Methoden moderner Röntgenphysik I + II: Struktur und Dynamik kondensierter Materie

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1 I + II: Struktur und Dynamik kondensierter Materie Vorlesung zum Haupt/Masterstudiengang Physik SS 2009 G. Grübel, M. Martins, E. Weckert, W. Wurth 1

2 Trends in Spectroscopy Wolfgang Caliebe Wolfgang Caliebe Ralf Röhlsberger Wolfgang Drube IXS IXS NRS XPES 2

3 Inelastic X-Ray Scattering Measure energy loss (or gain) of scattered photon High energy resolution Good momentum resolution S(q, ) ℏ 1 ℏ 2 k 2 2 k 1 1 q 3

4 Differences in experiments based on resolution and incident energy: non-resonant: mev: Phonons ev: Plasmons, band excitations, soft absorption edges, Compton scattering resonant (incident energy tuned to an absorption edge): Resonant inelastic scattering Resonant emission spectroscopy Literature: Winfried Schülke, Electron Dynamics by Inelastic X-Ray Scattering K. Sturm, Wechselwirkung von Licht und Materie, in 23. IFF-Ferienkurs FZ Jülich, Synchrotronstrahlung zur Erforschung kondensierter Materie 4

5 Today: Electronic Excitations: Plasmons, valence band excitations, excitons, etc. Double differential scattering cross section S(q,ω) dielectric function ε(q,ω), susceptibility χ(q,ω) Plasmons in metals band-gap in insulators instrumentation for IXS Examples plasmons in Lithium and Aluminum band-gap in diamond 5

6 Hamilton-operator for scattering Vector-potential of the electro-magnetic wave: sum of photon-creation and annihilation operators ℏ c 2 A j = k, a k, exp i k r k, a k, exp i k r V k k, k wave-vector of photon Polarization of the wave Polarization vector [ a k,, a k, ]=1 k =0 (Bose relation) Momentum-operator of the electron: p j Jackson, Electrodynamics 6

7 Hamilton-operator for scattering Hamilton-operators: H 1 in t = j H i 2n t = j H 3i n t = H 4 in t e 2 2 Aj 2 2mc e A p mc j j e ℏ s j x A j mc j e ℏ e 2 = s [ A j x A j ] 2 2 j 2 m c c j Radiation field Interaction of radiation field and electron Interaction of spin magnetic moment with em field Spin orbit interaction Jackson, Electrodynamics I> and F> are eigenstates of H0 with eigenenergies EI and EF, and I> and F> are orthogonal: <F I>=0 7

8 Fermi's Golden Rule T I F= F H i n t H i n t H i n t H i n t I ℏ Ignore all terms in second order in A Ignore the spin-terms 2 d d d 2 =r ℏ =ℏ 1 ℏ 2 F exp i q r j I I,F 2 j Polarization factor E F E I ℏ Conservation of energy Separate this equation into sample- and photonccontribution Thomson scattering S q, = F exp i q r I E E ℏ d d =r Th I,F j j F I Dynamic structure factor 8

9 Thomson Scattering Cross section d d =r Th Coupling of em-wave with electrons Polarization factor: depends on scattering geometry SR: usually polarized in the horizontal plane becomes 1 for vertical scattering geometry becomes 0 for horizontal scattering geometry at 90 Energy factor: usually smaller than 1 9

10 Dynamic Structure Factor 2 S q, = F exp i q r j I E F E I ℏ I,F j Electron Density Fluctuations S q, = 1 2 dt C r, t = r t 0 0 d r exp i q r t C r, t Van Hove 1954 Pair distribution function describes correlations in space and time of particles in sample (see also lecture by C. Gutt) 10

11 Dynamic Structure Factor S q, = 1 dt exp i t nq t n q Susceptibility (response function, see lecture C. Gutt) q, = i ℏ dt exp i i t [ n q t, n q 0 ] Thermal average, pair correlation function Fluctuation-Dissipation-Theorem S q, = 1 ℏ ℑ q, exp ℏ 1 11

12 Dielectric constant and susceptibility 1 4 e2 =1 2 q, q, q ℏ q2 1 1 S q, = ℑ 2 q, 4 e 1 exp ℏ exp ℏ Detailed balance: Electronic excitations: S q, =exp ℏ S q, 1 exp ℏ 1 =1 ℏ q2 1 S q, = ℑ 2 q, 4 e sum-rule: 2 ℏq d S q, = 2m 12

13 Plasmon Drude-model (semi-classical) Free electron gas oscillates: What is the energy of the excitation? = n d e N electrons E =2 2 4 n e 2p = m =4 n d e 2 = n d e N/Z ions 13

14 Randon Phase Approximation (RPA) & Self Consistent Field (SCF) Start with simple assumptions:- homogeneous electron gas - background by ion cores - electrons individual particles - single particle states: plane waves - energy eigenvalues: parabolic band Electric field: sum of external field and induced field: tot q, = ext q, ind q, Induced field and electron density variation connected via Poisson-equation: 4 e2 e ind q, = q 2 n q, 14

15 Local Density Approximation Correlation and exchange lead to additional terms e i n d q, =v H q n q, v xc q n q, Hartree Term: Exchange Term: 4 e2 v H q = 2 q d2 v xc q = 2 [ n xc n ]=const dn Exchange energy xc n Express Exchange Term as a product of the Hartree Term and 0 an appropriate function G0(q): v xc q =v H q G q 15

16 Dielectric Function in the Self Consistent Field SCF M =1 v H q 0 q, v H q G q q, Response function 2 8 e 1 q, = 2 q V 0 Fermi function f K f K q ℏ i K K K q f Energy of plane wave: K = ℏ 2 K 2 / 2m LDA: G 0 q q 2 0 G q =const. Non LDA: for large q 16

17 RPA for a real solid 4 e2 2 r l ' K exp i G, G ' q, = G, G ' l K q G q V l, l ', K q G f l K f l ' K q ' r l ' K l K q exp i q G ℏ i l K l ' K q Use this equation for calculation of bandstructure Extended models: interaction of hole with excited electron... Other Approach: Linear Combination of Atomic Orbitals 17

18 Crystal Spectrometer Best way to obtain high energy resolution! flat or bent crystal(s) close to backscattering geometry bending increases bandwidth source size contribution to resolution large solid angle Johann Geometry Typical dimensions: Analyzer diameter 100mm Rowland circle 1000mm 18

19 Why Backscattering-Geometry? How to calculate the energy resolution? Braggs Law: 2dsin = partial derivatives give: E/E= cotan + d/d 0 cotan =0 for r 0 2 = F q = V d Silicon, Germanium: 4r for h k l=4n = f q a h k l 4 2 for h, k,l all odd { } Energy-momentum relation different for photons and neutrons: Ex~k En~k2 Ex~1/ (Ex) f(q) n~f(q) 19

20 Energy Resolution of different Si-reflections at backscattering 20

21 Energy-Resolution (cntd.) Apply forces to thin crystal: model totally neglects that we have crystal with lattice parameters, crystal planes, bonding angles, etc. 21

22 Energy-Resolution (cntd.) Apply forces to thin crystal: Numerical solutions possible, use of the 'lamellar model': Crystal is divided into lamellae with thickness such that the angle between two successive lamellae is equal to the Darwin width of the reflection. Broaden intrinsic energy resolution from 45meV to 200meV (worse for larger crystals, material has to go somewhere, so worse figure, FE-calculations!) 22

23 Energy-Resolution (cntd.) Additional factors (geometry contributions): Johann Geometry finite source size spherical bending for 90o All contributions get worse with larger deviations from 90o Optimum range: 80o 88o good range: 70o 80o acceptable range: 60o 70o Problem: Backscattering geometry with limited number of Sireflections creates gaps in energy ranges with good energy resolution! Significant improvement in size and quality of Sapphire (Al2O3) and Quartz (research at APS and SPring8)! 23

24 Picture of a Spectrometer 24

25 What about the monochromator? Specifications: good energy resolution (similar to analyzer) energy close to backscattering energy of analyzer good tunability (few ev to hundreds of ev) high througput We cannot use a backscattering monochromator for a few ev! We have to use different tricks! Dispersive geometry miscut 25

26 Dispersive geometry Normal monochromator: Double-crystal monochromator two parallel crystals (+-) Whatever the first crystal reflects, is accepted by the second one just intrinsic energy resolution Dispersive monochromator: Both crystals reflect into the same direction (++) Red beam: correct Bragg-angle on both crystals Magenta beam: correct Bragg-angle only on 1st crystal 26

27 Miscut Lattice planes not parallel to surface Focussing miscut Collimating miscut Miscut angle α used to calculate miscut parameter b b=sin B /sin B Energy resolution: E = E 0 / b 27

28 Four-crystal Monochromator 4 reflections, dispersive geometry and miscut Equation for resolution: E 1 = s1 b1 s2 E tan B2 tan B2 b2 T. Matsushita et al., Handbook of Synch. Rad., Vol. 1, Chap. 4, ed. E.E. Koch Advantage: zero-offset monochromator (fixed exit) 28

29 Scientific Results Plasmon in Lithium Plasmon in liquid Lithium Temperature dependence of the Plasmon in Al Bandgap in diamond 29

30 Lithium W. Schülke et al., Phys. Rev. B (1986) J.P. Hill et al., Phys. Rev. Lett (1996) RPA and NLDA band-structure effects cause dip 30

31 Aluminium C. Sternemann et al., Phys. Rev. B (1998) Position does not change with temperature: Electron density destruction of long-range order has no effect, still short-range order in liquid dip: result of long-range order, disappears upon melting 31

32 Diamond bandgap and bandstructure LCAO G. Painter et al., Phys. Rev. B (1971) 1.6 X 1.3 X 1.0 X 0.7 X R. Roberts and W. Walker, Phys. Rev (1967) W. Caliebe, C.-C. Kao, unpublished W. Caliebe et al., Phys. Rev. Lett (2000) 32