Introductory lectures on Introductory lectures on AngleAngle--resolved photoemission resolved photoemission
spectroscopy (ARPES)spectroscopy (ARPES)and its application to the experimental study and its application to the experimental study
of the of the electronic structure of solidselectronic structure of solids
AndrAndréés Felipe s Felipe SantanderSantander--SyroSyroUniversitUniversitéé ParisParis--SudSud
andandEcoleEcole SupSupéérieurerieure de Physique et de Physique et ChimieChimie IndustriellesIndustrielles -- ParisParis
Resources
BOOKS
• S. Hüfner. Photoelectron Spectroscopy – Principles and Applications, third edition, Springer (Berlin), 2003.
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/0209476.• J. Braun. The theory of angle-resolved ultraviolet photoemission and its application to ordered materials. Rep. Prog. Phys. 59, 1267-1338 (1996).
INTERNET
• www-bl7.lbl.gov/BL7/who/eli/SRSchoolER.pdf(by Eli Rotenberg, Advanced Light Source, Berkeley)
• www.physics.ubc.ca/~quantmat/ARPES/PRESENTATIONS/ Lectures/Exciting2003.pdf(by Andrea Damascelli, UBC)
• ftp://ftp.espci.fr/shadow/bontemps/cargese2005.zip(ARPES lectures by Ralph Claessen, Augsburg)
1D
2D
3D
Many properties of solids are determined by electrons within a narrow energy slice (~kBT) around EF (dc conductivity, magnetism, superconductivity…)
Fermi Surface
Condensed (crystalline) matter in a nutshell
(at room temperature, kBT = 25 meV)Adapted from A. Damascelli’s Exciting-2003 lecture and E. Rotenberg’s lecture
Allowed electronic statesRepeated-zone scheme
-kF kF
ARPES: energy and momentum conservation
x
y
z
e-θ
ϕ
Detector
Sample
Kinetic energy analyzer
, Av
νhEkin, W, θ, φ Measured
, Av
νh Fixed during experiment
θsin2 kinvacuum mE=||
khBE
In the solidIn the solid
||k
Bkin EWhE −−= νConservation laws
solidvacuum||||
kk =
To make it work…
levels) core ng(interesti eV 1500
band) (Valence eV 10~
eV 52~
→
−
−
B
B
E
E
W
λesc = electron escape depth(Ekin~ 10 – 2000 eV)
Pull the electron out of its bound state
The electron has to make its way up to the sample’s surface
eV 200010~
0
−⇒
−−=<
ν
ν
h
EWhE Bkin
Fixed
Experimentally:
λ[10 – 2000 eV] ∼ 10 – 50 Å
PES is a surface technique: one needs clean surfaces and work under ultra-high vacuum
ARPES needs, furthermore, atomically-flat surfaces (for ideal conservation of surface-parallel momentum): prepare surfaces in-situ, cleave,…
SUDDEN APPROXIMATIONThe ejected electron should be fast enough to neglect its
interaction with the hole left behind
Furthermore, one has to make sure that the photoemission process itself does not modify
the electronic structure of the material…
e- - e- interactionP
ee ωπτ 2
≈
e- time of escapemEkin
esc2/λτ ≈
2
221
>>⇒<<
πωλττ Pesc
kinee mE1.0/
eV 1~≈⇒ ee
P
ττωh
At hν≈ 25 eV
Å 10~eV 20~
esc
kinEλ
For cuprates
ARPES gives direct access to the single-particle electronic structure of a crystal:Band structure Spectral line-shapes and widths: electron scattering rate Interactions
EB [eV]
kx ky
ΓM
X
Y
0.63 π/a
Å 82.3CuO ==a
hν = 22 eV
When all of this works…
Bi2Sr2CaCu2O8+δ
Instrumentation and implementation
x
y
z
e-θ
ϕ
Detector
Sample
Kinetic energy analyzer
, Av
νhLight source
• Sample’s surface preparation• Sample moving, rotating, cooling
Radiation sourcesLaboratory sources
• Gas discharge lamps (hν ~ 20-50 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 (Ekin ~ 1-2 eV)
☺ λesc ~ 50 – 100 ÅSudden approximation ?!Probes reciprocal space only near Γ-point
• Under development: UV and soft X-ray laser (IR laser + high-harmonic generation inside rare-earth gas)
Electron energy analyzer
Adapted from A. Damascelli’s Exciting-2003 lecture
Interaction effects on ARPES spectra
( ) ( ) ( ) ( )ωωνω ,,,, 0 kAkk AfII = A(k,ω) = Probability of adding or removing one electron at (k,ω)
Binding energy EB
Σ’’(k,ω)(life-time)
Σ’(k,ω)(renormalization)
EF
εk(ω) I(k,ω)
EF
εk
k
( ) ( )( )[ ] ( )[ ]22 ,,
,1,ωωεω
ωπ
ωkk
kkk Σ′′+Σ′−−
Σ′′−=A 1 , =≡ hhωBE
Σ ′′Σ′ Energy renormalization
Lifetime of dressed e-Many-body physics
Adapted from:T.-C. Chiang, Chemical Physics 251, 133-140 (2000)
Assuming Lorentzian line-shapes, the total (measured) width is given by:
ee
hhtot v
vΓ+Γ≈Γ
⊥
⊥
~ meV ~ eV
Spectrum dominated by final-state (photo-electron) line-widths, unless
⊥⊥ << eh vv
2D and 1D systems !
Adapted from R. Claessen’s Cargese-2005 lectures
Lifetime of the photo-electron and measured line-widths
Γtot
Spectra analysis: EDCsLine-shapes and widths many-body physics
( ) ( )( )[ ] ( )[ ]22 ,,
,1,ωωεω
ωπ
ωkk
kkk Σ′′+Σ′−−
Σ′′−=A
EDC: Lorentzian if and only if
oft independen and ωΣ ′′Σ′
Spectra analysis: MDCs
( ) ( )( )[ ] ( )[ ]22 ,,
,1,ωωεω
ωπ
ωkk
kkk Σ′′+Σ′−−
Σ′′−=A
MDC: Lorentzian if and only if
( )[ ]( ) 0
HWHM
0
/
/
F
FFc
vk
vkk
ω
ωω
Σ ′′=∆
Σ′−+=
Line-shapes and widths many-body physics
Many-body physics – Effects of the interactions on the band structure:
Example of surface states of Mo(110)
T. Valla et al., PRL 83, 2085 (1999)
Mo(110) band along Mo(110) band along ΓΓNNT = 70 KT = 70 K
Γee ~ ω2
Γe-ph Eliashberg
Γe-imp = const
Strongly-correlated electron systems
(brief recall)
Transition-metal oxides: solid-state
SCES
∆MH ~ 1-2 eVCu O Cu
Transition-metal oxidesStrongly-correlated electrons systems
New physics displaying exotic electronic states in a solid sample
Crystal unit-cell
Antiferromagnetic unit cell
Cuprates: antiferromagnetic insulators that become high-Tc superconductors upon doping!
Cuprates: (rough) phase diagram
Coexistence ?
Electron-doped cuprates:
Generalities
R2-xCexCuO4 : crystal structure
R/Ce
CuO2 plane
CuO2 plane
CuO2 plane
(R,Ce)2O2 block
(R,Ce)2O2 block
• H. J. Kang et al., Nature Materials 6, 224 (Feb. 2007).• L. Shan et al., cond-mat/0703256 (March 2007).
Electron-doped cuprates: effects of Ce-doping and annealing
Nd2-xCexCuO4 (x = 0.15) Tl2Ba2CuO6+d
M. Platé et al., PRL 95, 077001 (2005)N. P. Armitage et al., PRL 88, 257001 (2002)
Fermi surfaces, Brillouin zones and AF-zones: hole-doped vs electron-doped cuprates
S. R. Park et al., cond-mat/0612419 (Dec. 2006)
Antiferromagnetic-induced band-folding in underdoped Sm2-xCexCuO4 (x = 0.14)
Annealed
As-grown
Effects of annealing on band structure at optimal doping: the case of Pr1.85Ce0.15CuO4
P. Richard et al., cond-mat/0704.0453 (Apr. 2007)
Electronic structure and signatures of interactions in
Sm1.84Ce0.16CuO4
Coworkers Coworkers -- collaboratorscollaborators
Takeshi Kondo, Adam KaminskiTakeshi Kondo, Adam Kaminski(Ames Lab (Ames Lab -- Iowa)Iowa)
StStééphanephane PailhPailhèèss (PSI and LLB)(PSI and LLB)Johan Chang, Ming Shi, Luc Johan Chang, Ming Shi, Luc PattheyPatthey (PSI)(PSI)
AlexandreAlexandre ZimmersZimmers(CSR (CSR –– Maryland and INP Maryland and INP –– P6)P6)
Bing Bing LiangLiang, , PengchengPengcheng Li, Rick GreeneLi, Rick Greene(CSR (CSR -- Maryland)Maryland)
Γ
(π,π)
(π,0)
Γ
(π,π)
(π,0)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
Momentum along AFZB [2-½π/a]
-0.4
-0.3
-0.2
-0.1
0.0
Bind
ing
ener
gy [e
V]Min
Max
Sm2-xCexCuO4 : Fermi surface and doping
Tight-binding fit
Doping from FS volume:x = 0.16 ± 0.01
Single-band FS (no band-folding)Suppressed spectral weight at “hot-spots”
AN
N
-0.8
-0.6
-0.4
-0.2
0.0
Bin
ding
ene
rgy
[eV]
0.1 Å-1Min
Max
kAFZB
AN
-0.8
-0.6
-0.4
-0.2
0.0
Bin
ding
ene
rgy
[eV]
Min
Max
0.1 Å-1
kAFZB
N
Sm2-xCexCuO4 (x = 0.16) : Nodal vs anti-nodal ARPES spectra
NAN
Relative momentum
0.1 Å-1
ω = 0 meV
ω = -100 meV
ω = -200 meVAN
Relative momentum
0.1 Å-1ω = 0 meV
ω = -100 meV
ω = -200 meVN
Sm2-xCexCuO4 (x = 0.16) : Nodal and anti-nodal MDCs
Anti-nodal and nodal MDCs are LorentziansMDC peak maximum gives the quasi-particle dispersionMDC width is proportional to quasi-particle scattering rate (self-energy)
NAN
-0.8 -0.4 0.0Binding energy [eV]
AN
Anti-node• Peak-hump structure close to kF• Lorentzian EDCs at energies > 300 meV
Node• No clear peak-hump structure• EDCs are not Lorentzian
-0.8 -0.6 -0.4 -0.2 0.0 Binding energy [eV]
N
AN
AN
N
Hump
Sm2-xCexCuO4 (x = 0.16) : Nodal vs anti-nodal EDCs
Sm2-xCexCuO4 (x = 0.16) : Nodal vs anti-nodal dispersions
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
Bin
ding
ene
rgy
[eV]
Relative momentum
0.1 Å-1
NAN
AN
N
0.4
0.3
0.2
0.1
0.0
Im(Σ
) [eV
]
-0.5 -0.4 -0.3 -0.2 -0.1 0.0
Binding energy [eV]
Sm2-xCexCuO4 (x = 0.16) : Nodal vs anti-nodal line-widths
( ) HWHM0 kvband ∆×=Σ ′′ ω
From EDC-widthN
AN
ConclusionsWhen it works…
ARPES is a powerful technique for the study of the electronic structure of complex systems
Outlook (and dreams)
Detailed band structures and Fermi surfacesk-dependent Fermi velocity and effective massGapsMany-body effects in the QP dispersion
• Kinks• Fermi-surface nesting
Spin-resolved ARPESTime-resolved ARPESMicro-ARPES