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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
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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:
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Some basics about optical spectroscopy technique
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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
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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
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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)(
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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
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Reflectivity measurement
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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)
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C. C. Homes et al.
Applied Optics 32,2976(1993)
FT-IR spectrometer In situ evaporation
In-situ overcoating technique
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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.
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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
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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
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CuxTiSe2 system
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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)
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Ti: 3d24s2
Se: 4s24p4
Ti : 3d band
Se: 4p band
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No band crossing
Fermi level below CDW.
It is insulating!!
L
L
High T
Low T
2x2x2 superlattice
Se 4p
Ti 3d
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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
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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)
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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
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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
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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
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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
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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.
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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)
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Cu-doped CuxTiSe2
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Single band metal
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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)
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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
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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:
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G. Wu et al. PRB (07)
Hall coefficient is almost T-independent for
superconducting sample!!
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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)
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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):
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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
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Simple metal
Correlated metal
Correlation effect in optical conductivity
Correlation effect：reducing
the kinetic energy of electrons,
or Drude spectral weight.
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Q M Si, Nature Physics 2009 Qazilbash et al. Nature Physics 2009
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Hubbard U physics:
DFMT
V2O3
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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.
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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/ ()
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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
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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
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Hwang, Timusk, Gu,
Nature 427, 714 (2004)
Bi2212
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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
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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
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This part of spectral weight (originated
from the kinetic energy reduction)
contributes to the sc condensate.
Kinetic energy lowering with superconducting condensate
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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
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Superconductors
G. Li et al PRL 08
Cuprates, Homes data
Ba0.6K0.4Fe2As2
Structure of energy gap
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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？
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More about extended Drude model
——Extraction the boson (phonon) spectral function from optical conductivity
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• 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:
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Dip at +R
2F()
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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??
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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.
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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
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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
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normal state !
Marsiglio et al., Phys. Lett.
A 245, 172 (1998)
])(
1[
2
1)(
2
2
d
dW or
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Pb (actual, and
numerically inverted ---
these are indistinguishable
approximate (from (A) )
(A)
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K3C60 L. Pintschovius,
Rep. Prog. Phys.
57, 473 (1996)
Tc = 19 K
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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
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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
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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
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J. Hwang, T. Timusk, et al. cond-mat/0505302
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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
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kF
ARPES –Band Mapping and Fermi Surface
Energy Distribution Curve (EDC)
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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)
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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()
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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
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