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München
Ludwig
UniversitätMaximilians
Theoretical Description of Angle ResolvedPhotoemission in the X-ray Regime on the Basis
of the One-Step Model
H. Ebert, J. Minar and J. Braun
Department Chemie und Biochemie,
Physikalische Chemie, Universitat Munchen, Germany
In collaboration with:
N. Brookes - ESRF
C. Fadley - University of California Davis
C.M. Schneider - Research Center Jülich
haxpes09 – p.1/23
München
Ludwig
UniversitätMaximilians Outline
Introduction
Importance of surface contributions
One-step model of photo-emission
Reduction of surface sensitivityfor high photon energies (HAXPES)
HAXPES specific issuesphoton momentumthermal effectsnon-dipole contributions (not yet)
Summary
haxpes09 – p.2/23
München
Ludwig
UniversitätMaximilians Description of PES via three-step model
three-step model of photo-emission (ARPES)Berglund and Spicer (1964)
I
III
IIIII Escape to the vacuum D(E,ω)
II Transmission to surface T (E,~k)
I Excitation 〈n′~k|~p|n~k〉
I ∼ D(E, ω)∑
nn′
∫d3k T (E,~k) |〈n′~k|~p|n~k〉|2
×δ(E − En~k
− ω)δ(En′~k
− E)Θ(E − EF )Θ(EF + ω − E)
haxpes09 – p.3/23
München
Ludwig
UniversitätMaximilians PES calculations based on three-step model
PES of polycrystalline Cu for ~ ω = 8 − 17 eVTheory: Janak et al. (1975) Expt: Eastman and Grobman
central ingredients tothree step-modelcalculated from bandstructure data;e.g.: 〈n′~k|~p|n~k〉
life times enteras parameters
T (E,~k) = α(ω)l(E,~k)
1+α(ω)l(E,~k)
τphot → α(ω)
τel → l(E,~k) Expt. and Theor. photoemission distributionfor ~ω = 8 - 12 eV (left) and 13 - 17 eV (right)
haxpes09 – p.4/23
München
Ludwig
UniversitätMaximilians Ag/Au(111): surface state emission
475 meV 305 meV 237 meV 168 meV 93 meV 59 meV
kp
Ene
rgy
(eV
)
0.20.10.0-0.1-0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
kpE
nerg
y (e
V)
0.20.10.0-0.1-0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
kp
Ene
rgy
(eV
)
0.20.10.0-0.1-0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
kp
Ene
rgy
(eV
)
0.20.10.0-0.1-0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
kp
Ene
rgy
(eV
)
0.20.10.0-0.1-0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
kp
Ene
rgy
(eV
)
0.20.10.0-0.1-0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
482 meV 302 meV 243 meV 178 meV 92 meV 57 meV
H. Cercellier, D. Malterrre, F. Reinert et al. Phys. Rev. B 73, 1 (2006)
J. Braun, J. Minár, S. Bornemann and H. Ebert (2009).1haxpes09 – p.5/23
München
Ludwig
UniversitätMaximilians One-step model of photo-emission I
radiation sourcewave vector ~q
polarisation λ
photo electron detectorwave vector ~k
spin state ms
φ
’E
E
EEF
ii
f
ω
ω’
’φff
φf
photo-current (Fermi’s golden rule)
j ∝∑
i
|〈φf |H~qλrad|φi〉|
2δ(Ef − Ei − ω)
with final state φf = TRφLEED — time reversed LEED statehaxpes09 – p.6/23
München
Ludwig
UniversitätMaximilians One-step model of photoemission II
photo current
j ∝∑
i
〈φf |H~qλrad|φi〉〈φi|H
~qλ †rad |φf 〉 δ(Ef − Ei − ω)
∝ 〈φf |H~qλrad|Im Gi|H
~qλ †rad |φf 〉
initial state Green’s function
Im Gi(E) =∑
i
|φi〉〈φi| δ(E − Ei)
final stateφf = TRφLEED
= TR
[ei~kf~r +
∫d3r′G(~r, ~r ′, Ef ) V (~r ′) ei~kf~r ′
]
e.g. Caroli et al. (1973), Feibelmann and Eastman (1974)haxpes09 – p.7/23
München
Ludwig
UniversitätMaximilians One-step model of photoemisison III
Implementation via multiple scattering theoryAdaption of LEED formalism (Pendry et al. 1980)
single layer scattering matrix
M ττ ′ss′
gg′ = δττ ′ss′
gg′ +8π2
kk+gz
∑
κµ
∑
κ′µ′
∑
κ′′µ′′
∑
ii′
i−lCκµsYµ−sl (kτ
g)e−ikτgRi
× ti κµ
κ′′µ′′
(1 − X)−1iκ′′µ′′
i′κ′µ′
il′
Cκ′µ′s′Yµ′−s′
l′(kτ ′
g′)eikτ ′
g′Ri′
In-layer scattering
UV photoemission: X 6= 0
X-ray photoemission: X = 0
Relativistic formulation: e.g. Braun (1996) haxpes09 – p.8/23
München
Ludwig
UniversitätMaximilians Surface sensitivity: nMgO/Fe(001) at 1000eV
calculated ARPES intensities I(E,Θ)
Fe(001) MgO
1 ML MgO/Fe(001) 8 ML MgO/Fe(001)
increasing
⇒coverlayerthickness
Fe-related features get lost with increasing MgO coverage haxpes09 – p.9/23
München
Ludwig
UniversitätMaximilians Surface sensitivity for high photon energies
calculated ARPES intensities I(E,Θ)Fe(001) Fe(001)
8 ML MgO/Fe(001) 8 ML MgO/Fe(001)
Ephot = 1000eV ⇒ Ephot = 6000eV
Fe-related features recovered for high photon energies⇒ access to burried interfaces (SPIN HAXPES; Felser et al.) haxpes09 – p.10/23
München
Ludwig
UniversitätMaximilians Photon momentum effects on Ag(001)
Ag(001) photoemission intensities along ΓK with LCP-light at hν=552 eV
A
-1.5 -1 -0.5 0 0.5 1 1.5
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)
B
-1.5 -1 -0.5 0 0.5 1 1.5
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)qphoton ignored
ki = (k|| + g,
√2(E − iVi1) − |k|| + g|2)
kf = (k|| + g,
√2(E + ω − iVi2) − |k|| + g|2)
qphoton included
ki = (k|| − q|| + g,
√2(E − iVi1) − |k|| − q|| + g|2)
kf = (k|| + g,
√2(E + ω − iVi2) − |k|| + g|2)
Venturini et al. PRB 77, 045126 (2008)haxpes09 – p.11/23
München
Ludwig
UniversitätMaximilians How to minimise photon momentum effects?
q = 0,Θ = 0 q 6= 0,Θ = 0
q = 0,Θ = 0.7 q 6= 0,Θ = 0.7
A
-1.5 -1 -0.5 0 0.5 1 1.5
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)
B
-1.5 -1 -0.5 0 0.5 1 1.5
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)
C
-1.5 -1 -0.5 0 0.5 1 1.5
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)
D
-1.5 -1 -0.5 0 0.5 1 1.5
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)
haxpes09 – p.12/23
München
Ludwig
UniversitätMaximilians Photon momentum effects on Ag(001)
Experiment Theory
Γ XX
-1.4 -1 -0.6 -0.2 0.2 0.6 1 1.4
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)Γ XX
-1.4 -1 -0.6 -0.2 0.2 0.6 1 1.4
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)
Γ KK
-1.5 -1 -0.5 0 0.5 1 1.5
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)
Γ KK
-1.5 -1 -0.5 0 0.5 1 1.5
kparallel (A° -1)
-8
-7
-6
-5
-4
-3
-2
-1
0
Bin
ding
Ene
rgy
(eV
)
Venturini et al. PRB 77, 045126 (2008) haxpes09 – p.13/23
München
Ludwig
UniversitätMaximilians Thermal effects I
Thermal vibrations: fundamental limit to bandmapping as energy or temperature is raised
Superposition of direct and Non-direct transitions:
I(E, T ) = W (T )IDT (E) + (1 − W (T ))INDT (E)
Debye-Waller factor W (T ) ∝ exp(−∆k2〈u2〉T )
mean-square displacement 〈u2〉TShevchik (1977)
Experimental support:White and Fadley (1981)
T -dependent scattering matrix t withinLEED-formalism (Duke (1980))
t(T,~k,~k′) = t(0, ~k,~k′)W (T )haxpes09 – p.14/23
München
Ludwig
UniversitätMaximilians Thermal effects II
Based on Glaubert’s theorem (1955)
〈ei∆~k·~u〉T = e−1
2〈(∆~k·~u)2〉T
Transfer to 1-step model of PES and inclusion ofmatrix element effects: Larsson and Pendry (1981)
Forward focussing in VB-XPS:Osterwalder et al. (1990)
Improved treatment of phonon effects on LEEDstate - cluster implementation:Zampieri et al. (1996)
haxpes09 – p.15/23
München
Ludwig
UniversitätMaximilians W(110): Experiment and theory at 870eV
Plucinski, Minar et al., PRB 78, 035108haxpes09 – p.16/23
München
Ludwig
UniversitätMaximilians W(110) at 5954eV, T = 30K
Experiment Theory (T=0K)
Theory, wide angular scan (T=0K)
haxpes09 – p.18/23
München
Ludwig
UniversitätMaximilians W(110) at 5954eV, T = 100K
Direct contribution Non-direct contribution Total intensity
Effects of Phonons on final LEED state
Shevchik (1977): I(E, T ) = W (T )IDT (E) − (1 − W (T ))INDT (E)
Zampieri et al.(1996): Cluster approach, two site scattering
Current approach: full multiple scattering of final state
XPD limit (?)
haxpes09 – p.19/23
München
Ludwig
UniversitätMaximilians Outlook: Electron-phonon interaction
Eliasberg function of Ni
0 10 20 30 40 ω [meV]
0
0.05
0.1
α2 F(ω
, kF)
[meV
]
Γ − L Spectral function for Ni (kink)
KKR Phonon calculations: linear response
Self energy Σel−ph(E,~k) =R
2ΣEi(E, ω)α2F (ω,~k)dω: A. Eiguren, C. Ambrosch-Draxl
include Σel−ph(E,~k) into KKR via Dyson Equation + Photoemission
calculations for complex systems and alloys (high-Tc-materials)haxpes09 – p.20/23
München
Ludwig
UniversitätMaximilians Outlook: effect of thermal vibrations in HAXPES
Scattering theory for dislocated atoms
vibrations of lattice sitesuncorrelated
assume dislocations ∆~Rn
to depend on temperature T
with probability P (∆~Rn, T )
average over dislocationsusing CPA alloy theoryP (∆~Rn, T ) ⇔ concentration
Combine with PES-theoryfor alloys (Durham)
+ + + + ++ + + + +
+ + + + ++ + + + +
. . . . .. . . . ...... . . . . .CPA averaging ⇓
+ ++ + +
+ + + + ++ + + + +
++ + +
+
leads to proper Green’sfunction without artefacts(Heiglotz)!
haxpes09 – p.21/23
München
Ludwig
UniversitätMaximilians Summary
Extension of the one-step model of photo-emission allowto deal with all aspects of HAXPES
photon momentum
thermal effectsmore refined models necessary
relativistic effects
matrix elements effects(magnetic) dichroismnon-dipole contributions
spin-resolution (SPIN-HAXPES)
haxpes09 – p.22/23