Syro Université Paris-Sud and de Physique et Chimie Industrielles - Paris

Transcription

1 Introductory lectures on Angle-resolved photoemission spectroscopy (ARPES) and its application to the experimental study of the electronic structure of solids Andrés s Felipe Santander-Syro Syro Université Paris-Sud and Ecole Supérieure de Physique et Chimie Industrielles - Paris

2 Resources BOOKS S. Hüfner. Photoelectron Spectroscopy Principles and Applications, third edition, Springer (Berlin), REVIEW ARTICLES F. Reinert and S. Hüfner, New Journal of Physics 7, 97 (2005). A. Damascelli, Z.-X. Shen, S. Hussain, Rev. Mod. Phys. 75, 473 (2003). J. C. Campuzano, M. R. Norman, M. Randeria, cond-mat/ J. Braun. The theory of angle-resolved ultraviolet photoemission and its application to ordered materials. Rep. Prog. Phys. 59, (1996). INTERNET www-bl7.lbl.gov/bl7/who/eli/srschooler.pdf (by Eli Rotenberg, Advanced Light Source, Berkeley) Lectures/Exciting2003.pdf (by Andrea Damascelli, UBC) ftp://ftp.espci.fr/shadow/bontemps/cargese2005.zip (ARPES lectures by Ralph Claessen, Augsburg)

3 Condensed (crystalline) matter in a nutshell Allowed electronic states Repeated-zone scheme -k F k F Many properties of solids are determined by electrons within a narrow energy slice (~k B T) around E F (dc conductivity, magnetism, superconductivity ) Fermi Surface 3D 2D 1D Adapted from A. Damascelli s Exciting-2003 lecture and E. Rotenberg s lecture (at room temperature, k B T= 25meV)

4 ARPES: energy and momentum conservation hν, A v z Kinetic energy analyzer θ e - Detector hν, A v E kin, W, θ, φ Fixed during experiment Measured x Sample ϕ In the solid E B k y E hk Conservation laws kin E B vacuum = hν W vacuum k = k = solid 2mEkin sinθ

5 To make it work W E E ~ 2 5 ev ~ 0 1eV (Valence band) B B Pull the electron out of its bound state 1500 ev (interesting core levels) 0 < E kin = hν W E hν ~ ev B Fixed The electron has to make its way up to the sample s surface λ esc = electron escape depth (E kin ~ ev) Experimentally: λ[ ev] Å PES is a surface technique: one needs clean surfaces and work under ultrahigh vacuum ARPES needs, furthermore, atomically-flat surfaces (for ideal conservation of surface-parallel momentum): prepare surfaces in-situ, cleave,

6 Furthermore, one has to make sure that the photoemission process itself does not modify the electronic structure of the material SUDDEN APPROXIMATION The ejected electron should be fast enough to neglect its interaction with the hole left behind e - -e - interaction e - time of escape τ << τ ee E kin >> τ ee τ λ 1 2 2π ω esc P / λescωp m 2π 2 m E kin 2 At hν 25 ev E kin λ esc ~ 20 ev ~ 10 Å For cuprates hωp ~ 1eV τ / τ 0.1 ee

7 When all of this works ARPES gives direct access to the single-particle electronic structure of a crystal: Band structure Spectral line-shapes and widths: electron scattering rate Interactions E B [ev] a = CuO = 3.82 Å Γ Y M X 0.63 π/a k x ky hν = 22 ev Bi 2 Sr 2 CaCu 2 O 8+δ

8 Instrumentation and implementation Kinetic energy analyzer Light source hν, A v z θ e - Detector x Sample ϕ y Sample s surface preparation Sample moving, rotating, cooling

9 Radiation sources Laboratory sources Gas discharge lamps (hν ~ ev) X-ray tubes (hν ~ 1500 ev) Synchrotron radiation Tunable (hν ~ 10 ev 10 kev) Brilliant Polarized (linear and circular) Temporal structure (time-resolved experiments) Laser IR laser + 2*(frequency doubling): hν ~ 6 7 ev (E kin ~ 1-2 ev) λ esc ~ Å Sudden approximation?! Probes reciprocal space only near Γ-point Under development: UV and soft X-ray laser (IR laser + highharmonic generation inside rare-earth gas)

10 Adapted from A. Damascelli s Exciting-2003 lecture Electron energy analyzer

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12 Interaction effects on ARPES spectra ( k, ω ) 0( k, ν, A) f ( ω) A( k, ω) I = I Σ ( k, ω) [ ω ε Σ ( k, ω) ] 2 + [ Σ ( k, ω) ] 2 A(k,ω) = Probability of adding or removing one electron at (k,ω) 1 A( k, ω) = E hω, h = 1 π B k Σ Energy renormalization Σ Lifetime of dressed e - Many-body physics ε k ε k (ω) I(k,ω) E F Σ (k,ω) (renormalization) Σ (k,ω) (life-time) k Binding energy E B E F

13 Lifetime of the photo-electron and measured line-widths Assuming Lorentzian line-shapes, the total (measured) width is given by: Γ tot Γ tot Γ h + v v h e Γ e ~ mev ~ ev Spectrum dominated by finalstate (photo-electron) line-widths, unless v h << v e Adapted from: T.-C. Chiang, Chemical Physics 251, (2000) 2D and 1D systems! Adapted from R. Claessen s Cargese-2005 lectures

14 Spectra analysis: EDCs Line-shapes and widths many-body physics A ( k, ω) 1 = π Σ ( k, ω) [ ω ε Σ ( k, ω) ] 2 + [ ( k, ω) ] 2 k Σ EDC: Lorentzian if and only if Σ and Σ independent of ω

15 Spectra analysis: MDCs Line-shapes and widths many-body physics A ( k, ω) 1 = π Σ ( k, ω) [ ω ε Σ ( k, ω) ] 2 + [ ( k, ω) ] 2 k Σ MDC: Lorentzian if and only if k c k = k F HWHM + = [ ω Σ ( ω) ] 0 Σ ( ω) / v F / v 0 F

16 Many-body physics Effects of the interactions on the band structure: Example of surface states of Mo(110) Γ Mo(110) band along ΓN ee ~ ω 2 T = 70 K Γ e-ph Eliashberg Γ e-imp = const T. Valla et al., PRL 83, 2085 (1999)

17 Strongly-correlated electron systems (brief recall)

18 Transition-metal oxides: solid-state SCES Cu O Cu MH ~ 1-2 ev

19 Transition-metal oxides Strongly-correlated electrons systems New physics displaying exotic electronic states in a solid sample

20 Cuprates: antiferromagnetic insulators that become high-t c superconductors upon doping! Crystal unit-cell Antiferromagnetic unit cell

21 Cuprates: (rough) phase diagram Coexistence?

22 Electron-doped cuprates: Generalities

23 R 2-x Ce x CuO 4 : crystal structure CuO 2 plane R/Ce (R,Ce) 2 O 2 block CuO 2 plane (R,Ce) 2 O 2 block CuO 2 plane

24 Electron-doped cuprates: effects of Ce-doping and annealing H. J. Kang et al., Nature Materials 6, 224 (Feb. 2007). L. Shan et al., cond-mat/ (March 2007).

25 Fermi surfaces, Brillouin zones and AF-zones: hole-doped vs electron-doped cuprates Nd 2-x Ce x CuO 4 (x = 0.15) Tl 2 Ba 2 CuO 6+d. P. Armitage et al., PRL 88, (2002) M. Platé et al., PRL 95, (2005)

26 Antiferromagnetic-induced band-folding in underdoped Sm 2-x Ce x CuO 4 (x = 0.14) S. R. Park et al., cond-mat/ (Dec. 2006)

27 Effects of annealing on band structure at optimal doping: the case of Pr 1.85 Ce 0.15 CuO 4 Annealed As-grown P. Richard et al., cond-mat/ (Apr. 2007)

28 Electronic structure and signatures of interactions in Sm 1.84 Ce 0.16 CuO 4 Coworkers - collaborators Takeshi Kondo, Adam Kaminski (Ames Lab - Iowa) Stéphane Pailhès (PSI and LLB) Johan Chang, Ming Shi, Luc Patthey (PSI) Alexandre Zimmers (CSR Maryland and INP P6) Bing Liang, Pengcheng Li, Rick Greene (CSR - Maryland)

29 Sm 2-x Ce x CuO 4 : Fermi surface and doping 0.0 Γ (π,π) (π,0) Binding energy [ev] Max Min Γ (π,π) (π,0) Tight-binding fit Momentum along AFZB [2 -½ π/a] Doping from FS volume: x = 0.16 ± 0.01 Single-band FS (no band-folding) Suppressed spectral weight at hot-spots

30 Sm 2-x Ce x CuO 4 (x = 0.16) : Nodal vs anti-nodal ARPES spectra N 0.0 AN AN 0.0 N Binding energy [ev] k AFZB Max 0.1 Å -1 Min Binding energy [ev] Å -1 k AFZB Max Min

31 Sm 2-x Ce x CuO 4 (x = 0.16) : Nodal and anti-nodal MDCs N AN AN ω = -200 mev N ω = -200 mev ω = -100 mev ω = -100 mev 0.1 Å -1 ω = 0 mev 0.1 Å -1 ω = 0 mev Relative momentum Relative momentum Anti-nodal and nodal MDCs are Lorentzians MDC peak maximum gives the quasi-particle dispersion MDC width is proportional to quasi-particle scattering rate (self-energy)

32 Sm 2-x Ce x CuO 4 (x = 0.16) : Nodal vs anti-nodal EDCs AN Hump N AN AN N AN Binding energy [ev] N Binding energy [ev] Anti-node Peak-hump structure close to k F Lorentzian EDCs at energies > 300 mev Node No clear peak-hump structure EDCs are not Lorentzian

33 Sm 2-x Ce x CuO 4 (x = 0.16) : Nodal vs anti-nodal dispersions Binding energy [ev] AN N 0.1 Å -1 N AN -0.6 Relative momentum

34 Sm 2-x Ce x CuO 4 (x = 0.16) : Nodal vs anti-nodal line-widths ( ) = v band khwhm Σ ω Im(Σ) [ev] AN From EDC-width N Binding energy [ev]

35 Conclusions When it works ARPES is a powerful technique for the study of the electronic structure of complex systems Detailed band structures and Fermi surfaces k-dependent Fermi velocity and effective mass Gaps Many-body effects in the QP dispersion Kinks Fermi-surface nesting Outlook (and dreams) Spin-resolved ARPES Time-resolved ARPES Micro-ARPES