Introduction to Photoemission Spectroscopy

Transcription

1 Introduction to Photoemission Spectroscopy Michael Sing Physikalisches Institut and Röntgen Center for Complex Material Systems (RCCM) Universität Würzburg, Germany Outline: Basics PES theory I: (mainly) independent electrons PES theory II: many body picture Case studies towards higher photon energies

2 Photoemission basics

3 The photoelectric effect quantitative experiment: (H. Hertz 1886, W. Hallwachs 1888, P. Lenard 1902) measure kinetic energy of photoelectrons in retarding field light of frequency experimental observations: light intensity increases, but not (contrary to classical expectation) instead: depends on light frequency

4 The photoelectric effect Theoretical explanation by A. Einstein (1905): QUANTIZATION OF LIGHT Ann. d. Phys. 17, 132 (1905): A. Einstein Nobel prize 1921 max E kin h h Ekin Planck s constant photocathode workfunction

5 Photoelectron spectroscopy energy conservation: E kin h E B K.M. Siegbahn Nobel prize 1981 electron analyzer h Ekin

6 Photoelectron spectroscopy: ESCA energy conservation: E kin h E B Electron Spectroscopy for Chemical Analysis K.M. Siegbahn Nobel prize 1981 Fe 3 O 4 (magnetite) chemical composition

7 Photoelectron spectroscopy: ESCA energy conservation: E kin h E B Electron Spectroscopy for Chemical Analysis K.M. Siegbahn Nobel prize 1981 chemical shifts in C 1s spectrum of ethylfluoroacetate

8 Photoelectron spectroscopy: valence bands TiOCl O 2p / Cl 3p Ti 3d electron analyzer h E E kin kin, k single crystal

9 Angle resolved photoemission (ARPES) measure energy and momentum of the photoelectrons:,, with 1 2 vacuum conservation laws solid

10 Angle resolved photoemission (ARPES) measure energy and momentum of the photoelectrons: energy example: Au/Ge(111) k space band structure mapping: band dispersions, Fermi surface,

11 Angle resolved photoemission (ARPES) Example: Cuprate-High Tc superconductor - 2D Pb-BSCCO (Bi 2 Sr 2 CaCu 2 O 8+ ) momentum distr. curves energy distribution curves Angle (momentum) Binding energy Borisenko et al., Phys. Rev B 64, (2001)

12 PES theory I: (mainly) independent electrons

13 Fermi's Golden Rule starting point for theoretical description of PE process: effect of photon field is weak perturbation Hamiltonian: unperturbed system operator of time dependent perturbation (radiation field) unperturbed system (electrons in atom, solid): with known eigenstates and eigenenergies perturbation (photon field):

14 Fermi's Golden Rule time dependent perturbation theory transition rate from initial state to final state of the unperturbed system due to perturbation is: 2 transition matrix element: k conservation symmetry energy conservation!

15 Fermi's Golden Rule Perturbing radiation field: What is? describe by vector potential of a classical* electromagnetic plane wave:, electric field:,, magnetic field:,, N.B.:, div, 0, if photon wavevector true in vacuum and deep in the solid (transverse wave), but not necessarily at the surface due to discontinuity in dielectric constant surface photoemission see, e.g., Miller et al., PRL 77, 1167 (1996) *classical description ignores quantum nature of photon, justified for sufficiently low photon intensities ( VUV laser, FEL?)

16 Fermi's Golden Rule Perturbed electronic system (consider only single electron: independent particle picture!): canonical replacement in unperturbed Hamiltonian:, describes two photon processes, can be neglected for weak radiation fields 2 = 0, except possibly at surface! of the form to be used in Fermi's Golden Rule!

17 Fermi's Golden Rule back to Fermi s Golden Rule 2 for the transition matrix element we now obtain:, or expressed in "real" wave functions:

18 Matrix element and dipole approximation length scales: the matrix element can be viewed as spatial Fourier transform ( ) the wavefunctions (atomic orbitals or Bloch waves) oscillate rapidly on atomic dimensions (Å) the photon wave probes length scales of order 2 which for VUV radiation is large compared to atomic dimensions, e.g.: = 21.2 ev = 584 Å (VUV) kev = 8.3 Å (XPS) 6 kev = 2.0 Å (HAXPES) expansion of the plane wave (generates el./magn. multipole moments): 1 1, with ~ 2 1 for VUV radiation

19 Matrix element and dipole approximation simplified matrix element: Using the quantum mechanical identity the matrix element can be further transformed into: electrical dipole operator selection rules, polarization dependence dipole approximation valid only up to VUV energies at higher photon energies (XPS, HAXPES): el. quadrupole/magn. dipole contributions increasingly important!

20 Fermi s Golden rule and the one step model photoemission intensity determined by transition rate: 2 What are the initial and final states? One step model: final states: "time inverted LEED state" in vacuum: free electron wave in the solid: matched to high lying Bloch waves, damped by e e scattering energy and wavevector initial states in the solid: bulk Bloch waves energy and wavevector

21 Fermi s Golden rule and the one step model photoemission intensity determined by transition rate: 2 What are the initial and final states? One step model: more on final one step states: theory "time inverted of PES LEED state" in vacuum: free electron wave lecture in the solid: matched to high lying Bloch waves, by Jan Minar damped by e e scattering energy and wavevector initial states in the solid: bulk Bloch waves energy and wavevector

22 One step model vs. three step model courtesy of A. Damascelli

23 Step 1: Excitation in the solid 2 momentum conservation: k f k i G k photon only "vertical" transitions for VUV excitation

24 Step 2: Transport to the surface inelastic scattering of the photoelectron with other electrons (excitation of e h pairs, plasmons) phonons generation of secondary electrons "inelastic background" intensity intrinsic spectrum w background E kin

25 Inelastic background Shirley background background at energy proportional to intrinsic spectrum integrated over all energies : can be viewed as convolution with step like loss function : 0 Tougaard background loss function will generally have structure due to interband transitions, plasmons, etc. use phenomenological model or determine loss function experimentally (EELS) 0 1

26 Step 2: Transport to the surface inelastic scattering of the photoelectron with other electrons (excitation of e h pairs, plasmons) phonons generation of secondary electrons "inelastic background" intensity intrinsic spectrum w background E kin

27 Step 2: Transport to the surface inelastic scattering of the photoelectron with other electrons (excitation of e h pairs, plasmons) phonons generation of secondary electrons "inelastic background" loss of energy and momentum information in the photoelectron current: inelastic mean free path

28 Mean free path and PES probing depth "universal curve" h E kin typical PES energies ev (E kin ) = 2 20 Å PES probing depth: ~3 (95% of the signal) PES is surface sensitive on atomic length scales!

29 Step 3: transition to vacuum conservation of wavevector component parallel to surface, But: change of changes due to electron diffraction at surface barrier source: E. Rotenberg

30 The k problem electron wave matching at the surface E solid E kin vacuum surface potential step final state dispersion generally not known E ~ K 2 E vac E F h K (fixed K ll ) k (fixed k ll = K ll )

31 The k problem pragmatic solution: free electron final state model surface potential step E ~ K 2 V 0 K (fixed K ll ) "inner potential" k (fixed k ll = K ll )

32 The k problem pragmatic solution: free electron final state model E kin kinematic relations Kout k in V0 2m 2m k k in, ll in, K out, ll 2m ( E 2 kin 2m E 2 cos 2 kin sin out V out 0 ) uniquely determined from measured data:,, but need to know inner potential (from band theory, k periodicity)

33 Summary: ARPES in the indep. electron approx. measure energy and escape angle of the photoelectrons: get bandstructure (dispersions, Fermi surface, ) from conservation laws: energy: momentum: 2 sin not so straightforward

34 PES theory II: many body picture

35 Photoemission: many body effects Damascelli et al., Rev. Mod. Phys. 75, 473 (2003) non interacting electrons interacting electrons ARPES band structure ARPES spectral function,,

36 Photoemission: many body effects interacting electrons (Coulomb repulsion) h E kin photoemission process: sudden removal of an electron from particle system

37 Photoemission: many body effects interacting electrons (Coulomb repulsion) E kin photoemission process: sudden removal of an electron from particle system "loss" of kinetic energy due to interaction related excitation energy stored in the remaining 1 electron system!

38 Photoemission: many body effects electron phonon coupling h E kin photoemission process: sudden removal of an electron from particle system "loss" of kinetic energy due to interaction related excitation energy stored in the remaining 1 electron system!

39 Photoemission: many body effects electron phonon coupling E kin photoemission process: sudden removal of an electron from particle system "loss" of kinetic energy due to interaction related excitation energy stored in the remaining 1 electron system!

40 Fermi's Golden Rule for an electron system 2 initial state, 0 electron ground state with energy, (=0) final states 1,; electron excited state of quantum number and energy,, consisting of 1electrons in the solid and one free photoelectron with wavevector and energy transition operator in second quantization one electron matrix element, conserves wavevector:

41 Fermi's Golden Rule for an electron system, 1,;, 0,, initial state, 0 electron ground state with energy, (=0) final states 1,; electron excited state of quantum number and energy,, consisting of 1electrons in the solid and one free photoelectron with wavevector and energy transition operator in second quantization one electron matrix element, conserves wavevector:

42 Fermi's Golden Rule for an electron system, 1,;, 0,, Sudden Approximation: 1,;

43 Fermi's Golden Rule for an electron system, 1,;, 0,, Sudden Approximation: 1,; 1, factorization! photoelectron th eigenstate of remaining 1electron system physical meaning: photoelectron decouples from remaining system immediately after photoexcitation, before relaxation sets in

44 Fermi's Golden Rule for an electron system, 1,, 0,, Sudden Approximation: 1,; 1, factorization! photoelectron th eigenstate of remaining 1electron system physical meaning: photoelectron decouples from remaining system immediately after photoexcitation, before relaxation sets in

45 Fermi's Golden Rule for an electron system, 1,, 0,, Sudden Approximation: 1,; 1, factorization! photoelectron th eigenstate of remaining 1electron system physical meaning: photoelectron decouples from remaining system immediately after photoexcitation, before relaxation sets in

46 Sudden Approximation after some algebra (using the momentum conservation in and assuming that ~ in the energy and k range of interest) one obtains:, 1,,0,,

47 Sudden Approximation after some algebra (using the momentum conservation in and assuming that ~ in the energy and k range of interest) one obtains:, 1,,0,,, ) spectral function The ARPES signal, directly proportional to the removal part of the spectral function,, probability of removing (or adding) an electron at energy and momentum from (to) the system single-particle Green s function

48 Sudden Approximation after some algebra (using the momentum conservation in and assuming that ~ in the energy and k range of interest) one obtains:, 1,,0,,, ) spectral function The ARPES signal, directly proportional to the removal part of the spectral function,,

49 Spectral function,,,,,, Theoretical energy distribution curves (EDC)

50 Many body effects in photoemission example: photoemission of the H 2 molecule E E kin electrons couple to proton dynamics! H 2 L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970)

51 Many body effects in photoemission example: photoemission of the H 2 molecule E E kin electrons couple to proton dynamics! photoemission intensity: H 2 I( ) H2, s cˆ H2,0 ( E H, s E, 0) 2 s 2 2 H electronic vibrational eigenstates of H 2+ : L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970) H 2, s 1, v 1, v 0 1 1, v 2

52 Many body effects in photoemission example: photoemission of the H 2 molecule E E kin Franck Condon principle H 2 L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970) ħ 0 v = 0 v = 2 v = 1 eneregy v' = 0 proton distance

53 The k problem again: photoelectron damping ARPES signal is actually a convolution of photohole and photoelectron spectral function I( kll, ) dk Ah ( kll, k, h ) Ae ( kll, k, ) v e tot total width assuming lifetimebroadened Lorentzian lineshapes Γ Γ Γ energy h e ~ mev ~ ev spectrum dominated by photo electron linewidth unless v h v 1 e h slope v h h k low dim systems (e.g., surfaces) k vector in high symmetry planes k

54 Summary: The view from many body physics Photoelectron spectroscopy is ideal tool for the study of many body effects in the electronic structure photohole probes interactions between electrons and with other dynamical degrees of freedom energy shifts shake up satellites line broadening line shape (generalized Franck Condon effect) phonons spin excitations other electrons? ARPES signal proportional to single particle spectrum, (if photohole is localized surface!) facilitates direct comparison to many body theoretical description of interacting system

55 Low energy photoemission: Doping a one dimensional Mott insulator

56 TiOCl: a low dimensional Mott insulator t van der Waals-gap t c b a a b a configuration: Ti 3d 1 1e - /atom: Mott insulator local spin s=1/2? frustrated magnetism, resonating valence bond (RVB) physics?

57 Doping a Mott insulator Hubbard model description: non-interacting bandwidth W local Coulomb energy U band-filling n lower Hubbard band (LHB) n = 1 U > = W0 upper Hubbard band (UHB) electron removal (PES) electron addition (IPES)

58 Doping a Mott insulator Hubbard model description: non-interacting bandwidth W local Coulomb energy U band-filling n lower Hubbard band (LHB) n = 1 U > W upper Hubbard band (UHB) electron removal (PES) electron addition (IPES) bandfilling-controlled Mott transition? here: n-doping n = 1+x quasiparticle

59 Doping a Mott insulator Hubbard model description: non-interacting bandwidth W local Coulomb energy U band-filling n lower Hubbard band (LHB) n = 1 U > W upper Hubbard band (UHB) electron removal (PES) electron addition (IPES) bandfilling-controlled Mott transition? here: n-doping n = 1+x quasiparticle

60 TiOCl: doping by intercalation donors acceptors n: alkali metals p: organic acceptors (e.g., TCNQ) Li x (THF) y HfNCl Nature 392, 580 (1998)

61 XPS: in situ Na intercalation and n doping Ti2p 3/2 (+ Na) Na1s Ti 3+ Ti 2+ Ti 2+ intensity doping (arb. x (%) units) x = 37% 32% 23% 15% 10% 4% binding energy (ev) binding energy (ev) PRL 106, (2011) electron transfer Na Ti doping x from relative Ti 2+ and Ti 3+ weight

62 UPS: doping effects in valence band Ti 3d O 2p Cl 3p rigid band shift new spectral weight in gap PRL 106, (2011)

63 Doping a Mott insulator QP LHB UHB LHB UHB µ 0.6eV µ LHB 2eV UHB µ new peak in the Mott gap: UHB? absence of metallic quasiparticle (QP)?

64 Spectral weight transfer from LHB to UHB undoped: LHB UHB UHB remove LHB w = 1 doped: remove +U UHB w = 2x remove LHB w = 1-x new peak behaves as upper Hubbard band! PRL 106, (2011)

65 Spectral weight transfer from LHB to UHB undoped: LHB UHB UHB remove LHB w = 1 doped: remove +U UHB w = 2x remove LHB w = 1-x new peak behaves as upper Hubbard band! PRL 106, (2011)

66 Absence of metallic QP Molecular dynamics: Y. Z. Zhang, Phys. Rev. Lett. 104, (2010) GGA+U - DOS: Ti O Cl Na Na ions occupy specific sites close to one Ti-O layer in-gap states due to Ti sites closest to Na ions local doping into alloy band (AB) transfer of spectral weight LHB AB AB: all sites always doubly occupied fundamental gap between AB and UHB insulating for all doping levels

67 Hard x ray photoemission: Profiling the buried two dimensional electron system in an oxide heterostructure

68 Oxide heterostructures in a nutshell General idea: combine interface functionalities with intrinsic functionalities of oxides novel phases, tunability of interactions Paradigm material: LAO/STO both oxides: wide gap band insulators LAO thickness 4 unit cells (uc): formation of a high mobility interface 2D electron system (2DES) conductivity LaAlO 3 sheet carrier density (Hall) 2DES SrTiO 3 A. Ohtomo et al., Nature 427, 423 (2004) S. Thiel et al., Science 313, 1942 (2006)

69 Oxide heterostructures in a nutshell 2DES properties: conductivity tunable conductivity by electric gate field superconducting below 200 mk magnetoresistance sheet carrier density (Hall) coexistence of superconductivity and ferromagnetism Origin of 2DES (and its critical behavior)? O interface cation intermixing (La x Sr 1 x TiO 3 ) electronic reconstruction see also: D.G. Schlom and J. Mannhart, Nature Materials 10, 168 (2011)

70 Polar catastrophe and how to avoid it charge: AlO 2 LaO AlO 2 LaO AlO 2 LaO TiO 2 SrO TiO 2 SrO electrostatic energy increases with thickness of polar film polar catastrophe 1/ q = 1/ / AlO 2 LaO AlO 2 LaO AlO 2 LaO TiO 2 SrO TiO 2 SrO electronic reconstruction 0.5e per layer unit cell n 2D = cm 2 partial Ti 3d occupation Ti 3.5 (d 0.5 ) = Ti 3+ /Ti 4+ Nakagawa et al., Nature Mat. 5, 204 (2006)

71 El. reconstruction: The view from band theory 1/2 +1 AlO 2 LaO surface e surface LAO VBM q = 1/2e /2 0 AlO 2 LaO AlO 2 LaO TiO 2 SrO interface STO CBM 0 TiO 2 0 SrO Yun Li et al., PRB 84, (2011) Pentcheva and Pickett, PRL 102, (2009)

72 LAO/STO: Electronic reconstruction surface q = 1/2e 1/ / AlO 2 LaO AlO 2 LaO AlO 2 LaO TiO 2 SrO TiO 2 SrO ideal el. reconstruction scenario

73 Spectroscocpy of buried interfaces LaAlO 3 2DES SrTiO 3 Challenges and requirements: suitable probing depth: photons (10 nm microns) electrons ( nm) interface signal vs. background intensity from bulk spectroscopic contrast: symmetry element specificity chemical shift electronic configuration sufficiently high count rates Methods presented here: hard x ray photoelectron spectroscopy (HAXPES) resonant soft x ray angle resolved PES (SX ARPES)

74 HAXPES of LAO/STO: core levels >15 Å LaAlO 3 2DES SrTiO 3 Ti 3+ weight evidence for 2DES Ti 3+ /Ti 4+ ratio sheet carrier density angle dependence thickness PRL 102, (2009)

75 Model and quantitative analysis Angle dependence Exponential damping: Intensity ratio: Accessible parameters: PRL 102, (2009)

76 Quantitative analysis PRL 102, (2009)

77 Quantitative analysis: 2DEG thickness Sample 2 uc 4 uc 5 uc 6 uc e - θ e - d (uc*) 3 ±1 1 ±0.5 6 ±2 8 ±2 *lattice constant of STO unit cell (uc) = 3.8 Å d interface thickness < 3 nm consistent with CT AFM Basletic et al. (2008) TEM EELS Nakagawa et al. (2006) * density functional theory Pentcheva et al. (2009) 2D superconductivity Reyren et al. (2007) ellipsometry Dubroka et al. (2010) * HAXPES data taken at 300K! PRL 102, (2009)

78 Quantitative analysis: sheet carrier density Sample 2 uc 4 uc 5 uc 6 uc p n 2D (10 13 cm 2 ) el. reconstr corr. for photocarriers n 2D much smaller than for purely electronic reconstruction n 2D higher than Hall effect data photogeneration of extra Ti 3d electrons remaining excess due to additional localized Ti 3d electrons? (cf. Li et al. and Bert et al., Nature Phys. (2011): coexistence of superconductivity (free carriers) and magnetism (local moments))

79 Resonant angle resolved soft x ray photoemission: Direct k space mapping of the electronic structure in an oxide oxide interface

80 Soft x ray ResPES of the valence band HAXPES (h = 3.5 kev) SX ResPES (h460 ev) O2p 3.09eV? PRB 88, (2013) resonance enhancement at Ti L edge

81 Soft x ray ResPES of the valence band ResPES process SX ResPES (h460 ev) cf.: Drera et al., APL 98, (2011) Koitzsch et al., PRB 84, (2011) resonance enhancement at Ti L edge

82 LAO/STO: resonant photoemission at Ti L edge ResPES XAS two Ti 3d resonance features below (A) and at E F (B)

83 LAO/STO: resonant photoemission at Ti L edge ResPES XAS two Ti 3d resonance features below (A) and at E F (B) feature A: max enhancement at Ti 3+ e g resonance (cf. LaTiO 3 )

84 LAO/STO: resonant photoemission at Ti L edge ResPES XAS two Ti 3d resonance features below (A) and at E F (B) feature A: max enhancement at Ti 3+ e g resonance (cf. LaTiO 3 ) feature B: max enhancement delayed characteristic for localized (A) and delocalized (B) resonating states

85 LAO/STO: resonant photoemission at Ti L edge ResPES XAS two Ti 3d resonance features below (A) and at E F (B) feature A: max enhancement at Ti 3+ e g resonance (cf. LaTiO 3 ) feature B: max enhancement delayed characteristic for localized (A) and delocalized (B) resonating states features A and B also seen in O deficient STO (e.g., Aiura et al., Surf. Sci. 515, 61 (2002))

86 LAO/STO: resonant photoemission at Ti L edge ResPES XAS two Ti 3d resonance features below (A) and at E F (B) feature A: max enhancement at Ti 3+ e g resonance (cf. LaTiO 3 ) feature B: max enhancement delayed characteristic for localized (A) and delocalized (B) resonating states features A and B also seen in O deficient STO (e.g., Aiura et al., Surf. Sci. 515, 61 (2002)) A: charge carriers trapped in d orbitals of Ti ions surrounding oxygen vacancies B: mobile interface chargecarriers (2DES)

87 Photoemission of bare STO surface photoemission of fractured STO oxygen dosage "in gap" weight itinerant carriers O vacancies at surface cause n doping of Ti 3d states Aiura et al., Surf. Sci. 515, 61 (2002) itinerant carriers in conduction band in gap weight: localized electrons trapped next to oxygen vacancies

88 LAO/STO: k space mapping by SX ARPES dispersive Ti 3d interface band(s) at Γ schematic Ti 3d derived Fermi surface (Popovic et al., PRL 101, )

89 LAO/STO: k space mapping by SX ARPES good agreement with DFT band calculations individual quantum well states not resolved O 2p surface band not observed

90 LAO/STO: k space mapping by SX ARPES larger k F values along ΓX than ΓM also seen in experiment

91 LAO/STO: Fermi surface mapping h=460.35ev BL23SU, SPring 8 experiment does not show LAO surface hole pockets predicted by band theory! bare STO Phys. Rev. Lett. 110, (2013) cf. also: Cancellieri et al., arxiv:

92 LAO/STO: Fermi surface mapping h=460.35ev BL23SU, SPring 8 dynamical equilibrium? bare STO Phys. Rev. Lett. 110, (2013) cf. also: Cancellieri et al., arxiv: but: 2uc samples charge up no variation on changing photon flux band bending seen in other systems, e.g., LaCrO 3 /STO

93 Oxygen vacancies at LAO surface? LaO AlO 2 LaO AlO 2 LaO AlO 2 LaO TiO 2 SrO TiO 2 SrO TiO 2 SrO AlO 1.75 O vacancies induce localized hole states above E F Yun Li et al., PRB 84, (2011) cf. also: Zhong et al., PRB 82, (2010) Bristowe et al., PRB 83, (2011) Pavlenko et al., PRB 86, (2012) Yu and Zunger, arxiv: modified el. reconstruction scenario critical thickness, if for each O vac formation energy < discharge energy

94 Sources and further reading Most of the first part has been taken from the following sources: Internet wuerzburg.de/ep4/teaching/cargese2005/cargese.php and LesHouches2014 (coming soon) (by R. Claessen, U Würzburg) www bl7.lbl.gov/bl7/who/eli/srschooler.pdf (by E. Rotenberg, Advanced Light Source) (by A. Damascelli, U British Columbia) Books S. Hüfner, Photoelectron Spectroscopy Principles and Applications, 3rd ed. (Berlin, Springer, 2003) W. Schattke, M.A. van Hove (eds.), Solid State Photoemission and Related Methods Theory and Experiment (Weinheim, Wiley VCH, 2003) S. Suga, A. Sekiyama, Photoelectron Spectroscopy Bulk and Surface Electronic Structures (Berlin, Springer, 2014) Review articles F. Reinert and S. Hüfner, New Journal of Physics 7, 97 (2005) A. Damascelli, Physica Scripta T109, 61 (2004) S. Hüfner et al., J. Electron Spectrosc. Rel. Phen. 100, 191 (1999) For the second part look up references in the lecture notes DMFT at 25: Infinite Dimensions.