Report

INTERNATIONAL RESEACH SCHOOL AND WORKSHOP ON ELECTRONIC CRYSTALS ECRYS2011 August 15 -27, 2011 Cargèse, France Dirac electrons in solid Hidetoshi Fukuyama Tokyo Univ. of Science Acknowledgement Bi Yuki Fuseya (Osaka Univ.) Masao Ogata (Tokyo Univ.) α-ET2I3 Akito Kobayashi(Nagoya Univ.) Yoshikazu Suzumura (Nagoya Univ.) Dirac electrons in solids contents 1) “elementary particles” in solids <= band structure , locally in k-space 2) Band structure similar to Dirac electrons Examples: bismuth, graphite-graphene molecular solids αET2I3, FePn, Ca3PbO 4x4 (spin-orbit interaction), 2x2 (Weyl eq.) 3) Particular features of Dirac electrons small band gap => inter-band effects of magnetic field effects Hall effect, magnetic susceptibility Dirac equations for electrons in vacuum 4x4 matrix Equivalently, In special cases of m=0, 2x2 matrix Weyl equation for neutrino “Elementary particles in solids” band structures, locally in k-space Semiconductors , Carrier doping Si electron doping ->n type hole doping -> p type InSb electrons holes Dispersion relation＝＞effective masses and g-factors “elementary particles” Luttinger-Kohn representation (k・p approximation) LK vs. Bloch representation Bloch representation: energy eigen-states Ψnk(r)= eikrunk(r) : unk(r+a)=unk(r) Luttinger –Kohn representation “k・p method” [ Phys. Rev. 97, 869 (1955) ] Hamiltonian is essentially a matrix Χnk(r)= eikrunk0(r) k0 = some special point of interest Spin-orbit interaction If εn(k) has extremum at k0 LK vs. Bloch * LK forms complete set and are related to Bloch by unitary transformation * k-dependences are completely different, * in Bloch, both eikr and unk(r) , the latter being very complicated, while in LK only in eikr as for free electrons. * just replace k=> k+eA/c in Hamiltonian matrix once in the presence of magnetic field Dirac types of energy dispersion(1) *Graphite [ P. R. Wallace (1947),J.W. McClure(1957)] semimetal（ne=nh≠０） McClure(1957) *graphene: special case of graphite （ne=nh=０）Geim H = v( kxσx + kyσy ) Weyl eq. for neutrino Isotropic velocity Dirac types of energy dispersion(2) *Bi, Bi-Sb [M. H. Cohen and E. I. Blount (1960), P.A. Wolf(1964)]：semimetals strong spin-orbit interaction Tilted Dirac eq. *α-ET2I3：molecular solids S. Katayama et al.[2006] A. Kobayashi et al.(2006) This term is negligible Anisotropic masses and g-factors H = k・Vρσρ σ0 = 1, σα α= x,y,z Tilted Weyl eq. Anisotropic velocity Dirac types of energy dispersion(3) *FePn Hosono(2008) Ishibashi-Terakura(2008) DFT in AF states HF : JPSJ Online—News and Comments [May 12, 2008] * Ca3PbO : Kariyado-Ogata(2011)JPSJ Dirac electrons in solids Bulk *Bi *graphite-graphene *ET2I3 *FePn *Ca3PbO cf. topological insulators at surfaces Effective Hamiltonian Characteristics of energy bands of Dirac electrons *narrow band gap, if any *linear dependence on k (except very near k0) Gapless (Weyl 2x2) negligible s-o => effects of spins additive Finite gap(mass)(4x4) s-o => spin effects are essential Essence of Luttinger-Kohn representation Hamiltonian is a matrix H nn’ = [εn(k0)+ k2/2m] δ n,n’ + kαpαnn’ /m e.g. 2x2 Eg/2 + k2/2m kp/m H= kp /m -Eg/2 + k2/2m E= k2/2m ± √ (Eg/2 )2 +(kp)2 if (Eg/2 )2 >> (kp)2, E ± = ± Eg/2 + k2/2 m* ± : 1/2 m∗ ±=1/2m ± p2 /m2Eg Effective mass approximation Effective g-factors as well precise determination of parameters to describe electronic state => foundations of present semiconductor technology Luttinger-Kohn representation E= k2/2m ± √ (Eg/2 )2 +(kp)2 On the other hand, if (Eg/2 )2 << (kp)2 E ± ~ ± |kp| k-linear Particular features of Dirac electrons Narrow band gaps =>Inter-band coupling “ Inter-band effects” Different features form effective mass approximation in transport and thermodynamic properties. Especially , in magnetic field Hall effects, orbital magnetic susceptibility 10th ICPS (1970) - corresponds to the Peierls phase in the tight-binding approx. εn(k) => εn(k+eA/c) p・A : p has matrix elements between Bloch bands Landau-Peierls Formula χLP = 0 if DOS at Fermi energy =0 Orbital Magnetism in Bi Landau-Peierls Formula χLP = 0 if DOS at Fermi energy =0 Expt. Indicate importance of inter-band effects of magnetic field. Landau-Peierls formula (in textbooks) is totally invalid !! Diamagetism of Bi Strong spin-orbit interaction P.A. Wolff J. Phys. Chem. Solids (1964) Dirac electrons in solids! HF-Kubo: JPSJ 28 (1970) 570 Exact Formula of Orbital Susceptibility in General Cases In Bloch representation With Gregory Wannier ＠Eugene, Oregon (1973) Weak field Hall conductivity, σxy One-band approximation based on Boltzmann transport equation, General formula based on Kubo formula: HF-Ebisawa-Wada PTP 42 (1969) 494. Inter-band effects have been taken into account => Existence of contributions with not only f’(ε) but also f(ε) HF for graphene (2007) Weyl eq. A. Kobayashi et al., for α-ET2I3 (2008) Tilted Weyl eq. Y. Fuseya et al., for Bi (2009) Tilted Dirac eq. Bi Wolf(1964) Δ=EG/2 Assumption = isotropy of velocity “Isotropic Wolf” = original Dirac In weak magnetic field R=0 , but not 1/R=0 Fuseya-Ogata-HF, PRL102,066601(2009) Under strong magnetic field Isotropic Wolf model (original Dirac) Under magnetic field, k=> π=k+eA/c * Reduction of cyclotron mass = enhancement of g-factor => Landau splitting = Zeeman splitting both can be 100 times those of free electrons * Energy levels are characterized by j=n+1/2 +σ/2 orbital and spin angular momenta contribute equally to magnetization * Spin currents can be generated by light absorption Fuseya –Ogata-HF, JPSJ Molecular Solids ET2X layered structure ET molecule (ET=BEDTTTF) S S S S S S S S ET layers Anions layers ET2X=> ET+1/2 ET layers conducting X- closed shell ET2X Systems ET=BEDT-TTF S S S S S S S S Dirac cones α Spin Liquid Degree of dimerization (effectively ¼-filled for weak, ½ for strong) and degree of anisotropy of triangular lattice, t’/t Hotta,JPSJ(2003), Seo,Hotta,HF:Chemical Review 104 (2004) 5005. JPSJ 69(2000)Tajima-Kajita α-ET2I 3 p =19Kbar α-ET2I3 by charge order μeff T-indep. R under high pressure Kajita (1991,1993) μeff deduced by weak field Hall coefficient has very strong T-dep. n is also, since σ=neμ Hall coefficient in weak magnetic field depends on samples, some change signs at low temperature. Tight-binding approximation Massless Dirac fermion in α-(BEDT-TTF)2I3 Confirmed by DFT: Energy dispersion Katayama et al. (2006) Kino et al. (2006) Ishibashi (2006) Tilted Dirac cone エ ネ ル ギ ー H k v 0,1, 2, 3 fastest (eV) slowest Tilted Weyl Hamiltonian Kobayashi et at. (2007) k k0 k0 Hall effect: Tajima et al. (2008) Kobayashi et al. (2008) NMR：Takahashi et al. (2006) Kanoda et al. （2007） Shimizu et al.(2008) Interlayer Magnetoresistance Osada et al.(2008) Tajima et al.(2008) Morinari et al. (2008) 2d model Without tilting=graphene Transport properties: Hall effect Kobayashi et al., JPSJ 77(08)064718 Orbital susceptibility The conventional relation RH∝1/n is invalid. ------ typically, RH=0 at μ=0 ( neff=0 for semicoductors) sharp μ-dependence in narrow enegy range of the order of Γ. 1/Γ: elastic scattering time extremely sensitive probe! xx 0 xx K xx σμν=σ0 Kμν e2 xx X 0 2 conductivity Hall conductivity μ:chemical potential X=μ/Γ Effect of Tilting Kobayashi-Suzumura-HF,JPSJ 77, 064718(2008) Based on exact gauge-invariant formula X=ε/Γ speculations on T-dep. with μ=0 for T/Γ>1 σxx= Kxx σxx (T) =-∫dεf’(ε）σ(ε）～ Γ/T weak T dep. of σ => Γ ~ T, Then σxy= Kxy ~ 1/T 2 R ~ 1/T 2 α= 0 σ=neμ n~ T2 μ ~1/T2 Stronger T-dep In expts ? Possible sign change of Hall coefficient; A. Kobayashi et al., JPSJ 77(2008) 064718. Asymmetry of DOS relative to the crossing energy, ε0. Chemical potential crosses ε0 as T->0 if I3- ions are deficient of the order of 10-6 (hole-doped) Prediction, diamagnetism will be maximum, when Hall coefficient changes sign. Bulk 3d effects Cf. specific heat Hall coefficient can change sign, in accordance with expt. by Tajima et al. as below. Under strong perpendicular magnetic field p=18kbar α-(BEDT-TTF)2I3 H // c axis N. Tajima et al. (2006) T1 T0 T0 T1 A.Kobayashi et al, JPSJ78(2009)114711 *For tilted-cones, inter-valley scattering plays important roles. *Mean-filed phase transition(T0) to pseudo-spin XY ferromagnetic state. *Possible BKT transition at lower temperature. TKT Tc 1 4 Massless Dirac fermions under magnetic field Landau quantization E1 E1 EN sgn( N ) 2vc2eH N c T0 E0 E0 With tilting M. O. Goerbig et al. (2008) T. Morinari et al. (2008) E1 5meV T0 At H=10T Zeeman energy E1 E1 EZ 1 g B H 0.5meV T0 2 Effective Coulomb interaction e2 50 H[T ] I meV lB Electron correlation can play important roles! Kosterlitz-Thouless Transition in Strong Magnetic Field H H0 H H0 H c k , H0 0 k Kobayashi et at. (2007) ck Tilted Weyl Hamiltonian vk σ w0 k 0 EZ 0 Zeeman term H0 , vk σ* w0 k 0 EZ 0 pseudo-spin （valley) :spin ↑、↓ ：pseudo-spin （valley) R,L L R v: cone velocity w: tilting velocity H dr drV0 r rnr nr Long-range Coulomb interaction e2 V0 r 0r Katayama et al. (2006) To treat interaction effects, “Wannier function” for N=0 states Wave function of N=0 states (Landau gauge) 1 X (r) L x X 2 Xy exp-ik 0r exp-i 2 exp 2 l 2 l l 1 ΔX X-direction: localized Y-direction: plane wave 2l 2 ΔX l L Magetic length l c 100Å H 10T eH |Φ|2 Wannier functions (ortho-normal) can be defined on magnetic lattice Fukuyama (1977, in Japanese) L R r a b iXnb dX exp 2 X ma r l a / 2 a/2 Ri ma, nb i a b a b 2l 2 magnetic unit cell : a flux quantum Φ0 Effective Hamiltonian Effective Hamiltonian on the magnetic lattice Landau quantization (N=0)＋Zeeman energy＋long-range Coulomb interaction H EZ ci ci Vijklci c j ck cl Wijklci c j ck cl V term：intra-valley scattering W term：inter-valley scattering independent of tilting Induced by Tilting! SU(4) symmetric Breaking SU(4) symmetry q 2k0 L R ~ 2a w L W O V l L R 0.07 for α-(BEDT-TTF)2I3 H=10T ~ w 0.8 w v :tilting parameter Ground state of the effective Hamiltonian In the absence of tilting W 0 V-term ：symmetric in the spin and pseudo-spin space Only Ez-term breaks the symmetry L R Spin-polarized state In the presence of tilting W 0 W-term :Pseudo-spins are bound to XY-plane. V W EZ If the interaction is larger than Ez , the phase transition can occur at finite T in the mean-field approximation. Pseudo-spin ferromagnetic state Mean field theory (finite T) 4 Spin-polarized state Taking fluctuations of pseudo-spins in XY-plane, Tc/EZ ~ Si ci , L ci , R ~ Si Y 2 :Pseudo-spin operator X ~ ~ S S exp- i Pseudo-spin XY ferro 0 0 5 I/E Z ~ ~ S x Re S 10 ~ ~ S y Im S Tc ~ 0.5 I Effective “spin model” on the magnetic lattice H MF 2 EZ mI S jz 2 I ij j I ij Vijji Wiijj ij ~ ~ ~ ~ Si S j Si S j ：interactions between pseudo-spins Kosterlitz-Thouless transition Expanding the free energy from long-wavelength limit, F F0 J ij cos j i 1 cosi j 1 J ij 4 S i j 2 I00=I I ij b a The fluctuations are described by the XY model vortex and anti-vortex excitations Berenzinskii-Kosterlitz-Thouless transition nearest-neighbor interaction nearly isotropic if b 2a TKT 1.54J (J. M. Kosterlitz, J. Phys. C7 (1974) 1046. ) J J i ,i 1 0.0865I (in the present case) Tc~ 0.5 I TKT Tc 1 4 Under strong perpendicular magnetic field p=18kbar α-(BEDT-TTF)2I3 H // c axis N. Tajima et al. (2006) T1 T0 T0 T1 A.Kobayashi et al, JPSJ78(2009)114711 *For tilted-cones, inter-valley scattering plays important roles. *Mean-filed phase transition(T0) to pseudo-spin XY ferromagnetic state. *Possible BKT transition at lower temperature. TKT Tc 1 4 Graphenes Checkelsky-Ong,PRB 79(2009)115434 BKT transition T=0.3K at 30T K. Nomura, S. Ryu, and D-H Lee, cond-mat/0906.0159 Without tilting (W=0) : electron-lattice coupling Massless Dirac electrons in α-ET2X *Described by Tilted Weyl equation *Unusual responses to weak magnetic field Hall coefficient Inter-band effects of magnetic field (vector potential, A) are crucial. *Under strong magnetic field possible Berezinskii-Kosterlitz-Thouless transition * Further many-body effects ? Massless Dirac electrons in α-ET2X *Described by Tilted Weyl equation *Unusual responses to weak magnetic field Hall coefficient Inter-band effects of magnetic field (vector potential, A) are crucial. *Under strong magnetic field possible Berezinskii-Kosterlitz-Thouless transition * Further many-body effects ? Ca3PbO Kariyado-Ogata to appear in JPSJ Synthesis not yet. Similarity to and differences from Bi Dirac electrons in solids Summary * Examples: bismuth, graphite-graphene molecular solids αET2I3, FePn, Ca3PbO 4x4 (spin-orbit interaction), 2x2 (Weyl eq.) * Particular features are “small band gap” => inter-band effects of magnetic field effects Hall effect, magnetic susceptibility ~~ Targets Effects of boundary( surfaces, interfaces) Supplement FePn Superconductivity Prepared by JST Year 2008: New High-T “Fever” derived from Hosono’s Discovery c 1st International Symposium June 27-28, Tokyo HgCaBaCuO (High-Pressure) Tc (K) HgCaBaCuO TlCaBaCuO Hosono BiCaSrCuO 2008 YBaCuO Akimitsu LaSrCuO 1911 Pb Hg 2001 MgB2 SmFeAsO LaBaCuO 1986 Nb NbC NbN Nb3Ge 1st Proceedings LaFeAsO (High-Pressure) LaFeAsO LaFePO Vol. 77 (2008) Supplement C November 28 1st Focused Funding Program Year 1913 Physics Onnes 1987 Physics Bednorz Muller Transformative Research-Project on Iron Pnictides Call for proposal: July-August Start: October (till March 2012) World-wide Competition and Collaboration triggered by TRIP Prepared by JST Oct 2008 – Mar 2012 New priority program ‘High-temp. superconductivity in iron pnictides’ (SPP 1458) From 2010; 6 Yrs (3Yrs + 3Yrs) Collaboration Leader: Hide Fukuyama 24 Research Subjects 0.3-0.8 M$/ 3.5 Yrs International Workshop on the Search for New SCs Co-sponsored by JST-DOE-NSF-AFOSR May 12-16, 2009, Shonan Collaboration JST-EU Strategic Int. Cooperative Program on ‘Superconductivity’ (3-Yrs period) Under ex ante evaluation Leader: Hideo Hosono Mar 2010 – Mar 2013 Frontiers in Crystalline Matter Reported by National Academy of Sciences Oct 2009 P108-109 Box 3.1 Iron-Based Pnictide Materials: Important New Class of Materials Discovered Outside the United States A15-MgB2-Cuprates-FePn *A15 : BCS, structural change *MgB2 : BCS, strong ele-phonon, 2bands *Cuprates: strong correlation in a single band, Doped Mott, t-J model *FePn: strong correlation in multi bands structural change Journal of the Physical Society of Japan Vol. 77 (2008) Supplement C Proceedings of the International Symposium on Fe-Pnictide Superconductors Published in JPSJ online November 27, 2008 Preface Outline ＊Layered Iron Pnictide Superconductors: Discovery and Current Status Hideo Hosono ＊A New Road to Higher Temperature Superconductivity S. Uchida ＊Doping Dependence of Superconductivity and Lattice Constants in Hole Doped La1-xSrxFeAsO Gang Mu, Lei Fang, Huan Yang, Xiyu Zhu, Peng Cheng, and Hai-Hu Wen ＊Se and Te Doping Study of the FeSe Superconductors K. W. Yeh, H. C. Hsu, T. W. Huang, P. M. Wu, Y. L. Huang, T. K. Chen, J. Y. Luo, and M. K. Wu Total ~50 papers In 2011, Special Issue : Solid State Communications, to appear. FePn Phase diagram 1111 J. Zhao et al.: Nature Mater. 7 (2008) 953 Ort 111 122 Courtesy: Ono S. Nandi et al.: Phys. Rev. Lett. 104 (2010) 057006 Tet R. Parker et al.: Phys. Rev. Lett. 104 (2010) 057007 11 T-W Huang et al.: Phys. Rev. B82 (2010) 104502 Tet Ort TS>TN for x>0 No TN Courtesy: Ono 1111 J. Zhao et al.: Nature Mater. 7 (2008) 953 Ort Tet Basic difference from cuprates Parent compound Cuprates : Mott insulator (odd) 1 band FePn : semimetal (even) multi-band Importance of magnetism : spin-fluctuations Roles of many bands : Mazin, Kuroki Effects of crystal structure: Lee plot (Pn height-Kuroki) film MKWu Electronic inhomogeneity Phase separation C 66 Courtesy: Yoshizawa Ba122Co Minimum Analysis for softening in C66 of Ba(Fe1-xCox)2As2 C66 of Ba(Fe1-xCox)2As2 æ D ö C66 = C ç1÷ è T -Qø Q = -30K 0 66 Co ( % ) Θ(K) Δ(K) 3.7 % 75.5 5.4 6% 17.2 8.3 10 % - 30 15.6 M.Yoshizawa et al., arXiv:1008.1479v3 (Aug 2010) D = 15.6K Increasing of Co doping in Ba(Fe1-xCox)2As2 reduces Θ and enhances Δ. Constant Θ changes its sigh from + to – over quantum critical point. Temperature dependence in elastic constants of Ba(Fe0.9Co0.1)2As2 C66 reveals huge softening of 28% from room temperature down to Tsc=23K. Courtesy: Goto little change by H No sigh of softening in (C11–C12 ) / 2 and C44. Electric quadrupole of Ou is relevant A15 1d bands Labbe-Friedel:band Jahn Teller Gorkov:dimerization along chains 3d bands <= band calc. by Mattheiss Bhatt-McMillan, Bhatt: 2 close-lying saddle points based on dx2-y2 band Matheiss dz2 Tc Klein ele-phonon FePn: Coulomb interaction +el-ph interaction due to multi-orbit(multi-band) END