An introduction to the optical spectroscopy of solidsctcp/China_US_Workshop/Nanlin Wang.pdf ·...
Transcript of An introduction to the optical spectroscopy of solidsctcp/China_US_Workshop/Nanlin Wang.pdf ·...
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
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
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.
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
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
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
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
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