Electron and vibrational spectroscopy

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1 Electron and vibrational spectroscopy Stéphane Pailhès Institute of Light and Matter, CNRS and UCBLyon 1 Team (Nano)Materials for Energy

2 Phonons definition A phonon (i.e. a lattice wave) is described by : a wave vector, q : determines the direction of propagation a dispersion relation, ω q = F(q), relates the wave vector of a phonon, q, to its energy, ω q a polarization vector, e q gives the direction of atomic displacements. For each q, three types of waves : 1 longitudinal (compressional) for e q //q and 1 transverse (shear) e q q an occupation number given by the Bose-Einstein distribution : n ω q,j = 1/ exp ħω q,j /k B T 1 b* e 010 Schematic representation of a transverse wave q 100 a*

3 Lecture of Dr. S. Merabia Thermodynamic of phonons Bose-Einstein statistic Internal energy Specific heat 0 N q exp 1 ω(q) k B T 1 N 0 q q q 1 2 C V E T Phonons submitted to a thermal gradient (non equilibrium thermodynamics) N q N 0 q n q Q= -λ ΔT λ thermal conductivity Q = Σ q (energy in state q)*(group velocity of phonon q) Q q Experiment ( q) ( q) ( q) ( nq q) C v ( ) l( ) avec l( ) ( ) q q q (q) τ(ω)~ 1 Γ(ω) Γ ω : energy width of phonon peak

4 Which phonons 1Thz=10 3 GHz~4meV~48K Ex. : TA(100) 011 phonons in Ge ω TA,011 [100] = 26.5q 21.5q 2 v LA LA = lim dω q 0 [100] /dq ~4000 m. s 1 Experimental tools : Dynamical thermal conductivity Inelastic X-ray Scattering (IXS) Inélastic Neutrons Scattering (INS) λ T = λ ω, T dω

5 Binary cages materials Ba 8 Si 46 (silicon clathrate of type I) λ~1-2 W/mK hhl : l=2n 00l : l=2n - Regular arrangement of Si 20 (5 12,dodecahedra) and Si 24 cages (tetrakaidecahedra, ) - Cubic cell : Pm-3n (223), a~10.3 Å

6 X-ray and Neutron diffraction I Bragg ~ F(Q) 2 avec F(Q) = f i e iq R ie W i(q) i f i : atomic scattering factors depends on the interaction of X-ray/Neutrons with matter X-ray : interact with the electron clouds of an atom Neutron : interact with atomic nuclei (and magnetic fields from unpaired electrons) f i ~Z Z(Si) 2 = 194 Z(Ba) 2 = 3136 X-ray and Neutrons weighted Bragg intensities in the scattering plane (HH0)/(00L) f i ~b i b Si 2 ~ 2.16 b Ba 2 ~ 3.23

7 X-ray and Neutron phonon intensity Dynamical structure factor I phonon ~ F D Q 2 ω ph n ω ph Q = Q B + q Q B reciprocal lattice vector q phonon wave vector F D Q = i 1 M i i Q e ph f i e W i(q) e ( Q r i) Acoustic limit (ω ph 0) close to the zone center Q = q + G ~ G i Atoms are vibrating in phase : e j,q lim ω exp ħω k B T ~ 1 ω = e j,q = 1 Bragg 400 lim F D (Q) 2 2 Q F(Q) 2 B Q Q B Mω Q 2 B I Bragg ω 2 M

8 Energy (mev) X-ray vs Neutrons weighted phonons intensities Rayons X Z(Si) 2 = 6.9 M Si Z(Ba) 2 = 22.8 M Ba Neutrons b Si 2 M Si ~ b Ba 2 M Ba ~ RX N Wave vector Wave vector

9 Phonon polarization term Q e ph ( with Q = Q B + q ) e T1 q Transverse Q e ~ Q B006 e T e T2 q e L // q Longitudinal Q e ~ = Q B222 e L

10 Neutron triple axis spectrometer 1Thz=10 3 GHz~4meV~48K Neutron analyzer 18 cm * 10 cm Conservation equations : ħq = ħk i ħk f ħω = E i E f Desired incident (k i, E i ) and diffused (k f, E f ) neutrons are selected from the beam using single crystals (Bragg s law) Neutrons (~20 mev) Neutron (INS-3axes) Beam size ~cm 2 20GHz q 0.05nm 1

11 Q 2q Inelastic X-ray scattering spectrometer 1 curved crystals analyzer : crystal 0.6*0.6*3 mm 3 E/E 10-7 T-scan mk Monochromator Si (n,n,n) Undulators Mirror vertical focusing q B (Ei,Ef)~20 kev >> ħω Q = k i k f Q = 2k i sin (θ) 75 m High energy resolution backscattering cristals : ΔE/E~10 8 Δd/d=αΔT (α Si = K 1 ) Temperatures of monok and analyzer controlled in the mk-range

12 Samples Rayons X Neutrons V ~ 0.4 cm 3 V ~ mm 3 Ba 8 Si 46 (5 GPa et 1000 C, ILM Lyon) Ba 8 Ge 40 Ni 6 (Bridgman furnese, MPI Dresden) Array built of several single crystals for neutrons. V=350 mm 3

13 Neutron mapping of transverse phonons

14 Neutron mapping of longitudinal phonons

15 Phonon lifetime q=(6,h,h) en Å -1 S(Q, ω) = F j D Q ω q,j 2 2 ω 1 exp ħω k B T + Spectral function F Q (ω) F Q ω dω = 1 F Q ω = 1 π Γω q,j ω 2 ω q,j 2 2 Γ 2 ω 2 F Q (ω) Damped harmonic oscillator

16 Instrumental resolution in (Q,ω) space RX : Q and ω not coupled Neutron : Q and ω coupled pente~3000 m.s -1 I = R(Q, ω) Phonon(Q, ω) Measurements of phonon lifetime require : - Single grain with extremely good mosaicity (0.1 ) - A fine analysis in regard to the instrumental resolution

17 Phonons density of states (p-dos), g(ω) Number of phonons between ω et ω+dω All lattice properties (K) are sums over phonons states K= K ω n ω g(ω)dω Require polycristals R. Lortz et al., Phys. Rev. B 77, (2008)

18 Electron spectroscopy

19 Cours de L. Chaput, J. Heremans Thermodynamic of electrons Electronic band structure E k g ω = ds m = ħ 2 1 FS 4π 3 k E(k) 2 E(k) k 2 Non equilibrium thermodynamics τ(ω)~ 1 Γ(ω) 1 => transport integrals 0 N q Fermi-dirac statistic 1 E(k) - 1 exp k B T Experiment E(k) τ(ω)~ 1 Γ(ω) Γ ω : energy width of electron peak

20 Photoelectrons Analyseur hémisphérique Measured In the solid E é vacuum kin (E kin ) K é vacuum kin (K) E B é solid (E B ) k é solide (k)

21 Travel of the excited electron to the surface k i = k f ~ Å 1 k photon ~0.01Å 1 E F = ħ2 k F 2 2m e = E i + hθ A very surface sensitive technique

22 Escape of the photoelectron into vacuum Conservation of the component parallel to the surface K // = k // = 2m e ħ 2 E kin sin (θ out ) More delicate for the component perpendicular to the surface no conservation law Need the «free electron final state approximation» E F V 0 = ħ2 k 2 2 F + k F 2m e V 0 = E kin k = 2m e ħ 2 (E kin cos θ out 2 + V 0 )

23 SLS (Switzerland) Energie cinétique e -

24 Les conditions pour faire une expérience - Des monocristaux - Clivables (surface de bonne qualité et représentative du volume) - Pas trop isolant - Plus facile sur des matériaux bidimensionnels

25 Energy and momentum resolutions Momentum resolution k // = 2m e ħ 2 E kin cos(θ out ) θ out θ out ~0.2 k // ~0.01 Å -1 (@21 ev) < 1% of the Brillouin zone ARPES experiment are performed in UV for hν < 100 ev Energy resolution Synchrotron : ΔE ~ 7-8 mev Laser : ΔE ~ 0.5 mev Laser (ArF, 6.4 ev) based ILM, Lyon

26 Fermi surface mapping 1 line = 1 detector position Intensity map measured at E=0 Area enclosed by the Fermi surface = number of carriers

27 Energy dispersion curves, peak profile analysis I ARPES = I 0 k f ω A(k, ω) f ω : Fermi function I 0 k : dipole matrix element The one particle spectral function A( k, ) 1 [ Im ( k, ) 2 Re ( k, ) ] [Im ( k, )] 0 k 2 Σ k, ω : electron proper self-energy contains all the information on the energy renormalization and lifetime of an electron Non interacting case : Σ k, ω = 0 A k, ω = δ(ω ε k ) Voir A. Damascelli RMP 75,473 (2003)

28 Interacting case Interacting electrons (so-called quasiparticles) have a renormalized energy, ε k, and mass m* and a finite lifetime τ k = 1/Γ k Pines and Nozières (1966) (Fermi liquid) A k, ω = Z k Γ k /π (ω ε k ) 2 +Γ k 2 + A inch Fermi liquid (=normal metal) Σ FL ω = αω + iβ[ω 2 + (πk B T) 2 ] Z k =(1 Σ / ω) 1 ε k = Z k (ε k 0 + Σ ) Γ k = Z k Σ

29 Peak profile analysis : measurement of Σ For k near k F, the A(k,ω 0 ) is a Lorentzian Γ k (ω 0 ) = Σ (ω 0 )/v F Σ calculated from the Kramers-Kronig relation : ReΣ = 1 π P Σ (ω 0 ) ω 0 ω dω 0 Quasiparticle residue : Z k =(1 Σ / ω) 1

30 Matrix element in the dipole approximation Symmetry Analysis

31 XPS-UPS loupe de la structure électronique 8eV below Ef 1.2eV below Ef Réunion -- Déc : 1) Principe d une expérience de photoémission 43

32 Core levels studies Probing the electronic environment by means of core levels Probing the doping of Ag in Ba 8 Ag x Si 46-x