INTERNATIONAL RESEACH SCHOOL - PowerPoint Presentation

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

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Dirac electrons in solid　　　 Hidetoshi Fukuyama Tokyo Univ. of Science

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AcknowledgementBiYuki Fuseya (Osaka Univ.)Masao Ogata (Tokyo Univ.)α-ET2I3Akito Kobayashi(Nagoya Univ.)Yoshikazu Suzumura (Nagoya Univ.)

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Dirac electrons in solidscontents“elementary particles” in solids <= band structure , locally in k-spaceBand 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

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Dirac equations for electrons in vacuumEquivalently,In special cases of m=0,

Weyl

equation for neutrino

4x4 matrix

2x2 matrix

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“Elementary particles in solids”band structures, locally in k-spaceSi

InSb

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)

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LK vs. Bloch representationBloch representation: energy eigen-states Ψnk(r)= eikrunk(r) : unk(r+a)=unk(r) Luttinger –Kohn representation [ Phys. Rev. 97, 869 (1955) ]　　 Χnk(r)= eikru

nk0

(r)

k

0

= some special point of

interest

If

ε

n

(k) has

extremum

at k

0

Spin-orbit interaction

“k

・

p

　

method”

Hamiltonian is essentially a matrix

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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

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Dirac types of energy dispersion(1)*Graphite [ P. R. Wallace (1947),J.W. McClure(1957)] semimetal（ne=nh≠０）　　*graphene: special case of graphite

（

n

e

=

n

h

=

０

）

Geim

H = v(

k

x

σ

x

+ kyσy ) Weyl eq. for neutrino

Isotropic velocity

McClure(1957)

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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

・

V

ρ

σ

ρ

　

σ

0

= 1,

　

σα　α= x,y,z Tilted Weyl eq.

Tilted Dirac eq.

Anisotropic velocity

Anisotropic masses and g-factors

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*FePnHosono(2008)Ishibashi-Terakura(2008) DFT in AF states

HF : JPSJ

Online

—News and Comments [May 12, 2008

]

* Ca

3

PbO :

Kariyado

-Ogata(2011)JPSJ

Dirac types of energy

dispersion(3)

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Dirac electrons in solidsBulk*Bi*graphite-graphene*ET2I3*FePn*Ca3PbO cf. topological insulators at surfacesEffective Hamiltonian

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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

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Essence of Luttinger-Kohn representationHamiltonian is a matrix H nn’ = [εn(k0)+ k2/2m] δ n,n’ + kαpαnn’ /me.g. 2x2 Eg/2 + k2/2m kp

/m

H=

kp

/m -

Eg

/2

+

k

2

/2m

E= k

2

/2m

(Eg/2 )2 +(kp)2

if (Eg/2 )

2

>> (

kp

)

2

, E

=

Eg

/2 + k

2

/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

 

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Luttinger-Kohn representationE= k2/2m (Eg/2 )2 +(kp)2 On the other hand, if (Eg/2 )2 << (kp)2 E ~ |kp| k-linear

 

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Particular features of Dirac electronsNarrow 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

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10th ICPS (1970)- corresponds to the Peierls phase in the tight-binding approx.

ε

n

(k) => ε

n

(k+eA/c)

　

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Landau-Peierls Formula

χ

LP

= 0 if DOS at Fermi energy =0

p

・

A :

p

has matrix elements between Bloch bands

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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

χ

LP

= 0 if DOS at Fermi energy =0

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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

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Exact

Formula of Orbital

Susceptibility in General Cases

In Bloch representation

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With Gregory Wannier ＠Eugene, Oregon (1973)

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Weak field Hall conductivity, σxyOne-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-ET

2

I

3

(2008)

Tilted

Weyl

eq.

　

Y.

Fuseya

et al., for Bi (2009)

Tilted Dirac eq.

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BiWolf(1964)Assumption = isotropy of velocity“Isotropic Wolf”

Δ=E

G

/2

= original Dirac

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In weak magnetic fieldR=0 , but not 1/R=0

Fuseya

-Ogata-HF, PRL102,066601(2009)

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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

Under strong magnetic field

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Molecular Solids ET2Xlayered structureET 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

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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.

ET

2

X Systems

ET=BEDT-TTF

S

S

S

S

S

S

S

S

α

Spin Liquid

Dirac cones

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α-ET2I3

JPSJ 69(2000)Tajima-Kajita

T-indep. R under high pressure

Kajita (1991,1993)

p =19Kbar

μ

eff

deduced by

weak field Hall coefficient

has

very strong T-dep.

n

eff

is also

, since

σ=

neμ

μ

eff

α-ET

2

I

3

b

y charge order

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Hall coefficient in weak magnetic field depends on

samples,

some

change signs

at low

temperature.

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Tight-binding approximation

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fastest

slowest

エネルギー

(eV)

Energy dispersion

Massless Dirac fermion in α-(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)

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The conventional relation RH∝1/n is invalid. ------ typically, RH=0 at μ=0 (

n

eff

=0 for

semicoductors

)

sharp μ-dependence in narrow enegy range of the order of Γ.

1/

Γ

: elastic scattering time

extremely sensitive probe!

Orbital

susceptibility

conductivity

Hall conductivity

X=μ/Γ

Transport properties:

Hall effect

Kobayashi et al., JPSJ 77(08)064718

σ

μν

=σ

0

K

μν

μ:chemical potential

2d model Without tilting=

graphene

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Effect

of Tilting

Kobayashi-

Suzumura

-HF,JPSJ 77, 064718(2008)

Based on exact gauge-invariant formula

X=ε/Γ

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speculations on T-dep. with μ=0 for T/Γ>1σxx= Kxx

σ

xx

(T)

=-

∫

dεf

’(ε

）

σ(ε

）～

Γ/T

　

weak T dep. of σ => Γ ~ T,

Then σxy=

~ 1/T

2

R ~

1/T

2

K

xy

σ=

neμ

n~ T

2

μ

　

~1/T

2

α= 0

Stronger T-

dep

In

expts

?

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Possible sign change of Hall coefficient;

A. Kobayashi et al., JPSJ 77(2008) 064718.

Asymmetry of DOS

relative to the crossing energy, ε

0.

Chemical potential crosses ε

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

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Under strong perpendicular magnetic field

p=18kbar

α-(BEDT-TTF)

2

I

3

N. Tajima et al. (2006)

T

0

T

1

*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

T

1

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Landau quantizationMassless Dirac fermions under magnetic field

At H=10T

T

0

With

tilting

M. O.

Goerbig

et al. (2008)

T.

Morinari

et al. (2008

)

Electron correlation can play important roles!

Effective Coulomb interaction

Zeeman energy

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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)

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Wave function of N=0 states (Landau gauge)

Å

X-direction: localized

Y-direction: plane wave

Magetic length

magnetic unit cell :

a flux quantum

Φ

0

|Φ|

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

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Effective Hamiltonian on the magnetic latticeLandau 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 scattering

W

term

：

inter-valley scattering

for α-(BEDT-TTF)

2

I

3

H=10T

:tilting parameter

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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

,

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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

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Kosterlitz-Thouless transitionExpanding 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

n

earest-neighbor

interaction

nearly

isotropic

if

I

00

=I

Slide46

Under strong perpendicular magnetic field

p=18kbar

α-(BEDT-TTF)

2

I

3

N. Tajima et al. (2006)

T

0

T

1

*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

T

1

Slide47

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

Slide48

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 ?

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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 ?

Slide50

Ca3PbO

Synthesis not yet.

Similarity to and differences from Bi

Kariyado

-Ogata to appear in JPSJ

　

Slide51

Dirac electrons in solidsSummary* 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)

Slide52

SupplementFePn Superconductivity

Slide53

Year 2008: New High-Tc “Fever” derived from Hosono’s DiscoveryPbNb

NbC

NbN

Nb

3

Ge

MgB

2

Hg

Year

T

c

(K)

Onnes

1913

Physics

1911

LaBaCuO

LaSrCuO

YBaCuO

BiCaSrCuO

HgCaBaCuO

HgCaBaCuO

(High-Pressure)

1986

Bednorz

Muller

1987

Physics

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

T

ransformative

R

esearch-Project

on

I

ron

P

nictides

Call for proposal: July-August

Start: October (till March 2012)

TlCaBaCuO

2008

Prepared by JST

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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

‘Superconductivity’

(3-Yrs period)

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

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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

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Journal 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 PrefaceOutline＊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

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In 2011,Special Issue : Solid State Communications, to appear.

Slide58

S. Nandi et al.: Phys. Rev. Lett. 104 (2010) 0570061111R. Parker et al.: Phys. Rev. Lett. 104 (2010) 057007111

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

Slide59

1111Tet

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

Slide60

MinimumCourtesy: Yoshizawa

Ba122Co

Slide61

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

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)

Increasing of Co doping in

Ba(Fe

1-x

Co

x

)

2

As

2

reduces

Θ

and enhances

Δ

.

C

66

of

Ba

(

Fe

1-x

Co

x

)

2

As

2

Constant

Θ

changes its

sigh

from + to – over

q

uantum critical point.

Slide62

Temperature dependence in elastic constants of Ba(Fe0.9Co0.1)2As2C66 reveals huge softening of 28% from room temperature down to Tsc=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

Slide63

1d bands Labbe-Friedel:band Jahn Teller Gorkov:dimerization along chains3d bands <= band calc. by Mattheiss Bhatt-McMillan, Bhatt: 2 close-lying saddle points based on dx2-y2 band Matheiss dz2Tc Klein ele-phonon A15

Slide64

FePn: Coulomb interaction +el-ph interaction due to multi-orbit(multi-band)

Slide65

END