Effect of Acetylene Links on Electronic and Optical Properties of Semiconducting Graphynes.

The family of graphynes, novel two-dimensional semiconductors with various and fascinating chemical and physical properties, has attracted great interest from both scientific and industrial communities. Currently, the focus is on graphdiyne or graphyne-2. In this work, we systematically study the effect of acetylene, i.e., carbon-carbon triple bond, links on the electronic and optical properties of a series of graphynes (graphyne-n, where n = 1-5, the number of acetylene bonds) using ab initio calculations. We find an even-odd pattern, i.e., n = 1, 3, 5 and n = 2, 4 having different features, which has not been discovered in studying graphyne or graphdiyne alone. It is found that as the number of acetylene bonds increases, the electron effective mass increases continuously in the low-energy range because of the flatter conduction band induced by the longer acetylene links. Meanwhile, longer acetylene links result in a larger red shift of the imaginary part of the dielectric function, loss function, and extinction coefficient. In this work, we propose an effective method to tune and manipulate both the electronic and optical properties of graphynes for the applications in optoelectronic devices and photochemical catalysis.

Introduction

The large variety of <n class="Chemical">span class="Chemical">carbonn> allotropes, showing different physical and chemical properties, is due to the different <spn>an class="Chemical">carbon hybridizations, i.e., sp, <span class="Gene">sp2, and sp3. For example, the natural three-dimensional (3D) graphite and diamond are formed through sp2 and sp3 hybridizations of carbon atoms, respectively. Meanwhile, the sp2 hybridization occurs in some novel man-made carbon allotropes, such as fullerene,[1] carbon nanotube,[2] and graphene.[3] In 1987, the concept of sp–sp2 hybridized graphyne-n was theoretically proposed by Baughman,[4] where n indicated the number of carbon–carbon triple (acetylene) bonds in graphyne (see Figure ). Accordingly, there are several kinds of structures based on the polymerization mode, such as graphyne (n = 1), graphdiyne (n = 2), graphyne-3 (n = 3), and so on. After the successful synthesis of graphyne (n = 1) in the experiment, it has been of particular interest due to its unique semiconducting electronic structure and extensive applications in many fields, such as catalysis, sensor and transistor technologies, and energy storage (see reviews[5,6] and references therein). Simultaneously, engineering of tuning the electronic structure by simply constructing the acetylene bonds n has been attracted more and more attention to this kind of two-dimensional (2D) materials theoretically and experimentally.

Figure 1

Optimized geometric structures of unit cells of 2D graphynes. Each graphyne is named with an index “n”, which indicates the number of carbon–carbon triple bonds in a link, highlighted in red, between two adjacent hexagons: (a) graphyne-1, (b) graphyne-2, (c) graphyne-3, (d) graphyne-4, and (e) graphyne-5.

Optimized geometric structures of unit cells of 2D <span class="Chemical">graphynes. Each <span class="Chemical">graphyne is named with an index “n”, which indicates the number of carbon–carbon triple bonds in a link, highlighted in red, between two adjacent hexagons: (a) graphyne-1, (b) graphyne-2, (c) graphyne-3, (d) graphyne-4, and (e) graphyne-5.

Recently, n class="Chemical">2D semiconducting graphyne-2 has been synthesized on a copper surface by the cross-coupling reaction.[7−9] Soon, these types of 2D materials attracted great attention in many research fields, such as catalysis, energy storage, water purification, and optoelectronic devices, due to their large interlayer distance, unique porous structure, large specific surface area, and high conductivity.[10−19] Theoretical calculations reveal that graphyne-2 has higher electron mobility than graphene.[20,21] Kuang et al.[22] further pointed out that the electron mobility and photoconversion efficiency of perovskite solar cells with doped graphyne-2 were significantly improved, which paves the way for optoelectronic applications of graphyne-2. Wang et al.[17] synthesized graphyne-2 composites by the hydrothermal method, which exhibited excellent photocatalytic degradation of methylene blue. The π-conjugated structure in graphyne-2 makes it efficient to receive photogenerated electrons in the conduction band and to suppress the recombination of electrons and holes. Luo et al.[23] found that the multibody effect had a significant impact on the electronic structure and optical absorption of graphyne-2 both theoretically and experimentally. Due to the additional acetylene bond in graphyne-2 compared with graphyne-1, graphyne-2 has a larger porous and much softer structure than graphyne-1, which indicates that graphdiyne could easily form a hybrid with other materials for optical applications.[5,6] The difference in the electronic structures and mechanical properties between graphyne and graphdiyne as well as the resulting different potential applications has accelerated the engineering and application of the graphyne-n family, especially in the properties of graphyne-n with longer acetylene links beyond n = 1 and 2.

In this paper, the electronic and optical properties of five members in the <n class="Chemical">span class="Chemical">graphynen> family, i.e., <spn>an class="Chemical">graphyne-n (n = 1–5), are systemically investigated using ab initio calculations. It was found that the length of the <span class="Chemical">acetylene links greatly changes the feature of the energy bands near the Fermi level. Thus, both the electronic and optical properties of these types of 2D materials could be feasibly tuned and manipulated for optoelectronic devices and photochemical catalysis applications. This may open a way for exploring the extended graphynes in optoelectronic applications.

Computational Methods

In this work, we carried out ab initio calculations with the CASTEP module,[24] which was implemented in the framework of density functional theory (DFT)[25] using the generalized gradient approximation (GGA) in the parameterization of the Perdew–Burke–Ernzerhof (PBE) format exchange–correlation functional.[26] The Grimme[27] under din class="Chemical">spersion correction (DFT-D) was used to improve the calculation accuracy of the weak interaction in 2D graphynes. The electron–ion interactions were described by the Vanderbilt ultrasoft pseudopotentials (US-PP).[26] The convergence test and geometric optimization of the graphyne-n unit cell were performed first. The kinetic cutoff energy used for plane wave expansions was 650 eV. For graphyne-1, graphyne-2, and graphyne-3, the Monkhorst–Pack K-point meshes of 11 × 11 × 1 were used in the first Brillouin zone,[28] while for graphyne-4 and graphyne-5 with large unit cells, the Brillouin zone integrations were performed using a Monkhorst–Pack grid of 8 × 8 × 1. The vacuum layer thickness was set to 15 Å to eliminate the interlayer interaction. Each calculation was converged when the total energy changes during the geometry optimization process were less than 1 × 10–5 eV/atom, and the force per atom and the residual stress of the unit cell were less than 0.01 eV/Å and 0.05 GPa, respectively. The maximum displacement between cycles was less than 0.005 Å when the convergence reached.

Results and Discussion

Geometric Structures

The geometric structures of the unit cells of <n class="Chemical">span class="Chemical">graphyne-n>1, <spn>an class="Chemical">graphyne-2, <span class="Chemical">graphyne-3, graphyne-4, and graphyne-5 are shown in Figure . The size of the cavity of graphynes is proportional to the length of the acetylene linkages. The structural stability of graphynes can be estimated by the cohesive energy, which can be defined as follows.[29]where Ecoh is the cohesive energy of graphynes, n is the number of carbon atoms in a unit cell, and Eatom and Etotal are the energy of a single carbon atom and the total energy in a unit cell, respectively. The details of the lattice constant, cohesive energy, and comparison with other reports are shown in Table . As can be seen, the calculated lattice parameters in this work are in good agreement with previously reported works. Our cohesive energies are slightly higher than other results, which might be due to the different pseudopotentials used.

Table 1

Lattice Constants and Cohesive Energies of Graphynes

		graphyne	graphdiyne	graphyne-3	graphyne-4	graphyne-5
lattice constant (Å)	this work	6.872	9.436	12.011	14.576	17.592
	other works	6.86a, 6.877b, 6.89c	9.44a, 9.46c, 9.490c	12.02a, 12.04c, 12.43d	14.6a, 14.60c
cohesive energy (eV atom–1)	this work	8.635	8.513	8.450	8.419	8.397
	other works	7.95a, 7.21e	7.78a, 7.87e	7.70a	7.66a

Ref (31).

Ref (32).

Ref (33).

Ref (34).

Ref (29).

It is noted that the cohesive energy is the energy required for separating the neutral atoms in the ground state at 0 K.[30] Thus, the larger the <n class="Chemical">span class="Chemical">Ecohn>, the more stable the crystal structure. According to the calculated cohesive energies in Table , it can be found that the planar two-dimensional structure of <spn>an class="Chemical">graphyne-1 is the most stable. Meanwhile, the cohesive energy of <span class="Chemical">graphynes decreases gradually with the increase of the number of acetylene bonds (n).

Electronic Properties

Figure shows the band structures of <n class="Chemical">span class="Chemical">graphynesn>. It can be seen that all of them are direct-band-gap (Eg) semiconductors with band gaps of 0.446, 0.464, 0.548, 0.524, and 0.544 eV. It is worth noting that the band gap size is not simply linearly proportional to the number of <spn>an class="Chemical">acetylene bonds (n). Interestingly, the direct band gap of <span class="Chemical">graphyne-n with odd n, i.e., n = 1, 3, 5, is located at the M-point, while it is at the Γ-point for even n. Meanwhile, the energy-band dispersions at the bottom of the conduction band and at the top of the valence band are quite similar for all graphynes, which indicates that they have similar effective mass of both electrons and holes.

Figure 2

Energy-band structures of graphyne-n. Eg is the direct band gap. A and B indicate the possible electron excitation, hopping from the valence band to the conduction band.

Energy-band structures of <n class="Chemical">span class="Chemical">graphyne-n>n. Eg is the direct band gap. A and B indicate the possible electron excitation, hopping from the valence band to the conduction band.

The effective mass of electrons and holes in semiconductors is an essential parameter, which greatly affects the performance of electronic and/or optical devices. The effective mass of <n class="Chemical">span class="Chemical">graphynesn> is calculated by the following equation.[35]The values of the effective mass in the conduction band (mc, holes) and valence band (mv, electrons) are listed in Table . It is found that the effective mass is isotropic from M to K and Γ for <spn>an class="Chemical">graphyne-n with even (n = 2, 4) acetylene bonds, while it is anisotropic if n is odd. Our calculations show that the low energy levels of the conduction band are mainly contributed by the 2p state of carbon atoms, where the electrons have quite a small effective mass under the excitation of photons. This indicates that it facilitates the formation of the effective photogenerated electrons and the transferred charge carriers, while more effective photogenerated holes would be formed in the valence-band region.

Table 2

Values of the Effective Mass of Graphynes in the Conduction (mc) and Valence Band (mv) and the Band Gap (Eg)

	mc/m0		mv/m0
structure	Γ → M	M ← K	Γ → M	M ← K	Eg (eV)
graphyne-1	0.146(0.15a, 0.21b)	0.086(0.063a, 0.087b)	0.150(0.17a, 0.22b)	0.068(0.066a, 0.0901b)	0.446 at M (0.46c,0.47d,e)
graphyne-2	0.080(0.073a)		0.074(0.075)		0.464 at Γ (0.48,0.46f,)
graphyne-3	0.082(0.099a)	0.053(0.081a)	0.106(0.12a)	0.081(0.085a)	0.548 at M (0.56d)
graphyne-4	0.078(0.081a)		0.110(0.080)		0.524 at Γ (0.54d)
graphyne-5	0.101	0.091	0.133	0.120	0.544 at M

Ref (31).

Ref (32).

Ref (33).

Ref (36).

Ref (37).

Ref (38).

Ref (39).

It is found that all <n class="Chemical">span class="Chemical">graphynesn> have covalent bonds from the Mulliken population (MP) analysis (Table S1), implying their good structural stability. Moreover, the MP analysis shows that the electronic states of <spn>an class="Chemical">graphynes are mainly contributed by the <span class="Gene">C-2p state, which is consistent with the projected density of state (PDOS) analysis shown in Figure S1. The bond lengths and band populations of C1–C2 (sp2–sp2), C2–C3 (sp2–sp), C4–C5 (sp–sp), and C6–C7 (sp–sp) bonds of graphynes (see Figure ) remain constant with the increase of the number of acetylene bonds (n). However, the bond lengths (band populations) of C3–C4, C5–C6, and C7–C8 (sp≡sp) bonds (Figure ) are enlarged as n increases. These alternate bonding characteristics are also illustrated by the charge-density-difference calculations as shown in Figure S2. Such an alternate −C≡C–C≡C– structure can energetically stabilize atomic carbon chains or rings, which have been reported in many carbon allotropes.[40,41]

Optical Properties

The calculated band structures of <n class="Chemical">span class="Chemical">graphynesn> in Figure show that they all are direct-band-gap semiconductors, and the values of band gaps are close to those of <spn>an class="Chemical">silicon (0.57 eV GGA-PBE[42]). Thus, we next study the optical properties of <span class="Chemical">graphynes and their potential optoelectronic applications. It is well-known that standard GGA functionals like PBE underestimate the band gap. One of the major improvements in the band gap calculation is to use the hybrid functionals, such as HSE06.[43] However, both GGA-PBE and HSE06 give similar spectra except for an energy shift of graphynes,[44] so only GGA-PBE results are presented in this work.

The complex dielectric function ε(ω), a significant parameter to determine the polarization effect inside materials, is calculated aswhere the imaginary part of the dielectric function ε2(ω) describes the absorption of light and the real part of the dielectric function ε1(ω) represents the amplitude modulation, that is, the resonant absorption of the electron transition.[45]where m is the free electron mass, e is the free electric charge, ω is the incident photon frequency, BZ is the first Brillouin zone, |e·MCV(K)| is the momentum transition matrix element, K is the inverted lattice vector, k is the extinction coefficient, C is the conduction band, V is the valence band, and EC(K) and EV(K) are the intrinsic levels of the conduction and valence bands. Meanwhile, the absorption coefficient I(ω), conductivity σ(ω), refractive index n(ω), loss function L(ω), and reflectivity R(ω) can all be deduced from ε(ω) with the Kramers–Kronig din class="Chemical">spersion.[46−48]

Figure shows the calculated imaginary and real parts of the complex dielectric function, absorption function, and electrical conductivity of <n class="Chemical">span class="Chemical">graphynesn>. The peak position of ε2(ω) is determined by the band gap and the degree of nonlocalization in the low-energy regions. With the increase of the number of <spn>an class="Chemical">acetylene bonds, ε2(ω) of graphynes has red-shifted as shown in Figure a. For the real part of the dielectric function ε1(ω), the static dielectric constant (capacitance) is denoted ε1(0) at the zero frequency. The ε1(0) of graphynes is strongly dependent on the corresponding band gaps as shown in Table . Our results in Figure b show that ε1(ω) dramatically decays to zero with the increase of the photon energies, indicating a resonance of the energy transition between electrons and photons in graphynes. Figure b shows that ε1(ω) of graphyne-2–5 has negative values (positive for graphyne-1). According to the wave vector equation belowε1 < 0 means that the wave vector K is an imaginary number. Furthermore, this negative value region is red-shifted with the increase of the number of acetylene bonds (n) as shown in Figure b.

Figure 3

(a) Imaginary and (b) real parts of complex dielectric functions of graphynes. (c) Absorption function and (d) electrical conductivity of graphynes. The visible light region is labeled by two vertical dotted lines. The low-energy part is enlarged in the insets.

Table 3

Energy Corresponding to the Peak of the Imaginary Part of the Dielectric Function ε2(ω), Energy of Electron Interband Transition, Static Dielectric Constant ε1(0), and Absorption Edges Eop of Graphynes

structure	photon energy (eV)	transition (eV)	ε1 (0)	Eop (eV)
graphyne	A = 0.9	–0.20 → 0.70	12.9	5.37
	B = 6.1	–2.07 → 4.03
graphdiyne	A = 1.2	–0.36 → 0.84	10.3	3.57
	B = 4.1	–2.64 → 1.46
graphyne-3	A = 1.0	–0.21 → 0.79	8.9	2.65
	B = 3.0	–1.10 → 1.90
graphyne-4	A = 1.1	–0.27 → 0.84	9.6	2.15
	B = 2.4	–0.80 → 1.60
graphyne-5	A = 1.1	–0.25 → 0.85	8.9	1.79

(a) Imaginary and (b) real parts of complex dielectric functions of <n class="Chemical">span class="Chemical">graphynesn>. (c) Absorption function and (d) electrical conductivity of <spn>an class="Chemical">graphynes. The visible light region is labeled by two vertical dotted lines. The low-energy part is enlarged in the insets.

Figure c shows that the absorption coefficient in <n class="Chemical">span class="Chemical">graphynesn> increases first and consequently decreases as the photon energy goes up. The peak of the absorption coefficient of <spn>an class="Chemical">graphynes shifts to the low-energy region and induces a narrow absorption range. Further analysis on the absorption function of <span class="Chemical">graphyne-1 (Figure c) shows that there are four main absorption peaks at 1.95, 2.96, 6.63, and 12.03 eV, which are well consistent with the experimental values.[23] For graphyne-2 and graphyne-3, the highest peak is located within the ultraviolet region. Thus, they probably may have applications in ultraviolet protection or detection materials. The shift of the photoconductivity of graphynes with the photon energy is shown in Figure d. The conductivity peaks of graphyne-1 are at 1.46, 2.59, 6.24, and 11.97 eV. The profiles of the photoconductivity of graphynes shift to the low-energy region (Figure d), which approaches the energy range of the visible light.

The peak energy of the dielectric function ε2(ω), energy of electron interband transition, static dielectric constant ε1(0), and absorption edges are presented in Table . For <n class="Chemical">span class="Chemical">graphyne-n>1, the primary peaks of ε2(ω) are located at A (0.9 eV) and B (6.1 eV). Based on the band structure in Figure , peak A mainly originates due to the transition between the unoccupied states at 0.70 eV and the occupied states at −0.20 eV. Meanwhile, peak B primarily originates from the transition from −2.07 to 4.03 eV of the 2p electrons in the valence band of the C atoms. The details of transitions of two peaks in ε2(ω) of <spn>an class="Chemical">graphyne-2, graphyne-3, and graphyne-4 are summarized in Table . For graphyne-5, it is worth noting that the imaginary part of the dielectric function ε2(ω) gives rise to a bimodal pattern, which shows only one peak at 1.1 eV due to the band localization and red shift of the functional profile (see in Figure ). Furthermore, the transition between the occupied state at −0.25 eV and the empty state at 0.85 eV corresponds to peak A. Note that the peak in the imaginary part of the dielectric function ε2(ω) may not correspond to a single interband transition (Figure ), and other interband transitions at the same peak energy may also possibly occur in the band structure.[49,50]

The optical absorption edges of <span class="Chemical">graphynes are shown in Figure . The optical absorption band edge Eop is described by the following extrapolation relationship[51]where α represents the absorption spectrum, hν is the photon energy, and A is a function of the refractive index of the material, the reduced mass, and the speed of light in vacuum. For direct-band-gap semiconductors, n takes 0.5.

Figure 4

Optical absorption edges of graphynes, with the enlarged view shown in the inset.

Optical absorption edges of <n class="Chemical">span class="Chemical">graphynesn>, with the enlarged view shown in the inset.

The calculated values of the optical absorption band edge in <n class="Chemical">span class="Chemical">graphyne-n>n are listed in Table . It is found that the absorption band edge shifts to the low-energy region with the increase of n. Moreover, there is a deviation between the absorption band edge and the correspn>onding band gap. It is mainly due to the electron localization in the free energy level of band structures.

The relationship between the complex refractive index n(ω) and extinction coefficient k(ω) with the photon energy of <n class="Chemical">span class="Chemical">graphynesn> is shown in Figure a,b. At a frequency of zero, the static refractive indices of <spn>an class="Chemical">graphyne-n (n = 1–5) are located at 3.61, 3.22, 3.00, 3.01, and 2.99 eV, respectively. This indicates that the static index of refraction strongly depends on the band gap of graphynes. Furthermore, the complex refractive index of graphynes goes down first and then increases with an increase of the photon energy. The graphynes have distinct peaks of the energy loss in the calculated profile, as shown in Figure c. They correspond to the region of ε1(ω) = 0 and ε2(ω) < 0, which are the resonance peaks of the energy loss function in graphynes. The reflection spectrum of graphynes is shown in Figure d. Our results show that the reflectivity R(0) of graphyne-1 approaches 32.2 and diminishes with the increase of the photon energy. The maximum value is 12.4% at a photon energy of 7.20 eV. Furthermore, as the acetylene bonds (n) increase, the profiles of the energy loss function shift to the low-energy region, and the electron localization and maximum values are enhanced (Figure S3).

Figure 5

(a) Real and (b) imaginary parts of the complex refractive index of graphynes. (c) Loss function and (d) reflectivity of graphynes. The visible light region is labeled by two vertical dotted lines. The low-energy part is enlarged in the insets of (a) and (b).

(a) Real and (b) imaginary parts of the complex refractive index of <n class="Chemical">span class="Chemical">graphynesn>. (c) Loss function and (d) reflectivity of <spn>an class="Chemical">graphynes. The visible light region is labeled by two vertical dotted lines. The low-energy part is enlarged in the insets of (a) and (b).

Conclusions

In summary, we carried out ab initio studies on the geometric, electronic, and optical properties of n class="Chemical">2D graphyne-n sheets, a family of sp–<spn>an class="Gene">sp2 hybrid materials with acetylene bonds. The odd–even pattern of n of graphynes on the position of direct band gaps and the dispersion of the effective mass are revealed by the electronic-structure calculation. Our photoelectron transport results show a red shift of the imaginary part of the dielectric function with the increase of the number of acetylene bonds. Furthermore, the loss function and extinction coefficient move to the low-energy region with the increase of n. These findings show that the optical parameters could be tuned and manipulated by the number of acetylene bonds to fulfill different applications.