Varshni, Y P

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Imho"./,/>/iw. 79 (3), 235-237 (2005)$ U P  ?Effect of the parabolic confinement potential on the binding «nergy of a donor in a double-step barrier quantum dotY P Vard|»niDeparlnicnt o f  Physics. University o f C]btawa, Ottawa. C^inada KIN 6N5 Email ypvsj (o5||otlaw'a.caiR e c e iv e d  17  June 2004 , a c t ^ te < l  14 J an u ary  2 0 0 ‘i\b s tra c t "I'hc effect o f  the parabolic confinement potential due to electrons on the binding energy ol a donor impurity centered in a double-stepntioniial hairier quantum dot has been investigated within the effective mass approximation, as a function o f  the dot radius(/f) Calculations are urf.>rmed foi several states using a num eiical method both with and without the parabolic confinement potential The results show that as the dot Jills increases, the effect o f  the parabolic potential depends very much on the state and is quite complexcvwords Quantum dot, donor impurity, confined systems»\('vS Nos. 7 3 .2 1 .La, 78 67 He, 85.3.5 Be1. Introductionin iccent years, there has been a great deal o f interest in studying the properties of hydrogenic impurities in quantum dots. An understanding of the physics of impurity states in semiconductor quantum-dot struc tu res is im portan t for several device applications. A number of investigations on the binding energy hydrogenic impurities in spherical quantum dots have been reported during the last few years [ 1-141- Usually, the barrier pcneniial is taken to be infinite beyond the radius of the quantum dotRecently, Belancur, M ikhailov and O liveira [151 have calculated energies of the ground and some excited stales of t^ n-cemre donors (D^) in spherical quantum dots, within the dfective mass approximation, as functions of the dot radius iind tor different potential shapes. They used a trigonometric sweep method for a numerical solution of the radial Schrddinger quaiion in a quantum dot with any arbitrary spherical potential. One of the potential barriers considered by these authors is relevant for a structure consisting o f  a GaAs quantum dot ^urrounded by two concentric spherical layers o f Goq ^Al j^ 3 As Ga  ^s^AI^^jA s. Such structures have been discussed to a high optical nonlinearity for inter-band and inter-sub- pnd transitions [16]. For such a structure, Betancur et al [15]have represented the barrier potential by a double-.step barrier model which consists of a small rectangular potential inside a large rectangular potential. These authors have calculated the properties of the shallow donors in a quantum dot with such a double-step potential barrier and two peaks m the binding energy were found, which is quite interesting.The question of the confining potential of electrons in a quantum dot was considered by Kumar, l>aux and Stern 117] who self-consistently calculated the electron states in GaAs / AIGaAs h e tc ro stru c tu res  with confinem ent in all th ree  dimensions and they found that the evolution of levels with increasing magnetic field is similar to that ftmnd for a parabolic potential. Thus, the confinement potential due to electrons could be described by a simple one-parameter adjustable parabolic potential. Peeters [181 pointed out that this is a consequence of the generalized Kohn theorem which is valid for a harmonic confinement potential.It was of interest to examine the effect of the parabolic confinement potential due to electrons on binding energies in a structure like that o f  ^  GaAs quantum dot surrounded by two concentric spherical layers o f Ga ,^ ^ Alj, 3AS and Ga^ ^^ .AIq^^As , which has been considered by Betancur e ta l \15] and for which the double-step potential barrier m<xlcl is valid.236 y  P  Varshnt!n the present paper, wc first present results on the binding energies o f  several states o f  an on-centre hydrogenic donor in such a quantum dot structure as a function o f  the radius o f  the dot, taking into account the Coulomb potential and the double­step potential barrier. Next, we obtain results on the binding energies taking into account the parabolic confining potential due to electrons also. The difference between the two binding energies gives the contribution due to the parabolic potential. The behaviour o f this contribution is studied as a function o f  the radius o f  the dot.2. Theory and calculationsWc shall use atomic units such that the unit ot' length is the reduced Bohr radius j  ni * e ” , and the unit o f energyis reduced Rydberg Ry -  / H r  l'I = . Here, w*is the effective mass and is the dielectric constant o f  the material o f the quantum dot. In these units, the Hamiltonian can be written as/ /  -  -V* -  -  + y~r~ ^  I ( 1 )where ~h<Oj,j2 Ry, c o being  the harm onic o sc illa to r frequency. is the double-step spherical rectangular barrier potential used by Betancur ct t// [ 15J, which is as follows :To = 0  for r <  0 3 R  --40 Ry for 0 3 R <  r <R- 80 Ry for r > R. (2)We have c a lcu la ted  the e ig e n e n e rg ie s  by num erica l integration o f the SchrOdinger equation using Nurncrov's method and the logarithmic mesh for the T ,^ 2.y, 2p, 3 .v and 3/> states o f  the donor, fo see the effect o f  the confining potential due to electrons, two sets o f  calculations were carried out: without the /  “ term and with it. The value o f i s  usually considered to be less than 1. Here we have taken two values o f  y p  , namely0.4 and 0.8. The term was assum ed to be zero for r > R.Calculations were carried out for a large num ber o f  values of/?.3. R esults and  discussionThe binding energy o f  the impurity is defined as the energy o f  the system without the impurity present minus the energy with the impurity (£ )  ;(3)First we consider the case when there is no confinem ent due to electrons. In Figure I , we show the binding energies for the 15, 2s and 3a* states, and in Figure 2 for the 2p  and 3p states as a function o f /?. Betancur et a / f 15] have obtained results for \s , 2s and 2p states (their Figure 8). O ur results for these statesfully confirm theirs except on one important point. The cur^ t; labelled 2p  in their Figure 8 is actually for the 2 .s stiitc, similarly the one labelled 2s  is actually for the 2p state Figure 1 shows two peaks for the Tv and 2s states, hut th(. behaviour o f  the 3.v state is more com plicated. 7'herc are iw^  close peaks, followed by a minimum and a peak and then i',, binding energy continues to decrease. It will be noticed i- Figure 2 that 2p  and 3p states show a sim ilar behaviour, boii, having two peaks.Figure 1radius R the sake, Hmdmg enetgies lor Ihe l.y, 2.s and l.v states as a tunctn'ii ,>! i The curve for the 2 .^  state ha,s been drawn with bn>kcn liik" o f  clarilNFigure 2. Binding energies for the 2p and 3p states as l function i.f ii'radius /? fhe curve for the 3p  state has been drawn with broken line*' the sake o f  clarityE ffec t o f  the  p a ra b o lic  confinem ent p o ten tia l on the hindin}^ e n e r ^ ' etc 237Next, we consider the effect o f  the parabolic confinement potential due to electrons on the binding energy. We define the binding energy difference, A E , as follows:A E ^ E ,^ i n J ) ~ E , ( n ,  C). (4)uherc £ /,,,( /^ 0  b inding energy when the confiningpotential due to the electrons is included and () is thebinding energy without it.We show AE  as a function o f  R in Figure 3 for Fv and 2s slates for y p * 0.4, and in Figure 4 for the 2p  state for y  p 0.4.I'leiirc 3. Binding energy difference AE  for the l.v and 2.v slates as a of H for y /, — 0 4  I he curve for the 2s slate has been diawn 'Mill broken lines for the sake o f  claritvIt will be noticed from Figure 3 that l.v state shows a small maximum at /? -  1 and then AE  continues to increase with R. In the case o f  2.v state, AE  at first, shows a shallow minimum at /? -  0.9 followed by a sharp peak and then it continues to decrease. Figure 4 shows that for 2p stale, AE  shows a sharp and narrow minimum, followed by a sharp peak at R -  1.5 and then it continues to increase. 1 'hus, we see that the effect o f the parabolic potential depends very much on the state and is quite complex. The results for yp  0.8 qualitatively show a very  ^ similar behaviour; quantitatively they arc o f  course different.; 4. ConclusionsI^ We have investigated the effect o f  the parabolic confinement j potential due to electrons on the binding energies o f  l.v, 2 ,v and ; 2p states o f  a donor impurity centered in a double-step potential ' barrier quantum dot w ithin the eficclivc mass approximation, as a function o f  the dot radius. The results show' that as the dot radius increases, the effect o f  the parabolic potential depends very much on the stcite and is quite complexAcknowledgmentThis work was supported in part, by a research grant from the Natural Sciences and Fnginecrm g Research Council o f  Canada.Rc fere 11 CCSMl I2J.1 I /h ii , J .1 Xiong and B L Ciii /Vn.s Rev B4I 6001 (1090)I) S Chuu, C M lls ia o  iind W' N Mci Rhys R ev  B46 .?X9K(1 9 9 2 )I’ctcv-M crchancano R h y\ RevHgure 4. 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Effect of the Parabolic Confinement Potential on the Binding Energy of a Donor in a Double-Step Barrier Quantum Dot

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Effect of the Parabolic Confinement Potential on the Binding Energy of a Donor in a Double-Step Barrier Quantum Dot

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