Cylindrical quantum well of finite depth in external magnetic field.

Transcription

1 Cylidical quatu well of fiite depth i eteal agetic field Lobaova OR a Ivaov AI b ab Kaliigad State Uivesity Theoetical Physics Depatet Kaliigad Russia Abstact egy spectu of a electo cofied by fiite had-wall potetial i a cylide quatu dot placed i weak up to 00 koe hoogeeous eteal agetic field wee calculated usig the ethod of vaiatio of vecto potetial lectoic otio alog the cylide ais is liited by oediesioal ifiite potetial baie ad electoic otio o the plae pepedicula to the ais is liited by two-diesioal fiite potetial baie PACs: 35Md 3640Cg 730D Keywods: Quatu well lectoic stuctue Magetic field Itoductio Recet pogess i aotechology allows ceatig seicoducto aostuctues quatu dots QDs wells o wies [-3] - whee electos ae cofied i oe two o all thee diectios The electoic popeties of dots show ay paallels with those of atos Most otably the cofieet of the electos i all thee spatial diectios esults i a quatied eegy spectu [4-6] Quatu dots ae theefoe egaded as atificial atos Such aostuctues ae suitable fo epeiets that caot be caied out i atoic physics i paticula due to the high level of cotol This has give ise to the possibility of a ube of vey iteestig applicatios such as poducig sigle-electo tasistos [7] quatu dot lases [8] etc Spheelike cicula cylidelike o coal shaped eclosed stuctues wee widely ivestigated i aoete scale [9-5] It is especially iteestig to obseve the effect of a agetic field o the atoic-like popeties eegy spectu agetic susceptibilities etc [6-9] Coespodig autho: tel: ; fa: ail addess: olgalobaova@yahooco

2 ffective ethod of calculatio of weak eteal agetic field ifluece o eegy levels i quatu dot is ethod of vaiatio of vecto potetial offeed by TK Rebae [0] alie this ethod was successfully applied to calculatio of poto uclea agetic shieldig costats i diatoic olecules [] ad to calculatio of eegy levels of a electo cofied i a closed egio of space with cylidical geoety [6] I the peset wok we have ivestigated the eegy spectu of oe electo cicula cylide quatu dot with a had-wall cofieet i the pesece of a eteal agetic field Alog the cylide ais a electo is cofied by ifiite potetial baie wheeas o the plae pepedicula to the ais electo is cofied by potetial baie of fiite height I such stateet of the poble the Schoedige equatio fo cicula cylide QD i cylidical coodiates is sepaable as fo the etee case of a ifiite potetial well It is ecessay to ephasie that fo the aalogous cicula cylide poble with fiite baie i all thee diectios the Schoedige equatio is ot sepaable as istakely asseted i [4] Applied agetic field is diected pepedicula to the cylide ais ad cosideed to be weak up to 00 koe fied ad hoogeeous Calculatio ethod ad esults We coside a electo cofied i cicula cylide of adius R ad height l The electo iteacts with a potetial: V V 0 ρ > R 0; l 0 ρ 0; R 0; l < 0 > l Fo give cylidical QD the sepaatio of vaiables of Schoedige equatio i cylidical coodiates leads to two equatios: oe equatio cotais -vaiable ad ca easily be solved; othe equatio i XY-plae cotais agula ad adial pats Solutios of adial pat ae Bessel fuctios egy spectu i XY-plae y is obtaied fo the cotiuity coditio of logaithic deivative at the bouday of the well ρ R: J kr J kr K ℵR K ℵR whee J kr K ℵ R ae Bessel fuctios k y ℵ V0 y i au Coespodig eige-equatios ae 0 : kk > 0 : kk 0 ℵR J ℵR J kr ℵJ 0 kr ℵJ kr K kr K ℵR ℵR 3 Total eegy y Futhe give atificial ato is placed i eteal agetic field diected pepedicula to the cylide ais with vecto potetial А Vecto potetial А satisfies the coditio ot А This

3 coditio defies vecto potetial to withi the gadiet of abitay fuctio of electo coodiates ece all vecto potetials of the class A [ ] f 4 whee f - abitay doubly diffeetiable fuctio physically equal: they descibe the sae agetic field Because of gadiet ivaiace of Schoedige equatio electo eegy ad coectios to eegy ae ivaiat elatively the choice of fuctio f It is well kow that i case of weak agetic field coectios to eegy i the fist ad secod odes of petubatio theoy ae as follows: d d 0* 0* W W 0 p dv 0 dv p 0* W dv 5 d whee - diaagetic ad paaagetic coectios to eegy espectively p W W - liea ad quadatic elatively agetic field itesity i au petubatio opeatos W i A diva c W c A 6 I give poble upetubed wave fuctio 0 is eal hece value tus to eo W is iagiay To calculate quadatic coectio it is ecessay to kow coectio wave fuctio which ca be obtaied eve appoiately oly i siplest cases I ethod of vaiatio of vecto potetial [0 ] calculatio of coectio to wave fuctio i agetic field is eplaced by calculatio of optiu fuctio of gadiet tasfoatio f opt withi the vaiatio pocedue The optiu fuctio f opt is cosistet with optiu vecto potetial A opt [ ] f opt Withi this ethod calculatio of coectios to eegy caused by eteal agetic field is educed to calculatio of coespodig diaagetic quatities with A opt Diaagetic coectio to eegy coespodig to vecto potetial i au is [ ] f 0 J f dv c 7 Value Jf is a fuctioal elative to fuctio of gadiet tasfoatio f Fo statioaity coditio of fuctioal Jf δjf 0 8

4 it is possible to obtai optiu fuctio of gadiet tasfoatio f opt Miiu value of fuctioal Jf equals to total quadatic elatively agetic field itesity coectio to eegy i Jf 9 Quadatic coectio to eegy is copletely deteied by upetubed wave fuctio 0 Miiu of fuctioal Jf usually is defied appoiately vayig fuctio of gadiet tasfoatio f i soe liited class of fuctios egy levels of electo i cylide QD wee calculated i cicula ad elliptic appoiatios Magetic field was diected alog Y-ais: 0 0 I cicula appoiatio oe ay fid i au i f opt 0 whee 0 dv Theefoe quadatic elatively agetic field itesity coectio to eegy i cicula appoiatio has a fo c whee 0 dv 0 dv I elliptic appoiatio we have i au k i f opt Thus i elliptic appoiatio coectio to eegy is 4 c 3 lecto eegy spectu i the pesece of agetic field is show i Table vey eegy level is chaacteied by thee quatu ubes: 0 ; k ubes eos of Bessel fuctios; k ubes eos of sie It is coveiet to set obtaied eegy levels i ascedig ode of thei values I peset pape fist 0 low-lyig eegy levels ae show Paaetes used i these calculatios ae R 75 ; l 4 ; V 0 ev ; 00 koe

5 Table : Quatu dot eegy levels i ev i agetic field upetubed eegy levels; - eegy levels i agetic field calculated i cicula ad elliptic appoiatios espectively is ot a quatu ube ide ubes fist 0 low-lyig eegy levels i ascedig ode ad coespods to defiite cobiatio {k k } States with 0 ae cosideed Coclusios Calculatio of eegy spectu of oe electo cicula cylide quatu dot i weak hoogeeous eteal agetic field was ipleeted Alog the cylide ais a electo was cofied by ifiite potetial baie wheeas o the plae pepedicula to the ais electo was cofied by potetial baie of fiite height I the fist ode of petubatio theoy coectio to eegy liea elatively agetic field itesity tus ito eo Secod ode of petubatio theoy leads to shifts of eegy levels i agetic field Quadatic elatively agetic field itesity coectios to eegy obtaied i cicula ad elliptic appoiatios weakly diffe fo each othe Refeeces [] TChakaboty Coets Codes Matte Phys [] RTuto The Quatu Dot A Jouey ito futue icoelectoics Ofod Uiv Pess NY 995 [3] LJacak Pawylak AWojs Quatu Dots Spige-Velag Beli 997 [4] MMacucci Kess GJIafate PhysRevB [5] WDeiss RGNaitdiov PhysLettA ; PhysRevB [6] STaucha DGAustig Toda PhysRevLett [7] Zhuag Lei Guo Ligjie Chou SY ApplPhysLett [8] Guda M Physica [9] M F Coie C P Lut ad D M igle Sciece [0] Masuda ad K Fukuda Sciece [] D Peaso ad R J Toucci Sciece [] M Tadic FM Peetes KL Jasses M Kokusiski P awylak J Appl Phys [3] M Tadic FM Peetes KL Jasses Phys Rev B

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