Solid State Spectroscopy. Wilfried Wurth. Introduction. Synchrotron Radiation. Photoelectron Spectroscopy. Magneto optical spectroscopy.

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2 - Basic Idea Use a monochromatic light source to map the occupied electronic structure of a system to empty states above the vacuum level - measure electron distribution using an electron analyzer F. Reinert und S. Hüfner, New Journal of Physics 7, 97 (2005)

3 Literature Principles and Applications Hüfner, Stefan Springer Series in Solid-State Sciences 662 Seiten ISBN:

4 Photo Effect Discovery by Hertz 1887

5 Photo Effect First experiments by Hallwachs 1887

Energy of electrons independent of intensity of

6 Photo Effect light electrical effect Lenard 1902 Nobel prize for discovery of cathode rays 1905 Energy of electrons independent of intensity of incoming light below threshold frequency no emission of electrons

7 Photo Effect explanation by Einstein 1905 Nobel prize 1921 E kin = R N A βν P P work function R universal gas constant, N A Avogadro-constant β = h k B

8 Planck on Einstein

9 Photo Effect P. Lukirsky, S. Prilezaev, Zeitschrift für Physik 49, 236 (1928) 1 2 mv2 max hν measure for Planck constant and work function of different materials

10 Photo Effect Energy balance eui =0 + ΦA = hν eui =max = ΦC ΦA

11 Modern Goes back to Kai Siegbahn (Nobel prize in physics 1981) C. Nordling E. Sokolowski and K. Siegbahn, Phys. Rev. 105, 1676 (1957)

12 Modern photon source: undulator beamline with grating monochromator hemispherical electron analyzer to determine energy and emission angle of electrons

13 Hemispherical Electron Analyzers ideal performance: only electrons with a kinetic energy PE (pass energy) shall pass the analyzer on an orbit with R = R 0 (R 0 = (R 1 + R 2 )/2) and hit the detector

14 Electrostatic Potentials spherical capacitor V(r) = 1 4πε 0 q r + C V 1(R 1) = 1 4πε 0 q R 1 + C V 2(R 2) = 1 4πε 0 q R 2 + C voltage between hemispheres q R1 R2 = (V 1 V 2) 4πε 0 R 2 R 1 electric field E( r) = R1 R2 R 2 R 1 (V 1 V 2) 1 r 2ˆr

15 Pass Energy on the ideal orbit we require the centrifugal force to be equal to the electrostatic force mv 2 R 0 = ee(r 0) 2PE = ee(r 0) R 0 R1 R2 PE = e (V 1 V 1 R1 R2 2) = e R 2 R 1 2R 0 R2 2 (V 1 V 2) R2 1 relationship between pass energy and voltage between hemispheres solely determined by the geometry (sometimes called the spectrometer constant) usually obtained from calibration runs with photoemission lines where the energy is known

16 Resolution relative resolution E x1 + x2 = + δα 2 PE 2R 0 x i entrance and exit slit width (modern analyzers do not have an exit slit, instead a two-dimensional imaging detector (e.g. combination of microchannel plate, phosphor screen and CCD camera) is used to obtain energy position and emission angle simultaneously) δα entrance angle Since resolution E is proportional to pass energy, PE is kept constant in most experiments Retardation mode (see below)

17 Focusing deviations from ideal orbit: m d2 r dt 2 = mω2 r ee constant angular momentum: ωr 2 = ω 0r 2 0 m d2 (r 0 + r) dt 2 = mω2 0r 4 0 r 3 e c r 2 with m d2 r dt 2 = mω2 0r 4 0 r 3 0 m d2 r dt 2 1 (1 + r r 0 ) ec 3 r0 2 1 (1 + r r 0 ) 2 mω0r 2 0(1 3 r ) ec (1 2 r ) r 0 r 0 mω 2 0r 0 = ec r 2 0 d2 r dt 2 r 2 0 = ω 2 0 r Solution: Oscillations around ideal orbit with frequency ω 0 zero crossing (back to ideal orbit) for ω 0 t = π = 180 double focusing, i.e for a 180 -analyzer a point at the entrance slit will be imaged as a point at the exit slit

18 Energy Balance - Measurement reference level for solids - Fermi level of sample: hν ǫi,ef = Ekin,EF = Uret + ΦA + PE Note: The work function of the analyzer ΦA has to be determined to calibrate the energy scale

19 Energy Balance - Measurement Spectrum from Cu(110) measured using the He I line (hν=21.23ev) of a gas discharge lamp. The 4s and the 3d-bands can be seen together with a large cascade of so-called secondary electrons (see below). From the width of the spectrum the work function of the material can be determined. (taken form S. Hüfner, )

20 Energy Resolution - Determination At the Fermi edge of metals one can determine the energy resolution of an analyzer. The measured Energy Distribution Curve (EDC) is given by a convolution of the Fermi-Dirac-distribution (FDD 1 f (E) = ) and the exp((e E F )/k B T)+1 instrumental resolution (typically Gaussian). Example: poly-crystalline Ag measured using the He I line (hν=21.23ev) of a gas discharge lamp F. Reinert und S. Hüfner, New Journal of Physics 7, 97 (2005)

21 Photoemission - Energy Conservation initial state energy: hν + E 0(N) with the ground state energy of the N-electron system E 0(N) and the photon energy hν final state energy: E f (N 1) + E kin with the energy of the excited N-1-electron system E f (N 1) and the kinetic energy of the photoelectron E kin If the energy of the N-1 electron system does not change between initial and final state, we have: E f (N 1) E 0(N) = ǫ i = hν E kin where ǫ i is the energy Eigenvalue of the photoelectron in the initial state (Koopmans Theorem)

22 Transition Probability perturbation theory (Fermi s golden rule): W i f = 2π ψ f H int ψ i 2 δ(e f E i hν) operator: includes canonical momentum p (p + q A) (A vector potential) H int = e e2 (A p + p A) eφ + 2m 2m A A witha p + p A = 2A p + i ( A) and A A 0 (neglect 2-photon-processes) and φ = 0 (gauge) and A = 0 (result of translational invariance of a solid) H int = e m A p provides: intensities selection rules

23 Dipole Approximation Ansatz (plane waves): A = k A k e i k r if k r 1, i.e. λ r we obtain A = A 0 = const (dipole app.) it follows (using commutation relations): ψ f A p ψ i ψ f A V ψ i ψ f A r ψ i Photoemission requires V 0 (momentum conservation, no photoabsorption for free electrons (see below)) dipole selection rules (for A = const): ψ f e r ψ i 0

24 Cross Section Effects energy dependence of cross sections can be used to identify certain states (more localized ( higher l ) states are more prominent at higher photon energies)

25 Polarization Effects (from E. Dietz und F. Himpsel, Comm.30, 234 (1979)) full line: E parallel (1 10); dashed line: E parallel (001) The dipole matrix element depends on the orientation of the electric field vector with respect to high symmetry directions of a crystalline solid. This allows to determine the symmetry of certain bands.

$the solid 2 transport to surface 3 diffraction at the surface$

26 3-step Model of Photoemission in Solids 1 optical excitation in the solid 2 transport to surface 3 diffraction at the surface barrier

27 1 optical excitation in the solid assumption: absorption of a photon with energy hν leads to a transition between an initial state ψ i and a final state ψ f which are both states of the infinite solid (taken from S. Hüfner) Since the photon momentum is small k = 2π/λ such a (direct) transition is not possible for free electrons

28 1 momentum conservation in a crystalline solid ψ j = u k ( r)e i k r = u ( k G) ( r)e i( k G) r G reciprocal lattice vector From the transition probability: W i f ψ f A r ψ i 2 Bloch functions we obtain for momentum conservation in the solid ki = k f G

29 1 momentum conservation in a crystalline solid from S. Hüfner taken

30 2 scattering - mean free path small mean free path of electrons - universal energy dependence - almost independent of material photoelectron is surface sensitive Propagating electrons in the solid will be inelastically scattered changing the electron energy gives rise to background in photoemission

$3 Diffraction at the surface potential step kin, = k out, 2m 2m sinθ in$

31 3 Diffraction at the surface potential step kin, = k out, 2m 2m sinθ in E 2 in = sinθ out 2 Eout sinθ in,max = Escape cone or Mahan cone Eout E in

32 Model: free-electron final states match free electron wave functions inside and outside the solid 2m 2m k in = 2 E in = 2 (E kin,e V,out + V 0 ) 2m 2m k out = 2 E out = 2 (E kin,e V,out)

33 Reciprocal lattices and selected points in k-space fcc bcc hcp

34 How to determine band structures with angle resolved photoemission (ARPES) - free electron final states Surface of constant kin. energy is equivalent to the surface of a sphere in k-space k f,in = 2m/ 2 (E kin + V 0) for constant angle of emission we obtain kf,in = k 2 f,in=out /sin2 Θ out + 2mV 0/ 2 k 2 f,in=out = k 2 f,out /tan2 Θ out + 2mV 0/ 2

35 How to determine band structures with angle resolved photoemission (ARPES)- free electron final states Optics 19, 3964 (1980) F. J. Himpsel, Applied

36 Band structure of Nickel spectra for normal emission resulting band structure F. J. Himpsel et al., Physical Review B 19, 2919 (1979)

Band structure of Ni using x-rays Hard x-ray photoemission (HAXPES) Higher photon energies result in enhanced bulk sensitivity due to higher kinetic energy

37 Band structure of Ni using x-rays Hard x-ray photoemission (HAXPES) Higher photon energies result in enhanced bulk sensitivity due to higher kinetic energy of electrons band mapping requires higher angular resolution due to large k-values of outgoing electrons N. Kamakura et al., Physical Review B 74, (2006)

38 Band structure using x-rays N. Kamakura et al., Physical Review B 74, (2006)

39 Band structure of copper after Ibach-Lüth

40 Band structure copper - high-resolution ARPES experiments single set of data for one point in k-space (k x ) Data set varying k x, gray shading according to photoemission intensity after E. Rotenberg, Advanced Light Source Berkeley

41 Fermi surface mapping after E. Rotenberg, Advanced Light Source Berkeley

42 Cu-Fermi surface see F. Reinert und S. Hüfner, New Journal of Physics 7, 97 (2005)

43 Band structure of a semiconductor: GaAs sp-derived valence bands spin-orbit splitting (heavy holes - light holes) T.C. Chiang et al., Physical Review B 21, 3513 (1980)

44 Band gap of a conventional super conductor Photoemission data for V 3 Si (T c =17.1 K): Quasi particle-density of states: see F. Reinert und S. Hüfner, New Journal of Physics 7, 97 (2005)

45 Band gap of a high-t c -super conductor Photoemission data of Bi 2 Sr 2 CaCu 2 O 8+δ (T c =87 K): M. Randeria et al., Physical Review Letters 74, 4951 (1995)

lattice - CuO-Ebenen anisotropy of band gap

46 band gap as function of T and k LEM=leading-edge midpoint, SCP=superconducting peak k B T crystal structure and reciprocal lattice - CuO-Ebenen anisotropy of band gap d-wave super conductor after A. Damascelli et al., Review of Modern Physics 75, 473 (2003)

47 2D-electron systems The simplest 2D-electron systems are surface states after E. Rotenberg, Advanced Light Source Berkeley electrons are fully localized at the surface fully delocalized in k-space in the direction of k z no dispersion in k z

48 surface states true surface states exist in band gaps of the solid, exponential decrease of the wave function inside and outside the solid Tamm-states electronic states which are localized on the surface atoms, e.g. p-orbitals of semiconductors (dangling bonds), d-orbitals of transition state metals and f-orbitals of lanthanides Shockley-states parallel to the surface delocalized, free electron-like states, e. g. s-p-orbitals of metals, d-orbitals of lanthanides

Fermi-edge 3d-derived Tamm-state above the d-bands

49 Cu-surface states Fermi-surface of Shockley state - circle 4sp-derived Schockley-state close to the Fermi-edge 3d-derived Tamm-state above the d-bands from F. Reinert und S. Hüfner, New Journal of Physics 7, 97 (2005)

50 Quantum well states in metallic thin films Ag on Au walls of the quantum well are on the one side vacuum, on the other side a band gap of the substrate T.-C. Chiang / Surface Science Reports 39 (2000) 181

51 Spin-polarized quantum well states Quantum well states exist only for one spin orientation because of the exchange-split bands of the substrate C. Carbone et al., Comm. 100, 749 (1996)

, Physical Review Letters 95, 205504 (2005)

52 1D-states Nano wires on surfaces: Gd on Si: STM-pictures H. W. Yeom et al., Physical Review Letters 95, (2005) Electrons in 2-dimensions localized Dispersion only in the direction parallel to the wires

53 Graphene Graphene corresponds to one layer of graphite start material for carbon nano tubes or fullerenes

54 Graphene-band structure Linear dispersion of bands constant velocity 1 de dk c 300 mass less Dirac-Fermions A. Bostwick et al., Nature Physics 3, 36 (2007)

55 spectral function 1 Transition probability in photoemission in sudden approximation, i.e. under the assumption that the outgoing photoelectron is so fast that its interaction with the (N-1)-electron system can be neglected: final state: Ψ f (N) = Φ f,ekin ( k f ) s Ψ f,s(n 1) the Ψ f,s (N 1) are all possible states of the (N-1)-electron final state initial state: Ψ i (N) = Φ i ( k i )Ψ i (N 1) photo current: I s Φ f,e kin ( k f ) r Φ i ( k i ) 2 Ψ f,s (N 1) Ψ i (N 1) 2 with the spectral function: A( k, E) = s Ψ f,s(n 1) Ψ i (N 1) 2 in second quantization: A( k, E) = s N 1, s c k N 2 c k, c + k annihilation and creation operators

56 spectral function 2 single-particle-greens-function: G( k 1, k 2, E) = Ψ i (N) c k2 1 E + H iδ c+ k1 Ψ i (N) G is the probability that a particle in a state k 1 is found in a state k 2 after a scattering process with energy transfer E under the influence of the interaction operator H(δ is a infinitesimal small positive number) For photoemission we have k 2 = k 1 = k one can show that the spectral function follows: A( k, E) = 1 π ImG( k, E) (e.g. R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, Dover (1992))

57 spectral function 3 non-interacting fermi gas: G 0 ( k, E) = 1 E E 0 ( k) iδ = 1 E 2 k 2 iδ 2m A 0 ( k, E) = 1 π δ(e E 0 ( k)) after A. Damascelli et al., Review of Modern Physics 75, 473 (2003)

58 spectral function 4 Fermi gas with interaction: perturbation theory G( 1 k, E) = E E 0 ( k) Σ( k, E) with Σ( k, E) = ReΣ + iimσ the so-called self energy A( k, E) = 1 ImΣ π (E E 0 ( k)) ReΣ) 2 + (ImΣ) 2 ReΣ describes a renormalization of the energy of the quasiparticles due to their interaction ImΣ a finite lifetime of the final state

59 spectral function 5 Fermi gas with interaction: Fermi-liquid theory Σ( k, E) = a E + ib E 2 = Σ + iσ after A. Damascelli et al., Review of Modern Physics 75, 473 (2003) The spectral function has in general a coherent (quasiparticle) and an incoherent (discrete excitations satellites ) part

60 Lifetime 1 Σ( k, E) = i 2 A( k, E) = 1 /2 π (E E 0 ) /4 Lorentz-curve with FWHM ( full width half maximum ) equivalent to a wave function with an imaginary part of the energy mit Φ e i( E +i 2 ) t n(t) = Φ Φ e t τ τ =

61 Lifetime 2 electron-electron-scattering leads to electron-hole pair excitations quadratic increase of the scattering rate with distance to the Fermi-edge - due to increasing phase space for e-h-pair excitations electron-phonon scattering for an Einstein-mode threshold behaviour for excitations with discrete energies

62 Exp. of line width after P.M. Echenique et al., Surface Science Reports 52 (2004) τ = evs

63 Electron-electron vs. electron-phonon scattering Line width of surface states as function of temperature- comparison with a Debye-Model after P.M. Echenique et al., Surface Science Reports 52 (2004)

64 Kinks in the dispersion High-T c-super conductor-bi2212 Kinks in the dispersion (real part of the self energy) as indication for coupilg to a boson (A. A. Kordyuk, S. V. Borisenko, Low Temp. Phys. 32, 298 (2006), Phys. Rev. Lett. 97, (2006)) - spin-fluctuations?

65 Kondo-effect experimentally for some metals an increase of the resistivity at low temperatures is observed spin-dependent scattering of free electrons at localized (impurity-) spins many body effect preferred anti-parallel orientation of spins - bound singlet-state localized state right above the Fermi-edge cerium: prototype system for Kondo-effect configuration 4f 1 (6s5d6p) 3 localized 4f-spin

66 Kondo-effect spectral function without f-valence interaction with f-valence interaction photoemission: F. Reinert und S. Hüfner, New Journal of Physics 7, 97 (2005)

67 Spinons and holons separation of spin- and charge excitations in one-dimensional, anti-ferromagnetic chains - SrCuO 2 after B.J. Kim et al, Nature Physics 2, 397 (2006)

Angle-resolved photoemission spectroscopy (ARPES) Overview-Physics 250, UC Davis Inna Vishik

Angle-resolved photoemission spectroscopy (ARPES) Overview-Physics 250, UC Davis Inna Vishik Outline Review: momentum space and why we want to go there Looking at data: simple metal Formalism: 3 step model