Electronic Excitation in Atomic Collision Cascades

COSIRES 2004, Helsinki

C. Staudt

Andreas Duvenbeck

Zdenek Sroubek

Filip Sroubek

Andreas Wucher

Barbara Garrison

kinetic excitation

• atomic motion in collision cascade• electronic excitation in inelastic collisions

• electron emission, charge state of sputtered particles

,eT r t

space and time dependent electron temperature ?

excitation model (1)

• energy transfer

– kinetic energy electronic excitation

– electronic stopping power (Lindhard):

vKdx

dE kinEAvKdt

dE 2

Sroubek & Falcone 1988

i

ikin

source

el trEAtrdt

dE,,

total energy fed into electronic system :

electronic friction ?

kinE

elE

kinA E dt

• ab-initio simulation of H adsorption on Al(111)

(E. Pehlke et al., unpublished)

Lindhard formula works well for low energies

excitation model (2)

• diffusive transport

– diffusion coefficient may vary in space and time

• „instant“ thermalization

– electronic heat capacity depends on Te !

2 ,el elel

source

E dED E r t

t dt

2, ,e elT r t E r t

C21

2e

e e B eF

TC n k C T

T

instant thermalization ?

• ab-initio simulation of H adsorption on Al(111)

(E. Pehlke et al., unpublished)geometry

electronic states

Fermi-like electron energy distribution at all times !

diffusion coefficient

• fundamental relation :

• electron mean free path :

• relaxation time :

• lattice disorder :

1

3 F eD v

Fermi velocity

e F ev

21 1 1e L

e e e e ph

a T b T

22

1 1

3 Fe L

D vaT bT

lattice temperatureelectron temperature

2

2

1,

3 , ,F

e L

vD r t

aT r t bT r t

te interatomic distance

numerics

• Green's function

• explicit finite differences

21

3 20

1( , ) ( , ) exp

44 ( )

im k

el k i kin m ni nn m i n

E t A t E tD t tD t t

r rr r

1

3

,2

1

, ,

, , 2 ,, ,

el k i el k i

el k i el k i el k ie L kin k i

E t E t

t

E re t E re t E tD T T A E t

r

r r

r r rr

,e k iT tr ,L k iT tr

solution of diffusion equation by

crystallographic order (rk,ti)

0elE

boundary conditions

0elE

2 2 0elE y

y

x

z

42 Å

0elE

• finite differences• Green's function

y

x

z

0elE

0elE

0elE

MD Simulation

5 keV Ag Ag(111)

• trajectory 952 • trajectory 207

Ytot = 16 Ytot = 48

4500 atoms

lattice temperature

• N atoms in cell

0 100 200 300 400 500 600 700 800100

101

102

103

104

105

106

tem

per

atu

re (

K)

time (fs)

traj. 207

1

1 Ni

kin kini

E EN

2

3kin

LB

ET

k

220D cm s

even at Te = 0 !

averaged over entire surface

calculated TL

limitation of D

constant diffusivity

– differences at small times (< 100 fs)– same temperature variation at larger times

0 100 200 300 400 500 600 7000

5000

10000

15000

20000

25000

30000

Neumann boundary at surface

traj. 207

Tel(K

)

time(fs)

r = 0 Å r = 3 Å r = 6 Å r = 9 Å r = 12 Å r = 15 Å r = 18 Å

D = 0.5 cm2/s

0 100 200 300 400 500 600 7000

5000

10000

15000

20000

25000

30000

D = 0.5 cm2/s

traj. 207

Te (

K)

time (fs)

r = 0 r = 4 Å r = 8 Å r = 12 Å r = 16 Å r = 20 Å

Green's function

finite differences

5t fs 310t fs

0 100 200 300 400 500 600 7000

5000

10000

15000

20000

traj. 207

D = D(Te(r,t),T

L=104K)

r = 0 Å r = 3 Å r = 6 Å r = 9 Å r = 12 Å r = 15 Å r = 18 Å

Te(

K)

time(fs)

electron temperature dependence

0 100 200 300 400 500 600 7000

5000

10000

15000

20000

traj. 207

D = 5.4 cm2/s

r = 0 Å r = 3 Å r = 6 Å r = 9 Å r = 12 Å r = 15 Å r = 18 Å

Te(

K)

time(fs)

D = const (TL = 104 K)

Te variable, TL = const

• Te - dependence small for t > 100 fs

0 100 200 300 400 500 600 7000

20

40

60

80

100

traj. 207

D = D(Te(r,t),T

L(r,t))

r = 0 Å r = 3 Å r = 6 Å r = 9 Å r = 12 Å r = 15 Å r = 18 Å

time(fs)

Te(

103 K

)

full temperature dependence

0 100 200 300 400 500 600 7000

20

40

60

80

100

traj. 207

D = D(Te(r,t),T

L=104K)

r = 0 Å r = 3 Å r = 6 Å r = 9 Å r = 12 Å r = 15 Å r = 18 Å

Te(

103 K

)

time(fs)

TL constant, Te variable TL variable, Te variable

• TL dependence strong ! • Te < 1000 K for t > 100 fs

0 100 200 300 400 500 600 7000.0

0.2

0.4

0.6

0.8

1.0

total crystal volume partial crystal volume

time (fs)

atomic disorder

0 2 4 6 8 10 12 14 16 18 20

traj. 207

interatomic distance (Å)

6 fs 100 fs 200 fs 300 fs 500 fs 2500 fs

time dependence of crystallographic order (traj. 207)

pair correlation function order parameter

1

3 x y z

• N atoms in cell

•

•

1

21cos

Ni

xxi

x

N a

0 100 200 300 400 500 600 7000

1000

2000

3000

4000

5000

6000

traj. 207

Te(

K)

time(fs)

r=0 Å r=3 Å r=6 Å r=9 Å r=12Å r=15Å r=18Å

D = 20 --> 0.5 cm2/s in 300fs

order dependence

0 100 200 300 400 500 600 7000

1000

2000

3000

4000

5000

6000

traj. 207

D = 20 --> 0.5 cm2/s in 300 fs

Te (

K)

time (fs)

r = 0 Å r = 4 Å r = 8 Å r = 12 Å r = 16 Å r = 20 Å

• linear variation of D between 20 and 0.5 cm2/s within 300 fs

Green's function finite differences

lattice disorder extremely important !

e-ph coupling

0 100 200 300 400 500 600 700 800102

103

104

105

D=20 --> 0.5 cm2/s in 300 fs

tem

per

atu

re (

K)

time (fs)

lattice electrons

traj. 207

0 100 200 300 400 500 600 700 80010-6

10-5

10-4

10-3

10-2

10-1

100

101

D=20 --> 0.5 cm2/s in 300 fs

ener

gy

den

sity

(eV

/Å3 )

time (fs)

electronic kinetic

traj. 207

surface energy density surface temperature

negligible back-flow of energy from electrons to lattice !

• two-temperature model :

e LE

const T Tt

Summary & Outlook

• MD simulation– source of electronic excitation

• diffusive treatment of excitation transport– include space and time variation of diffusivity by

• temperature dependence• lattice disorder

• MD simulation

– calculate Eel and Te as function of

– position

– time of emission t

• Calculate excitation and ionization probability individually for every sputtered atom

,tr

r

of sputtered atoms

0 200 400 600 800 1000 12000.0

0.5

1.0

P(t

) /

P(t

=0

)

time (fs)

r = 4.1 A r = 5.1 A r = 5.8 A r = 6.5 A

Diffusion Coefficient

• peak value vs. time (normalized)

• time dependent diffusion coefficient :

D

r

0 200 400 600 800 1000 12000.000

0.003

0.006

0.009

0.012

0.015

0.018

D = 14.4 ---> 0.88 in 150 fs

traj. 952

Pex

c

time (fs)

0 4 8 12 16 20

excitation probability

Excited atoms emitted later in cascade

excitation probability electronic energy density

r (A)

Time Dependence

0 100 200 300 400 5000.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

r (A) =

D = 14.4 a.u.

traj 207

en

erg

y d

en

sity

Ee

(eV

/ A

3 )

time (fs)

0 4 8 12 16 20

r

Electron Temperature

0 100 200 300 400 5000

1000

2000

3000

4000

r (A) =

D = 14.4 a.u.

traj 207

ele

ctro

n t

em

pe

ratu

re T

e (K

)

time (fs)

0 4 8 12 16 20

r

electron temperature

0 200 400 600 800 1000 12000

1000

2000

3000

4000

5000

D = 14.4 ---> 0.88 in 150 fs

traj. 952

average electron temperature

T

e (K

)

t (fs)

Energy Spectrum

• excitation probability time dependent

– small for t < 300 fs

– large for t > 300 fs

First (crude) estimate :

• simulation of energy spectrum

– no account of excitation

– count all atoms for ground state

– count only atoms emitted after 300 fs for excited state

simulationexperiment

Summary & Outlook

• MD simulation

– calculate Eel and Te as function of

– position

– time of emission t

• Qualitative explanation of

– order of magnitude

– velocity dependence of excitation probability (Ag* , Cu*)

• Calculate excitation and ionization probability individually for every sputtered atom

• Quantitative correlation between order parameter and electron mean free path

tr ,

r

of sputtered atoms

Electron Energy Distribution

• 3 x 3 x 3 Å cell grid

• numerical solution of diffusion equation

• variable diffusion coefficient D

– Te dependence

– TL dependence

– lattice disorder

electron energy density at the surface

,,el i

el kin ii

E tD E t A E

t

r

r r r

Excitation

Co atoms sputtered from Cobalt

ground state

excited state

population partition

V. Philipsen, Doctorate thesis 2001

Excitation

Ni atoms sputtered from polycrystalline Nickel by 5-keV Ar+ ions

ground state

excited state

velocity distribution

V. Philipsen, Doctorate thesis 2001