Athermal and thermal relaxation of high density electron-hole plasma ...

Physica 134B (1985) 394-398 North-Holland.Amsterdam

394

ATHERMAL AND THERMAL RELAXATION OF HIGH DENSITY ELECTRON-HOLE PLASMA IN GaAs 3acques COLLET, Thierry AMAND Laboratoire de Physique des Solides, associ~ au CNRS, INSA, Avenue de Rangueil 31077 Toulouse-Cedex (France) We develop a model calculation to investigate the electron-hole plasma dynamics in GaAs under subpicosecond excitation during the very first athermal regime. We also report on an experimental study of the plasma relaxation at high density at room temperature in thermalized conditions.

I) the electron distribution function fc(k)

1. ATHERMAL REGIME subpicosecond

2) the hole distribution function f (k) v 3) the optical phonon distribution function b(q)

laser pulse (typically a 100 fs laser pulse) with a

The discussion of more general treatments is

We

develop

investigate

first

the

a

model

interaction

of

calculation a

to

deferred to the end of this section. We restrict

direct gap semiconductor. Because

of

the

selection

electron-hole pairs created have almost

the

rules,

all

the

in the material, do

same kinetic

energy, so that

neither the electron distribution function nor the hole

distribution

function

internal carrier the

at

band.

each

the

Fermi

like.

distribution

pumping

kinetic

will

exhibit

energy

in

a

each

maximum intensity of these peaks . the conduction (respectively fc max and f v max m efficiency

processes

of

and

phonon-carrier intense 10

to

can

materials

a

be by

spatially

easily

the on

the

coupling.

enough, rise

absorption,

to i.e.

an

It

is

investigation

on

the

during

with this

be

athermal

quantitatively,

system

composed

of

the

very

the of

three

the

of

requires

regime.

To

choose

which material

clear

interaction

we

condition

fullfilled, of

increase

semiconductors

regime

of

following

can

transmission. pulse

intensity

saturation to

scattering

Provided the pumping is

the

(fv max + fcmaX)

gives

carrier-carrier

sample

that

any

the

laser

calculations

approach to

transport

homogeneous plasma

generated

means of

two

in

direct

gap

photon absorption.

Under this assumption, the kinetic equations read : d (1) ~

Ic(K)~-t-fl +~'t-t f l +~-t-fl +gc(t) = cv cc c,LO

(2) ~d - fv(K) = ~__~.f ]

The

band and valence band) will depend critically on the

study

which

No

temperature can be defined. On

contrary~

peak

are

our

this

solve

a

equations

which describe the kinetics of • 0378-4363/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

+ ~__~..flvc + ~--t" f Iv,LO + gv (t)

vv

d b(q) d ~ b +~(3) ~ = LO,v LO,c Equation

(3)

included

to

account

equilibrium center.

relative

us

different

terms

side

the

represents under

the

df l

is

which first

the

influence

rate

of the

must

generation the

the

the

The

rate

of

collisions

zone

of

the

right

hand

df Ic,v

term the

be

of non-

Brillouin meaning

compose

scattering

~

electrons

with

holes.

rate due to df I%LO is collisions. The term ~-~

rate

emission.

generation

the

phonons

equation.

similary

scattering

phonon

LO

near

clarify

c,c electron-electron the

for

phonons

Let

of

to

~"LO(q)

scattering

resulting

Finally, by the

from

gc(t)

laser

is

the the

optical electron

pulse. For equations

2 and 3, the right hand side terms have similar

J. Collet and T Amand / High density electron-hole plasma in GaAs

meanings.

For example the scattering term df ~ I ~ reads :

carrier-carrier

side

395

of

equations

(1, 2, 3).

plasma,

screening

is

In

a

well

high

density

described

in

the

frame of the random phase approximation (RPA),

aTd

L;

=',q,

so

eIl

that

it

would

be

natural

to

use

the

corresponding dielectric function ~ RPA(q,~o) to (K) + E ~ (K')-E~

screen all the interactions. Unfortunately, such a

E

general approach is impossible for

x(f

(l_f

-

because of

f

system

of

coupled

integro-differential

complicated

analytically

as

in

thermodynamic authors

(4)

a

equilibrium.

have attempted

solution

to

exclude

the

cannot

regime

the

cost

solved

from

Nevertheless,

to

force

of

analysis

be

far

many

an analytical

approximations

of

high

the

density

that

effects

static

approximation

assuming

function

As

cas

collision integrals

be

are

seen

in

we

with

LO

carriers

assumed

interaction

is

approximation

to

be

always also

conveniently

of

(l,2,3)

to

chose the

function

over

variation

of

In

this

work,

we of

the

functions

for

distribution

equal to the

approximations is assumed, so that

a

energy

numerical

energy plotted due

the system

figure (Ia))

used

a

be

the

Now

the at

physics

is now

figures the

from

(1)

are

distribution

case,

namely

of

the

mean kinetic

in

figure

(2)

shows that

under

to

screening

and

LO

phonon

estimated

when

t ~

as

0.8 ps)

follows : electrons

a temperature T~/ LO-phonon

In are

335 K.

occupation

corresponding to this temperature is b(33$)A/ 0.35. just

the

In of

to the

ranges

extreme

thermalized

this

all

plasma

of

decay

simply

to numerically solve the transport equations. In approach,

¢ MW/cm 2

reabsorption. The amplitude of this reabsorption can

standard fifth order predictor-corrector technique kinetic

half

strong excitation, kinetic energy is released more

mentioned

We

energy : at

The

The

plasmas.

from

kinetics

quasi-thermalized to

degenerate

pulse width

broadening

3 x 1017 cm -3

the

of

the

equations

photon

ranging

(1, 2, 3) will be investigated for the first time in case

apply

the excitation pulse we

electron-hole

simplifies

above

For

to

transport

The corresponding density

the

consider the

chose

of the

generated

slowly

None

We

powers

to

optical phonon. approach.

excitation

1016 cm -3

an energy interval

Coulomb

height "- 85 fs ;

displayed

the

the

following specifications-

half

I$0 MW/cm 2, slow

both

GaAs.

this

exchange

calculations. a.3) Very

s~e~ned

resolution

and

The

dielectric

and LO phonons have been computed for several

scattering

equal.

this

(4))

formula

complicated)

With

phonons.

numerical

l$0 MW/cm 2. direct

/q2

height : 6 meV. The kinetics of electrons, holes

always handily simplified

and

q is

(i.e.

under this assumption a.2) Exchange

dielectric

interaction and the Fr61ich interaction of carriers

We point out the "most-standard" approximations : l-f0((K) N I).

the

that cO= 0 and that

RpA(q,0) = t + q2

hv ° = 1.61 eV ;

assumption

of

small. In these conditions) ~ RPA reduces to =

at

degeneracy

So we

ourselves to a simplified approach, i.e.

the

(degenerate plasma, state filling saturation) ...). a.1) Non

the moment,

computers.

to

equations such as (1, 2, 3) with collision integrals as

slowness of

resign function, A

the

peak of this

located in the matrix elements of the different

wavelength

collision integrals which compose the right hand

short

the

phonon distribution

time 0.$ ps

reaches

q N 2 x 10 6 cm -I

calculation,

it

turns

0.25

(Fig. lc). out

that

at After

b(q) the a

phonon

396

J. Collet and T Amand /High density electron-hole plasma in GaAs

I 11i ~ . ~ ~P/~~40Mw/cm~

16'-'~---

'

r/',"

emission

up

to

< 5 1 0 6 c m - I ) . So the accumulation

phonons

near

part

the

['~

point

plays

a

in slowing down the transfer of

the plasma kinetic energy to the lattice.

Id(

, /,i. 0

, (?

50

100

~ 10

150- e V m

hole

q,,,

~10 ~

1ii ,6' li \

50

~ IOO~/~