Off-Plane Dielectric Screening of Few-Layer Graphdiyne and Its

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Off-Plane Dielectric Screening of Few-Layer Graphdiyne and Its Family Jahyun Koo,† Li Yang,‡ and Hoonkyung Lee*,† †

Department of Physics, Konkuk University, Seoul 05029, Korea Department of Physics, Washington UniversitySt. Louis, St. Louis, Missouri 63136, United States

‡

S Supporting Information *

ABSTRACT: We performed first-principles calculations on few-layer graphdiyne (GDY) and its family, sp−sp2 hybrid carbon atomic layers, for an off-plane, static dielectric screening. The vertical dielectric constants of semiconducting GDY structures are finite and independent of the thickness. However, unlike the widely accepted wisdom that the static metallic screening is infinite, those of metallic GDY structures are finite and dependent on their thickness. Furthermore, the vertical dielectric screening can be tuned by varying the interlayer distance. We also studied the dielectric properties of heterostructures of GDY/its family; the vertical dielectric constant has an equivalent value from the two distinct values of the two distinct monostructures. The dielectric screening behaviors are well described by the uniform dielectric slab model. In addition, the band gaps can be widely tuned from 0 to 0.8 eV, by varying the thickness and electric field. Our results provide a method for engineering the dielectric constant and band gap of GDY and its family for applications of supercapacitors and nanodevices. KEYWORDS: graphdiyne, sp−sp2 carbon allotropes, off-plane dielectric constant, van der Waals materials, density functional theory, linear response theory

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INTRODUCTION Recent advances in various exfoliation techniques have produced van der Waals (vdW) nanosheets of various twodimensional (2D) materials1 including graphene,2,3 boron nitride,4,5 metal dichalcogenides,6−8 and black phosphorus (BP).9−11 Owing to their planar geometry, 2D structures have been naturally implemented into field-effect transistor structures,12−14 and they have shown intriguing properties such as high performance and integration because of their small size of a few nanometers.15 Since their capabilities were demonstrated for new possibilities and applications in 2D nanodevices,16−19 vdW materials have been an actively pursued application in nanodevices. The dielectric screening of the materials can determine the physical properties of materials such as the band gap, Fermi level, doping density, manyelectron effects (i.e., excitons and trions), and carrier mobility.20,21 Thus, the gate switching is dependent on the dielectric screening of the materials, which results in changes in the performance of the devices. Therefore, an understanding of the dielectric screening under a gate field is necessary for predicting the physical properties of materials and designing future ultrathin devices. The vertical (off-plane) dielectric screening of vdW semiconductors is widely believed to be sensitive to the sample thickness because of the obvious quantum confinement effect and the corresponding variations in the band gaps.22,23 © XXXX American Chemical Society

However, recently, experiments on few-layer BPs and MoS2 showed that the dielectric constants of 2D semiconductors are independent of the thickness.24,25 Moreover, the first-principles calculation based on the linear response theory demonstrated that the dielectric constant of few-layer vdW semiconductors is independent of the thickness within the weak-field limit, and this result is applied for a wide range of vdW materials such as MoS2, BP, and SnS.26 Therefore, if edge effects are neglected, the vertical dielectric screening of few-layered vdW materials is approximately a constant value, which is comparable with the bulk value, and the quantum confinement effect will not explicitly influence the vertical dielectric screening with the thickness. On the other hand, the tight-binding (TB) model with the constant vertical dielectric screening predicted the band gap results of BP and MoS2, which were highly consistent with not only the density functional theory (DFT) result but also the GW results,25,26 in which many-electron effects are included. Therefore, the first-principles calculations based on Special Issue: Graphdiyne Materials: Preparation, Structure, and Function Received: January 17, 2018 Accepted: February 23, 2018

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DOI: 10.1021/acsami.8b00877 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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Figure 1. (a) Atomic structure of N-GDY. The red round circle indicates the acetylene linkage, and N denotes the number of acetylene linkages (−CC−). (b) Calculated energy band gaps of N-GDY as N varies. The calculated band structures from DFT for monolayers (c) 2-GDY and (d) 3-GDY.

can be widely tuned from 0 to 0.8 eV, by varying the thickness and electric field. Our findings provide a way for engineering the dielectric constant and band gap of GDY and its family.

the linear response theory are efficient in predicting the dielectric properties of new vdW materials. Graphdiyne (GDY), which is a carbon atomic layer structure consisting of sp carbon chains (−CC−CC−) and sp2 carbon atoms, has recently been synthesized in the form of bulk powders, large-area films (∼4 cm2), and flakes27−29 using a selfcatalyzed vapor−liquid−solid growth process.30 Recently, GDY and its family, which has the same symmetry as that of GDY but longer sp carbon chains, are semiconductors with a band gap of ∼0.5 eV, and they present the possibility of application for new 2D device materials.31−33 An understanding of the dielectric screening properties of GDY and its family is necessary for predicting the performance of devices and interesting physical phenomena such as excitons. In addition, because GDY and its family have an exotic geometrical structure such as a porous texture, they have theoretically been predicted for use as energy storage materials in supercapacitor electrodes and lithium- or sodium-ion battery anodes because of its large surface area.34−37 Actually, recent experiments confirmed that GDY could be used for energy storage applications.38−40 In order to predict how GDY and its family respond under a gate field and to investigate whether the vertical dielectric screening is dependent on their thickness, in this study we carried out first-principles calculations based on the DFT on few-layer GDY and its family. We found that the vertical dielectric constants of semiconducting materials are independent of the thickness, while those of metallic materials are dependent on the thickness. Furthermore, the dielectric constants decrease as the interlayer distance increases. We also studied the dielectric properties of heterostructures of GDY/its family; the vertical dielectric constant has a value equivalent with two distinct values in two distinct monostructures. These dielectric screening behaviors are described by the uniform dielectric slab model. In addition, the band gaps

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COMPUTATIONAL DETAILS

All of our calculations were performed using the first-principles method based on DFT.41 We developed and implemented our calculations in the Vienna Ab initio Simulation Package code with a projector-augmented-wave method.42 For the exchange-correlation energy functional, we used generalized gradient approximation in the Perdew−Bruke−Enzerhof scheme.43 The kinetic energy cutoff was taken as 500 eV. To describe interlayer stacking energy such as the vdW interaction accurately, we employed the PBE-D2 method.44 The vdW interaction was empirically included using Grimme’s approach. According to our previous study,26 the calculated vertical dielectric constants of few-layer vdW semiconductors obtained by PBE-D2 were in good agreement with the experimental values. Therefore, we used the PBE-D2 method for the dielectric screening of few-layer GDY and its family. The first Brillouin zone integration was carried out using the Monkhorst−Pack scheme.45 The 9 × 9 × 1 k-point sampling was conducted for a 1 × 1 unit cell of each material. Geometry optimization of the structure was carried out until the Hellmann− Feynman force acting on each atom became smaller than 0.01 eV/Å. The effect of the electric field was evaluated using a sawtooth potential, while taking the dipole correction into consideration. Because our calculations on the vertical (static) dielectric screening were carried out using the periodic boundary conditions, the in-plane size of GDY is infinite. Thus, our calculations do not consider the effects of the edges of GDYs on the vertical dielectric screening.

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RESULTS AND DISCUSSION We considered GDY and its family, consisting of hexagons and sp carbon chains, as shown in Figure 1a. We define them by NGDY, where N denotes the number of acetylenic linkages (which is a finite positive integer), and, therefore, the sp carbon chain is expressed as (−CC−)N. For instance, 0-GDY, 2GDY, and ∞-GDY indicate graphene, GDY, and carbyne, B


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Figure 2. (a) Structure of the stacked bilayer 1-GDY under an external electric field that is vertical to the plane of GDY. (b) Difference in the electric potential between when the vertical external electric field is 0.1 V/Å and when the field is 0.0 V/Å, where the shaded lines indicate the position of the bilayer GDY and the inter electric field is calculated from the slope of the potential with linear approximation. (c) Potential difference according to the strength of the external electric field. (d) Calculated dielectric constants of the bilayer from the ratio of the external electric field to the internal electric field according to the number of acetylenic linkages N.

respectively. We first performed calculations for monolayer NGDY systems, as N was varied from 0 to ∞, to figure out their intrinsic electronic properties. Our calculations showed that the energy band gaps of N-GDY increase from 0.46 up to ∼0.8 eV (Figure 1b). The band gaps of 1-GDY and 2-GDY are in good agreement with the previous studies.46−48 Thus, monolayer GDY and its family are semiconducting. Unlike the consistent tendency by which the band gap becomes larger when N is larger, there is a difference in the electronic structure of GDY, depending on whether N is odd or even. It has a direct band gap regardless of whether N is odd or even; however, when N is odd, the position of the band gap on the symmetry line is located at the M point, whereas when N is even, it exists at the Γ point (Figure 1c,d). Next, we studied how N-GDY responds with an increase in N under an electric field that is vertical with respect to the plane of GDY. Because, for monolayer GDYs, the effects of the electric field are negligible because the electrons are strongly bounded in the plane, our discussion about the dielectric response of N-GDY starts from bilayer N-GDY. We considered the bilayer 1-GDY under an electric field, as shown in Figure 2a. To evaluate the dielectric constant of the bilayer 1-GDY, we obtained the difference in the potential with or without the gated field: ΔV(z) = Vgated(z) − Vungated(z). According to the linear response theory,25 the dielectric constant is given by the ratio of the external electric field to the internal electric field between the bilayers, that is

εr = Eext / E in

shown in Figure 2b. For the bilayer 1-GDY, the dielectric constant was calculated as 3.3 by using eq 1. We also checked the dependence of the dielectric constants on the strength of the electric field (Figure 2c). It turned out that the dielectric constant does not depend on the strength of the electric field if the electric field is smaller than ∼1 V/nm, until the band gaps are zero (Figure 2d). We refer to this regime of constant dielectric constants as the weak-field limit. Hereinafter, we only consider the vertical dielectric screening of materials at the weak-field limit. Figure 2d concludes the dielectric constant of N-GDY, calculated from the bilayer. The dielectric constant decreases as N increases, as shown in Figure 2d. We also carried out calculations on few-layer 1-GDY as the number of the layers increases up to 5 (Figure 3a), in order to investigate the layer-number dependence of the dielectric screening of few-layer GDYs. As shown in Figure 3b, the difference in the potential was calculated as the number of layers was varied. Importantly, the slope of the internal potential is the same regardless of the number of layers, which means that the average internal electric field is independent of the number of layers. This is consistent with the results of other vdW materials.26 From this result, we can conclude that the vertical dielectric constant of vdW materials does not depend on the number of layers and the strength of the electric field in a weak gate field regime. To identify the origin of the exotic feature of dielectric screening that does not depend on the number of layers or the strength of the electric field, we investigated the microscopic charge distribution according to the electric field. We obtained the redistribution of charges from the difference in the charge density with or without the electric field (Figure 3c,d). The negative and positive charges are accumulated below and above, respectively, with respect to each layer. The charge density is largest in the top and bottom layers (with different charge

(1)

where Eext and Ein denote the external and internal electric fields, respectively. The dielectric constant εr was calculated from the average of the internal electric field, i.e., ⟨Ein⟩, where the linear approximation for the potential between the bilayers was used to calculate the average of the internal electric field, as C


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Figure 3. (a) Few-layer 1-GDY under a vertical external electric field, where z indicates the axis that is vertical with respect to the plane of GDY. (b) Calculated potential difference with or without the electric field as the number of layers varies from 2 to 5. (c) Calculated difference in the charge density with or without the electric field when the number of layers is 5. (d) Charges projected along the z direction. (e) Schematic of a dielectric inserted into an electrode with a gate field, where σ and σind denote the surface charge densities of the electrode and dielectric, respectively. (f) Projected charges as the number of layers varies.

kσ and σind → kσind, where k is a constant (the electric field is proportional to the charge density). From this model, the surface charge density is determined according to the property of the materials, and therefore the density determines the values of the dielectric constants of few-layer vdW materials. We believe that the dependence of the vertical dielectric screening of N-GDY on N can be explained by the dependence of the induced charge density, σind, at the top or bottom layers on N. The decrease in the dielectric constants of few-layer N-GDY as N increases (Figure 2d) is ascribed to the fact that σind becomes lower when the band gaps are increased (Figure 1b). Therefore, the uniform dielectric slab model well describes the vertical dielectric screening of few-layer vdW materials and can be utilized for predicting the properties of the dielectric screening of vdW materials. Now, we consider the effects of stacking of the layers on the vertical dielectric screening of vdW materials. We defined three and six stacked structures for the bilayers 1-GDY and 2-GDY, respectively (see Figures S1 and S2). The vertical dielectric constant for the bilayers 1-GDY and 2-GDY with different stacking was calculated from the internal electric field (Figure 4) as the interlayer distance was varied. An important point is

signs), and the negative and positive charges inside are symmetric (Figure 3c,d). According to Gauss’s law for electrostatics, ∮ D⃗ da ⃗ = Q enclose , the charge inside is canceled out and the net charge in the bottom or top layer contributes to the enclosed charges, Qenclose, in the Gaussian surface. Here, we made a uniform dielectric slab model using Gauss’s law to explain the origin of the feature of the dielectric constants (Figure 3e). Therefore, the electric field induced by the external field is approximately given by E⃗ ind = −σind/ε0ẑ, where σind denotes the induced surface charge density and ẑ denotes the unit vector along the z axis when the charge densities in the top and bottom layers are uniform. Thus, the net electric field inside is E⃗ in = E⃗ ext + E⃗ ind (E⃗ ext = σ/ε0ẑ, E⃗ in = σ/εrε0ẑ), leading to σ/ε0εr = σ/ε0 − σind/ε0, that is εr = σ /(σ − σind)

(2)

Because the induced surface charge density (σind) in the top or bottom layers is constant irrespective of the number of layers (Figure 3f), the dielectric constant is independent of the number of layers. Furthermore, we can interpret that the dielectric constant is independent of the electric field when σ → D


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Figure 4. Calculated dielectric constants of the bilayers (a) 1-GDY and (b) 2-GDY with different stackings as the interlayer distance varies.

that the values of the vertical dielectric constants are the same regardless of the stacking modes, except for the AA stacking, and the dielectric constants decreased as the interlayer distance increased. This tendency is consistent with the previous study.25 In addition, the equilibrium interlayer distance (∼3.2 Å for 1-GDY and ∼3.7 Å for 2-GDY) for the AA-stacked bilayer is longer than that for the others (∼3.3 Å for 1-GDY and ∼3.3 Å for 2-GDY). The energy of the AA-stacked bilayer is higher than that of the others by a few millielectronvolts per unit cell, while the other is energy degenerate. Furthermore, we also performed calculations for a few-layer GDYs with a variety of stackings as the number of layers increases. We found that AA-, AAA-, and AAA-stacked structures are metallic, while the other stacked ones are semiconducting with almost the same band gaps. In addition, the dependence of the band gaps on N is negligible (within the order of a few millielectronvolts). Therefore, their off-plane dielectric screenings are almost the same. This result is consistent with that of a recent study on few-layer vdW semiconductors such MoS2 and BP.26 In general, the vertical dielectric constants of the AA-stacked 1-GDY and 2-GDY bilayers are larger than those of the others. This is ascribed to the fact that the AA-stacked GDY bilayer is metallic (Figure S3), while the others are semiconducting (Figure S4). We believe that the strong hybridization of pz orbitals between the interlayers in AA-stacked GDY makes it metallic, leading to a larger dielectric constant. An attractive feature is that, although they are metallic, the dielectric constant of AA-stacked 1-GDY is a finite value, 4.0, 7.0, 16.6, and 26.6 when the number of layers is 2, 3, 4, and 5, respectively, as shown in Figure 5. This screening property of vdW conductors is distinct from that of conventional conductors with infinite dielectric constant values. The fundamental reason for the unusual screening property of few-layer vdW metals could be associated with the very thin thickness, which is less than the screening length of the metals. Therefore, we believe that this is a unique screening property of ultrathin films. The increase of the dielectric screening according to the number of layers is caused by the increase of the density of states at the Fermi level (Figure S3). Importantly, this is in sharp contrast to the semiconducting GDY and few-layer vdW materials, in which the dielectric constant does not depend on the number of layers; the electrons that screen the field remain constant with and increase in the number of layers. Therefore, the dielectric constants of metallic few-layer vdW materials depend on the thickness, whereas the dielectric constants of semiconducting ones do not depend on the thickness. Using our uniform dielectric model εr = σ/(σ − σind), we can explain this result on AA-stacked GDY; a higher number of electrons in the Fermi level causes an increase in the induced surface

Figure 5. Calculated potential difference for AA-stacked 1-GDY with or without the external electric field with respect to z, when the number of layers varies: (a) 2; (b) 3; (c) 4; (d) 5. The inset shows the dielectric constants calculated from the ratio of the external to internal slopes.

density, σind, resulting in larger dielectric constants as the number of layers increases. Therefore, we expect that the dielectric constant of the multilayer tends to infinity because the density of states at the Fermi level is continuously increased as the number of layers increases. This interpretation can also explain the increase in the dielectric constant of graphene as the number of layers increases49 and that of electron- or holedoped few-layer BP as the electron or hole concentration increases.50 Next, we expanded our dielectric constant for few-layerstacked GDY. We applied our dielectric constant from the bilayer result for the prediction of the band gap from the electric field and number of layers. To obtain the band gap as a function of the number of layers and electric field, we used a simple TB Hamiltonian. The effective Hamiltonian of the bandedge energies of gated n-layer vdW structures can be written as ⎛E t1 t2 ⎜ ⎜ t1 E + Δ ⋯ t1 ⎜ t1 E + 2Δ ⎜ t2 ⎜ ⋮ ⋱ ⎜ ⎜t tn − 2 ⋯ E + ⎝ n tn − 1

⎞ ⎟ ⎟ tn − 1 ⎟ tn − 2 ⎟ ⎟ ⋮ ⎟ (n − 1)Δ ⎟⎠ nxn tn

(3)

The applied field will generate a potential drop Δ = Eextd/εr. The off-diagonal elements t1 and t2 represent the interlayer couplings of the nearest neighbor and next-nearest neighbor, respectively (Figure 6a). In Figure 6b, we have plotted the band gaps with the DFT result as dots and the TB results as lines. The fitted parameters of the TB Hamiltonian for the pentalayer 1-GDY are presented in Table S1. These two independent calculations are highly consistent with each other. The band gap of the other N-GDY systems can be explained by the vertical dielectric constant and interlayer hopping parameters. To investigate the dielectric screening of heterostructured vdW materials, we suggest a heterostructure with the N-GDY system, consisting of a 1-GDY 2 × 2 cell and a 4-GDY 1 × 1 cell. The stacked heterostructure is shown in Figure 7a, in which the interlayer distance is 3.36 Å, which is comparable to that of monostructured N-GDYs. We found that there are two E



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CONCLUSION In conclusion, based on first-principles calculations with linear response theory, we present the vertical dielectric screening of GDY and its family. By investigating few-layer vdW GDYs, we conclude the following: (1) Unlike bulk metals with infinite screening, the screening of metallic ones is finite and dependent on the thickness, while that of semiconducting ones are independent of the thickness. (2) The dielectric constants of the heterolayer structures are approximately equivalent with the values of the monolayers. These findings can be utilized for understanding the vertical dielectric screening of the recently discovered few-layer vdW semiconductors such as transitionmetal dichalcogenides and tin selenide. From an application perspective, our findings can be applied for engineering of the dielectric constants of supercapacitors and nanocapacitors for energy storage. The dielectric constants can be tuned by controlling the interlayer distance, sp carbon length, and stacking modes. The heterostructures can also be used for engineering the dielectric screening. On the other hand, we also found that the properties of the dielectric screening are strongly associated with the band gaps. Therefore, the band gap can be tuned by controlling the dielectric constants. We suggest GDY and its family for developing materials whose dielectric constant and band gap can be engineered, which is feasible in the field of nanotechnology.

Figure 6. (a) Schematic of the band splitting of the bilayer by a vertical field. (b) Calculated band gaps of few-layer 1-GDY as a function of the field. The dots and lines indicate the calculated values and the fitted curves from the TB Hamiltonian (eq 3), respectively.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsami.8b00877. Geometric stackings and band structures of bilayer GDY and its family, band structures, and fitting parameters of a TB Hamiltonian PDF)

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Figure 7. (a) Geometry of a heterostructure with 1-GDY and 4-GDY, where the blue and red layers indicate 1-GDY and 4-GDY, respectively. (b) Calculated potential difference of the bilayer of the heterostructure. (c and d) Schematics of the dielectrics stacked serially and in parallel, respectively.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (H.L.). ORCID

Hoonkyung Lee: 0000-0002-6417-1648 slopes in the potential of the heterostructure (Figure 7b), which is the same as the slope in monostructured 1-GY and 4-GY, respectively. This indicates that heterostructured vdW materials can be interpreted by a dielectric made up of two distinct dielectrics, as shown in Figure 7c. Therefore, the equivalent dielectric constant can be calculated through a serial capacitor model. If two distinct vdW materials with different thicknesses are serially stacked as shown in Figure 7c, the equivalent dielectric constant is given by the formula εeq = ε1ε2(d1 + d 2)/(ε1d 2 + ε2d1)

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by WTU joint Research Grants of Konkuk University. L.Y. is supported by the National Science Foundation CAREER Grant DMR-1455346 and the Air Force Office of Scientific Research Grant FA9550-17-1-0304.

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(4)

where ε1 and ε2 denote the dielectric constants of the two distinct monostructured vdW materials with thicknesses d1 and d2, respectively. If they are stacked in parallel with the lengths L1 and L2 (Figure 7d), the dielectric constant is calculated by εeq = (ε1L1 + ε1L1)/(L1 + L 2)

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Therefore, the dielectric constants of heterostructured vdW materials can be calculated from the values of the dielectric constants of monostructured vdW materials using the capacitor model. F


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