INTERNATIONAL RESEACH SCHOOL - PowerPoint Presentation

INTERNATIONAL RESEACH SCHOOL
INTERNATIONAL RESEACH SCHOOL

INTERNATIONAL RESEACH SCHOOL - Description


          AND WORKSHOP               ON        ELECTRONIC CRYSTALS            ECRYS2011                 ID: 796828 Download

Tags

spin magnetic 2008 dirac magnetic spin dirac 2008 field band effects hall electrons strong interaction energy weyl inter kobayashi

Download Section

Please download the presentation from below link :


Download - The PPT/PDF document "INTERNATIONAL RESEACH SCHOOL" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Embed / Share - INTERNATIONAL RESEACH SCHOOL


Presentation on theme: "INTERNATIONAL RESEACH SCHOOL"— Presentation transcript


Slide1

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

Slide2

Dirac electrons in solid    Hidetoshi Fukuyama Tokyo Univ. of Science

Slide3

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

Slide4

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

Slide5

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

Weyl

equation for neutrino

4x4 matrix

2x2 matrix

Slide6

“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)

Slide7

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

Slide8

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

Slide9

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

=

Geim

H = v(

k

x

σ

x

+ kyσy ) Weyl eq. for neutrino

Isotropic velocity

McClure(1957)

Slide10

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

Slide11

*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)

Slide12

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

Slide13

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

Slide14

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

 

Slide15

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

 

Slide16

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

Slide17

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

ε

n

(k) => ε

n

(k+eA/c)

 

Slide18

Landau-Peierls Formula

χ

LP

= 0 if DOS at Fermi energy =0

p

A :

p

has matrix elements between Bloch bands

Slide19

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

Slide20

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

Slide21

Exact

Formula of Orbital

Susceptibility in General Cases

In Bloch representation

Slide22

With Gregory Wannier @Eugene, Oregon (1973)

Slide23

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.

Slide24

Slide25

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

Δ=E

G

/2

= original Dirac

Slide26

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

Fuseya

-Ogata-HF, PRL102,066601(2009)

Slide27

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

Slide28

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

Slide29

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

Slide30

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

Slide31

Hall coefficient in weak magnetic field depends on

samples,

some

change signs

at low

temperature.

Slide32

Tight-binding approximation

Slide33

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)

Slide34

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

0

K

μν

μ:chemical potential

2d model Without tilting=

graphene

Slide35

Effect

of Tilting

Kobayashi-

Suzumura

-HF,JPSJ 77, 064718(2008)

Based on exact gauge-invariant formula

X=ε/Γ

Slide36

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

?

Slide37

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

Slide38

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

Slide39

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

Slide40

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)

Slide41

Wave function of N=0 states (Landau gauge)

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

Slide42

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

Slide43

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

,

Slide44

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

Slide45

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 ?

Slide49

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

Slide54

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

Slide55

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

Slide56

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

Slide57

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

Shom More....