complex electronic materials (I) rHgCb@g ÿ ÷Rÿ l } · D. N. Basov, Richard D. Averitt, Dirk van der Marel, Martin Dressel, Kristjan Haule . Rev. Mod. Phys., 2011 . Optical spectroscopy

Optical spectroscopy of

complex electronic materials (I)

Nan-Lin Wang

International Center for Quantum Materials

(ICQM)

School of Physics, Peking University

2015 Winter School on Superconductivity @ HPSTAR

版权所有，请勿转载

Introduction about optical spectroscopy (Optical constants, Kramers-Kronig transformation, intra- and interband

transitions, Drude model of simple metal, Spectroscopy of correlated

electrons, superconducting condensate…)

Some applications (CDW, high Tc supercondutors,

heavy fermions, 3D massless Dirac fermions)

Pump-probe and time domain THz spectroscopy

Outline:


Some basics about optical spectroscopy technique


D. N. Basov, Richard D. Averitt, Dirk van der Marel, Martin Dressel, Kristjan Haule

Rev. Mod. Phys., 2011

Optical spectroscopy of solids

Units： 1 eV = 8065 cm-1 = 11400 K

1 THz = 33 cm-1 ~ 4 meV


The electrodynamical properties of solids can be

described by a number of so-called “optical constants”:

complex refractive index, or complex dielectric constants,

or complex conductivity. The charge excitations of an

electronic system is directly reflected in those “constants”.

Those optical constants could be probed either directly

(ellipsometry, ultrfast laser-based time domain terahertz

spectroscopy,…) or indirectly (reflectance

measurement over broad frequencies). 版权所有，请勿转载

Optical constants

Consider an electromagnetic wave in a medium

index refractive :)n( ),c/n(/q vwhere

)(

0

)/(

0

)(

0

t

c

nxi

tvxitqxi

y eEeEeEE

If there exists absorption,

K: attenuation factor

)(

0

tc

nxi

c

Kx

y eeEE

c

Kx

y eEE

2

2

0

2I

))(

(

0 t

c

xNi

y eEE

)()()( iKnN Introducing a complex refractive index:

x

y E

Intensity 版权所有，请勿转载

Reflectivity

If n, K are known, we

can get R, ; vice versa.

1

2

)1(

)1(|)(||/|

)1(

)1(

1

1

)(

22

22

2222

)(

22

22

)(

Kn

Ktg

Kn

KnrEER

eKn

Kn

iKn

iKn

errE

E

inref

i

i

in

ref

1/ 2

1/ 2

1/ 2

1

1 2 cos

2 sin

1 2 cos

Rn

R R

Rk

R R


Dielectric function

)()(2)(

)()()(

))()(()()()(

)()(

)()0,(,0,

),(),(),(

2

22

1

2

21

Kn

Kn

iKni

N

qqphoton

qEqqD

0 /a~1Å-1

Infrared

q=2/~10-4 Å-1

1 2

1 2

2 2

1

2 2

1

1( ) ( ) ( )

2

1( ) ( ) ( )

2

n

k

or 版权所有，请勿转载

conductivity

In a solid, considering the contribution from ions

or from high energy electronic excitations

)(i41)( amics,electrodynBy

)()( 21

)(i4)(


Now, we have several pairs of optical

constants:

n(), K()

R(), ()

1(), 2()

1(), 2()

Usually, R() can be obtained from

measurement. 版权所有，请勿转载

High- extrapolations:

R~ -p (p~0.5-1, for intermedate region)

R~ -n (n=4, above interband transition)

Low- extrapolations:

Insulator: R~ constant

Metal: Hagen-Rubens

Superconductor: two-fluids model

2 20

ln ( ')'

'

RP d

For optical reflectance

( ) ( )

ln ( ) (1/ 2) ln ( )

ir R e

r R i

Kramers-Kronig relation

-- the relation between the real and

imaginary parts of a response function.

21 2 2

0

' ( ')d '2( )

'P

12 2 2

0

( ')d '2( )

'P


Reflectivity measurement


From R() to 1()

Si-beamsplitter for 113v Bruker 113, 66v, 80v, and

grating spectrometers

Energy range: 17 -50000 cm-1 (2 meV~6 eV)


C. C. Homes et al.

Applied Optics 32,2976(1993)

FT-IR spectrometer In situ evaporation

In-situ overcoating technique


Δk

Δω

X

k,E e- k’,E

k,E

k’,E’

q,ωq e-

(a) Impurity-assisted absorption

(b) boson-assisted absorption

Holstein process, if phonons are involved.

Intraband transition

Eg h

Ec(k)

Ev(k)

Interband transition

| , , ( )

| , , ( )

v

c

v k E k

c k E k

( ) ( ) ( )c v cvE k E k E k

3( )

(2 ) | ( ) |k cv

V dsJ E

E k

2 22

2 2 2

8( ) ( ) | ( ) |vc

eJ p

m

Kubo-Greenwood formula

)(4

1)( 21

Infrared light cannot be absorbed

directly by electron-hole excitation.


Drude model

--free carrier response

2

01 2 2 2 2

2

02 2 2 2 2

1( )

1 4 1

( )1 4 1

p

p

0( )1 i

22

0 * 4

pne

m

0 200 400 600 800 10000

4000

8000

12000

16000

1 (

-1 c

m-1)

(cm-1)

1

2

p=10000 cm

-1

1/=100 cm-1

1/版权所有，请勿转载

2

0 1( )

1 4 1/

D

i i

4( ) ( )

i

2

1 2 21/

p

'2

2 '2 2 2 2

'

/1Im{ }

( ) ( )

/

p

p

p p

2

1

0

( )8

pd

Schematic picture for Drude response

R

1

p’

1/

1/

dc-1

1()

(m*/m)Neff

Im(-1/ )

2

1 2 2

1/( )

4 (1/ )

D

Screened plasma frequency


Drude-Lorentz model

Lorentz model: localized electron

Drude model: free electron gas, =const.

2

1 2 2

1( )

4 1

p

2 2

1 2 2 2 2 2

0

( )4 ( )

p

Phenomenological Drude-Lorentz model


CuxTiSe2 system


Parent compound 1T-TiSe2

• 1T-TiSe2 was one of

the first CDW-bearing

materials

• Broken symmetry at

200 K with a 2x2x2

superlattice

0 50 100 150 200 250 3000.0

0.5

1.0

1.5

2.0

2.5

3.0

mc

m

T (K)


Ti: 3d24s2

Se: 4s24p4

Ti : 3d band

Se: 4p band


No band crossing

Fermi level below CDW.

It is insulating!!

L

L

High T

Low T

2x2x2 superlattice

Se 4p

Ti 3d


Th. Pillo, et al. PRB (2000)

metallic picture

L

L

High T

Low T

2x2x2 superlattice

Se 4p

Ti 3d

But does not satisfy the charge neutrality!! X


Issue

• ARPES experiments did not resolve

conclusively whether the compound is a

semimetal or semiconductor with a small

indirect gap.

• The mechanism of the CDW transition:

not due to the Fermi Surface nesting 版权所有，请勿转载

0 100 200 300 400 500 6000.0

0.2

0.4

0.6

0.8

1.0

300K

225K

200K

150K

080K

050K

010K

Reflectivity

Frequency(cm-1)

TiSe2

0 1000 2000 3000 4000 50000.2

0.4

0.6

0.8

1.0

300K

225K

200K

150K

080K

050K

010K

Reflectivity

Frequency(cm-1)

TiSe2 single

crystal

G. Li et al., PRL (07a)


100 1000 100000.00

0.02

0.04

0.06

0.08

0.10

1132298

347

300 K

225 K

200 K

150 K

80 K

50 K

10 KIm[-

1/]

(cm-1)

' /p p


0.15 eV

CDW gap Drude

Free carriers with very long

relaxation time exist in the

CDW gapped state FS is not fully gapped??

0 2000 4000 60000

1000

2000

3000

4000

300 K

225 K

200 K

150 K

80 K

50 K

10 K

1 (

S/c

m)

(cm-1)

0.4 eV


100 1000 100000.2

0.4

0.6

0.8

1.0 300 K

10 K

dash curves: fit with Drude

+ two Lorentz

0 50 100 150 200 250 3000

2000

4000

6000

8000

0

200

400

600

800

1000

p (

cm

-1)

T (K)

1/

(cm

-1)

R


1T-TiSe2 is a semimetal with very low carrier density at

all T

Carrier density changes with T, decreases from room T

to 150 K then increases slightly with further decreasing T

Development of an energy gap ~0.15 eV below 200 K

Dramatic different carrier damping at different T

Semimetal to semimetal transition for CDW


Excitonic Phases W. Kohn, PRL 67

The electron-hole coupling acts to mix the electron band and hole

band that are connected by a particular wave vector.


Exciton-driven CDW

)(4 22

e

e

h

hp

m

n

m

ne

0 50 100 150 200 250 3000.0

0.5

1.0

1.5

2.0

2.5

3.0

mc

m

T (K)

G. Li et al., PRL (07a)



Cu-doped CuxTiSe2


Single band metal


0 1000 2000 3000 4000 5000

0.4

0.6

0.8

1.0

R

(cm-1)

10 K

50 K

80 K

150 K

200 K

225 K

300 K

100 1000 100000.2

0.4

0.6

0.8

1.0

R

10 K

50 K

80 K

150 K

200 K

225 K

300 K

0 2000 4000 6000 80000.00

0.02

0.04

0.06

0 50 100 150 200 250 30013000

14000

15000

16000

17000

Im[-

1/]

(cm-1)

10 K

50 K

80 K

150 K

200 K

220 K

300 K

p (

cm

-1)

T(K)

X=0.07

Plasma frequency increases

with decreasing T!!

Rarely seen phenomenon!!

G. Li et al., PRL (07b)


0 1000 2000 3000 4000 50000.4

0.6

0.8

1.0

100 1000 100000.2

0.4

0.6

0.8

1.0

R

(cm-1)

300 K

10 K

R

(cm-1)

300 K


Plasma frequency increases

with decreasing T??

• n increases with decreasing T??

• m* decreases with decreasing T, undressing effect??

*

22 4

m

nep

2 2

* *4 ( )e h

p

e h

n ne

m m

In terms of two bands:


G. Wu et al. PRB (07)

Hall coefficient is almost T-independent for

superconducting sample!!


Possible shift of chemical potential with T

=0[1-(T/TF)2],

EF~80-100 meV at 300 K

heavy mass at L point due to very

strong electron-phonon scattering,

(linked with lattice instability in

parent compound). smaller mass at Fermi

crossing point due to reduced

scattering (away from L point).

L

(k) changes along dispersive band!

G. Li et al., PRL (07b)


0 1000 2000 3000 4000 50000.0

0.2

0.4

0.6

0.8

1.0

R

(cm-1)

10 K

100 K

200 K

300 K

LaSb

2000 2500 3000 3500 40000.0

0.1

0.2

0.3

0.4

0.5

Im(-

1/

)

(cm-1)

300 K

200 K

100 K

10 K

LaSb

Recently, we found similar phenomena in a number of low-

carrier density metals, e.g. LaSb (semimetal):


0 10000 20000 30000 400000.0

0.2

0.4

0.6

0.8

1.0

R

(cm-1)

gold

0 5000 10000 15000 200000.0

0.2

0.4

0.6

0.8

1.0

R (cm

-1)

YBCO

Bi2212

Simple metal High-Tc cuprates

From simple metal to correlated systems


Simple metal

Correlated metal

Correlation effect in optical conductivity

Correlation effect：reducing

the kinetic energy of electrons,

or Drude spectral weight.


Q M Si, Nature Physics 2009 Qazilbash et al. Nature Physics 2009


Hubbard U physics:

DFMT

V2O3


Sum rule

f-sum rule:

It has the explicit implication that at energies higher than the total

bandwidth of a solid, electrons behave as free particles.

Kubo partial

sum-rule:

The upper limit of the integration is much larger than the bandwidth

of a given band crossing the Fermi level but still smaller than the

energy of interband transitions. For , the Kubo sum

rule reduces to the f-sum rule.

WK depends on T and on the state of the system because of nk---

"violation of the conductivity sum rule", first studied by Hirsch.

kinetic energy

in the tight

binding model

In reality, there is no true violation: the change of the spectral weight of a given band would be

compensated by an appropriate change in the spectral weight in other bands, and the total

spectral weight over all bands is conserved.


Non-Drude spectra of strongly correlated electrons

General feature: a sharp peak at =0

+ a long tail extending to high energies

Two possible interpretations

1()

1()

Drude

MIR

1/ ()


2

2

*2

*

1( , )

4 ( , )

1

4 1/ ( , ) [1 ( , )]

1

4 1/ ( , )

p

p

p

TM T i

T i T

T i

21

( )4 1

p

i

( , ) 1/ ( , ) ( , )M T T i T 1/,T: Frequency dependent scattering rate

: Mass enhancement m*=m(1+

Drude Model

Extended Drude Model

Let

e.g. Marginal Fermi Liquid model:

2 2

( , ) 1/ ( , ) ( , )

(0) ln2 c

M T T i T

xg N x i

Where x=max(||,T),

or x=(2+(T)2)1/2

2

2

1/ ( , ) ( / 4 ) Re[1/ ( , )]

* / 1 ( ) ( / 4 ) Im[1/ ( , )]

p

p

T T

m m T


The extended Drude model in terms of optical

self-energy

21

( , )4 ( ( , ) )

pT

T i

According to

Littlewood and

Varma, 1 2

( ) 2

2 [ ( ) ( )]

opi

i i

1 2

*

2 1

( ) 1/ ( ) 2

( ) (1 / ) 2m m

Optical self-energy

Relation to the 1/() and m*/m


Hwang, Timusk, Gu,

Nature 427, 714 (2004)

Bi2212


2Δ / TDW ~ 3.5 under the weak coupling BCS theory

Energy gaps for symmetry broken states：

Coherence factors

Coherent factors play an extremely important role in determining the energy

gap structures of broken symmetry states

From the text book of Tinkham

Superconductor vs density wave state

Density wave

superconductivity


Optical spectra of a superconductor

T=0, London electrodynamics gives

)(41

or )(81

4/)(8

12221

0122

22

cd

ci

L

sn

L

psps

clean limit: <l 2>

Absorption starts at 2+.

dirty limit: >l 2<

Absorption starts at 2.

∵pippard coherence length

=vF/, =1/=vF/l

Ferrell-Glover-Tinkham sum-rule: missing area

is equal to the superconducting condensate. Clean limit dirty limit


This part of spectral weight (originated

from the kinetic energy reduction)

contributes to the sc condensate.

Kinetic energy lowering with superconducting condensate


Structure of energy gap Density wave

0 100 200 300 400 5000

3000

6000

9000

12000

20 K

300 K

70 K

20 K

15 K

8 K

1 (

-1cm

-1)

(cm-1)

300 K

200 K

100 K

70 K

50 K

dc values

URu2Si2


Superconductors

G. Li et al PRL 08

Cuprates, Homes data

Ba0.6K0.4Fe2As2

Structure of energy gap


Coherent peak below Tc

O.Klein et al PRB 94

T. Fischer et al PRB

2010 R. V. Arguilar et al,

arXiv:107.3677

THz data are different from NMR？S+- pairing？


More about extended Drude model

——Extraction the boson (phonon) spectral function from optical conductivity


• Mode coupling effect in ARPES

and tunneling

• Feature in IR spectra of high-Tc

superconductors

Extraction the

boson (phonon)

spectral function

Mode coupling effect in infrared spectra of high-T_c cuprates

Mode coupling?? 版权所有，请勿转载

I. Giaever, H.R. Hart, Jr., and K. Megerle, PR 126, 941 (1962)

McMillan and Rowell,

Superconductivity, ed.

By R.D. Parks (1969)

BCS

data Pb

requires Eliashberg theory:

• phonon dynamics (retardation) taken into account

• gap is a function of frequency

• density of states is modified:



Dip at +R

2F()


YBCO C C Homes

D N Basov

T. Timusk

R()

1()

1/()

Tl-2212 thin

film Tc=108 K

N. L. Wang et al.,

PRB03

0 500 1000 1500 20000

2000

4000

6000

0 200 400 6000.0

0.5

1.0

(b)

1 (

-1 c

m-1)

Wave number (cm-1)

0.8

0.9

1.0

(a)

R

10 K

95 K

120 K

200 K

300 K

R

Wave number (cm-1)

0 500 1000 1500 20000

1000

2000

(b)

1/

(cm

-1)

Wave number (cm-1)

120 K

10 K

0.95

1.00

1.05

1.10

(a)R

10K/R

120K

Rs/R

n

Due to mode

coupling

Mode coupling in

optical spectra??


Δk

Δω

(a) Impurity-assisted absorption

(b) boson-assisted absorption

Intraband transition

Infrared light cannot be

absorbed directly by

electron-hole excitation.

In the clean limit, we must

consider the boson-assisted

electron-hole excitations.


The electron-boson (phonon) interaction

T=0 K

P.B.Allen 1971 )()()(

2/1

0

2

Fd tr

0 1000 2000 30000

1200

2400

3600

4800

0.0

1.5

3.0

4.5

6.0

F

1

F

At 0K, based on Allen's formula

1c

m1

Wave number (cm-1)

_0=500 cm-1

=100 cm-1

_p^2=50000 cm-2

2 2

2

2 2 2 2 2

0

( )( )

pF


0 1000 2000 30000

2000

4000

6000

0.0

2.5

5.0

7.5

F

300K

200K

100K

50K

10K

0K

F

1c

m1

Wave number (cm-1)

Electron-phonon coupling at finite temperature

Formula based on the

Kubo formula for the

conductivity

by Shulga 1991


normal state !

Marsiglio et al., Phys. Lett.

A 245, 172 (1998)

])(

1[

2

1)(

2

2

d

dW or


Pb (actual, and

numerically inverted ---

these are indistinguishable

approximate (from (A) )

(A)


K3C60 L. Pintschovius,

Rep. Prog. Phys.

57, 473 (1996)

Tc = 19 K


Coupling to 41 meV mode

J.P. Carbotte, E. Schachinger

and D.N. Basov, Nature 401,

354(1999)

The peak in W(): +

Because of d-wave pairing,

the peak is shifted by (not

2!)

2

2

1 1( ) [ ]

2 ( )

dW

d


Schasinger and

Carbotte, PRB 03

A. Abanov, et al.

Phys. Rev. B 63,

180510 (R) (2001)

The peak in W(): +

Because of d-wave pairing,

the peak is shifted by (not

2!)

2

2

1 1( ) [ ]

2 ( )

dW

d

2 + 版权所有，请勿转载

problem

the bosonic spectral function can not be

negative. The negative values are linked with

the overshoot in 1/().

0 1000 2000 30000

1200

2400

3600

4800

0.0

1.5

3.0

4.5

6.0

F

1

F

At 0K, based on Allen's formula

1c

m1

Wave number (cm-1)

A mode is unable

to cause a

overshoot in 1/

0 500 1000 1500 20000

1000

2000

(b)

1/

(cm

-1)

Wave number (cm-1)

120 K

10 K

0.95

1.00

1.05

1.10

(a)R

10K/R

120K

Rs/R

n


S. V. Dordevic, et al. PRB 71, 104529 (2005)

22

2

20

2 41/ 1d F E

Allen’s formula for the scattering

rate in the superconducting state

E(x) is the second kind

elliptic integral

P.B.Allen 1971


J. Hwang, T. Timusk, et al. cond-mat/0505302


Angle-Resolved Photoemission Spectroscopy (ARPES)

X

Y

Z hv

e-

f

Electron detector

Crystal

2 2

Im ( , )1( , )

[ Re ( , )] [Im ( , )]k

kA k

k k

PEAK POSITION: Dispersion (velocity; Effective mass;etc.)

PEAK WIDTH: Im or 1/ scattering rate


kF

ARPES –Band Mapping and Fermi Surface

Energy Distribution Curve (EDC)


P. D. Johnson, et al., PRL 87, 177007 (2001)

( Re / )FE 版

权所有，请勿转载

Dispersion of LSCO along the (0,0)-(,) Nodal Direction

Fine structures

X. J. Zhou et al., PRL (05)


Extraction of Bosonic Spectral Function from Re

In metals, the real part of self-energy is related to the bosonic spectral function by:

0

2Re ( , , ()) ( ; , , )F k KT dkT

kkT

where

2 2

2 '( , ') ( )

'

yK y y dx f x y

x y

with f(x) being the Fermi-Dirac distribution Function

Maximum Entropy Method 2F()


Comparison of the Extracted Spectral Function with Known Structure

Hayden et al.,

PRL 76(1996) 1344.

Magnetic

McQueeney et al.,

PRL 87(2001) 077001

Phonon

(1). el-ph

(2). multiple

phonons


Thanks!


Documents

complex electronic materials (I) rHgCb@g ÿ ÷Rÿ l } · D. N. Basov, Richard D. Averitt, Dirk van der Marel, Martin Dressel, Kristjan Haule . Rev. Mod. Phys., 2011 . Optical spectroscopy