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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)

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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?

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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

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pP rr−=

Mm

mp

MPE ×==Δ

22

22 rr

220001

=Mm

keV82

2

≈m

preV!36.0≈ΔE

Recoil effect is observable！

(carbon atom)

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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.

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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

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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

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[ ])(exp)( )( tGeetF ikutiku =≡ −

( )( ){ }titntG qqqq

q λλλλ

λ ωωα sin1cos12)( 2 −−+=∑

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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

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Excitation energy dependence （normal emission）

meV(HWHM)80=Γ

M)120meV(FWHresolutionalExperiment =

Y. Takata et al. Phys. Rev. B 75, 233404 (2007).

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Emission angle dependence (7940eV, RT）

grazing

normal

Y. Takata et al. Phys. Rev. B 75, 233404 (2007).

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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 !

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Which atom(s) receives the recoil momentum?

Recoil effect in Bloch electrons?

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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

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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.

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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

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( ) ( ) ( ) ( )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

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-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!)

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Next TargetHow is it in compounds?

Recoil splitting ?

BM

AM

BE

BE

Soft X-ray

Hard X-ray

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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

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( ) ( )111 1 Xxxx −=≡ ϕϕ

( ) ( )222 2 Xxxx −=≡ ϕϕ

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21

βα

αβ

+=

+−=

b

a

a

b

Adiabatic approx.

0⊗⊗⊗=Ψ gbhi ν

mKkvacf ⊗⊗⊗=Ψ

( )mKkbm K

k EhbgpkKmI εεενδε −−−+=∑∑ 2,,0,,)(

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termscross)()()( 22

22

12

12 ++= kkk III εςβεςαε

( )xdxdiedx i

ikx

i ϕς ⎟⎠⎞

⎜⎝⎛−=

−

∫

kε0

kε0

+

relativeEΔ CMEΔ

CMEΔ

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2 ςα×

22

2 ςβ×

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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!

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How about amorphous materials?Which atom(s) receive recoil momentum?

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Yokohama CU, Yamada Lab.

LiCoO2

Recoil effect as a probe of local atomic environment

Lithium ion battery

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R

Valence level

Second Order Effect of Recoil

Valence band Auger

Valence band X-ray emission

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νh

νh

Recoil Spectroscopy of Local Vibration

site selective

angle selective

level selective

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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?