Contrastive band gap engineering of strained graphyne nanoribbons with armchair and zigzag edges

Abstract

The band gap engineering of nanostructures is the key point in the application of nanoelectronic devices. By using first-principles calculations, the band structures of graphyne nanoribbons with armchair (a-GNRs) and zigzag (z-GNRs) edges under various strains are investigated. A controllable band gap of a strained narrow a-GNR (1.36–2.85 eV) could be modulated almost linearly under an increasing strain in the range of −5% to 16%. In contrast, the band gap of a strained narrow z-GNR (2.68–2.91 eV) is relatively insensitive to −16% to 16% strain. This contrastive band gap engineering of narrow GNRs is attributed to different structure deformation of the specific graphyne structure including two kinds of carbon atoms different from those of graphene. For wider strained GNRs, the band gap (depending on its width and edge morphology) generally decreases as tensile strain increases, similar to 2D graphyne sheet. The charge density distributions of key states around Fermi level are presented to investigate the reason for band gap variation.

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Introduction

Graphyne, a fantastic 2D carbon nanostructure like graphene, attracts tremendous attention due to its superior properties, such as large carrier mobility,¹ relatively low in-plane Young’s modulus (162 N m⁻¹),² anisotropic optical property and Poisson ratio.³ Furthermore, different graphyne structures possess diverse electrical characteristics, like good electrical conduction⁴ and native band gap.^5–7 This is important since band gap engineering is the key point in the application of nanoelectronic devices. Many strategies to tune the band gap of nanostructures have been studied intensively. For example, a finite value band gap of graphene is obtained by tailoring the 2D graphene sheet into nanoribbons with different widths^8–11 and edges,^12–14 the band gap of silicon nanowire is modified by saturating the silicon surface¹⁵ and uniaxial strain,¹⁶ and the band gap of phosphorus is modulated by stacking order¹⁷ and tailoring into nanoribbons along three orientations with various strains.^18,19

Meanwhile, similar approaches have been also applied to modulate the band gap of graphyne, including tailoring a graphyne sheet into a graphyne nanoribbon (GNR),²⁰ loading strain, stacking multilayer graphyne,²¹ and doping with boron or nitrogen atoms.²² In particular, the band gap of the z-GNR shows a unique “step effect” as the width increases.²⁰ The band gap of the 2D graphyne sheet is influenced by homogeneous biaxial strain.³ Similarly, graphdiyne is cut to obtain a graphdiyne nanoribbon whose band gap decreases monotonically as the width increases.^20,23 The tunable band gap of graphdiyne monotonously increases with uniform increasing strain.²⁴ Plenty of previous works investigating the strain influence on the mechanical and electronic properties of graphyne, show tremendous potential applications for novel strain-tunable nanoelectronic and optoelectronic devices.^25–28 However, the response of the band gap of GNRs to the uniaxial tensile and compressive strains has little been reported. To gain insight into this, a systematic investigation of strain effect on the band gap of GNRs is desperately needed.

In this paper, we investigate the critical compressive strain of GNRs and the response of the band gap of graphyne nanoribbons to the homogeneous uniaxial strain, by first principles calculations. To reveal the effect of edge morphology and carbon hybrid orbitals, GNRs with narrow width and wide width are investigated separately. The band structures, charge density distributions and projected density of states (PDOS) are calculated to analyze the law of band gap variation.

Computational methods

All calculations are carried out by SIESTA code²⁹ with the framework of density functional theory (DFT).^30,31 Local density approximation described by Ceperley–Alder³² is chosen as the exchange correlation potential, and the Troullier–Martins scheme norm-conserving pseudopotentials³³ are employed to represent the interaction between localized pseudoatomic orbitals and ionic cores. The double-ζ basis plus polarization orbital set is adopted to ensure a good computational convergence. The energy cutoff is 120 Ry. The Brillouin zone sampling is performed using a Monkhorst-Pack special k points grid.³⁴ All structures are fully relaxed with a force on each atom of less than 0.05 eV Å⁻¹. A vacuum layer of 10 Å in the supercell is designed to render the interaction between the GNR layer and its periodic image negligible.

Results and discussions

The graphyne structure is optimized and the lattice constant of 6.92 Å is in agreement with previous results.^3,20 The graphyne sheets are cut along the X direction and the Y direction to obtain a-GNRs and z-GNRs, respectively, as shown in Fig. 1. The nanoribbon edge is passivated with hydrogen atoms as in similar works.²⁰ In our work, a series of a-GNRs (n = 1–5) and z-GNRs (n = 1–4) are investigated with a maximal width of about 28.89 Å; here n indicates the number of repeated chains of benzene rings. Each nanoribbon is strained within the range of the critical compressive strain to 16% tensile strain by scaling the lattice constant.

Fig. 1 Schematic representation of graphyne nanoribbons of two edges. (a) a-GNR cut along the Y (armchair) direction, and (b) z-GNR cut along the X (zigzag) direction.

For very thin films, the large compressive strain will induce out-of-plane buckling and undulation, and it is much smaller than the critical tensile strain for fractures, such as 2D graphene sheets for which the critical compressive strain is several orders of magnitude smaller than the critical tensile strain.³⁵ So, before investigating the variation of strained GNRs, the critical compressive strain should be identified to ensure stable and realistic structure. The structures of strained GNRs are initial buckling or undulation, by moving atoms out-of-plane artificially (Fig. S1, ESI†), and the structures with an initial plane are also calculated. At a specific compressive strain, the difference value of energy between the two initializations is changed, from positive to negative (Fig. S3, ESI†), and this strain is chosen as the critical compressive strain. The critical compressive strain of all GNRs ascertained is presented in Table SI†. In particular, the z-GNR of n = 1 withstands larger compressive strain than 16%, and the critical compressive strain (2–3%) of other z-GNRs shows a zigzag variation as the width increases. The maximal compressive strain of a-GNRs is about 4%.

a-GNRs and z-GNRs with narrow widths

We firstly focus on the strained GNRs with narrow widths, and take width n = 1 as the example here. Their band structures around Fermi level and corresponding tendency of band gap variation under strain are given in Fig. 2(a). The band structure detail of zero-strain a-GNR and z-GNR is also shown in Fig. 2(b and d), and their calculated band gaps are in good agreement with the tendency of a previous result.²⁰ It was found that a controllable band gap of strained a-GNR in a range of 1.36–2.85 eV could be modulated almost linearly under an increasing strain. Contrasting with the a-GNR spectacularly, the band gap of the strained z-GNR (2.68–2.91 eV) with narrow widths is little influenced by strain.

Fig. 2 (a) Band gap variation of the strained a-GNR and z-GNR with widths n = 1; inset is a plot of the strained band structure of the a-GNR and z-GNR. (b) Band structures and (c) charge density distribution (isovalue = 0.008 e Å⁻³) for different states of a-GNR (n = 1) without strain. (d) Bands structure and (e) charge density distribution for different states of z-GNR (n = 1) without strain.

The band structure of the unstrained a-GNR (in Fig. 2(b)) shows a direct band gap at X point, and from the insets in Fig. 2(a), we could find that the modulated band gap is due to the fall of the valence band maximum (VBM) energy and the rise of the conduction band minimum (CBM) energy under increasing strain. The charge density distributions (isovalue = 0.008 e Å⁻³) at the CBM and VBM states of the a-GNR shown in Fig. 2(c) visualize the main contribution from the bonds in the a-GNR structure. Similarly, the z-GNR has a direct band gap at Γ point. Its band gap is little influenced by strain due to VBM and CBM being retained at Γ point with little variation under various strains. The corresponding charge density distributions of these key states are also visualized in Fig. 2(e).

Subsequently, the variations of bond lengths and bond angles in strained GNR structures are listed in Fig. 3. For the a-GNR, the lengths of C1–C2 (1.37–1.45 Å) and C3–C4 (1.37–1.74 Å) bonds increase remarkably along with growing strain. The interaction of atoms connected with these stretched bonds will have a large change, and lead to the rise of c_a state energy and the fall of v_a state energy, resulting in the enlarging of the band gap.

Fig. 3 Structure variations of the strained (a) a-GNR and (b) z-GNR of n = 1 under various strains.

On the contrary, the length of bonds in the z-GNR shows little variation, especially under small strains. It is interesting that strain primarily influences the bond angle of C3–C4–C5, due to the unparallel strain direction with acetylenic bonds in the z-GNR structure. The primary change occurring in bond angle instead of length results in little variation of CBM and VBM states, which determines an almost invariable band gap of around 2.7 eV as shown in Fig. 2(b). Under larger strain, the acetylenic bond will get close to the strain direction, and the length of bond, instead of the bond angle, shows a perceptible variation. Meanwhile the band gap of the z-GNR has a gradual increase under +8% to +12% strains, and the strain, primarily influencing bond length, shifts CBM from c_z1 state to c_z2 state, resulting in a wider band gap.

It is obvious that the band gaps of the a-GNR and of the z-GNR have different responses to various strains. Strain primarily influences bond lengths of the a-GNR, which results in an almost linear increase of the band gap. On the contrary, due to bond angle variation leading to little variation of bond lengths, the band gap of the z-GNR has an insensitive response to strain.

a-GNRs with wide widths

So far, the calculations have provided conclusions about band gap variation with narrow width under strain. Next, the influence of strain on the band gap of a-GNRs as width widens are investigated. The band gaps of a-GNRs decrease as the width increases with the same strain (see Fig. 4(a)) which is in good agreement with previous work.²⁰ It is obvious that for wider strained a-GNRs, the band gap (depending on its width and edge morphology) generally decreases as tensile or compressive strain increases, similar to 2D graphyne sheets.³⁶

Fig. 4 Band gap variation of a-GNRs and z-GNRs with different widths under strain.

In Fig. 5(a), the unstrained band gaps of the band structures which are almost symmetric about Fermi level are shown in detail. Due to the similar band structures and variation tendencies of the band gap under various strains (see Fig. 4(a)), a-GNRs of n = 3 are taken as an example to reveal the tendency of the band gap with various strains, which is different from that of a-GNR (n = 1).

Fig. 5 (a) Band structures of a-GNRs with different widths (n = 1, 2 and 3). (b) Band structures of the a-GNR of n = 3 under typical strains (4% and 16%). Here the Fermi level (red dashed line) is at zero. (c) Charge density distributions for different states in the a-GNR of n = 3 with zero strain, (isovalue = 0.008 e Å⁻³). (d) 2p orbital PDOS of two different carbon atoms.

Under compressive or tensile strain (−4% to +16%), the band gap of the a-GNR (n = 3) possesses a monotonic decrease within the range from 0.84 to 0.96 eV. With approximate width, the band gap of the armchair graphene nanoribbon (0.2–1.2 eV) demonstrates a zigzag behavior under −16% to +16% strain,³⁷ while the band gap of armchair phosphorene nanoribbons (0.4–1.0 eV) monotonously increases under −10% to +10% strain¹⁸ and the band gap variation of [110] silicon nanowire is similar to a-GNR (n = 3) within the range from 1.3 eV to 1.8 eV under −5% to +5% strain.³⁸

The band gap of a-GNRs of n = 3 is calculated according to c_Γ, v_Γ, c_X and v_X states (Fig. 5(a)) on a couple of bands at Γ and X points under different strains, as shown in Fig. 5(b). Under compression or zero strain, the band gap is determined at c_X and v_X states. Their charge distributions presented in Fig. 5(c) primarily locate at edge atoms, which is similar to the a-GNR of n = 1. Similar charge distributions result in similar incremental variation in the band gap. As the strain turns tensile, the band gap of the a-GNR is determined at c_Γ, v_Γ, c_X and v_X states, and their charge distributions primarily locate at non-edge atoms, similar to that of pure graphyne,³⁶ resulting in a declining variation in the band gap. The energies of these CBM and VBM states have different responses to strain, leading to a shift of band structure from direct to indirect band gap, under +4% to +12% strains.

In particular, the band gap variation tendency under tensile strain is primarily due to the inner atoms situated in the body of a-GNRs. This result verifies the declining tendency of the band gap as width widens. Owing to an enhancing proportion of non-edge charge distribution, the variation tendency of the band gap of GNR with wider width will tend to the same as graphyne,³⁶ because the nanoribbon will return to a 2D structure. Furthermore, these inner atoms are divided into two types: one atom linked by an acetylenic bond and another residing within a benzene ring. The 2p orbital PDOS of two different carbon atoms in Fig. 5(d) proves that the carbon atom residing within a benzene ring primarily influences the band gap, in good agreement with previous work.^3,20

z-GNRs with wide widths

As the width of the z-GNR increases, the band gap of unstrained z-GNRs shows a step decrease in Fig. 4(b) and 6(a), which is in good agreement with previous work,²⁰ and their band gap shows a declining tendency under tensile strain, as with a-GNRs.

Fig. 6 (a) Band structures of z-GNRs with widths ranging from n = 1 to n = 3. Here the Fermi level is at zero. Charge density distributions for different states in unstrained z-GNRs of (c) n = 2.5 and (d) n = 3 (isovalue = 0.008 e Å⁻³).

We take two z-GNRs (n = 2.5, 3) as examples for the following investigations, due to their consistent band gap variation. Under strain (−2% to +16%), the band gap of the z-GNR (n = 2.5) possesses a monotonic decrease within the range of 0.52 to 0.88 eV. With approximate width, the band gap of zigzag graphene nanoribbon (0.2–0.3 eV) monotonously increases under −15% to +15% strain,³⁹ while the band gap of zigzag phosphorene nanoribbons (0.7–1.4 eV) increases firstly, then decreases under −10% to +10% strain,¹⁸ similarly to z-GNRs, and the band gap of zigzag boron nitride nanoribbon (2.3–4.0 eV) decreases monotonously under 0–12% strain.⁴⁰

To investigate the band gap variation, the band structures of unstrained z-GNRs were calculated and are presented in Fig. 6(a). The calculated band structures reveal that the energy of v_X state obviously rises, while the other states around Fermi level remain almost the same under different strains. The band structure remains direct gap (v_X state and c_X state at X point in Fig. 6(a)) under positive strain.

The step decrease of the band gap of z-GNRs could be explained by comparing Fig. 6(b) with 6(c). The z-GNRs with widths n = 2.5 and n = 3 have similar charge distributions at c_X, c_Γ and v_Γ states, except for the edge of v_X state. The similar charge distributions result in similar band gap variation of z-GNRs (n = 2.5, 3) under strain. It was also found that the charge distributions at the inner region of z-GNRs are similar to those of a-GNRs in Fig. 5(b). The non-edge charge distributions are primarily contributed by inner benzene ring carbon (as in the a-GNRs’ case) leading to a similar tendency to a-GNRs, shown in Fig. 4(b). For example, the band gap of the z-GNR and a-GNR of n = 3 reduces as tensile strain increases. Furthermore, the edge morphology gives rise to different submarginal charge distribution, which will cause a slight difference in variation of the band gap between a-GNRs and z-GNRs.

Conclusions

In summary, a series of band gaps of strained a-GNRs (n = 1–5) and z-GNRs (n = 1–4) are investigated by using first-principles calculations. The z-GNR of n = 1 can be taken to a wide range of strain (−16% to +16%). A controllable band gap of the strained a-GNR (n = 1) in the range of 1.36–2.85 eV could be modulated almost linearly, due to the lengths of C1–C2 (1.37–1.45 Å) and C3–C4 (1.37–1.74 Å) bonds increasing remarkably under an increasing strain. The band gap of the strained z-GNR (2.68–2.92 eV) with width n = 1 is slightly influenced by strain owing to the fact that strain primarily influences the bond angle of C3–C4–C5 instead of bond length. The band gap of the a-GNRs experiences a direct-to-indirect gap transition with sufficient strain. For wide widths, the band gap of GNRs (depending on the width and edge morphology) generally decreases as tensile strain increases. With width of the z-GNR increasing, the band gap of strained z-GNRs shows a step decrease. The band gap of GNRs has a wide controllable range from 0.05 to 2.92 eV. Thus, our work provides an efficient method, including edge morphology, width and strain, for modulating the band gap of GNRs which shows great potential for the application of GNRs in novel strain-tunable nanoelectronic devices.

Acknowledgements

This research work was supported by Fundamental Research Funds for the Central Universities (HUST 15A71).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra10372j

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