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Polaronic effects on laser dressed donor impurities in a quantum well N. Radhakrishnan a, A. John Peter b, a b

Department of Biotechnology, IIT Madras, Chennai 600 036, India Department of Physics, Government Arts College, Melur 625 106, India

a r t i c l e in fo

abstract

Article history: Received 24 January 2009 Received in revised form 28 July 2009 Accepted 29 July 2009 Available online 13 August 2009

The effect of laser ﬁeld on the binding energy in a GaAs/Ga11xAlxAs quantum well within the single band effective mass-approximation is investigated. Exciton binding energy is calculated as a function of well width with the renormalization of the semiconductor gap and conduction valence effective masses. The calculation includes the laser dressing effects on both the impurity Coulomb potential and the conﬁnement potential. The valence-band anisotropy is included in our theoretical model. The 2D Hartree–Fock spatial dielectric function and the polaronic effects have been employed in our calculations. We investigate that reduction of binding energy in a doped quantum well due to screening effect and the intense laser ﬁeld leads to semiconductor–metal transition. & 2009 Elsevier B.V. All rights reserved.

Pacs: 71.70.Di 72.20.Ht 73.20.Hb 73.21.b Keywords: Quantum wells excitons Laser ﬁeld

1. Introduction Superlattices and quantum wells (QW) have received a great deal of attention with recent advances in technological applications. Among them GaAs/Ga1xAlxAs quantum wells are the most extensively studied semiconductor heterostructures. The physical properties of electrons in quantum wells are very different from those in the bulk. As a consequence of the conﬁnement, energy levels are discrete. The GaAs/AlxGa1xAs nanostructures have been the subject of recent research because they have a direct-gap band structure, abrupt spatial transition in the energy gap and the lattice matched structures. These nanostructures have been the subject of extensive experimental and theoretical researches because of their possible applications in optoelectronic devices and laser systems [1]. Reduction in dimensionality makes the electron to be localized more, which in turn leads to the enhancement of its binding energy eventually changing its optical absorption spectrum. Recently, the effect of an intense laser ﬁeld on the impurity binding energy has been investigated in undoped semiconductors [2,3]. It has been observed that the impurity binding energy dramatically decreases with the increase of laser ﬁeld amplitude and it leads to a metal–insulator transition under certain circumstances [4]. Recently, not only the dressed band approach

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E-mail address: [email protected] (A.J. Peter). 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.07.012

[5,6] but also the conﬁnement potential and the direction of external ﬁelds [7,8] to treat the interaction of a laser ﬁeld with the semiconductor quantum wells have been proposed. Former suggested a model in which the effect of the laser–semiconductor interaction takes into account through the renormalization of the semiconductor band-gap and the electron/hole effective masses whereas the latter investigated the laser dressing effects on both impurity Coulomb potential and the graded conﬁnement potential. The electronic and excitonic states in low-dimensional semiconductors have attracted much attention in recent years [9,10]. The binding energy usually increases when going from bulk to low-dimensional semiconductor systems due to the conﬁnement. So also the interaction between electron and hole is enhanced by the conﬁnement it is due to the overlap of the electron and the hole wave functions in these structures. Usually the binding energy is deﬁned as the difference in energy between the electron and hole pair interacting by the Coulomb attraction and the uncorrelated one. However, the binding energies in these structures have been investigated to reach values in the range of 2.5–3.5 times the bulk Rydberg [11]. The simplest exciton complex is created when an electron and a hole of effective mass m*e and m*h, respectively trapped by a charged impurity and they can be used as a test for the theoretical description of exciton impurity interaction [12]. In bulk semiconductors, the binding energy of excitons generally will be low because their existence sometimes depends on speciﬁc stability conditions [13] but in quantum wells, due to the overlap of wave

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functions of electrons and holes, they are more bound. Excitons ˚ in GaAs are hydrogen like systems with Bohr radii of order 100 A, and binding energies of order 10 meV. In particular, there have been a great deal of research works dedicated to the study of electron–phonon interaction in quantum wells (QW) [14,15]. There are a great deal of works devoted to the study of the electron–phonon interactions in QWs [16,17]. It is well known that the electron-optical phonon interaction plays an important part in the physical properties of polar crystals, especially in lowdimensional quantum systems [18] and it is believed that the electrons-longitudinal phonons are in the strong coupling regime leading to polaronic effect [19]. Guo and Chen presented the polaron effects on the second-harmonic generation in quantum well with an electric ﬁeld [20]. The effect of the bulk LongitudinalOptical (LO) phonon on the binding energy is investigated for a shallow donor impurity in a superlattice in the effective massapproximation by using the variational approach recently by Tayebi et al., [21]. They have obtained the results as a function of parameters which characterize the superlattice and the position of the impurity center. Since the optical, electrical and transport properties of semiconductor materials are strongly inﬂuenced by doped impurities the knowledge of effect of conﬁned potential barriers on the laser ﬁeld intensities and the binding energy of impurities is important. Applications of external electric, magnetic and laser ﬁelds on semiconductor systems will pay the way to have better performance of optoelectronic switches and modulators. In particular, the excitons which are created by the interaction of laser ﬁeld with the nanostructures should be theoretically taken into account. Very recently the effect of the high-frequency laser ﬁeld on the subband structure and on polarizabilities of the shallow donors in a GaAs/ GaAlAs inverse V-shaped quantum well for different Al concentrations at the structure center is investigated by Burileanu et al., [22]. In the present work, we study the effect of the laser ﬁeld on the binding energy of a hydrogenic donor and excitons using the variational method, within the single band effective mass-approximation. We compute the exciton binding energy as a function of the well width for different ﬁelds of laser intensity. The spatial dielectric function and the polaronic effects have been employed for a GaAs/ AlGaAs quantum well. The valence-band anisotropy is included in our theoretical model by using different hole masses in different spatial directions. In Section 2, we brieﬂy describe the method and the quantum well model used in our calculation. And the results and discussion are presented in Section 3. A brief summary and results are presented in the last section.

2. Theory and model In the effective mass-approximation, the Hamiltonian of a hydrogenic donor impurity in a GaAs/GaAlAs quantum well is given by, Hðr; j; zÞ ¼

‘2 2m

r2

e2 þ VðzÞ er

ð1Þ

where r is the distance between the carrier and the impurity ion, m* the effective mass of GaAs, e the static dielectric constant and e the electron charge. We assume that the band-gap discontinuity [23,24] in a GaAs/ Ga1xAlxAs quantum well is distributed about 40% on the valence band and 60% on the conduction band with the total band-gap difference DEg(x) (in eV) between GaAs and Ga1xAlxAs given as a function of the Al concentration as

DEg ðxÞ ¼ 1:155x þ 0:37x

2

ð2Þ

which is the variation of the energy gap difference thus the value of barrier potential, V(z) turns out to be 195 meV. The eigen function of the Hamiltonian with the impurity is given by, ( N1 cosðk1 jzjÞexpðaðr2 þ z2 ÞÞ; jzjoL=2 ð3Þ cðr; zÞ ¼ N2 expðk2 jzjÞexpðaðr2 þ z2 ÞÞ jzjZL=2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 where k1 ¼ 2m Es =‘ , k2 ¼ 2m ðVðzÞ Es Þ=‘ , Es is the subband energy. Here N1 and N2 are normalization constants and a is the variational parameter. The subband pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃenergy ﬃ iscalculated by solving the transcendental equation Es =VðzÞ ¼ cos pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ðL 2mm Es =‘ Þ numerically. This ﬁxes the values of k1 and k2 for the lowest values of Es. By matching the wave functions and their derivatives at boundaries of the well and along with the normalization, we ﬁx all the constants except the variational parameter. 2.1. Laser ﬁeld intensity ¨ The time dependent Schrodinger equation describing the interaction dynamics is transformed by Kramers [25]

‘2 2mm

r2 Cð~ r ; tÞ þ Vð~ r þ~ a ðtÞÞCð~ r ; tÞ ¼ i‘

@Cð~ r ; tÞ @t

ð4Þ

here ~ a ðtÞ ¼ ea0 sinðotÞ; a0 ¼

eA0 mm o

ð5Þ

describes the motion of the electron in the laser ﬁeld and a0 is the laser dressing parameter given by

a0 ¼ ðI1=2 =o2 Þðe=mm Þð8p=cÞ1=2

ð6Þ

where e, m*, c and A0 are the charge and effective mass of the electron, velocity of the light, the amplitude of the vector potential and the frequency of applied ﬁeld respectively. In the high-frequency limit [26] the laser dressed eigenstates ¨ are the solutions of the time independent Schrodinger equation

‘2 2m

r2 þ Vd ð~ r; ~ a 0 Þf ¼ Ef

r; ~ a 0 Þ is the laser dressed potential given by where Vd ð~ Z 2p=o o r; ~ a0Þ ¼ Vð~ r þ~ a ðtÞÞ dt Vd ð~ 2p 0

ð7Þ

ð8Þ

where Vd ð~ r; ~ a 0 Þ is the dressed potential which depends on o and I only through a0 [11]. Eq. (8) is obtained from Eq. (4) assuming that otb1, t being electron transit time in the well. In this limit, the electron motion is dominated by the oscillation of the well due to the laser ﬁeld. For a deep well t is on the order ˚ of 2m*Lw2/p_ Therefore, for GaAs quantum wells with Lw ¼ 100 A, if oZ1014 s1. At this limit, the inﬂuence of the high-frequency laser ﬁeld is entirely determined by the dressed potential r; ~ a 0 Þ. Vd ð~ We consider the z-axis to be along the growth direction of the well. The Hamiltonian of the system consisting of an electron bound to a donor ion inside the quantum well in the presence of an intense high-frequency laser ﬁeld given by Hðr; j; zÞ ¼

‘2 2mm

r2 þ Vc ðz; a0 Þ

ð9Þ

where Vc(z, a0) is the dressed conﬁnement potential has the form Vc ðz; a0 Þ ¼ 12V yðz2 L2EW Þ

ð10Þ

y(z) is the step function, LEW ¼ LW/2a0 half of the effective well width and Lw the width of the quantum well. For the Coulomb potential case, V(r¯) ¼ (e2/e|r|), the dressed potential has

ARTICLE IN PRESS N. Radhakrishnan, A.J. Peter / Physica E 41 (2009) 1841–1847

the form [27], Vðr; z; a0 Þ ¼

0

with the Hamiltonian

1

e2 B 1 1 C @qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃA; 2e 2 2 2 2 2 2 r þz þa r þz a 0

H ¼ Heh þ Vint ðrÞ; ð11Þ

0

where e is the dielectric constant of GaAs. ¨ Since an exact solution of the Schrodinger equation with the Hamiltonian in Eq. (1) is not possible, a variational approach has been adopted. The wave function of the laser dressed hydrogenic donor is given by ( N3 cosðk1 jzjÞexpðbðr2 þ z2 þ a20 ÞÞ; jzjoL=2 cðr; z; a0 Þ ¼ N4 expðk2 jzjÞexpðbðr2 þ z2 þ a20 ÞÞ jzjZL=2 ð12Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 where k1 ¼ 2mm Es =‘ , k2 ¼ 2mm ðVc ðz; a0 Þ Es Þ=‘ , Es is the subband energy. Here N3 and N4 are normalization constants energy ﬃ is and b is the variational parameter. The subband pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ calculated by solving the transcendental equation Es =Vc ðz; a0 Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 cosðL 2mm Es =‘ Þ numerically. This ﬁxes the values of k1 and k2 for the lowest values of Es. By matching the wave functions and their derivatives at boundaries of the well and along with the normalization, we ﬁx all the constants except the variational parameter. The expression for renormalized effective mass given by [6] " # 1 1 M L20 M ð13Þ ¼ 1þ b7 P m m7 2M 3m d with 1/M ¼ 1/mG6+1/mG8 and b ¼ (m68/m67)(1/d)+(8Ep0/3)(m68/ m0)((8/e20)+(2/e0 20)+(2/e0e0 0)), where b ¼ (4Ep0/3)(m68/m0)((8/e20)+ (2/e0 20)+(2/e0e0 0)). The k-dependent semiconductor energy gap is dressed by laser effects and is given by the difference between the above renormalized conduction and valence bands, vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !2 u u8L2 L2 4L20 ‘ 2 k2 þ ð14Þ EðkÞ ¼ e0 d þ t 0 d þ 0 þ 3 3d 3L1 2m with 1/m ¼ 1/m+1/m.Corresponding values in the above expressions are taken from Ref. [6]. We follow the variational scheme within a renormalized laser dressed effective mass approach using the envelope wave function Eq. (12) and the oncentre donor binding energies can be obtained as a function of well width. ¨ The Schrodinger equation is solved variationally by ﬁnding /HSmin and the binding energy of the donor in a quantum well is given by the difference between the energy with and without Coulomb term. The binding energy of ground state is given by Eb ¼ Es /Hðr; j; zÞSmin ;

1843

ð15Þ

ð17Þ

where Heh ¼

‘2 2mm

2

e

@2 ‘ @2 þ Ve ðze Þ þ Vh ðzh Þ þ VeB ðr e Þ 2 2mm h @z2h @ze

ð18Þ

where Ve(h)(ze(h)) is the conduction (valence) band offset, and m*e m*h the effective mass of electron and hole respectively and VeB(re) refers the diamagnetic terms m*Je and m*?e the parallel and perpendicular effective masses of the electron. The anisotropy and the corrections due to the conduction band non-parabolicity are important [28] since the subband energy due to the z-conﬁnement is around the order of 100 meV. The ﬁrst order nonparabolicity approximation [29] is given by m?e ¼ me ð1 þ tEÞ

ð19Þ

mJe ¼ me ½1 þ ð2t þ dÞE

ð20Þ

where m*e is the bulk electron mass of the material, t and d the non-parabolicity parameter (Table 1) and E the ground state energy of the electron. The parallel mass determines the electron energy in the xy plane and perpendicular mass determines the quantization energy of the electron in the z-plane [30]. The strain is not included in the calculation of heavy and light holes because of the experimental growth conditions [31]. In the valence band of semiconductors, there are heavy-hole and light-hole bands which are degenerate at the top of the valence band. The valence-band anisotropy is described in by assuming different hole masses in different directions. The mass in each direction of the same corresponding to the curvature of the hole bands around the G point, i.e., for the heavy-hole and light-hole respectively m0 ¼ g1 2g2 ; ð21Þ mm hh m0 ¼ g1 þ 2g2 ; mm hh

ð22Þ

where g1 and g2 are Luttinger parameters (Table 1) and m0 ¼ is the free hole mass. ¨ The Schrodinger equation is solved variationally by ﬁnding /HSmin and the binding energy of the donor in a quantum well is given by the difference between the energy with and without Coulomb term. The binding energy of the exciton system is deﬁned as Eexc ¼ Ee; h /Hmin S

ð23Þ

where Ee, h is the sum of the free electron and the free hole selfenergies in the same QW.

where Es is the subband energy. Table 1 Material parameters used in the calculations.

2.2. Exciton energy Consider a Wannier-Mott exciton localized in a freestanding cylindrical quantum well placed in vacuum, with a radius R. We take the problem of an exciton in a GaAs/GaAlAs quantum well grown along the z-axis within the single band effective massapproximation. The conduction and valence bands are centered ¨ for around the G valley of GaAs and AlxGa1xAs. The Schrodinger the exciton is then given by Hcðre ; rh Þ ¼ Ecðre ; rh Þ

ð16Þ

Parameter

GaAs

Eg me mh

1.519 0.067 0.058 6.98 2.06 0.64 0.70 12.9

g1 g2 t d

e

Parameters taken from Ref. [30].

Ga1xAlxAs 1.956 0.096

Unit eV (m0)

(eV)1 (eV)1 11.9

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2.3. Dielectric function

Vb ðqÞ ¼

2pe2 e0 qL2

ð35Þ

It is well known that Hartree–Fock dielectric function in 3D is given by [32] 4pe2 F3D ðqÞ q2

e3D ðqÞ ¼ 1 þ

ð24Þ

Vs ðqÞ ¼

with F3D ðqÞ ¼

3N 4Ef

(

) 4k2f q2 2kf þ q log 1þ 2kf q 4kf q

e2D ðqÞ ¼ 1 þ where

F2D ðqÞ ¼

then,

e2D ðqÞ ¼

ð26Þ

> > > > > :

1 2 !1=2 2kf A 1 q

q42kf

> > > > > :

qo2kf

0

e0 ‘ 2 q

where kf is the Fermi wave vector. In 2D, 4pN 1=2 kf ¼ A

Vb ðqÞ eðqÞ

ð30Þ

where Vb(q) is the bare potential seen by a test charge in a free electron gas due to the immersion of a positive point charge. The screening function is in general related to q and o. In the present work, we are interested only in the static screening functions. The bare potential of the 2D hydrogen atom in real space reads Vb ðrÞ ¼

e2

ð31Þ

re0

Taking the Fourier transform of bare potential in real space, we get, Z 1 Vb ðqÞ ¼ 2 eiq r Vb ðrÞ dq ð32Þ L

¼

1 L2

Z

1 Z 2p 0

0

e2

re0

eiqr cos y dy dr

In y part, Z 2p eiqr cos y dy ¼ 2pJ0 ðqrÞ

Z

e0

2kf 0

J0 ðqrÞ 2

1 þ ð2m e2 =e0 q‘ Þ

0

dq

J0 ðqrÞ 1þ

2

e0 q‘ Þð1 ð1 ð4kf =q2 ÞÞ1=2 Þ

ð2m e2 =

From Eqs. (37) and (38), we have, Z 2kf 1 qJ0 ðqrÞ ¼ re0 dq e2D ðrÞ 0 e0 q þ 2m Z 0 q2 J0 ðqrÞ dq: þ 2 1=2 2 2kf ðe0 q þ 2mm q 2mm ðq2 4kf Þ

dq

ð39Þ

ð40Þ

ð29Þ

where N/A ¼ n is the concentration of electrons. Electron–electron interaction leads to a screening function such that Vs ðqÞ ¼

e2

þ

ð28Þ

ð37Þ

Taking Fourier transform of Vs(q) and using Eq. (36), we obtain, Z iq r L2 e Vb ðqÞ dq ð38Þ Vs ðrÞ ¼ 2p e2D ðqÞ

2kf

q42kf

2m e2

e2

e0 rL2 e2D ðrÞ

Z

qo2kf

8 1=2 ! > 4kf 2m e2 > > >1 þ 1 1 2 > < q e ‘ 2q 1þ

Vs ðrÞ ¼

Vs ðrÞ ¼ ð27Þ

2m L2 ‘ 2 2p

ð36Þ

Screened coulomb potential in real space reads

2pe2 F ðqÞ e0 L2 q 2D

0 8 > > 2m L2 @ > > > < ‘ 2 2p 1

2pe2 e0 qL2 e2D ðqÞ

ð25Þ

where N is the number of electrons per unit volume, Ef the Fermi energy, and kf the Fermi wavevector. In 2D the same theory leads to a dielectric function,

0

This is the coulomb potential. Then the screened potential in Fourier space is given by

ð33Þ

ð34Þ

¨ Since an exact solution of the Schrodinger equation with the Hamiltonian in Eq. (1) is not possible, a variational approach has been adopted. The wave function of the hydrogenic donor is given by ( N3 cosðk1 jzjÞexpðbðr2 þ z2 þ a20 ÞÞ; jzjoL=2 cðr; z; a0 Þ ¼ N4 expðk2 jzjÞexpðbðr2 þ z2 þ a20 ÞÞ jzjZL=2 ð41Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 where k1 ¼ 2mm Es =‘ , k2 ¼ 2mm ðVc ðz; a0 Þ Es Þ=‘ , Es is the subband energy in the heavy-hole and light-hole exciton calculations and similarly. Here N5 and N6 are normalization constants and g is the variational parameter. The subband energy is calculated by solving the transcendental equation qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 Es =Vc ðz; a0 Þ ¼ cosðL 2m Es =‘ Þ numerically. This ﬁxes the values of k1 and k2 for the lowest values of Es. By matching the wave functions and their derivatives at boundaries of the well and along with the normalization, we ﬁx all the constants except the variational parameter. The LO phonon Hamiltonian is given by [33] X 1 HLO ¼ ‘ oq aþ a þ ; ð42Þ q q 2 a where aq+(aq) is the creation (annihilation) operator of a LO phonon with the wave vector q and frequency oq. For GaAs, we take _oq ¼ _oLO 36.75 meV, the value of LO phonon energy at 4.2 K [34]. The electron–phonon interaction if given by X iq r Hph ¼ ðVq aq eiq r þ Vq aþ Þ; ð43Þ qe q

ARTICLE IN PRESS N. Radhakrishnan, A.J. Peter / Physica E 41 (2009) 1841–1847

where Vq is the Fourier coefﬁcient of the electron–phonon interaction given by jVq j2 ¼

4pa ð‘ oLO Þ3=2 O q2

ð44Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ with O the crystal volume and a ¼ Rm =‘ oLO ððe0 =e1 Þ 1Þ the ¨ hlich coupling constant and _oLO the high-frequency standard FrO dielectric constant of the material. In our calculation we have taken the value of a ¼ 0.068 for GaAs [35]. ¨ The Schrodinger equation is solved variationally by ﬁnding /HSmin and the binding energy of the donor in a quantum well is given by the difference between the energy with and without Coulomb term. The binding energy of ground state with phonons is given by Eph esc ¼ Es /HSmin ;

ð45Þ

where Es is the subband energy. 3. Results and discussion Our numerical computation is carried out for one of the typical semiconducting materials, GaAs, as an example with the material parameters shown in the following: eo ¼ 13.13; and m* ¼ 0.067me; where me is the single electron bare mass. With these values, the value of effective Rydberg is R*y ¼ 5.29 meV and ˚ Numerical calculations the effective Bohr radius, R* ¼ 103.7 A. have been carried out to investigate the effects of laser dressing on donor states in semiconductor quantum states in semiconductor quantum well with the renormalized approach. Exciton binding energy is calculated as a function of well size with the renormalization of the semiconductor gap and conduction valence

effective masses. We take into account the laser dressing effects on both the impurity Coulomb potential and the conﬁnement potential. These effects are carried out with the inclusion of polaronic effect and the dielectric function. We have chosen GaAs and AlGaAs because of the same crystal structures and lattice constants [36]. The material parameters are eN ¼ 10.9, _o ¼ 36.75 meV, heavy-hole effective mass ¼ 0.34m0 and the light-hole effective mass ¼ 0.095m0. The other material parameters are given in Table 1. Fig. 1 shows the variation of binding energy of an exciton with the well width of GaAs with laser ﬁeld amplitude 20 A˚ for a laser detuning d ¼ 0.05e0. The binding energy has been calculated with and without the inclusion of dielectric function and polaronic effect. In all the cases, the binding energy increases with a decrease of well width, reaching a maximum value and then decreases when the well width still decreases. As the well size approaches zero the conﬁnement becomes negligibly small, and in the ﬁnite barrier problem the tunneling becomes huge. The binding energy still reduces. The effect of dielectric function and polaronic enhances the binding energy for all the well size and it is pronounced more for narrow wells [37]. It has been observed that both the effects of dielectric function and polaronic increase ˚ the binding energy by 2 meV for 90 A. We present the variation of impurity binding energies of GaAs/ Ga1xAlxAs QW with the laser ﬁeld amplitude a0 for a well width 100 A˚ in Fig. 2. The effect of spatial dielectric function and the polaronic has been included in the calculation. It is clear from ﬁgure that the binding energy of an impurity ﬁrst increases with a0, reaches a maximum value and then decreases for large values of a0. The donor binding energy increases ﬁrst as the geometric conﬁnement of the QW decreases eventually increasing the conﬁnement of donor electron and the impurity ion. After the

20

20 (1)

with Dielectric and Polaronic effect

(2)

without Dielectric and Polaronic effect

(1)

Binding Energy (meV)

(1)

(2) 10

10

(1)

with Dielectric and Polaronic effect

(2)

without Dielectric and Polaronic effect

5

5

0

100

200 300 Well Width (Å)

400

500

Fig. 1. Variation of binding energy of an exciton with the well width of GaAs/ ˚ The binding energy has been calculated AlGaAs with laser ﬁeld amplitude 20 A. with and without the inclusion of dielectric function and polaronic effect.

L =100 Å

(2)

15

α0 = 20 Å

15 Binding Energy (meV)

1845

α0 = 20 Å

0

10

20

30

40

50

α0 = (Å) Fig. 2. The impurity binding energy is shown as a function of the laser ﬁeld ˚ The binding energy has been calculated with and amplitude for a well width 100 A. without the inclusion of dielectric function and polaronic effect.

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20

7 α0 = 20 Å 6 L =100 Å

Binding Energy (meV)

Critical Screening Wavevector (1\R*y)

(1)

15

(2)

10

with Dielectric and Polaronic effect without Dielectric and Polaronic effect

(1) (2)

(1)

without Polaronic effect

(2)

with Polaronic effect α0 = 20 Å

5

4 Metal (2) 3 (1)

2 Insulator 1

5

0 0

106

108

1010

1012

1014

0

Concentration (cm-2)

100

200 300 Well Width (Å)

400

500

Fig. 3. Variation of binding energy with the concentration for the laser ﬁeld ˚ The binding energy has been calculated with and without the amplitude 20 A. ˚ dielectric function and the polaronic effect for the well width 100 A.

Fig. 4. The critical screening wavevector is plotted against the well width for the ˚ It has been displayed with and without the inclusion of laser ﬁeld amplitude 20 A. dielectric function and polaronic effect.

˚ the donor electron becomes more critical value of a0 ¼ 30 A, energetic and the conﬁnement becomes negligibly and in the ﬁnite barrier problem the tunneling becomes huge and hence penetrate into the potential barrier easily. Also it is obvious from ﬁgure that the binding energy of the impurity for narrow well is more sensitive to the laser ﬁeld amplitude due to the compression of wave function. For larger values of laser amplitude due to the weakening of the impurity Coulomb potential the binding energy decreases. These results are in good agreement with Ref. [7], however, they have investigated the same effect for QW case without the inclusion of spatial dielectric function and the polaronic effect. Furthermore, the laser dressed impurity Coulomb potential decreases with a0, and the binding energy decreases consequently. Also the binding energy enhances with the inclusion of spatial dielectric function and the polaronic effect. Variation of binding energy with the concentration for the laser ﬁeld amplitude 20 A˚ is shown in Fig. 3. It has been displayed with and without the inclusion of dielectric function and ˚ Electron–electron polaronic effect for the well width 100 A. interaction is included through the Hartree–Fock 2D dielectric function and the electron-impurity ion interaction will be screened by the potential reducing the binding energy. Thus it follows that the binding energy decreases when the concentration increases. The results are not reliable for the concentrations beyond 1012 cm2. We also obtain complex numbers for binding energy when the concentration is beyond 1012 cm2, which leads to some unphysical results for higher concentrations. It is shown that the phase transition, the bound electron entering into the conduction band whereby Eb ¼ 0, is impossible beyond this concentration. Our results are in a good agreement with the other investigators [38]. From the experimental point of view, no other work on GaAs/GaAlAs quantum well systems is available. However, for a 2D GaAs/GaAlAs system a value of Nc ¼ 3 1010 cm2 has been reported for holes [39].

In Fig. 4, we present the critical screening wavevector against ˚ It has been the well width for the laser ﬁeld amplitude 20 A. displayed with and without the inclusion of dielectric function and polaronic effect. The results show that the metal (insulator) phase lies above (below) the curve in ﬁgure which is a very good agreement with Ref. [4] where they have investigated the same effect without the polaronic effect. Our investigations show that the dielectric screening effect in the background of electron– phonon interaction should be taken into account for a narrow QW.

4. Summary We have investigated the effect of laser ﬁeld intensity on exciton binding energy and laser dressed donor binding energy in the GaAs quantum well within the single band effective mass-approximation. Exciton binding energy has been calculated as a function of well width with the renormalization of the semiconductor gap and conduction valence effective masses. We have taken into account the laser dressing effects on both the impurity Coulomb potential and the conﬁnement potential. The valence-band anisotropy is included in our theoretical model by using different hole masses in different spatial directions. The spatial dielectric function and the polaronic effects have been employed for a GaAs/AlGaAs quantum well. The numerical calculations reveal that the inﬂuences of laser ﬁeld on donors are considerable and should not be neglected especially for narrow wells. The binding energy of the impurity for the narrow well is more sensitive to the laser ﬁeld amplitude. Not only the renormalization of the semiconductor gap and conduction valence effective masses but also the laser dressing effects on both the impurity Coulomb potential and the conﬁnement potential have been included in this calculation. Our calculations reveal that the strong localization of the electronic and impurity states in the quantum well enhances due to application of laser conﬁnement resulting the

ARTICLE IN PRESS N. Radhakrishnan, A.J. Peter / Physica E 41 (2009) 1841–1847

interpretation of metal–insulator transition. This may cause a useful manipulation of electronic and donor states especially in potential applications in infrared detectors and modulates and in some proposed solid state based quantum computers. However, we hope that these results would provide important improvements in device applications in modulators especially for narrow wells with a suitable choice of ﬁelds experimental efforts are expectant to provide support to our calculations.

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