Electronic Excitation in Atomic Collision Cascades COSIRES 2004, Helsinki C. Staudt Andreas...
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Transcript of Electronic Excitation in Atomic Collision Cascades COSIRES 2004, Helsinki C. Staudt Andreas...
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(E. Pehlke et al., unpublished)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(E. Pehlke et al., unpublished)geometry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