FINITE ELEMENT ANALYSIS OF QUANTUM STATES IN LAYERED QUANTUM SEMICONDUCTOR STRUCTURES WITH BAND NONPARABOLICITY EFFECT

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1 FINITE ELEENT ANALYSIS OF QUANTU STATES IN LAYERED QUANTU SEICONDUCTOR STRUCTURES WITH BAND NONPARABOLICITY EFFECT Khai Q. L Dpartmnt of Information Tchnology, Ghnt Univrsity-IEC, Sint- Pitrsniuwstraat 41, 9000 Ghnt, Blgium; Corrsponding author: ronaldokhai@yahoo.com Rcivd 24 April 2008 ABSTRACT: A simultanous calculation of ignstats in th layrd quantum smiconductor structurs using th finit lmnt mthod was dvlopd. W proposd th approximation to linariz th nonlinar ignvalu problm du to nonparabolicity ffct and found th application rang for th approach. Th finit lmnt calculation rsults showd an xcllnt agrmnt with th publishd rsults obtaind by othr authors. In addition, our calculatd rsults confirm that th ffct of nonparabolicity on th transition nrgy shift is considrably larg for highr subbands and should b takn into account in th simulation and dsign of light mittrs basd on layrd quantum smiconductor structurs Wily Priodicals, Inc. icrowav Opt Tchnol Ltt 51: 1 5, 2009; Publishd onlin in Wily IntrScinc (www. intrscinc.wily.com). DOI /mop Ky words: band structur calculation; layrd quantum smiconductor structurs; quantum cascad lasrs; nonparabolicity ffct; finit lmnt mthod 1. INTRODUCTION Quantum nginring of lctronic nrgy stats and wav functions using nanoscal layrs of smiconductor compounds allows th dsign and th obsrvation of quantum phnomna which ar typically obsrvd in atomic structurs. Furthrmor, th advancs in growth tchnology of quantum wll htrostructurs hav allowd tailoring of th optical and transport proprtis of smiconductor matrials, rsulting in a considrabl incras in rsarch activity dirctd toward th dvlopmnt of novl usful optolctronic dvics. On of th xcllnt xampls of how quantum nginring can b usd to dvis fficint light mittrs in midinfrard to THz rgion is th quantum cascad (QC) lasr [1]. To undrstand th physical proprtis of th htrostructur dvics, on nds to solv th ignvalu problm of carrir in th quantum wll structur. Howvr, th prsnc of th conduction band nonparabolicity ffct () [2] maks it complicat. Undr this ffct, it bcoms a nonlinar ignvalu problm. Thrfor, th problm in gnral cannot b solvd xactly. Th approximat calculation nds to b dvlopd to find th clos solution. In addition, sinc th oprating wavlngth of and th carrir dynamics in th QC lasrs ar dtrmind mainly by a quantum wll structur rathr than bandgaps of matrials, an accurat thortical modling is vry important in th QC lasr dsign. Thr hav bn various mthods usd to calculat th band profils in th quantum wll structurs, which includ th matrix approach [3], th transfr matrix mthod [4], th finit diffrnc mthod (FD) [5, 6], and th finit lmnt mthod (FE) [7, 8]. Th FD and FE ar quit straightforward mthods and advantagous in that all th subbands can b calculatd simultanously. Howvr, this advantag of simultanous calculation of all subbands hardly can b takn if th is includd. Thus, in th prvious work, to includ th, subbands wr calculatd squntially, and in ach subband calculation, a good guss of an ignnrgy was to b mad [9]. In our rcnt work [10], w proposd th nw band structur calculation mthod basd on th FD, in which all subbands wr calculatd simultanously vn with taking into account th. Th ky lmnt of th nw approach was th using of th first-ordr Taylor approximation of th nrgy dpndncy of th lctron ffctiv mass. th proposd mthod, w hav simultanously calculatd all th subband dg nrgy lvls and th corrsponding wav functions for QC lasr structurs. Th finit diffrnc calculation rsults showd good agrmnt with th prviously rportd rsults in spcifid rang. Howvr, in som cass of THz QC lasrs [11] in which an accurat oprating wavlngth calculation is strongly rquird bcaus of thir low transition nrgy, a mthod mor accurat than FD may b rquird. In this articl, w prsnt a mor accurat approach sharing th sam ida of using th first-ordr Taylor approximation of th nrgy dpndncy of th lctron s ffctiv mass to tak into account th. Th mthod is basd on th Galrkin s finit lmnt procdur, and first-ordr lin lmnt is usd for finit lmnts. Th proposd mthod has bn vrifid by bing applid to calculat th conduction subband structurs of th layrd quantum smiconductor structurs including th singl quantum wll with various wll widths [8] as wll as th prviously rportd QC lasrs [12 16]. 2. FORULATION 2.1. Basic Equation th ffctiv mass approximation, th nvlop function of an lctron is obtaind by solving th 1D Schrödingr quation: 2 d 1 d x 2 dxm*e, x dx V x x E x, (1) whr Vx, E, x, and m*e,x ar th potntial du to th xtrnal and th built-in lctric fild, ignnrgy, wav function corrsponding to th ignnrgy E, and nrgy- and positiondpndnt lctron ffctiv mass, rspctivly. For a singl quantum wll, th nrgy-dpndnt lctron ffctiv masss in th wll and th barrir including th ffct of conduction band nonparabolicity ar spcifically givn by [8] m* w E m* w1 E E w, (2) m* b E m* b1 E E b V E b, (3) whr E w, E b, and V ar th nrgy gaps btwn th conduction and th light-hol valnc bands in th wll and barrir matrials, and th nrgy barrir hight at th intrfacs, rspctivly. Th nonparabolicity cofficint is givn by 2 i, i w,b. (4) 2m* i E i For a QC lasr structur, th nrgy- and position-dpndnt lctron ffctiv mass with taking into account th can b writtn as [17] m*e, x m* b x1 E Vx (5) E g x E g x, ICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 1, January

2 whr m* b x and E g x ar th ffctiv mass of conduction band bottom and nrgy bandgap, rspctivly. To tak th advantag of th simultanous calculation of all subbands, w propos th nw approximation of th nrgy- and position-dpndnt ffctiv lctron mass basd on th first-ordr Taylor approximation: 1 m*e i,x 1 m* b x1 E Vx if E Vx E E g x E g x g x. (6) By substituting Eq. (6) into Eq. (1), w obtaind d 2 dx 1 m* b x1 E Vx E g x E g x dx dx Vxx Ex. (7) 2 To rduc th numbr paramtrs, th coordinat x, th potntial V(x), th band gap E g x, th lctron ffctiv mass m* b x, and th nrgy E ar normalizd as x/l, V Vx/E 1, E g E g x/e 1, m* m* b x/m* b 0, and E/E 1, rspctivly, whr E /2m* b 0L 2 with th solution domain lngth of L, and m* b 0 is th ffctiv mass of th lctron at x 0. this notation, th Schrödingr quation (7) is rducd to d 1 d d V d d m d d m E g d d 1 d d me g d 2 2 V 0. (8) 2.2. Finit Elmnt thod W us th Galrkin s finit lmnt procdur [18, 19] to solv Eq. (8). This quation could b rducd to [6] 1 d 1 1 m 1 d 1 d V m d 1 d 1 1 m 1 E g1 d V d m E g d 1 d 1 1 m 1 E g1 d 1 d m E g d P S 2 Q 2 R H, (9) R N N T d, (14). (15) Assuming th Dirichlt condition ( 0) or th Numann condition (d/d 0 ) at th lftmost Nod 1 and th rightmost nod, w obtain from Eq. (9) th simpl ignvalu quation: P S 2 Q 2 R H. (16) Th abov quation can b transformd into th following ignvalu matrix quation: A B, (17) whr [A] and [B] ar th N N symmtric and spars matrixs (N is th total numbr of nods), is an ignvalu, and 1, 2,..., N T is th corrsponding ignvctor. Equation (17) is th matrix ignvalu problm. Thr hav bn various mthods dvlopd to solv this kind of problm. Evn som numrical libraris hav bn stablishd. In this work, LA- PACK packag was usd. 3. NUERICAL RESULTS AND DISCUSSION As shown prviously, th approximation for th nrgy-dpndnt lctron ffctiv mass using th first-ordr Taylor xpansion is only valid whn th approximation condition is satisfid. Thrfor, to vrify th validity of our proposd approximation and find th application rang of this approach, w hav calculatd th subband dg nrgis in th singl quantum wll for various wll widths as wll as th subband structur for various typs of QC lasrs and compard with thos obtaind by othr authors. W first calculatd th subband dg nrgis of th singl quantum wll considrd in [8]. Th paramtrs usd for th calculation of th subband dg nrgy using FE ar th sam as in thir work. In Tabl 1, th calculatd subband dg nrgy E (mv) and th nrgy shift E s du to th ar listd. It can b clarly sn that our rsults ar in good agrmnt with valus in th litratur [8]. In a singl quantum wll, it is known that th nrgy shift du to th bcoms substantial for highr subband dg for larg wll widths. whr P S H 1 m V dn dn T d, (10) d d dn dn T d, (11) m E d d g 1 dn dn T d, (12) m E d d g Q V N N T d, (13) TABLE 1 Wll Width and Associatd Eignstat Enrgis and Enrgy Shifts From Nonparabolicity Vrsus Quantum Numbr of th Eignstat n 1 n 2 n 3 n 4 n 5 Wll width (nm) 5 E FE Rf E s FE Rf E FE Rf E s FE Rf E FE Rf E s FE Rf ICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 1, January 2009 DOI /mop

3 TABLE 2 atrial Paramtrs Usd in th Calculation of th GaAs/Al 0.33 Ga 0.67 As-Basd QC Lasr atrial Paramtrs GaAs Al 0.33 Ga 0.67 As GaAs/Al 0.33 Ga 0.67 As Band offst 300 E c (mv) Effctiv mass m o 0.094m o Band gap (V) For th QC lasr, to invstigat th ffct of nonparabolicity on nrgy shift and vrify th validity of our FE calculation, w hav prformd th subband structur calculation of various typs of QC lasrs with and without th. W first invstigatd th application of th mthod to th GaAs/Al 0.33 Ga 0.67 As-basd QC lasrs proposd by Sirtori t al. [12] and Kruck t al. [13]. Th matrial paramtrs of ths structurs ar givn in Tabl 2, whil th calculatd valus [20] of th transition nrgy using FE as wll as FD ar listd in Tabls 3 and 4. (Th ffct of doping concntration was ignord in all calculations. Th bandgaps of matrials ar givn in th work of Vurgaftman t al. [20].) Th calculatd wav functions ar dpictd in Figurs 1 and 2, rspctivly. It can b clarly sn that th finit lmnt analysis rsults ar in xcllnt agrmnt with valus prsntd in Rf. [12, 13] and provid closr stimation of th transition nrgis than our prvious work using FD. On can also s that th transition nrgy shift du to th is small for th lowr subbands, whras it is considrably larg for th highr subbands. Th proposd mthod is nxt applid to calculat th most common QC lasr basd on Al 0.48 In 0.52 As and Ga 0.47 In 0.53 As [14]. Th matrial paramtrs of th structur ar givn in Tabl 5. Th calculatd wav functions ar dpictd in Figur 3. In Rf. [14], th authors rportd th QC lasr s xprimntal mission wavlngth of 11.2 m (E mv) and thir thortical calculatd transition nrgy E mv. Our calculatd transition nrgy using FE with and without ar E mv and E mv, rspctivly. Our calculation including is in good agrmnt with thir calculation. Again, w can not that th inducs a considrabl chang in th transition nrgy of th QC lasrs, and for an accurat modling of QC lasrs, should b includd. Nxt, w calculatd th subband structur for th QC lasr basd on AlAs and GaAs matrial which has a largr band offst. Th matrial paramtrs of th structur ar givn in Tabl 6. Th calculatd lasr transition nrgy rportd by Bckr t al. [15] is E 32 Figur 1 Schmatic conduction band diagram of a portion of th GaAs/Al 0.33 Ga 0.67 As-basd QC lasr [12] undr an lctric fild of 48 kv/cm. Th solid curvs rprsnt th moduli squard of th rlvant wav functions calculatd by our FE. [Color figur can b viwd in th onlin issu, which is availabl at mv ( 11.4m ). Our calculatd rsult with and without ar E mv and E mv, rspctivly. Th band profils and rlvant wav functions of this QC lasr ar shown in Figur 4. It is clarly sn that our FE calculation is also in vry good agrmnt with th rsult obtaind by Bckr t al. W now turn our attntion to th THz QC lasr structur proposd by Kumar t al. [16]. Th matrial paramtrs of th structur ar givn in Tabl 7, whil th calculatd valus of th transition nrgy and th band profil appar in Tabl 8 and Figur 5, rspctivly. Sinc th transition nrgy of THz QC lasrs is quit low, an accurat thortical calculation is strongly rquird in TABLE 3 Calculatd Enrgy in th GaAs/ Al 0.33 Ga 0.67 As-Basd QC Lasr [12] Enrgy (mv) FE FD (with ) Rfrnc (with ) E E TABLE 4 Calculatd Enrgy in th GaAs/ Al 0.33 Ga 0.67 As-Basd QC Lasr [13] FE Enrgy (mv) FD (with ) Rfrnc (with ) E E Figur 2 Schmatic conduction band diagram of a portion of th GaAs/Al 0.33 Ga 0.67 As-basd QC lasr [13] undr an lctric fild of 44 kv/cm. Th solid curvs rprsnt th moduli squard of th rlvant wav functions calculatd by our FE. [Color figur can b viwd in th onlin issu, which is availabl at DOI /mop ICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 1, January

4 TABLE 5 atrial Paramtrs Usd in th Calculation of th Al 0.48 In 0.52 As/Ga 0.47 In 0.53 As-Basd QC Lasr atrial Paramtrs Ga 0.47 In 0.53 As Al 0.48 In 0.52 As Band offst E c (mv) Ga 0.47 In 0.53 As/ Al 0.48 In 0.52 As 520 Effctiv mass 0.043m m 0 Band gap (V) th THz QC lasr dsign. In that cas, for dtrmination of accurat transition nrgy, th FE is xpctd to provid mor accurat rsult than th FD as sn in Tabl 8. Howvr, for th QC lasrs basd on th matrials whos band paramtrs violat th approximation condition, th calculatd rsults ar incorrct. To prov this statmnt, w carrid out th band structur calculation of th QC lasr basd on Ga 0.47 In 0.53 As and AlAs 0.56 Sb 0.44 [21]. In this work, authors rportd th thortical calculatd transition nrgy E mv, whil our FE with and without is and mv. Th calculatd wav functions ar dpictd in Figur 6. Th dvlopd mthod was mployd to calculat th subband structur of som common QC lasrs basd on various typs of matrials. Th calculatd rsults show an xcllnt agrmnt with thos obtaind by othr authors whr thir band structur calculation wr mad by taking into account th in diffrnt mannr. From th calculation, w confirm that th transition nrgy shift du to th is considrably larg for th highr subbands, and for an accurat dsign of QC lasr dvics, th ffct of nonparabolicity should b takn into account. Figur 4 Schmatic conduction band diagram of a portion of th AlAs/ GaAs-basd QC lasr [15] undr an lctric fild of 39 kv/cm. Th solid curvs rprsnt th moduli squard of th rlvant wav functions calculatd by our FE. [Color figur can b viwd in th onlin issu, which is availabl at Up to now, w hav bn considring th ignvalu problm whr th prsnc of th changs it to a nonlinar ignvalu problm. Thrfor, this work provids th approach to linariz th ignvalu problm. As shown arlir, th approximation for th nrgy-dpndnt lctron ffctiv mass using th first-ordr Taylor xpansion is only valid whn th variabl is small nough. By calculating th subband structur of th most common QC lasrs, w found that this approach is valid for th QC lasrs basd on th matrials whr th dirct conduction band discontinuity of th htrostructurs must b smallr than th bandgap of ach matrial (E c E g ). 4. CONCLUSIONS A FE has bn stablishd for th analysis of quantum stats in singl quantum wll and various typs of QC lasrs. Th proposd approximation of th nrgy-dpndnt lctron ffctiv mass shows an xcllnt agrmnt with prviously rportd rsults. W can simultanously calculat all subband stats of singl quantum wll and QC lasr structurs vn including th conduction band. W dvlopd a numrical tchniqu using th first-ordr Taylor xpansion of th nrgy-dpndnt lctron ffctiv mass TABLE 7 atrial Paramtrs Usd in th Calculation of th GaAs/Al 0.15 Ga 0.85 As-Basd QC Lasr Figur 3 Schmatic conduction band diagram of a portion of th Al 0.48 In 0.52 As/Ga 0.47 In 0.53 As-basd QC lasr [14] undr an lctric fild of 39 kv/cm. Th solid curvs rprsnt th moduli squard of th rlvant wav functions calculatd by our FE. [Color figur can b viwd in th onlin issu, which is availabl at TABLE 6 atrial Paramtrs Usd in th Calculation of th GaAs/Al 0.33 Ga 0.67 As-Basd QC Lasr atrial Paramtrs GaAs AlAs GaAs/AlAs Band offst E c (mv) 1000 Effctiv mass m m 0 Band gap (V) atrial Paramtrs GaAs Al 0.15 Ga 0.85 As Band offst E c (mv) GaAs/ Al 0.15 Ga 0.85 As Effctiv mass m m 0 Band gap (V) TABLE 8 Calculatd Enrgy in th GaAs/Al 0.15 Ga 0.85 As-Basd QC Lasr [16] Enrgy (mv) FE FD Rfrnc E E ICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 1, January 2009 DOI /mop

5 Figur 5 Schmatic conduction band diagram of a portion of th GaAs/Al 0.15 Ga 0.85 As-basd QC lasr [16] undr an lctric fild of 8.4 kv/cm. Th solid curvs rprsnt th moduli squard of th rlvant wav functions calculatd by our FE. [Color figur can b viwd in th onlin issu, which is availabl at mbddd in th FE and applid it to various quantum wll structurs to vrify its validity. Our calculations show good agrmnts with th prviously rportd rsults. Th finit lmnt calculatd rsults also showd an obvious improvmnt compard to our prvious work using FD. Sinc all th boundd subbands in QC lasrs can b simultanously calculatd with good accuracy using th proposd mthod, w bliv that this mthod is vry usful for QC lasr dsign. Espcially, for th QC lasrs basd on th bound-tocontinuum structur [22] whr many quantum stats should b considrd, th proposd mthod will provid a powrful dsign tool. To improv th accuracy of th finit lmnt analysis of ignstats in QC lasrs, w can apply th scond or highr ordr lin lmnt for finit lmnts, which will b rportd lswhr. Figur 6 Schmatic conduction band diagram of a portion of th In 0.53 Ga 0.47 As/AlAs 0.56 Sb basd QC lasr [21] undr an lctric fild of 130 kv/cm. Th solid curvs rprsnt th moduli squard of th rlvant wav functions calculatd by our FE. [Color figur can b viwd in th onlin issu, which is availabl at REFERENCES 1. J. Faist, F. Capasso, D.L. Sivco, A.L. Hutchinson, and A.Y. Cho, Quantum cascad lasr, Scinc 64 (1994), D.F. Nlson, R.C. illr, and D.A. Klinman, Band nonparabolicity ffcts in smiconductor quantum wlls, Phys Rv B 35 (1987), A.K. Ghatak, K. Thyagarajan, and.r. Shnoy, A novl numrical tchniqu for solving th Schrodingr quation using matrix approach Application to quantum wll structurs, IEEE J Quantum Elctron 24 (1988), B. Jonsson and S.T. Eng, Solving th Schrodingr quation in arbitrary quantum wll potntial profils using th transfr matrix mthod, IEEE J Quantum Elctron 26 (1990), P. Harrison, Quantum wlls, wirs and dots: Thortical and computational physics, 2nd d., Wily, Chichstr, UK, K. Kawano and T. Kitoh, Introduction to optical wavguid analysis Solving axwll s quation and Schrödingr s quation, Wily, Nw York, 2001, pp K. Nakamura, A. Shimizu,. Koshiba, and K. Hayata, Finit-lmnt analysis of quantum wlls of arbitrary smiconductors with arbitrary potntial profils, IEEE J Quantum Elctron 25 (1989), H. Hayata,. Koshiba, K. Nakamura, and A. Shimizu, Eignstat calculation of quantum wll structurs using finit lmnts, Elctron Ltt 24 (1988), Y. Li, O. Voskoboynikov, C.P. L, and S.. Sz, Computr simulation of lctron nrgy lvls for diffrnt shap InAs/GaAs smiconductor quantum dots, Comput Phys Commun 141 (2001), L.Q. Khai and S. Kim, A novl band structur calculation for th quantum cascad lasrs with conduction band nonparabolicity ffct, Prsntd at th 9th Intrnational Confrnc on Intrsubband s in Quantum Wlls, Amblsid, Cumbria, UK. 11. R. Kohlr, A. Trdicucci, F. Bltram, H.E. Br, E.H. Linfild, A.G. Davis, D.A. Ritchi, R.C. Lotti, and F. Rossi, Trahrtz smiconductor hrtrostructur lasr, Natur 417 (2002), C. Sirtori, P. Kruck, S. Barbiri, O. Collot, J. Nagl,. Bck, J. Faist, and U. Ostrl, GaAs/Al 0.33 Ga 0.67 As quantum cascad lasrs, Appl Phys Ltt 73 (1998), P. Kruck, H. Pag, C. Sirtori,. Stllmachr, and J. Nagl, Improvd tmpratur prformanc of GaAs/Al 0.33 Ga 0.67 As quantum cascad lasrs with mission wavlngth at 11m, Appl Phys Ltt 76 (2000), C. Sirtori, J. Faist, F. Capasso, D.L. Sivco, A.L. Hutchinson, and A.Y. Cho, Long wavlngth infrard ( 11m ) quantum cascad lasrs, Appl Phys Ltt 69 (1996), C. Bckr, C. Sirtori, H. Pag, G. Glastr, V. Ortiz, X. arcadt,. Stllmachr, and J. Nagl, AlAs/GaAs quantum cascad lasrs basd on larg dirct conduction band discontinuity, Appl Phys Ltt 77 (2000), S. Kumar, B.S. Williams, Q. Hu, and J.L. Rno, 1.9 THz quantum cascad lasrs with on-wll injctor, Appl Phys Ltt 88 (2006), J. Bai and D.S. Citrin, Suprsymmtric optimization of scond harmonic gnration in mid-infrard quantum cascad lasrs, Opt Exprss 14 (2006), K. Kawano and T. Kitoh, Introduction to optical wavguid analysis Solving axwll s quation and Schrödingr s quation, Wily, Nw York, 2001, pp O.C. Zinkiwics, Th finit lmnt mthod, 3rd d., cgraw-hill, Nw York, Vurgaftman t al., J Appl Phys 89 (2001), C.A. Evans, V.D. Jovanovic, D. Indjin, Z. Ikonik, and P. Harrison, Dsign and simulation of InGaAs/AlAsSb quantum-cascad lasrs for short wavlngth mission, Appl Phys Ltt 87 (2005), J. Faist, D. Hofstttr,. Bck, T. Alln,. Rochat, and S. Blasr, Bound-to-continuum and two-phonon rsonanc quantum-cascad lasrs for high duty cycl, high-tmpratur opration, IEEE J Quantum Elctron 38 (2002), Wily Priodicals, Inc. DOI /mop ICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 1, January