ARTICLE IN PRESS

0921-4526/$ - se

doi:10.1016/j.ph

�Correspondifax: 00212 56 50

E-mail addre

(A. Nougaoui).

Physica B 370 (2005) 178–185

www.elsevier.com/locate/physb

Hydrostatic stress dependence of the exciton–phonon coupledstates in cylindrical quantum dots

A. El Moussaouya, D. Briab, A. Nougaouia,�

aLaboratoire de Dynamique et d’Optique des Materiaux, Departement de Physique, Faculte des Sciences, Universite Mohamed I,

B.P. 524, 60000 Oujda, MoroccobInternational Centre for Theoretical Physics ICTP, Trieste Italie

Received 10 June 2005; received in revised form 14 September 2005; accepted 14 September 2005

Abstract

We investigate theoretically the effects of compressive stress on the binding energy of an exciton in a cylindrical

quantum dot (QD) using a variational procedure within the effective mass approximation. The stress was applied in the

z direction and the interaction between the charge carriers (electron and hole) and confined longitudinal optical (LO)

phonon modes was taken into account. Specific applications of these results are given for GaAs QDs embedded in a

Ga1�xAlxAs semiconductor. The result shows that the binding energy and the polaronic correction increases linearly

with increasing stress. Moreover, we obtain the binding energy and the polaronic contribution in the limit in which the

QD turns into a quantum well.

r 2005 Elsevier B.V. All rights reserved.

PACS: 71.35.�y; 71.38.�k; 68.65.Hb

Keywords: Quantum dots; Exciton; Exciton–phonon; Pressure effect

1. Introduction

Semiconductor technology makes it possible togrow quantum dots (QDs), quantum well wires(QWWs), and quantum wells (QWs) structures inwhich carriers confinement is achieved in one, two,

e front matter r 2005 Elsevier B.V. All rights reserve

ysb.2005.09.008

ng author. Tel.: 00212 56 50 06 01;

06 03.

ss: nougaoui@sciences.univ-oujda.ac.ma

and three spatial dimensions. These systemsprovide many surprising properties that havenever been observed in the bulk materials andhave many device applications [1–3]. Excitonicstates play an important role for the photolumi-nescence and absorption spectra of low-dimen-sional systems [4–9].

The problem of stress influence on opticalproperties of low-dimensional systems is very im-portant from both fundamental and technical pointsof view. Previous theoretical and experimental

d.

ARTICLE IN PRESS

A. El Moussaouy et al. / Physica B 370 (2005) 178–185 179

investigations on the effect mostly considereddirect optical transitions between valence- andconduction-band states. Using the effective massapproximation, Gil et al. [10] have showed theimportance of an accurate theoretical treatment ofthe Coulomb interaction, including intersubbandmixing of valence wave functions, to describesubband-to-subband transition energies and oscil-lator strengths in asymmetrical GaAs–(Ga,Al)Asdouble quantum wells under an uniaxial stress.The authors in Ref. [11] have found that theabsorption of light with polarization parallel to theheterointerface in uniaxially stressed GaAs=Ga1�xAlxAs heterostructures may be sensitive tothe direction of light polarization. Rau et al. [12]have investigated both theoretically and experi-mentally the mixing between excitonic statescaused by valence-band and have analyzed howthis mixing is modified by stress in GaAs=Ga1�xAlxAs QWs. Elabsy [13] has calculated theeffect of hydrostatic pressure on the bindingenergy of donor impurities in QW heterostructure,and found that the binding energy increases withincreasing external hydrostatic pressure for a givenQW thickness. Using a variational techniquewithin the effective-mass approximation, Oyokoet al. [14] have shown that donor binding energyincreases with increasing uniaxial stress anddecreasing size of the QD.

It is well known, in our previous work [15], thatthe exciton–longitudinal optical (LO) phononcoupling affects the excitonic energy levels. InRefs. [16,17], the authors observed the Ramanspectra and the optical phonons energies inGa1�xAlxAs bulk material under applied stress.Till now, a theoretical study for the excitonsbinding energies in a QD system with considera-tion of both the confined LO phonons contribu-tion and pressure effects has not been given.

The aim of the present report is to study thepressure dependence of the exciton binding en-ergies in a CdTe cylindrical QD’s of radius R andheight H placed in Cd1�xZnxTe matrix. Thequantum confinement induced by the barriermaterial is described by a finite potential well inthe z direction and the interaction of the chargecarriers with LO phonon modes is included. Theexciton binding energies are calculated within the

variational approach in the effective mass approx-imation. The work is organized as follows. After abrief introduction, we present in Section 2 ourtheoretical framework, in Section 3 we give ournumerical results and discussions, and finally inSection 4, we summarize by a conclusion.

2. Formalism

Let us consider an exciton which is confinedperfectly in a cylindrical QD of radius R andheight H ¼ 2d embedded in a barrier materialsemiconductor. Within the framework of theeffective mass and non-degenerated-band approx-imations, the exciton Hamiltonian versus pressurecan be written as

HðPÞ ¼ HexðPÞ þHLOðPÞ þHex�LOðPÞ, (1)

where Hex is the excitonic Hamiltonian which, inthe absence of LO phonons modes, is given by

Hexðre; rh;PÞ

¼ �_2

2m�e ðPÞr2

e �_2

2m�hðPÞr2

h �e2

e0ðPÞjre � rhj

þ V ewðre;PÞ þ Vh

wðrh;PÞ, ð2Þ

where m�e and m�h are the effective masses of theelectron and hole, respectively, and re ¼ ðre; zeÞand rh ¼ ðrh; zhÞ are the spatial coordinates of theelectron and hole in cylindrical frame, respectively.�e2=ðe0ðPÞjre � rhjÞ is the Coulomb potential,where e0ðPÞ is the static dielectric constant atpressure P.

V ewðre;PÞðV

hwðrh;PÞÞ is the corresponding elec-

tron (hole) confining potential at pressure P:

Viwðri;PÞ ¼

0 if ripR et jzijpd;

V iðPÞ if ripR et jzijXd;

1 if riXR:

8><>: (3)

We can write the expressions of V eðx;PÞ andVhðx;PÞ as follows [14]:

V eðx;PÞ ¼ 0:658DEgðx;PÞ and

Vhðx;PÞ ¼ 0:342DEgðx;PÞ,

where

DEgðx;PÞ ¼ DEgðx; 0Þ � Pð1:3� 10�3Þx. (4)

ARTICLE IN PRESS

A. El Moussaouy et al. / Physica B 370 (2005) 178–185180

DEgðx; 0Þ can be expressed as

DEgðx; 0Þ ¼ EgGað1�xÞAlðxÞAsðxÞ � EgGaAsðxÞ

¼ 1:155xþ 0:37x2. ð5Þ

In the presence of the LO phonon modes, theCoulomb potential will be taken as

�1

e1ðPÞe2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2eh þ ðze � zhÞ2

q ,

where e1ðPÞ is the high-frequency dielectric con-stant.In Eq. (1), HLOðPÞ and Hex�LOðPÞ are respec-

tively, the Hamiltonian operator for confined LOphonon and the exciton–LO phonon interaction.

HLOðPÞ ¼Xl;n1

_oLOaþln1aln1, (6)

aþln1ðaln1Þ are creation (annihilation) operators forthe LO phonon of the ðl; n1Þth mode, withfrequency oLO and wave vector ðK== ¼

wn1=R;Kz ¼ lp=2dÞ where wn1 is the n1th root ofthe zero-order Bessel function.

Hex�LOðPÞ ¼ He�LOðPÞ þHh�LOðPÞ, (7)

Hi�LO ¼ �X

n1

J0wn1

Rri

� �

�X

l¼1;3::

Vln1 coslpi

2dzi

� �ðaln1 þ aþln1Þ

"

þX

l¼2;4::

V ln1 sinlpi

2dzi

� �ðaln1 þ aþln1Þ

#

ði ¼ e;hÞ, ð8Þ

where

V2ln1ðPÞ ¼

1

V

4pe2_oLO

ðwn1

RÞ2J2

1ðwn1Þ½1þ ðlpR2dwn1Þ2�

1

e1�

1

e0

� �,

(9)

with V ¼ 2pR2d is the volume of cylindrical dot.To deal with the Hamiltonian of this system, we

shall adopt the variational treatment for quasi-zero-dimensional systems developed in our pre-vious work [15]. The effective Hamiltonian, in theatomic units system (a�ex ¼

e0ð0Þ_2

me2 excitonic Bohr

radius and Rydberg energy R�ex ¼me4

2e20ð0Þ_2

), reads

Heff

¼ �1

1þ sm�e ð0Þ

m�e ðPÞ

q2

qr2eþ

1

re

qqre

�

þr2eh þ r2e � r2h

rereh

q2

qreqrehþ

q2

qz2e

�

�s

1þ sm�hð0Þ

m�hðPÞ

q2

qr2hþ

1

rh

qqrh

�

þr2eh þ r2h � r2e

rhreh

q2

qrhqrehþ

q2

qz2h

�

�q2

qr2ehþ

1

reh

qqreh

� ��

e0ð0Þe1ðPÞ

�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2eh þ ðze � zhÞ2

q þ V ewðze;PÞ þ Vh

wðzh;PÞ

þ V e�LOðre; ze;PÞ þ Vh�LOðrh; zh;PÞ

þ V exce�LO�hðre; ze;rh; zh;PÞ, ð10Þ

where s ¼ m�e=m�h. V e�LOðre; ze;PÞðVh�LOðrh;zh;PÞÞ and V exc

e�LO�hðre; ze;rh; zh;PÞ are, respec-tively, the effective potentials induced by theelectron–(hole)–phonon LO coupling and theelectron–hole exchange potential via an LOphonon. These expressions are determined byusing unitary transformation.

V i�LOðri; zi;PÞ

¼ �X

n1

J20ð

wn1

RriÞ

_oLO

�X

l¼1;3::

V 2ln1 cos

2 lpi

2dzi

� �"

þX

l¼2;4::

V 2ln1 sin

2 lpi

2dzi

� �#; ði ¼ e;hÞ, ð11Þ

and

V exce�LO�hðre; ze;rh; zh;PÞ

¼X

n1

2J0ðwn1

RreÞJ0ð

wn1

RrhÞ

_oLO

ARTICLE IN PRESS

A. El Moussaouy et al. / Physica B 370 (2005) 178–185 181

�X

l¼1;3::

V2ln1 cos

lpe2d

ze

� �cos

lph2d

zh

� �"

þX

l¼2;4::

V 2ln1 sin

lpe2d

ze

� �sin

lph2d

zh

� �#.

ð12Þ

In order to calculate the exciton binding energy,we choose the following wave function [18]:

cexðre; rh; ze; zh;PÞ ¼ F eðre; ze;PÞFhðrh; zh;PÞ

�F ehðreh; jze � zhjÞ, ð13Þ

with

F ehðreh; jze � zhjÞ ¼ expð�arehÞ expð�gðze � zhÞ2Þ,

(14)

and

Fiðri; zi;PÞ ¼ f iðri;PÞgiðzi;PÞ; ði ¼ e;hÞ. (15)

Respectively, the corresponding 2D (lateral direc-tion) and 1D (longitudinal direction) effectivemass Schrodinger equations are:

�_2

2m�ir2

i þ V iwðri;PÞ

� f iðri;PÞ

¼ Eiðri;PÞf iðri;PÞ; ði ¼ e;hÞ, ð16Þ

�_2

2m�ir2

i þ V iwðzi;PÞ

� giðzi;PÞ

¼ Eiðzi;PÞgiðzi;PÞ; ði ¼ e;hÞ, ð17Þ

with solutions of the form

f iðriÞ ¼ J0 y0ri

R

� �, (18)

giðzi;PÞ

¼cosðpiðPÞ

zi

2dÞ for jzijpd

BiðPÞ expðkiðPÞjzijÞ for jzij4d

(ði ¼ e;hÞ.

ð19Þ

J0 and K0 are the modified Bessel functions of 0thorder. piðPÞ, kiðPÞ, and BiðPÞ are determined fromthe following equations, which are obtained fromthe boundary conditions at the interface jzij ¼ d:

BiðPÞ ¼ cosðpiðPÞ=2Þ= expð�kiðPÞdÞ, (20)

tanðpiðPÞ=2Þ ¼ kið2d=piðPÞÞ. (21)

In Eq. (14), a and g are variational parameters.The pressure dependence of variational bindingenergy of the exciton system is given by

EBðPÞ ¼ ElðPÞ � EGðPÞ, (22)

where ElðPÞ and EGðPÞ are, respectively, thepressure dependence of the variational lowestenergy and the exciton ground-state energy whichcan be written as

ElðPÞ ¼ Ee þ Eh, (23)

EGðPÞ ¼ mina;ghcexjHeff jcexi, (24)

EeðPÞ and EhðPÞ are the free electron and holeenergies. The lowest energy ElðPÞ is obtained bysolving the Schrodinger equation by taking a and gequal to zero and omitting the Coulombic inter-action term.

3. Numerical results and discussions

In what follows, we study the effect of appliedstress onto the binding energy of the ground stateand the polaronic correction to the binding energyof the exciton in cylindrical QD made of GaAsembedded in Ga1�xAlxAs material. The pressuredependence of the binding energy, by including theinteraction between the electron and hole with(LO) phonons, has been obtained numerically byevaluating different kinds of triple and doubleintegrals over the GaAs QD and the Ga1�xAlxAsmatrix spaces. The integrals are evaluated with astep adaptative iterative method. Then, the valueof the pressure was set up and the correspondingenergies are found by iterations until a conver-gence better than 0.1 was attained. The values ofthe effective masses corresponding to the polarcrystal GaAs are obtained from references[19,20] m�e ¼ ð0:067þ 0:00055PÞm0, m�h ¼ ð0:087þ0:00064PÞm0. The pressure dependence of thestatic dielectric and high-frequency dielectric are[19,21] e0ðPÞ ¼ 13:1� 0:0088P, e1ðPÞ ¼ 10:89þ0:0219P. Our numerical results are presented inunits of the effective Rydberg energy R�ex ¼ 3meVand the effective Bohr radius a�ex ¼ 183:431, whichare evaluated at zero pressure. The variation of dotsize ðHÞ with pressure is taken into account using

ARTICLE IN PRESS

P (kbar)0 10 20 30 40

EB (

R* e

x)3

4

5

6

7

H=2a*ex

H=1a*ex

H=0.5a*ex

H=2a*ex

H=0.5a*ex

H=1a*ex

R=1a*ex

Fig. 2. Variation of the exciton binding energy in a cylindrical

GaAs=Gað1�xÞAlðxÞAs QD of radius R ¼ 1a�ex, as a function of

pressure P with (solid lines) and without (dotted lines) LO

A. El Moussaouy et al. / Physica B 370 (2005) 178–185182

[21] da1dP¼ �2:6694� 10�4a0, where a0 is the lattice

constant. In all our calculations, we work with thealuminum concentration, x ¼ 0:3.In Fig. 1, we present our theoretical results for

the binding energy of the exciton in a cylindricalQD of the radius R ¼ 1a�ex, as a function of theheight H of the cylinder considering three differentvalues of the stress (P ¼ 0, 20, and 40 kbar) alongthe z direction with and without including LOphonon modes. We note that the binding energydiminishes as H increases. This is due to the factthat exciton wave function is more spread alongthe axial direction because the potential barriersare far away. Additionally, it is expected that ateach value of the height H, the binding energyincreases with increase in the compression due tothe applied stress; this effect is stronger when theheight H is around 0:3a�ex and when including LOphonon correction.In Fig. 2, we display our calculated results for

the exciton binding energy in GaAs=Gað1�xÞ

AlðxÞAs QD of the radius R ¼ 1a�ex, as a function

H (a*ex)

0.0 0.5 1.0 1.5 2.0 2.5

EB (

R* ex

)

3

4

5

6

7

20 kbar

R=1a*ex

40 kbar

P=0

Fig. 1. Variation of the exciton binding energy in a cylindrical

GaAs=Gað1�xÞAlðxÞAs QD, of radius R ¼ 1a�ex, as a function of

the height H with (solid lines) and without (dotted lines) LO

phonon modes for three values of the pressure (P ¼ 0, 20, and

40kbar).

phonon modes for three values of the height (H ¼ 0:5a�ex, 1a�ex,

and 2a�ex).

of the stress applied in the z direction of thecylinder and consider different values of the heightof the QD (H ¼ 0:5a�ex, 1a�ex, and 2a�ex). Theexciton binding energy shows a nearly linearincrease with applied stress. Furthermore, it isclear that as the height of the structure increases,the slope of the curve diminishes. This is becausethe exciton wave function does not feel the smallcompression in the structure when the size is verylarge. The effect of the stress is more importantwhen including LO phonon modes than withoutphonons.

In order to understand more the pressure effecton the LO phonon modes contribution to theexciton binding energy in GaAs QD, we haveplotted in Fig. 3, the polaronic correction ðDEBÞ tothe binding energy in QD with radius R ¼ 1a�exconsidering different applied stress (P ¼ 0, 20, and40 kbar) as a function of the height H, where DEB

is defined as the difference between the bindingenergy in the presence and absence of the LOphonon modes. The behavior we observe in thecurves is similar to those in Fig. 1. It is shown that

ARTICLE IN PRESS

H (a*ex)0.0 0.5 1.0 1.5 2.0 2.5

∆EB (

R* e

x)

0.8

1.0

1.2

1.4

P=0

P=20 kbar

P=40 kbar

R=1a*ex

Fig. 3. Polaronic correction ðDEBÞ to the binding energy in a

cylindrical GaAs=Gað1�xÞAlðxÞAs QD of radius R ¼ 1a�ex, as a

function of the height H for three values of the pressure (P ¼ 0,

20, and 40kbar).

P (kbar)

0 10 20 30 40∆E

B (

R* e

x)0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

H=2a*ex

H=1a*ex

H=0.5a*ex

R=1a*ex

Fig. 4. Polaronic correction ðDEBÞ to the binding energy in a

cylindrical GaAs=Gað1�xÞAlðxÞAs QD of radius R ¼ 1a�ex, as a

function of the pressure P for three values of the height

(H ¼ 0:5a�ex, 1a�ex, and 2a�ex).

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

0

10

20

40

0.00.51.01.52.0

EB (

R* e

x)

P (kba

r)

H (a*ex)

Fig. 5. Variation of the exciton binding energy in a cylindrical

GaAs=Gað1�xÞAlðxÞAs QD as a function of the cylinder height H

and pressure P for fixed radius R ¼ 1a�ex, with and without LO

phonon modes.

A. El Moussaouy et al. / Physica B 370 (2005) 178–185 183

the LO phonon correction increases as the heightdecreases and reaches a maximum and thendiminishes to the value of quasi-QWW limit. InFig. 4, we display the polaronic correction to theexciton binding energy in GaAs=Gað1�xÞAlðxÞAsQD of the radius R ¼ 1a�ex, as a function of theapplied stress for different heights(H ¼ 0:5a�ex,1a�ex, and 2a�ex). As was mentioned in Fig. 2, weremark a linear enhancement of the polaroniccorrection versus pressure. Additionally, as theheight of the structure increases, the slope of thecurve diminishes. This behavior is because theconfined LO phonon modes contribution de-creases and does not feel the small compressionin the structure when the size is very large.

In order to have a global picture about thecombined effect of the geometrical confinement(height of the cylinder) and applied stress on theexciton binding energy, we have presented in Fig.5, the variation of the binding energy in GaAs QDas a function of both the height H and pressure P

with and without LO phonon contribution. Thisrepresentation leads us to evaluate clearly thecombined effect of continuous height and pressure.It can be seen that at each value of the height H,

the binding energy increases linearly with pressureand the polaronic correction is stronger at highstress value, which enhances the stability of

ARTICLE IN PRESS

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0

10

20

40

0.00.51.01.52.0

EB (

R* e

x)

P (kba

r)

H (a*ex)

Fig. 6. Variation of the exciton binding energy in a cylindrical

GaAs=Gað1�xÞAlðxÞAs heterostructure as a function of the

cylinder height H and pressure P for fixed radius R ¼ 20a�ex,

with and without LO phonon modes.

A. El Moussaouy et al. / Physica B 370 (2005) 178–185184

exciton. Furthermore, the effect of applied stressdiminishes for the large dot sizes (height, H).It is well known that by varying the radius and

height of the QD, we obtain various limitingsituations of the bulk, the QW, the QWW, and theQD. Here, we present in Fig. 6, the exciton bindingenergy in GaAs=Gað1�xÞAlðxÞAs heterostructure forR ¼ 20a�ex as a function of both H and appliedcompressive stress P with and without includingLO phonon modes. Due to the large value ofradius R, our here-considered structure is practi-cally in the QW limit. The relatively same behaviorof the binding energy versus H and P illustratedabove in Fig. 5 is to be noted. Our results arequalitatively similar to those obtained in literature[13,14] for the donor impurity without includingphonons.

4. Conclusions

Summing up, we have calculated the pressuredependence of the exciton binding energy corre-sponding to the ground state for an exciton inGaAs cylindrical QD following a variational

procedure within the effective mass approxima-tion. Investigations were performed by using afinite potential deep in z direction and by includingLO phonon modes. Our results show that theexciton binding energy in a QD increases almostlinearly with applied stress and diminishes with thesize of the structure. Also, we have found that thepolaronic correction to the exciton binding energyenhances with applied stress, it increases linearlywith increasing pressure which allows the excitonto become more stable.

We expect that the present work will be usefulfor a good understanding of experimental studieson GaAs–(Ga,Al)As QD such as exciton opticalabsorption spectra including LO phonon modesunder stress effects.

References

[1] A. Imamoglu, D.D. Awschalom, G. Burkard, D.P. DI

Vincenzo, D. Loss, M. Sherwin, A. Small, Phys. Rev. Lett.

83 (1999) 4204.

[2] F. Troiani, U. Hohenester, E. Molinari, Phys. Rev. B 62

(2000) R2263.

[3] K.I. Kolokolov, A.M. Savin, S.D. Beneslavski, N.Ya.

Minina, O.P. Hansen, High Press. Res. 18 (2000) 69.

[4] A.N. Forshaw, D.M. Whittaker, Phys. Rev. B 54 (1996)

8794.

[5] J. Bellessa, V. Voliotis, R. Grousson, X.L. Wang,

M. Ogura, H. Matsuhata, Phys. Rev. B 58 (1998) 9933.

[6] V.A. Fonoberov, E.P. Pokatilov, V.M. Fomin, J.T.

Devreese, Phys. Rev. Lett. 92 (2004) 127402.

[7] J. Perez-Conde, A.K. Bhattacharjee, M. Chamarro,

P. Lavallard, V.D. Petrikov, A.A. Lipovskii, Phys. Rev.

B 64 (2001) 113303.

[8] A. Feltrin, R. Idrissi Kaitouni, A. Crottini, M.-A.

Dupertuis, J.L. Staehli, B. Deveaud, V. Savona, X.L.

Wang, M. Ogura, Phys. Rev. B 69 (2004) 205321.

[9] D.H. Feng, Z.Z. Xu, T.Q. Jia, X.X. Li, S.Q. Gong, Phys.

Rev. B 68 (2003) 035334.

[10] B. Gil, P. Lefebvre, P. Bonnel, H. Mathieu, C. Deparis,

J. Massies, G. Neu, Y. Chen, Phys. Rev. B 47 (1993) 1954.

[11] K.I. Kolokolov, S.D. Beneslavski, N.Ya. Minina, A.M.

Savin, Phys. Rev. B 63 (2001) 195308.

[12] G. Rau, A.R. Glanfield, P.C. Klipstein, N.F. Johnson,

G.W. Smith, Phys. Rev. B 60 (1999) 1900.

[13] A.M. Elabsy, Phys. Scr. 48 (1993) 376.

[14] H.O. Oyoko, C.A. Duque, N. Porras-Montenegro,

J. Appl. Phys. 90 (2001) 819.

[15] A. El Moussaouy, D. Bria, A. Nougaoui, R. Charrour,

M. Bouhassoune, J. Appl. Phys. 93 (2003) 2906.

[16] J. Nakahara, T. Ichimori, S. Minomura, H. Kulklmoto,

J. Phys. Soc. Japan 56 (1987) 1010.

ARTICLE IN PRESS

A. El Moussaouy et al. / Physica B 370 (2005) 178–185 185

[17] M. Holtz, M. Seon, O. Brafman, R. Manor, D. Fekete,

Phys. Rev. B 54 (1996) 8714.

[18] S. Le Goff, B. Stebe, Phys. Rev. B 47 (1993) 1383.

[19] A. Sukumar, K. Navaneethakrishnan, Phys. Stat. Sol. 5b

(158) (1990) 193.

[20] K. Reimann, M. Holtz, K. Syassen, Y.C. Lu, E. Bauser,

Phys. Rev. B 44 (1991) 2985.

[21] Sr. Gerardin Jayam, K. Navaneethakrishnan, Solid State

Commun. 126 (2003) 681.