Dirac electrons in solids contents 1)elementary particles in
solids inter-band effects of magnetic field effects Hall effect,
magnetic susceptibility
Slide 5
Dirac equations for electrons in vacuum Equivalently, In
special cases of m=0, Weyl equation for neutrino 4x4 matrix 2x2
matrix
Slide 6
Elementary particles in solids band structures, locally in
k-space SiInSb electrons holes Semiconductors, Carrier doping
electron doping ->n type hole doping -> p type Dispersion
relation effective masses and g-factors elementary particles
Luttinger-Kohn representation (k p approximation)
Slide 7
LK vs. Bloch representation Bloch representation: energy
eigen-states nk (r)= e ikr u nk (r) : u nk (r+a)=u nk (r) Luttinger
Kohn representation [ Phys. Rev. 97, 869 (1955) ] nk (r)= e ikr u
nk0 (r) k 0 = some special point of interest If n (k) has extremum
at k 0 Spin-orbit interaction k p method Hamiltonian is essentially
a matrix
Slide 8
LK vs. Bloch * LK forms complete set and are related to Bloch
by unitary transformation * k-dependences are completely different,
* in Bloch, both e ikr and u nk (r), the latter being very
complicated, while in LK only in e ikr as for free electrons. *
just replace k=> k+eA/c in Hamiltonian matrix once in the
presence of magnetic field
Slide 9
Dirac types of energy dispersion(1) *Graphite [ P. R. Wallace
(1947),J.W. McClure(1957)] semimetal n e =n h *graphene: special
case of graphite n e =n h = Geim H = v( k x x + k y y ) Weyl eq.
for neutrino Isotropic velocity McClure(1957)
Slide 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 *-ET 2 I 3 molecular solids S.
Katayama et al.[2006] A. Kobayashi et al.(2006) H = k V 0 = 1, =
x,y,z Tilted Weyl eq. Tilted Dirac eq. Anisotropic velocity
Anisotropic masses and g-factors
Slide 11
*FePn Hosono(2008) Ishibashi-Terakura(2008) DFT in AF states HF
: JPSJ OnlineNews and Comments [May 12, 2008] * Ca 3 PbO :
Kariyado-Ogata(2011)JPSJ Dirac types of energy dispersion(3)
Slide 12
Dirac electrons in solids Bulk *Bi *graphite-graphene *ET 2 I 3
*FePn *Ca 3 PbO cf. topological insulators at surfaces Effective
Hamiltonian
Slide 13
Characteristics of energy bands of Dirac electrons *narrow band
gap, if any *linear dependence on k (except very near k 0 ) Gapless
(Weyl 2x2) negligible s-o => effects of spins additive Finite
gap(mass)(4x4) s-o => spin effects are essential
Slide 14
Essence of Luttinger-Kohn representation
Slide 15
Luttinger-Kohn representation
Slide 16
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
Slide 17
10 th ICPS (1970) - corresponds to the Peierls phase in the
tight-binding approx. n (k) => n (k+eA/c)
Slide 18
Landau-Peierls Formula LP = 0 if DOS at Fermi energy =0 p A : p
has matrix elements between Bloch bands
Slide 19
Orbital Magnetism in Bi Landau-Peierls formula (in textbooks)
is totally invalid !! Expt. Indicate importance of inter-band
effects of magnetic field. Landau-Peierls Formula LP = 0 if DOS at
Fermi energy =0
Slide 20
HF-Kubo: JPSJ 28 (1970) 570 Diamagetism of Bi P.A. Wolff J.
Phys. Chem. Solids (1964) Dirac electrons in solids! Strong
spin-orbit interaction
Slide 21
Exact Formula of Orbital Susceptibility in General Cases In
Bloch representation
Slide 22
With Gregory Wannier Eugene, Oregon (1973)
Slide 23
Weak field Hall conductivity, xy 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() but also f() HF for graphene (2007) Weyl eq. A.
Kobayashi et al., for -ET 2 I 3 (2008) Tilted Weyl eq. Y. Fuseya et
al., for Bi (2009) Tilted Dirac eq.
Slide 24
Slide 25
Bi Wolf(1964) Assumption = isotropy of velocity Isotropic Wolf
=E G /2 = original Dirac
Slide 26
In weak magnetic field R=0, but not 1/R=0 Fuseya-Ogata-HF,
PRL102,066601(2009)
Slide 27
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
Slide 28
Molecular Solids ET 2 X layered structure ET layers Anions
layers S S S S S S S S ET molecule (ET=BEDTTTF) ET 2 X - => ET
+1/2 ET layers conducting X- closed shell
Slide 29
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. ET
2 X Systems ET=BEDT-TTF S S S S S S S S Spin Liquid Dirac
cones
Slide 30
-ET2I3 JPSJ 69(2000)Tajima-Kajita T-indep. R under high
pressure Kajita (1991,1993) p =19Kbar eff deduced by weak field
Hall coefficient has very strong T-dep. n eff is also, since =ne
eff -ET 2 I 3 by charge order
Slide 31
Hall coefficient in weak magnetic field depends on samples,
some change signs at low temperature.
Slide 32
Tight-binding approximation
Slide 33
fastest slowest Energy dispersion Massless Dirac fermion in
-(BEDT-TTF) 2 I 3 Katayama et al. (2006) Tilted Dirac cone
Confirmed by DFT : Kino et al. (2006) Ishibashi (2006) 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) Tilted Weyl Hamiltonian Kobayashi et at.
(2007) Hall effect: Tajima et al. (2008) Kobayashi et al.
(2008)
Slide 34
The conventional relation R H 1/n is invalid. ------ typically,
R H =0 at =0 ( n eff =0 for semicoductors) sharp -dependence in
narrow enegy range of the order of . 1/: elastic scattering time
extremely sensitive probe! Orbital susceptibility conductivity Hall
conductivity X=/ Transport properties: Hall effect Kobayashi et
al., JPSJ 77(08)064718 = 0 K :chemical potential 2d model Without
tilting=graphene
Slide 35
Effect of Tilting Kobayashi-Suzumura-HF,JPSJ 77, 064718(2008)
Based on exact gauge-invariant formula X=/
Slide 36
speculations on T-dep. with =0 for T/>1 xx = K xx xx (T) =-
df( ( /T weak T dep. of => ~ T, Then xy = ~ 1/T 2 R ~ 1/T 2 K xy
=ne n~ T 2 ~1/T 2 = 0 Stronger T-dep In expts ?
Slide 37
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->0 if I 3 - ions
are deficient of the order of 10 -6 (hole-doped) Hall coefficient
can change sign, in accordance with expt. by Tajima et al. as
below. Prediction, diamagnetism will be maximum, when Hall
coefficient changes sign. Bulk 3d effects Cf. specific heat
Slide 38
Under strong perpendicular magnetic field p=18kbar -(BEDT-TTF)
2 I 3 N. Tajima et al. (2006) T0T0 T1T1 *For tilted-cones,
inter-valley scattering plays important roles. *Mean-filed phase
transition(T 0 ) to pseudo-spin XY ferromagnetic state. *Possible
BKT transition at lower temperature. A.Kobayashi et al,
JPSJ78(2009)114711 T 0 T1T1
Slide 39
Landau quantization Massless Dirac fermions under magnetic
field At H=10T T0T0 With tilting M. O. Goerbig et al. (2008) T.
Morinari et al. (2008) Electron correlation can play important
roles! Effective Coulomb interaction Zeeman energy
Slide 40
Kosterlitz-Thouless Transition in Strong Magnetic Field
Long-range Coulomb interaction :spin pseudo-spin valley) R,L Tilted
Weyl Hamiltonian v: cone velocity pseudo-spin valley) Katayama et
al. (2006) Zeeman term w: tilting velocity Kobayashi et at.
(2007)
Slide 41
Wave 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
defined on magnetic lattice Fukuyama (1977, in Japanese) To treat
interaction effects, Wannier function for N=0 states
Slide 42
Effective Hamiltonian on the magnetic lattice Landau
quantization (N=0) Zeeman energy long-range Coulomb interaction
Effective Hamiltonian SU(4) symmetric independent of tilting
Breaking SU(4) symmetry Induced by Tilting! V term intra-valley
scatteringW term inter-valley scattering for -(BEDT-TTF) 2 I 3
H=10T :tilting parameter
Slide 43
Ground state of the effective Hamiltonian In the absence of
tilting Spin-polarized state the 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
space In the presence of tilting Pseudo-spin ferromagnetic state
Only E z -term breaks the symmetry If the interaction is larger
than E z,
Slide 44
Mean field theory (finite T) :Pseudo-spin operator interactions
between pseudo-spins Taking fluctuations of pseudo-spins in
XY-plane, Spin-polarized state Pseudo-spin XY ferro Effective spin
model on the magnetic lattice Tc ~ 0.5 I
Slide 45
Kosterlitz-Thouless transition Expanding the free energy from
long-wavelength limit, The fluctuations are described by the XY
model Berenzinskii-Kosterlitz-Thouless transition (J. M.
Kosterlitz, J. Phys. C7 (1974) 1046. ) (in the present case) vortex
and anti-vortex excitations Tc~ 0.5 I nearest-neighbor interaction
nearly isotropic if I 00 =I
Slide 46
Under strong perpendicular magnetic field p=18kbar -(BEDT-TTF)
2 I 3 N. Tajima et al. (2006) T0T0 T1T1 *For tilted-cones,
inter-valley scattering plays important roles. *Mean-filed phase
transition(T 0 ) to pseudo-spin XY ferromagnetic state. *Possible
BKT transition at lower temperature. A.Kobayashi et al,
JPSJ78(2009)114711 T 0 T1T1
Slide 47
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
Slide 48
Massless Dirac electrons in -ET 2 X *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 ?
Slide 49
Massless Dirac electrons in -ET 2 X *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 ?
Slide 50
Ca 3 PbO Synthesis not yet. Similarity to and differences from
Bi Kariyado-Ogata to appear in JPSJ
Slide 51
Dirac electrons in solids Summary * Examples: bismuth,
graphite-graphene molecular solids ET 2 I 3, FePn, Ca 3 PbO 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)
Slide 52
Supplement FePn Superconductivity
Slide 53
Year 2008 : New High-T c Fever derived from Hosonos Discovery
Pb Nb NbC NbN Nb 3 Ge MgB 2 Hg Year T c (K) Onnes 1913Physics 1911
LaBaCuO LaSrCuO YBaCuO BiCaSrCuO HgCaBaCuO HgCaBaCuO(High-Pressure)
1986 BednorzMuller 1987Physics 2001 Akimitsu LaFePO LaFeAsO
LaFeAsO(High-Pressure) SmFeAsO Hosono 1 st International Symposium
June 27-28, Tokyo 1 st Proceedings Vol. 77 (2008) Supplement C
November 28 1 st Focused Funding Program Transformative
Research-Project on Iron Pnictides Call for proposal: July-August
Start: October (till March 2012) TlCaBaCuO 20082008 Prepared by
JST
Slide 54
World-wide Competition and Collaboration triggered by TRIP Oct
2008 Mar 2012 Leader: Hide Fukuyama 24 Research Subjects 0.3-0.8
M$/ 3.5 Yrs Collaboration Leader: Hideo Hosono Mar 2010 Mar 2013
Outcome New priority program High-temp. superconductivity in iron
pnictides (SPP 1458) From 2010; 6 Yrs (3Yrs + 3Yrs) Collaboration
Collaboration JST-EU Strategic Int. Cooperative Program on (3-Yrs
period) Superconductivity (3-Yrs period) Under ex ante evaluation
Under ex ante evaluation International Workshop on the Search for
New SCs Co-sponsored by JST-DOE-NSF-AFOSR May 12-16, 2009, Shonan
Collaboration 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 Prepared by JST
Slide 55
A15-MgB 2 -Cuprates-FePn *A15 : BCS, structural change *MgB 2 :
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
Slide 56
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 Layered Iron Pnictide Superconductors:
Discovery and Current Status Hideo Hosono A New Road to Higher
Temperature Superconductivity A New Road to Higher Temperature
Superconductivity S. Uchida Doping Dependence of Superconductivity
and Lattice Constants in Hole Doped La 1-x Sr x FeAsO Doping
Dependence of Superconductivity and Lattice Constants in Hole Doped
La 1-x Sr x FeAsO Gang Mu, Lei Fang, Huan Yang, Xiyu Zhu, Peng
Cheng, and Hai-Hu Wen Se and Te Doping Study of the FeSe
Superconductors 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
Slide 57
In 2011, Special Issue : Solid State Communications, to
appear.
Slide 58
S. Nandi et al.: Phys. Rev. Lett. 104 (2010) 057006 1111 R.
Parker et al.: Phys. Rev. Lett. 104 (2010) 057007 111 FePn Phase
diagram Tet Ort T S >T N for x>0 T-W Huang et al.: Phys. Rev.
B82 (2010) 104502 Tet Ort J. Zhao et al.: Nature Mater. 7 (2008)
953 122 No T N 11 Courtesy: Ono
Slide 59
1111 Tet Ort J. Zhao et al.: Nature Mater. 7 (2008) 953
Courtesy: Ono 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
Slide 60
Minimum Courtesy: Yoshizawa Ba122Co
Slide 61
Analysis for softening in C 66 of Ba(Fe 1-x Co x ) 2 As 2 Co (
% ) ( K ) ( K ) 3.7 %75.55.4 6 %17.28.3 10 %- 3015.6 M.Yoshizawa et
al., arXiv:1008.1479v3 (Aug 2010) Increasing of Co doping in Ba(Fe
1-x Co x ) 2 As 2 reduces and enhances . C 66 of Ba(Fe 1-x Co x ) 2
As 2 Constant changes its sigh from + to over quantum critical
point.
Slide 62
Temperature dependence in elastic constants of Ba(Fe 0.9 Co 0.1
) 2 As 2 C 66 reveals huge softening of 28% from room temperature
down to T sc =23K. No sigh of softening in (C 11 C 12 ) / 2 and C
44. Electric quadrupole of O u is relevant Courtesy: Goto little
change by H
Slide 63
1d bands Labbe-Friedel:band Jahn Teller Gorkov:dimerization
along chains 3d bands
