAPP-TPA and TAPP-TCA both feature the coexistence of square and parallelogram network configurations. For all the structures shown here, gray, pink, red, yellow, blue, and white spheres represent C, B, O, S, N, and H atoms, respectively.

II. METHODS A. Density functional theory

All the DFT calculations are performed using a planewave basis as implemented in the VASP package.15 The exchange-correlation (xc) functional is approximated within the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) description.16 Projector augmented wave (PAW) pseudopotentials are used.17,18 A cutoff energy of 400 eV is employed for the plane-wave basis set and the geometries are fully relaxed until all residual forces were less than 0.04 eV/Å. The relaxed in-plane lattice constants (x and y directions) range from 21 to 36 Å, and a vacuum region of at least 14 Å in the z direction is maintained to avoid spurious interactions between periodic images. For such large unit cells, 4 × 4 × 1 Γ-point-centered k-point sampling is sufficient to converge the total energy to within 1 meV/atom. For systems supported on the h-BN substrate, van der Waals (vdW) interactions19 are included within the DFT-D2 approach of Grimme.20 B. Many-body methods

For all free-standing COFs, the quasiparticle (QP) corrections to DFT Kohn-Sham eigenvalues are calculated within the GW0 approximation as implemented in VASP.21–24 The two important ingredients for GW calculations, the polarizability

and the GW self-energy, usually involve the summation over a very large number of empty conduction bands, which can be rather time-consuming and can lead to poor convergence with respect to the empty states. Some specific approaches to improve the convergence have been proposed,25,26 and an efficient method to optimize the computational procedure has also been used in the VASP implementation.21,22 The polarizability and self-energy matrices at each frequency can be obtained by the spectral representation of the involved matrices and their subsequent Hilbert or Kramers-Kronig transforms. Evaluation of the spectral function at a given frequency is rather efficient, since the only states that must be included in the summation are those that satisfy a certain criteria (more detail in Ref. 21). In other words, many empty states are not required in the summation. According to VASP,21,22,27 reasonable results can be usually obtained if the number of bands is chosen to be roughly twice or thrice that of occupied bands. In our calculations, the number of bands is thus chosen to be more than double the number of occupied bands. To finally obtain the selfenergy, a special non-equally spaced frequency grid is adopted in VASP in accordance with the spectral representation, and then the frequency integration is carried out using the finite element basis set.21 Since GW calculations are computationally expensive and free-standing COFs have large unit cells with about 100 atoms, QP energies are iterated only three times, the energy cutoff for response function is fixed at 60 eV, and the number of frequency points is set to 54. The last two variables

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are two key parameters that dominate the GW computational cost. Their further increase would lead to excessive memory requirement and is beyond current computational capability. Fortunately, according to VASP benchmarks and our previous works, the GW band gaps converge quickly with respect to the employed numerical parameters and the error in GW gaps is expected to be on the order of 0.2 eV.21,22,24,28 More importantly, the main objective is to reveal the importance of many-body effects in the significant corrections to DFT gaps, instead of obtaining absolutely precise GW gaps. We also note the recent work by Klimeš et al. which suggests that QP energies in VASP can potentially converge to the wrong values using the PAW method particularly for 3d states, and the error can be avoided by adopting norm-conserving partial waves.29 However, this work, here, only involves lower Z elements (H, B, C, N, O, S) with no 3d states, so the QP energies in this work are expected to have no such convergence issue.28 Finally, the systems made up of COFs supported on h-BN contain more than 300 atoms and are too large to allow a full GW treatment. In this case, we estimate the substrate polarization-induced band gap reductions with a semi-empirical image charge model (more details are provided below). This method has proven to be a satisfactory alternative approach to account for substrate polarization corrections to the adsorbate’s band gap, so long as the substrate/adsorbate interaction is weak.10,11,30–32 III. RESULTS AND DISCUSSIONS

Both free-standing and substrate-supported 2D COFs are considered. For the free-standing COFs, we focus on exploring effects of altering chain molecules on electronic properties. Here, T-COF-4 and TAPP-TCA are chosen to represent the TCOF-n and TAPP-x families, respectively. For the monolayer films supported on the h-BN substrate, we concentrate on comparing the electronic properties of systems deposited on h-BN substrate with those of their free-standing counterparts. The h-BN layer is slightly stretched to adopt the lattice constant of T-COF-4 and TAPP-TCA, with a maximum lattice mismatch of 2.3%. In order to differentiate these systems on the h-BN substrate from their free-standing counterparts, hereafter we will use the notation T-COF-4/h-BN and TAPPTCA/h-BN to refer to the systems deposited on h-BN.

J. Chem. Phys. 142, 184708 (2015)

A. Free standing COFs: Structural properties

Figure 1 shows that the T-COF-n and TAPP-x families share the same structure/property tuning strategy which consists in varying the composition of chain molecules while keeping the same node molecule (a three-connected HHTP node for T-COF-n family and a four-connected TAPP node for TAPP-x family), except that T-COF-2, as a second product of T-COF-1, arises from a different topology with boroxine rings formed by self-condensation of the boronic acids similar to those formed in COF-1.1 Here, T-COF-n must have threefold symmetry to yield the formation of a 2D crystal.7 The situation becomes more complicated with the TAPP-x family since their structural variations are proven to depend not only on the chemical composition of chain molecules but also on the linkage conformations of two adjacent -C==N- bonds connected to one chain molecule,8 as shown in Figures 2(a) and 2(b). Based on the characteristics of the linking unit, the two configurations are named square and parallelogram mesh, respectively.8 Among the three members in the TAPP-x family considered here, TAPP-BCA is the only one we find to consistently exhibit a square mesh topology while TAPPTPA and TAPP-TCA are observed to feature the coexistence of square and parallelogram configurations. In addition, while T-COF-n members are shown to be absolutely planar networks, TAPP-x members display non-negligible out-ofplane corrugation since the phenyl rings connected to the node rotate up to 60◦ after full relaxation and small rotations also happen to the chain molecules. This means TAPP-x members form rippled 2D sheets, with expected observable differences in friction properties and micro-mechanical behavior. For example, TAPP-TCA presents an out-of-plane corrugation (∆z) around 3.5 Å for the free-standing TAPP-TCA. This corrugation is slightly decreased to about 3.3 Å when the structure is deposited on a h-BN substrate.

B. Free standing COFs: Electronic properties

We compute the band structures of TAPP-TPA and TAPP-TCA for two different configurations to study how variations in the linkage conformation affect the electronic properties of TAPP-x systems (Figs. 2(c) and 2(d)). The electronic bands for the two configurations are very similar,

FIG. 2. (a) and (b) Co-existing square and parallelogram network configurations in TAPP-TPA due to different linkage conformations. TAPP-TCA features the similar coexistence of square and parallelogram mesh configurations. (c) and (d) Band structures for TAPPTPA and TAPP-TCA in the two different configurations, blue solid lines corresponding to the square configuration and the black dashed lines to the parallelogram one.

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especially around the Fermi energy. In other words, our calculations show that the variation of linkage conformation from square to parallelogram causes no significant difference in their electronic properties such as band gaps (∆Eg = 0.004 eV/−0.012 eV for TAPP-TPA/TCA). Since no obvious configuration-dependent behavior is observed, we will only discuss the square configuration and adopt it as the default configuration for all further results shown in this paper to illustrate properties of all TAPP-TPA and TAPP-TCA structures. Figures 3(a) and 3(b) show the band structures of freestanding T-COF-n and TAPP-x systems. Solid blue lines represent bands computed within DFT while dashed black lines represent GW bands. The corresponding detailed values of band gaps Eg are listed in Table I. As shown in Fig. 1, free-standing T-COF-1, -3, -4 structures share a number of similar features resulting from the use of the same nodal molecule HHTP and same topology. Their band gaps vary

J. Chem. Phys. 142, 184708 (2015)

only slightly (Fig. 3(a) and Table I), due to the change in chain molecule composition. Since DFT tends to underestimate band gaps of low-dimensional nanostructures,33,34 many-body GW calculations are also performed to obtain more accurate values of band gaps. As expected, GW yields systematically larger band gaps compared to DFT (Table I). T-COF-2 is a second product of T-COF-1 with different topologies and thus the two share very close band gap values at both DFT and GW levels, as shown in Table I. Turning to the free-standing TAPP-x family, various chain molecules are used from TPA to BCA to TCA, with the same nodal molecule TAPP. They also share some similar bands at energy values around EF (Fig. 3(b)). They also have comparable band gaps varying slightly from 1.23 eV to 1.32 eV while GW band gap values vary between 3.70 eV and 4.03 eV, respectively (Table I). The shape of the GW bands remains essentially the same as those computed within DFT, albeit with a systematic rescaling.

FIG. 3. (a) Electronic band structures of free-standing T-COF-1, -2, -3, -4 and T-COF-4 deposited on the h-BN substrate. Solid blue (dashed black) lines represent bands at DFT (GW) level. The red dots superimposed on the bands correspond to projected bands due to TCOF-4. (b) Electronic band structures of free-standing TAPP-TPA, -BCA, -TCA and TAPP-TCA deposited on the h-BN substrate. Solid blue (dashed black) lines represent bands at DFT (GW) level and red circles superimposed on the bands correspond to projected bands due to TAPP-TCA.

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TABLE I. DFT and GW band gaps Eg of the T-COF-n and TAPP-x systems shown in Fig. 1. The band gap reductions of T-COF-4 and TAPP-TCA due to the h-BN substrate at both DFT and GW levels are given between parentheses in the last column. Note that the band gap reductions by the substrate at the GW level are computed using the image charge model.

(eV) EDFT g EGW g (eV)

(eV) EDFT g EGW g (eV)

T-COF-1

T-COF-2

T-COF-3

T-COF-4

T-COF-4/h-BN

2.42 6.30

2.41 6.30

2.16 5.47

2.32 6.08

2.25 (−0.07) 5.01 (−1.07)

TAPP-TPA

TAPP-BCA

TAPP-TCA

TAPP-TCA/h-BN

1.32 4.03

1.25 3.99

1.23 3.70

1.07 (−0.16) 2.52 (−1.18)

C. COFs on h-BN substrate

Up to now, we have discussed the band structures of TCOF-n and TAPP-x systems in their free-standing form. The drawback of this form is that they can only be simulated in theoretical studies but not be used in most experiments or applications since a substrate is usually needed for applications involving 2D network films, unless a suspended sample is specifically prepared. For this reason, we constructed two atomistic models using a h-BN substrate: T-COF-4/h-BN and TAPP-TCA/h-BN. The DFT band structures of these two systems are shown in Figs. 3(a) and 3(b). The red dots superimposed on the bands correspond to projected states due to T-COF-4 and TAPP-TCA, respectively. From the clear coincidence of solid lines and red dots at energy ranges around EF, we conclude that the electronic properties of T-COF-4/h-BN and TAPP-TCA/h-BN are dominated by contributions from T-COF-4 and TAPP-TCA; the existence of h-BN substrate, which is a wide-gap insulator, is to cause only a very weak perturbation to the bands around EF, due to the limited chemical hybridization between the two subsystems. Compared to free-standing T-COF-4, T-COF-4/hBN (Fig. 3(a)) has a band gap reduced by ∆Eg = −0.07 eV (see Table I). Similar band gap decrease is found for the TAPPTCA/h-BN system (Fig. 3(b)) with ∆Eg = −0.16 eV. These substrate-induced band gap reductions for adsorbates have been extensively studied in the literature.10,11,30,31,35–38 Polarization effects occur at the adsorbatesubstrate interface due to image charge formation for weak coupling or dynamic charge transfer for strong coupling.36 Such many-body polarization effects are expected to screen the Coulomb potential experienced by the adsorbate to effectively reduce the band gap compared to that of the corresponding isolated adsorbate.10,11,30 However, since this process is inherently an expression of many-body effects, most conventional DFT approximations such as local-density approximation (LDA) or GGA cannot properly describe the substrate polarization effects and hence usually yields too small or no band gap reductions.11,35 Full GW calculations have been found to yield more accurate values compared to experiments,10,35,37 but these calculations are very computationally demanding, especially for large systems such as those considered here. A unit cell of T-COF-4/h-BN and TAPP-TCA/h-BN contains more than 300 atoms, and thus it is not practical to use a full GW approach to accurately evaluate the band gap reductions in this case. However, we showed in a previous study that a semi-

empirical image charge model constitutes a computationally affordable alternative and yields excellent estimation of band gap reductions of graphene nanoribbons on a gold substrate.11 This model has also been successfully used to evaluate band gap reductions in graphone on a h-BN substrate.10 We will therefore employ this approach for the weakly coupled T-COF4/h-BN and TAPP-TCA/h-BN systems. In the image charge model,10,11,30 the QP energy of a substrate-supported adsorbate is given by E QP j;supported QP = E QP + ∆P , where E is the QP GW energy of state j j;free j;free | j⟩ of the free-standing film (T-COF-4 or TAPP-TCA here), and ∆P j is a correction term introduced to account for substrate polarization. Although conventional DFT accurately models the short-range surface potential, it fails to depict the potential far from the surface. In the long-range limit, effective potential due to the surface is better described by an image potential, Vim.39 Hence, ∆P j can be approximated as dx dy dz[Vim(z) − Vxch-BN(x, y, z)]ρ j (x, y, z) ∆P j = z >z 1 ≈ dz[Vim(z) − V¯xch-BN(z)] dxd y ρ j (x, y, z) z >z 1 = dz[Vim(z) − V¯xch-BN(z)] ρ¯ j (z), (1) z >z 1

where Vim(z) = −e2Q/4|z − z0| is the image potential; of the h-BN surface; Vxch-BN(x, y, z) is the DFT xc potential V¯xch-BN(z) = dxd yVxch-BN(x, y, z)/ dxd y is the corresponding plane-averaged xc potential; ρ j (x, y, z) is the charge density for state | j⟩ of the free-standing adsorbate; ρ¯ j (z) = dxd y ρ j (x, y, z) is the corresponding charge density integrated over the x y plane. z0 is the image plane position, and z1 is where the transition from short-range to long-range limit takes place and Vim(z) starts to deviate from V¯xch-BN(z), as shown in Figure 4(a).39 Note that Eq. (1) is valid since Vxch-BN(x, y, z) has much weaker dependence on in-plane coordinates x and y than out-of-plane coordinate z (i.e., the substrate is homogeneous in the x y plane). Thus, the energy correction term ∆P j does not depend on x and y and consequently does not depend on the size of the system, which is periodic in the x y plane. To further understand the physical picture of the image charge model, we consider a point charge q above the substrate surface for illustration. The induced image charge below the surface is q ′ = −q(ε − 1)/(ε + 1), where ε is the relative dielectric constant of the substrate. With the image

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FIG. 4. (a) Plane-averaged DFT xc potential of a clean h-BN surface and its image potential. The origin is set at the h-BN surface, and z 0 and z 1 are fitted by setting the intersection of the two curves at z 1. (b) Plane-integrated charge densities for the CBM and the VBM of the free-standing TAPP-TCA.

plane position at z0, the image potential is given by Vim(z) =

ε − 1 q2 q2 qq ′ =− = −Q , 4|z − z0| ε + 1 4|z − z0| 4|z − z0|

(2)

which describes the electrostatic interaction between the charge above the surface and the polarization image charge below the surface and also constitutes the basic term of the GW self-energy from the substrate (Q is the ratio given by Q = (ε − 1)/(ε + 1)).35,39–41 Therefore, as a simplified approximation, Vim(z) is the GW self-energy term from the substrate and Eq. (1) is essentially the GW correction term due to the substrate. According to Eq. (2), the larger ε is, the stronger the interaction (or substrate polarization effect) is. For a metal substrate (essentially ε = ∞), q ′ = −q and the interaction or polarization is the strongest. For an anisotropic material such as √ h-BN, the dielectric constant is ε = ε ∥ ε ⊥, where ε ∥ and ε ⊥ are in-plane and out-of-plane dielectric constants, respectively.10 From the literature, the experimental static dielectric constant ε ≈ 5.89 is used here and thus Q = 0.71.42,43 To calculate ∆P j , simulations must be performed separately for the two subsystems, using atomic positions determined in the optimized T-COF-4/h-BN or TAPP-TCA/h-BN systems.11 The isolated h-BN surface is first treated to obtain the xc potential Vxch-BN(x, y, z) (Fig. 4(a)); the free-standing T-COF-4 or TAPP-TCA is then studied to obtain the DFT charge densities at the conduction band minimum (CBM) and the valence band maximum (VBM) (Fig. 4(b)). Finally, the QP band gap of the substrate-supported T-COF-4 or TAPPQP QP TCA can be obtained using Egap;supported = Egap;free + ∆PCBM − ∆PVBM, where ∆PCBM and ∆PVBM are corrections to the CBM and VBM, respectively. For the TAPP-TCA/h-BN system, we determine the intersection of the xc potential and image potential of the h-BN substrate to find z0 and z1 to be 1.28 Å and 2.58 Å, respectively (Fig. 4(a)). Using Eq. (1), for TAPP-TCA supported on hBN, ∆PCBM = −0.60 eV and ∆PVBM = 0.58 eV. Note that ∆PCBM and ∆PVBM are almost equal but with opposite signs, since ρ j (x, y, z) is positive for conduction bands and negative for valence bands, as shown in Fig. 4(b).41 This energy

level renormalization due to substrate polarization leads to a band gap reduction of TAPP-TCA as −1.18 eV. As a result, the GW band gap of TAPP-TCA adsorbed on h-BN is reduced to 2.52 eV compared to the isolated counterpart’s GW band gap 3.70 eV (Table I). Similarly, the band gap reduction ∆PCBM − ∆PVBM is calculated as −1.07 eV for the T-COF-4/h-BN system, and thus the GW band gap of TCOF-4 adsorbed on h-BN is reduced to 5.01 eV, as shown in Table I. In short, band gaps of both TAPP-TCA and TCOF-4 are reduced by about 1.0 eV due to the h-BN substrate polarization, consistent with previous works.10,37 As discussed above, the corresponding band gap reductions estimated by conventional DFT are only around −0.1 eV (Table I), clearly indicating that DFT fails to quantitatively capture band gap reductions by substrate polarization while the semi-empirical image charge model offers a satisfactory and computationally less demanding (compared to a full GW treatment) tool to estimate such band gap reductions.11 Finally, we note that while GW accounts for significant increase in band gaps for the isolated structures, the many-body substrate polarization effects account for a large band gap reduction. It may so appear that these two effects could quantitatively cancel each other, leading to the tempting claim that DFT can yield excellent agreement with experimentally obtained band gaps in deposited structures. However, it is important to note that it is only the fortuitous cancellation of two unrelated effects that can yield such agreement, especially in metallic substrates where the effect is largest.38 We note that in the present study, the substrate polarization effect is substantial but still too small to cancel the many-body effects in the adsorbate itself. To establish that the substrate-induced band gap reductions are not limited to h-BN, we also chose graphene as the substrate and studied the adsorption of TAPP-TCA on it. Graphene has a very similar lattice and structure to hBN but with significantly different electronic properties.44 The intrinsic semi-metallic behavior of graphene due to the absence of a gap makes it a good alternative substrate to study the adsorbate’s band renormalization, compared to insulating h-BN. Our calculations found that the equilibrium separation between TAPP-TCA and graphene is very close to that between TAPP-TCA and h-BN, owing to similar vdW interactions. According to Fig. 5(a), the xc potentials of hBN and graphene (black and blue lines) are very similar as well. Therefore, when the dielectric constant ε of graphene is also chosen as the h-BN experimental value 5.89, the image plane position z0 and intersection position z1 are practically the same between h-BN and graphene substrates, thereby the same image potential. Consequently, the band gap reduction of TAPP-TCA adsorbed on graphene using Eq. (1) is calculated to be 1.31 eV, similar to but slightly larger than that of TAPP-TCA adsorbed on h-BN (1.18 eV). The gap reduction difference can be largely attributed to the minor difference of the substrate xc potentials far from the surface (see the inset in Fig. 5(a)). Note that ε of graphene is set as the h-BN experimental value 5.89 for comparison purpose, while its ε has been reported to vary from 2 to 16 by different experiments.45,46 For h-BN, different experiments have also found its ε changing from 3 to 6.42,43,47 The substrate ε can be sensitive to a variety of

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FIG. 5. (a) Plane-averaged DFT xc potentials of h-BN (black) and graphene (blue) monolayers and the image potential (red). For comparison purpose, both h-BN and graphene surfaces are positioned to 0 Å, and their dielectric constants ε are set as 5.89 (the experimental value of h-BN). The inset highlights the difference between xc potentials of h-BN and graphene near 0 eV. (b) The band gap reductions of TAPP-TCA due to h-BN (black) and graphene (blue) substrate as a function of the substrate’s dielectric constants ε. The gap reductions at the upper limit of ε = ∞ are also computed for comparison. Note that the negative signs represent reductions of band gaps.

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factors such as its thickness, the presence of an external electric field, the surrounding medium, etc. To investigate the influence of ε on the gap reduction of the adsorbate, we considered ε of h-BN and graphene in the range of 2-16 and computed the corresponding gap reduction of TAPP-TCA. As indicated in Eqs. (1) and (2), with increasing ε, Vim (or substrate GW self-energy term) further deviates from the substrate DFT xc potential, and z0 and z1 become smaller. As a result, the gap reduction is increased with increasing ε as shown in Fig. 5(b), which can be fundamentally attributed to the stronger substrate polarization effect. More specifically, by increasing ε from 2 to 16 (towards ∞), the gap reduction by h-BN increases from 0.26 to 1.55 (reaching the limit of 1.82) eV, while the gap reduction by graphene increases from 0.45 to 1.71 (reaching the limit of 1.96) eV. The gap reductions by h-BN (black line in Fig. 5(b)) are systematically slightly smaller than that by graphene (blue line in Fig. 5(b)), as already explained above. Clearly, the gap reduction of the adsorbate shows strong dependence on the substrate ε, and the choice of ε is hence very important. Compared to the previous value 5.89, if the lower limit of experimental value is chosen for h-BN (i.e., ε = 3),47 the gap reduction of TAPP-TCA is decreased to 0.72 eV; if the average experimental value is chosen for graphene (i.e., ε = 9),45,46 the gap reduction of TAPP-TCA is increased to 1.51 eV. Numerous previous works based on full GW calculations have found similar or larger band gap reductions of adsorbates by the vdW-interacting substrates.10,35,37,40,48 For example, for the polymer poly(para-phenylene) physisorbed on graphene, a gap reduction of 1.25 eV is present.40 For benzene adsorbed on graphite, a more than 3 eV gap reduction was predicted.35 For graphone supported on graphite, it has a

FIG. 6. Spatial distributions of partial charge densities for CBM (yellow) and VBM (green) of free-standing T-COF-n family, TAPP-x family, and the two example systems on the h-BN substrate.

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gap reduction of 1.32 eV, while its gap reduction by h-BN is 0.86 eV.48 The gap reductions estimated by our semi-classical image charge model here follow into the right trend compared to the numbers obtained using full GW calculations for similar systems in the literature. Therefore, it is expected that the image charge model holds reasonable accuracy and the band gap reduction of TAPP-TCA should be around 1 eV by the hBN substrate while higher than 1 eV by the graphene substrate. D. Spatial charge densities

We conclude this study with an examination of the spatial distribution of charges. The partial charge densities for conduction band minimum and valence band maximum of free-standing and deposited systems are shown in Fig. 6. For both COF families, CBM’s electronic densities (yellow) are seen to be primarily localized along the chain molecules while VBM’s electronic densities (green) are mainly localized at the node molecules, consistent with previous reports.49,50 Such separation of CBM and VBM could be important for the applications of photovoltaic devices, since it indicates that photo-generated electrons and holes could be spatially separated.50 In addition, due to the spatial localization of charge densities of CBM and VBM at the chain and node molecules, respectively, the band width dispersions of CBM and VBM are small, as shown in Fig. 3. The small band width dispersions for many bands in Fig. 3 are probably also due to spatial localization of their charge densities. For TCOF-4/h-BN and TAPP-TCA/h-BN, the distribution of partial charge densities for CBM and VBM is quasi-unaffected by the substrate compared to their free-standing counterparts, thereby accounting for the minor differences noted between their band structures. In addition, the fact that both CBM and VBM charge densities of TAPP-TCA reach zero rapidly as they approach the h-BN surface indicates weak coupling at the interface, thereby further confirming the required condition for the applicability of the image charge model used in Sec. III C.

IV. SUMMARY AND CONCLUSIONS

First-principles calculations have been carried out to study the electronic and structural properties of two kinds of monolayer 2D COFs. For free-standing 2D COFs, we investigated how changes of chain molecules affect their electronic properties at both DFT and GW levels. DFT tends to underestimate band gaps compared to GW. Computed band gaps of T-COF-1, -2, -3, -4 systems indicate that they are wide band-gap semiconductors and are not significantly affected by changing chain molecules. TAPP-TPA,-BCA,-TCA systems, despite not being exactly planar structures, are found to be potential candidates for photovoltaics and energy-storage materials. We explored how the presence of a h-BN substrate causes band gap reductions due to the substrate polarization effects for two chosen systems. For the h-BN substrate with the dielectric constant ε = 5.89, the band gap reductions of T-COF-4 and TAPP-TCA are estimated to be around 1.07 eV and 1.18 eV, respectively. Similar band gap reduction of TAPPTCA has been found by a graphene substrate with the same ε.

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Moreover, our calculations found that the gap reductions are very sensitive to the substrate ε, which is a measure of the strength of the substrate polarization effect. By choosing the lower limit of experimental value for h-BN (ε = 3), the gap reduction of TAPP-TCA is decreased to 0.72 eV; by choosing the average experimental value for graphene (ε = 9), its gap reduction is increased to 1.51 eV. These results suggest that the choice of substrate constitutes an attractive band gap engineering means for wide-gap COFs. ACKNOWLEDGMENTS

L.L. is supported by New York State under NYSTAR Contract No. C080117. V.M. is supported by the Office of Naval Research. All calculations were performed in the Center for Computational Innovation at Rensselaer Polytechnic Institute. 1A.

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