Abstract

Periodic π-vacancies in graphene superlattices (GSLs) provide a symmetry-based route to band gap opening in graphene by modifying the π-band dispersion. Yet, the symmetry conditions that determine whether a vacancy motif can open a band gap remain unclear. Here, we investigate periodic π-vacancy GSLs using a nearest-neighbor tight-binding (TB) model with one pz orbital per carbon site to identify the symmetry requirements for band gap opening. π-vacancies, resembling functionalized, substituted, or missing carbon sites, are modeled as site deletions in the π basis, and the corresponding hopping matrix elements to and from the deleted sites are set to zero. We focus on π-vacancy motifs with C₂ and C₃ point groups. A 3n× 3n GSL, where n= 1, 2, 3, … is the integer scaling factor multiplying the honeycomb primitive-cell vectors, is required to fold K and K′ to Γ, and thereby opens a band gap. For C₃-type vacancies, the Dirac cones are pinned at the high-symmetry points, thus remaining at Γ in the folded 3n GSLs. However, those C₂-type vacancies which reduce the global point group of the GSL to D2h by preserving a pair of perpendicular mirror symmetries (σv⊥σd), can constrain the positions of Dirac cones at Γ. The absence of the σv and σd planes allows the cones to shift around Γ (±Δq, ±Δq) in the 3n superlattice. The C₃-type vacancies open a relatively larger band gap compared to the C₂-type vacancies due to twofold degeneracy at Γ, while the C₂-type vacancies leave the GSL with nondegenerate bands. The energy contours near the Dirac cone for C₃-symmetric vacancies remain isotropic, while the absence of C₃ generates anisotropic cones. The band gap openings in 3n superlattices scale linearly with defect concentration for each vacancy motif, with the D6h motif yielding the largest gap, reaching 314 meV at only 3.7% defect concentration. A displacement of the C₃-type vacancies shifts the Dirac cones, lifts the degeneracy, and suppresses the gap.

Abstract

Periodic π-vacancies in graphene superlattices (GSLs) provide a symmetry-based route to band gap opening in graphene by modifying the π-band dispersion. Yet, the symmetry conditions that determine whether a vacancy motif can open a band gap remain unclear. Here, we investigate periodic π-vacancy GSLs using a nearest-neighbor tight-binding (TB) model with one pz orbital per carbon site to identify the symmetry requirements for band gap opening. π-vacancies, resembling functionalized, substituted, or missing carbon sites, are modeled as site deletions in the π basis, and the corresponding hopping matrix elements to and from the deleted sites are set to zero. We focus on π-vacancy motifs with C₂ and C₃ point groups. A 3n× 3n GSL, where n= 1, 2, 3, … is the integer scaling factor multiplying the honeycomb primitive-cell vectors, is required to fold K and K′ to Γ, and thereby opens a band gap. For C₃-type vacancies, the Dirac cones are pinned at the high-symmetry points, thus remaining at Γ in the folded 3n GSLs. However, those C₂-type vacancies which reduce the global point group of the GSL to D2h by preserving a pair of perpendicular mirror symmetries (σv⊥σd), can constrain the positions of Dirac cones at Γ. The absence of the σv and σd planes allows the cones to shift around Γ (±Δq, ±Δq) in the 3n superlattice. The C₃-type vacancies open a relatively larger band gap compared to the C₂-type vacancies due to twofold degeneracy at Γ, while the C₂-type vacancies leave the GSL with nondegenerate bands. The energy contours near the Dirac cone for C₃-symmetric vacancies remain isotropic, while the absence of C₃ generates anisotropic cones. The band gap openings in 3n superlattices scale linearly with defect concentration for each vacancy motif, with the D6h motif yielding the largest gap, reaching 314 meV at only 3.7% defect concentration. A displacement of the C₃-type vacancies shifts the Dirac cones, lifts the degeneracy, and suppresses the gap.

Abstract

Periodic π-vacancies in graphene superlattices (GSLs) provide a symmetry-based route to band gap opening in graphene by modifying the π-band dispersion. Yet, the symmetry conditions that determine whether a vacancy motif can open a band gap remain unclear. Here, we investigate periodic π-vacancy GSLs using a nearest-neighbor tight-binding (TB) model with one pz orbital per carbon site to identify the symmetry requirements for band gap opening. π-vacancies, resembling functionalized, substituted, or missing carbon sites, are modeled as site deletions in the π basis, and the corresponding hopping matrix elements to and from the deleted sites are set to zero. We focus on π-vacancy motifs with C₂ and C₃ point groups. A 3n× 3n GSL, where n= 1, 2, 3, … is the integer scaling factor multiplying the honeycomb primitive-cell vectors, is required to fold K and K′ to Γ, and thereby opens a band gap. For C₃-type vacancies, the Dirac cones are pinned at the high-symmetry points, thus remaining at Γ in the folded 3n GSLs. However, those C₂-type vacancies which reduce the global point group of the GSL to D2h by preserving a pair of perpendicular mirror symmetries (σv⊥σd), can constrain the positions of Dirac cones at Γ. The absence of the σv and σd planes allows the cones to shift around Γ (±Δq, ±Δq) in the 3n superlattice. The C₃-type vacancies open a relatively larger band gap compared to the C₂-type vacancies due to twofold degeneracy at Γ, while the C₂-type vacancies leave the GSL with nondegenerate bands. The energy contours near the Dirac cone for C₃-symmetric vacancies remain isotropic, while the absence of C₃ generates anisotropic cones. The band gap openings in 3n superlattices scale linearly with defect concentration for each vacancy motif, with the D6h motif yielding the largest gap, reaching 314 meV at only 3.7% defect concentration. A displacement of the C₃-type vacancies shifts the Dirac cones, lifts the degeneracy, and suppresses the gap.

Abstract

Periodic π-vacancies in graphene superlattices (GSLs) provide a symmetry-based route to band gap opening in graphene by modifying the π-band dispersion. Yet, the symmetry conditions that determine whether a vacancy motif can open a band gap remain unclear. Here, we investigate periodic π-vacancy GSLs using a nearest-neighbor tight-binding (TB) model with one pz orbital per carbon site to identify the symmetry requirements for band gap opening. π-vacancies, resembling functionalized, substituted, or missing carbon sites, are modeled as site deletions in the π basis, and the corresponding hopping matrix elements to and from the deleted sites are set to zero. We focus on π-vacancy motifs with C₂ and C₃ point groups. A 3n× 3n GSL, where n= 1, 2, 3, … is the integer scaling factor multiplying the honeycomb primitive-cell vectors, is required to fold K and K′ to Γ, and thereby opens a band gap. For C₃-type vacancies, the Dirac cones are pinned at the high-symmetry points, thus remaining at Γ in the folded 3n GSLs. However, those C₂-type vacancies which reduce the global point group of the GSL to D2h by preserving a pair of perpendicular mirror symmetries (σv⊥σd), can constrain the positions of Dirac cones at Γ. The absence of the σv and σd planes allows the cones to shift around Γ (±Δq, ±Δq) in the 3n superlattice. The C₃-type vacancies open a relatively larger band gap compared to the C₂-type vacancies due to twofold degeneracy at Γ, while the C₂-type vacancies leave the GSL with nondegenerate bands. The energy contours near the Dirac cone for C₃-symmetric vacancies remain isotropic, while the absence of C₃ generates anisotropic cones. The band gap openings in 3n superlattices scale linearly with defect concentration for each vacancy motif, with the D6h motif yielding the largest gap, reaching 314 meV at only 3.7% defect concentration. A displacement of the C₃-type vacancies shifts the Dirac cones, lifts the degeneracy, and suppresses the gap.