Analysis of a Finite Quantum Well

Transcription

1 alysis of a Fiit Quatu Wll Ira Ka Dpt. of lctrical ad lctroic girig Jssor Scic & Tcology Uivrsity (JSTU) Jssor-748, Baglads ika94@uottawa.ca Or ikr_c@yaoo.co Joural of lctrical girig T Istitutio of girs, Baglads Vol. 7, No. II, Dcbr, bstract I tis papr o disioal (D) quatu cofit i a Fiit Quatu Wll (FQW) is aalyzd troug a siulator usig MTB. particl bavior isid a FQW is discussd ad aalyzd. T ffct of various paratrs suc as wll boudary tickss, dpt of t wll ad widt of t wll ar discussd. T rsults ar copard wit t Ifiit Quatu Wll (IQW). Diffrt typs of pottial structur s bavior ca b aalyzd by usig tis siulator wic is vry usful bfor fabricatio. Kywords Fiit quatu wll, ifiit quatu wll, quatu cofit, quatu tulig. N I. INTRODUCTION ow a day, t buzzig word is t quatu cofit. Quatu ffct tat is dsigd to trap carrirs witi a vry sall spac is kow as quatu cofit. For crtai applicatio or rsarc w d to cag t lctrical or optical proprty of a atrial ad t fficit way to do so is t quatu cofit. W t diatr of a particl is t sa as t agitud of t lctro wav fuctio oly t t quatu ffct is obsrvd. W t siz of t cofiig structur is coparabl wit t wavlgt of t particl t lctroic ad optical proprtis ar cagd. Quatu cofiig ca b do i tr diffrt ways suc as tr disioal (D) w cofid i a quatu dot, two disioal (D) w cofid i a quatu wir ad o disioal (D) w cofid i t quatu wll. D quatu wll (QW) is wll discussd tortically i []. I tis papr t particl (lctro) baviour i a fiit quatu wll is aalysd quatitativly troug siulatios. Hr diffrt paratrs of a D fiit quatu wll suc as t tickss, dpt ad widt ar varid ad t baviour is obsrvd. Ts paratrs variatios ar do quatitativly, wic is vry usful to cosidr prior to ay fabricatio. Fially t rsults ar copard wit t ifiit quatu wll. II. QUNTUM W (QW) pottial wll avig oly discrt rgy valus is kow as a quatu wll (QW). D cofit is possibl i QW. W t QW tickss is coparabl to t carrir wavlgt oly t t cofit is possibl.. Ifiit Quatu Wll (IQW) W t dpt of t pottial wll is ifiit it is calld ifiit quatu wll (IQW). IQW ca b dfid (Fig.) atatically as-, x, () U ( x) =, < x <,, x. Figur. Ifiit Quatu Wll []. ifiit QW is sow i Fig. wr ad ar t statioary rgy stats, ψ ad ψ ar t corrspodig wav fuctios ad t QW is ifiit i dpt. Fro t dfiitio of a QW w kow tat t lctros i t pottial wll or QW av oly crtai discrt valus of allowd rgis. Ts rgis ca b foud troug t forula as []- k π = = = () 8 Wr is t lctro rgy, is t ass of t lctro, is t widt of t wll, is t lctro rgy stat. T wav fuctio of t lctro i QW is dfid as []- πx ψ ( x) = / si ()

2 Ka : J. lc. gg., Ist. grs., Baglads, 7(II), Dcbr B. Fiit Quatu Wll (FQW) W t dpt of t pottial wll is fiit it is calld fiit quatu wll (FQW). FQW ca b dfid (Fig.) atatically as- U, x, (4) U ( x) =, < x <, U, x. III. SIMUTION & RSUTS T siulatio is do by calculatig t statioary stats for a lctro particl wit a ffctiv ass of % of t rst ass wit crtai widt ad dpt by usig MTB. T algorit for tis siulatio is sow i Fig.. Figur. Fiit Quatu Wll []. fiit QW is sow i Fig. wr ad ar t statioary rgy stats, ψ ad ψ ar t corrspodig wav fuctios ad t QW is fiit i dpt. I cas of t FQW t discrt rgy stats ca b rprstd as i []- π (5) U ( + /( )) ltratly w ca rprst t quatio as i []- U ta (6) ( ) = ( U ) Wr- Outsid of t wll t wav fuctio is ot zro but for ifiit cas it is zro. So w av- x fiit > d fro t ucrtaity pricipl w av- p fiit x < p x s dscribd i [] for FQW t avrag valu of otu is lss ta IQW. s a cosquc t kitic rgy isid t wll is lss for FQW ta IQW. Morovr, du to t o-zro valu of wav fuctio outsid of t FQW tr xist t possibility to fid t particl tr ad tis is t rsult of tulig. Figur. Siulatio algorit flowcart.

3 Ka : J. lc. gg., Ist. grs., Baglads, 7(II), Dcbr QW tr will b a corrspodig trasissio. Hr to fid t corrspodig rsoac pak, bisctio tod was usd. s t trasfr atrix approac is usd for tis aalysis, so t approxiatio of a arbitrary pottial fild is do troug stp wis approxiatio. T ti idpdt Scrodigr quatio wit a costat pottial ( V ) is- Figur 4. alyzd Fiit Quatu Wll. Figur 5. Rsoacs & trasissios. For t siulatio purpos first of all t pottial structur is dfid as FQW. T discrtizatio of t structur is do for calculabl trasfr atrix as a product of idividual propagatio atrix as wll as t itrfac atrix. Hr it is cosidrd tat t particl wav (uit wav fuctio) is coig fro rigt to lft. ftr tat t local axia for rlativ rgy dtctio was do by bisctio tod. Fially t rsoacs ad trasissios ar dtctd. T stps of t algorit ar giv i Fig.. T rgy valus corrspodig to t local axias of t trasissio ar cosidrd as t statioary stats. T structur tat was aalyzd is giv i Fig.4. I Fig.4 t aalyzd fiit QW is sow wr t fiit dpt of t wll is V ad t widt is i siz. Tis FQW structur was varid i dpt as wll as t boudaris to cck t ffct o rgy stats. I Fig.5 t rsoacs ad corrspodig trasissios ar sow. I t FQW w tr is a rsoac isid t H Ψ = Ψ ψ + Vψ = Ψ (7) x quatio (7) is a ordiary diffrtial quatio ad t caractristic quatio is- λ = ( V ) Wr is t rgy, is t ass of t particl. So tr ar two coditios to cosidr for t solutio- > V ad < V. T gral solutios will b- ikx Ψ = + ( ) V Wr, k = ad Wr, k = Ψ = ( ) V ikx B (8) k x + B k x I (8) t first tr of t rigt ad sid is calld forward propagatig wav ad t scod tr is calld t backward propagatig wav. Siilarly for (9) t rigt ad sid s first tr is kow as forward dcayig fild ad scod tr is kow as backward dcayig fild rspctivly. I tis aalysis w ar cosidrig ultilayrd structurs. Figur 6. Matrix foratio of t layr structur. (9)

4 Ka : J. lc. gg., Ist. grs., Baglads, 7(II), Dcbr Cosidrig Fig. 6, usig t boudary coditios ad t cotiuity at t itrfac it ca writt- = D + + wit D is t itrfac atrix. Now i cas of wav propagatio i ultilayrd structur agai w cosidr siilar two coditios as it was i costat pottial cas. Now t pottial is ultilayrd wic is dotd as V so for first cas ( > V ) ikd ikd = ad B = B Wr d is t tickss of t layr. If w writ t abov quatio i atrix for w gt- = P wit P Siilarly for t scod cas ( = P B wit P = < ) B V k d = ikd ikd k d Wr P is t propagatio atrix. So for t coplt structur w ca writ- = T Wr trasfr atrix- T + + = PD P DP D... P + D+ P + () So t trasfr atrix is t cobiatio of propagatio atrix ad itrfac atrix. W t rgy is dtrid, usig t trasfr atrix w obtaid t oralizd squard odulus of wav fuctios of diffrt ods wic ar sow i Fig. 7. T od ubr ad tir corrspodig rgis ar giv i Tabl I. It is obsrvd tat w t od ubr is icrasig at t sa ti t rgis ar also icrasig. Ts rgis also dpd o t QW structur. T ffcts of various paratr of t wll will b discussd i t followig sctio. TB I. Mod Nubr MODS & NRGIS rgy (V) Figur 7. Noralizd squard odulus of wav fuctios of diffrt ods.. ffct of Boudaris Tickss For tis aalysis t valus of dpt ad widt of t FQW was fixd i.. dpt & widt wr costat. T rsult of tis boudaris tickss cag is sow i Tabl II. Boudaris Tickss (): ig Valus (V) TB II. THICKNSS VRITION Fro t rsult it is clar tat t cag of boudaris tickss affct a lot o rgy stats. For t iitial stats t rgy is icrasig but for igr rgy stats t rgy is dcrasig. B. ffct of Wll s Dpt For t xt aalysis t boudaris tickss ad widt wr costat ad t dpt was varid. T rsult is giv i Tabl III. TB III. DPTH VRITION Dpt (V): 5 ig Valus (V)

5 Ka : J. lc. gg., Ist. grs., Baglads, 7(II), Dcbr W t dpt is just V, oly o rgy stat xists. But wit t icrt of t dpt t ubr of rgy stats is also icrasig. C. ffct of Wll s Widt Fially t boudaris tickss ad dpt wr ad costat ad t widt of t FQW was varid. T rsult is sow i Tabl IV. TB IV. WIDTH VRITION Widt (): 5 ig Valus (V) Fro t aalysis w ca s tat t sa typ of ffct is obsrvd for t widt variatio as it was for dpt variatio. But t diffrc is tat tr is a cag i t valu of t rgy. T rsult is siilar as it is dscribd i []. D. Copariso For t copariso purposs w av copard t FQW wit IQW as tiod i Tabl V. Fro t copariso w ca s tat for t IQW ac stat s rgy is igr ta t FQW. d t rsult is siilar tat was foud i []. O or copariso is t quatu tulig ffct. T quatu caical poo wr a carrir or a particl tuls troug a barrir wic is ot xplaiabl by classical pysics is kow as t quatu tulig (as xapl t workig pricipl of t tul diod). For t cas of IQW tr is o quatu tulig but for FQW tr is quatu tulig. I Fig. t quatu tulig is sow for FQW. Morovr t wav fuctios of t FQW ar or sprad ta t wav fuctios i IQW [4]. Tis is aotr cosquc of t quatu tulig. TB V. COMPRISON: FQW & IQW Wll: Boudaris Tickss (): FQW (Siulatd) IQW (Calculatd).5 Dpt (V): Widt (): 5 5 ig Valus (V) Wll: FQW (Siulatd) IQW (Calculatd) IV. PPICTIONS By usig tis siulator a lot of quatu wll basd dvics ca b siulatd bfor t fabricatio. Higly flxibl ipltatio of diffrt structurs ca b ralizd troug tis siulator. V. CONCUSION T siulatio is do troug MTB. Fro t aalysis of FQW w av t followig obsrvatiosicrasig t tickss of t boudaris t ig rgis cags, by icrasig t dpt, t valus of boud rgis icras ad by icrasig t widt, t ig rgis icras but tir valus dcras. s t wol aalysis is do quatitativly it is vry uc usful to cosidr bfor ay fabricatio. Bcaus t fabricatio procss of ay dvic basd o QW is so difficult ad costly. So it will b a grat lp for t to av a ida, wat apps if t paratrs ar varid i cas of fiit quatu wll ad wat ar t ffcts du to tis. s a rsult t fabricatio or dsig of ay QW basd dvic ca b do prcisly. CKNOWDGMNT T autor would lik to tak Prof. C. Dbas for is lp. utor would also lik to tak. Dakal ad M. Olszko for tir support. RFRNCS [] V.V.Miti, D.I.Stsov, N.Z.Vagidov, Quatu Mcaics for Naostructurs, Cabridg Uivrsity Prss,. [] V. idbrg, Fiit Squar Wll, vailabl: ttp:// [] B.R.Nag, Pysics of Quatu Wll Dvics, Kluwr cadic Publisrs,. [4] C..Tag, Fudatals of Quatu Mcaics for Solid Stat lctroics ad Optics,Cabridg Uivrsity Prss, 5. 4