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Possibility of a strongly correlated Dirac metal in kagome lattice

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Ryutaro Okuma
Ryutaro Okuma

A brief review of Dirac cone are given with such materials as graphene and Ba(FeAs)2. A recently published paper which asserts that symmetry protected Dirac cone in kagome lattice be realized by substituting Ga for Zn in Herbertsmithite are introduced. Although, the proposed compound there would be necessarily regarded as just fantasy, newly synthesized transition metal oxides consisting of perfect kagome network are more appropriate to the search for the novel phase.

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Possibility of a strongly correlated Dirac metal in kagome lattice Hiroi Lab. M1 Ryutaro Okuma 1010/2014
Outline Dirac fermion Interesting phenomena caused by massless fermion Dirac cone in strongly correlated system A theoretical candidate for a Dirac metal: Ga- Herbertsmithite Structural stabilities, ferromagnetism, and f-wave superconductivity Promising compounds to be kagome Dirac metal
Dirac equation Classical equation describes relativistic motion of a free fermion If m = 0 dispersion becomes linear like photon m=0 m>0
Dirac fermion in solid state physics: graphene dispersion relation in tight binding model At Femi level energy bands crosses near K and K point (Dirac cone protected by chiral symmetry ) Zero gap semiconductor ( D(EF)=0 )
Half integer quantum Hall effect and SdH oscillation at room temperature In 2D systems, Hall conductivity xy takes on the quantized values Ne2/h at LOW T & in STRONG B The Hall conductivity of graphene: xy = (N+1/2)e2/h B = 14T, T = 4K Nature 438, 197-200(2005) Shubnikov de Haas Oscillation is observed at room temperature related to Berry Phase
Application: Transistor and Transparent Electrode Solid State Commun. 146, 351355 (2008) ACS Nano, 4 (1), pp 4348(2010) Dirac fermion: m = 0 High mobility 亮~200,000cm2/V-1s-1(Si~1400) Strong and Flexible Transparent Electrode LED, solar battery, touch panel
Graphene families: silicene, germanene, stanene Honeycomb lattice consisting of only Si, and Ge was synthesized recently Larger LS coupling leads to topological insulator Stanene(Sn graphene) would be zero on the edge. STM image of silicene on Ag(111) layer PRL 108, 155501 (2012) 77 structure of germanene New J. Phys. 16 095002 (2014) PRL 111, 136804 (2013)
Dirac cone in strongly correlated system: Ba(FeAs)2 Antiferromagnetic metal (Tw = 132K) By e- doping superconducting (max Tc = 56K) Dirac point due to the node of SDW gap Ba(FeAs)2 I4/mmmPRL 106, 217004 (2011) Magnetoresistance linear to B(usually B2) Dirac cone + strongly correlated e- Novel Physics?
A necessary condition for existence of Dirac point Generalized von Neumann Wigner Theorem nd=nu-m2+1-nc nd:dimension of the Dirac point (usually 0 point) nu: unknown parameters of Hamiltonian m: degeneracy at Dirac point (usually 2) nc: number of constraints by symmetry operation Phys. Rev. B 83, 245125 (2011). If theres a band crossing and nc=m2-1 thats symmetry protected Dirac cone.
Flat band and symmetry protected Dirac point on a kagome lattice Flat band from n=0 to n=1/3 Due to the properties of line graph Symmetry protected Dirac cone zero gap semiconductor Energy band of nearest neighbor Hubbard model on kagome lattice
Organic Kagome Dirac Metal: (EDT-TTF- CONH2)6[Re6Se8(CN)6] dimer TTF molecules form S=1/2 kagome lattice electron per site: 4/6*2=4/3 Dirac metal MI transition at 150~200K R-3 P1 [Re6Se8(CN)6]4- EDT-TTF- CONH2 SG:R-3 J. Am. Chem. Soc., 127 (33), 1178511797(2005)
A structurally perfect kagome mineral Herbertsmithite (ZnCu3(OH)6Cl2) S=1/2 antiferromagnetic Mott insulator Cu2+ forms uniform kagome network A candidate for a spin liquid Synthesized by hydrothermal method ZnCu3(OH)6Cl2 SG: R-3m Photo Copyright 息 Steve Rust
Ga substituted Herbertsmithite: GaCu3(OH)6Cl2 A theoretically proposed compound isostructural to Herbertsmithite (Nature Comm. 5 54261 (2014)) Valence of Cu is +1.67 (n=4/3) Dirac cone shifts between K and X (symmetry protected) Ion radius for Zn2+(4-fold): 0.47; Ga3+: 0.60 BZ and Femi surface of Ga-Hebertsmithite
Structural stability of Ga-Herbertsmithite Stability was checked by Extended Hubbard model U = 5-7eV, t = 0.3 eV, V = 0.11 eV Kagome structure is stable against weak distortion Strucual data for Ga-Hebertsmithite element x y z Ga(Zn) 0 0 0 Cu 1/3 1/6 1/6 O 0.1244 (0.1265) 0.24882 (0.2529) 0.09852 (0.1050) H 0.198 (0.192) 0.397 (0.384) 0.066 (0.084) Cl 0 0 0.31342 (0.30521) R-3m: 留=硫=90属, 粒=120属 a=6.9779(6.8342), c=13.4589(14.0320)
Weak ferromagnetism Nagaokas theorem asserts that t/U<<1 and half filled+1 electron should cause ferromagnetism Dynamical Cluster Approximation calculation suggests weakly polarized itinerant ferromagnetism at 1n4/3 At n > 4/3, ferromagnetism favored by suppression of super-exchange interaction. Phys. Rev. 147, 392(1966)
f-wave superconductivity Itinerant ferromagnetism induces BCS-superconductivity Wave function belongs to B1u D6h Tc as high as 60K??? B1u symmetry
Transition metal oxides with kagome network KM3Ge2O9: M=V3+(S=1, r=0.64), Mn3+(S=2, r=0.645) SG: P63mmc Substitution of M3+ for N2+ is more plausible than hydroxide compounds If M = Ti3+ (6-coordination, r=0.67) exists, it can be promising Dirac metal compounds by e- doping.
Possibility of dehydrated Vesignieite: BaCu3V2O9 BaCu3(VO4)2(OH)2 SG: R-3m BaCu3V2O9 Isostructural to KMn3Ge2O9 sharing V5+ ion leads to -H2O Substitution of Ba2+ for La3+ might realize Dirac metal
Summary Dirac fermion creates novel physics Dirac cone is protected by kagome symmetry at n=4/3 Ferromagnetism and related f-wave superconductivity might appear on kagome Dirac metal Kagome oxides are candidate for a kagome Dirac metal

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Possibility of a strongly correlated Dirac metal in kagome lattice

  • 1. Possibility of a strongly correlated Dirac metal in kagome lattice Hiroi Lab. M1 Ryutaro Okuma 1010/2014
  • 2. Outline Dirac fermion Interesting phenomena caused by massless fermion Dirac cone in strongly correlated system A theoretical candidate for a Dirac metal: Ga- Herbertsmithite Structural stabilities, ferromagnetism, and f-wave superconductivity Promising compounds to be kagome Dirac metal
  • 3. Dirac equation Classical equation describes relativistic motion of a free fermion If m = 0 dispersion becomes linear like photon m=0 m>0
  • 4. Dirac fermion in solid state physics: graphene dispersion relation in tight binding model At Femi level energy bands crosses near K and K point (Dirac cone protected by chiral symmetry ) Zero gap semiconductor ( D(EF)=0 )
  • 5. Half integer quantum Hall effect and SdH oscillation at room temperature In 2D systems, Hall conductivity xy takes on the quantized values Ne2/h at LOW T & in STRONG B The Hall conductivity of graphene: xy = (N+1/2)e2/h B = 14T, T = 4K Nature 438, 197-200(2005) Shubnikov de Haas Oscillation is observed at room temperature related to Berry Phase
  • 6. Application: Transistor and Transparent Electrode Solid State Commun. 146, 351355 (2008) ACS Nano, 4 (1), pp 4348(2010) Dirac fermion: m = 0 High mobility 亮~200,000cm2/V-1s-1(Si~1400) Strong and Flexible Transparent Electrode LED, solar battery, touch panel
  • 7. Graphene families: silicene, germanene, stanene Honeycomb lattice consisting of only Si, and Ge was synthesized recently Larger LS coupling leads to topological insulator Stanene(Sn graphene) would be zero on the edge. STM image of silicene on Ag(111) layer PRL 108, 155501 (2012) 77 structure of germanene New J. Phys. 16 095002 (2014) PRL 111, 136804 (2013)
  • 8. Dirac cone in strongly correlated system: Ba(FeAs)2 Antiferromagnetic metal (Tw = 132K) By e- doping superconducting (max Tc = 56K) Dirac point due to the node of SDW gap Ba(FeAs)2 I4/mmmPRL 106, 217004 (2011) Magnetoresistance linear to B(usually B2) Dirac cone + strongly correlated e- Novel Physics?
  • 9. A necessary condition for existence of Dirac point Generalized von Neumann Wigner Theorem nd=nu-m2+1-nc nd:dimension of the Dirac point (usually 0 point) nu: unknown parameters of Hamiltonian m: degeneracy at Dirac point (usually 2) nc: number of constraints by symmetry operation Phys. Rev. B 83, 245125 (2011). If theres a band crossing and nc=m2-1 thats symmetry protected Dirac cone.
  • 10. Flat band and symmetry protected Dirac point on a kagome lattice Flat band from n=0 to n=1/3 Due to the properties of line graph Symmetry protected Dirac cone zero gap semiconductor Energy band of nearest neighbor Hubbard model on kagome lattice
  • 11. Organic Kagome Dirac Metal: (EDT-TTF- CONH2)6[Re6Se8(CN)6] dimer TTF molecules form S=1/2 kagome lattice electron per site: 4/6*2=4/3 Dirac metal MI transition at 150~200K R-3 P1 [Re6Se8(CN)6]4- EDT-TTF- CONH2 SG:R-3 J. Am. Chem. Soc., 127 (33), 1178511797(2005)
  • 12. A structurally perfect kagome mineral Herbertsmithite (ZnCu3(OH)6Cl2) S=1/2 antiferromagnetic Mott insulator Cu2+ forms uniform kagome network A candidate for a spin liquid Synthesized by hydrothermal method ZnCu3(OH)6Cl2 SG: R-3m Photo Copyright 息 Steve Rust
  • 13. Ga substituted Herbertsmithite: GaCu3(OH)6Cl2 A theoretically proposed compound isostructural to Herbertsmithite (Nature Comm. 5 54261 (2014)) Valence of Cu is +1.67 (n=4/3) Dirac cone shifts between K and X (symmetry protected) Ion radius for Zn2+(4-fold): 0.47; Ga3+: 0.60 BZ and Femi surface of Ga-Hebertsmithite
  • 14. Structural stability of Ga-Herbertsmithite Stability was checked by Extended Hubbard model U = 5-7eV, t = 0.3 eV, V = 0.11 eV Kagome structure is stable against weak distortion Strucual data for Ga-Hebertsmithite element x y z Ga(Zn) 0 0 0 Cu 1/3 1/6 1/6 O 0.1244 (0.1265) 0.24882 (0.2529) 0.09852 (0.1050) H 0.198 (0.192) 0.397 (0.384) 0.066 (0.084) Cl 0 0 0.31342 (0.30521) R-3m: 留=硫=90属, 粒=120属 a=6.9779(6.8342), c=13.4589(14.0320)
  • 15. Weak ferromagnetism Nagaokas theorem asserts that t/U<<1 and half filled+1 electron should cause ferromagnetism Dynamical Cluster Approximation calculation suggests weakly polarized itinerant ferromagnetism at 1n4/3 At n > 4/3, ferromagnetism favored by suppression of super-exchange interaction. Phys. Rev. 147, 392(1966)
  • 16. f-wave superconductivity Itinerant ferromagnetism induces BCS-superconductivity Wave function belongs to B1u D6h Tc as high as 60K??? B1u symmetry
  • 17. Transition metal oxides with kagome network KM3Ge2O9: M=V3+(S=1, r=0.64), Mn3+(S=2, r=0.645) SG: P63mmc Substitution of M3+ for N2+ is more plausible than hydroxide compounds If M = Ti3+ (6-coordination, r=0.67) exists, it can be promising Dirac metal compounds by e- doping.
  • 18. Possibility of dehydrated Vesignieite: BaCu3V2O9 BaCu3(VO4)2(OH)2 SG: R-3m BaCu3V2O9 Isostructural to KMn3Ge2O9 sharing V5+ ion leads to -H2O Substitution of Ba2+ for La3+ might realize Dirac metal
  • 19. Summary Dirac fermion creates novel physics Dirac cone is protected by kagome symmetry at n=4/3 Ferromagnetism and related f-wave superconductivity might appear on kagome Dirac metal Kagome oxides are candidate for a kagome Dirac metal