Electronic Energies in Delta Doped AlGaAs/GaAs Heterostuctures

Transcription

1 Amcn Jounl of Appld Scncs 5 (4: 45-49, 8 ISS Scnc Publctons Elctonc Engs n Dlt Dopd AlGAs/GAs Htostuctus A. Mfth, H. Ajln, A. Mouk, R. Chtouou nd M. Ouslt Fculty of Scncs of Tuns, Unvsty cmpus, 9 El Mn, Tuns tonl Insttut of Scntfc nd Tchncl Rsch Abstct: Th bnd bndng of th potntl t th AlGAs/GAs ntfc ncss wth th lctons concntton n th GAs chnnl. Ths bnd bndng s oftn gnod n th thotcl clculton of th lctonc ngs nd ln potntl ppoxmton ws usd. Ths ld to mpcs ngy vlus nd occupd subbnd confgutons. W hv tkn ccount of th bnd bndng n th subbnd ngs clculton. Th potntl xpsson ws consdd s sum of ln nd qudtc tms. Th qudtc tm ws consdd s tm ndpndnt ptubton nd th lctonc ngs w coctd to th fst od. W hv clcultd th Fm ngy fo dffnt lctonc concnttons. Th lton btwn th Fm ngy nd th lctonc concntton t th ntfc lds to th dsd confguton of th occupd subbnds. Whn th fundmntl subbnd s only occupd, th chctstcs of thos htostuctus bst nhncd. Th cospondng lctonc concntton ws clcultd. Th thotcl sults w usd fo th dtmnton of th ctv lcton concntton n two AlGAs/GAs htostuctus. Kywods: AlGAs/GAs ntfc, lctonc ngs. ITRODUCTIO In AlGAs/GAs htostuctus, two dmnsonl lcton gs (DEG s fomd n th GAs chnnl n th AlGAs/GAs ntfc []. In th pln of th DEG, lctons hv hgh moblty nd hgh concntton chng cm []. Thos lcton chctstcs usd n th fbcton of hgh spd optolctonc dvcs. Mo sophstclly dvc ngnng qus n ccut knowldg of subbnds stuctu nd Fm s lvl loclston. In AlGAs/GAs htostuctus, lctonc stuctus oftn numclly clcultd usng slf consstnt soluton of Schödng nd Posson qutons [-8] o usng SILVACO Mcuy softw. In thotcl bnd stuctu clculton, th GAs conducton bnd bndng n th AlGAs/GAs ntfc s oftn gnod, nd tngul potntl s consdd [9]. Schubt nd ll hv ppoxmtd th GAs bnd dg by polygonl cuv [] nd th lcton ngy s clcultd by mtchng th lcton D Bogl wv lght nd th wdth of th tngul wll. In ths pp, w show tht th tngul potntl ntoducd n th cunt dnsty quton gvs s to sptl dpndnc of th lctonc concntton whch cn b ppoxmtd by ln functon ntfc. Th soluton of th Posson quton gvs s to n ppoxmtd potntl V ( b. Th ln tm s fst consdd n th Schödng quton nd th lctonc ngs nd wv functons clcultd n th bsc of th ffctv mss ppoxmton. Th qudtc tm s ttd s tm ndpndnt ptubton nd th lctonc ngs coctd to th fst od. Th confgutons of occupd subbnds ltd to th Fm ngy. Ths ngy s clcultd fo dffnt concntton of th two dmnsonl lcton gs. Elctcl fld nd potntl xpssons n th AlGAs/GAs ntfc: th AlGAs/GAs ntfc, ln potntl V ( s oftn consdd. By wtng th cunt-dnsty nd th Posson qutons, w wll cy th lctcl fld nd potntl xpssons. Th cunt-dnsty quton fo lctons n th GAs chnnl s: n DEG E µ D ndeg E s th lctcl fld. n DEG,, µ nd D lcton concntton, chg, moblty nd dft coffcnt spctvly. Th soluton of quton gvs: dpndng of th dstnc to th AlGAs/GAs Cospondng Autho: Abdl MEFTAH, Fculty of Scncs of Tuns, Unvsty cmpus, 9 El Mn, Tuns 45

2 Am. J. Appld Sc., 5 (4: 45-49, 8 µ D n DEG ( ndeg ( Fo wk vlu of, on cn wt: n DEG ( α ( µ Wh ndeg ( D α nd s th D µ flt bnd bgnnng bscs. In th two dmnsonl lcton gs ppoxmton, ll lctons supposd to b n th ntfc pln wh th lcton concntton s n DEG. W cn wt: n DEG n DEG ( d n Ths gvs: α DEG Th soluton of th Posson quton gvs: α E( ( α nd V ( ( 6 Wh s th GAs dlctc-pmttvty constnt To th fst od ppoxmton w hv n V α ( DEG, thn ndeg. To th scond od ppoxmton, th potntl s: V ( b V ( s constnt fo, thn b Subbnd ngs n th tngul potntl: In th bsc of th ffctv mss ppoxmton, th Schödng quton n th GAs chnnl s: P ( // m// P m GAs E g ( E ( ψ ( ψ ( P //, m // nd P, m lcton momntum nd ffctv mss n th dctons plll nd GAs ppndcul to th ntfc spctvly. E g s th GAs bnd gp. In lttu, no dffnc btwn th ffctv msss m // nd m w potd, w thn wt: m // m m Th clculus of th wv functons nd ngs gvn n th ppndx. Th wv functon s ψ k // S // ϕ(, wh k // nd // ( th wv nd poston vctos plll to th ntfc spctvly. m ϕ ( s th Ay functon (( A ( h GAs h k// nd th ngy s E Eg wth m π h ndeg ( ( (.75 ( m Wh,,,... Fo m.67 * m nd *8.85* SI s gvn by:, th ngy ( mv 6 ( mv.997 * ndeg (.75 (4 Wh n DEG s xpssd n cm W msmtch th subbnd ngs clcultd by th xpsson (4 nd thos found n lttu, fo th DEG concntton n DEG 5. cm. Th ngs nd of th fundmntl nd th fst xctd lvls spctvly gvn n tbl. Engy cocton: Th fst od cocton of th ngy s: 46

3 Am. J. Appld Sc., 5 (4: 45-49, 8 Tbl : Engs nd of th fundmntl nd fst xctd lvls n th tngul potntl of th AlGAs/GAs htostuctu. Th fst ow shows ngs gvn by ths wok. Th oth ows show ngs found n lttu. Rfncs Engs Ths wok [] [9] [] [] [,] [4] ( mv ( mv Tbl : α nd β vlus, nd fundmntl nd 5 fst xctd lvls ngs bfo nd ft coctons fo n DEG 5* cm nd 5. Α 4 5 α β (mv ( ( mv ϕ b ϕ (5 ϕ ϕ m Wh ( Z ( h { n DEG α n DEG { h β ( m } h α ( m I I Wh α, β, I A I I dz, ZA ( Z dz nd I I, ltd to I t, I Z A dz I clcultd by comput. n DEG nd by: s β.77* n } DEG, n DEG nd xpssd n mv spctvly. Fo n DEG 5* cm nd 5 Α Α, cm nd th α nd β coffcnts nd th coctd ngs fo th fundmntl nd th fv fst xctd lvls gvn n tbl. Th volutons wth n DEG of th ngs nd fo, nd psntd n fgu. Fo ch lvl, th ngy cocton s ngtv, ts bsolut vlu ncss wth ncsng n DEG. 47

4 Am. J. Appld Sc., 5 (4: 45-49, 8 Engy (mv F,E 4,E 6,E 8,E n DEG Fg. : Evoluton wth n DEG of th fundmntl, two fst xctd nd Fm lvls ngs bfo nd ft th cocton. Th volutons wth th wll wdth of th ngy cocton fo th th lvls psntd n fgu. Fo wk vlu, th bnd bndng s mo ponouncd n th AlGAs/GAs ntfc, th ln potntl ppoxmton s nsuffcnt nd th ngy cocton fo ch lvl s mpotnt. Fo lg vlu, th potntl s goously ln n th AlGAs/GAs ntfc nd th ngy cocton s wk. Engy (mv (b (c ( F m n ( DEG Log (6 Wh T nd K tmptu nd Boltmn constnt. n DEG could b wttn: n DEG F m Log( ( m LogA π (7 Wh A ( F (...( Wth nd s th ngy of th upp confnd lvl Th condton > s oftn stsfd, thus s wk nd w cn ppoxmt A to th scond od n nd wt: A (... W put: α j nd (8 β ( (9 j j Th qutons (8 nd (9 gv: ndeg m α β A Th soluton of gvs: Z (Angstom Fg. : Evoluton wth th potntl-wll wdth of th ngy coctons of th fundmntl (, th fst (b nd scond (c xctd lvls Th Fm ngy: Th ov ll lcton concntton n th two dmnsonl lcton gs s gvn by [5] : 48 n j DEG m ( 4( ( j j F Log j ( j j Th Fm ngy F s clcultd by comput. Th voluton of F wth n DEG s shown n fgu. Th fundmntl lvl s popultd fo

5 Am. J. Appld Sc., 5 (4: 45-49, 8 n DEG > 6.* cm. Th fst xctd lvl s popultd fo n DEG > 4.8* cm. Th n DEG cn b ltd to th dopng concntton, whch choc lds to th dsd confguton of th occupd subbnds. COCLUSIO In GAs chnnl nd n th AlGAs/GAs ntfc, th potntl s ppoxmtd by th xpsson V ( b. Th qudtc tm s wk compd to th ln on. To fst ppoxmton th potntl s tngul. Th subbnd ngs nd wv functons xctly clcultd n th bsc of th ffctv mss ppoxmton. Th qudtc tm s consdd s tm ndpndnt ptubton, nd ngs coctd to th fst od. Th clcultd subbnd ngs g wll wth tht found n lttu. Th Fm ngy s clcultd fo dffnt concnttons n DEG of th two dmnsonl lcton gs. Th concntton n DEG cn b ltd to dopng concntton, thus th dsd confguton of th occupd subbnds wll b obtnd by th choc of th dopng concntton. REFERECES. Stöm, L. H., R. Dngl, A. C. Gossd, W. Wgmn nd M. D. Stug, 98. TWO DIMESIOA L ELECTRO GAS AT SEMICODUCTOR-SEMICODUCTOR ITERFACE. Sold. Stt. Com., 9: Shubt, E. F., J. E. Cunnghm nd W. T. Tsng, 987. Elcton-Moblty Enhncmnt nd Elcton concntton Enhncmnt n δ-dopd n- GAs t K. Sold. Stt. Com., 6: Stn, F.nd S. Ds Sm, 984. Elcton ngy lvls n GAs-G -x Al x As htojunctons. Phys. Rv. B, : Tn, I. H., G. L Snd, L. D. Chng, nd E. L Hu, 99. A slf-consstnt soluton of Schödng- Posson qutons usng nonunfom msh. J. App. Phys., 68 (8: Boun, L., L. Sfx, H. Sgh.nd H. Mf, 999., Impovmnt of th lcton dnsty n th chnnl of n AlGAs/GAs htojuncton by ntoducng S dopng n th quntum wll. J. Appl. Phys., 85 : Aloulou, S., H. Ajln, A. Mfth, M. Ouslt, L. Sfx nd H. Mf,. Elcton confnmnt n pln dopd htostuctu G - xal x As:S/GAs. Mtl Scncs nd ngnng B, 96: Sngh, R., C. M. Snowd, 999. A chg contol HEMT modl ncopotng dp lvl ffcts,.sold. Stt.. Elctoncs, 4: Ajln, H., A. Mfth, R. Chtouou, M. Ouslt nd H. Mf, 6. Photolumnscnc studs of confnd stts n AlGAs/GAs symmtc quntum wll. Physc E, : Ando, T., A. B. Fowl nd F. Stn, 98. Elctonc popts of two-dmnsonl systms. Rv. Mod. Phys., 54 (: Schubt, E. F., 985. Elcton Subbnd Stuctu n Slctvly Dopd n-al x G -x As/GAs Htostuctus. IEEE Tns. Elcton Dvcs ED-, 9: Ando, T., 98. Slf-Consstnt Rsults fo GAs/Al x G -x As Htojuncton Subbnd Stuctu nd Lght-Scttng Spct. J. Phys. Soc. Jpn, 5: Stn, F., 98. Dopng consdton fo htojunctons. Appl. Phys. Ltt., 4: Dlgbuduf, D. nd. T. Lhn, 98. Mtl- (n AlGAs-GAs two-dmnsonl lcton gs FET. IEEE Tns. Elcton Dvc, ED-9: Mthu, H., 998. Physqu ds smconductus t ds composnts élctonqus. Edton MASSO, Ps, Qutèm édton, pp:5. 49