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

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

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

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

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

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

7. Outline Energy/t Toy model of Coulomb interaction in armchair flat band Propose and test Jastrow-correlated wavefunctions: crystals and liquids Energy/t Model Coulomb interaction in zig-zag flat band Prediction: Ferromagnetic quantum crystals Wang and Scarola, arxiv:1108.0088 Wang and Scarola, PRB ‘11 Overview of flat bands V. Scarola Virginia Tech

8. 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

9. Classical vs. Quantum Flat Bands 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) 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) site (no hopping) V. Scarola Virginia Tech Classical Quantum

10. 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

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

12. Outline Energy/t Toy model of Coulomb interaction in armchair flat band Propose and test Jastrow-correlated wavefunctions: crystals and liquids Energy/t Model Coulomb interaction in zig-zag flat band Prediction: Ferromagnetic quantum crystals Wang and Scarola, arxiv:1108.0088 Wang and Scarola, PRB ‘11 Overview of flat bands V. Scarola Virginia Tech

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

14. N=12 particles N x =34 unit cells Smooth transition from crystal to a uniform liquid as basis states delocalize. Ground State in a Flat-Band of an Armchair Ribbon Density     V. Scarola Virginia Tech Wang and Scarola, PRB ‘11

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

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

17. Outline Energy/t Toy model of Coulomb interaction in armchair flat band Propose and test Jastrow-correlated wavefunctions: crystals and liquids Energy/t Model Coulomb interaction in zig-zag flat band Prediction: Ferromagnetic quantum crystals Wang and Scarola, arxiv:1108.0088 Wang and Scarola, PRB ‘11 Overview of flat bands V. Scarola Virginia Tech

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

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

20. (Coulomb units e 2 /  a ~ 27 eV) Matrix Elements from Single-Particle Wannier Functions V. Scarola Virginia Tech Wang and Scarola, arxiv:1108.0088

21. Flat Band Projection Flat band Coulomb model of zig-zag ribbons New projected operators are delocalized V. Scarola Virginia Tech Wang and Scarola, arxiv:1108.0088 FBR

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

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

24. 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 V. Scarola Virginia Tech H. Wang (Virginia Tech-> U. Hong Kong) Hao Wang and V. W. Scarola Phys. Rev. B 83, 245109 (2011) Hao Wang and V. W. Scarola arxiv:1108.0088