An introduction to the optical

spectroscopy of solids

Nan Lin Wang

Institute of Physics

Chinese Academy of Sciences

Outline

1. Optical constants and their relations

(Optical constants, Kramers-Kronig

transformation, intra- and interband transitions)

2. Drude model of simple metal (example of

application)

3. Spectroscopy of correlated electrons

4. THz time domain spectroscopy (Particularly

optical pump, THz probe)

References

• C. Kittle, Solid State Physics

• F. Wooten, Optical properties of solids (Academic Press,

1972)

• M. Dressel and G. Grüner, Electrodynamics of solids

(Cambridge University Press, 2002)

Units：1 eV = 8065 cm-1 = 11400 K

1.24 eV=10000 cm-1

D. Basov, et al.

Rev. Mod. Phys.

(2011)

Optical spectroscopy is a primary tool to probe the charge

dynamics and quasiparticle excitations in a material.

I. Optical constants and their relations

The electrodynamical properties of solids

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

complex refractive index, or complex dielectric constants,

or complex conductivity.

Those optical constants could be probed either directly

(ellipsometry, ultrfast laser-based time domain terahertz

spectroscopy,…) or indirectly (reflectance

measurement over broad frequencies).

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 eEE2

2

0

2I

))(

(

0 t

c

xNi

y eEE

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

x

y E

Intensity

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

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, only R() can be measured

experimentally.

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

FTIR

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.

4( ) ( )

i

22

2

12

22

2

1

1

14

/1

p

p

R

1

p’

1/

1/ dc

-1

1()

(m*/m)Neff

energy

Im(-1/ )

'2

2 '2 2 2 2

'

/1Im{ }

( ) ( )

/

p

p

p p

2

1

0

( )8

pd

2

0 1( )

1 4 1/

D

i i

简单金属：Drude model

举例：Parent compound 1T-TiSe2

• 1T-TiSe2 was one of the first CDW-bearing materials

• Broken symmetry at 200 K with a 2x2x2 superlattice

• Semiconductor or semimetal?

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 fully empty ?

Se: 4p band fully occupied?

related to the CDW mechanism

Ti: 3d24s2

Se: 4s24p4

Ti : 3d band

Se: 4p band

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.

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)

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

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)

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

Simple metal

Correlated metal

Electron correlations reflected in optical conductivity

Correlation effect：reducing

the kinetic energy of electrons,

or Drude spectral weight.

Q M Si, Nature Physics 2009

Basov and Chubakov, Nature Physics 2011

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.

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

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

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

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

Coherent factors and characteristic spectral

structures in density wave or superconductors

Gap feature in density waves

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

Gap structure in superconductors

G. Li et al PRL 08

Cuprates, Homes data

Ba0.6K0.4Fe2As2

Coherent peak below Tc

O.Klein et al PRB 94

T. Fischer et al PRB

2010 R. V. Arguilar et al,

arXiv:107.3677

铁基超导体的THz数据与NMR不同？S+-配对？

Optical measurement under magnetic field

(10 Tesla split coils from Cryomagnetic Inc.) Bruker 113v spectrometer

Liquid

He

Josephson Plasma edge

R.H. Yuan et al, Scientific Reports (2012)

Josephson Plasma edge in

K0.75Fe1.75Se2 from nanoscale

phase separation

1000 1500 2000 2500 30000

10

20

30

40

50

60

70

80

90

100

Frequency (cm -1)

Ref

lect

ivit

y (

%)

EuB 6 T=20 K

0 T

1 T

2 T

3 T

4 T

5 T

6 T

7 T

Degiorgi et al., Phys. Rev. Lett. 79, 5134 (1997)

Effect of magnetic field:

Magneto-Optical Reflectivity in EuB6

Quasiparticle relaxation

after optical pump

Pump-probe experiment based on

femtosecond laser

飞秒激光

计算机

锁相放大器

样品、恒温器或磁体系统

太赫兹天线

分光镜

平移台

ZnTe

/4 渥拉斯棱镜

平衡探测器

太赫兹时域光谱系统原理图

硅分光镜

光阑

THz time domain spectroscopy

Optical (e.g. midinfrared) pump, THz probe !

Femtosecond laser can selectively

excite certain modes of correlated

electronic systems, and controllably

push materials from one ordered

phase to another.

Manipulation and control

Science 2011

Thanks！