Effect of QWR Shape on the Induced Elastic and Piezoelectric Fields

Transcription

1 Effect of QW Sae o te Iduced Eatc ad Peoeectrc Fed E. Pa ad. Ja btract It of reat mortace to udertad te factor tat cotrbute to te tra ad eectrca dtrbuto wc are duced by te mft tra betwee a bured quatum wre QW ad t urroud matr. Oe of te mortat factor te ae or eometry of cro ecto of te QW. Ut a recet eact coed-form outo [Pa ] we tudy te mode ytem of QW wt dfferet ae ad cacuate bot te urface ad tera eatc ad eoeectrc fed duced by QW embedded emcoductor Ga ubtrate by roery ett te e ad ocato of te QW. Te effect of te QW ae a we a te matera oretato ad free urface of te ubtrate o te eatc ad eoeectrc fed are ceary demotrated. Key cocuo are ao draw. keyword: mft tra QW ae effect eact coedform outo eatc ad eoeectrc fed. Itroducto Muc reearc effort a bee made dur recet year o quatum eterotructure were quatum mecac effect aear tu rod rc oortute for te de of oe dece. For eame emcoductor aer coud be mroed by u two dmar matera a acte ad cadd ayer [L ] for rom acato a emcoductor t-emtt ad aer dode. U quatum aotructure amey quatum we QW quatum wre QW ad quatum dot QD a acte ayer ead to ee better aer erformace [L ; Grudma Ster ad Bmber 99; Fu ao Ferdo Sade Wa ad Laro ; Sm o Jeo V Ho Su Lee Km ad Hwa ; Martet Duertu eardt Bao Kao Ster Grudma ad Bmber ]. ecety te traed aotructure ae draw coderabe atteto ce te mft tra ca ead to te cae te coducto/aace bad wc tur toeter wt te dmeo cofemet ater te eectroc ad otca roerte of te emcoductor tructure [otom Hammerber Wema ojma Suura Okamoto Tamamura ad Potemk 99; Leare otat Lefer odo Lakoe Martet udra ad Kao 999; Medero-bero ; Martet Duertu eardt Bao Kao Ster Grudma ad Bmber ; otat Martet Lefer udra Leare Kao Gayra ad Gerard ]. For tace te traed QW ca acqure a waeet rae ot accebe wt attce-matced matera [Grudma Ster ad Bmber 99]. For QW tructure muc work a bee doe to eore te effect of QW ae o arou roerte. Varamd ad Vaaade etated te deedece of te quated coductace of eectro traort o te ae of QW cro ecto wt te etece of defect or murte. Ter work tmuated by a reou reearc of Boacek Scerbako ad Ladma 997 wo took bot te area ad ae of te QW cro-ecto or aowre to coderato. We toy ad Key 99 tuded aaytcay te effect of te cro-ecto eometry of te QW o cofemet eere Taam 997 erformed aay of te QW cro-ecto o te ecto bd eery. Teter ad Trber aayed te effect of QW ae o te roud tate QW tructure u fte dfferece metod wt Oawa Ito ad Myo 99 ued fte eemet metod to tudy te effect of eometrc ae o te aece bad of QW. Sae effect a bee teey tuded ot oy umercay but ao eermetay. By meaur te otoumecece PL of four dfferet eometre.e. rectauar trauar traeoda ad oyoa oe Sm o Jeo V Ho Su Lee Km ad Hwa foud tat te et t emo effcecy ad bet reroducbty tety ad waeet were aocated wt traeoda QW. Lkewe eo o Su ad Lee roed tat traeoda QW ece rectae QW a acte matera to mroe erformace Det. of Eeer Te Uerty of kro kro OH 9. E-ma: a@uakro.edu

2 otoeectroc dece. e eed Scaff ad Eatma 99 obered tat te tra dtrbuto wt IGa/Ga QW of rectauar ae ot uform wc woud cotrbute to te de of te traed QW tructure. Dacemet ad tra dtrbuto wt QW were ao aayed by Uyaeko Iaba Mkuk Darowk Omote Petc Greer ad Force u -ray dffracto. V-aed [Duertu Ober ad Kao ] T-aed [Grudma Ster ad Bmber 99] traeoda [Uyaeko Iaba Mkuk Darowk Omote Petc Greer ad Force ] ad trauar [Go ad Freud 99] QW were ao tuded. So far oweer a rorou ad comreee aay o te duced tra ad eectrc fed t uaaabe. Uder te aumto of otroc eatcty Fau Dowe ad O ey 997 obtaed te tra fed for bot rectauar ad crecet QW embedded te fu ae. Very recety Ga reeted reut o te tra dtrbuto de ad outde te QW wt te ae of crce trucated cyder ad crecet otroc emcoductor. U te coforma ma metod u dered a aaytca outo for QW of arbtrary ae wt aotroc eoeectrc meda. More recety Pa dered a eact coed-form outo for te QW duced eatc ad eoeectrc fed emoy te Gree fucto ad equaet body-force metod. Te QW ca be of arbtrary ae ad embedded wt te emcoductor af ae wt eera aotroc eatc ad eoeectrc roerte. I a ery recet work Pa ad Ja owed tat matera oretato coud ae a fcat fuece o te tra ad eectrc dtrbuto de te traeoda QW. I t aer we cocetrate o te effect of te QW ae o te duced eatc ad eectrc fed u te eact coed-form outo dered recety [Pa ]. I wat foow we w e a bref reew o t eact outo ad te ay t to te robem o QW wt dfferet ae te Ga af-ae tructure. oc eame are reeted to furter roe te accuracy ad effcecy of te eact outo. Effect of te QW ae are reeaed ad cocuo are draw wc coud be mortat to emcoductor dece de. aytca outo Let u uoe tat tere a eera mft tra de te QW doma V wc embedded te < af-ae ubtrate F.. um ao tat te mft tra wt te QW uform te te duced fed ca be foud aaytcay u te Gree fucto metod ad te equaet body-force cocet a a bee doe recety by Pa. We te detaed derato ca be foud Pa for te ake of eay referece we brefy reet te ma reut wt defto for te oed yca quatte. It wa ow tat te mft tra-duced eatc dacemet ad eoeectrc oteta ca be ereed a a tera o te boudary of te QW wt te Gree fucto be te terad [Pa ]: K u u ; ds K J J V K wc u J ; te J-t Gree eatc dacemet/eectrc oteta at due to a e-force/ecare te K-t drecto aed at. J te eteded matera coeffcet matr defed a foow: JK jk ej ek ε J K J J ; ; K J K K were jk e j ad ε are te eatc teor eoeectrc coeffcet ad deectrc teor reectey. For te boudary art made of trat-e emet te boudary tera ca be carred out aaytcay [Pa ]. Tu for a QW wt a arbtrary oyoa ae we um u te cotrbuto from a te e emet of te QW boudary. Furtermore te crcuar ae ca be aromated wt ecewe trat-e emet. Terefore te fa ereo of te duced eteded dacemet due to a arbtrary oyoa cuo wt -de [Pa ]: u J J K J QK ν K were te -t outward orma comoet o te -t e emet. te et of te -t e emet ad ca be ereed a by ett te eerc e emet from ot to ot. I do o te outward orma comoet are cotat e by: / ; / o Eq. te eematr correod to te Stro eeaue J e T 99 ad Pa. Matr Q defed a: Q B B - S Sν ν were matr B te ecod eematr reated to te Stro eeaue J wt te uercrt - deot ere matr ad a oerbar te cojuate matr. Te frt term cota Eq. tad for te cotrbuto from te fu-ae Gree fucto ad te oe o for te cotrbuto from te urface boudary codto of te af ae. ad ca be ereed a fucto of :

3 ]} [ ] {[ dt t ]} [ ] {[ dt t ν υ ad eetuay ae ect ereo a: ] [ 7a ] [ 7b U te bac tra-dacemet ad eectrc fed-eectrc oteta reato te tra ad eectrc fed ca be obtaed te eact coed form [Pa ]: { } { } J J J J J J J J Q Q.... β β β ν β β a { } J J J J Q.. b { } J J J J Q E c umerca aay S dfferet QW ae are tuded amey crce quare dowward ad uward traeod ad dowward ad uward equatera trae F.. QW ae are aumed to ae te ame cro-ecto area S.7 m ad to be ocated at te ame cetrod m for te uroe of comaro. Furtermore a QW are ocated ymmetrcay about te -a te Ga af ae < F. ad ae te ame uform mft ydrotatc tra tem.e...7 yy Fure : ro-ecto arameter of te QW wt dfferet ae bured Ga ubtrate <. te commo eometrc ceter. Te eatc roerte for Ga are 9 /m 9 /m ad 9 9 /m [Pa a b]. Te eoeectrc ad reate deectrc cotat for Ga are reectey e -./m ad ε r. [Pa a b]. Two oretato are codered: Oe Ga wc te oba coordate y ad are cocdet wt te crytae ae [] [] ad [] ad te oter Ga were te -a ao [-] y-a ao [- ] ad -a ao [] drecto of te crytae [Pa a b]. Te boudary codto o te urface of te ubtrate aumed to be tracto-free ad uat [Pa a b]. Te QW ae effect o te urface fed of Ga ubtrate etated frt a ow F. -F.. Fure a ow te ydrotatc tra ao te free urface of te ubtrate Ga werea F. b ot tat of te ubtrate Ga. Wat tee two fure ae commo te equece of te eak aue of QW for dfferet ae.e. uward trae a te aret eak aue foowed by crce ad quare wt amot te ame aue; Dowward trae come after foowed by uward ad dowward traeod ubtrate Ga a ad ubtrate Ga b. oter teret ot tat ao te free urface te ydrotatc tra of crce ad quare are amot te ame ad t ao te cae for bot uward ad dowward traeod QW were oy t dfferece ca be ee te mdde. T me tat term of te duced eatc fed ao te urface quare amot detca to crce ad o te dowward traeod to uward traeod. Look to deta of F. a we fd tat te mamum aue for dfferet QW are a reaced te mdde wt a mamum aue of. duced by te uward trae wc ½ of te mft tra. otce ao tat F. a a te cure are ymmetrc about te mdde ot a eected.7 a.9 a. a...

4 .. rce Square Traeod Dowward Traeod Uward Trae Dowward Trae Uward a.. rce Square Traeod Dowward Traeod Uward Trae Dowward Trae Uward b. H ydrotatc S tra.. Hydrotatc Stra m m Fure : Hydrotatc tra ao te free urface of te ubtrate duced by te dfferet QW. Subtrate Ga a ad Ga b. Ga. omar te ydrotatc tra cae of Ga ad Ga we furter otce tat we tere a obou ymmetry wt reect to te -a for te cae of Ga te tra fed Ga o oer ymmetrc wc ceary due to te effect of matera oretato. Furtermore te matude of te tra fed ao atered a te oretato cae wt te eak aue of ydrotatc tra from dfferet QW Ga be eeray arer ta ter couterart Ga. For tace te mamum ydrotatc tra F. b rouy / of te mft tra. ½ of te mft tra for te Ga cae F. a. It ao teret to ot out tat te duced eak aue of te ydrotatc tra from te dfferet QW Ga are ced ad are rouy ocated te ame trat e F. b daed e. Te duced tota eectrc fed E E 7 V/m ao E te free urface of ubtrate Ga ow F.. It aready kow tat for ubtrate Ga te eectrc fed detcay ero tu we oy eed to tudy te eectrc fed Ga urface. Te equece of te eak eectrc fed for te dfferet QW rema te ame a tat of te eatc tra fed tou te tred ao te urface ted to be more rreuar for te eectrc fed. a te mamum aue o te urface of Ga are rouy ocated ao a ced trat e. Te eatc ad eectrc fed duced wt te ubtrate are otted F. -F.. otour of te ydrotatc fed for te dfferet QW bured ubtrate Ga are frt ow F.. Geeray te ydrotatc aue de te QW muc arer ta tat outde. For eame for te crce cae te data rae de te QW aromatey from.7 to.7 werea outde te Tota Eectrc Fed E rce Square Traeod Dowward Traeod Uward Trae Dowward Trae Uward - - m Fure : Tota eectrc fed E E 7 V/m E ao te free urface of ubtrate Ga duced by dfferet QW.

5 a - - b m m m m c -. - d m m m m e f m m m m Fure : otour of ydrotatc tra wt Ga for dfferet QW. rce a; quare b; dowward traeod c; uward traeod d; dowward equatera trae e; ad uward equatera trae f. QW t rouy from -. to.7. D cematc ow e F. a & b for te crcuar QW were te tra dfferece de ad outde te QW ca be ceary obered. Te cocetrato ear te ceter of te free urface ca be ao ee from F. a wc cotet wt te reut ow F. a. We ae ao marked te ero aue wt a tck e wc te terface of te tee ote aue ad comree eate aue doma. I ummary te mot trk feature of F. are a foow:

6 a b Fure : Iduced ydrotatc tra of crcuar QW wt ubtrate Ga. a. wt te ubtrate; b. wt te QW. Tere aear a cocetrato de quare QW wc totay dfferet from te oter QW ae. omared wt te data rae de QW of oter ae more uform tra dtrbuto de crce obered wc mt be attrbute to te ymmetrca ature of te crce ae. Te duced tra fed wt te crce embedded a fu ae uform a we kow. o te outde boudary of te crce mamum ad mmum aue aear ateratey F. a & b wt for oter QW tere eera a oartmc uarty at eac corer [Pa ]. We ao comared our reut wt ome reou oe e.. te aaytca outo by Ga for ear otroc eatcty dered ad memeted trou a eeat roce. F. a ad F. 7a are reectey cotour of tra comoet ad for a crcuar QW Ga ubtrate ad F. b F. 7b are toe Ga reectey. It teret tat for Ga te cotour ae F. a F. 7a are ery mar to Ga otroc reut [Ga ]. Howeer we ao ot out tat te ced cae te matera oretato woud ae a fcat fuece o te duced fed comar F. a wt F. b F. 7a wt F. 7b o tat ot oy te matude of te tra fed caed aroud % but ao te dtrbuto or te ae of te cotour bot de ad outde QW caed wt a roouced o of ymmetry. Eatc tra fed duced by a trae QW Ga ubtrate ow F. wc a rarey bee tuded before. otce aa tat te de aue muc arer ta tat outde te QW wt te cocetrato due to te corer uarty. Fay te eectrc fed wt ubtrate Ga are otted F 9 ad F.. Fure 9 utrate te eectrc fed E 7 V/m duced by te dfferet QW ubtrate Ga were eera feature are mar to te correod eatc tra fed:. Te eectrc fed de crcuar QW more uform ta tat de oter QW;. Mamum ad mmum aue aear ateratey ao te outde boudary of crcuar QW; but te cotour are o oer ymmetrc due to te cae of matera oretato F. 9a F.. Furtermore te cotour of te dowward ad uward trae F. 9e 9 f are wort of certa detaed dcuo: o te de marked wt etter of te trae

7 7 a - a m m m m b - b m m m Fure : otour of tra comoet of crcuar QW wt te ubtrate. a ad b correod to QW Ga ad Ga reectey. Fure 7: otour of tra comoet of crcuar QW wt te ubtrate. a ad b correod to QW Ga ad Ga reectey. a b m m m m - - m Fure : otour of ydrotatc tra for te trauar QW wt ubtrate Ga : dowward equatera trae a ad uward equatera trae b.

8 a b - - m m m m c - d m m m e - f B B m B m m B - - m - - m Fure 9: otour of tota eectrc fed E E E 7V/m wt ubtrate Ga duced by QW of dfferet ae: crce a; quare b; dowward traeod c; uward traeod d; dowward equatera trae e; ad uward equatera trae f. te duced eectrc fed are amot cotuou from de to outde F. 9e 9f. Howeer o te oter two de marked wt etter B tey are cometey dcotuou! T feature eem to be aocated wt te ubtrate oretato cae ad terefore furter demotrate te mortace of te matera oretato. We ao otce tat F. 9e ad f te duced eectrc fed aromatey ymmetrc wt reect to a e about ± to -a te

9 9 dae F. 9 e ad f ad tat for te crcuar cae tere are two eak aue o eac de of t e ao te outde boudary of te QW F. 9a F.. foow te arumet of Fau Dowe ad O ey 997. recet tudy by Ya ad Pa baed o te omoeety aumto.e. u te buk matera roerty of I for te QW ad QD a ow tat te matude of te duced eatc fed coud be dfferet from tat baed o te cuo aumto atou tey are ome commo feature. Terefore order to ay te eact coed-form outo reeted t artce or te umerca metod deeoed reouy to te ractca de of QW tructure oe woud eed to frt determe te matera roerty de te QW wc actuay a fucto of te mft attce tra [Eaway ad Fau ; u ad amburu ]. Ietato o t teret eteo w be reorted te ear future. ckowedemet Te autor woud ke to ere ter rattude to te Uerty of kro for uort t work uder rat # ad to te reewer for cotructe commet. eferece Fure : Tota eectrc fed E E 7 V/m E outde crcuar QW wt ubtrate Ga. ocud emark Te eact coed-form outo dered recety by Pa uted to oe te QW duced eatc ad eectrc fed ubtrate Ga ad Ga. Te duced fed due to dfferet QW.e. crce quare dowward ad uward traeod ad dowward ad uward equatera trae are eamed certa deta. Tee cude te reoe ao te urface of te ubtrate de te QW a we a outde te QW. From our tude te foow mortat feature are obered: Te eatc ad eectrc fed o te free urface of te af ae are ery coe to eac oter for te quare ad crcuar QW. Te eatc ad eectrc fed wt te crcuar QW are more uform ta toe wt QW of oter ae. ocetrato ear te erte ot of te oter-tacrce ae ceary obered ad terefore ar corer oud be aoded dur fabrcato. For te duced ydrotatc tra ad tota eectrc fed a cocetrato de te quare QW obered. Matera oretato ay a ery mortat roe te duced eatc ad eectrc fed. We fay remark tat te reut reeted are baed o te cuo aumto. Tat te matera roerty wt te QW of I te ame a t ubtrate Ga toy. J.; Key M. J. 99: Te effect of cro ecto o te eectro tate quatum wre aer. Semcoductor Scece ad Tecooy o. 9.. Boacek E..; Scerbako. G.; Ladma U. 997: Sae effect o coductace quatato treedmeoa aowre: Hard eru oft oteta. Pyca eew B o... Bocatto P. G.; Proetto.. 999: Sae ad deectrc mmatc effect emcoductor quatum dot. Pyca eew B o e Y. -P.; eed J. D.; Scaff W. J.; Eatma L. F. 99: ouform dtrbuto of tra IGa/Ga quatum wre. J Vacuum Scece Tecooy B o.. 9. eo M. G.; o. J.; Su E. -K.; Lee H. J. : Ba effect o te umecet roerte of rectauar ad traeoda quatum-we tructure. ed Pyc Letter o... u P. W.; amburu.. : O a formuato for a mutcae atomtc-cotuum omoeato metod. It. J. Sod Struct. o... otat.; Martet E.; Lefer K.; udra.; Leare F.; Kao E.; Gayra B.; Gerard J. M. : Straed V-rooe quatum wre mutdmeoa mcrocate. Matera Scece ad Eeer B o. 7.. Duertu M..; Ober. D. Y.; Kao E. : omut ecto V-aed quatum wre cud bad-tructure ad deectrc effect: bd eere ad oarato aotroy of te brt B ecto. I: Tecca Proceed of te Secod Iteratoa

10 oferece o omutatoa aocece ad aotecooy. 7. Eaway S. W.; Fau D.. : Effecte eatc tffee of I uder uform tra. J of ed Pyc o Fau D..; Dowe J..; O ey E. P. 997: aytca outo for tra dtrbuto quatum-wre tructure. J ed Pyc o Fu Y.; ao Q..; Ferdo F.; Sade M.; Wa S. M.; Laro. : Stra ad otca trato I quatum dot o Ga. Suerattce ad Mcrotructure o... Ga F. : Eatc reaato of a trucated crcuar cyder wt uform datatoa eetra a af ace. Pyca Statu Sod b o Go T. J.; Freud L. B. 99: crtca tcke codto for trauar traed quatum wre row V- rooe o a attered ubtrate. cta Materaa o... Grudma M.; Ster O.; Bmber D. 99: Symmetry break eudomorc V-rooe quatum wre. Pyca eew B o Grudma M.; Het.; Ledeto.; Ster O.; Bmber D.; Uto V. M.; Ko e P.S; fero. I.; umo S. S.; Werer P.; Goee U.; Heyderec J. 99: Eectroc tructure ad eery reaato traed I/Ga quatum yramd. Suerattce ad Mcrotructure o. 9.. Grudma M.; Ster O.; Bmber D. 99: Eectroc tate a traed ceaed-ede-oerrowt quatum wre ad quatum dot. Pyca eew B o Leare F.; otat.; Lefer K.; odo.; Lakoe V.; Martet E.; udra.; Kao E. 999: Effect of dum ereato o otca roerte of V- rooe IGa/Ga traed quatum wre. ed Pyc Letter o. 7.. L G. : Stude o Semcoductor Quatum Structure Laer. Doctora Dertato. atoa ao Tu Uerty Tawa a. Martet E.; Duertu M..; eardt F.; Bao G.; Kao E.; Ster O.; Grudma M.; Bmber D. : Searato of tra ad quatum-cofemet effect te otca ectra of quatum wre. Pyca eew B o.. 9. Medero-bero G. : Etaa rowt of traed aocryta. Pyca Statu Sod o... otom M.; Hammerber J.; Wema H.; ojma S.; Suura H.; Okamoto M.; Tamamura T.; Potemk M. 99: Dmeoaty effect o tra ad quatum cofemet attce-mmatced IP-/IP quatum wre. Pyca eew B o.. 7. Oawa M.; Ito M.; Myo T. 99: ay of aece-ubbad tructure a quatum wre wt a arbtrary cro-ecto. Pyca B o. 7.. Pa E. a: Eatc ad eoeectrc fed aroud a quatum dot: Fuy coued or emcoued mode? J ed Pyc o Pa E. b: Eatc ad eoeectrc fed ubtrate Ga ad Ga due to a bured quatum dot. J ed Pyc o Pa E. : Eeby robem of oyoa cuo aotroc eoeectrc fu- ad af-pae. J. Mec. Py. Sod re. Pa E.; Ja. : aytca tudy of te eatc ad eoeectrc fed Ga quatum-wre wtructure. ubmtted. u. Q. : Eeby robem for two-dmeoa eoeectrc cuo of arbtrary ae. Proc.. Soc. Lod. o... Sm H. W.; o. J.; Jeo S. M.; V Le Va; Ho. -H Su E.-K.; Lee H. J.; Km Y. W.; Hwa Y. G. : Ifuece of te quatum-we ae o te t emo caractertc of IGa/Ga quatum-we tructure ad t-emtt dode. ed Pyc Letter o... Taam. 997: Effect of eometrca ae te cro ecto of quatum wre o ecto bd eery. J ed Pyc o T T..T. 99: otroc Eatcty Oford Uerty Pre Oford. Teter M.; Trber G. P. : tudy of te roud tate of quatum wre u te fte dfferece metod. Suerattce ad Mcrotructure o Uyaeko.; Iaba K.; Mkuk P.; Darowk.; Omote K.; Petc U.; Greer J.; Force. : - ray dffracto ad refectty aay of Ga/IGa freetad traeoda quatum wre. J Pyc D: ed Pyc o Varamd V.; Vaaade O. : Sae effect o catter tree-dmeoa quatum wre. J ed Pyc o e W.; e. 999: Se ad ae effect of quatum dot o tree-eectro ectra. Pyca B o Ya B.; Pa E. : Eatc aay of a omoeeou quatum dot mutayered emcoductor u a boudary eemet metod. J of ed Pyc o. 9..