The dependence of the optical characteristics on light intensity in a semiconductor

(1)

HAL Id: jpa-00210973

https://hal.archives-ouvertes.fr/jpa-00210973

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The dependence of the optical characteristics on light intensity in a semiconductor

Nguyen Ba An, Nguyen Trung Dan

To cite this version:

Nguyen Ba An, Nguyen Trung Dan. The dependence of the optical characteristics on light intensity in a semiconductor. Journal de Physique, 1989, 50 (9), pp.1009-1013.

�10.1051/jphys:019890050090100900�. �jpa-00210973�

(2)

1009

PHYSIQUE

Short Communication

The dependence ^{of the} ^optical characteristics on light intensity ⁱⁿ ^a

semiconductor

Nguyen ^{Ba An} ^and Nguyen Trung ^Dan

Centre for Theoretical Physics, Academy of Sciences of Vietnam, Nghia Do, ^Tu Liem, ^Ha Noi, ^Viet-

nam

(Reçu ^le 3 janvier 1989, accepté ^{le 2} ^mars 1989)

Résumé.

²⁰¹⁴

Nous étudions les caractéristiques optiques ^non linéaires de semiconducteurs au voisi- nage de la résonance d’absorption ^de ^deux photons par ^un biexciton. Nous montrons que ^la bistabilité

optique apparaît au-dessous de cette résonance à très faibles intensités de lumière.

Abstract.

²⁰¹⁴

The nonlinear optical characteristics of semiconductors are studied near the two-photon

biexciton resonance. Optical bistability ^{is shown} ^to happen ^below ^this ^resonance ^at very ^low light

intensities.

J. Phys. ^{France 50} (1989) ^1009-1013 ^ler ^MAI ¹⁹⁸⁹

Classification

Physics ^Abstracts

78.20 D

The Maxwell equation ^forx-axis propagation ^{of a} light ^wave ⁱⁿ ^a semiconductor is of the form :

(in ^{the unit} system ^{with A} ⁼ ^c ⁼ 137e 2/47r ⁼ 1)

where E and P are the electric vector on the light ^wave ^{and the} ^induced polarization ^{of the} ^matter.

Conventionally, ône separates P into linear and nonlinear parts ând ât ^low light intensity, ône

restricts oneself only ^to ^the ^former part ^whose perturbative corrections are the latter. However,

in many semiconductors ^like CuCI, CuBr,

^...

large optical non-linearities have been observed in the vicinity ôf ^the two-photon ^biexciton ^resonance [1-12]. ^With ^such â huge ^nonlinear response the perturbational approach ^must ^not ^be ^{valid and} ône ^has ^to develop â nonperturbational theory

wich requires considering ^both parts ^{of P} ât ^the ^same ^time ând ôn ^the ^same footing.

Let us define the nonperturbative susceptibility x ^and ^the complex dielectric function - which

are dependent ôn ^the light intensity Î = IEI2 âs ^{follows :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019890050090100900

(3)

1010

Then, équation (1) ^{becomes :}

A possible ^solution ^of equation (3) ^is

which yields ^the dispersion ^{law :}

with n and x being ^the refractive index and the extinction coefficient Denoting ^the absorption

coefficient by ^A ^we ^{have the} following relations :

Note that (6-8) ^are formally ^the ^{same as} ⁱⁿ ^the ^linear approximation, ^but ⁱⁿ ^{fact here} they ^are

the nonperturbative ^ones describing ^the optical ^nonlinear response ^of ^the semiconductor to all

perturbation ^orders, ^without distinguishing between the effects caused by ^different ^orders of x :

X(1), X(3),

^...

^and ^{all of} ^them ^are intensity-dependent.

As we are interested in the spectral région ^near ^the biexcitonic two-photon absorption ^we ^can

write for a coherent photon-biexciton system ^the following ^effective Hamiltonian [2-11] :

where _ck and b2k ^are ^bosonic operators ^{for the} photon ^with frequency ^cvk, ^wave vector k and the

(4)

and V-the volume of the crystaL ^The optical characteristics dépend ^on ^the photon density ^{p =}

tr ^{(c+ c)} ^inside ^the ^crystal ^{which in} ^turn ^{is driven} ^by the external classical light ^wave. ^The coupling

between the external driving ^field ând ^the ^field ^mode (the înside photon) îs ^described by [12] :

with Ek = Eo e ikx ^{and w -} wk . The total Hamiltonian then is

Taking ^the functional derivative of -L ^(Hext) ^with ^respect ^to the external field we could get ^P

which together ^with (2) ^would give :

Setting up ^from (9-11) ^the equations ôf ^motion ^for (c), (b) ând (c+c) ând solving ^them ^some-

how [12] ^{for the} stationary regime ^we ^have ôbtained ^the analytic expressions ôf (c) ând (c+ c) ^which

cast (12) ^{into :}

where

In (13,14) ^r-1 (y-1 ) ^{and T} ^are ^the ^biexciton (photon) dephase time and the photon depopulation

time. It is provable [12] ^{that when} ^A > ,v’3 ^{and 2w} 2wc ⁼ ^{Q -} ^r (y ⁺ Av’3) ^/ (A - ,v’3)

all the optical characteristics become three-valued functions of I in certain intensity intervals, ^Le.

optical bistability ^occurs. ^For illustration, ^we ^draw ⁱⁿ ^the figure ^the dependence of the normalized

dimensionless quantities p - 0.02g2 p/I‘y, Él - ê1/103, l2 = ê2/103, ae, ⁼ 0.1r., fi ⁼ ^O.ln and Â ⁼ 0.02A/Ç2 ^on ⁱ ⁼ 2wg2 TI/104,r ^{in the} intensity region ^where optical bistability ^takes place. ^For ^{CuCI with} ^the parameter = 5,n ⁼ ^6.3725 eV, A ⁼ ^2.76 meV, ^r ⁼ ^0.2 meV,

,y = 0.1 meV, T-1 = 1 mev, wc ⁼ ^3.18606 ^eV and g2 ⁼ ^10-21 ^eV2 ^cm3. The threshold of

the driving light intensity ^{for the} ^occurence ^of optical bistability estimated here for CuCI is about

13 w/cm2 ^{which is} very ^low (see [1, 5]).

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1012

Fig. ^{1.- The} 1 - dependences of p, £1, e2, K, fi ^and A for: w ⁼ 3186.16 meV (solid ^{curves ;} ^no bistability) ; w ⁼ ^wc ⁼ 3186.06 meV (dotted curves ; critical case) ; w ⁼ ^3185.87 ^meV (dashed

curves ; bistable case) ^and ⁼ ^3185.69 ^meV (dashed - dotted curves ; bistable case, too).

(6)

tion due to their interaction with the crystal ^valence êlectron (~ A) ^such problems âs ^the êxistence

of double or strange poles ⁱⁿ ê [3-5, 8-10] ând ^the necessity of treating ^the ^resonator [1, 5, 7] ôr ^the

local field effect [3-6] ^do ^not appear ⁱⁿ ^our approach (for ^details ^see [12]).

Acknowledgements.

The authors are deeply grateful ^to ^Prof. Nguyen ^{Van Hieu} ^and ^{Prof. Dao} Vong ^Duc ^{for their}

continuous supports in this work.

References

[1] ^KOCH S.W and HAUG H., Phys. Rev. Lett. 46 (1981) ^450.

[2] ^KHADZI ^P.I., MOSKALENKO S.A., ^ROTARU A.Kh., BELKIN S.N. and KISELEVA E.S., ^Fiz Tverd. Tela 24 (1982) ^1624.

[3] ^ABRAM ^{I. and} ^MARUANI ^A., Phys. ^{Rev. B26} (1982) ^4759.

[4] ^ABRAM ^I., Phys. ^{Rev. B28} (1983) 4433; ^B29 (1984) ^4480.

[5] ^SUNG C.C. and BOWDEN C.M., Phys. ^{Rev. A29} (1984) ^1957.

[6] ^DROR SARID, PEYGHAMBARIAN N. and GIBBS H.M., Phys. ^Rev. ^B31 (1985) ^4031.

[7] ^GRUN ^J.B., ^HONERLAGE B. and LEVY R., J. ^Lumines. ³⁰ (1985) ^217.

[8] ^HAUG H., J. ^Lumines. ³⁰ (1985) ^171.

[9] HENNEBERGER F., Phys. ^Status ^Solidi. (B) ¹³⁷ (1986) ^371.

[10] ^NGUYEN ^BA ÂN, ÎCTP Preprint ÎC/86/82 (1986).

[11] ^NGUYEN ^BA ^AN ^and ^NGUYEN ^TRUNG DAN, ^{Czech. J.} Phys. ^B39 (1989) ⁱⁿ print.

[12] ^NGUYEN ^BA ÂN ând ^NGUYEN ^TRUNG ^DAN, ^to ^be published êlsewhere.

The dependence of the optical characteristics on light intensity in a semiconductor

HAL Id: jpa-00210973

https://hal.archives-ouvertes.fr/jpa-00210973

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The dependence of the optical characteristics on light intensity in a semiconductor

Nguyen Ba An, Nguyen Trung Dan

To cite this version:

Nguyen Ba An, Nguyen Trung Dan. The dependence of the optical characteristics on light intensity in a semiconductor. Journal de Physique, 1989, 50 (9), pp.1009-1013.

�10.1051/jphys:019890050090100900�. �jpa-00210973�

1009

PHYSIQUE

Short Communication

The dependence of the optical characteristics on light intensity in a

semiconductor

Nguyen Ba An and Nguyen Trung Dan

Centre for Theoretical Physics, Academy of Sciences of Vietnam, Nghia Do, Tu Liem, Ha Noi, Viet-

nam

(Reçu le 3 janvier 1989, accepté le 2 mars 1989)

Résumé.

Nous étudions les caractéristiques optiques non linéaires de semiconducteurs au voisi- nage de la résonance d’absorption de deux photons par un biexciton. Nous montrons que la bistabilité

optique apparaît au-dessous de cette résonance à très faibles intensités de lumière.

Abstract.

The nonlinear optical characteristics of semiconductors are studied near the two-photon

biexciton resonance. Optical bistability is shown to happen below this resonance at very low light

intensities.

J. Phys. France 50 (1989) 1009-1013 ler MAI 1989

Classification

Physics Abstracts

78.20 D

The Maxwell equation forx-axis propagation of a light wave in a semiconductor is of the form :

(in the unit system with A = c = 137e 2/47r = 1)

where E and P are the electric vector on the light wave and the induced polarization of the matter.

Conventionally, one separates P into linear and nonlinear parts and at low light intensity, one

restricts oneself only to the former part whose perturbative corrections are the latter. However,

in many semiconductors like CuCI, CuBr,

large optical non-linearities have been observed in the vicinity of the two-photon biexciton resonance [1-12]. With such a huge nonlinear response the perturbational approach must not be valid and one has to develop a nonperturbational theory

wich requires considering both parts of P at the same time and on the same footing.

Let us define the nonperturbative susceptibility x and the complex dielectric function - which

are dependent on the light intensity I = IEI2 as follows :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019890050090100900

1010

Then, équation (1) becomes :

A possible solution of equation (3) is

which yields the dispersion law :

with n and x being the refractive index and the extinction coefficient Denoting the absorption

coefficient by A we have the following relations :

Note that (6-8) are formally the same as in the linear approximation, but in fact here they are

the nonperturbative ones describing the optical nonlinear response of the semiconductor to all

perturbation orders, without distinguishing between the effects caused by different orders of x :

X(1), X(3),

and all of them are intensity-dependent.

As we are interested in the spectral région near the biexcitonic two-photon absorption we can

write for a coherent photon-biexciton system the following effective Hamiltonian [2-11] :

where ck and b2k are bosonic operators for the photon with frequency cvk, wave vector k and the

and V-the volume of the crystaL The optical characteristics dépend on the photon density p =

tr (c+ c) inside the crystal which in turn is driven by the external classical light wave. The coupling

between the external driving field and the field mode (the inside photon) is described by [12] :

with Ek = Eo e ikx and w - wk . The total Hamiltonian then is

Taking the functional derivative of -L (Hext) with respect to the external field we could get P

which together with (2) would give :

Setting up from (9-11) the equations of motion for (c), (b) and (c+c) and solving them some-

how [12] for the stationary regime we have obtained the analytic expressions of (c) and (c+ c) which

cast (12) into :

where

In (13,14) r-1 (y-1 ) and T are the biexciton (photon) dephase time and the photon depopulation

time. It is provable [12] that when A &#x3E; ,v’3 and 2w 2wc = Q - r (y + Av’3) / (A - ,v’3)

all the optical characteristics become three-valued functions of I in certain intensity intervals, Le.

optical bistability occurs. For illustration, we draw in the figure the dependence of the normalized

dimensionless quantities p - 0.02g2 p/I‘y, Él - ê1/103, l2 = ê2/103, ae, = 0.1r., fi = O.ln and Â = 0.02A/Ç2 on i = 2wg2 TI/104,r in the intensity region where optical bistability takes place. For CuCI with the parameter = 5,n = 6.3725 eV, A = 2.76 meV, r = 0.2 meV,

,y = 0.1 meV, T-1 = 1 mev, wc = 3.18606 eV and g2 = 10-21 eV2 cm3. The threshold of

the driving light intensity for the occurence of optical bistability estimated here for CuCI is about

13 w/cm2 which is very low (see [1, 5]).

1012

Fig. 1.- The 1 - dependences of p, £1, e2, K, fi and A for: w = 3186.16 meV (solid curves ; no bistability) ; w = wc = 3186.06 meV (dotted curves ; critical case) ; w = 3185.87 meV (dashed

curves ; bistable case) and = 3185.69 meV (dashed - dotted curves ; bistable case, too).

tion due to their interaction with the crystal valence electron (~ A) such problems as the existence

of double or strange poles in e [3-5, 8-10] and the necessity of treating the resonator [1, 5, 7] or the

local field effect [3-6] do not appear in our approach (for details see [12]).

The dependence ^{of the} ^optical characteristics on light intensity ⁱⁿ ^a

Nguyen ^{Ba An} ^and Nguyen Trung ^Dan

Centre for Theoretical Physics, Academy of Sciences of Vietnam, Nghia Do, ^Tu Liem, ^Ha Noi, ^Viet-

(Reçu ^le 3 janvier 1989, accepté ^{le 2} ^mars 1989)

Nous étudions les caractéristiques optiques ^non linéaires de semiconducteurs au voisi- nage de la résonance d’absorption ^de ^deux photons par ^un biexciton. Nous montrons que ^la bistabilité

biexciton resonance. Optical bistability ^{is shown} ^to happen ^below ^this ^resonance ^at very ^low light

J. Phys. ^{France 50} (1989) ^1009-1013 ^ler ^MAI ¹⁹⁸⁹

Physics ^Abstracts

The Maxwell equation ^forx-axis propagation ^{of a} light ^wave ⁱⁿ ^a semiconductor is of the form :

(in ^{the unit} system ^{with A} ⁼ ^c ⁼ 137e 2/47r ⁼ 1)

where E and P are the electric vector on the light ^wave ^{and the} ^induced polarization ^{of the} ^matter.

Conventionally, ône separates P into linear and nonlinear parts ând ât ^low light intensity, ône

restricts oneself only ^to ^the ^former part ^whose perturbative corrections are the latter. However,

in many semiconductors ^like CuCI, CuBr,

large optical non-linearities have been observed in the vicinity ôf ^the two-photon ^biexciton ^resonance [1-12]. ^With ^such â huge ^nonlinear response the perturbational approach ^must ^not ^be ^{valid and} ône ^has ^to develop â nonperturbational theory

wich requires considering ^both parts ^{of P} ât ^the ^same ^time ând ôn ^the ^same footing.

Let us define the nonperturbative susceptibility x ^and ^the complex dielectric function - which

are dependent ôn ^the light intensity Î = IEI2 âs ^{follows :}

Then, équation (1) ^{becomes :}

A possible ^solution ^of equation (3) ^is

which yields ^the dispersion ^{law :}

with n and x being ^the refractive index and the extinction coefficient Denoting ^the absorption

coefficient by ^A ^we ^{have the} following relations :

Note that (6-8) ^are formally ^the ^{same as} ⁱⁿ ^the ^linear approximation, ^but ⁱⁿ ^{fact here} they ^are

the nonperturbative ^ones describing ^the optical ^nonlinear response ^of ^the semiconductor to all

perturbation ^orders, ^without distinguishing between the effects caused by ^different ^orders of x :

^and ^{all of} ^them ^are intensity-dependent.

As we are interested in the spectral région ^near ^the biexcitonic two-photon absorption ^we ^can

write for a coherent photon-biexciton system ^the following ^effective Hamiltonian [2-11] :

where _ck and b2k ^are ^bosonic operators ^{for the} photon ^with frequency ^cvk, ^wave vector k and the

and V-the volume of the crystaL ^The optical characteristics dépend ^on ^the photon density ^{p =}

tr ^{(c+ c)} ^inside ^the ^crystal ^{which in} ^turn ^{is driven} ^by the external classical light ^wave. ^The coupling

between the external driving ^field ând ^the ^field ^mode (the înside photon) îs ^described by [12] :

with Ek = Eo e ikx ^{and w -} wk . The total Hamiltonian then is

Taking ^the functional derivative of -L ^(Hext) ^with ^respect ^to the external field we could get ^P

which together ^with (2) ^would give :

Setting up ^from (9-11) ^the equations ôf ^motion ^for (c), (b) ând (c+c) ând solving ^them ^some-

how [12] ^{for the} stationary regime ^we ^have ôbtained ^the analytic expressions ôf (c) ând (c+ c) ^which

cast (12) ^{into :}

In (13,14) ^r-1 (y-1 ) ^{and T} ^are ^the ^biexciton (photon) dephase time and the photon depopulation

time. It is provable [12] ^{that when} ^A > ,v’3 ^{and 2w} 2wc ⁼ ^{Q -} ^r (y ⁺ Av’3) ^/ (A - ,v’3)

all the optical characteristics become three-valued functions of I in certain intensity intervals, ^Le.

optical bistability ^occurs. ^For illustration, ^we ^draw ⁱⁿ ^the figure ^the dependence of the normalized

,y = 0.1 meV, T-1 = 1 mev, wc ⁼ ^3.18606 ^eV and g2 ⁼ ^10-21 ^eV2 ^cm3. The threshold of

the driving light intensity ^{for the} ^occurence ^of optical bistability estimated here for CuCI is about

13 w/cm2 ^{which is} very ^low (see [1, 5]).

Fig. ^{1.- The} 1 - dependences of p, £1, e2, K, fi ^and A for: w ⁼ 3186.16 meV (solid ^{curves ;} ^no bistability) ; w ⁼ ^wc ⁼ 3186.06 meV (dotted curves ; critical case) ; w ⁼ ^3185.87 ^meV (dashed

curves ; bistable case) ^and ⁼ ^3185.69 ^meV (dashed - dotted curves ; bistable case, too).

tion due to their interaction with the crystal ^valence êlectron (~ A) ^such problems âs ^the êxistence

of double or strange poles ⁱⁿ ê [3-5, 8-10] ând ^the necessity of treating ^the ^resonator [1, 5, 7] ôr ^the

local field effect [3-6] ^do ^not appear ⁱⁿ ^our approach (for ^details ^see [12]).

The authors are deeply grateful ^to ^Prof. Nguyen ^{Van Hieu} ^and ^{Prof. Dao} Vong ^Duc ^{for their}

[1] ^KOCH S.W and HAUG H., Phys. Rev. Lett. 46 (1981) ^450.

[2] ^KHADZI ^P.I., MOSKALENKO S.A., ^ROTARU A.Kh., BELKIN S.N. and KISELEVA E.S., ^Fiz Tverd. Tela 24 (1982) ^1624.

[3] ^ABRAM ^{I. and} ^MARUANI ^A., Phys. ^{Rev. B26} (1982) ^4759.

[4] ^ABRAM ^I., Phys. ^{Rev. B28} (1983) 4433; ^B29 (1984) ^4480.

[5] ^SUNG C.C. and BOWDEN C.M., Phys. ^{Rev. A29} (1984) ^1957.

[6] ^DROR SARID, PEYGHAMBARIAN N. and GIBBS H.M., Phys. ^Rev. ^B31 (1985) ^4031.

[7] ^GRUN ^J.B., ^HONERLAGE B. and LEVY R., J. ^Lumines. ³⁰ (1985) ^217.

[8] ^HAUG H., J. ^Lumines. ³⁰ (1985) ^171.

[9] HENNEBERGER F., Phys. ^Status ^Solidi. (B) ¹³⁷ (1986) ^371.

[10] ^NGUYEN ^BA ÂN, ÎCTP Preprint ÎC/86/82 (1986).

[11] ^NGUYEN ^BA ^AN ^and ^NGUYEN ^TRUNG DAN, ^{Czech. J.} Phys. ^B39 (1989) ⁱⁿ print.

[12] ^NGUYEN ^BA ÂN ând ^NGUYEN ^TRUNG ^DAN, ^to ^be published êlsewhere.