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Theory of Recoil Effect in HAXPES and Its Prospect as a Tool for Material Research
Y. Kayanuma
aGraduate School of Engineering,Osaka Prefecture University
Japan
HAXPES 2009 (NSLS), 20 May 2009
Collaborators:
Y. Takata (RIKEN/SPring-8) , S. Tanaka (OPU), S.Oshima(OPU)
2
1. Introduction
Recoil effect of atoms, molecules and solids
2. Core level X-ray photoemission in graphites
3. Valence band X-ray photoemission
4. Outlook
Recoil spectroscopy?
3
Anomalies in HXPES for 1s core-level in graphiteY. Takata et al.
Normal angle emission
RT
1. Shift of the binding energy
2. Anomalous broadening depend on X-ray energy
3. Asymmetric line shape
4
pP rr−=
Mm
mp
MPE ×==Δ
22
22 rr
220001
=Mm
keV82
2
≈m
preV!36.0≈ΔE
Recoil effect is observable!
(carbon atom)
5
Energy momentum relation
10keV 20keV
Mom
entu
m
chpX /ν=
mEpe 2=
Eh orν
1.0/ keV,10at ≈= eX pphν 01.0/ ≈ΔΔ∴ eX EE
Momentum of photon is negligible.
But momentum of photoelectron is NOT.
6
A guy with weight 70kg dived from a ship of 1500t , and you observed the ship to move!
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Doppler Broadening
νh
pr
M M
Energy conservation
pPirr
−
iPr
MpP
mp
MPh ii
c 2)(
22
222 rrrr−
+=++ νε
MpP
MPh
mpE ii
cobs 2)(
22
222 rrrr−
−++== νε
MpP
Mph i
c
rrr⋅
+−+=2
2
νε
( ) ( ) TkEpM
P
MpPEE B
iiobsobs Δ==
⋅=− 2
32
2
2
2
22 r
rrr
EMp
Δ=2
2r
0=⋅
MpPirr
m
In crystal, ωh→TkB
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We need quantum mechanics
ωhMpE
2
2
=ΔsolidM
M
)(1 ωh<<Δ<< ES
ωh/ES Δ=
)(1 ωh>>Δ>> ES Observed peak shift EΔ=
Observed peak shift 0=
Se−
Mossbauer Effect
Asymmetric line shape quantum effect of phonons
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Recoil effect comes out from only one assumption.
)( Rrr cc
rrr−=ψψAdiabatic
approximation
Translational symmetry
Momentum conservation
Initial state
Atomic position is a dynamic variable.
Rr
Final state ( )rkiV
r kvrr⋅= exp1ψ
Fermi’s golden rule
( )∑ Γ+−−−+
ΨΨΓ=
f mcn
iIf
hE
HEI
22
2
)(εενεπ
10
[ ])(exp)( )( tGeetF ikutiku =≡ −
( )( ){ }titntG qqqq
q λλλλ
λ ωωα sin1cos12)( 2 −−+=∑
22
2 λλ
λ ηω
α qq
q kNM
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛=
rh
Wave vector of electronAtomic mass
Displacement vector of atom
)(21)( || tFedtEI itEt∫
∞
∞−
−Γ−= hh
π
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Phonon dispersion of graphite
L. Wirtz and A. Rubio, Solid State Commn., 131,141 (2004)
bending
stretching
Equi-frequency surface
//k
Anisotropic Debye Model
12
Excitation energy dependence (normal emission)
meV(HWHM)80=Γ
M)120meV(FWHresolutionalExperiment =
Y. Takata et al. Phys. Rev. B 75, 233404 (2007).
13
Emission angle dependence (7940eV, RT)
grazing
normal
Y. Takata et al. Phys. Rev. B 75, 233404 (2007).
14
D
ES,λ
λ ωhΔ
=
ES D Δ=,λλ ωh
DDDD ES ,,,, λλλλ ωωω ∝Δ= hh
Peak shift =
1if >>λS
Width =
Effective electron-phonon coupling const.
Why?
coordinate space Momentum space
Doppler broadening is larger for high ω
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Recoil effect is universal Neutron scatteringsElectron scatteringsHX photoelectron emission
....etc
But,the photoelectron carries also information on the electronic state it has occupied initially.
The recoil effect in valence level XPS will be much more interesting !
16
Which atom(s) receives the recoil momentum?
Recoil effect in Bloch electrons?
17
Y. Takata et al.
Au=197Al=27
Striking recoil shift does exist!
-1-0.500.51
AuAl
Intensity (ar
b. units)
Binding Energy (eV)
hν=7940eVT=20K
18
Al (pseudopotential)Free electron (FCC)
existence of HX photoemission existence of Umklapp process
electrons see lattice
Even nearly free electrons see lattice structure!
27eV11
eV108 3
≈×
=Fermi
ronphotoelect
pp
Momentum is not conserved.
19
Recoil Effect of Bloch Electrons
∑ +−=ji
jijie aaRRtH,
)(rr
)(0)( iii Rrarrrrrr
−=≡ + ϕϕ iii uRR rrr+= 0
)(2 ji
jij
jL RRV
MP
Hrr
r
−+= ∑∑≠
0+∑= ii
ik acψ
01iikR
i eN
c =)()(0
ii
ikRk Rrer i
rrv −=∑ ϕψ
⎟⎠⎞
⎜⎝⎛ += +∑ 2
1,,
,, λλ
λλω qq
qqL bbH h
Wannier function
Bloch function
20
( ) ( ) ( ) ( )tFedteKkI jj
RkKithEit jk ∑∫ ⋅−∞
∞−
−−−Γ−=0
21,
rrrhh rrr εν
π
( ) ( ) 0uKtuKij eetF j
rrrr⋅−⋅=
( ) ( ) ( )tFdteEI tkhEitKk 02
1, ∫∞
∞−
−−−Γ−= hrhrr
εν
πε
It seems that a single atom receives the recoil momentum.
Shrinkage of the wave function by the measurement?
Off diagonal terms are negligible.
dynamical structure factor
21
-1-0.500.51
Au(Experiment)Al(Experiment)Au(Theory)Al(Theory)
Intensity (arb. units)
Binding Energy (eV)
hν=7940eVT=20K
Recoil effect of photoelectrons in the Fermi edge of simple metalsY. Takata et al., Phys. Rev. Lett, 101, 137601(2008)
-1-0.500.51
Au(Experiment)Al(Experiment)Au(Theory)Al(Theory)
Intensity (arb. units)
Binding Energy (eV)
hν=880eVT=50K
Origins of broadening
1. Temperature (Fermi distribution)
2. Resolution (Gaussian broadening)
3. Recoil effect (New!)
22
Next TargetHow is it in compounds?
Recoil splitting ?
BM
AM
BE
BE
Soft X-ray
Hard X-ray
23
pp−
12
b
a
( )12212211 21 +Δ++= εεeH
( )212
22
1
21
22XXV
MP
MPHa −++=
( )dxdipAApH I h−=+= + ,
ae HHH +=
( ) ( )111 1 Xxxx −=≡ ϕϕ
( ) ( )222 2 Xxxx −=≡ ϕϕ
Model 1 diatomic molecule
24
( ) ( )111 1 Xxxx −=≡ ϕϕ
( ) ( )222 2 Xxxx −=≡ ϕϕ
21
21
βα
αβ
+=
+−=
b
a
a
b
Adiabatic approx.
0⊗⊗⊗=Ψ gbhi ν
mKkvacf ⊗⊗⊗=Ψ
( )mKkbm K
k EhbgpkKmI εεενδε −−−+=∑∑ 2,,0,,)(
25
termscross)()()( 22
22
12
12 ++= kkk III εςβεςαε
( )xdxdiedx i
ikx
i ϕς ⎟⎠⎞
⎜⎝⎛−=
−
∫
kε0
kε0
+
relativeEΔ CMEΔ
CMEΔ
21
2 ςα×
22
2 ςβ×
26
Model 2 composite metal
Heavy atom and light atom
In a unit cell
( ) ( ){ } jRki
j
BjB
AjAk eRrwkCRrwkC
Nr
0
21 )()(1)(rrrrvrrrr ⋅∑ −+−=ψ
Bloch function
Atomic positions are dynamic variables (not parameters).
Au
Al
27
)()()( εεε BA DDD +=
( ) ( ) ( )tFedtEI AthEi
A ∫∞
∞−
−−−= hεν
πε
21,
( )( ){ }⎥⎥⎦
⎤
⎢⎢⎣
⎡−−+= ∑ titntF qqq
q
AqA λλλ
λλ ωωα sin1cos12exp)(
2
22
2 λλ
λ ηω
α qqA
Aq k
NM⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛=
rh
( ) ( ) ( ) ( ) ( ) ( ){ }∫∞
∞−+= εεςεεςεε ,, 22 EIDEIDfdEI BBBAAA
Local DOS
Contribution from A site
Contribution from B site BA→
Valence band PES may split due to recoil effect!
28
How about amorphous materials?Which atom(s) receive recoil momentum?
29
Yokohama CU, Yamada Lab.
LiCoO2
Recoil effect as a probe of local atomic environment
Lithium ion battery
30
R
Valence level
Second Order Effect of Recoil
Valence band Auger
Valence band X-ray emission
31
νh
νh
Recoil Spectroscopy of Local Vibration
site selective
angle selective
level selective
32
Message of this work1. Recoil effect in x-ray photoemission is a purely kinematic effect.
Only the adiabatic approximation is needed theoretically.
2. Line shape in core level recoil spectra carries information of the
rigidity of the atomic environment. (Doppler broadening)
3. Even Bloch electrons show recoil effect. (From which atom was it ejected?)
4. Valence spectra may split due to recoil effect.
Recoil Spectroscopy, a new tool?
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