Abstract:A nodal-line
semimetal (NLSM) is suppressed in the presence of
spin–orbit coupling unless it is protected by a nonsymmorphic
symmetry. We show that two-dimensional (2D) materials can realize
robust NLSMs when vacancies are introduced on the lattice. As a case
study we investigate borophene, a boron honeycomb-like sheet. While
the Dirac cones of pristine borophene are shown to be gapped out by
spin–orbit coupling and by magnetic exchange, robust nodal
lines (NLs) emerge in the spectrum when selected atoms are r… Show more
“…This behavior should be distinguished from that in Ref. 18 , where the inversion-point combines with a last local constraint on the Hamiltonian to generate NLs at . Within the present framework, the influence of the inversion-point symmetry alone, i.e.…”
Section: Resultsmentioning
confidence: 58%
“…These nodal points, which were not unveiled in Ref. 18 , are the precursors of symmetry-enforced NLs. Figure 3 ( a ) Doubly-degenerate eigenvalues of the glide-plane matrix winding around the half-unit circles on the complex plane, one pair of eigenvalues from to i through 1, and the other pair from i to through − 1.…”
Section: Resultsmentioning
confidence: 78%
“…Quite surprisingly, besides symmetry-enforced NLs, vacancy-engineered materials exhibit also accidental NLs which survive under very strong SOC, with strength even exceeding the experimentally attainable values. These additional NLs were first noted in our previous report 18 where their robustness was conjectured to be symmetry enforced since accidental NLs had so far been known to be easily destroyed by perturbations. However, these NLs can be moved inside the Brillouin zone (BZ) by varying the parameters of the material, contradicting a known feature of symmetry-enforced NLs, namely, that they are pinned at high-symmetry momenta in the BZ.…”
Section: Introductionmentioning
confidence: 64%
“…Turning to the time-reversal transformation T , not considered in Ref. 18 (where the focus was on breaking time-reversal symmetry by magnetic proximity effect), the invariance relation is given by where since is spinless.…”
Symmetry-enforced nodal-line semimetals are immune to perturbations that preserve the underlying symmetries. This intrinsic robustness enables investigations of fundamental phenomena and applications utilizing diverse materials design techniques. The drawback of symmetry-enforced nodal-line semimetals is that the crossings of energy bands are constrained to symmetry-invariant momenta in the Brillouin zone. On the other end are accidental nodal-line semimetals whose band crossings, not being enforced by symmetry, are easily destroyed by perturbations. Some accidental nodal-line semimetals have, however, the advantage that their band crossings can occur in generic locations in the Brillouin zone, and thus can be repositioned to tailor material properties. We show that lattice engineering with periodic distributions of vacancies yields a hybrid type of nodal-line semimetals which possess symmetry-enforced nodal lines and accidental nodal lines, with the latter endowed with an enhanced robustness to perturbations. Both types of nodal lines are explained by a symmetry analysis of an effective model which captures the relevant characteristics of the proposed materials, and are verified by first-principles calculations of vacancy-engineered borophene polymorphs. Our findings offer an alternative path to relying on complicated compounds to design robust nodal-line semimetals; one can instead remove atoms from a common monoatomic material.
“…This behavior should be distinguished from that in Ref. 18 , where the inversion-point combines with a last local constraint on the Hamiltonian to generate NLs at . Within the present framework, the influence of the inversion-point symmetry alone, i.e.…”
Section: Resultsmentioning
confidence: 58%
“…These nodal points, which were not unveiled in Ref. 18 , are the precursors of symmetry-enforced NLs. Figure 3 ( a ) Doubly-degenerate eigenvalues of the glide-plane matrix winding around the half-unit circles on the complex plane, one pair of eigenvalues from to i through 1, and the other pair from i to through − 1.…”
Section: Resultsmentioning
confidence: 78%
“…Quite surprisingly, besides symmetry-enforced NLs, vacancy-engineered materials exhibit also accidental NLs which survive under very strong SOC, with strength even exceeding the experimentally attainable values. These additional NLs were first noted in our previous report 18 where their robustness was conjectured to be symmetry enforced since accidental NLs had so far been known to be easily destroyed by perturbations. However, these NLs can be moved inside the Brillouin zone (BZ) by varying the parameters of the material, contradicting a known feature of symmetry-enforced NLs, namely, that they are pinned at high-symmetry momenta in the BZ.…”
Section: Introductionmentioning
confidence: 64%
“…Turning to the time-reversal transformation T , not considered in Ref. 18 (where the focus was on breaking time-reversal symmetry by magnetic proximity effect), the invariance relation is given by where since is spinless.…”
Symmetry-enforced nodal-line semimetals are immune to perturbations that preserve the underlying symmetries. This intrinsic robustness enables investigations of fundamental phenomena and applications utilizing diverse materials design techniques. The drawback of symmetry-enforced nodal-line semimetals is that the crossings of energy bands are constrained to symmetry-invariant momenta in the Brillouin zone. On the other end are accidental nodal-line semimetals whose band crossings, not being enforced by symmetry, are easily destroyed by perturbations. Some accidental nodal-line semimetals have, however, the advantage that their band crossings can occur in generic locations in the Brillouin zone, and thus can be repositioned to tailor material properties. We show that lattice engineering with periodic distributions of vacancies yields a hybrid type of nodal-line semimetals which possess symmetry-enforced nodal lines and accidental nodal lines, with the latter endowed with an enhanced robustness to perturbations. Both types of nodal lines are explained by a symmetry analysis of an effective model which captures the relevant characteristics of the proposed materials, and are verified by first-principles calculations of vacancy-engineered borophene polymorphs. Our findings offer an alternative path to relying on complicated compounds to design robust nodal-line semimetals; one can instead remove atoms from a common monoatomic material.
“…We proceed to demonstrate that the proposed mechanism can coexist with another vacancy engineering principle proposed recently, namely the nodalline semimetals caused by 2D nonsymmorphic vacancy configurations. [50][51][52] In these nonsymmorphic configurations, every two carbon atoms map to each other under glide plane operation, resulting in nodal lines at the BZ boundary regardless the details of the Hamiltonian, which serves as a vacancy-engineering principle to ob-tain symmetry-enforced nodal lines. [53][54][55][56] As an example, in Fig.…”
Section: Coexistence Of Zefbs and Nodal Linesmentioning
A bipartite lattice with chiral symmetry is known to host zero energy flat bands if the numbers of the two sublattices are different. We demonstrate that this mechanism of producing flat bands can be realized on graphene by introducing periodic vacancies. Using first-principle calculations, we elaborate that even though the pristine graphene does not exactly preserve chiral symmetry, this mechanism applied to holey graphene still produces single or multiple bands as narrow as ∼ 0.5eV near the Fermi surface throughout the entire Brillouin zone. Moreover, this mechanism can combine with vacancy-engineered nonsymmorphic symmetry to produce band structures with coexisting flat bands and nodal lines. A weak coupling mean-field treatment suggests the stabilization of superconductivity by these vacancy-engineered narrow bands. In addition, superconductivity occurs predominantly on the majority sublattices, with an amplitude that increases with the number of narrow bands.
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