We  studied the simultaneous of the magnetic and electric field  			effects on the binding energy for a shallow donor confined to move  			in spherical Quantum Dot – Quantum Well (GaAs-GaAlAs). Calculations  			are performed in the framework of the effective mass approximation  			using the Hass variational approach. We describe the effect of the  			quantum confinement by a infinite deep potential. The result shows  			that the corrections due to the magnetic and electric field are very  			important and cannot be neglected or ignored. We have demonstrated  			the existence of a critical value (a/b)cri which can be  			used to distinguish the three dimensions confinement from the  			spherical surface confinement and it’s may be important for the  			nanofabrication techniques.We   studied the simultaneous of the magnetic and electric field  			 effects on the binding energy for a shallow donor confined to move  			 in spherical Quantum Dot – Quantum Well (GaAs-GaAlAs). Calculations  			 are performed in the framework of the effective mass approximation  			 using the Hass variational approach. We describe the effect of the  			 quantum confinement by a infinite deep potential. The result shows  			 that the corrections due to the magnetic and electric field are very  		 	important and cannot be neglected or ignored. We have demonstrated  			 the existence of a critical value (a/b)cri which can be  			 used to distinguish the three dimensions confinement from the  			 spherical surface confinement and it’s may be important for the  			 nanofabrication techniques

RAHMANI, K.

ZORKANI, I.

English

We   studied the simultaneous of the magnetic and electric field  			 effects on the binding energy for a shallow donor confined to move  			 in spherical Quantum Dot – Quantum Well (GaAs-GaAlAs). Calculations  			 are performed in the framework of the effective mass approximation  			 using the Hass variational approach. We describe the effect of the  			 quantum confinement by a infinite deep potential. The result shows  			 that the corrections due to the magnetic and electric field are very  		 	important and cannot be neglected or ignored. We have demonstrated  			 the existence of a critical value (a/b)cri which can be  			 used to distinguish the three dimensions confinement from the  			 spherical surface confinement and it’s may be important for the  			 nanofabrication techniques.We  studied the simultaneous of the magnetic and electric field  			effects on the binding energy for a shallow donor confined to move  			in spherical Quantum Dot – Quantum Well (GaAs-GaAlAs). Calculations  			are performed in the framework of the effective mass approximation  			using the Hass variational approach. We describe the effect of the  			quantum confinement by a infinite deep potential. The result shows  			that the corrections due to the magnetic and electric field are very  			important and cannot be neglected or ignored. We have demonstrated  			the existence of a critical value (a/b)cri which can be  			used to distinguish the three dimensions confinement from the  			spherical surface confinement and it’s may be important for the  			nanofabrication techniques

Revues Scientifiques Marocaines

MAGNETIC AND ELECTRIC FIELD EFFECTS ON THE BINDING ENERGY OF A SHALLOW DONOR IN QUANTUM DOT–QUANTUM WELL

M.J.CONDENSED.MATER                                       VOLUME  11, Number 2                                       July 2009MAGNETIC AND ELECTRIC FIELD EFFECTS ON THE BINDING ENERGY OF A SHALLOW DONOR IN QUANTUM DOT–QUANTUM WELLK. RAHMANI1, I. ZORKANI11Groupe de Photoélectronique, LPS, Département de Physique,Faculté des Sciences Dhar Mehrez, B.P. 1796 Fes, MoroccoWe studied the simultaneous of the magnetic and electric field effects on the binding energy for a  shallow  donor  confined  to  move  in  spherical  Quantum  Dot  –  Quantum  Well  (GaAs-GaAlAs). Calculations  are  performed  in  the  framework  of  the  effective  mass  approximation  using  the  Hass variational approach. We describe the effect of the quantum confinement by a infinite deep potential. The result shows that the corrections due to the magnetic and electric field are very important and cannot be neglected or ignored. We have demonstrated the existence of a critical value cribawhich can be used to distinguish the three dimensions confinement from the spherical surface confinement and it’s may be important for the nanofabrication techniques.• key words: Quantum Dot - Quantum Well, Magnetic Field, Electric Field* kh.rahmani@enssup.gov.maI.IntroductionIn the past ten years, studies on the impurity states  in  nanostructure  semiconductors,  such  as quantum well (QW), quantum well wire (QWW) and quantum  dots  (QD),  have  become  subjects  of extensive  investigations  both  in  basic  and  applied researches [1-5]. Recently,  a  new class  of  quantum dots called quantum-dots quantum-well (QDQW) or Inhomogeneous  quantum  dots  ’’IQD’’,  have  been studied both theoretically and experimentally [6-12]. QDQW  are  composed  of  two  semiconductor materials one of which, with the smaller band gap, is embedded  between  a  core  and  outer  shell  of  the material with the larger band gap.  We show in Fig.1 a schematic diagram of quantum dots – quantum well structure where a and b are the inner and the outer radius of the QDQW respectively.  For more details on  the  chemical  fabrication  of  this  new  artificial structure,  we  refer  to  references  [13-22]: CdS/HgS/CdS  [13,  15,  17],  InAs/GaAs  [16], CdSe/HgSe/CdSe  [17],  ZnS/CdS/ZnS  [18,  21]  and InAs/ZnSe  [22].  These  structures  could  exhibit somme remarkable and interesting phenomena. The original characteristic of these structures is that their physical properties can be controlled and can be tuned by changing the core radius, the thickness of the well and the size of the outermost shell.Fig.1 : Schematic diagram of quantum-dots quantum-well  structure.  a  and  b are  the inner  and the outer radius of the QDQW respectively11     35   ©2009  The Moroccan Physical and Condensed Matter Society     a         b                         rr0rbaθCoreWellShellV(r)36                                                                  K. RAHMANI1, I. ZORKANI 11Understanding  the  Impurity  states  in  these structures is an important problem in semiconductors technology. Many works have been devoted to the studies  of  impurity  states  in  the  quantum  well, quantum  wire  and  quantum  dots  [1-5].  Recently, J.Silva  et  al [2]  have  calculated  the  donor  binding energy as a function of the donor position in QD with an infinite spherical well and for different radius of the  structure.  They  found  that  the  donor  binding energy decreases when the donor position increases reaching a minimum, as the donor position is equal to the radius of the QD. For QDQW, to the best of our knowledge no work has been done to treat the effect of the donor's position  and magnetic  field  on the binding energy. Cheng-Ying  Hsieh  [6]  have  calculated  the  ground and excited states of hydrogenic impurity located at the  center  of  a  quantum-dots  quantum-well  of  a spherical  core  (GaAs)  coated  by  a  spherical  shell (Ga1-xAlxAs) and then embedded in a  bulk material (Ga1-yAlyAs).The quantum confined stark effect have been the subject of numerous experimental and theoretical investigations [3-5, 7, 10] which  demonstrated that the binding energy are shifted to low energies in the presence of an uniform electric field. Recently, Asaid et  al [4]  have  analyzed  the  effect  of  an  uniform electric field on the energy of a shallow donor placed anywhere  in  quantum  crystallite.  They  found  that when the donor is  placed  at  the center,  the energy level is shifted to low energies, the energy shifted is proportional to the square of electric field. When the donor  is  placed  anywhere,  the  energy  level  shift depends  on  the  direction  of  the  electric  field. Khamkhami  et  al [7]  have  analyzed  the  uniform electric field effects on the excitons, in QDQW, they found that the stark effect appears even for very small size and the energy shift is more significant when the exciton is placed on the spherical surface.In  the  present  work,  we  use  a  variational method  to  calculate  the  hydrogenic  donor  binding energy  in  QDQW  in  the  strong  and  the  moderate confinement regimes. We will examine the effect of the  magnetic  and  the  electric  field  on  the  binding energy. This paper is organized as follows: in section 2, we present the general formalism, we deduce the expression  of  the  donor  binding  energy  in  the presence of the uniform magnetic and electric field. The numerical results and discussions are presented in section 3.II. General formalism  We consider a donor Impurity located at the position  0r of  a  quantum-dots  quantum-well  made out  of  [GaAlAs  (Core)  /  GaAs  (Well)  /  GaAlAs (Shell)]. The confining potential well is assumed to an infinitely deep well.In  the  effective  mass  approximation,  the Hamiltonien of the system is written as:              H= H0 + Vw + W + M               (1)Where the unperturbed Hamiltonien is given by:H0 = T + V = 0022*2 rrem −−∆−ε        (2)Where m* is the electron band effective mass and ε0 is the dielectric constant.  r  is the electron position relative to the donor located at 0r position.With :| 0rr − | = θCosrrrr 0202 2−+        (3)We assumed that the confinement potential energy is modelized by:      〉〈∞〈〈=brandarbraVw0             (4)Where a and b denote the inner and outer radius of the QDQW, respectively (See Fig-1)The energy due to the external electric field  F  is given by:)cos( 0rreFW −= θ          (5)In  the  present  study,  we  do  not  take  into account the possible spin-orbit coupling as well as the Zeeman  effect,  restricting  ourselves  to  the diamagnetic  contribution. Therefore,  our result  may be  interpreted  as  “mean”  result  independently  of  a possible  splitting.  In  these  conditions,  the diamagnetic contribution M due to the magnetic field B reads, using the coulomb gauge, θ222222sin*2rBcmeM =               (6)In the following, all expressions will be given in the effective  units:   202e*m*aε= for  length  and 20242**εemR =  for  energy.  Furthermore,  we 11                               MAGNETIC AND ELECTRIC FIELD EFFECTS ON …. 37introduce  the  dimensionless  parameters **RFaef =  and *Rcωγ =  characterizing the strength  of  the  electric  and  magnetic fields. cmeBc *=ω  is the effective cyclotron frequency.In  order  to  solve  numerically,  the Schrödinger  equation, we use a variational  method. The wave function given by :ψ = ψ0 [1+β f (r cos θ - r0 )]                  (7)Where β is a variational parameter (which takes into account the presence of the electric field) and  ψ0 is the wave function in the absence of the electric field (f = 0) given by:0)]([0rrerarKSin −−−= αψ    (8)   α is a variational parameter and abK−= π . The  exponential  factor  exp- 0rr−α describes  the coulomb spatial interaction. The  energy  is  obtained  by  the  minimization  with respect to the variational parameter α :Et = αminψψψψ H           (9)The binding energy Eb of the Impurity is given by:Eb = ESub – Et                    (10)Where Et and ESub represents the state energy of an electron  in  QDQW  with  and  without  the  impurity binding respectively.III. Results and DiscussionWe will examine the effect  of  the electric and magnetic fields.Two extreme cases that present them selves : a = 0 (i.e homogeneous quantum dot ’’HQD’’ of b radius ) and a → b for b fixed wich corresponds to an infinitely thin spherical layer.III.1 Effect of the electric fieldIn  fig.2,  we  present  the  binding  energy  Eb as  a function of the ratio a/b for two different confinement regimes b = 1 a* and b = 2 a* and for two several values of the electric field f = 0, 0.4 and 0.8. We notice  that  when  the  electric  field  increases  the binding  energy  decreases  and  we  remark  that  the effect  of the electric  field becomes more important when the ratio a/b offers toward 1 or the outer radius of the QDQW increases. We also remark that the binding energy Eb presents a minimum  for  a  critical  value  of  the  ratio  (ba)crit depending of the value of b. For a/b equal zero, Eb recovers the limit corresponding to a HQD.FIG.  2:  Variations  of  the  binding  energy  Eb as  a function of the radio a/b for two different values of the outer radius of the QDQW, b = 1 a* and 2 a* and two values of the electric field f = 0.4 and 0.8. The solid  curve  corresponds  to  the  binding  energy variation in the absence of electric field.In fig.3, we fix the outer and the inner radius of the QDQW respectively at b = 3 a* and a = 1 a* and we plot  the  binding  energy  bE  as  a  function  of  the donor  position  0r  for  two  different  values  of  the electric  field  f  =  0  and  0.4.  We  remark  that  the electric field reduced the binding energy. Its effect is more pronounced when the impurity is placed to the center of the spherical layer ( 20bar += ) and decreases when  the  donor  moves  toward  extremities  of  the spherical layer.38                                                                  K. RAHMANI1, I. ZORKANI 11FIG. 3 : Variations of the binding energy as a function of the impurity position for two different values of the electric field.III.2 Effect of the magnetic fieldIn  fig.4,  we  present  the  binding  energy  Eb as  a function of the ratio a/b for two different confinement regimes b = 1 a* and b = 2 a* and for various values of the magnetic field γ = 0, 0.4 and 0.8. We observe that  the  binding  energy  increases  as  the  magnetic field increases.0,0 0,2 0,4 0,6 0,8 1,01,82,02,22,42,62,83,03,23,43,63,84,0b=2b=1 E b (R*)Ratio a/bFIG.  4:  Variations  of  the  binding  energy  Eb as  a function of the radio a/b for two different values of the outer radius of the QDQW, b = 1 a* and 2 a* and various values of the magnetic field γ =0, 0.4 and 0.8.In fig.5, we fixed the outer and the inner radius of the QDQW respectively at b = 3 a* and a=1a* and we plotted  the  variation  of  the  binding  energy  bE  associated  to  two  different  values  of  the  magnetic field  γ =  1  and  0.3  as  a  function  of  the  impurity position 0r . We remark that when the magnetic field increases the binding energy increases too. Its effect is more pronounced when the impurity is placed to the  center  of  the  spherical  layer  ( 2ba0r += )  and decreases when the donor moves toward extremities of the spherical layer. We can explain this result by the fact that the geometric confinement increases.1,0 1,2 1,4 1,6 1,8 2,00,120,140,160,180,200,22a=1 and b=2E b(B=1)-Eb(B=0.3)donor position r0FIG.  5 :  variation  of  the  binding  energy  bE  associated  to  two  different  values  of  the  magnetic field  γ =  1  and  0.3  as  a  function  of  the  impurity position 0r .IV. Conclusion In  conclusion,  we have  studied  the  donor binding  energy  in  QDQW.  The  calculation  was performed  within  the  effective  mass  approximation and using the variational method. An infinitely deep potential well have been used. The results show that the binding energy depends strongly on the core and the shell radius. We have demonstrated the existence of a critical value (ba)crit which may be important for the nanofabrication techniques. This value may also be  used  to  distinguish  the  three  dimensional confinement from the spherical surface confinement. 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MAGNETIC AND ELECTRIC FIELD EFFECTS ON THE BINDING ENERGY OF A SHALLOW DONOR IN QUANTUM DOT–QUANTUM WELL

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Revues Scientifiques Marocaines

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