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