Electrostatic Surface Waves on Semi-Bounded Quantum Electron-Hole Semiconductor Plasmas

Transcription

1 Commun. Thor. Phys Vol. 67 No. 3 March 07 Elctrostatc Surfac Wavs on Sm-Boundd Quantum Elctron-Hol Smconductor Plasmas Afshn Morad Dpartmnt of Engnrng Physcs Krmanshah Unvrsty of Tchnology Krmanshah Iran Rcvd May 0 06; rvsd manuscrpt rcvd Dcmbr 06 Abstract Th lctrostatc surfac wavs on sm-boundd quantum lctron-hol smconductor plasmas ar studd wthn th framwork of th quantum hydrodynamc modl ncludng th lctrons and hols quantum rcol ffcts quantum statstcal prssurs of th plasma spcs as wll as xchang and corrlaton ffcts. Th dsprson charactrstcs of surfac lctrostatc oscllatons ar nvstgatd by usng th typcal valus of GaAs GaSb and GaN smconductors. Numrcal rsults show th xstnc of on low-frquncy branch du to th mass dffrnc btwn th lctrons and hols n addton to on hgh-frquncy branch du to charg-sparaton ffcts. PACS numbrs: 4.0.Cv 7.35.E DOI: 0.088/053-60/67/3/37 Ky words: lctron-hol smconductor plasma dsprson rlaton Introducton Th propagaton of plasma wavs n quantum lctronhol smconductor QEHS plasmas hav bn of consdrabl ntrst n rcnt yars. Msra [] studd th propagaton of surfac lctromagntc wavs n a magntzd QEHS plasma usng quantum hydrodynamc QHD modl. H found a forward propagatng mod by ncludng th quantum tunnlng ffct. Moslm t al. [] nvstgatd th propagaton of nonlnar acoustc wavs n a QEHS plasma by mans of QHD thory. Thy studd th contrbutons of th tunnlng of dgnrat plasma spcs through th Bohm potntal barrr th xchang ntractons and corrlatons and th statstcal quantum prssurs of dgnrat lctron and hol fluds on th acoustc puls profl. Thn by mployng QHD modl Zba t al. [3] nvstgatd th lctron-hol two-stram nstablty n a QEHS plasma. Furthrmor Yaha t al. [4] studd th bhavor and nstablty of th lctrostatc prturbatons n a QEHS plasma pumpd by nrgtc lctron bam takng nto account th xchang-corrlaton ffcts th quantum statstcal prssurs and th quantum rcol ffcts of th dgnrat lctrons and hols as wll as th lctron bam paramtrs. Thn th nvstgaton of th quantum ffcts n th proprts of lnar and nonlnar quantum lctrostatc acoustc wavs n a QEHS plasma has bn followd by Wang and Elasson [5] usng th QHD thory. In ths way Wang and Lu [6] dscrbd th modulatonal nstablty of th quantum acoustc wavs n a QEHS plasma takng nto account th combnd ffcts of th quantum rcol th xchang-corrlaton potntal standng for spn th dgnrat prssur ffcts as wll as th lctronphonon hol-phonon collson. Also Amn [7] rportd E-mal: a.morad@kut.ac.r c 07 Chns Physcal Socty and IOP Publshng Ltd th nonlnar dsprsons of th wav propagatng n th QEHS plasmas. Howvr dspt ths walth of thortcal nformaton about th optcal proprts of th unboundd QEHS plasmas thr s no avalabl thortcal account of surfac wavs propagaton on th sm-boundd QEHS plasmas wth nonlocal charactr whl n ralty undr laboratory condtons or n mcro- and nanolctronc dvcs plasmas ar spatally lmtd. Thrfor t s mportant to study th boundary ffct on th natur of th QEHS plasmas. On th othr hand n our prvous nvstgatons th propagaton of lctrostatc surfac wavs on a smboundd quantum mtallc plasma on-spcs plasma systm n th absnc [8] and prsnc of an xtrnal magntc fld [9] was studd by usng th on-flud QHD modl. For smconductor plasmas two-spcs plasma systm w may trat th lctron and hol componnts as two dstnct but ntractng fluds. [0 ] In ths way w may xpct to fnd two typs of oscllatons. On of thm s th usual plasma oscllaton. [8] Th othr s an oscllaton assocatd wth a dnsty wav smlar to a sound wav n gass. In lght scattrng xprmnts Pnczuk t al. [] hav obsrvd th acoustc mod n bulk GaAs sampls. In th prsnt work motvatd by th prvous paprs [ 7] and our nw fndngs for th proprts of th surfac and bulk wavs on th lctron-hol smconductor nano-sphrs [0] and nano-wrs [] w wsh to study th lctrostatc surfac wavs on th sm-boundd QEHS plasmas. In ths way by usng th lnarzd QHD thory Posson quaton and applyng th approprat boundary condtons w obtan a gnral analytcal xprsson for

2 38 Communcatons n Thortcal Physcs Vol. 67 th dsprson rlaton of surfac wavs on sm-boundd QEHS plasmas. W not that for smconductor quantum dvcs workng wth th lctrons and hols n nanoscal szs t s mportant to study th quantum ffcts on th dynamcs of th chargd carrrs. Th quantum ffcts.g. quantum rcol ffcts dgnrat prssurs of th plasma spcs as wll as th xchang and corrlatons ar xpctd to play an mportant rol n th collctv bhavor of th QEHS plasma. [] Formulaton of Problm W consdr a homognous unmagntzd twocomponnt QEHS plasma occupyng th half-spac z > 0 and boundd by vacuum.. z < 0 n a Cartsan coordnat systm wth R = r z whr r = x y. Th lctrostatc surfac wavs ar supposd to propagat paralll to th ntrfac z = 0 along th x- and y-drcton. Thus th QEHS plasma wth th dnsty n 0 pr unt volum whr = for lctrons and = h for hols wll b prturbd by th surfac wavs and may b rgardd as two chargd flud wth thr-dmnsonal scalar dnsty fld n R t and vctor vlocty fld u R t. Th nonlnar dynamcs of such dsturbancs ar govrnd by th followng st of QHD quatons: [] n R t + [n R tu R t] = 0 u R t + [u R t u R t] = q m ΦR t m V xc + m m n R t P n R t n R t ΦR t = 4π [n R t n h R t] 3 whr s th rlatv dlctrc constant of th systm s th lmnt charg m s th ffctv mass q = for lctrons and q = + for hols and Φ s th slf-consstnt lctrostatc potntal. Th scond trm n th rght-hand sd of Eq. s du to th non-rlatvstc dgnrat partcl prssur P and s gvn by [3] P = 3 π /3 π n 5/3 R t 5 3 whr s th Planck constant dvdd by π. Th thrd trm n Eq. rprsnts th lctron and hol xchangcorrlaton forc btwn th dntcal partcls whn thr wav functons ovrlap du to th hgh numbr dnsty. Th xchang-corrlaton potntal V xc for th lctrons and hols ar gvn by: [4 6] V xc = n /3 R t [ a B n/3 R t m ln a Bn /3 R t ] whr a B = /m. Th last trm n th rght-hand sd of Eq. rprsnts th quantum rcol forc assocatd wth th Bohm potntal du to th lctrons/hols tunnlng through a potntal barrr and t contans th quantum mchancal ffcts of th prsnt modl. [7 8] Hr w consdr only th frst ordr prturbatons by lttng n = n 0 + n wth n 0 n ; u = u ; and Φ = Φ whr n u and Φ ar th prturbd quantts. As a rsult w obtan th followng lnarzd quatons: n R t + n 0 u R t = 0 4 u R t = q m Φ R t α γ λ n 0 n R t + β n 0 n R t 5 Φ R t = 4π [n R t n h R t] 6 whr γ = 0.38 n /3 0 /m λ = 0.05 n /3 0 / a Bn /3 0 m α = 3π n 0 /3 / 3m β = /m. By lmnatng th vlocty fld u from Eqs. 4 and 5 and usng Eq. 6 on can obtan th followng quatons n R t = ωp[n R t n R t] n h R t + α γ λ + β n R t 7 = ω ph[n R t n R t] + α h γ h λ h + β h n h R t 8 whr ω p = 4π n 0 /m /. In th followng w dfn th tm-spac Fourr transform Ãk z ω of an arbtrary functon AR t by dkdω AR t = π 3 Ãk z ω k r ωt 9 and apply t to Eqs. 6 8 whr k = k x ê x +k y ê y. Now w xpand th prturbd dnsty n R t nto n k z ω and lctrostatc potntal Φ R t nto Φk z ω accordng to Eq. 9 and consdr k / z th vry slow nonlocal varatons ar nglctd [8] whr k = kx + ky. W fnd: n z z δ + z δ n h z = 0 0 Φz whr δ ± = k + σ + σ h ± σ σ h + ω p ψ ω ph ψ h and ψ = α η λ + β k σ = ω p ω /ψ. Not that w hav droppd th xplct dpndncs of Φ and n on k and ω. For th prsnt systm w look for a soluton of Eq. 0 of th forms

3 No. 3 Communcatons n Thortcal Physcs 39 A kz z 0 Φz = B kz + C + δ+ k δ +z + C δ k δ z z 0 whr { 0 z 0 n z = D + δ+z + D δ z z 0 { 0 z 0 n h z = D h+ δ +z + D h δ z z 0 ω p D ± = 4π ω + ψ δ± k C ± D h± = [ ω ] p 4π ω + ψ δ± k C ±. 3 In ths stag w nd four boundary condtons at z = 0 to dtrmn th four coffcnts A B and C ± n th abov quatons. Thy ar: Φ n z z=0 = Φ out z z=0 4 z Φ n z = z=0 z Φ out z z=0 5 Φ n z n n 4π ω z p = ψ 6 z z=0 z z=0 Φ n z n n 4π ω h ph = ψ z h 7 z z=0 z z=0 whr n and out rfr to nsd and outsd th QEHS plasma half-spac rspctvly. Applyng th abov boundary condtons aftr far amount of algbra w obtan a rlaton btwn ω and k as [δ + + ψ ][ ω /δ+ k k + ψ δ + k δ k [ δ δ + k + ψ ][ ω /δ k k + ψ + ψ h ω p ψ ωph + ψ ] h ωph δ δ + + k + ψ h ω p ψ ωph + ψ ] h ωph δ + = 0. 8 Th abov dsprson rlaton of th lctron-hol wav oscllatons s th orgnal rsult of th prsnt work. In th followng scton w show that Eq. 8 rvals two branchs of solutons for lctrostatc wavs vz. a low frquncy and a hgh frquncy mod. It may b mntond that n th lmt m h and = th low frquncy branch of th abov quaton gts supprssd and th othr branch gvs th man rsult n Rfs. [9] and [0]. 3 Numrcal Rsults and Dscusson In ordr to study th bhavor of th propagatng mods numrcally t s convnnt to transform th dsprson rlatons Eq. 8 n a mor sutabl form by ntroducng th followng dmnsonlss varabls: Ω K H N M η ϑ Ξ Γ Λ Σ Ψ and ± s appndx. Thus n trms of ths varabls w can wrt Eq. 8 as: [ + + Ψ ][ ϑ Ω / + η K ηk + Ψ + η K η K [ + ηk + Ψ ϑ Ω / η K ηk + Ψ + Ψ h M Ψ N ][ + + ηk + M N Ψ h ] + Ψ h M + M ] Ψ N N Ψ h + = 0. 9 To s clarly bhavour of th surfac wavs dsprson curvs of th sm-boundd QEHS plasmas w plot dmnsonlss frquncy Ω vrsus dmnsonlss varabl K from Eq. 9 and rsults ar dpctd n Fg.. In ths way w us th GaAs smconductor wth typcal paramtrs [ 3] n 0 = n 0h = cm 3 m = 0.067m g m h = 0.5m g =.8 and GaSb smconductor wth typcal paramtrs [ 3] n 0 = n 0h = cm 3 m = 0.047m g m h = 0.4m g and = Also w consdr m 0 = 0.m g n 0 = 0 6 cm 3 = 0. From Fg. t s clar that mdum supports two mods of lctrostatc surfac wavs a low frquncy mod acoustc mod and a hgh frquncy mod. It s asy to fnd that th low frquncy mod shows saturaton bhavor for larg wavnumbrs and for small wavnumbrs th long wavlngth lmt th acoustc mod could propagat on surfac of th sm-boundd QEHS plasmas du to th mass dffrnc btwn th lctrons and hols. Ths rsult s smlar to surfac plasma oscllatons n quantum lctron-on plasma half-spac. [] On th othr hand th hgh-frquncy surfac plasma oscllatons ar prmarly du to charg sparaton whr th lctrons and hols oscllat aganst ach othr wth oppost phass. On can fnd that th hgh frquncy mod has a lowr cutoff and ts lowr cutoff frquncy s ω = ωp + ωph / +. Now w xamn th ffct of th lctron dnsty concntraton on th dsprson rlaton of plasma mods

4 30 Communcatons n Thortcal Physcs Vol. 67 as shown n Fg.. Hr w us th GaN smconductor wth typcal paramtrs [ 3] m = 0.3m g m h =.3m g =.3 and varabl numbr dnsty n 0 = n 0h = 0 7 cm 3 n 0 = n 0h = 0 7 cm 3 and n 0 = n 0h = cm 3. On ncrasng th dnsty concntraton frquncy of th surfac wavs ncrass whr curvs sparaton s mor sgnfcant for hgh-frquncy mod. Fnally lt us nvstgat th xchang-corrlaton and quantum rcol ffcts on th dsprson rlaton of surfac mods. W fnd n th prsnc of th quantum rcol ffct th frquncy of hgh-frquncy surfac plasma mod shfts toward hgh frquncy not shown hr whl th dsprson rlaton s almost unchangabl by th xchang-corrlaton ffcts. Fg. Color onln Dsprson curvs ω/ω 0 vs. th varabl kα 0/ω 0 for GaAs blu ln and GaSb rd ln smboundd smconductor plasmas. Fg. Color onln Dsprson curvs ω/ω 0 vs. th varabl kα 0/ω 0 for GaN sm-boundd smconductor plasmas wth varabl numbr dnsty of n 0 = n 0h = 0 7 cm 3 blu ln n 0 = n 0h = 0 7 cm 3 rd ln and n 0 = n 0h = cm 3 grn ln. 4 Concluson To summarz w hav studd th lctrostatc surfac wavs on sm-boundd QEHS plasmas. In ths rgards w hav usd two-flud hydrodynamc thory n conjuncton wth th Posson quaton and approprat boundary condtons. W hav drvd th dsprson charactrstcs of surfac lctrostatc oscllatons of smboundd QEHS plasmas. Numrcal rsults show th xstnc of on hgh-frquncy branch du to chargsparaton ffcts and on low-frquncy branch du to th balanc btwn prssur and nrta of th lctrons and hols. Th rsults may b usful for undrstandng th dsprson proprts of quantum surfac wavs n smconductor plasmas whch may b obsrvd xprmntally. Appndx Th dmnsonlss varabls Ω and K ar gvn by Ω = ω/ω 0 and K = kα 0 /ω 0 whr w hav α 0 = 3π n 0 /3 / 3m 0 and ω 0 = 4π n 0 / m 0 /. Also th dmnsonlss varabls H N M η ϑ Ξ Γ Λ Σ Ψ and ± ar gvn by H = β ω p /α N = n h /n M = m h /m and η = m0 m 0 6 n0 n 0 ϑ = Ξ = a B n/ a Bh n/3 0h 0 m m 0 n 0 n 0 Γ = m n /3 0 3π Γ /3 h = M N Γ / m Λ = n /3 0 3π / a B n/3 0 Λ h = Ξ M N /3 Λ Σ = ϑ Ω Ψ Σ h = M N /3 N/M ϑ Ω Ψ h Ψ = Γ Λ + η H K Ψ h = N /3 M M N /3 Γ Ξ M ± = η K + Σ + Σ h ± N Λ /3 + Σ Σh η N /3 H K + MN /3 Ψ Ψ h.

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