Intensity-dependent absorption coefficient and refractive index near the band gap of highly excited semiconductors

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Intensity-dependent absorption coefficient and refractive

index near the band gap of highly excited

semiconductors

Nguyen Ba An

To cite this version:

1

LE

JOURNAL

DE

_PHYSIQUE

Short Communication

Intensity-dependent absorption

coefficient and

_refractive

index

near

the band

_gap

of

_highly

excited

semiconductors

Nguyen

Ba An

Centre for Theoretical

_{Physics, Academy}

of Sciences of

Vietnam,

Nghia

Do,

Tu

Liem,

Ha

Noi,

Viet-nam

(Reçu

le

_{10 juillet}

1989,

accepté

le 25 octobre

₁₉₈₉₎

Résumé. - Nous étudions la

_dépendance

du coefficient

_{d’absorption}

03B1 et de l’indice de réfraction n

en fonction de l’intensité de lumière incidente au

_voisinage

de la résonance

_excitonique.

Le

comporte-ment

_{non-bosonique}

des excitons _peut constituer un mécanisme de non-linéarité

_optique

conduisant à des formes bistables de 03B1 et n au-dessus de cette résonance.

Abstract. 2014 The

dependence

of

_absorption

coefficient 03B1 and refractive index n on the

_intensity

of incident

_light

are studied near the exciton resonance. The non-boson behaviour of excitons _may serve as a mechanism

_{of optical nonlinearity leading}

to bistable

_shapes

of 03B1 and n above the resonance.

1

_Phys.

France 51

₍₁₉₉₀₎

1-4 ler JANVIER

1990,

Classification

Physics

Abstracts 71.35-78.20D-78.90

1Wo

_important

branches of research of

_optical

_properties

in

_highly

laser-excited

semiconduc-tors near the

_absorption

_edge

are the derivation’of the

_dependence

of the dielectric function E

on exciton

_density.

_{[1, 2]}

and the determination of the conditions for the occurrence of

_density

bistability (DB) [3-5].

As the

_intensity

I of incident

_pumping

laser

_is,

_among other

_quantities,

an

experimentally

controlable

_parameter,

it is reasonable and even _necessary to have concrete

opti-cal characteristics

_dependent

_directly

on I rather than on o-. The aim of this communication is to

study

such a

_dependence

which in a sense can be seen as a

_{bridge connecting}

the two branches of research mentioned above. Similar task has been done

_recently

in the

_{spectral region}

near the

two-photon

biexciton resonance

_[6].

Let us start from an effective bosonic Hamiltonian of Hanamura

_[7]

_describing

a

_system

of many

interacting

excitons

placed

under the action of an

_externally

driven monochromatic classical

2

laser field with

_{frequency w : (h}

= c = 1

being

used

hereafter)

where

Ek

_(t)

the electric field

_strength

of the radiation inside the

_crystal,

_at

the creation

_operator

of an exciton with wave vector _{p and energy}

_{wxp, V}

the

_sample

_volume,

d the matrix element of

dipole

moment, w =

267r Ebr3 /3, U

=

7r-1/2r-3/2@ À

=

77r,-3

with

Eb and

r

_being

the

_binding

energy and radius of an exciton. The dielectric function

_(£00

the

_background

dielectric

_constant)

can be derived from the

_knowledge

of the

_averaged

_(over

the

_eigenstate

of

_H)

value of the

polar-ization

_{operator Pk}

which within the Hartree-Fock

_{approximation}

is of the form :

Using (1)

to set

_up

the

_equations

of motion for ak and

a+ k

which will then be solved in the

frame-work of the Hartree-Fock

_{approximation}

for a

_{stationary regime. Putting}

the solutions of ak and

a+ k

into

₍₃₎

and then

₍₃₎

into

₍₂₎

we obtain :

where

IIcv

the interband matrix element of the wave-vector

_operator,

e and m the free electron

charge

and mass

_{and y}

the exciton inverse

_dephase

time

_being

_{phenomenologically}

introduced. The

_absorption

coefficient a and the refractive index n of CdS obtained from

₍₄₎

are

_plotted

as

functions of x = w

_/

_wX in

_figures

1 and

_2,

_resp., for 0’ =

0 ;

9 x

1015

and 1.8 x

1016

cm- 3 .

The

Fig.l.

Fig.2.

Fig.1 - Absorption

coefficient as functions of x.

used

_parameters

for CdS are _£00 =

7.6,

_WX = 2.5528

eV, Eb

= 32.9

meV, r

= 25.5 x

10-8

_cm, me

= 0.205 _{m, mh} = 1.348 _m,

_Eg

_{(band gap)}

= 2.5857 eV and

y-

1 = 7

ps.

We see that the

_peaks

of the curves are

_shifting

to

_{higher frequency}

_region

for

_increasing

exciton

_density.

This is due

to the blue shift of the exciton _energy level which in turn is caused

_by

the non-boson character of excitons

_(w

=

267rE br3/3

is the contribution of

_purely

kinematical exciton-exciton

interactions ;

the

_dynamical

ones vanish in the k - 0 limit used here. For details see

_{[7]). As}

a matter of

fact,

the excitons are

_generated

_optically

and a must be

_governed

_by

the

_input

laser

_intensity

I =

IEI2

/T with

T = 1 -

1 -

qi]

/

] l +

qi]

being

the transmition coefficient at the

_{vacuum-crystal}

boundary

surface.

_{Thus, a}

and n as functions of I

_{depend parametrically}

on a. The

(cr- I ) -relation

is followed from the balance condition in a

_stationary

situation :

where E2 = lm E and T the exciton life time

_equal

to about 0.5 ns for CdS. As

_T,

c2 and n are all

dependent

on cr

_{equation (5)}

becomes a _very

_complicated

one

_stronly

nonlinear in u. The

depen-dences of cr on I are

_numerically

evaluated for CdS and

_represented

in

_figure

3 which show the

Fig.3 -

Exciton

_density

as functions of I.

occurrence of DB for x > 1.0024. The threshold

_intensity

for the DB is of the order of about 40

w/cm2 .

Figures

4 and 5

_display

a and n as functions of I for several values of x.

_According

to

_figure

Fig.4.

Fig.5.

Fig.4 - Absorption

coefficient as functions of I.

4

3 the behaviour of a and n is bistable for x >

1.0024,

too. Note that x > 1.0024 means

_{(w - wx)}

_>

6.4 meV which is well smaller than

_{(Eg -}

_wx)

= Eb =

32.9 meV This makes

_{possible experimental}

measurements in the exciton resonance

_{spectral region.}

Theoretical curves of CdS for the

_intensity

of reflected beam

Ir

=

(1 -

T ) I

versus the

_pumping

_intensity

I are drawn in

_figure

6 as a

_possible

reference for

_{experimentalists.}

Fig.6 -

Reflected

_intensity

vs. incident

_intensity.

Acknowledgements

.

The author thanks Academicians

_Nguyen

Van Hieu and Dao

_Vong

Duc for their constant

_supports

in this direction of research.

References

[1]

MANZKE

G.,

MAY

_{V ,}

HENNEBERGER

_K.,

_Phys.

Status Solidi b 125

₍₁₉₈₄₎

693.

[2]

BOLDT

F.,

HENNEBERGER

_K.,

MAY

_V,

_Phys.

Status Solidi b 130

₍₁₉₈₅₎

675.

[3]

HOANG XUAN

_NGUYEN,

ZIMMERMANN

R.,

Phys.

Status Solidi b 124

₍₁₉₈₄₎

191.

[4]

HENNEBERGER K. et

_al.,

_Physica

138A

₍₁₉₈₆₎

557.

[5]

NGUYEN BA

_AN,

_Phys.

Status Solidi b 150

₍₁₉₈₈₎

845.

[6]

NGUYEN BA

_AN,

NGUYEN TRUNG

_{DAN, J.}

_Phys.

France 50

₍₁₉₈₉₎

1009.