Theory of triangulene two-dimensional crystals

Abstract

Abstract Equilateral triangle-shaped graphene nanoislands with a lateral dimension of n benzene rings are known as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext>[n]</mml:mtext> </mml:mrow> </mml:math> triangulenes. Individual <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext>[n]</mml:mtext> </mml:mrow> </mml:math> triangulenes are open-shell molecules, with single-particle electronic spectra that host n − 1 half-filled zero modes and a many-body ground state with spin <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:math> . The on-surface synthesis of triangulenes has been demonstrated for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> </mml:math> and the observation of a Haldane symmetry-protected topological phase has been reported in chains of [3]triangulenes. Here, we provide a unified theory for the electronic properties of a family of two-dimensional honeycomb lattices whose unit cell contains a pair of triangulenes with dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> . Combining density functional theory and tight-binding calculations, we find a wealth of half-filled narrow bands, including a graphene-like spectrum (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> ), spin-1 Dirac electrons (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:math> ), <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> </mml:math> -orbital physics (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:math> ), as well as a gapped system with flat valence and conduction bands (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> ). All these results are rationalized with a class of effective Hamiltonians acting on the subspace of the zero-energy states that generalize the graphene honeycomb model to the case of fermions with an internal pseudospin degree of freedom with C 3 symmetry.