INTERNATIONAL RESEACH SCHOOL AND WORKSHOP ON ELECTRONIC CRYSTALS ECRYS2011 August 15 -27, 2011 Cargèse , France

1. INTERNATIONAL RESEACH SCHOOL AND WORKSHOP ON ELECTRONIC CRYSTALS ECRYS2011 August 15 -27, 2011 Cargèse, France

2. Dirac electrons in solid Hidetoshi Fukuyama Tokyo Univ. of Science

3. Acknowledgement Bi Yuki Fuseya (Osaka Univ.) Masao Ogata (Tokyo Univ.) α-ET2I3 Akito Kobayashi(Nagoya Univ.) YoshikazuSuzumura (Nagoya Univ.)

4. Dirac electrons in solidscontents • “elementary particles” in solids <= band structure , locally in k-space • Band structure similar to Dirac electrons Examples: bismuth, graphite-graphene molecular solids αET2I3,FePn, Ca3PbO 4x4 (spin-orbit interaction), 2x2 (Weyl eq.) • Particular features of Dirac electrons small band gap => inter-band effects of magnetic field effects Hall effect, magnetic susceptibility

5. Dirac equations for electrons in vacuum 4x4 matrix Equivalently, In special cases of m=0, Weyl equation for neutrino 2x2 matrix

6. “Elementary particles in solids”band structures, locally in k-space Semiconductors , Carrier doping electron doping ->n type hole doping -> p type Si InSb electrons holes Dispersion relation＝＞effective masses and g-factors “elementary particles” Luttinger-Kohn representation (k・p approximation)

7. LK vs. Bloch representation Bloch representation: energy eigen-states Ψnk(r)= eikrunk(r) : unk(r+a)=unk(r) Luttinger –Kohn representation [ Phys. Rev. 97, 869 (1955) ] Χnk(r)= eikrunk0(r) k0 = some special point of interest “k・pmethod” Hamiltonian is essentially a matrix Spin-orbit interaction If εn(k) has extremum at k0

8. 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

9. Dirac types of energy dispersion(1) *Graphite [P. R. Wallace (1947),J.W. McClure(1957)] semimetal（ne=nh≠０） *graphene: special case of graphite （ne=nh=０）Geim H = v( kxσx + kyσy ) Weyl eq. for neutrinoIsotropic velocity McClure(1957)

10. 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 This term is negligible *α-ET2I3：molecular solids S. Katayama et al.[2006] A. Kobayashi et al.(2006) H = k・Vρσρ σ0 = 1, σαα= x,y,z Tilted Weyl eq. Tilted Dirac eq. Anisotropic masses and g-factors Anisotropic velocity

11. Dirac types of energy dispersion(3) *FePn Ishibashi-Terakura(2008) DFT in AF states HF : JPSJ Online—News and Comments [May 12, 2008] Hosono(2008) * Ca3PbO : Kariyado-Ogata(2011)JPSJ

12. Dirac electrons in solids Bulk *Bi *graphite-graphene *ET2I3 *FePn *Ca3PbO cf. topological insulators at surfaces Effective Hamiltonian

13. 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

14. Essence of Luttinger-Kohn representation • Hamiltonian is a matrix • H nn’ = [εn(k0)+ k2/2m] δn,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* =1/2m • Effective mass approximation • Effective g-factors as well • precise determination of parameters to describe • electronic state • => foundations of present semiconductor technology

15. 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

16. Particular features of Dirac electrons Narrow band gaps =>Inter-band coupling “ Inter-band effects” Different features form effective mass approximationin transport and thermodynamic properties. Especially , in magnetic field Hall effects, orbital magnetic susceptibility

17. 10th ICPS (1970) - corresponds to the Peierls phase in the tight-binding approx. εn(k) => εn(k+eA/c)

18. p・A : p has matrix elements between Bloch bands Landau-Peierls Formula χLP = 0 if DOS at Fermi energy =0

19. Orbital Magnetism in Bi Landau-PeierlsFormula 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 !!

20. 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

21. Exact Formula of Orbital Susceptibility in General Cases In Bloch representation

22. With Gregory Wannier ＠Eugene, Oregon (1973)

23. 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.

25. Bi Wolf(1964) Δ=EG/2 Assumption = isotropy of velocity = original Dirac “Isotropic Wolf”

26. In weak magnetic field Fuseya-Ogata-HF, PRL102,066601(2009) R=0 , but not 1/R=0

27. 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

28. S S S S S S S S Molecular Solids ET2Xlayered structure ET molecule (ET=BEDTTTF) ET2X- => ET+1/2 ET layers  ET layers conducting X- closed shell Anions layers

29. ET2X Systems ET=BEDT-TTF Dirac cones S S S S α S S S S 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.

30. JPSJ 69(2000)Tajima-Kajita α-ET2I3 p =19Kbar α-ET2I 3 by charge order μeff T-indep. R under high pressure Kajita (1991,1993) μeffdeduced by weak field Hall coefficient has very strong T-dep. neffis also, since σ=neμ

31. Hall coefficient in weak magnetic field depends on samples, some change signs at low temperature.

32. Tight-binding approximation

33. Massless Dirac fermion in α-(BEDT-TTF)2I3 Confirmed by DFT: Kino et al. (2006) Ishibashi(2006) Energy dispersion Katayama et al. (2006) Tilted Dirac cone エネルギー(eV) fastest slowest Tilted Weyl Hamiltonian Kobayashi et at. (2007) Hall effect: Tajima et al. (2008) Kobayashi et al. (2008) Interlayer Magnetoresistance Osada et al.(2008) Tajima et al.(2008) Morinari et al. (2008) NMR：Takahashi et al. (2006) Kanoda et al. （2007） Shimizu et al.(2008)

34. 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 σμν=σ0Kμν extremely sensitive probe! conductivity Hall conductivity μ:chemical potential X=μ/Γ

35. Effect of Tilting • Kobayashi-Suzumura-HF,JPSJ 77, 064718(2008) • Based on exact gauge-invariant formula X=ε/Γ

36. 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= ~ 1/T 2 R ~ 1/T 2 Kxy α= 0 σ=neμ n~ T2 μ~1/T2 Stronger T-dep In expts ?

37. Possible sign change of Hall coefficient; A. Kobayashi et al., JPSJ 77(2008) 064718. Hall coefficient can change sign, in accordance with expt. by Tajima et al. as below. Chemical potential crosses ε0 as T->0 if I3- ions are deficient of the order of 10-6 (hole-doped) Asymmetry of DOS relative to the crossing energy, ε0. Prediction, diamagnetism will be maximum, when Hall coefficient changes sign. Bulk 3d effects Cf. specific heat

38. Under strong perpendicular magnetic field p=18kbar α-(BEDT-TTF)2I3 N. Tajima et al. (2006) T1 T 0 T0 T1 *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. A.Kobayashi et al, JPSJ78(2009)114711 

39. Massless Dirac fermions under magnetic field With tilting M. O. Goerbig et al. (2008) T. Morinari et al. (2008) Landau quantization T0 At H=10T Zeeman energy Effective Coulomb interaction Electron correlation can play important roles!

40. Kosterlitz-Thouless Transition in Strong Magnetic Field Kobayashi et at. (2007) Tilted Weyl Hamiltonian Zeeman term pseudo-spin （valley) :spin ↑、↓ ：pseudo-spin （valley)R,L v:cone velocity w: tilting velocity Katayama et al. (2006) Long-range Coulomb interaction

41. To treat interaction effects, “Wannier function” for N=0 states Wave function of N=0 states (Landau gauge) X-direction: localized Magetic length Y-direction: plane wave |Φ|2 Wannier functions (ortho-normal) can be defined on magnetic latticeFukuyama (1977, in Japanese) Å magnetic unit cell : a flux quantum Φ0

42. Effective Hamiltonian Effective Hamiltonian on the magnetic lattice Landau quantization (N=0)＋Zeeman energy＋long-range Coulomb interaction Vterm：intra-valley scattering Wterm：inter-valley scattering independent of tilting Induced by Tilting! SU(4)symmetric Breaking SU(4) symmetry for α-(BEDT-TTF)2I3 H=10T :tilting parameter

43. Ground state of the effective Hamiltonian In the absence of tilting V-term ：symmetric in the spin and pseudo-spin space Only Ez-termbreaks the symmetry Spin-polarized state In the presence of tilting W-term :Pseudo-spins are bound to XY-plane. If the interaction is larger than Ez , Pseudo-spin ferromagnetic state the phase transition can occur at finite T in the mean-field approximation.

44. Mean field theory (finite T) Taking fluctuations of pseudo-spins in XY-plane, Spin-polarized state :Pseudo-spin operator Pseudo-spin XY ferro Tc ~ 0.5 I Effective “spin model” on the magnetic lattice ：interactions between pseudo-spins

45. Kosterlitz-Thouless transition Expanding the free energy from long-wavelength limit, I00=I The fluctuations are described by the XY model nearest-neighbor interaction nearly isotropic if vortex and anti-vortex excitations Berenzinskii-Kosterlitz-Thoulesstransition (J. M. Kosterlitz, J. Phys. C7 (1974) 1046. ) (in the present case) Tc~ 0.5 I

46. Under strong perpendicular magnetic field p=18kbar α-(BEDT-TTF)2I3 N. Tajima et al. (2006) T1 T 0 T0 T1 *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. A.Kobayashi et al, JPSJ78(2009)114711 

47. GraphenesCheckelsky-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

48. 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-Thoulesstransition * Further many-body effects ?

49. 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-Thoulesstransition * Further many-body effects ?

50. Ca3PbO Kariyado-Ogata to appear in JPSJ Synthesis not yet. Similarity to and differences from Bi