Strong Correlation Effects in Fullerene Molecules and Solids

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1 Strong Correlation Effects in Fullerene Molecules and Solids Fei Lin Physics Department, Virginia Tech, Blacksburg, VA 2461 Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 1 / 3

2 Collaborators: Erik Sorensen, McMaster University, Canada Catherine Kallin, McMaster University, Canada John Berlinsky, McMaster University and Perimeter Institute for Theoretical Physics, Canada Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 2 / 3

3 Outline: Introduction to Fullerene molecules and solids. Effective tight-binding Hubbard model for the molecules. C 2 molecule and solid: pair-binding energies; magnetic properties; density of states, etc. C 6 molecule and solid: Single-particle excitation spectrum for solid; molecular orientations; comparison with ARPES. Some recent experimental developments. Summary. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 3 / 3

2D: planar sheet structure: graphene (Nobel prize 21) Fei Lin

4 Introduction Carbon based nano materials D: cage structures: C 6, C 2, etc. (Nobel Prize in 1996) 1D: rod structures: nanotube. 2D: planar sheet structure: graphene (Nobel prize 21) Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 4 / 3

5 Introduction Intensive researches in the past 21, derivatives of fullerene found by the end of 22; 23, papers published by the end of 22; Amazing properties: superconducting, ferromagnetic, antiferromagnetic, spin density wave,... Crystal structures seems relatively simpler than cuprates; However, after 2 years research superconducting mechanism still unclear. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 5 / 3

A consequence of Euler s polyhedral formula: V E +F = 2.

6 Introduction. What are Fullerene molecules? Carbon cage molecules with exactly 12 pentagons and varying number of hexagons. A consequence of Euler s polyhedral formula: V E +F = 2. Three examples: C 2 : V = 2, E = 3, F = 12; C 36 : V = 36, E = 54, F = 2; C 6 : V = 6, E = 9, F = 32. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 6 / 3

7 Introduction. Synthesis and properties. C 6 discovered by Kroto, Curl, and Smalley [Nature, 1985] (Nobel Prize in 1996). K 3 C 6 molecular superconductor (Tc=18K) found by Hebard et al [Nature, 1991]. Pentagon separation rule, Kroto [Nobel lecture, 1997]. Solid C 36 first produced by Zettl group [Nature, 1998]. Gas phase C 2 first produced by Prinzbach et al [Nature, 2]. Solid C 2 synthesized by Wang et al [Phys. Lett. A, 21] and Iqbal et al [Eur. Phys. J. B, 23]. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 7 / 3

8 Outline: Introduction to Fullerene molecules and solids. Effective tight-binding Hubbard model for the molecules. C 2 molecule and solid: pair-binding energies; magnetic properties; density of states, etc. C 6 molecule and solid: Single-particle excitation spectrum for solid; molecular orientations; comparison with ARPES. Some recent experimental developments. Summary. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 8 / 3

9 Effective Hamiltonian Approach Importance of Carbon radial p r orbitals Fullerene molecules electronic properties are mainly determined by the electrons from Carbon atom radial p r orbitals. Eg. for C 6 molecule, 94% of bonds around Fermi energy are made of p r orbitals as shown by Satpathy [Chem. Phys. Lett., 1985]. Conjugated π bonds: electron delocalization One-band tight-binding Hamiltonian description can be successful. Effective Hamiltonian: Hubbard Model H = ij σ t ij (c iσ c jσ +h.c.)+u i n i n i, (1) What are U/t values? Solve by quantum Monte Carlo (QMC) and exact diagonalization (ED). Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 9 / 3

10 Effective Hamiltonian Approach Huckel energy level diagrams For U = (non-interacting), we can easily diagonalize Hamiltonian: Molecular Orbitals of C C36 Molecular Orbitals Energy / t C2 Molecular Orbitals T 2u Gg G u T1u Ag Hg Energy / t LUMO HOMO Energy (t) t 1g t 1u h u h g Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 1 / 3

11 Effective Hamiltonian Approach Huckel energy level diagrams: electron dopings Energy (t) Molecular Orbitals of C 6 t 1g t 1u h u h g C 6 : quasi 1D lattice, phase transition at 5K from 1D conductor to spin density wave ground state; Chauvet et al [Phys. Rev. Lett. 1994], Stephens et al [Nature 1994]. C 2 6 : no; not yet realized. : yes; discuss this later. C 3 6 C 4 6 C 5 : insulator; Fleming et al [Nature 1991] 6 : metal monolayer; Crommie group [Science 25, Phys. Rev. Lett. 27] C 6 6 : insulator; Zhou et al [Nature 1991], Stephens et al [Phys. Rev. B 1992] Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 11 / 3

12 Effective Hamiltonian Approach Parameters Determination Hopping t ij can be determined by fitting Huckel energy levels around Fermi energy to those from DFT calculations. U can be determined by comparison to some experiments: Electron affinity energy Wang et al [J. Chem. Phys., 1999] for C 6 Prinzbach et al [Nature, 2] for C 2. X-ray scattering Chow and Friedman [Phys. Rev. B, 28] Dynamic structure factor: S(k, ω). AE = E(C N ) E(C N ). (2) Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 12 / 3

13 Effective Hamiltonian Approach Parameters for C 2 molecule By matching energy gaps from DFT and Huckel energy levels, we can determine the hopping t 1.36 ev. By matching AE = E(2) E(21) to experimental value of 2.25 ev, we find U/t 7.1. Consistent with U values assumed in some literatures. AE (ev) cage,qmc cage, ED bowl,qmc ring, QMC U/t Parameters for C 6 molecule t 2.5 ev U/t 4 Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 13 / 3

14 Outline: Introduction to Fullerene molecules and solids. Effective tight-binding Hubbard model for the molecules. C 2 molecule and solid: pair-binding energies; magnetic properties; density of states, etc. C 6 molecule and solid: Single-particle excitation spectrum for solid; molecular orientations; comparison with ARPES. Some recent experimental developments. Summary. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 14 / 3

15 Single C 2 molecule Magnetic properties Magnetic from Huckel energy diagram; With ED and QMC: Spin triplet (magnetic) for U/t < U c /t 4.1 (3-fold orbital degeneracy); Spin singlet for U > U c (non-degenerate); Hund s rule violation; Jahn-Teller is suppressed for U > U c. Above for I h symmetry; similar for D 3d symmetry (except at U = and U c 4.19) E T 2u G g G u H g T 1u A g _ I h ] ] ] ε= ε=1 ] D 3d (a) (b) (c) Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 15 / 3

16 Single C 2 molecule Pair-binding energy b (N +1) = E(N +2) 2E(N +1)+E(N). (3) Importance of mesoscale structure in electronic mechanism of superconductivity, Chakravarty and Kivelson [Science, 1991; Phys. Rev. B, 21]; b (21)/t (a) PQMC, I h ED, I h ED, D3D, ε=1 (b) b (21)/t b always positive absence of pair binding U/t t/u.5. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 16 / 3

17 Single C 2 molecule Jahn-Teller distortion of C 2 with I h symmetry D 3d has 4 bond lengths: a ab = 1.464Å, a bc = 1.469Å, a cc = 1.519Å, and a cc = 1.435Å, Yamamoto et al [Phys. Rev. Lett., 25]; t a /t = 1 ε(a ā)/ā, where I h (ε = ) and D 3d (ε = 1); Jahn-Teller suppressed for U > U c. b (21)/t (b).5 1 ε U=2 U= ε E (a) Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 17 / 3

C 2 solid Lattice structure Solid C 2 synthesized by Wang et al [Phys. Lett. A, 21] and Iqbal et al [Eur. Phys. J. B, 23]. fcc structure; 1 additional carbon between 4 C 2 molecules.

18 C 2 solid Lattice structure Solid C 2 synthesized by Wang et al [Phys. Lett. A, 21] and Iqbal et al [Eur. Phys. J. B, 23]. fcc structure; 1 additional carbon between 4 C 2 molecules. t intra-molecular hopping; t inter-molecular hopping. Cluster perturbation theory, with exact diagonalization (ED) by Sénéchal et al [Phys. Rev. Lett., 2]; with QMC by Lin et al [HPC6, p.27]. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 18 / 3

19 C 2 solid Metal-insulator transition Magnetic to nonmagnetic transition at U = U c ; Metal to insulator transition at U = U c, too; Appearance of gap in DOS. N(ω) N(ω) U=2t, QMC U=5t, ED 1.39t Single C 2 Molecule FCC, t =-t (a) (b) (ω µ)/t Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 19 / 3

20 C 2 solid Metal-insulator transition Spectral dispersions: A(k,ω) A(k,ω) U=2t, t =-t, QMC U=5t, t =-t, ED (a) (b) K Γ L W X Γ K Γ L W X (ω µ)/t Γ Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 2 / 3

21 Outline: Introduction to Fullerene molecules and solids. Effective tight-binding Hubbard model for the molecules. C 2 molecule and solid: pair-binding energies; magnetic properties; density of states, etc. C 6 molecule and solid: Single-particle excitation spectrum for solid; molecular orientations; comparison with ARPES. Some recent experimental developments. Summary. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 21 / 3

22 Single C 6 molecule Pair-binding energy Perturbation calculations Chakravarty et al [Science 1992]; Ground state of C 2 6 is in spin-singlet channel at U/t > 3; b < for U/t > 3.3. QMC calculations Ground state of C 2 6 is in spin triplet channel; b > for U/t 4.5 b (61)/t L=1,S=1 -.1 L=,S= PQMC U/t Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 22 / 3

23 Single C 6 molecule Density of states (DOS) (a) C 6, N=6 Neutral C 6 molecule is insulating; Three electron doped C 6 molecule is metallic. Intensity (arb. units) (b) C 6, N= (E-E F )/t Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 23 / 3

24 C 6 monolayer Monolayer setup 2D hexagonal C 6 superlattice: Z.X. Shen s group [Science, 23; Phys. Rev. B, 25] Orientation dependent band dispersion. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 24 / 3

25 C 6 monolayer Intermolecular hopping matrix V = (t α Ii,Jj +t ind Ii,Jj )(c Iiσ c Jjσ +h.c.), (4) Ii,Jj σ where direct hoppings between NN molecules are given by α Ii,Jj = [V σ (d) V π (d)](ˆr Ii ˆd)(ˆR Jj ˆd)+V π (d)(ˆr Ii ˆR Jj ), (5) Satpathy et al [Phys. Rev. B, 1992], and indirect hoppings via K + ions are given by t ind Ii,Jj = γ t Ii,γ t Jj,γ ǫ C ǫ K, (6) Gunnarsson et al [Phys. Rev. B, 1998]. t Ii,γ = 1.84D 2 md 2e 3(d R min ). (7) Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 25 / 3

26 C 6 monolayer Band dispersion matching Intermolecular hopping t and molecular orientation angle θ are variational parameters. Cluster perturbation theory. Best fit gives t =.3t, θ = 64. E-E F (ev) E-E F (ev) k // (1/Angstrom) o M Γ k // (1/Angstrom) Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 26 / 3

27 E-E F (ev) Θ= o M Γ E-E F (ev) Θ=1 o M Γ k // k // E-E F (ev) Θ=2 o M Γ E-E F (ev) Θ=3 o M Γ E-E F (ev) k // k // k // (1/Angstrom) E-E F (ev) Θ=4 o M Γ E-E F (ev) Θ=5 o M Γ E-E F (ev) o M Γ k // k // k // (1/Angstrom) E-E F (ev) Θ=6 o M Γ E-E F (ev) Θ=7 o M Γ k // k // Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 27 / 3

28 C 6 monolayer Monolayer DOS Evolution of DOS with U and electron doping. Comparison with experimental PES. (a) C 6 monolayer, U= (a) C 6 monolayer exp QMCPT Intensity (arb. units) (b) C 6 monolayer, U=4t (c) K 3 C 6 monolayer, U=4t Intensity (arb. units) (b) K 3 C 6 monolayer exp QMCPT (E-E F )/t E-E F (ev) Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 28 / 3

29 Recent developments in experiments Future projects K 4 C 6 monolayer, Crommie group [Science, 25]; K x C 6 monolayers 3 x 5, Crommie group [Phys. Rev. Lett., 27]; Why is K 4 C 6 insulating? Jahn-Teller? Role of orientational ordering? Cs 3 C 6 superconductor, Rosseinsky group [Science, 29] T c = 38K; disorder-free parent state: AF insulator; pressure-inudced SC unconventional superconductor Single molecule transistors: Winkelmann et al [Nature Physics 29], Yu et al [Nano Lett. 24], etc. Can we calculate transport property? Carbon is light. Intramolecular vibration frequency up to.2 ev; E F =.3 ev. Can we include Carbon dynamics in QMC? Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 29 / 3

30 Summary Effective tight-binding Hubbard model can capture most of the physics of Fullerene molecules; U/t increases with molecular curvature: C 2 is the most correlated molecule; C 2 molecule/solid undergoes magnetic metal to nonmagnetic insulator transition at U c /t 4.1; Neutral C 2 is an insulator; Jahn-Teller distortion is suppressed in C 2 molecule since U/t 7.1; Neutral C 6 is an insulator; K 3 C 6 a metal; Not able to find favorable pair binding from b calculations. Able to compare with ARPES for molecular orientation dependent band dispersion. Fei Lin (Virginia Tech) Correlations in Fullerene SESAPS 211, Roanoke, VA 3 / 3

arxiv:cond-mat/ v2 [cond-mat.str-el] 24 Feb 2006

arxiv:cond-mat/ v2 [cond-mat.str-el] 24 Feb 2006 Applications of Cluster Perturbation Theory Using Quantum Monte Carlo Data arxiv:cond-mat/0512406v2 [cond-mat.str-el] 24 Feb 2006 Fei Lin, Erik S. Sørensen, Catherine Kallin and A. John Berlinsky Department