Ultrafast Phenomena in Condensed Ma3er : basics Prof. P.Ruello Ins;tut des Molécules et Matériaux du Mans, UMR 6283 CNRS-Université du Maine.

Ultrafast Phenomena in Condensed Ma3er : basics Prof. P.Ruello

Ins;tut des Molécules et Matériaux du Mans, UMR 6283 CNRS-Université du Maine.

h ν

pump probe

phonons

Pascal.ruello@univ-lemans.fr

Ultrafast Phenomena in Condensed Matter

Enseignant : P. Ruello (IMMM) Chargé de TD : P. Ruello (IMMM)

Nombre d’heures : 10h cours/10h TD/5h TP Nombre ECTS : 3

Langue d’enseignement : English

Prérequis : Solid State Physics, Quantum Mechanics, Electrodynamics.

Evaluation : 2 examinations

In this lecture we introduce the fascinating world of ultrafast phenomena in condensed matter where the motion of electrons and atoms can be somehow viewed in the time domain. After a general introduction and some reminder about fundamentals of optics and electrodynamics, we will discuss the experimental setup permitting such studies (femtosecond lasers). We will then present the general electron-electron and electron- phonon collision processes. In particular, we will see how it is possible to measure the electron-phonon coupling involving, relaxation of electron and phonon with recent time- resolved optical setup.

-1 Introduction on the ultrafast physics in condensed matter : history, goals, applications : -2 Light-matter interaction at equilibrium :

- Classical electrodynamics : reminder on Maxwell equations and the classical dielectric response in a metal and in an insulator.

- Quantum electrodynamics : description of the dielectric constant within the time- dependent Shrödinger equation.

-3 Introduction on experimental time-resolved studies :

- principle of a pump-probe method : detectors bandwidth, stroboscopic regime of measurement, lock-in amplifier detection, basics of ultrafast laser technology -4 Properties of electron and phonons at the thermodynamic equilibrium

- reminder on the Sommerfel model, the band theory of electron and the classical lattice dynamics (phonons)

- quantum origin of the electron-electron and electron-phonon coupling (deformation potential, Fröhlich interaction)

-5 Properties of non-equilibrium photoexcited carriers - Two-Temperatures Model for metals

- Boltzman equation applied to photoexcited semiconductors -6 Optical and acoustic phonons ultrafast photogeneration processes

- Optical phonons : stimulated Raman process, displacive excitation (deformation potential).

- Acoustic phonon : deformation potential, thermoelasticity

-7 Applications of picosecond acoustics : evaluation of elasticity at the nanoscale of

nanostructures (echography of nanostructures by laser optoacoustics, example coming from the industry and labs).

Dynamic of electron and phonon at thermodynamic equilibrium

•  Spectrum of Phonons

•  Characteris;c ;mes of electron dynamics and electron-electron sca3ering

•  Characteris;c ;mes of electron-phonon sca3ering

Dynamic of electron and phonon at thermodynamic equilibrium

•  Spectrum of Phonons

•  Characteris;c ;mes of electron dynamics and electron-electron sca3ering

•  Characteris;c ;mes of electron-phonon sca3ering

Phonons characteristics

kHz MHz GHz THz

1 - 5µ m 1- 5 nm

1 - 5 mm 1 - 5 m λ   (phonon

wavelength) Laser pulse duration

µs ns fs

ms Heat

Hypersound

ultrasound sound

frequency

(Incoherent, coherent)

To control/study the GHz-THz coherent phonons with light à femtosecond laser

APPLIED PHYSICS REVIEWS

Nanoscale thermal transport

David G. Cahill^a)

Department of Material Science and Engineering and the Frederick Seitz Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801

Wayne K. Ford

Intel Corporation, 5200 NE Elam Young Parkway, Hillsboro, Orgeon 97124 Kenneth E. Goodson

Department of Mechanical Engineering, Stanford University, Palo Alto, California 94305 Gerald D. Mahan

Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802 Arun Majumdar

Department of Mechanical Engineering, University of California, Berkeley, California 94720 Humphrey J. Maris

Department of Physics, Brown University, Providence, Rhode Island 02912 Roberto Merlin

Department of Physics, University of Michigan, Ann Arbor, Michigan 48109 Simon R. Phillpot

Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439

!Received 28 January 2002; accepted 1 August 2002"

Rapid progress in the synthesis and processing of materials with structure on nanometer length scales has created a demand for greater scientific understanding of thermal transport in nanoscale devices, individual nanostructures, and nanostructured materials. This review emphasizes developments in experiment, theory, and computation that have occurred in the past ten years and summarizes the present status of the field. Interfaces between materials become increasingly important on small length scales. The thermal conductance of many solid–solid interfaces have been studied experimentally but the range of observed interface properties is much smaller than predicted by simple theory. Classical molecular dynamics simulations are emerging as a powerful tool for calculations of thermal conductance and phonon scattering, and may provide for a lively interplay of experiment and theory in the near term. Fundamental issues remain concerning the correct definitions of temperature in nonequilibrium nanoscale systems. Modern Si microelectronics are now firmly in the nanoscale regime—experiments have demonstrated that the close proximity of interfaces and the extremely small volume of heat dissipation strongly modifies thermal transport, thereby aggravating problems of thermal management. Microelectronic devices are too large to yield to atomic-level simulation in the foreseeable future and, therefore, calculations of thermal transport must rely on solutions of the Boltzmann transport equation; microscopic phonon scattering rates needed for predictive models are, even for Si, poorly known. Low-dimensional nanostructures, such as carbon nanotubes, are predicted to have novel transport properties; the first quantitative experiments of the thermal conductivity of nanotubes have recently been achieved using microfabricated measurement systems. Nanoscale porosity decreases the permittivity of amorphous dielectrics but porosity also strongly decreases the thermal conductivity. The promise of improved thermoelectric materials and problems of thermal management of optoelectronic devices have stimulated extensive studies of semiconductor superlattices; agreement between experiment and theory is generally poor. Advances in measurement methods, e.g., the 3#method, time-domain thermoreflectance, sources of coherent phonons, microfabricated test structures, and the scanning thermal microscope, are enabling new capabilities for nanoscale thermal metrology. ©2003 American Institute of Physics. $DOI: 10.1063/1.1524305%

JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 2 15 JANUARY 2003

(@RT)

LaXce dynamics : phonons

1.1 Propriétés électroniques et ioniques

1.1.2 Le réseau ionique

a) Vibrations réticulaires : relation de dispersion des phonons

La maille élémentaire des métaux est constituée d’un seul atome ce qui conduit unique- ment à des phonons acoustiques, les branches de phonons optiques nécessitant au moins deux atomes par maille élémentaire. Les branches acoustiques sont constituées de deux branches transverses (notéesT1 etT2) et d’une branche longitudinale (notéeL). Pour chacune des trois branches, N_i modes existent dans la première zone de Brillouin (où N_i est le nombre d’ions qui forment le réseau).

La relation de dispersion des phonons, liant la pulsation ω_s au vecteur d’onde ⃗k (où s = 1,2,3 est un indice relatif à la branche considérée : transverse ou longitudinale), peut être mesurée par diffusion inélastique de neutrons. La figure 1.4 présente le résultat dans le cas de l’or [15]. Le long des axes cristallins de forte symétrie (ΓL, ΓX), les deux branches transverses sont dégénérées. Des relations de dispersion similaires sont obtenues pour l’argent [16] et le cuivre [17], signature d’un réseau cubique à faces centrées.

Figure 1.4 – Structure de bandes des phonons dans l’or [15]. T₁ et T₂ (notées T lorsqu’elles sont dégénérées) correspondent aux branches transverses et L à la branche lon- gitudinale. ζ est le vecteur d’onde réduit défini par ζ =ka/2π.

Deux modèles simples permettent d’obtenir des expressions analytiques qui fournissent une description approchée de la relation de dispersion réelle : le modèle de Debye et le modèle sinusoïdal. Dans le modèle de Debye, on fait l’approximation d’une dispersion linéaire des branches acoustiques. Elle est souvent associée à une hypothèse d’isotropie. La relation de dispersion est alors :

ω(⃗k) =vϕk , (1.7)

avec vϕ la moyenne de la vitesse de phase des trois branches.

10

3.3. Propri´et´es dynamiques et vibratoires d’un r´eseau lin´eaire 21

3.3.1 Vibration d ?une chaˆıne lin´eaire monoatomique : ´equation du mouvement Figure :

md²u⇥_n

dt² = ⇥F _{n+1 n} +⇥F _{n 1 n} (3.11)

Compte des d`Eplacements arbitraires choisis et repr`Esent`Es sur la figure ci-dessus, l ?`Equation du mouvement prend la forme alg`Ebrique suivante

md²u_n

dt² = C(un+1 un) k(un un 1) (3.12)

!"#$%"&'()*+,$-$./0'

1' u!_n !

u_n+1

F!_n+1!n

2' 2'

3' 3' 3'

3' 2' 3'

u!_n!1

2'

F!_n!1"n 444' 444'

15&6' 157&89:6' 157&49:6'

Figure 3.4 – cha ´One monotoatomique 1D. Forces s ?exer ´Aant sur la masse m d ?indice n pour des d`Eplacements des voisins arbitraires tels que u_n+1 u_n u_{n 1}.

Relation de dispersion

Par invariance de translation (cristal 1D infini), tous les atomes sont identiques du point de vue des ondes qui vont se propager le long de cette cha ´One. Aussi, la forme g`En`Erale du d`Eplacement longitudinal de l ?atome d ?indice n sera : u<sub>n</sub> = Ae<sup>i(kxn t</sup> = Ae<sup>i(kna t</sup>. Le d`Eplacement de chaque atome est le m´Ime et est caract`Eris`E par l ?amplitude alg`Ebrique A. Seul un d`Ephasage des oscillations appara ´Ot entre les diffErents atomes et est quantifi`` E par le terme e^ika pour deux atomes s`Epar`Es de la distance na. Cette onde m`Ecanique harmonique sera donc solution de l ?`Equation du mouvement si elle v`Erifie l ?`Equation suivante :

m ² = C(e^ika 1) C(1 e ^ika) = 2C + 2Ccos(ka) (3.13) Sachant que cos(ka) = 1 2sin²(ka/2) , nous obtenons donc :

2 = 4C

m sin²(ka

2 ) (3.14)

Cette relation entre et k est appel`Ee relation de dispersion. Nous pouvons d`Ej ? constater que

ω

_L

= 4C

_L

m sin(qa / 2) ω

_T

= 4C

_T

m sin(qa / 2)

FCC laXce BCC Brillouin zone laXce

V

_L

= 4C

_L

a

²

m

Sound speed (zone center)

V

_T

= 4C

_T

a

²

m

6

Interatomic bonds strength

V

_L

= 4C

_L

a

²

m = 3200 m / s a=0.28nm, m=3.10

^-25

kg C

_L

=10 N/m

0.4nm Example : gold

Please make you sure that you know how to es;mate rapidly the

concentra;on of

atoms in a solid !

Hafnium (2 atoms per unit cell)

260 ON THE LATTICE DYNAMICS OF hcp HAFNIUM Vol. 35, No. 3

4 ^- HAFNIUM I 7 ^- 296K ^-

(001)

____ -—-—- —---..-

2 - / / ~

// ~ ~ ^~ // ^/7

3.

/1

^//

//

^{2 ‘~ .//////~ .— ~‘ -}

// ~ ^- ~1/1^/ /

,~7 V~-~ ^{0 295K} ,7 ^{// -‘ ——- ZIRCONPJM}

// ^{~2 ~ 300K I —--— HAFNIUM -}

——TECHP Tk1~l

I I ~ ^{I I}

00 0.1 0.2 0.3 04 05 0 0.1 0.2 0.3 0.4 0.5

REDUCED WAVEVECTOR REDUCED W~EVECTOR

Fig. 1 Room temperature and 1300 K measurements Fig. 2 Comparison of the c—axis dispersion of the dispersion curves of hcp Hf along the curves of Hf, Zr,4 Ti,5 and Tc.7 The transverse

[001] symmetry direction, branches are not shown for the sake of clarity.

of the superconducting transition elements of In this picture the temperature dependence of the IV column of the Periodic Table may be the zone center [OOl]LOmode arises from the characteristic of all hcp superconducting tran— disorder and thermal repopulation which at sition elements. In fact the [0Ol]L0branch of higher temperatures diminishes the effectiveness technetium7 (see Fig. 2) also exhibits anomalous of the band splitting in lowering the electronic dispersion and similar temperature dependence8 energy.

as found in Zr, Ti, and Hf. It should also be pointed out that the Hf Since similar anomalies were observed in room temperature data (as well as the Ti data) the dispersion curves of Hf, Zr, and Ti it is in the [0011 direction can be fitted9 very well natural to attribute their origin to the by a phenomenological model’° proposed by

electronic band structure of these metals. This Wakabayashi which includes the coupling of dipo—

is consistent with the results of a recent for— lar charge fluctuations to the longitudinal pho—

mulation~- of the lattice dynamics of transition nons. The success of this model is due to the metals and compounds proposed by Varma and Weber. fact that in these metals a favorable situation

In this formulation phonon anomalies arise when— occurs for induced dipolar charge fluctuations.

ever a lattice distortion causes a reduction in In fact our band calculations show that the the electronic energy. Of course this reduction states of the fifth and sixth bands near the in electronic energy is particulary significant symmetry point H and just below EF, contain a if the lattice distortion opens up a local gap large admixture of p electronic character. In in the electronic structure at the Fermi surface, response to a lattice distortion these states This interpretation is supported by our band can admix with the d States above HF and give

theoretical calculations for Zr in which the rise to a dipolar charge density response. Thus nuclei have been frozen at the positions (fur— in the case of the superconducting transition

thest from equillibrium) which they assume for elements of the fourth column of the Periodic the zone center [OO1]LOmode. Our calculations Table the Varma—Weber formalism and models em—

show that this mode Is particularly effective in phasizing the role of real space charge density splitting the doubly degenerate bands In the AJIL distortions are not mutually exclusive. Quanti—

!"#$%&'"()*+($%,"-./0(#/-&,$%1234%

!"#$%"5*+($%,"-./0(#/-&,%1264%

789:&%

& ;<&=-$%&(%>$5")%

789:&%

& ;<&=-$%&(%>$5")%

!"#$%&

%

'()*&%+(,-./,*&01%2-3*1-*&4567&

'()*&(8./,*&01%2-3*1-*&4597&

LA

LO

TA

TO LA

LO

TO

TA

8 Stassis et al Solid. State Comm. 1980

Characteris;c ;mes :

At thermodynamic equilibrium :

e+

e+

λ

_e

= (2 π × 3.10

⁸

m .s

⁻¹

) / ω

_e

= 400 − 500nm

e-

C : elas;c constant C ~1-10 N/m

(determined thanks to sound speed measurements)

m =9.1.10

^-31

kg

C

C

Case of an electrosta;c interac;on

Characteris;c frequency C

ω

_ion

= C

m

_ion

= 3.10

¹³

rad.s

⁻¹

ω

_e

≈ C

m

_e

≈ 3.10

¹⁵

Hz

λ

_ion

= (2 π × 3.10

⁸

m . s

⁻¹

) / ω

_ion

= 60 µ ^m

Equivalent wavelength of the electomagne;c wave

Equivalent wavelength of the electomagne;c wave ca;on

electron Very simple

model !

Dynamic of electron and phonon at thermodynamic equilibrium

•  Spectrum of Phonons

•  Characteris;c ;mes of electron dynamics and electron-electron sca3ering

•  Characteris;c ;mes of electron-phonon sca3ering

Plasma oscilla;on

+e +e

+e +e

+e +e

+e +e

+e

+e -e

-e -e

-e

-e

-e

-e

-e

-e -e

Let’s consider there is a collec;ve mo;on of the electronic cloud (on the leià, so that there is a macroscopic electric field that arises (spa;al separa;on of the center of charge –e and +e)

- - -

+ + + M

At any point M, N, O, P, etc an electron is then submi3ed to a macroscopic electric field …

By taking into account the Coulomb force, show that the electron M follow a mo;on equa;on of an oscillator

N

O

P

^!!! _{+ !}

!!! = 0

Maxwell equa;on and Maxwell ;me

div  E ( 

r , t ) = ρ ^{( r  , t )} ε

₀

div 

B( 

r , t ) = 0

rot     E ( 

r, t ) = − ∂  B( 

r , t )

∂ t

rot     B( 

r , t ) = µ

₀

^{J  ( r  , t )} + µ

₀

ε

₀

^∂

E  (  r, t )

Eq Gauss

(loi locale du théorème de Gauss)

Eq. Maxwell-Faraday (loi de Lenz)

Eq. Maxwell-Ampère Eq Gauss

(conserva;on du flux magné;que, les charges magné;ques n’existent pas)

Formes locales

Charge conserva;on and Maxwell ;me

!"# !"#! = 0 = !

_!

!"# ! + !

_!

!

_!

!!

!!

!"#! + !!

!! = 0

!"#! = !

!

_!

with and ! = !!

! !

!

_!

+ !!

!! = 0

We arrive to

! ! = !

_!

!

^{! !!! !}

!

!"#$%&&

= !

_!

!

t

τ

_Maxwell

Maxwell ;me

!

!"#$%&&

= 1

!

_!^!

!

_!!!!

With the Drude

conduc;vity ! = !!

^!

!

_!!!!

!

And the plasma frequency !

_!^!

= !!

^!

!!

_!

Maxwell ;me : ;me necessary to come back to the equilibrium aier a local fluctua;on of a charge (ini;al condi;on ρ

₀

)

! _! ^! ~10 ^!" ! ^!!

orders

! _{!! !!} ~100!"

!

!"#$%&&

~ 10

^!!"

!

+ +

0.1nm !

_!

~ 0.1!"

10

^!

( ! ) = 10

^!!"

!

Electrons in metals : quantum theory

Sommerfeld model : electrons are described by a plane wave and follows the Fermi- Dirac sta;s;c.

!"#$%&'()*+,*(-./

0*12(*34567/

f(E)= 1 e

E!E_F kT +1

8"#$/

#_&// #/"'7'(9:$///

!^(E)= V

2"²⁽ 2m_e^*

!² )^3/2 E

#7'(9:/;(63'/#_;(63'//

# // #//

Quan;za;on of the wave vector

N = 4

3 π ^k

_F³

× V

8 π

³

^{× 2}

Number of electrons in the Fermi sphere

k 

E ( 

k ) = 

²

k

²

2 m

^*

Energy of the electron

N / V = ρ ^{(E ) dE}

O E_F

∫ ^{≈ 5.10}

²⁸

^m

⁻³

^E

^F

^{= (} _V ^3N

π

²

⁾

2/3



²

2 m *

h3p://www.s.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/

r2_1_1.html

Electron dynamics : orders

Chapitre 1 : Propriétés des systèmes métalliques massifs et confinés

La densité d’états électronique s’écrit alors : ρ(E ) = dn

dE = m

^3/2e

π

²

!

³

√ 2E , (1.2)

où n est le nombre de niveaux électroniques.

Les électrons étant des fermions, leur nombre d’occupation f

_FD

est régi par la statistique de Fermi-Dirac :

f

_FD

(E ( ⃗ k)) = 1 exp

! E( ⃗ k) − E

_F

k

_B

T

_e

"

+ 1

, (1.3)

pour des électrons à une température T

_e

et où l’on a assimilé le potentiel chimique à l’énergie de Fermi car T

_e

≪ T

_F

, k

_B

étant la constante de Boltzmann. À température nulle, f

_FD

(E ) = 1 si E ≤ E

_F

et f

_FD

(E ) = 0 si E > E

_F

.

Cette énergie de Fermi E

_F

est définie comme l’énergie maximale des états occupés à température nulle [13]. Elle s’écrit :

E

_F

= !

²

k

_F²

2m

e

, (1.4)

avec k

_F

le vecteur d’onde de Fermi. On peut définir la vitesse de Fermi v

_F

= ! k

_F

/m

_e

et la température de Fermi T

_F

= E

_F

/k

_B

. Les valeurs de ces différentes constantes sont résumées dans le tableau 1.2 pour le cuivre, l’argent et l’or.

Métal m

_e

/m E

_F

#

eV $

v

_F

#

m.s

⁻¹

$

k

_F

#

m

⁻¹

$

T

_F

# K $ Cuivre 1,5 4,67 1,05 × 10

⁶

1,36 × 10

¹⁰

54400 Argent 1 5,49 1,39 × 10

⁶

1,20 × 10

¹⁰

63000 Or 1 5,53 1,40 × 10

⁶

1,21 × 10

¹⁰

64200

Table 1.2 – Propriétés des électrons de conduction pour les métaux nobles. m : la masse de l’électron. E

_F

: l’énergie du niveau de Fermi. v

_F

: la vitesse de Fermi qui cor- respond à la vitesse des électrons au niveau de Fermi. k

_F

: le vecteur d’onde de Fermi. T

_F

: la température de Fermi.

Les propriétés du gaz d’électrons peuvent être obtenues à partir de la statistique de Fermi- Dirac. La densité d’énergie volumique contenue dans le gaz d’électrons est notamment donnée par :

u

_e

=

%

+∞

0

Eρ(E )f

FD

(E )dE . (1.5)

On peut alors calculer la capacité thermique volumique du gaz d’électrons qui est égale,

Δ t ≈ l

V

_F

≈ 10

⁻¹⁰

m

10

⁶

m .s

⁻¹

= 10

⁻¹⁶

s

If an electron moves towards a ca;on, it will occupy a state that another electron has to leaves (Pauli rule). This ;me corresponds to the electron-electron sca3ering ;me.

N = ρ ^{(E ) dE}

O E_F

∫ ^{≈ 5.10}

²⁸

^m

⁻³

E

_F

= ( 3N V π

²

⁾

2/3



²

2 m * = 1

2 mV

_F²

16

(V. Juvé PhD, France)

How long do an electron stay around a ca;on ? Δ E Δ t ≈ 

Electronic dynamics :

Δ E ≈ 1 − 5eV

Extension of bands

Chapitre 1 : Propriétés des systèmes métalliques massifs et confinés

Énergie

K Γ

L W

X Γ

E_F

(a) Structure de bandes calculée pour les électrons libres dans un réseau cfc [13].

(b) Structure de bandes électroniques de l’or calculée à l’aide de la méthode « relati- vistic augmented-plane-wave » [14]. Les 5 bandes d, peu dispersées, occupent la partie grisée. Échelle de gauche en eVet échelle de droite en Rydberg (0,1 Ry = 1,36 eV).

Figure 1.3 – Structures de bandes calculées seulement pour les électrons libres dans un réseau cfc (1.3(a)) et en prenant en compte les bandes internes d et les effets relativistes (1.3(b)).

Gold (Au)

Δ t ≈ 10

⁻¹⁶

s

Typical ;me that an electron stays « close » to a ca;on before moving to another one.

17

(Christensen & Seraphin PRB 1971)

Electron-electron sca3ering

Γ

_!^!_!^!_!!^!!_!^!

= 2!

ℏ !

_!

, !

_!

! !

_!

, !

_! ^!

!

_!!

!

_!!

(1 − !

_!!

)(1 − !

_!!

)!(!

_!!

+!

_!!

− !

_!!

− !

_!!

) Full

Fermi sphere

2 1 3 kT

4

k

_x

k

_y

W=Coulomb

interac;on

Γ

_!^!_!^!_!!^!!_!^!

= 2!

ℏ !

_!

, !

_!

! !

_!

, !

_! ^!

!(!) Golden Fermi rule D(ε) is the density of final states (eV

^-1

)

When the collision involves 4 states :

! = 1 4!!

_!

!

^!

!

_!

− !

_!

(without the Thomas-Fermi

screening )

Electron sca3ering process and Pauli principle

Electron 1 with an energy E

_F

+kT>ε

₁

>E

_F

can be sca3ered only by an electron 2 whose energy is E

_F

>ε

₂

>E

_F

-kT (else electron 1 will be sca3ered into a state below Fermi level that it is already occupied= impossible = Pauli principle).

First statement :

2Ef+kT>ε

₁

+ε

₂

>2Ef = conserva;on of energy (ε

₂

comes from the full Fermi sphere – ε

₂

<Ef- and the final states cannot exceed 2Ef+kT )

So ε

₂

>2Ef-ε

₁

, but the maximun of e1 is Ef+kT so ε

₂

>Ef-kT

The electrons 1 and 2 can be sca3ered then into states 3 and 4 whose energy is above E

_F

but below E

_F

+kT due to energy conserva;on. Electron 2, 3 and 4 must have energy above E

_F

for energy conserva;on.

Another way : If ε

₂

<Ef-kT, then ε

₁

+ε

₂

<2Ef -- > impossible !! (Pauli) Second statement :

e1+e2=e3+e4 so since 2Ef+kT>e1+e2>2Ef then 2Ef+kT>e3+e4>2Ef

The number of electrons in state 1 that can be sca3ered by electron 2 is propor;onal to

The final states are contained within a similar volume

g(E

_F

) = 3 2

n kT

_F

dN = kT × g(E

_F

) = 3

2

nkT kT

_F

1

τ ^{∝ dN}

2

∝ kT

kT

_F

"

# $ %

&

'

So, the probability of sca3ering is

2

propor;onal to the product of the number of possible states 1 and possible states 2.

n ≈ 10

²⁸

m

⁻³

With :

Density of states

5.85 5.90 5.95 6.00 6.05 6.10 6.15 6.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

!_"#$%&'

()*%&' +#()),-'

f(E)= 1 e

E!E_F kT +1

"%.*/'0123452'67'()),'

State 1

State 2

1

τ ^{∝ dN}

2

∝ kT

kT

_F

"

# $ %

&

'

2

τ

_ee

= τ

₀

^E

^F

kT

!

"

# $

% &

2

Excess of energy kT

τ

_ee

Sca3ering ;me.

τ

₀

≈ n

^−1/3

V

_F

≈ 10

^−28/3

10

⁶

≈ 1 fs

n = 10

²⁸

m

⁻³

, V

_F

= 10

⁶

m / s

With

Aschroi & Mermin, Saunders College Publishing

The electron is sca3ered by the other

electron as fast as its energy departure

from E

_F

is large.

Example : life;me of photoexcited excited electrons in Rh and Mo.

Zhukov et al Phys. Rev. B 2004

τ

_ee

= τ

₀

^E

^F

kT

!

"

# $

% &

2

τ

_ee

= τ

₀

^E

^F

Δ E

"

# $ %

&

'

2

f(E)

=Fermi-Dirac distribu;on

f (E ) = 1 e

E−E_F kT

+ 1

E

_F

τ

_ee

Dynamic of electron and phonon at thermodynamic equilibrium

•  Spectrum of Phonons

•  Characteris;c ;mes of electron dynamics and electron-electron sca3ering

•  Characteris;c ;mes of electron-phonon sca3ering

Electron-phonon interac;on : Drude model

•  Characteris;c ;me of electron-phonon coupling from the DC conductvity measurement

σ = ne

²

τ m

^*

For a metal

σ ≈ 10

⁸

S / m n ≈ 10

²⁸

m

⁻³

m

^*

≈ 0.5m

₀

τ ≈ 100 fs = 0.1.10

⁻¹²

s = 0.1 ps

electron ca;on

v(t + τ ⁾ − v(t ) eE τ

τ Time between two electron-phonon shocks

ne

²

τ

h3p://en.wikipedia.org/wiki/Drude_model

Beyond the Born-Oppenheimmer approxima;on

25

The electrons are considered as so fast in their dynamics (see slides 13-16) that they can follow any kind of mo;on of the ca;onic laXce

< x(t) > = X

Cjh ⁰ | xˆ | ^jie ^i✏j~✏0t +X

C_j^⇤h ^j | xˆ | ⁰ie^+i✏j~✏0t

= X

< x_0j > e ^i✏j~✏0t ⇥C_j(t) +X

< x_j0 > e^i✏j~✏0t ⇥C_j^⇤(t) (18) with hx_0ji = h ⁰ | xˆ | ^ji and hx_j0i = h ^j | xˆ | ⁰i = hx_0ji^⇤. In the following we note

!j = ^✏j_~^✏0.

At this level we inject the Eq. 11 to obtain :

< x(t) > = X

[ e < xj0 > E_x⁰ 2 4eⁱ

✏j0 ✏0+~!

~ t 1

✏j⁰ ✏0 +~! + eⁱ

✏j0 ✏0 ~!

~ t 1

✏j⁰ ✏0 ~! 3

5]h ⁰ | xˆ | ^jie ^i✏j~✏0t

+ X

[ e < x_j0 >^⇤ E_x⁰

"

e ^{i✏j ✏~0+~!t} 1

✏_j ✏₀ +~! + e ^{i✏j ✏~0 ~!t} 1

✏_j ✏₀ ~!

#

]h ^j | xˆ | ⁰ie^+i✏j~✏0t

< x(t) > = X

< x_j0 >² [ eE_x⁰ 2 4eⁱ

✏j0 ✏0+~!

~ t 1

✏j⁰ ✏0 +~! + eⁱ

✏j0 ✏0 ~!

~ t 1

✏j⁰ ✏0 ~! 3

5]e ^i✏j~✏0t

+ X

< x_j0 >² [ eE_x⁰

"

e ^{i✏j ✏~0+~!t} 1

✏_j ✏₀ +~! + e ^{i✏j ✏~0 ~!t} 1

✏_j ✏₀ ~!

#

]e^+i✏j~✏0t

(19) with h ⁰ | xˆ | ^ji = h ^j | xˆ | ⁰i^⇤ =< x_j0 >^⇤ and < x_j0 >< x_j0 >^⇤=< x_j0 >²=< x_0j >²

Vee = 1 4⇡✏0

e² ree

V_ii = 1 4⇡✏₀

(Z_ie)² r_ii Vie = 1

4⇡✏₀

Ze⇥e r_ei T_ee = P_e²

2m Tii = P_i²

2M

(20)

6

The hamiltonian depends only on the coordinate of the electrons. The ions are considered as

fixed. The constant A can be put to zero for convenience. We say that this approxima;on is the adiaba;c approxima;on

< x(t) > = X

C

_j

h

⁰

| x ˆ |

^j

i e

^i✏j~✏0t

+ X

C

_j^⇤

h

^j

| x ˆ |

⁰

i e

^+i✏j~✏0t

= X

< x

0j

> e

ⁱ

✏j ✏0

~ t

⇥ C

j

(t) + X

< x

j0

> e

ⁱ

✏j ✏0

~ t

⇥ C

_j^⇤

(t) (18) with h x

_0j

i = h

⁰

| x ˆ |

^j

i and h x

_j0

i = h

^j

| x ˆ |

⁰

i = h x

_0j

i

^⇤

. In the following we note

!

j

=

^✏j _~^✏0

.

At this level we inject the Eq. 11 to obtain :

< x(t) > = X

[ e < x

_j0

> E

_x⁰

2 4 e

ⁱ

✏j0 ✏0+~!

~ t

1

✏

j⁰

✏

0

+ ~ ! + e

ⁱ

✏j0 ✏0 ~!

~ t

1

✏

j⁰

✏

0

~ ! 3

5 ] h

⁰

| x ˆ |

^j

i e

^{i✏j ~✏0t}

+ X

[ e < x

_j0

>

^⇤

E

_x⁰

"

e

^{i✏j ✏~0+~!t}

1

✏

_j

✏

₀

+ ~ ! + e

^{i✏j ✏~0 ~!t}

1

✏

_j

✏

₀

~ !

#

] h

j

| x ˆ |

0

i e

⁺ⁱ

✏j ✏0

~ t

< x(t) > = X

< x

_j0

>

²

[ eE

_x⁰

2 4 e

ⁱ

✏j0 ✏0+~!

~ t

1

✏

j⁰

✏

0

+ ~ ! + e

ⁱ

✏j0 ✏0 ~!

~ t

1

✏

j⁰

✏

0

~ ! 3

5 ]e

^{i✏j ~✏0t}

+ X

< x

j0

>

²

[ eE

_x⁰

"

e

^{i✏j ✏~0+~!t}

1

✏

_j

✏

₀

+ ~ ! + e

^{i✏j ✏~0 ~!t}

1

✏

_j

✏

₀

~ !

# ]e

⁺ⁱ

✏j ✏0

~ t

(19) with h

⁰

| x ˆ |

^j

i = h

^j

| x ˆ |

⁰

i

^⇤

=< x

_j0

>

^⇤

and < x

_j0

>< x

_j0

>

^⇤

=< x

_j0

>

²

=< x

_0j

>

²

H = H

_e

+ H

_i

+ H

_{i e}

= T

_ee

(~ r) + V

_ee

(~ r) + V

_ie

(~ r, R) + ~ T

_ii

( R) + ~ V

_ii

( R) ~ H = H

e

+ H

i

+ H

i e

= T

ee

(~ r) + V

ee

(~ r) + V

ie

(~ r, R ~

f ixed

) + A

(20)

V

ee

= 1 4⇡✏

₀

e

²

r

_ee

V

ii

= 1

4⇡✏

₀

(Z

i

e)

²

r

_ii

V

_ie

= 1

4⇡✏

₀

Ze ⇥ e r

_ei

T

_ee

= P

_e²

2m T

_ii

= P

_i²

2M

(21)