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Conductivité OptiqueConductivité Optique
Ricardo Lobocar LÉcole Supérieure de Physique et Chimie Industrielles
de la Ville de Paris
lobo@espci.frlobo@espci.fr
Nichols, Phys. Rev. 1, 1 (1893)., y , ( )
A few optical excitations in solidsA few optical excitations in solids
~ 2 cm-1 10 000 cm-1 20 000 cm-1 30 000 cm-1 40 000 cm-11 eV~ 0.25 meV 2 eV 3 eV 4 eV 5 eV
~ 5 mm 1 m 500 nm 250 nm333 nmSuperconducting gap
Plasmon SemiconductorsPhonons
Molecular Vibrations (biology and chemistry)Molecular Vibrations (biology and chemistry)
Density-wave gapPolarons
Gap Semiconductors
Density-wave gap
Plasmon oxidesGap Semiconductors
Oxides Charge TransferPlasmon metals
OutlineOutline
The basics of optical conductivityThe optical conductivity of metalsp y fThe f-sum rule and gapsFermi liquid signaturesFermi liquid signaturesKinetic energyPh s d h s t siti sPhonons and phase transitionsDisordered Materials
Preview: YBCO from an insulator to a conductorPreview: YBCO from an insulator to a conductor
1.0O7 - conducteur
0.8vi
tyO7 con uct ur
0 4
0.6
flec
tiv
0.2
0.4
Re
O6 - isolant
100 1000 100000.0
0.2
100 1000 10000
Frequency (cm-1)
Optical C d i i
Optical C d i iConductivity
B ildi BlConductivity
B ildi BlBuilding Blocs Building Blocs
Light scatteringLight scatteringSource
Black Box
Detector
Optical conductivityPhoton in Photon out
Detector
Optical conductivityHence the misnomer "center of zone" technique
B tt " t d" t h iBetter name: "momentum averaged" technique
Elastic light scattering
The interaction with the electric field dominates
Broadband (whitelight) spectroscopy
Maxwell Equations – The Message in the "Photon Out"
Maxwell Equations – The Message in the "Photon Out"
I Maxwell Equations in matter In vacuum
Wave with speed "c"
In matter
Ohm's law
In matter
Wave with renormalized speedPlane wave
Wave with renormalized speed
The common optical functionsThe common optical functions
Refraction index: light propagation and tt tiattenuation
Dielectric function: microscopic polarizability
Optical conductivity: high-frequency electrical transport
Inside the Black Box – Fourier Transform Spectroscopy
Inside the Black Box – Fourier Transform Spectroscopy
Fixed mirror
Moving mirrorFellgett advantageMultiplex
Jacquinot advantageSlitless spectrometer
Nyquist theoremDiscrete FT is fine
A Strange Choice for UnitsA Strange Choice for Units
W b 1/ 2 / 1Wavenumber = 1/ = 2c/ cm-1
Energy E = h = hcgy1 eV ~ 8000 cm-1
Wavelength = 1/Wavelength = 1/1 µm = 10000 cm-1
Temperature T = hc/kB10000 K ~ 7000 cm-1
What else is in the black box?What else is in the black box?
I()
TR
R(ω) and T(ω) ~ |E|2
are real functions.R
Reflected Power (Real)
Kramers-Kronig, simulation; fit; R & T inversion... Other techniques using the same process:
Complex Optical Function
the same process:EllipsometryMach-ZehnderCoherent THzCoherent THz
Kramers-Kronig relations & the ReflectivityKramers-Kronig relations & the Reflectivity
Cauchy theorem0
Hilbert transform
Kramers-KronigCausality:Causality:
Reflectivity:
But we need R() at all energies
The optical The optical conductivity in conductivity in
metals metals
Drude ModelDrude Model
Free electron gas
No electron-electron interactionsNo electron electron nteract ons
No electron-phonon interactions
Scattering through elastic collisions
Drude, Ann. Physik 306, 566 (1900)
Drude Modelg g
Drude in lightly doped SiDrude in lightly doped Si
R Re 1 Hole dopedn ~ 10-15 cm-3
Electron doped
Im 1
n ~ 4 x 10-14 cm-3
Im 1
Dressel and Scheffler, Ann. Phys. 15, 535 (2006)
Transparent conducting oxidesTransparent conducting oxides
0.8
1.0CdO
mis
sion
0 4
0.6
& Tr
ansm
0.2
0.4
efle
ctiv
ity &
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0R
e
Energy (eV)
Keep n small (tweak materials) Increase (improve materials) Increase (improve materials) Play with m* (create materials)
Metz et al., JACS 126, 8477 (2004), , ( )
Drude in metals (Silver)Drude in metals (Silver)
1.0
0.8
30000
35000
40000
0.6
efle
ctiv
ity
20000
25000
30000
-1 c
m-1]
0.4Re
0 2 5000
10000
15000
1 [
S. Biermann
0 0
0.2
0 2000 4000 6000 80000
5000
Frequency [cm-1]0
0.010000 20000 30000
Frequency [cm-1]
Frequency [cm ]
But Drude fails in correlated matter…But Drude fails in correlated matter…
8000
60001/DC
4000cm-1)
2000 1 (
-1
YBCO – 100 K
0
Drude
0 500 1000 1500 2000 2500 30000
Fréquence (cm-1)
Ignoring the elephantIgnoring the elephant
3500.0 0.1 0.2 0.3
500
Energy [eV]
200
250
300
300
400
2000
3000m
-1]
Bi2Sr2CaCu2O8
T = 100 K
m-1]
cm-1]
50
100
150
100
200
1000
1 [-1 c
m
[
-1 c
m
1/
[c
0 50 100 150 200 250 3000
50
00 1000 2000 3000
0
Frequency [cm-1] Temperature [K]
Coherent peak described by Drude
Mid IR response
The Drude temperature dependent scattering rate
Quijada et al PRB 60 14917 (1999)
Mid-IR response simulated as Lorentz oscillators
follows the resistivity
Quijada et al., PRB 60, 14917 (1999)
The optical conductivity and multiband pnictidesThe optical conductivity and multiband pnictides
0 00 0 25 0 50 0 75 1 00
Energy [eV]
60000.00 0.25 0.50 0.75 1.00
Ag (divided by 100)Bi2212 (Quijada et al)
4000Bi2212 (Quijada et. al) Ba(Fe,Co)2As2
cm-1]
2000
1 [
-10 2000 4000 6000 8000
0
Frequency [cm-1]
Singh, PRB (2008).
Frequency [cm ]
Multiband – pnictides DresselMultiband – pnictides Dressel
Using two Drude terms [two bands (??)] solves the problems
Wu et al., PRB 81, 100512 (2010)
Can we take a look at the elephant?Can we take a look at the elephant?
8000
60001/DC
4000cm-1)
2000 1 (
-1
YBCO – 100 K
0
Drude
0 500 1000 1500 2000 2500 30000
Fréquence (cm-1)
Patching the Drude ModelPatching the Drude ModelDrude, Ann. Physik 306, 566 (1900)
Allen, PRB 3, 305 (1971)
Drude Model Extended Drude Model
S tt i t MassScattering rate Massenhancement
"Optical self-energy"Optical self energy
Is this extended Drude mambo-jambo of any use?Is this extended Drude mambo-jambo of any use?
0 00 0 25 0 50 0 75 1 000 00 0 05 0 10 0 15 0 20
Energy [eV]Energy [eV]
30.00 0.25 0.50 0.75 1.000.00 0.05 0.10 0.15 0.20
1000
Bi2212 - 300 Knon Fermi liquid (ω)
Correlations
1
2
500m-1]
Bi2212 300 K
= 1
+ YBCO 100 K
0
1500
1/
[cm
Cr - 320 K
m /
m* =
Fermi liquid (ω2)
Free electron
0 2000 4000 6000 8000-1
0 500 1000 1500 20000
Fermi liquid (ω )Interband
Frequency [cm-1] Frequency [cm-1]
Basov et al., PRB 65, 054516 (2002) Puchkov et al., JPCM 8, 10049 (1996)Basov et al., PRB 65, 054516 (2002) Puchkov et al., JPCM 8, 10049 (1996)
One of the multiple versions of the cuprates phase diagram
One of the multiple versions of the cuprates phase diagram
TW i d t lW i d m t l
TW i d t lW i d m t l
PsPs
Weird metalWeird metal
PsPs
Weird metalWeird metal
“F i h”t
ins.
tt in
s.
Pseudoga
Pseudoga “F i h”t
ins.
tt in
s.
Pseudoga
Pseudoga “Fermish”liquid
“Fermish”liquidM
otM
ot
gapgap “Fermish”
liquid“Fermish”
liquidMot
Mot
gapgap
SCSCAFAF SCSCSCSCSCSCAFAF
Hole concentrationHole concentration
From a Nodal Metal to a Bad Metal to a Fermi Liquid
From a Nodal Metal to a Bad Metal to a Fermi Liquid
1/: Fermi liquidness increases with doping
T
ins.
ins.
Pseudo
Pseudo
Weird metalWeird metalT
ins.
ins.
Pseudo
Pseudo
Weird metalWeird metal
Hwang, Nature 427, 714 (2004)
m*: Correlations decrease with increasing dopingSCSC
“Fermish”liquid
“Fermish”liquid
AFAF
Mot
t M
ott
dogapdogap
SCSCSCSCSCSC
“Fermish”liquid
“Fermish”liquid
AFAF
Mot
t M
ott
dogapdogap
Ma, PRB 73, 144503 (2006). Hole concentrationHole concentration
Heavy Fermions and LF DrudeHeavy Fermions and LF Drude
300 neV 3 µeV 30 µeV
UPd2Al3
Awasthiet al. PRB 48, 10692 (1993).
Mass enhancement leads to scattering rate decrease
Dressel and Scheffler, Ann. Phys. 15, 535 (2006), ( )
Gaps andGaps andGaps and the sum rule
Gaps and the sum rulethe sum rule the sum rule
The Optical Conductivity Sum RuleThe Optical Conductivity Sum Rule
Sum rule
1() 1()
nedT2
)(
Sum rule
mdT
0 1 2),(
Fréquence (cm-1) 0 Fréquence (cm-1) 0 q ( )q ( )
The charge transfer gap of (La,Sr)2CuO4The charge transfer gap of (La,Sr)2CuO4
Uchida et al. PRB 43, 7942 (1991).
Density-wave like gapDensity-wave like gap
P4W14O50
CDW transition at T~140 K
CDW 1400 1 CDW gap at 1400 cm-1
Spectral weight transfer from low to high frequencies from low to h gh frequenc es
Zh l PRB 65 214519 (2002)Zhu et al., PRB 65, 214519 (2002)
Hidden order in URu2Si2Hidden order in URu2Si2
Phase transition at 17K to an Phase transition at 17K to an unknown order parameter
Possible renormalization of th b d t tthe band structure
Order closely related to magnetismg
Gap opening at the Fermi surface
Bonn, PRL 61, 1305 (1988)E h d k h h Even when we do not know what the crap is the gap, it still creates a spectral weight transfer
And let's go superconductingAnd let's go superconducting
0 5 10 15 20
Energy [meV]
15NbN18 K
103
-1 c
m-1]
5
1 [x10
9 K
0 50 100 1500
Frequency [cm-1]
Is the sum rule violated???
Superconducting gap – Back to DrudeSuperconducting gap – Back to Drude
Area transfered toth ( ) functi n
1.0
b.)
the () function = 0 not accessible
to optics
The "Lost area" 0.6
0.8
vity
(arb
The Lost area gives superfluid density
0 2
0.4
Con
duct
iv
1/
0 5 10 150.0
0.2C
Energy / 2 Energy /
Manifestation of the superconducting gap in the infrared
Manifestation of the superconducting gap in the infrared
s-wave (Pr,Ce)2CuO4 - Tc = 21 KZimmers et al., PRB 70, 132502 (2004)
d-wave
NbN – Tc = 16.5 KSomal et al., PRL 76, 1525 (1996)
Mazin, Nature 464, 183 (2010).s-multiband
Pnictides:s ? d? Multiband?
MgB2 – Tc = 39 KPerucchi et al., PRL 89, 097001 (2002).
s±? d? Multiband?Extended s?
The gap in the cupratesThe gap in the cuprates
Tc changes by a factor 3 and the thing that looks like the gap stays put ?!?!?!?!
Clean and DirtyClean and Dirty
Clean superconductor
0 75
1.00N
0.50
0.75
dc NS
In a clean superconductor you do
0.25
1 /
SN
In a clean superconductor you do not see the gap because there is no optical conductivity left at that frequency
0 2 4 60.00
SS
Nq y
The important feature in the optical conductivity is the determination of the superfluid
Energy / determination of the superfluid density.
The gap is a bonus!!!
The gap in the cupratesThe gap in the cuprates
Tc changes by a factor 3 and the thing that looks like the gap stays put ?!?!?!?!Yep! Cause it ain't the gap, stupid!!
Pnictides Gap ReviewPnictides Gap Review
Claim: 2 gap BCS H t l Xi 0912 0636 1van Heuman et al., arXiv:0912.0636v1
Claim: Good agreement with s-wave BCSGorshunov et al., PRB 81, 060509 (2010)
Claim: 3 gap BCSKim et al., arXiv:0912.0140
s ± gap: Interband scattering is pair breakings± gap: Interband scattering is pair breaking
Interband scattering annihilates Cooper pairsp p
Residual Drude peak in the superconducting state
Lobo et al (2010)Lobo et al. (2010)
The infamous The infamous cuprate
dcuprate
dpseudogap pseudogap
The pseudogapThe pseudogapT
. PsPs
Weird metalWeird metalT
. PsPs
Weird metalWeird metal
“Fermish”liquid
“Fermish”liquidM
ott
ins.
Mot
t in
s .
Pseudogap
Pseudogap “Fermish”liquid
“Fermish”liquidM
ott
ins.
Mot
t in
s .
Pseudogap
Pseudogap
SCSC
Hole concentration
liquidliquidAFAF SCSCSCSCSCSC
Hole concentration
liquidliquidAFAF
Nd2-xCexCuO4Bi-2212
95 K
Alloul, PRL 63, 1700 (1989)Hole concentrationHole concentration
120 K
95 K
Armitage et al PRL 81 257001 (2002) Norman et al. Nature 392, 157 (1998).
180 K
Armitage et al. PRL 81, 257001 (2002). , ( )
Restricted Spectral Weight or Partial Sum RuleRestricted Spectral Weight or Partial Sum Rule
nedT2
0 1 2),(
Sum rule
m0 2
1() 1()
Restricted spectral i ht
W (T)
[0, 2.5/2 cm-1]W (300K)
weight
1.3
1 15
[ , 2 ]
dTW c
c )(),,0(0 1
1.15
1
0 50 100 150 200 250 300
TempératureFréquence (cm-1) 0 Fréquence (cm-1) 0
c
The Sum Rule when a Gap OpensThe Sum Rule when a Gap Opens
Ec0 8
1.0
Wei
ght
0.6
0.8
uctiv
ity
Spec
tral
W
0.4Con
du
S
0.0
0.2
Temperature0.0 0.2 0.4 0.6 0.8 1.0
Energy
(Pr,Ce)2CuO4 - A well behaved normal state (pseudo) gap
(Pr,Ce)2CuO4 - A well behaved normal state (pseudo) gap
Energy [eV]
0.00 0.25 0.50 0.75
3 Pr2-xCexCuO4
Energy [eV]
Partial Sum Rule
2 25 K 300 K
1 ]
1.8
H,3
00K) 0-500 cm-1
0.17
0
1
x 10
3 -1cm
-1
1.4
1.6
/ RSW
(L,
(a)0.15
0
1[x
1.2
W (
L,H,T
) /
0.13
0
0 2000 4000 60000
(b) 0 100 200 300
1.0
RSW
Temperature (K)
Zimmers et al EPL 70 225 (2005)
Wavenumber [cm-1]
Zimmers et al., EPL 70, 225 (2005).
Bi-2212 – Where is the pseudogap?Bi-2212 – Where is the pseudogap?
6 80 K
Optical conductivity Partial sum rule
T 70 K 7 7
4
6 80 K 150 K 250 K
Tc = 70 K
Bi2Sr2CaCu2O8+δ7.6
7.7
cm
)
Over Tc = 63K Under Tc = 70KOver T = 67K
0
2
3 -1
cm
-1]
2 2 2 8+δ
7.5
ht (
106
.c Over Tc 67K
Under Tc = 66K
1 [103
491 K 143 K 300 K
Tc = 86 K
7.3
7.4
Santander-Syro et al PRL '02ectra
l wei
gh0 500 1000 1500
0
2
50 100 150 200 250 3007.2
Santander Syro et al., PRL 02 Molegraaf et al., Science '02Sp
eT (K)0 500 1000 1500
Frequency (cm-1)
Santander Syro et al PRL 88 097005(2002)
Temperature (K)
Santander-Syro et al., PRL 88, 097005(2002).
Why the difference?Why the difference?
The ab-plane optical probes mostly the nodal ( ) directions() directions
Bi-221295 K
120 K
95 K
(La,Sr)2CuO4
Armitage et al PRL 81 257001 (2002)
Nd2-xCexCuO4
Norman et al. Nature 392, 157 (1998).
180 K
Armitage et al. PRL 81, 257001 (2002). , ( )
The pseudogap in the scattering rateThe pseudogap in the scattering rate
2500.0 0.1 0.2 0.3
3
Energy [eV]
200
(b)
K]
Bi2223
1
2
3
300 K(a)
200 K
100
150
pera
ture
[K
Bi22120
0
1
x103 c
m-1]
1
150 K
73 K
50
100
Tem
p
Bi2201
1 /
[x
0
1
20 K
0
1
73 K
0.1 0.2 0.30
Bi2212 - UD - Tc = 67 K
0 1000 2000 30000
1
0
Doping
Lobo et al. (2009)
Frequency [cm-1]
Hwang et al., Nature 427, 714 (2004)
The pseudogap along the c-axis of YBCO (Tc = 63 K)
The pseudogap along the c-axis of YBCO (Tc = 63 K)
Spectral weight decrease
Knight shift
Homes et al., PRL 71, 1645 (1993).Homes et al., RL 7 , 6 5 ( 993).
AND NOW FOR SOMETHING AND NOW FOR SOMETHING AND NOW FOR SOMETHING COMPLETELY DIFFERENT
AND NOW FOR SOMETHING COMPLETELY DIFFERENT
Light and Matter interaction in an Insulator
Light and Matter interaction in an Insulatorin an Insulatorin an Insulator
Phonons and the Harmonic ApproximationPhonons and the Harmonic Approximation
And for many phonons
Phonons & the f-sum rulePhonons & the f-sum rule
The f-sum rule (particle conservation):
The f-sum rule for phonons:
The f-sum rule for decoupled phonons:
Soft Mode & Phase TransitionsSoft Mode & Phase TransitionsQ is a dynamic variable of the system
Order parameter = < Q > = 0, T > Tc 0 T T
Order parameter Q ≠ 0, T < Tc
<Q>Energie libre QgT > Tc
TTc
T ~ Tc
T < Tc
= < Q0 > (paramètre d’ordre)
c
TT
Q (variable dynamique)
TTc
SrTiO3 – A wannabe ferroelectricSrTiO3 – A wannabe ferroelectric
1001.0
60
80
m-1)
0.6
0.8
vity
20
40
WTO
(1) (
c
0.4
0.6
Ref
lect
i
20 K 180 K 250 K 300 K
0 50 100 150 200 250 3000
20W
Temperature0 200 400 600 800
0.0
0.2
F ( -1) TemperatureFrequency (cm )
Incipient soft mode driven f l ferroelectric transition
TC ~ -15 K
Soft mode and FerroelectricsSoft mode and Ferroelectrics
BaTiO3100
Vogt et al, PRB 1982.
m-1)
50
(c
m
Merz, PR 91, 513 (1953)0300 500 700300 500 700
T(K)
The Multiferroic Materials TotemThe Multiferroic Materials Totem
Coexistence of at least two ferroic orders [ferromagnetic,
P
ferroelectric, ferroelastic, ferrotoroidal (??)] *
P
F l t iMagnetoelectricsPiezoelectrics
Schmid, Ferroelectrics 162, 317 (1994).
Ferroelectric
FerromagneticFerroelastic
Magnetostriction
Magneto-electric MultiferroicsMagneto-electric MultiferroicsEerenstein et al., Nature 442, 759 (2006).
Multiferroics
FM FEMagnetoelectricsMagnetoelectric
Magnetict i l
Electricl
Multiferroics(Control of either
polarization materials materials
pby either field)
*antiferromagnets also accepted.
Small coupling – 4 state memory Large coupling – “E” write / “M” read
also accepted.
Ying & YangYing & YangFerroelectricity & Magnetism coexist
Magnetism causes Ferroelectricity
BiFeO3Large moments High temperature transitions
TbMnO3Small momentsLow temperaturesHigh temperature transitions
Weak couplingmp
Strong coupling
Lebeugle et al. APL 2007 Kimura, Nature 2003
Yang (TbMnO3) is more fun!Yang (TbMnO3) is more fun!
Kimura Nature 2003Kimura, Nature 2003.
TbMnO3 Phonon Spectra (T = 5 K)TbMnO3 Phonon Spectra (T = 5 K)
0.8
1.0
115
116
117
E // a
0.4
0.6
112
113
114
204
cm-1)
0.0
0.2
vity
E // a
0.4201
202
203
est T
O m
ode
(c
E // b
Ref
lect
iv
0 0
0.2
E // b199
200
201
eque
ncy
of lo
we
168
0.0
0.2
0.4
E // c
Fre
165
166
167
168
0 200 400 600 800 10000.0
Frequency (cm-1)
E // c
0 50 100 150 200 250 300
164
165
Temperature (K)
E // c
q y ( )
So, where is the action?So, where is the action?
Pimenov et al., Nat. Phys, 2006.
M ti it ti
Senff et al. PRL 2007
Magnetic excitation
Activated by electric field of light only
Suppressed by an external magnetic field Suppressed by an external magnetic field
Magnons and PhononsMagnons and Phonons
Stronger electromagnon at 60 cm-1 coupled to phonon at 110 cm-1
Takahashi et al PRL 101 187201 (2008)
Stronger electromagnon at 60 cm coupled to phonon at 110 cm
Takahashi et al. PRL 101, 187201 (2008)
Phonon and magnon optical responsePhonon and magnon optical response
1 0 0 6
0.8
1.0
TbMnO3 E // a
0 4
0.5
0.6
n
50K 25K5K
0.4
0.6
flect
ivity
0.3
0.4
smis
sion 5K
simulation 5K
0.2
Ref
6.5 K 60 K 0.1
0.2
Tran
s
0 50 100 150 200 2500.0
Wavenumber (cm-1)
0 50 100 150 2000.0
Wavenumber (cm-1)( )
Phonon spectral weigthsPhonon spectral weigths
2 )
100 1 2 3
1 0
K) (
1 cm2
-100
0
0.8
1.0
E // aT = 5K
100
200
T)-W
(60K
4 7 5 86 9
0 4
0.6
efle
ctiv
ity
T 5K
0
100
(T) =
W(T 6 9
0.2
0.4Re
-200
-100
W (
0 200 400 600 800 10000.0
Frequency (cm-1)
0 20 40 60Temperature (K)
The electromagnon is built from two phononsThe electromagnon is built from two phonons
200
0
100
-1cm
-2)
All phonons MagnonPhonons 1 & 3
200
-100
T)-W
(50K
) ( Phonons 1 & 3
-300
-200
W (T
) = W
(T TNTC
0 10 20 30 40 50 60-500
-400W
0 10 20 30 40 50 60Temperature (K)
DiSordEredDiSordEredDiSordEred MEdiA
DiSordEred MEdiAMEdiAMEdiA
Disorder in (Y,Pr)Ba2Cu3O7Disorder in (Y,Pr)Ba2Cu3O7
8000
(P Y )B C O
4000
6000 (PrxY1-x)Ba2Cu3O7
100 K
)
2000
4000
x = 0 0-1cm
-1
Pr is an underdoping agent
Empties CuO2 planes
0
40 K x = 0 4
x = 0.0
1 ( p p
Localizes charges along chains
0
040 K
50 K
x 0.4
x = 0.50 500 1000 1500
0
Frequency (cm-1)Lobo et al. PRB 65, 104509 (2002).
Effective Medium TheoriesEffective Medium Theories
Gold evaporated onto hot (600K)
hi b t tsapphire substrates
Cranberry glass (Ruby gold glass)Cranberry glass (Ruby gold glass)
Maxwell-Garnett Theory
insul.
metal
J. C. Maxwell Garnett, Colours in Metal Glasses and in Metallic Films Philos Trans R Soc 203 385 (1904) Metallic Films, Philos. Trans. R. Soc. 203, 385 (1904).
Superconducting MoGe: Field Dependent Transmission
Superconducting MoGe: Field Dependent TransmissionTransmissionTransmission
2.0
K)
Far IR
1.5
K,H
)/T(8
B
1.0atio
T(3
K
0 5ssio
n R
a
0 T 6 T 1 T 7 T 2 T 8 T3 T 9 T0.5
Tran
smi 3 T 9 T
4 T 10 T 5 T
10 20 30 400.0
T
Frequency [cm-1]Frequency [cm ]
S d t2.5
Br ggeman
Effective Medium Models for a Superconductor with Vortices
Effective Medium Models for a Superconductor with Vortices
1.5
2.0
Superconductor& Normal metalin parallel
o T S(
f)/T N
1 5
2.0
BruggemanEffective MediumApproximation
T S(f)/T
N
0.5
1.0
Tran
smis
sion
Rat
io
= f +(1 f ) 1.0
1.5
smis
sion
Rat
io
10 20 300.0
0.5T
Frequency [cm-1]
= f N+(1-f ) S
0 10 20 300.0
0.5
Tran
s
super
normal
Frequency [cm-1]
2.0
2.5Garnett TheoryNormal withSuperconductinginclusions
f)/T N
1.0
1.5
mis
sion
Rat
io T
S(f
0 10 20 300.0
0.5Tran
sm super
normal
Frequency [cm-1]
Effective Medium Models for a Superconductor with Vortices
Effective Medium Models for a Superconductor with Vortices
2.5
Garnett TheorySuperconductor
/ N l i l i2.0 w/ Normal inclusionsT S(
f)/T N
1.5
on R
atio
T
1.0
ansm
issi
o
super0.5Tr
a super
normal
10 20 300.0
Frequency [cm-1]Frequency [cm ]
SummarySummary1 01 0
0.6
0.8
1.0
ctiv
ity
O7 - conducteur
0.6
0.8
1.0
ctiv
ity
O7 - conducteur Optics is mostly an electrical measurement
Momentum averaged technique 0.2
0.4
Refl
ec
O6 - isolant0.2
0.4
Refl
ec
O6 - isolant
measurement
100 1000 100000.0
Frequency (cm-1)
100 1000 100000.0
Frequency (cm-1)
The optical conductivity specgtral weight is always conservedalways conserved
RappelsRappels
Technique moyennée en impulsionq y pMesure de la conductivité électrique aux hautes
fréquencesf q Pic à fréquence nulle = charge mobile Pic à fréquence finie charge localisée Pic à fréquence finie = charge localiséeAccès aux états électroniques et à leur distribution
é ien énergieRègle de somme de la conductivité
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