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