ELECTRONIC TRANSMISSION THROUGH QUANTUM WIER CONTAINING HETERO-JUNCTIONS

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1 Jurnal f The Islamic f Gaza(Natural Sciences Series Vl.3, N., P03-, 005 ISSN , ELECTRONIC TRANSISSION THROUGH QUANTU WIER CONTAINING HETERO-JUNCTIONS H.. Fayad, and.. Shaat Physics Department, Al-Aqsa University, Gaza, Gaza Strip, Palestine. Physics Department, The Islamic University, Gaza, P. O. B 08, Gaza Strip, Palestine. Astract: By means f mde-matching and transfer matri methds, e calculate the transmissin f an electrn passing thrugh quantum ire cnsisting f several arrier ptentials. We take int accunt the spacedependent electrn effective mass and the cupling eteen cmpnents f the mtin f an electrn in directins parallel and perpendicular t the interfaces f arriers. Our results sh that the cupling effect leads t significant changes n the transmissin spectra. In additin, the variatin f the gemetrical dimensins is investigated. Keyrds: Quantum transprt; esscpic systems; Electrn aveguides. INTODUCTION With the rapid prgress in nanfaricatin technlgy, semicnducting nanires have een etensively studied fr the ptential applicatins n nanelectrnics. Tsu and Esaki [] realized the theretical significance f the resnant-tunneling prlem in semicnductr heterstructure. Dulearrier resnant-tunneling structures have fund ide applicatins in many ne devices, such as pht-detectrs, dides, transistrs, l-per lgic circuits, and quantum integrated circuits [,3]. There mainly eist t theretical framerks fr dealing ith resnant tunneling prlems. One is the tight-inding mdel [4,5]. The ther is the paralic-and effective-mass apprimatin, in hich, the cherent tunneling mdel [6] and the sequential tunneling mdel [7] have een prpsed and develped fr investigating resnant-tunneling characteristics in multiarrier heterstructure [8]. In the cherent tunneling mdel, an electrn keeps its phase memry in the tunneling prcess, hich is regarded as a cherent transmissin prcess and descried y the eigenstate f the electrn in the ttal structure. In the sequential tunneling mdel, the electrn lses its phase memry in the tunneling prcess, hich is regarded as a series f successive transitin prcesses ith phase incherence. The effective-mass apprimatin is restricted t the assumptin that the cmpnents f the mtin f the electrn in the directins nrmal and parallel t the layers are decupled. Hever, this assumptin is incrrect hen the difference f the

2 ELECTRONIC TRANSISSION effective mass f the electrn in the arrier and ell materials is taken int accunt. It arises ecause f the flling reasn. When the effective mass is space dependent, the cnservatin f mmentum f the electrn in the parallel directin des nt imply cnservatin f the kinetic energy cmpnent in the directin parallel t the layers. As a result less r mre energy is availale fr the mtin perpendicular t the arrier, hich hinders r helps the tunneling prcess. With the mlecular-eam-epitay technlgy, it is pssile t cnstruct t-dimensinal arrier structures such as GaAs-GaAlAs-GaAs in hich the idth f each layer is cntrlled ith cnsiderale accuracy. The effective mass f the electrn in such a structure is psitin dependent as a cnsequence f its value changing in GaAlAs as a functin f aluminum cncentratin hile remaining cnstant in GaAs. As shn y Lee [9], the effective mass difference in the t layers is sufficient t ring aut significant changes in the tunnel time and transmissin cefficient. In this paper, e eplre the effect f cupling eteen the parallel mtin and the nrmal mtin f the electrn n resnant-tunneling characteristics in quantum ire cntaining multi-arrier heterstructure (sectin. With the transfer matri methd, the electrnic transmissin thrugh the prpsed quantum ire is discussed (sectin 3.. TRANSFER ATRIX ETHED OF QUANTU WIRE We cnsider a dule arrier heterstructure (ne segment f ur quantum ire cnsisting f AlAs (arrier layers and GaAs (ell layer as shn in Fig.. AlAs GaAs AlAs Fig.. Schematic illustratin f a dule arrier heterstructure. Let us cnsider a eam f electrns ith kinetic energy E and effective mass m t e incident n a arrier frm the left. We descrie the prlem using the paralic-and effective-mass apprimatin. Because f the spatial symmetry in all directins parallel t the layers, the ave vectr k = ( k y, k z f the electrn is cnserved in the y-z plane and the crrespnding electrn ave functin is epressed y a plane ave. The ttal ave functin ψ ( r, fr the electrn can e ritten as 04

3 Hisham Fayad, hammed Shaat ψ ( r, = ep( ikr ϕ( ( here ϕ( satisfies the ne-dimensinal (D Schrödinger equatin ith psitin-dependent electrn effective mass, represents the grth directin f heterstructure, and r = (y,z is the psitin crdinate f the electrn in the plane f the arrier. The D Schrödinger equatins are h d ϕ( m ( d in the ell regin, and = E ϕ( ( h d ϕ( + U ( E ( ϕ = ϕ m ( d (3 in the arrier regin, here E = E E and E = E E are the lngitudinal energies f the electrn in the ell and arrier, respectively, m ( and m ( are the psitin-dependent electrn effective mass in the h k h k ell and arrier, respectively. Here E = and E = are, m m respectively, the transverse kinetic energy f the electrn in the ell and the arrier. Thus it can e seen that the psitin-dependent electrn effective mass effects the ave functin ϕ ( and intrduces a cupling eteen the ave functin in the parallel and perpendicular directins. As a cnsequence the ave functin ϕ ( is influenced nt nly y the idth ( and the arrier height ( U ut als y the ave vectr k f the plane ave parallel t the layers. Accrdingly, e intrduce an effective arrier height U k fr tunneling f electrns alng the -directin as [0] ( h k U ( k = U ( β (4 m ( m here β =. Taking int accunt Eq. (4, the D Schrödinger equatin m in the arrier regin ecmes: 05

4 ELECTRONIC TRANSISSION h d ϕ( + U ( k m ( d ϕ( = E ϕ( (5 Thus, the cupling can e interpreted as meaning that the effective arrier height felt y the electrn is n lnger a cnstant U. N, e need the transfer matri fr the th unctin, and then e shuld generalize the result fr N-unctins. This matri relates the cefficients f the ave functin at ne end f the unctin t thse at the ther ne, s that, the ave functins are ritten as ϕ ( = A ep( ik + B ep( ik (6 fr the th ell, and ϕ ( = A ep( k + B ep( k (7 fr the th arrier here ( ( h k = m E, k [ ( ( ( h ] = m U k E, and E is the lngitudinal kinetic energy f the incident electrn. By matching the cntinuity f the ave functin ϕ( and its dϕ( apprpriately nrmalized derivative at the undaries, e m( d derive a matri frmula that relates the cefficients A and B ith A + and B +, that is: A B = ik ik ik ik A B + + (8 here is the th transfer matri crrespnding t the th unctin ritten as: = ( ( a ( + (9 here and a are the idths f the th arrier and th ell, respectively, ( and ( a crrespnd t the transfer matrices fr the th arrier and th ell respectively, 06

5 Hisham Fayad, hammed Shaat and csh( k ( = βk sinh( k ( βk sinh( k csh( k (0 cs( k a ( k sin( k a ( = a ( k sin( k a cs( k a Setting A =, B = r, A N = t, and B N = 0, then the ttal transfer matri is epressed as the cascading f a series f the individual arrier and ell transfer matrices in sequence ith = r ik ik ik ik 0 t tt ( N tt = ( ( a ( N (3 = Frm Eq. (, it is readily fund that the transmissin amplitude t is given y t = + + i( k k (4 here i is the elements f the ttal transfer matri. Thus, the transmissin cefficient T is n defined as the rati f the transmitted particle flu divided y incident particle flu, and depends upn th the transfer ave numer and the lngitudinal kinetic energy f the incident electrn. This definitin leads directly t the flling epressin: T ( k, E = t (5 3. RESULTS AND DISCUSSION In this sectin e present ur results tained numerically fr the transmissin cefficient acrss the quantum ire cntaining multi-arrier heterstructure cnsisting f GaAs/Ga -z Al z As, here z represents the cncentratin f Al. In this structure the cnductin-and ffset and the 07

6 ELECTRONIC TRANSISSION effective mass f the electrn are determined as a functin f the Al cncentratin z y the flling standard apprimatin [] U 0.75z ( ev (0 p z 0.45 = 0.75z ( z 0.45 ( ev (0.45 z (6 and m m = ( z m (0 p z, = 0.067m e e (7 here m e is the free-electrn mass. The dependence f the transmissin cefficient T n the lngitudinal kinetic energy and the transverse ave numer f the electrn is shn in Fig. fr the dule arrier heterstructure ith an Al cncentratin f z = 0.50, a arrier idth f = 60 Å, and a ell idth f a = 50 Å, here e have pltted the lnt vs E fr fur different values f k. Curves slid, dashed, dtted, and dted-dashed are crrespnd t different k = 0.00, 0.03, 0.05, and 0.07 Å -, respectively. Frm the plt in Fig., e find that t sharp peaks shift tard the l-energy regin ith the increase f the transverse ave numer k. rever, the shift f the secnd peak is much greater than that f the first peak. In additin, it is als nted that ith an increase f k the idth f resnant peaks radens, and the peaks reduced ith respect t the valley in the transmissin spectrum. It is evident that the influence f the transverse mtin f the electrn n higher resnant states fr resnant tunneling is mre remarkale than n the ler resnant states. 08

7 Hisham Fayad, hammed Shaat ln T FIG.. Natural lgarithm f the transmissin cefficient T as a functin f the lngitudinal kinetic energy E fr dule arrier heterstructure ith parameters: a = 50 Å, = 60 Å, and z = Curves slid, dashed, dtted, and dted-dashed are crrespnd t the cases f k = 0.00, 0.03, 0.05, and 0.07 Å -, respectively. The results as shn in Fig. can e interpreted frm the epressin f the effective arrier height in Eq. 4. The increase f the transverse ave numer leads t a decrease f the effective arrier height. Cnsequently, effects such as the shift f resnant peaks, idth radening, and the reductin f peaks are natural cnsequences f the decrease f the effective arrier height. Draing the transmissin cefficient T as functin f the arrier idth ( prvides sme further insight int the influence f transmissin cefficient y the cupling effects. This is shn in Fig. 3 fr the triple arrier structure ith an Al cncentratin f z = 0.3, lngitudinal kinetic energy E = 400 mev, arrier idth is equal t the ell idth, and fr t different values f k. Curves slid and dashed crrespnd t t different values f k = 0.00, and 0.05 Å -, respectively. We nte that, ith the increase f k the cnductin ands mve tard the l-idth regin and the shifts f 09 E ( ev

8 ELECTRONIC TRANSISSION the cnductin ands at higher idths are much larger than thse f the ler nes T (Angstrm FIG. 3. Transmissin cefficient T as a functin f the arrier idth fr triple arrier heterstructure. The parameters are as fll a =, E = 400 mev, and z = Slid, and dashed curves crrespnd t the cases f k = 0.00, and 0.05 Å -, respectively. T cnclude this sectin, Eqs. (6 and (7 sh that the increase f z results in th a lift f U and an increase f m. Frm Eq. (4 it can e fund that the lift f U leads t an increase f the effective arrier height, hile the increase f m play an ppsite rle [8], leading t a decrease f the effective arrier height. It flls that, the lift f U shuld eaken the cupling effect, hile the increase f m shuld strengthen this cupling effect. In cnclusin, e have shn in this paper that the transmissin cefficient is significantly altered y the cupling eteen cmpnents f the mtin f the electrn in directins parallel and nrmal t the interfaces. In particular, these effects are larger at higher energies. 4. References. R. Tsu and L. Esaki, Appl. Phys. Lett., 56 (973.. Y. Cui et al., Nan Lett. 3, 49 (003. 0

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