Graphene nanoribbon band-gap expansion: Broken-bond-induced edge strain and quantum entrapment

Abstract

An edge-modified tight-binding (TB) approximation has been developed, enabling us to clarify the energetic origin of the width-dependent band gap (E_G) expansion of the armchaired and the reconstructed zigzag-edged graphene nanoribbons with and without hydrogen termination. Consistency between the TB and the density-function theory calculations affirmed that: (i) the E_G expansion originates from the Hamiltonian perturbation due to the shorter and stronger bonds between undercoordinated atoms, (ii) the combination of the edge-to-width ratio with a local bond strain up to 30% and the associated 152% potential well depression determines the width dependent E_G change; and, (iii) hydrogen termination affects insignificantly the band gap width as the H-passivation minimizes the midgap impurity states.

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1. Introduction

Unrolling the single-walled carbon nanotubes (SWCNT) triggers many intriguing properties that cannot be seen from the CNT or large graphene sheet.^1,2 In addition to the observed edge Dirac fermions states^3,4 with ultrahigh electrical and thermal conductivity,⁵ and unexpected magnetization,^6,7 the band gap (E_G) of graphene nanoribbon (GNR) expands monotonically with the inverse of ribbon width,^2,8 bare or hydrogen ended.⁹ Despite the magnitude difference between measurements¹⁰ and theoretical calculations,^6,11,12 the energetic origin for the width-dependent E_G expansion is under debate. The E_G expansion is attributed either to the carrier confinement^6,13,14 or to the edge energy pinning.^12,15,16 If the hopping overlap integrals employed in band structure calculations for the edge and for the interior were identical, no E_G expansion would be observed.¹⁵ The hopping integral is certainly greater at edge than that at the GNR interior.¹³ It is our opinion that the density and energy of carriers in GNR play important roles in the transport dynamics, but the E_G is determined intrinsically by the crystal potential and hence the Hamiltonian matrix.

Furthermore, theoretical reproduction¹⁷ of the measured elastic stiffness^18,19 (the product of Young's modulus Y and the wall (bond) thickness t, Yt = 0.3685 TPa ·nm) and the melting point²⁰ (T_m = 1593 K) of the open end of the SWCNT has led to optimization of the thickness and the length of a C–C bond of SWCNT to be 0.142 nm and 0.125 nm, respectively. A 18.5% bond contraction leads to a spontaneous 68% increase of the bond energy with respect to those of diamond.^21,22 A recent experimental discovery²¹ shows that the minimal energy required for breaking a C–C bond of a 3-coordinated carbon atom in the graphene (5.67eV/bond) is lower than that (7.5 eV/bond) for a 2-coordinated carbon near atomic vacancy. Theoretical reproduction of the measured C 1s energy shift of GNR,^23,24 graphite,^24,25 diamond,²⁶ has also led to the quantification of coordination-number-resolved C 1s binding energy. These confirm consistently the Goldschmidt-Pauling's rule of bond contraction^27,28 and the bond order-length-strength correlation (BOLS) theory expectation that the under-coordination shortens the C–C bond length and enhances its strength.

The broken bonds induced local bond contraction and bond energy gain, potential well depression with an association of densification of local charge and energy surrounding the under-coordinated atoms (Quantum trap) have been confirmed by our previous work on gold nanoclusters.²⁹ Stimulated by the aforementioned findings, we developed an edge-modified TB approach to examine the E_G expansion of bare ended arm-chaired GNR (AGNR) and reconstructed zigzag GNR (recZAGNR) reported recently³⁰(Fig.1), as discussed as follows.

Fig. 1 Crystal structure of an infinitely long GNR shows bare ended (a) AGNR and (b) recZGNR with one edge reconstructed unit cells (gray lines) as well as indication width N, atom positions A–C and the corresponding t_ij.

2. Principles and algorithm

As far as the π and π^*orbitals of the C 2p_z electrons are concerned, the Hamiltonian matrix elements are where V is the crystal potential and V_i is the ith intra-atomic potential; ε_2pzis the eigen energy of the 2p_z orbital of carbon atom; α is the onsite exchange integral of negative value; t_ij is the overlap hopping integral of the nearest neighbor jth atom.

In the conventional TB,^11,16,31,32 the t_ij was set uniformly as well, i.e. t = t₁ = t₂. According to the BOLS correlation,¹⁷ the shorter and stronger bonds between under-coordinated atoms, and the associated exchange and overlap integrals can be formulated as:

where z and d are the effective coordination number (ECN) and the bond length. As shown in Fig. 1, z = 2 of the bare edge atom and 3 for the interior atom. The H-end edge bond contraction coefficient, C_H, is defined as the bond length ratio between GNR edge and interior, and taken as 96% according to DFT relaxation results.⁶ Subscripts i and B correspond to atoms in the ith atomic site and in the bulk, respectively. The superscript m = 2.56 is optimized from BOLS reproduction of the elastic modulus,^18,19 the melting point^20,22 of CNT and the C 1s binding energy shift of carbon allotropes.¹ In conducting the edge-modified TB calculations, the t_ij are enlarged by C(z)^−2.56 according to eqn (2) for under-coordinated atoms. All the parameters are listed in Table 1.

Table 1 The BOLS-TB parameters for AGNR. The strain is set with respect to bond length of diamond. The bond positions A, B, and C are indicated in Fig. 1

Bond position	Bare A–B	Bare A–C	H-end A–C	Inner graphene	Graphite^23	Diamond^23
Effective z	2	2.5	2.125	3	5.335	12
Strain (C−1, %)	−30.2	−23.6	−11.5	−18.5	−7.8	0
t ij	1.49t	1.17t	1.11t	t = −2.4eV

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Fig. 2(a) illustrates the integrals enhancement due to the local strain and the associated potential well depression near the edge. According to eqn (2), as the ECN imperfection at the edge, the edge bond will be shortened and strengthened; and the neighbor potential trap to the bonding electron will be closer in real space and deepened in energy space. The t_ij will be enlarged proportional to the bond energy by C(z)^−m. The t_ij enhancement has actually been verified using density functional theory (DFT) calculations,³³ as compared in Fig. 2(b). The BOLS correlation and DFT calculation are in good accordance despite the tiny divergence at the shorter end. It should be noted that limitations exist to DFT in dealing with the low dimensional systems and edge states.

Fig. 2 (a) Schematic illustration of the GNR edge bond strain and potential trap. The shorter and stronger edge bonds cause the potential well depression, which enhances the overlap integral according to eqn (2). (b) Correlation between the t_ij and the bond length d derived from the DFT results³³ in comparison to the BOLS prediction.

3. Calculation details

The Hamiltonian matrix for AGNR and reconstructed ZGNR are established. As shown in the Fig. 1(b), the angle between bond A–C and the vertical line is 30.8° determined by the bond contraction at edges according to Table 1. Considering the edge dangling bond, the off-diagonal terms of the Hamiltonian between the dangling bond state and the p_z orbital are zero, because they have different symmetries in the z-direction.¹⁶ Thus, we only consider the diagonal term Δ_D, the dangling bond energy.

Both the conventional and the BOLS-modified Hamiltonian matrix were diagonalized to obtain the electronic structures. Meanwhile, the DFT calculations of GNR were performed in CASTEP code.³⁴ The GNR is modeled with a 1 nm vacuum slab in the y and the z direction. The PBE³⁵ functional form in general gradient approximation is chosen with plane wave basis set energy-off: 280eV. The geometry optimization are converged with the tolerances of 10⁻⁵ eV/atom energy, the maximum force of 0.03 eV/Å, maximum stress of 0.05 GPa and the maximum displacement of 0.001 Å.

4. Results and discussion

Fig. 3 compares the band structure of an AGNR-11 obtained using (a) the conventional TB, (b) the BOLS-TB, and (c) the DFT, with the dangling bond Δ_D as 1 eV compared with the DFT result. A high degeneracy of density of states at E = ±2.7 eV in (a) suggests that the system is unstable. This frustration can be avoided upon the modified TB or DFT in (b) and (c). Since there are 22 unpaired p_z electrons 22 double-degenerated bands in the AGNR-11 unit cell, the Fermi level lays at the middle of the E_G. The E_G opening has also been observed by BOLS-TB (b) and DFT (c), and the dangling bonds provide impurity level closing to the Fermi energy, which does not determine the generation of the band gap.^9,36

Fig. 3 The band structures of the AGNR-11 calculated based on (a) the classical TB, (b) the BOLS-TB method with parameters listed in Table 1 and (c) DFT. The dangling bonds provide impurity states near the Fermi energy, indicated by D. An E_G is generated in (b) and (c), and the dangling bond state does not affect the E_G generation.

Fig. 4 shows the E_G of both bare and H-end AGNR dependence on the ribbon width N. The conventional TB results show that the E_G remains to be zero when N = 3p + 2 (p is a natural number); while the modified BOLS-TB results shows the E_G opening, being consistent with observations, which is consistent with reported DFT^6,12 and experimental observations.¹⁰ The edge absorption of H do not affect the E_G generation and the expansion trend significantly. Upon Wang and co-worker's work,¹⁵ we further explain the causes and reasons of the overlap integral enhancement and determine the quantity of the enhancement not only at edges but also at other sites with under-coordinated atoms such as vacancies and adsorption atoms.

Fig. 4 Comparison of the width dependence of the E_G in bare and H-end AGNRs shows the absence of E_G expansion for the N = 3p + 2 in the classical TB approach, which is untrue. The BOLS-TB results have the same trend to that of DFT.⁶

The band structures of recZGNR-8 with bare and H-end calculated using the conventional and the BOLS-TB and DFT method are compared in Fig.5. A very small E_G (∼0.1 eV) is generated near Γ point by the BOLS-TB and DFT method. Thus, the E_G indeed originates from the edge Quantum trap depression. The dangling bond contributes to an additional state at the Fermi level. The nonbonding electron locally pinned and polarized by the densely trapped bonding electron, which is crucial to explain the extraordinary phenomena of nanomaterial such as local magnetism presence and quantum friction.³⁷

Fig. 5 The band structure of a recZGNR-8 obtained by (a) conventional TB with bare end; (b) BOLS-TB with bare end; (c) DFT with bare end; (d) TB with H-end; (e) BOLS-TB with H-end and (f) DFT with H-end. BOLS-TB and DFT method open a gap near Γ point.

5. Conclusion

In conclusion, we have developed a BOLS-TB algorithm to encode the BOLS correlation mechanism for the first time, which has enabled us to clarify the energetic origin of the width dependence of the GNR E_G expansion. It has been confirmed that the shorter and stronger bonds between the under-coordinated edge atoms and the associated edge potential trap depression provide the perturbation to the Hamiltonian and initiate the E_G opening. The developed approach could be useful and the findings could stimulate more interest in the impact of the shorter and stronger bonds between under-coordinated atoms in low-dimensional systems.

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