The splitting of the bands of semiconductors under laser irradiation

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The splitting of the bands of semiconductors under laser irradiation

Nguyen Vinh Quang

To cite this version:

Nguyen Vinh Quang. The splitting of the bands of semiconductors under laser irradiation. Journal

de Physique, 1982, 43 (1), pp.113-116. �10.1051/jphys:01982004301011300�. �jpa-00209367�

The splitting ^{of the bands of} semiconductors under laser irradiation

Nguyen ^Vinh Quang

Laboratory ^of Theoretical Physics, ^{Institute of} Physics ^{Centre for} Scientific Researchs of Vietnam, Hanoi, Vietnam (Rep ^le 19 juin 1981, accepté ^{le 17} septembre 1981)

Résumé.

²⁰¹⁴

La variation du spectre ^{de bandes de} semiconducteurs

sous

irradiation _par laser, ^{est étudiée} près ^{de la}

résonance

avec

la bande interdite

en

tenant compte ^{de la} dégénérescence ^de la bande de valence. On ^montre que la

dégénérescence à p

⁼

^{0 est} levée dans le

cas

où l’on est à la résonance,

^ou

proche de la résonance pour les transitions à

un

photon.

Le calcul numérique, pour les semiconducteurs de structure blende de zinc, ^montre que le dédoublement

a une

valeur très appréciable ^et peut ^donc être observé expérimentalemenl ^Le dédoublement

ne se

produit pas dans le

cas

d’une résonance à plusieurs photons.

Ce dédoublement peut être considéré

comme une

généralisation ^des gaps induits _{par le} champ (effet ^Franz-Kel- dysh), ^calculé

^en

^tenant compte ^{de la} dégénérescence de la bande de valence.

Abstract.

²⁰¹⁴

The variation of the two-band spectrum of semiconductors under laser irradiation is investigated

in the

resonance

approximation taking ^{into account the} degeneracy of the valence band. It is shown that in the

quasienergy spectrum ^the degeneracy ^at p

⁼

0 is removed in both exact and

near

one-photon

resonance cases.

Numerical estimate for the zinc-blende structure semiconductors shows that the splitting is essential and, therefore, could be observed experimentally. ^{In the} multi-photon

resonance cases

the splitting ^{is absent.}

The splitting ^{could be} considered

as

extended type ^{of the} field-induced-gaps ^and high-frequency Franz-Keldysh

effect calculated when taking ^{into account the} degeneracy of the valence band.

Classification

Physics ^Abstracts

^,

71.70

^-

78.20B

^-

78.20J

1. Introduction.

^-

In recent years the quasi- energy [1, 2] spectrum ^of crystals placed ^{in a} strong laser field was extensively ^studied [3, 14]. ^{The authors}

of references [3-5] ^have established the band picture

of the electron in the periodic crystal potential ⁱⁿ

the presence of electromagnetic ^{wave. In.} these articles the periodic potential ^of

^a

crystal ^{was considered} explicitly, ^{on an} equal footing ^{with an} external field.

In [6-14] ^{the variation of the} already ^known crystal

electronic spectrum ^{under the influence of radiation}

was investigated only ^{in the cases of two} nondegene-

rate bands. In semiconductors, however, the valence band is often degenerate. The purpose of this paper is to study the effects of a strong electromagnetic ^wave

on the degeneracy ^{of the valence band.}

2. Theory.

^-

^{As in} [12] ^{we determine}

^a

^{state of the} crystal electron in the laser field

from the time-dependent Schrbdinger equation

where m, e and p ^{are the mass,} charge ^{and the momen-}

tum operator ^{of the} electron, respectively; V(r), ^the periodic potential ^{of the} crystal.

First, ^we discuss in detail a model of the direct band gap semiconductor, in which conduction (c), heavy-hole (h) ^and light-hole (1) ^{bands are not} degene-

rate, but ^at the centre of Brillouin zone there is the

degeneracy ^{due to the contact of the} heavy-hole ^and

light-hole ^bands. It is natural to assume that the optic

transition between conduction and one of the hole- bands is allowed but between hole-bands is forbidden.

Consider the one-photon ^{resonance case, when}

(o ⁼

EG, EG being ^band gap.

In the resonance approximation [6-10, 12-14] ^the

set of equations ^for defining coefficients an p(n : ^{{ c, h, 1 ) )}

in the expansion ^of 03C8(r, t) ^{into set of} eigenfunctions

in the vicinity ^{of the} point p

⁼

^{0 is :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004301011300

114

where

J 1 is the Bessel function of order 1 ; En

^--_

En (p) being

the dispersion relations of the n-band. In obtaining

the set (3) ^{the term e2 A} ’(t)12 m ^is omitted, because its role in the quasienergy spectrum ^{is not essential} [12].

The solution of equations (3) ^{can be searched in} the form :

It is not difficult to see that A, B, C, a, [3, ^y satisfy ^the following ^{set of} equations :

That means that the problem ^of solving ^{the set} (3) formally ^{can be reduced to the} eigenvalue problem.

Due to the hermiticity of the matrix on the left side the values of

a

must be real. Equation ^for defining

^a

^{is :}

Using ^the completeness ^{of the} steady ^solutions (r, t) [1, 15-17] ^{it is not difficult to} prove that equation (6)

has to have 3 different roots.

We note that in the vicinity ^{of the} point p

⁼

0 this conclusion can be also obtained by ^another method, considering the discriminant of the Cardano solutions of equation (6) [18]. ^In the zero-field limit the roots

become 0,

^-

6,, - bh ^{and for} definiteness we put

Accordingly, ^{each of} the P, ^{y, A, B,} C has 3 different values. First equation ^{of the set} (3) gives 3 different solutions anp(t). ^Hence, in the field of the resonance wave 3 different stationary ^states

are replaced by ^{3 different} orthonormalized quasi- stationary ^states [1] ^functions Houston-Volkov :

where

Ni ^are determined by ^the normalization condition of the wave function t/ljp(r, ^t).

At p # ^{0 in the} zero-field limit 03C8 1p(r, ^t), 02p(r, t), t/I 3p(r, ^{t) tend} to 03C8cp(r, ^t), t/I?p(r, ^t), 03C8hp(r, ^t) respectively.

At the degenerate point p

⁼

0 when Ao - ^0, 03C8jp(r, ^t)

approache ³ different linear combination of the functions 0,0,p(r, ^{t). That is in analogy with the degene-}

rate case of the time-independent perturbation theory.

Corresponding ^to the definition of the quasienergy [1] J ^{there are 3 different} dispersion ^{relations of the}

reduced [15] quasienergy :

That means that in the quasienergy spectrum ^the degeneracy ^at p

⁼

0 is removed The splitting ^{is :}

From group theory points of view it is connected to the fact that the states with definite values of the quasi-

energy ^must be classified according ^to the represen- tations of the group H(t)

⁼

H(t ⁺ T) ; ^T

⁼

² nlw [1 ].

Then the degeneracy ^{of each level is} varied, ⁱⁿ general.

Let us investigate ^the splitting ^{in the} zinc-blende

structure semiconductors. In this case the states on

each of the _c, h, ^{I bands are} double-degenerate ^{and at}

p

⁼

0 they ^are transformed according ^to represen- tations F6(2), F8(4) respectively. ^The explicit expression

of the wave function and the selection rule for the momentum operator ^{can be} found, ^for example, ⁱⁿ [19].

By repeating ^the above discussions it is not difficult to see, that at p

⁼

0 there are 3 different quasienergy values, each of which is double-degenerate, ^{i.e. the} 4-multiple degenerate ^level splits ^{into two double-} degenerate ^{ones. The wave} polarization ^{has no}

influence on the quasienergy ^values and, therefore, ^on

the splitting, ^{which is :}

Using ^{the band} parameters given ⁱⁿ [20], for the value of the field intensity Eo

⁼

⁷

^x

¹⁰⁴ V/cm ^{we obtain the} following values of the splitting (see ^Table I).

Table I.

It is evident that the splitting is essential and could be observed experimentally.

The case of the diamond structure semiconductors

can be solved analogously.

For completeness, ^we consider the zinc-blende

structure without taking ^{into account} spin-orbit coupling. ^Then the conduction band is not degenerate [r 1(1)] and the valence band is triple-degenerate [r 15(3)]. ^{It could} be shown that at each point p there

are 3 different quasienergy ^{values, one} of which is

double-degenerate, ^{i.e. the} degeneracy ^is partly ^remov-

ed The splitting ^at p

⁼

0 is

In the nearby ^{resonance cases, when}

repeating ^{the formalism} just developed ^{above we can}

obtain the analogous results. As the splitting ^occurs

not only ^when

^m

> EG but also when cv EG ^it

could be considered as a extended type ^{of the field-} induced-gap ^effect [3-14], firstly shawn in reference [6]

for one-photon resonance case when

m

> EG.

Since _p

⁼

0 is a particular point ^{of the} type So [12]

o

in the multi-photon resonance cases nw

⁼

EG; n > t.

the splitting is absent in the resonance approximation.

It is useful to note that together ^{with the} splitting

there are the opposite ^shifts of the C-band and 1-band

edges (9), ^{in the} same sense as discussed by ^Wautelet

and Laude [5]. These shifts are equal ^{to the} splitting only ^{in exact} resonance case. It should be emphasized

that the quasienergy picture, in which there ^are field-

induced-gaps ^{and the} splitting qualitatively agrees with the picture described in references [3-5] by

another method

3. Discussion.

^-

Due to the significance ^{of its size,}

the splitting ^is important ^{in the} theory ^of generation

of

a

strong ^{field in} semiconducting ^lasers [21].. ^This splitting ^could play ^{an essential} role in the resonance

electronic Raman scattering processes (RERS), ^where

incident laser light ^has frequency ^{close to the band gap.}

In studying ^the absorption of the additional weak

wave [11, 22-24] ^the splitting ^{could be} considered ^as

the High-frequency Franz-Keldysh ^effect [25-27] ^cal-

culated in the two-band approximation ^{with the} dege-

neracy of the valence band taken into account

Acknowledgments.

^-

The author would like to

thank Prof Nguyen ^van Hieu for his valuable advises and stimulating discussions.

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