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

2

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

qq

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

),,(),,(