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 ) 2vc2eH 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
E1
E1
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  drV0 r  rnr nr
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-ik 0r exp-i 2  exp

2
l
2
l


 l


1
ΔX
X-direction: localized
Y-direction: plane wave
2l 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  2l 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  cosi   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