Stablty of an Exton bound to an Ionzd Donor n Quantum Dots by S. Baskoutas 1*), W. Shommrs ), A. F. Trzs 3), V. Kapakls 4), M. Rth 5), C. Polts 4,6) 1) Matrals Sn Dpartmnt, Unvrsty of Patras, 6500 Patras, Gr. ) Forshungszntrum Karlsruh, Hauptabtlung Informatons-und Kommunkatonsthnk, 7601 Karlsruh, Grmany 3) Physs Dpartmnt, Unvrsty of Patras, 6500 Patras, Gr. 4) Engnrng Sn Dpartmnt, Unvrsty of Patras, 6500 Patras, Gr. 5) Forshungszntrum Karlsruh, Insttut für Matralforshung I, POBox 3640, 7601 Karlsruh, Grmany 6) Forshungszntrum Karlsruh, Insttut für Nanothnolog, Forshungszntrum Karlsruh, 7601 Karlsruh, Grmany Abstrat Total nrgy, bndng nrgy, rombnaton rat (of th ltron hol par) for an xton (X) bound n a parabol two dmnsonal quantum dot by a donor mpurty loatd on th z axs at a dstan d from th dot plan, ar alulatd by usng th Hartr formalsm wth a rntly dvlopd numral mthod (PMM) for th soluton of th Shrödngr quaton. As our analyss ndats thr s a rtal dot radus R suh that for R< R th omplx s unstabl and wth an nras of th mpurty dstan ths rtal radus nrass. Furthrmor, thr s a rtal valu of th mass rato σ = m m h suh that for σ < σ th omplx s stabl. Th apparan of ths stablty ondton dpnds both on th mpurty dstan and th dot radus, n a way that wth an nras of th mpurty dstan w hav an nras n th maxmum dot radus whr ths stablty ondton appars. For dot rad gratr than ths maxmum dot radus (for fxd mpurty dstan) th omplx s always stabl. PACS: 73.3. Ps; 73.35. J; 71.55. ; 73.0. Dx; 71.55. Eq Kywords: Quantum Dots; Extons; Impurts; Bound Extons *Corrspondng author: S. Baskoutas -mal: bask@ds.upatras.gr, Tl.: 30610996370, Fax: 30610996370
1. Introduton Th thr-partl omplx (D, X) onsstng of an ltron and a hol bound to an onzd donor s th smplst possbl bound xton omplx. Sn Lamprt [1] frst suggstd th possblty of th bndng of an xton (X) to nutral (D 0 ) and onzd (D ) shallow mpurts n smondutors, a grat dal of ffort has bn dvotd to th study of ths ompounds both n bulk (3D) [] and onfnd ompound smondutors [3]. In bulk smondutors, t has bn provn [4] that th (D, X) omplx s only stabl whn th ltron to hol fftv mass rato σ = m mh s lss than th rtal valu σ 3D = 0.46. Howvr, mor favorabl ondtons for th dtton of th (D, X) omplx ar xptd [5,6] for lowr dmnson systms. In th prsnt papr w onntrat our study on th stablty ondtons of th barrr (D, X) omplx n a quantum dot (QD) wth parabol onfnmnt, usng th Hartr formalsm wth a rntly dvlopd numral mthod for th soluton of th Shrödngr quaton, whh s alld Potntal Morphng Mthod PMM [7,8]. W alulat th total nrgy of th omplx, th ground stat nrgy of th nutral donor, th bndng nrgy of th omplx and th rombnaton rat (wavfunton ovrlappng) of th ltron hol par for varous dot lngths, varous rats σ and dffrnt mpurty postons along th z axs. In our analyss w assum also that th onsdrd quantum dot systm s a two dmnsonal systm, whr ts hght z << R (R s th radus of th dot) [9]. Thrfor, n S. w prsnt th alulaton mthod and w obtan th total nrgy of th omplx, th bndng nrgy as wll as th wavfunton ovrlappng (rombnaton rat). In S. 3 w prsnt numral rsults and w dsuss thm. Fnally S. 4 s dvotd to onludng rmarks.. Thory a. Bndng Enrgy Wthn th fftv mass approxmaton, th Hamltonan of th (D,X) omplx n a QD, onsstng of an ltron and hol bound to onzd donor mpurty (D ) loatd at a fxd dstan d along th z axs, subjtd to a parabol onfnmnt potntal an b wrttn as [10]
3 H p 1 * p 1 * = mω 0 r mhω 0 rh V ( r ) V ( rh ) V ( r, rh ) (1) m m * h * h wth V () d ( r ) = - ε r and [11] V ( ) (3) d r h = ε rh ( r, r ) = ε ( r r z ) 1 ε ( r r ( z δ ) ) 1 h h h V (4) whr ε s th stat dltr onstant and ω 0 s th strngth of th onfnmnt. Furthrmor, th frst trm n q. (4) orrsponds to th ltron hol ntraton wth a ut - off at th fnt xtnt of th wavfunton z along th z axs (whh dnots a vry strong onfnmnt along th z axs) and th sond trm orrsponds to th ltron ndud hol harg ntraton wth a ut - off at z δ, whr δ s th wdth of an ntrfa surroundng th dot and whh tnds to zro for nfnt onfnmnt [8,1]. Th gnfuntons of th Hamltonan (1) ar postulatd n th Hartr form ( r r ) = Φ ( r ) Φ ( r ) Ψ, h h h (5) In suh an approxmaton th quaton for th gnstats of (1) s quvalnt to th followng systm of quatons p m U ( r ) Φ ( r ) = E Φ ( r ) ( =, h) (6) whr th slf onsstnt fftv fld that ats on th ltron, s gvn by th formula
U ( r ) V ( r ) 4 ( r ) Φ ( r ) 1 Φh h h h = C mω0 r drh 1 drh ε ε r rh z r rh ( z and th slf - onsstnt fftv fld that ats on th hol U h ( r h ) s obtand by hangng th nds n th abov formula: nto h and v vrsa. In ordr to fnd gnvalus and gnfuntons of q. (6) wth th PMM [7,8] w nd a rfrn systm wth wll known gnfuntons and gnvalus, as for xampl th smpl harmon osllator n dmnsons ( U HO ( r )). Th ssntal pont now s that th transton from th known systm to th unknown systm (U ( r )) an b prformd by mans of th tm dpndnt Shrödngr quaton (7) δ 1 ) p ( ) h Φ r, t t = Ut ( r ) Φ ( r, t) (8) m wth U t ( r ) σ ( t) U ( r ) [ σ ( t) ] U ( r ) = 1 (9) HO whr σ () t has th proprty: ( t) = 0 σ, for t ta and ( t) = 1 σ for t t b. For ta t t b th funton σ () t may hav any shap but should nras monotonally. A smpl ho whh w hav usd n our alulatons n th prsnt papr s σ () t = a( t t ) wth a = 1 /( t ). a For t t b th ( ) r t b Φ of q. (6) s gvn by a ( r ) = Φ ( r, t ) Φ (10) b and th nrgy E (q. 6) s obtand, usng q. (10) as follows E = Φ p m ( ) r U ( r ) Φ ( r ) dr (11)
5 Thn th total nrgy of th (D,X) omplx aftr th onvrgn n th Hartr shm has th form ( D, X ) = E Eh E (1) and th bndng nrgy s [10] 0 E E( D ) E E( b = h - D, X ) (13) ( 0 ) whr E D s th ground stat nrgy of th nutral donor and E s th lowst lvl of a hol n th QD wthout th Coulomb potntal. Aordng to th orrspondng stablty ondtons [10,13], th (D,X) omplx s stabl whn > 0. h E b b. Rombnaton rat Th wavfunton ovrlappng s dfnd as follows [14] ( r ) Φ ( r ) h f = Φ dr dr (14) h h h whr th abov wavfuntons of th ltron and hol ar obtand wthn th PMM produr (q. (10)), aftr th onvrgn n th Hartr shm. Assumng that th day tm s nvrsly proportonal to th rombnaton rat [14], w an study th day bhavor of th h par n th prsn of th mpurty harg. 3. Rsults Solvng th systm of quatons (6) slf onsstntly n th Hartr shm w tak as ntal wavfuntons (q. (5)) ths of th usual D harmon osllator [15]. Evry tm w nd a soluton of (6), w us th PMM shm [7,8] wth rfrn systm that of th D harmon osllator, and tm ntrval t = 10-4 (n unts m R h whr R s n m) and dx = dy = 0.05 (n unts of R (nm)). In vry
6 traton stp of th Hartr shm w obtan wth PMM both th nrgs and th orrspondng wavfuntons whh ar normalzd. Th onvrgn n th slf onsstnt Hartr produr s obtand n thr or fv rounds. W onsdr th as of GaAs smondutor and w hoos as matral paramtrs ε = 1.4, m =0.067 m, σ = 0.707 [10] and z = 0. R [8,9]) and δ = 10-5 R [8] *. Th fxd δ valu s spfd as th dstan blow whh th man valu of th Coulomb potntal ( ) V dos not hang. Obvously, for vry low δ valus th r, r h man valu of th potntal gvs unphysal rsults du to omputatonal round off rrors. Furthrmor, for gvn valu of whh th man valu of th Coulomb potntal ( r r ) z ( z <<R) orrsponds a valu of δ blow V, s th sam as abov. Workng n dmnsonlss unts w dfn th radus of th QD as [8] ( h mω ) 1 and w alulat ( D, X ) h R =, E as a funton of th QD radus for varous postons of th mpurty along th z axs. As s larly dptd n Fg. 1, E ( D, X ) drass wth th nras of R, but taks largr valus as th mpurty movs away from th QD plan. Ths rsult, whh s n agrmnt wth th orrspondng rsult of rf. [14], ssntally dnots that th smallr th radus of th dot s and/or th largr th mpurty dstan from th dot plan s, th hghr th dffulty for th xton to bound to th hargd mpurty. Furthrmor n Fg. w dpt th bhavor of th 0 nutral donor ground stat nrgy E ( D ) 0 sn ( D ) 0 obvous that as th mpurty movs away from th QD ntr ( D ), as a funton of th QD radus. As s larly E nrass as th dot radus drass (onfnmnt fft) [16]. It s also E taks largr valus. Furthrmor n Fg. 3 w prsnt th bhavor of th bndng nrgy funton of th QD radus. As s larly sn thr s a rtal valu R < E b as a R, suh that for R th (D, X) omplx boms unstabl [10]. Ths rtal valu dpnds strongly on th poston of th mpurty and as w an s from Fg. 3, whn th mpurty dstan from th QD plan nrass. R nrass * Th sam valu of δ has bn obtand n a smlar two dmnsonal quantum dot systm [8] and du to th larg onfnmnt along th z axs, w xpt that t tnds to zro [1].
7 Now n ordr to s th day bhavor of th h par n th prsn of th mpurty, w hav plottd n Fg. 4 th wavfunton ovrlappng f h as a funton of th QD radus for varous mpurty postons along th z axs. Takng nto aount that th day tm s nvrsly proportonal to th wavfunton ovrlappng, w an s that wth th nras of th QD radus th wavfunton ovrlappng drass th day tm nrass.g. th ltron hol par s mol stabl. Furthrmor, f h taks largr valus as th dstan of th mpurty from th QD plan nrass. Ths rsult whh s n agrmnt wth th orrspondng rsult of rf. [14], s n agrmnt also wth th rsults dptd n Fgs. 1 and 3, takng nto aount that largr ( D X ) E, and smallr mans largr dffulty for th xton to bound to th hargd mpurty E b mans that th hol s mor fr to mov away from th omplx and thus th systm s mor unstabl and v vrsa. In Fg. 5 w hav plottd th bhavor of th bndng nrgy as a funton of th mass rato σ for R=7 nm and th rsult w hav obtand ponts out that thr s a σ 3D D rtal valu σ (0.46= < σ < σ =0.88 [13]) suh that for σ < σ th systm s stabl. Th apparan of ths bhavor dpnds from th mpurty dstan as wll as from th dot radus. As s sn wth th nras of th mpurty dstan drass and also nrass th maxmum dot radus n whh ths bhavor appars. For dot rad gratr than ths maxmum radus (for fxd mpurty dstan) th omplx s stabl (Fg. 6). Fnally, n Fg. 7 w dpt th bhavor of th wavfunton ovrlappng as funton of th mass rato σ for R=7 nm and w s that as σ nrass f h nrass and furthrmor as th dstan of th mpurty nrass, f h also nrass whh s n agrmnt wth Fg. 5. σ 4. Conlusons In th prsnt work w hav studd th stablty of (D, X) omplx n a two dmnsonal GaAs QD wth a hght z << R. Usng th Hartr formalsm wth PMM for th soluton of th Shrödngr quaton, w hav obtand th total nrgy of th omplx E ( D, X ), th bndng nrgy E and th rombnaton rat f th ltron hol par. As our rsults ndat thr s a rtal radus for R < b h of R, suh that R th (D, X) omplx boms unstabl. Ths rtal radus dpnds on th
8 mpurty dstan from th dot plan n a way that wth an nras of th mpurty dstan th rtal radus nrass. Ths bhavor as rgards th stablty s n agrmnt wth th bhavor of th rombnaton rat of th ltron hol par, whh drass wth th nras of R and also taks largr valus as th mpurty dstan from th dot plan nrass. Fnally, for a fxd dot radus (R=7nm) w hav obtand a rtal valu of th mass rato σ, suh that for σ < σ th systm s stabl. As s sn wth th nras of th mpurty dstan drass and rtanly nrass th maxmum dot radus n whh ths stablty ondton appars. For dot rad gratr than ths maxmum radus (for fxd mpurty dstan) th omplx s always stabl. σ Aknowldgmnts Ths work has bn supportd by th oopraton btwn th Forshungszntrum Karlsruh, Grmany and th Unvrsty of Patras, Gr and also by th Intrnatonal Off of th Bundsmnstrum für Bldung und Forshung, Grmany. Th authors would lk to thank Mr. Frank Shmtz, from Hauptabtlung Informatons-und Kommunkatonsthnk (HIK), Forshungszntrum Karlsruh, Grmany for hs valuabl hlp wth th VPP Supromputr. Th authors S. B., V. K., and C. P., would lk to thank also th Rsarh Commtt of Unvrsty of Patras, Gr, for fnanal support undr th projt Karathodors: Synthss, Charatrzaton and Proprts of Nanostruturd Smondutors.
9 Rfrns 1. M. A. Lamprt, Phys. Rv. Ltt. 1 (1958) 450.. X. Lu, A. Ptrou, B. D. MComb, Phys. Rv. B 38 (1988) 85. 3. D. C. Rynolds, C. E. Lak, K. K. Bajaj, C. E. Stutz, R. L. Jons, Phys. Rv. B 40 (1989) 610. 4. T. Skttrup, M. Suffzynsk, W. Gorzkowsk, Phys. Rv. B 4 (1971) 51. 5. B. Stébé, L. Stauffr, Suprlatt. Mrostrut. 5 (1989) 451. 6. L. Stauffr, B. Stébé, Phys. Rv. B 39 (1989) 5345. 7. M. Rth, W. Shommrs, S. Baskoutas, to b publshd n Intrn. J. Mod. Phys. B (00). 8. S. Baskoutas, M. Rth, A. F. Trzs, V. Kapakls, C. Polts, to b publshd n Intrn. J. Mod. Phys. B (00). 9. W. X, Sold Stat Comm. 118, 305 (001); Physa B 79 (000) 53. 10. W. X, Phys. Ltt. A 70 (000) 343. 11. P. L. MEun, E.B. Foxman, J. Knart, U. Mrav, M. A. Kastnr, N. S. Wngrn, S. J. Wnd, Phys. Rv. B 45 (199-I) 11419. 1. P. G. Bolatto, C. R. Protto, J. Phys.:Condns. Mattr 13 (001) 319. 13. I. Essaoud, B. Stébé, A. Anan, M. Saabr, Physa E 14 (00) 336. 14. E. X. Png, V. Dalal, Sold Stat Commun. 8 (199) 749. 15. A. W. l. Jaak, P. Hawrylak, Quantum Dots, Sprngr, Brln (1998). 16. U. Woggon, Optal Proprts of Smondutor Quantum Dots, Sprngr Vrlag Brln Hdlbrg (1997).
10 Fgur Captons Fg. 1: Th nrgy E ( D, X ) of th xton omplx as a funton of th QD radus R, for varous mpurty dstans (d) from th QD plan and for σ = 0.707. 0 Fg. : Th nutral donor ground stat nrgy ( D ) E as a funton of th QD radus R, for varous mpurty dstans (d) from th QD plan and for m =0.067 m. Fg. 3: Th bndng nrgy E b of th xton omplx as a funton of th QD radus R, for varous mpurty dstans (d) from th QD plan and for σ = 0.707. It s larly sn th rtal dot radus R suh that for R< R th omplx s unstabl. Furthrmor t s obvous that ths rtal dot radus nrass as th mpurty dstan from th dot plan nrass. Fg. 4: Th wavfunton ovrlappng f h as a funton of th QD radus R for varous mpurty postons (d) along th z axs and for σ = 0.707. Fg. 5: Th bndng nrgy of th omplx as a funton of th mass rato = m mh σ for varous mpurty postons (d) along th z axs and for R=7 nm. It s larly sn th rtal valu σ of th mass rato for ah mpurty dstan suh that for σ < σ th omplx s stabl. It s also obvous that ths rtal valu drass as th mpurty dstan from th dot plan nrass. σ Fg. 6: Th bndng nrgy of th omplx as a funton of th mass rato = m mh σ for mpurty dstan d=0.nm and R=7nm, R=8nm and R=0nm. Fg. 7: Th wavfunton ovrlappng f h varous mpurty postons (d) along th z axs and for R=7nm. as a funton of th mass rato σ for
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