Flat Band Nanostructures Vito Scarola www phys vt

Flat Band Nanostructures Vito Scarola www. phys. vt. edu/scarola

Graphene • Graphene: honeycomb sheets of carbon atoms that allow electrons to hop from atom to atom Novoselov, K. S. et al. PNAS ’ 05 • Low energy electronic band structure of infinite graphene has Dirac cones P. R. Wallace, Phys. Rev. ‘ 47 Castro Neto et al. , RMP ‘ 09 V. Scarola Virginia Tech

Graphene Nano-Flakes J. H. Chen et al. , Nature Nanotechnology ‘ 08 • Real graphene nanostructure samples have disorder from substrates, defects, edge roughness, etc. • Processing goal: reduce imperfections V. Scarola Virginia Tech

Rough Graphene Nanoribbons C. Stampfer et al. PRL ‘ 09 Han et al. PRL ‘ 09 500 nm • Transport in rough nanoribbons dominated by substrate disorder and edge roughness V. Scarola Virginia Tech

Graphene Nanoribbons from Nanotubes • Less rough graphene nanoribbons from unzipped carbon nanotubes L. Jiao et al. , Nano Res ‘ 10 • STM reveals magnetic properties C. Tao et al. , Nature ‘ 09 V. Scarola Virginia Tech

Graphene Nanostructures: Interesting Topics of Study • Transport properties • Device Applications • Disorder Effects • Edge Roughness • Surface and Edge Chemistry • Band Effects • Strong Interaction This Talk • Magnetism V. Scarola Virginia Tech

Outline • Toy model of Coulomb interaction in armchair flat band • Propose and test Jastrow-correlated wavefunctions: crystals and liquids Energy/t • Overview of flat bands • Model Coulomb interaction in zig-zag flat band • Prediction: Ferromagnetic quantum crystals Energy/t Wang and Scarola, PRB ‘ 11 Wang and Scarola, arxiv: 1108. 0088 V. Scarola Virginia Tech

Example Zero-Field Flat-Band Nanostructures Pressed carbon nanotubes Graphene edges Graphene quantum dots/antidots Graphene nanoribbons • Interactions are the de facto dominant energy scale • Absence of conventional screening V. Scarola Virginia Tech

Classical vs. Quantum Flat Bands (no hopping) site Classical • Particles in well separated sites have small kinetic energy (flat bands) because they are classically localized • Interaction effects will then be classical because charge is localized (commuting density operators) Quantum • Particles in “flat-band lattices” interfere quantum mechanically to give no net kinetic energy (flat bands) • Interactions can lead to quantum effects because single-particle charge is delocalized (non-commuting density operators) V. Scarola Virginia Tech

Fractional Quantum Hall Models: Quantum Flat Bands Pan et al. PRL ‘ 06 • Coulomb interaction leads to a superstructure in flat kinetic energy bands: Landau levels • Low energy physics captured by composite fermion wavefunctions Jain PRL ‘ 89 V. Scarola Virginia Tech

Zero-Field Flat Bands in Ribbons Nearest neighbor hopping on honeycomb lattices: Zig-Zag ribbons Energy/t Armchair ribbons K. Nakada et al. PRB ‘ 96 V. Scarola Virginia Tech

Outline • Toy model of Coulomb interaction in armchair flat band • Propose and test Jastrow-correlated wavefunctions: crystals and liquids Energy/t • Overview of flat bands • Model Coulomb interaction in zig-zag flat band • Prediction: Ferromagnetic quantum crystals Energy/t Wang and Scarola, PRB ‘ 11 Wang and Scarola, arxiv: 1108. 0088 V. Scarola Virginia Tech

A Flat Band Model of Armchair Honeycomb Ribbons Second quantized Hamiltonian of polarized electrons in an interacting flat band: Toy Gaussian model of single-particle basis states: Exact Diagonalization (eigenstate) Wang and Scarola, PRB ‘ 11 V. Scarola Virginia Tech

Ground State in a Flat-Band of an Armchair Ribbon Wang and Scarola, PRB ‘ 11 Density b=2 n=1/3 b=3. 5 N=12 particles Nx=34 unit cells b=5 • Smooth transition from crystal to a uniform liquid as basis states delocalize. V. Scarola Virginia Tech

Jastrow Correlated Ansatz States • A general scheme for attaching correlation holes with first quantized wavefunctions Lattice filling Wannier Function Variational Parameter • mc correlation holes attached to particles in state yn* Wang and Scarola, PRB ‘ 11 V. Scarola Virginia Tech

Verification of Ansatz Wavefunctions Wang and Scarola, PRB ‘ 11 Δρc (basis state width) • Ansatz wave function accurately captures exact ground state in all regimes V. Scarola Virginia Tech

Outline • Toy model of Coulomb interaction in armchair flat band • Propose and test Jastrow-correlated wavefunctions: crystals and liquids Energy/t • Overview of flat bands • Model Coulomb interaction in zig-zag flat band • Prediction: Ferromagnetic quantum crystals Energy/t Wang and Scarola, PRB ‘ 11 Wang and Scarola, arxiv: 1108. 0088 V. Scarola Virginia Tech

Recent Work on Graphene Ribbons C. Tao et al. Nature Phys. ’ 11 • Mean field theory with the Hubbard model: ferromagnetic coupling along edges but antiferromagnetic coupling perpendicular to edges. O. Yazyev et al. , PRB ‘ 11 V. Scarola Virginia Tech

Interactions Dominate at Low Filling u Energy/t H=Coulomb + Corrections d u d H=Kinetic + Corrections • Focus on flat bands at low filling: Interactions dominate V. Scarola Virginia Tech

Matrix Elements from Single-Particle Wannier Functions Wang and Scarola, arxiv: 1108. 0088 (Coulomb units e 2/ea~ 27 e. V) V. Scarola Virginia Tech

Flat Band Projection FBR • New projected operators are delocalized • Flat band Coulomb model of zig-zag ribbons Wang and Scarola, arxiv: 1108. 0088 V. Scarola Virginia Tech

Predictions from Model at 1/3 Filling of Upper Band Ground State • Ferromagnetic Quantum Crystals Spin Excitations • Spin Waves J~400 K V. Scarola Virginia Tech

A Broad Connection Quantum Hall Zero-Field Flat-Band Lattices Experiments? V. Scarola Virginia Tech

Summary and Outlook • Jastrow-correlated wavefunctions capture physics of quantum liquids and crystals in flat band lattices • Models of zig-zag ribbons at low fillings • Future Work: Explore new models and wavefunctions for novel ground states and excitations Hao Wang and V. W. Scarola Phys. Rev. B 83, 245109 (2011) Hao Wang and V. W. Scarola arxiv: 1108. 0088 H. Wang (Virginia Tech-> U. Hong Kong) V. Scarola Virginia Tech