OPTICAL OSCILLATOR STRENGTHS FOR THE ELECTRON QUANTUM TRANSITIONS IN ELLIPTIC NANOTUBES

Transcription

1 OPTICAL OSCILLATOR STRENGTHS FOR THE ELECTRON QUANTUM TRANSITIONS IN ELLIPTIC NANOTUBES V. A. HOLOVATSKY, O. M. VOITSEKHIVSKA, V. I. GUTSUL Chrnivtsi Natinal Univrsity, Chrnivtsi, 58012, Ukrain, Rcivd Sptmbr 26, 2007 Th lctrn nrgy spctrum and th wav functins in smicnductr lliptic nantub cnstructd n th bas f GaAs crystal ar calculatd within th ffctiv mass apprximatin. Th dpndncs f ptical scillatr strngths fr th quantum intrsubband transitins in dipl apprximatin n th fcus distanc ar btaind and analyzd. Thr ar stablishd th slctin ruls and plarizatin ffcts fr intrsubband ptical transitins fr lctrns in lliptic nantub ar prdictd. Ky wrds: lliptic smicnductr nantub, ptical scillatr strngths, lctrn nrgy spctrum. 1. INTRODUCTION Th dvlpmnt f mdrn smicnductr lctrnics and transitin t th nanlctrnics ar tightly bund t th utilizatin f smicnductr naatrials and nantchnlgis. It is blivd that thir us in nanlctrnics wuld allw t crat nanstructur micrprcssrs, trabit mmry schms and incras th passing ability f cnnctin channls. Th dvlpmnt f nanlctrnics anticipats th phna f siz quantizatin f charg carrirs nrgy spctra in nw smicnductr dvics. Rcntly, th grat intrst f th rsarchs has bn attractd by n dimnsinal smicnductr nanstructurs, studid bth thrtically and xprimntally. Th mdrn tchnlgis allw grwing smicnductr quantum wir and nantubs with diffrnt shap f transvrsal crss. Th thrtical invstigatins ar mainly prfrmd fr th cylindrical quantum wirs fr which thr ar alrady btaind th xact slutins f Schrdingr quatin within Bssl functins. Th smicnductr quantum wirs and nantubs with cmplicatd shap f transvrsal crss, rcntly succssfully grwn by diffrnt mthds [1, 2] ar thrtically studid nt nugh. Th thry f quasiparticl spctra in lliptic quantum wirs (EQW) is stablishd in rf. [3 5]. Papr prsntd at th 8th Intrnatinal Balkan Wrkshp n Applid Physics, 5 7 July, 2007, Cnstanþa, Rmania. Rm. Jurn. Phys., Vl. 53, Ns. 7 8, P , Bucharst, 2008

2 834 V. A. Hlvatsky, O. M. Vitskhivska, V. I. Gutsul 2 It th papr it is prfrmd th invstigatin f lctrn nrgy spctra and wav functins and ar calculatd th scillatr strngths f its quantum transitins in lliptic smicnductr nantubs (ESN). 2. HAMILTONIAN OF THE SYSTEM AND SOLUTIONS OF SCHRODINGER EQUATION It is studid th lliptic smicnductr nantub GaAs, cnfind by innr and utr cylindr with smi axs a 1, b 1 and a 2, b 2, rspctivly, mbddd int th dilctric matrix r vacuum. Th crdinat systm is chsn in such a way that Oz axis is dirctd alng th tub, Ox and Oy alng smi axs a and b, rspctivly. An lctrn is frly mving in th dirctin alng nantub with th nrgy Ez = 2 k z 2 /2 μ, whr μ its ffctiv mass. Th nrgy, causd by transvrsal mvmnt f quasiparticl is btaind as a slutin f Schrdingr quatin 2 ΔΨ ( x, y) + U( x, y) Ψ ( x, y) = EΨ( x, y ), (1) 2μ whr U(x, y) rctangular ptntial f siz quantizatin. Eq. (1) in th lliptic crdinat systm (ξ, η, z), using th rlatinships is rwrittn as ξ x = f csh ξ cs η, 0 ξ< y = f sinhξ sin η, 0 η< 2π, (2) z= z, < z<+ f 2k2 + + (csh22ξ cs22 η) Ψ( ξ, η ) = 0 2, (3) η whr f = a b = a b fcus distanc f innr and utr llips and k = 2 μe /. Whn th nantub walls ar impntrabl fr th lctrn, th variabls in th wav functin can b sparatd, rf. [4] Ψ ( ξ, η ) = R ( ξ) θ ( η). (4) m Th radial R ( ξ ) and angular θm ( η) parts f wav functin satisfy Mathiu quatins 2 2 m m 2 2 θ ( η)/ η + ( c 2q cs2 ηθ ) ( η= ) 0, (5) R ( ξ)/ ξ ( c 2q csh2 ξ) R ( ξ ) = 0 (6) whr q = f 2 k 2 /4, c th sparating cnstant.

3 3 Elctrn quantum transitins in lliptic nantubs 835 Th dtail analysis f angular and radial Mathiu quatins and thir gnral slutins hav bn prfrmd in rfs. [3, 4]. Th slutins f q. (5), satisfying th pridical cnditins, ar vn cm( q, η) and dd sm( q, η) Mathiu functins f th first kind. Th slutins f radial q. (6) is linar cmbinatin f vn and dd mdifid Mathiu functins f th first and scnd kind: R A (, ξ ) + mjm q BmNm( q, ξ), vn stats, ( q, ξ ) = Am Jm ( q, ξ ) + BmNm ( q, ξ), dd stats. (7) Th rlatinships btwn A m, A m, B m, B m cfficints and lctrn nrgy spctrum ar dfind by th bundary cnditins: A (, ξ 1) + mjm q BmNm( q, ξ 1) = 0, A (, ξ 2) + mjm q BmNm( q, ξ 2) = 0 A (, ξ 1) + mjm q BmNm( q, ξ 1) = 0, (8) A (, ξ 2) + mjm q BmNm( q, ξ 2) = 0 whr ξ 1 = arctanh( b1/ a 1), ξ 2 = arctanh( b2/ a2). Th systms f qs. (8) rspctivly A m, A m, B m, B m cfficints hav th nn zr slutins nly at th magnituds q ( ), satisfying crrspnding disprsin quatins J ( q, ξ ) N ( q, ξ ) N ( q, ξ ) J ( q, ξ ) = 0, (9) m 1 m 2 m 1 m 2 J ( q, ξ ) N ( q, ξ ) N ( q, ξ ) J ( q, ξ ) = 0. (10) m 1 m 2 m 1 m 2 μ Th magnituds q ( ) = 2 f E ( ), 2 2 satisfying disprsin qs. (9, 10) dtrmin th infinit st f quasiparticl discrt nrgy lvls E ( ), whr n = 1, 2 main quantum numbr dnting numbr f th rt f th rspctiv quatin. Th scillatr strngths fixing th prbabilitis f ptical quantum transitins ar givn by th xprssins 2μ 2 Fnl nl = ( E ), 2 2 E d (11) whr E and E th nrgis f th first and last stats, d pratr f dipl mmnt prjctin at th plarizatin dirctin. 3. RESULTS OF CALCULATIONS AND THEIR DISCUSSION Th cmputr calculatins f th lctrn nrgis wr prfrmd fr th lliptic nantub GaAs with impntrabl walls at k = 0. Th ffctiv masss f

4 836 V. A. Hlvatsky, O. M. Vitskhivska, V. I. Gutsul 4 lctrn and lattic cnstant f bulk smicnductrs GaAs ar: μ = 0.067m 0 (m 0 th mass f fr lctrn) and a GaAs = 5.62 Å, rspctivly. In Fig. 1 thr ar shwn th nrgy dpndncs f vn E (slid curvs) and dd E (dash curvs) lctrn stats n th magnitud f innr llips big smi axis (a 1 ) at f = 4,5 a GaAs and Δa = a 2 a 1 = 5 a GaAs. In cas a 1 f, b 1 0, th innr llips dgnrats int th sgmnt with a 1 lngth. Its incrasing at th cnstant magnitud f fcus distanc brings t th rapid dgnratin f lliptic nantub int th circl n, what is prvd by th rlatinships a 1 /b 1, writtn at th tp axis f th figur. As a rsult, th nrgis f vn and dd stats, btaind frm qs. (9, 10), bcm clsr t ach thr and apprach th valus f th nrgis in cylindrical nantub. Th furthr incrasing f a 1 causs th dgnratin f th nrgis vr th quantum numbr m and it is btaind th lctrn nrgy spctrum in plan smicnductr layr with thicknss Δa. Fig. 1 Dpndnc f lctrn nrgy spctrum in lliptic nantub n a 1 magnitud at th cnstant thicknss Δa = 5 a GaAs.

5 5 Elctrn quantum transitins in lliptic nantubs 837 At th Fig. 1 n can s that at th small magnituds f magntic quantum numbr th nrgis f vn stats (n, m) ar cls t th nrgis dd stats (n, m + 1). Fr th big m it is bsrvd th rapprchmnt f nrgis f (n, m) and (n, m) stats. Such bhavir f th spctrum can b xplaind analyzing Fig. 2, whr it is shwn th dpndnc f lctrn nrgy in nantub n th magnitud f th fcus distanc (th magnitud f = 4.5 a GaAs, at which th graphics prsntd at Fig. 1 ar calculatd, is shwn by dash-dttd vrtical lin). Frm Fig. 2 it is clar that at f = 0 (llips dgnrats int th circl) th nrgis f vn and dd stats cincid. Th incrasing f fcus distanc at th cnstant a 1 and Δa valus causs th shift f all nrgy lvls int th rgin f smallr nrgis du t th siz quantizatin ffct and th incrasing f Δb magnitud. Hrin, th nrgy lvls ar splitting and th nrgis f th dd stats ar dcrasing fastr, appraching th nigbur vn stats with th smallr (by n) magnitud f quantum numbr m. At th incrasing f lattr th magnitud f th nrgy lvls splitting bcms smallr. In limit cas (f = a 1 ) th innr llips dgnrats int a sgmnt and lliptic nantub bcms similar t th lliptic quantum wir with a 2 = a 1 + Δa and b 2 = ( a2 2 1/2 2 f ) smi axs. Its nrgy spctra is frmd in th fllwing way: th grund nrgy lvl is btaind frm th cinciding Fig. 2 Dpndnc f lctrn nrgy spctrum in GaAs nantub n th magnitud f fcus distanc ( f ) at a 1 =10 a GaAs. Slid curvs vn stats, dash dd stats.

6 838 V. A. Hlvatsky, O. M. Vitskhivska, V. I. Gutsul 6 lvls (1, 0) and (1, 1). Th nrgy lvls f quantum wir vn stats (n, m) ar btaind frm th crrspnding nrgy lvls f lliptic nantub and th nrgis f dd stats (n, m) frm th stats (n, m+1) f lliptic nantub. Such nrgy spctrum ds nt cincid t th nrgy spctrum f lliptic quantum wir with a 2 and b 2 smi axs, calculatd accrding t th thry dvlpd in rfs. [3 5]. Fig. 3 Distributin f prbability dnsity f lctrn lcatin in th stats (1, 0), (1, 1), (1, 1) and (1, 2) in ESN at a 1 = 10 a GaAs, Δa = 5 a GaAs and diffrnt fcus distancs f = 9 a GaAs, 9.99 a GaAs, 10 a GaAs and EQW at a = 15 a GaAs and f = 10 a GaAs.

7 7 Elctrn quantum transitins in lliptic nantubs 839 Th diffrnc in lctrn nrgy spctra in dgnratd lliptic nantubs and lliptic quantum wir can b xplaind studying th distributin f prbability dnsity f lctrn lcatin in ths nansystms, prsntd in Fig. 3. Fig. 3 prvs that th nrgis f th dd stats f dgnratd nantub ar ttally qual t th nrgis f th rspctiv stats in th lliptic wir. Fr th vn stats f nantub th prsnc f dgnratd innr llips, in all pints f which th wav functin is qual t zr, brings t th ssntial incrasing f quasiparticl nrgy. Analysis f d n m matrix lmnt magnitud, dfining th scillatr strngth, pns th ability t dtrmin th slctin ruls fr th allwd intrsubband transitins in dipl apprximatin. Th calculatins shw that th matrix lmnt is nt qual t zr nly fr th transitins with Δm = ±1. Th sam slctin ruls xist als fr th cylindrical quantum wirs and nantubs. Dpndnc f scillatr strngth f svral allwd quantum transitins n th magnitud f fcus distanc is prsntd in Fig. 4. Fig. 4 prvs that th transitins btwn quantum stats with diffrnt pairing ar y-plarizd and th transitins btwn th stats with qual pairing x-plarizd. Th scillatr strngth f y-plarizd light is dcrasing and scillatr strngth f x-plarizd light is incrasing fr th biggr fcus distanc f nantub. Sinc, th intnsity f light, mittd in th dirctin f small llips smi axis at th big magnituds f fcus distanc is mr than th intnsity f Fig. 4 Dpndnc f scillatr strngth f intrsubband quantum transitins n th nantub fcus distanc.

8 840 V. A. Hlvatsky, O. M. Vitskhivska, V. I. Gutsul 8 light mittd alng th big smi axis. Such plarizatin pculiaritis f nantubs can find th practical utilizatin fr th mdrn ptlctrnic dvics. Frm th Fig. 4 n can als s that at f = 0, whn th lliptic nantub dgnrats int th cylindrical n, th intnsity f x- and y-plarizd light mitting cincid, as it must b fr th systm, istrpic in radial dirctin. 4. CONCLUSION Th pculiaritis f nrgy spctra and distributin f prbability dnsity f th lcatin f quasiparticl in nanhtrsystm ar invstigatd. It is stablishd that in ESN th dgnratin f nrgy spctrum is takn ff. In limit cass th btaind nrgy spctra fr ESN ar th sam as th knwn spctra f quasiparticls in cylindrical nantubs and lliptic quantum wirs. It is shwn that th magnitud f th splitting f nrgy lvls f vn and dd stats and als th scillatr strngths f intrsubband transitins dpnd n th llipticity f nantub. Such prprtis f nantubs can b usd in ptical dvics f n axis prssur and th thr ptlctrnics instrumnts f mdrn nantchnlgis. It is stablishd that th light, mittd du t th quantum transitin btwn th stats with diffrnt pairing, is linarly plarizd alng th small llips axis and du t th transitins btwn th stats with qual pairing alng th big llips smi axis. This prprty can b utilizd in light mitting dvics. REFERENCES 1. J. Nbrisaka, J. Mthisa, S. Hara, T. Fukui, Appl. Phys. Ltt. 87, (2005). 2. P. Mhan, J. Mthisa, T. Fukui, Appl. Phys. Ltt., 88, (2006). 3. M. van dn Brk, F. M. Ptrs, Physica E., 11, 345 (2001). 4. V. A. Hlvatsky, V. I. Gutsul, O. M. Makhants, Rmanian Jurnal f Phys., 52, 305 (2007). 5. V. Hlvatsky, V. Gutsul, Jurnal f Optlctrnics and Advancd Matrials, 9, 1437 (2007).