theoretical modeling of short pulse laser interaction with condensed matter tzveta apostolova...
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Theoretical modeling of short pulse laser interaction with
condensed matter
Tzveta Apostolova
Institute for Nuclear Research and Nuclear Energy,
Bulgarian Academy of Sciences
ELI Nuclear Physics Workshop’2010
Short pulse nanostructuring
Nanostructures in Si – Lebedev Physical Institute 2009 – unbuplishedNumber of incident pulses ~103Energy density ~50-300 mJ/cm2Wavelength 744 nmPulse duration (FWHM) ~100 fsPulse energy <0.5 mJPulse repetition rate 10 Hz
Formation of filaments
Single pulse filaments in PMMA, energy density 0.50-25 mJ/cm2– Lebedev Physical Institute 2009
Similar could be obtained for Si if irradiated with mid-IR
Laser radiation
electron
hole
Conduction band
Valence band
Forbidden band
semiconductor
nmnm 1100500
kHzkHzf 201
21 cmTWI
Laser radiation parameters
Laser radiation
Wavelength
Pulse duration
Repetition rate
Intensity
psfsL 125
nm870
GaAs
psL 1
• The damage of materials can be described by a critical density of photo-excited electrons in the conduction band
Non-uniform laser radiation• spatially and temporally dependent laser field inside the material
LL
L
L
GPILtransL nk
k
i
t
ki
nc
ES
k
i
z 022
02'
02''
02
00
02
0 222
0,0,00 LL
gvztt '
22
000 LncI
pL 2
•Optical damage
er
ep m
ne
*00
2
LLth IdttIF )(
•Electrical damage
electcI nN
Tk
ENNN
B
GvcI 2
exp232
, )2*(2, TkmvNc Bhe
•Structural damage
317107.8 cmn electc
KTm 1512for GaAs
BGkkkkkk VEdEfdEfEE
00
eL
nim
eInnn
0
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2020 2
tzkiL
L Letzyxk
trE
0,,,2
, 00
0
)()()( 0 tHtHtH I
)(ˆ)(ˆ)(ˆ)(ˆ)(0 tbtbtataEtH qqq
qkkk
e
k
)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)( *
,
tatdFtdtaFtbtbtataCtHkkkkk
kkqqkqk
qkqI
trtrAc
ep
mt
tri ,,
2
1),(2
*
ttitkitatc LLLxkk 2)2sin(expcos1exp)(ˆ)(ˆ 10
mean traveling length of electron in one period of laser radiation
induced energy due to the interaction of electrons with the laser field
Theoretical model
xLLL etcEtA
sin)( 0
2*00 LL meE
3*201 8 LL meE
tESftEAE
tEftED
E
tEftEV
t
tEf e
ek
eee
,,,
,,
,,
2
2
Terms:Photon ionizationImpact ionizationAuger recombination
Energy loss to phonons term
Conduction-Electron Energy
Joule HeatingCurrent
Spontaneous Phonon Emission Current
Diffusion Current
Dis
trib
uti
on
Fu
nct
ion
Ionization due toCoulomb Scattering
Recombination due toPhonon Scattering
Pumping due to Photon Absorption
• The distribution function is a product of the density-of-states and the state-occupation probability
• Electrons can flow to higher energies by gaining power from an incident pulsed laser field
• Electrons can also flow to lower energies by emitting spontaneous phonons
• The competition between different currents in the energy space can be described by a Fokker-Planck equation
1 pL
)cos()(0
tt LLL
0)( t
tL
0)(2 t
tL
local fluctuation of electron kinetic energy is included in the kinetic Fokker-Planck equation under the condition
but
20
20 3
)(3
1, LL
pTLLFT EAEVEVtEV
20
20 3
)(3
2, LL
pT
ekLLFT EVEEDEDtED
22*2 1 pLeeLc me Drude ac conductivity
qqkk
phqqqkk
phq
qkT
EENEEN
CEV
1
2 2
LqqkkLqqkkqkkphq
LqqkkLqqkkqkkphq
Lq
qF
EEEEEEN
EEEEEEN
mtqeCV
1
.2 22*2
dependence on magnitude and frequency of the field
22
2
4
22
kGhk
ekL
kkabs
FEEE
FFS
322
10
0*0
0
20
22
G
GG
eL
Lk E
EE
m
m
m
eF
Single photon excitation
0.0 0.2 0.4 0.6 0.8 1.0 1.20.0
0.2
0.4
0.6
Dis
trib
utio
n f
e
k (
10
23 c
m-3 /
eV
)
Energy Ee
k / E
G
With Recombination Without Recombination
0.0 0.2 0.4 0.6 0.8 1.0 1.20.00
0.05
0.10
0.15
Dis
trib
utio
n f e
k
( 10
23 c
m-3 /
eV )
Energy Ee
k / E
G
With Thermal Emission Without Thermal Emission
k -q
k
(a)
2q -k
k -q
U (q)eff
k -q
k
(a)
2q -k
k -q
U (q)eff
k
k + q
(b)
q -k
k
U (q)eff
k
k + q
(b)
q -k
k
U (q)eff
k -q
k
(a)
2q -k
k -q
U (q)eff
k -q
k
(a)
2q -k
k -q
U (q)eff
k
k + q
(b)
q -k
k
U (q)eff
k
k + q
(b)
q -k
k
U (q)eff
scattering-in scattering-out scattering-in scattering-out
k - q
k
q
(a)k - q
k
q
(a)k
k - q
q
(d)k
k - q
q
(d)
Effects of thermal emission and recombination
1.0L
t
1.0L
t
0.0 0.2 0.4 0.6 0.8 1.0 1.20.00
0.05
0.10
0.15
(a) t / L= -0.5
Dis
trib
utio
n f e
k (
102
3 cm
-3 /
eV )
Energy Ee
k / E
G
EG=1.50 eV (100 K)
EG=1.42 eV (300 K)
0.0 0.2 0.4 0.6 0.8 1.0 1.20.0
0.1
0.2
0.3
0.4
(b) t / L=0.1
Dis
trib
utio
n f
e
k (
102
3 cm
-3 /
eV
)Energy Ee
k / E
G
EG=1.50 eV (100 K)
EG=1.42 eV (300 K)
Effects of lattice temperature
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.0
0.1
0.2
0.3
0.4
(a)Ele
ctro
n D
ensi
ty n
e (
102
3 cm
-3 )
Time t / L
IL=8x1015 W/m2
IL=8x1014 W/m2
0.0
0.2
0.4
0.6
0.8
1.0
Inte
nsity
Pro
file
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.05
0.10
0.15
0.20
0.25
0.30
(b)
Ave
rage
Kin
etic
Ene
rgy
< E
e k >
/ E
G
Time t / L
IL=8x1015 W/m2
IL=8x1014 W/m2
Time dependence of the electron density and average kinetic energy for different intensities and different bandgap widths
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.0
0.1
0.2
0.3
0.4
(a)
EG=1.50 eV (100 K)
EG=1.42 eV (300 K)
Ele
ctro
n D
ensi
ty n
e (
102
3 cm
-3 )
Time t / L
0.0
0.2
0.4
0.6
0.8
1.0
Inte
nsity
Pro
file
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.05
0.10
0.15
0.20
0.25
0.30
0.35
(b)
Ave
rage
Kin
etic
Ene
rgy
< E
e k > /
EG
Time t / L
EG=1.50 eV (100 K)
EG=1.42 eV (300 K)
0.0 0.2 0.4 0.6 0.8 1.0 1.20.00
0.05
0.10
0.15
t / L= -0.5(b)
Dis
tribu
tion
f e
k (
1023
cm
-3 /
eV )
Energy Ee
k / E
G
d=400 meV
d=200 meV
0.0 0.2 0.4 0.6 0.8 1.0 1.20.0
0.2
0.4
0.6
t / L=0.55(c)
Dis
trib
utio
n f e
k (
1023
cm
-3 /
eV )
Energy Ee
k / E
G
d=400 meV
d=200 meV
L
0.0 0.2 0.4 0.6 0.8 1.0 1.20.000
0.003
0.006
0.009
t / L= -1.25(a)
Dis
trib
utio
n f
e
k (
10
23 c
m-3 /
eV )
Energy Ee
k / E
G
d=400 meV
d=200 meV
0.0 0.2 0.4 0.6 0.8 1.0 1.20.00
0.05
0.10
0.15
(a)t / L= -0.5
Dis
trib
utio
n f e
k (
102
3 cm
-3 /
eV )
Energy Ee
k / E
G
With Joule Heating Without Joule Heating
0.0 0.2 0.4 0.6 0.8 1.0 1.20.0
0.1
0.2
0.3
0.4
(b)t / L=0.1
Dis
trib
utio
n f e
k (
1023
cm
-3 /
eV )
Energy Ee
k / E
G
With Joule Heating Without Joule Heating
Effects of joule heating
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.0
0.1
0.2
0.3
0.4
Ele
ctro
n D
ensi
ty n
e (
102
3 cm
-3 )
Time t / L
With Joule Heating Without Joule Heating
0.0
0.2
0.4
0.6
0.8
1.0
(a)In
tens
ity P
rofil
e
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.05
0.10
0.15
0.20
0.25
0.30
(b)
Ave
rage
Kin
etic
Ene
rgy
< E
e k >
/ E
G
Time t / L
With Joule Heating Without Joule Heating
Effects of joule heating
Conclusions• We derive quantum-mechanically a generalized Fokker-Planck type
equation in energy space for conduction electrons in semiconductors irradiated by laser field in which the effect of the laser field treated classically is included via
-energy drift term for conduction electrons-joule heating effect
-renormalization of the electron-phonon scattering by the field- effect of free carrier absorption
• We include source and sink terms (up to 2nd order in perturbation theory)
• We investigate numerically the factors influencing the laser-induced damage to semiconductors
Results
• For , in GaAs we obtain
• The calculations show that for the chosen parameters the semiconductor GaAs is electrically damaged but not optically damaged.
• We find out that the average conduction electron kinetic energy is smaller than the bangap energy at all times which means that for the given parameters of the laser field the semiconductor GaAs is structurally stable.
215108 mWI L psL 1
Gke EE
323317 103.1107.8 cmnnncm eopt
ecrelec
• Work was done in collaboration with
Dr. Danhong Huang
Dr. David Cardimona
Dr. Paul Alsing
at the United States Air Force Research Laboratory
•In the final result include the spatial dependence by taking the envelope of the electric field as a function of the spatial coordinates regarded as parameters.
•Spatial dependence of Source terms
E
trEf
E
D
E
trEftrEDtrEftrEVJ
e ,,2
,,),,(,,,,0
trEAtrEVEVtrEV LLp
TLLFT ,3
)(,3
1,, 2
020
trEVEtrEDEDtrED LLp
TekLLFT ,
3)(,
3
2,, 2
020
22
2
4
22
kGhk
ekL
kkabs
FEEE
FFS
322
1,,,
0
0*0
0
20
22
G
GG
eL
Lk E
EE
m
m
m
tzyxeF
Single photon excitation
Generalized non-local Fokker Planck equation
x
trEftrED
E
trEftrEDtretrEftrEvJ xxL
dx
,,),,(,,),,(,,,,, 0
y
trEftrED
E
trEftrEDtreJ xxLy
,,),,(,,),,(,0
Following E. Bringuier J. Appl. Phys 86, 6847, (1999) we can augment the energy –space equation by adding the spatially dependent currents on the LHS
z
trEftrED
E
trEftrEDtreJ xxLz
,,),,(,,),,(,0
trEftrEStrEftrEAE
trEf
E
D
z
J
y
J
x
J
E
J
t
trEf zyx
e,,,,,,,,
~,,2
,, 0
pxD 2F
Spectral drift velocity and diffusion coefficient at energy in coordinate spaceE
dEtrEvtrv dd
0
,,,
dEtrEDtrD xx
0
),,(),(
pEeF
00 pEE 0t
220 00
21 tEtEEtE pFpF
2200 0
211exp EOtttc pF
energy change before any collision occurs is
electron with at
the probability for the electron starting from not to experience collisions between 0 and t is
0p
t
pdttc0
'exp
first order expansion
00 p
tEepp
0
E.Bringuier Phys. Rev. B 52, 1995
00 00 pFFpF EEEtE
0
0
312
1
20
2
0
pF
pF
t
t
0
0 pgFgF
vpvEedt
dE
gpE
gpgFE
dvdS
vdSpvEv
0
0
0
00 EvEeEV dF
averages over a collision free flight
The average energy gained over a flight is (to order ) is 2E
0000
pgFg vpvEeEtE
ensemble average over a constant energy surface
Number of states in a surface element in spacepdS p
pEpvg
The position-space diffusion coefficient is related to the field dependent diffusion coefficient in energy space by
20
),,(),,(
L
Fx
e
trEDtrED
The drift velocity in real space is related to the field induced drift velocity in energy space by
LF
xd e
trEVtrEv
0
),,(),,(