ECRYS

INTERNATIONAL RESEACH SCHOOLAND WORKSHOPONELECTRONIC CRYSTALSECRYS2011

August 15 -27, 2011Cargse, France

Dirac electrons in solid Hidetoshi Fukuyama Tokyo Univ. of ScienceAcknowledgementBiYuki Fuseya (Osaka Univ.)Masao Ogata (Tokyo Univ.)

-ET2I3Akito Kobayashi(Nagoya Univ.)Yoshikazu Suzumura (Nagoya Univ.)

Dirac electrons in solidscontentselementary particles in solids inter-band effects of magnetic field effects Hall effect, magnetic susceptibilityDirac equations for electrons in vacuum

Equivalently,In special cases of m=0,Weyl equation for neutrino

4x4 matrix2x2 matrixElementary particles in solidsband structures, locally in k-space

Si

InSbelectronsholesSemiconductors , Carrier dopingelectron doping ->n typehole doping -> p typeDispersion relationeffective masses and g-factorselementary particlesLuttinger-Kohn representation (kp approximation)6LK vs. Bloch representationBloch representation: energy eigen-states nk(r)= eikrunk(r) : unk(r+a)=unk(r) Luttinger Kohn representation [ Phys. Rev. 97, 869 (1955) ] nk(r)= eikrunk0(r) k0 = some special point of interest

If n(k) has extremum at k0

Spin-orbit interactionkpmethodHamiltonian is essentially a matrixLK 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)] semimetalne=nh

*graphene: special case of graphite ne=nh=Geim H = v( kxx + kyy ) Weyl eq. for neutrino Isotropic velocity

McClure(1957)9Dirac types of energy dispersion(2)*Bi, Bi-Sb [M. H. Cohen and E. I. Blount (1960), P.A. Wolf(1964)]semimetalsstrong spin-orbit interaction

This term is negligible *-ET2I3molecular solids S. Katayama et al.[2006] A. Kobayashi et al.(2006) H = kV 0 = 1, = x,y,z Tilted Weyl eq.

Tilted Dirac eq.

Anisotropic velocityAnisotropic masses and g-factors

10

*FePnHosono(2008)Ishibashi-Terakura(2008) DFT in AF statesHF : JPSJOnlineNews and Comments [May 12, 2008] * Ca3PbO : Kariyado-Ogata(2011)JPSJDirac types of energy dispersion(3)Dirac electrons in solidsBulk*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 additiveFinite gap(mass)(4x4) s-o => spin effects are essential Essence of Luttinger-Kohn representationLuttinger-Kohn representation

Particular features of Dirac electronsNarrow band gaps =>Inter-band coupling Inter-band effectsDifferent 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.n(k) => n(k+eA/c) 17

Landau-Peierls FormulaLP = 0 if DOS at Fermi energy =0 pA : p has matrix elements between Bloch bands18

Orbital Magnetism in Bi

Landau-Peierls formula (in textbooks) is totally invalid !!Expt. Indicate importance of inter-band effects of magnetic field.

Landau-Peierls FormulaLP = 0 if DOS at Fermi energy =0 19

HF-Kubo: JPSJ 28 (1970) 570Diamagetism of BiP.A. WolffJ. Phys. Chem. Solids (1964)Dirac electrons in solids!Strong spin-orbit interaction

20

Exact Formula of Orbital Susceptibility in General CasesIn Bloch representation21With Gregory Wannier Eugene, Oregon (1973)

22Weak field Hall conductivity, xyOne-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() but also f() HF for graphene (2007) Weyl eq. A. Kobayashi et al., for -ET2I3 (2008) Tilted Weyl eq. Y. Fuseya et al., for Bi (2009) Tilted Dirac eq.

23

BiWolf(1964)

Assumption = isotropy of velocityIsotropic Wolf

=EG/2= original Dirac

In weak magnetic field

R=0 , but not 1/R=0Fuseya-Ogata-HF, PRL102,066601(2009)

Isotropic Wolf model (original Dirac)Under magnetic field, k=> =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 +/2 orbital and spin angular momenta contribute equally to magnetization* Spin currents can be generated by light absorption

Fuseya Ogata-HF, JPSJ Under strong magnetic field

Molecular Solids ET2Xlayered structure

ET layers Anions layersSSSSSSSSET molecule(ET=BEDTTTF)ET2X- => ET+1/2ET layers conductingX- closed shell

Degree of dimerization (effectively -filled for weak, for strong) and degree of anisotropy of triangular lattice, t/t Hotta,JPSJ(2003), Seo,Hotta,HF:Chemical Review 104 (2004) 5005. ET2X SystemsET=BEDT-TTFSSSSSSSSSpin LiquidDirac cones29-ET2I3

JPSJ 69(2000)Tajima-KajitaT-indep. R under high pressure Kajita (1991,1993) p =19Kbareff deduced by weak field Hall coefficienthas very strong T-dep.neff is also, since =neeff-ET2I 3by charge order30

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

Tight-binding approximation32

fastestslowest(eV)

Energy dispersionMassless Dirac fermion in -(BEDT-TTF)2I3 Katayama et al. (2006)

Tilted Dirac coneConfirmed by DFT: Kino et al. (2006)Ishibashi (2006)NMRTakahashi et al. (2006) Kanoda et al. 2007Shimizu et al.(2008)Interlayer Magnetoresistance Osada et al.(2008) Tajima et al.(2008) Morinari et al. (2008)

Tilted Weyl Hamiltonian Kobayashi et at. (2007) Hall effect: Tajima et al. (2008) Kobayashi et al. (2008)33

The conventional relation RH1/n is invalid. ------ typically, RH=0 at =0 ( neff=0 for semicoductors)sharp -dependence in narrow enegy range of the order of .1/: elastic scattering time

extremely sensitive probe!

Orbital susceptibilityconductivityHall conductivityX=/Transport properties: Hall effectKobayashi et al., JPSJ 77(08)064718=0 K:chemical potential

2d model Without tilting=graphene34

Effect of TiltingKobayashi-Suzumura-HF,JPSJ 77, 064718(2008) Based on exact gauge-invariant formulaX=/35speculations on T-dep. with =0 for T/>1

xx= Kxx xx (T) =-df(( /T weak T dep. of => ~ T, Then xy= ~ 1/T 2 R ~ 1/T 2

Kxy=nen~ T2~1/T2= 0Stronger T-depIn expts ?36

Possible sign change of Hall coefficient;A. Kobayashi et al., JPSJ 77(2008) 064718.Asymmetry of DOS relative to the crossing energy, 0. Chemical potential crosses 0 as T->0if I3- ions are deficient of the order of 10-6 (hole-doped)Hall coefficient canchange sign,in accordance with expt.by Tajima et al. as below.

Prediction,diamagnetism will be maximum, when Hall coefficient changes sign.

Bulk 3d effectsCf. specific heat 37Under strong perpendicular magnetic field

p=18kbar -(BEDT-TTF)2I3 N. Tajima et al. (2006) T0T1

*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

T 0T1Two-step increase of resistivity has been observed in the ZGS of -(BEDT-TTF)2I3 in in the presence of magnetic field by Tajima-san.With decreasing $T$, the resistivity increases below the onset temperature $T_0$ about 40K, and is saturated with the plateau-like structure in lower temperature region.And then another increase around Tl, about 10K leading to apparent saturation as T goes to zero.Both $T_0$ and $T_l$ increase with increasing magnetic field

The electronic state in the presence of magnetic field is shown here. The energy spectrum of the massless Dirac fermions becomes discrete by the Landau quantization. Each Landau state is split into two states with up and down spins by the Zeeman energy.In the present case, the energy scale of $T_0$ is much smaller than $E_1 \cong 0.013$eV at H=15T but is much larger than the Zeeman gap, \cong 0.0017$eV at 15T.Then it can not be explained in the absence of the electron correlation effects.

38Landau quantizationMassless Dirac fermions under magnetic field

At H=10T

T0With tiltingM. O. Goerbig et al. (2008)T. Morinari et al. (2008)Electron correlation can play important roles!

Effective Coulomb interactionZeeman energyThe Landau quantization of orbital motion of massless Dirac electrons in magnetic field induces discrete energy spectrum, $E_N={\it sgn} (N) \sqrt{2 \hbar v^2 eH |n|/c}$, where $v$ is the velocity of the massless Dirac fermion, $H$ is the vertical component of magnetic field, and $N$ is an integer.\cite{Ando?}Each Landau state has large degeneracy proportional to $H$, and is spllitted into two states with up and down spins.At $H=15$Tesla, the Zeeman gap $g\mu_{\rm B}$H =0.0018$eV is smaller than $E_1 =0.013$eV, thus the energy scale of $T_c =0.005$eV (which corresponds to $50$K approximately) is located between those values, where we take the g-factor $g=2$ and the Boltzman factor $k_B =1$, $v=***$ in the absence of tilting of the Dirac cone.Recently, the renormalization effect of tilting on Landau quantization was investigated and it was suggested that $E_1$ becomes half by the expected value of tilting in $\alpha$-(BEDT-TTF)$_2$I$_3$.\cite{Montambaux2008,Thoyama2008}It is assumed that the renormalized $E_1$ is still a little larger than the energy scale of $T_c$.Thus we concentrate the two states with up and down spins of $N=0$ in the present study.The effects of $N \ne 0$ states will be investigated in the next stage.It is expected that narrow width and large degeneracy of the Landau states bring strong electron correlation and then long-range order can exists.39

Kosterlitz-Thouless Transition in Strong Magnetic Field

Long-range Coulomb interaction

:spin pseudo-spin valley)R,LTilted Weyl Hamiltonianv: cone velocitypseudo-spin valley)Katayama et al. (2006)

Zeeman termw: tilting velocityKobayashi et at. (2007) Hamiltonian of the ZGS in the presence of magnetic field is shown here.It is based on the tilted Weyl equation at the two valleys tau=+ and - .But in the present study, tilting is ignored.We treat the long-range Coulomb interaction.We concentrate on the physics for the N=0 states because the onset temperature T0 is smaller than E1.40Wave function of N=0 states (Landau gauge)

X-direction: localized Y-direction: plane wave

Magetic length

magnetic unit cell : a flux quantum 0||2

Wannier functions (ortho-normal) can be definedon magnetic lattice Fukuyama (1977, in Japanese)

To treat interaction effects, Wannier function for N=0 states41

Effective Hamiltonian on the magnetic latticeLandau quantization (N=0)Zeeman energylong-range Coulomb interaction

Effective HamiltonianSU(4) symmetricindependent of tiltingBreaking SU(4) symmetryInduced by Tilting!V termintra-valley scatteringW terminter-valley scattering

for -(BEDT-TTF)2I3 H=10T

:tilting parameter42Ground state of the effective HamiltonianIn the absence of tiltingSpin-polarized statethe phase transition can occur at finite T in the mean-field approximation.

W-term :Pseudo-spins are bound to XY-plane.V-term symmetric in the spin and pseudo-spin spaceIn the presence of tiltingPseudo-spin ferromagnetic state

Only Ez-term breaks the symmetry

If the interaction is larger than Ez ,

43Mean field theory (finite T)

:Pseudo-spin operator

interactions between pseudo-spinsTaking fluctuations of pseudo-spins in XY-plane,

Spin-polarized statePseudo-spin XY ferroEffective spin model on the magnetic lattice

Tc ~ 0.5 I44Kosterlitz-Thouless transitionExpanding the free energy from long-wavelength limit,

The fluctuations are described by the XY modelBerenzinskii-Kosterlitz-Thouless transition

(J. M. Kosterlitz, J. Phys. C7 (1974) 1046. )(in the present case)

vortex and anti-vortex excitationsTc~ 0.5 I

nearest-neighbor interaction nearly isotropic if

I00=IThen we obtain the free energy in the excitonic state.In the presence of finite amplitude,fluctuation of the phase with long wavelength (>>a, b) remainsBy expanding the spatial fluctuation of the phases, we obtain this free energy.Thus The phase fluctuations can be described by the XY Heisemberg model with J=0.0865V in the present case.leading to Kosterlitz-Thouless transitionThe Kosterlitz-Thouless transition temperature is given by TKT=1.54J by the renormalization group analysis.Thus in the present case, we obtain the ratio over =0.266.

45Under strong perpendicular magnetic field

p=18kbar -(BEDT-TTF)2I3 N. Tajima et al. (2006) T0T1

*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

T 0T1Two-step increase of resistivity has been observed in the ZGS of -(BEDT-TTF)2I3 in in the presence of magnetic field by Tajima-san.With decreasing $T$, the resistivity increases below the onset temperature $T_0$ about 40K, and is saturated with the plateau-like structure in lower temperature region.And then another increase around Tl, about 10K leading to apparent saturation as T goes to zero.Both $T_0$ and $T_l$ increase with increasing magnetic field

The electronic state in the presence of magnetic field is shown here. The energy spectrum of the massless Dirac fermions becomes discrete by the Landau quantization. Each Landau state is split into two states with up and down spins by the Zeeman energy.In the present case, the energy scale of $T_0$ is much smaller than $E_1 \cong 0.013$eV at H=15T but is much larger than the Zeeman gap, \cong 0.0017$eV at 15T.Then it can not be explained in the absence of the electron correlation effects.

46GraphenesCheckelsky-Ong,PRB 79(2009)115434

BKT transition T=0.3K at 30TK. Nomura, S. Ryu, and D-H Lee, cond-mat/0906.0159Without tilting (W=0) : electron-lattice couplingMassless Dirac electrons in -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 ?

48Massless Dirac electrons in -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 ?

49Ca3PbO

Synthesis not yet.Similarity to and differences from BiKariyado-Ogata to appear in JPSJ Dirac electrons in solidsSummary* Examples: bismuth, graphite-graphene molecular solids ET2I3, 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~~TargetsEffects of boundary( surfaces, interfaces)SupplementFePn Superconductivity

Year 2008: New High-Tc Fever derived from Hosonos DiscoveryPbNbNbCNbNNb3GeMgB2HgYearTc (K)

Onnes1913Physics1911LaBaCuOLaSrCuOYBaCuOBiCaSrCuOHgCaBaCuOHgCaBaCuO(High-Pressure)1986

BednorzMuller1987Physics

2001AkimitsuLaFePOLaFeAsOLaFeAsO(High-Pressure)SmFeAsO

Hosono

1st International SymposiumJune 27-28, Tokyo1st Proceedings

Vol. 77 (2008) Supplement CNovember 281st Focused Funding Program

Transformative Research-Project on Iron PnictidesCall for proposal: July-AugustStart: October (till March 2012)TlCaBaCuO2008Prepared by JST53World-wide Competition and Collaboration triggered by TRIPOct 2008 Mar 2012Leader: Hide Fukuyama24 Research Subjects0.3-0.8 M$/ 3.5 Yrs

CollaborationLeader: Hideo HosonoMar 2010 Mar 2013

Outcome

New priority program High-temp. superconductivityin iron pnictides (SPP 1458) From 2010; 6 Yrs (3Yrs + 3Yrs)

CollaborationCollaborationJST-EU Strategic Int. CooperativeProgram onSuperconductivity (3-Yrs period) Under ex ante evaluationInternational Workshop on the Search for New SCsCo-sponsored by JST-DOE-NSF-AFOSRMay 12-16, 2009, Shonan

Collaboration

Frontiers in Crystalline MatterReported byNationalAcademy ofSciences Oct 2009P108-109 Box 3.1 Iron-Based Pnictide Materials: Important New Class of Materials Discovered Outside the United StatesPrepared by JST 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 changeJournal of the Physical Society of JapanVol. 77 (2008) Supplement CProceedings of the International Symposium on Fe-Pnictide Superconductors Published in JPSJ online November 27, 2008 PrefaceOutlineLayered 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.

S. Nandi et al.: Phys. Rev. Lett. 104 (2010) 0570061111

R. Parker et al.: Phys. Rev. Lett. 104 (2010) 057007111FePn Phase diagramTetOrtTS>TN for x>0T-W Huang et al.: Phys. Rev. B82 (2010) 104502

TetOrt

J. Zhao et al.: Nature Mater. 7 (2008) 953122

No TN11Courtesy: Ono58

1111TetOrtJ. Zhao et al.: Nature Mater. 7 (2008) 953Courtesy: OnoBasic difference from cupratesParent compoundCuprates : Mott insulator (odd) 1 bandFePn : semimetal (even) multi-bandImportance of magnetism : spin-fluctuationsRoles of many bands : Mazin, KurokiEffects of crystal structure: Lee plot (Pn height-Kuroki) film MKWuElectronic inhomogeneityPhase separation59

MinimumCourtesy: YoshizawaBa122Co

Analysis for softening in C66 of Ba(Fe1-xCox)2As2

Co ( % ) ( K ) ( K )3.7 %75.55.46 %17.28.310 %- 3015.6M.Yoshizawa et al., arXiv:1008.1479v3 (Aug 2010)Increasing of Co doping in Ba(Fe1-xCox)2As2 reduces and enhances .C66 of Ba(Fe1-xCox)2As2

Constant 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.No sigh of softening in (C11C12 ) / 2 and C44.Electric quadrupole of Ou is relevant Courtesy: Goto little change by H

1d bands Labbe-Friedel:band Jahn Teller Gorkov:dimerization along chains3d bands