N. G. Mensah Department of Mathematics and Statistics, University of Cape Coast, Ghana

Transcription

1 AMPFCAON OF ACOUSC WAE N GaN N HE PRESENCE OF SOWY CHANGNG PEROC EECRC FE N. G. Mnsah armn of Mahmais and Saisis, Univrsiy of Ca Coas, Ghana Absra: Aousi wav roagaion in bulk GaN Smionduor in h rsn of a slowly hanging a. lri fild and a onsan lri fild has bn sudid. Analyial rssion for h anuaion (amlifiaion) offiin has bn obaind. is shown ha h a. lri fild is aing as a modulaor and ha h aousolri gain inrass o a maimum valu and hn fall off as h aousi wav frquny is inrasd. Also for a a. lri fild of 4 4 CM h aousolri gain rmains zro unil h aousi wav frquny is abou 5.8 Hz hn a rsonan amlifiaion ak aars. f h a. lri fild is inrasd a sond rsonan ak aurs a 8.7 Hz. is hrfor suggsd ha h saml an b usd as a masr. PACS od: 43.5 Ed; 43.4 ; q; 84.4k Kyworks: Aousi wav, aousolri gain, izolri, masr. NROUCON h masur of h anuaion of aousi wav rmi on o sudy h influn on h roagaion bhavior of any suh rory of h solid ha is suffiinly wll ould o h lai; for aml Elron honon inraion [ 5], hrmo-lasi or haing ff [6 8], Magno-lasi loss ff in frromagni marial [9 ], Phonon-honon inraion [,5], Aousolri ff [ 6], Aousomagnolri ff [7-], aousohrmal ff [3]. As a rsul of h aformniond rasons h sudy of anuaion of aousi wavs has rivd a lo of anion [4 9]. his hnomnon an b laind as follows: Whn aousi wav asss hrough a marial i may inra wih various lmnary iaions. n suh an inraion h aousi wav may los nrgy or gain nrgy undr rain irumsans. h lar is known as amlifiaion and h formr as anuaion of aousi wav. h ida of aousi wav amlifiaion was horially rdid in 956 by olygo and Uriskii [3] and by Winrih [4] and rimnally obsrvd in CdS by Hason al [5] and in n-g by Pomran [3]. Mor rn works ondud on surlai marial rvald a lo of inrsing rsuls [3]. Work has also bn don on a wo-dimnsional sysm by Govorov al [9] and on GaN [7, 8]. his ar is aimd a sudying how h anuaion amlifiaion is ffd by a slowly hanging lri

2 fild on h drif vloiy of h lrons in bulk GaN. Currnly, h Ga N rlad marial sysms ar riving muh anion baus of rsn and onial dvis aliaions, suh as blu lasrs, U dors and high owr high mraur fild ff ransisors. is our oinion ha i will also find imoran aliaion in aousolronis.. MACROSCOPC HEORY n disussing h inraion of aousi wavs wih mobil arrirs, wo limiing ass, ql << and ql >> ar onvninly onsidrd ( q π λ, whr λ aousi wav lngh and l is h man fr ah of h arrirs). h aousi wavlngh λ is usually larg omard wih a ulrasoni frqunis so ha ql is onsidrably lss han uniy. his ondiion allows h ramn of h aousi wav as a lassial wav whih rurbs h arrir disribuion, and hn maks h hory marosoi in naur. Considr a lan aousi wav, ihr ur shar or ur longiudinal, roagaing along h dirion of an orhogonal s of o-ordinas. f u is h aril dislamn along (longiudinal wav) or in h y,z lan (shar wavs), hn h srain S is u. As a rsul, only longiudinal izolri filds nd o b onsidrd. hus rduing h roblm ssnially o on-dimnsional as. Using h lromhanial quaions of a izolri smionduor, h dislamn is givn by h quaion S ε E () and h srss has h form S E () whr is h lasi onsan a onsan alid lri fild E and is izolri onsan. S and ε ar h srain and h dilri onsan a onsan srain. For h izolri quaions, h wav quaion in an lasi mdium an b wrin as X u u E ρ (3) X whr ρ is h lai dnsiy. h urrn dnsiy is givn as n j η n ) µ E n ( (4) whr µ is h lron, mobiliy, n is h diffusion offiin, n is h quilibrium dnsiy of onduion lron and n is h variaion ausd by h aousi wav. Eq(4) ould wih h Poisson and oninuiy quaions i.. X j n n d (5) (6) lad o marial dislamn and lri fild E, quaions [ i( k )] E E E s (7) [ i( k ) ] s (8)

3 3 h E in Eq(7) rrsns an rnal d lri fild alid o rodu a sady lri drif. h linar small-signal hory imlis ha h rm.e η in Eq(4) is ngld. Combining Eq(4) o Eq(6) and liminaing E from Eq(4) afr umbrsom maniulaion givs a nw rlaion u u ρ (9) whr C i C C ε () hr n is h lron diffusion frquny, E d µ is h drif aramr and ε δ ε µ η is h onduion rlaaion frquny. No ha is h sd of sound in marial. h soluion for Eq(9) an now b sough as [ ] ) ( k i u u s () hr h roagaion onsan ; i k s bing h anuaion offiin. h anuaion or absorion offiin and h aousi wav vloiy s ar hn found as ( ) m ρ () ( ) R s ρ (3) hn ε ρ (4) his quaion an b wrin in rms of lromhanial ouling onsan as v K (5) whr ( ) K ε lρ is h lromhanial ouling onsan whih is a masur for h srngh of izolriiy. EERMNAON OF N HE PRESENCE OF CHANGNG PEROC EECRC FE E() Considr aousi wav moving hrough a saml only on a (ngling rflion a h dgs) undr h influn of hanging riodi fild E() wih frquny saisfying h rlaion π ( is h lngh of h saml). h innsiy of aousi wav ), ( will saisfy h following quaion X ) ( (6) whr ) ( is h offiin of anuaion (amlifiaion) of sound a im. h boundary ondiion is givn as ), ( (7)

4 is asy o show ha h soluion of Eq(6) wih Eq(7) as h boundary ondiion Ι ( ) Ι ( τ ) X is givn as, d τ (8) ( ) ( ) () Hn h dndn of on h rnal riodi fild a h nd of h saml boms onsan and is qual o. Eq(8) rrsn h innsiy of sound a h nd of h saml. s assum h fligh im akn by sound o mov hrough h saml o b π n θ. Whr n,, 3, and is h frquny of h rnal lri fild. No ha saisfy h ondiion π and θ saisfy π < θ <. hn solving h ingral in h onn for >> w obain O { [ ] θ } ( ) Ι ( ) Ι, (9) whr π π ( τ ) d τ () is h avrag of h anuaion (amlifiaion) offiin ovr h riod of h rnal fild. And if θ η θ l () ( ) d θ of h saml boms hn h aousi wav innsiy a h nd For maimum amliud rlaiv o h fluuaion a h nd of h saml boms < ( ma min ) for suffiinly small riod suh ha ( ) (3) ma min << (4) h innsiy of h aousi wav a h nd of h saml boms almos onsan and is givn by h Eq(). For a slowly hanging rnal fild (i.. τ << ) E ) E E sin( ) w rla h onsan ( lri fild E in Eq(4) wih E E sin( ) and avrag h rsul ovr h riod as indiad in Eq().h following rssion is hn obaind whr ab π π ( Sin ) ( Sin ) π d b (5) ρ ; ; ; a b ε µ E µ E ; 4

5 o ingra Eq(5) w mak h following ransformaion sin( ) hn d ab π hr d b. ( ) ( ) and Eq(5) boms. d (6) Baus of h oml naur of h ingral in Eq(6). W furhr rform anohr ransformaion i.. d d ; whih ransforms Eq(6) o h following rssion ab d (7) π ( ) Afr umbrsom maniulaion w obain for h rssion ab 4 A ; ( A ) A l w v ρ ε k s A A A ( A ) A ( A ) (8) whr A ( ) RESUS, SCUSSON AN CONCUSON n ordr o alula h anuaion and vloiy of h roagaing ulrasoni wav in GaN, h lromhanial ouling offiin has o b valuad using h aramrs of Ridly [9, ], O Clok and uffy [], and Shimada al [] who hav abulad h lri and lasi onsans for GaN. h ffiv izolri onsans for h wurzi sruur for A and A mods ar [3] 3 ( ) Sin θ Cosθ Cos, (9) θ 3 ( ) Cos θ Sinθ θ (3) Sin whr θ is h angl bwn h dirion of h roagaion and h -ais. h lromhanial ouling offiin is [9, ] χav (3) ε ε whr and ar h angular avrags off lasi onsans dsribing h roagaion of A and A wavs, rsivly [3] and ar h, shrial avrags of izolri onsans rsivly, whih ar and 4 8 ( ) ( ) 5 (3)

6 35 ( ) ( ) 5 (33) W analysis h rsul of q(7) for bulk GaN in a onsan lri fild and slowly hanging a. lri fild using h valu of χ in [8]. Firsly h gain (db m )..5 as whr h a. lri fild is absn, i is obsrvd ha as h onsan lri fild inrass h ak of h aousolri gain also inras. his is du o h fa ha h mobiliy of h lron inrass wih h inras in h onsan lri fild (rf Eq (5)) s fig.. gain (db m ) A grah of gain (dbm) rsus frquny (Hz) Frquny (Hz) Fig.. E.56 m E.56 m E 3.56 m his rsul has also bn obaind in [8]. Sondly, h as whn h a. fild is alid o h sysm, and d. is swihd off is also onsidrd in fig frquny (Hz) Fig. : h lo of aousolri gain agains frquny. is obsrvd ha as h amliud of h a. fild inrass h aks of h gain drass. ik h as of h d. fild, h aks our a sifi frqunis. hirdly, h siuaion whr boh h d. and a. fild ar rsn is onsidrd (s fig. 3). gain (db m) A grah of gain(dbm) agains frquny (Hz) Frquny (Hz) Fig. 3: 6

7 is inrsing o no ha whn h aousolri gain is lod agains h frquny of h aousi wav for givn valus of h a. lri fild ( ) h maimum gain dros as inrass. his is baus in h rsn of a. lri fild lron bunhing is slowd down and hn inraion wih aousi wav is rdud. h a. lri fild is hrfor modulaing h dir urrn. Anohr id rsul is h as whn h d. lri fild is k a abou.4 3 m and h aousolri gain is lod agains h aousi wav frquny for a givn a. lri fild (s fig. 4). ransarn. is our oinion ha h saml an b usd as masr in his rgion of frqunis. n onlusion w hav sudis h anuaion (amlifiaion) of aousi wav in GaN undr a dir urrn and a slowly hanging a. lri fild and suggs h us of his marial as masr. Rfrns []. C, Caroli, R Combso, P, Nozirs and Sain Jams J. Phys. C: Solid Sa Phys. 5, (97) []. W. Wbr, Phys. Rv 58, 54 (987) [3]. Y. Kong, O.. olgov, O. Jsn, and O. K. Andrsn. Phys. Rv. B 64, 5 (R) () - [4].. Hrl and G. Moos Phys. Rv. 84, 5 55 () gain (db m ) [5]. Prbinos, J. rsoff and P. Avouris, Phys. Rv.. 94, 868 (5) frquny (Hz) [6] F.G. Rammrsorfr,.F. Fishr, W. Mir, K. J. Bah and M.. Snydr. Comur and Sruurs ol 3 ssu (98) Fig. 4: Aousolri gain agains frquny whn E 4* m. is obsrvd ha h saml boms ransarn ovr wid rang of frquny hn suddnly hr is hug amlifiaion a around 5.8 Hz. Wha is inriguing is ha as h a. lri fild inrass a sond rsonan ak is obsrvd a around 8.7 Hz. wih furhr inras of h a. fild, h gain boms zro and h saml boms omlly [7] Y. M. Shabana and N. Noda. Comosi Par B: Enginring ol 3, ssu, - () [8] H.C. ong and C.M. Wayman Aa Mllurgia ol, ssu 7, (974) [9] M. Bamann, J.. Soubyrou, R. Barr,. Fruhar, R. Zah, S. Niol and R. Fruhar Journal of Magnism and Magni Marials ol 34, ssu, 59-67, (994) 7

8 [] Yu. N. Podilhuk and.n. rshhnko nrnaional Alid Mhanis, ol 38, No5, () [] R. ondon Advans in Physis. () [] R.H. Parmnr, Phys. Rv. B. 89, 99 (953) [] S.Y. Mnsah, F.K.A. Alloy and S.K. Agyong, J. Phys: Condns. Mar 8, 35 (996) [3] S.Y. Mnsah and G.K. Kangah J. Phys.: Condns. Mar. 3, 45 (99) [4] G. Winrih Phys. Rv. 4, 3 (956) [3] F.A. Maa and Y. Galrim Phys. Rv. B 56, 48 (997) [5] A. R. Huson, J.H. Mf and.. Whi. Phys. Rv. l (96) [4] S.Y. Mnsah, F.K.A. Alloy and S.K. Agyong, J. Phys: Condns. Mar 6, 6793 (994) [6] S.Y. Mnsah, F.K.A. Alloy, N.G. Mnsah and.w. Elloh, Surlai and Mirosru (4) 453 (3) [5] S.Y. Mnsah, F.K.A. Alloy, N.G. Mnsah, J. Phys: Condns. Mar, 55 () [6] S.Y. Mnsah, F.K.A. Alloy, N.G. Mnsah, H. Akrobou and G, Nkrumah, Surlai and Mirosruur 37, 87 (5) [7] S.Y. Mnsah, N.G. Mnsah,.W. Elloh, G.K. Banini, F. Sam and F.K.A. Alloy Physia E ol 8 ssu (5) [8] S.K. Abdlrahm,.P.Blyh and N. Balkan, Phys. Sas. Sol.(a) 85, () [7] A.A. Grinbrg and N.. Kramr Sov. Phys. oklady ol 9, No7, 55 (965) [8]. Yamda J. Phys. So Jaan, 44 (965) [9] A Govorov, A.. Kalamsiv, M. Ror, K.H. Hoffmann and N. Bokin, Phys. Rv. B 6, 659 () [9] E.M. Eshin and Yu.. Gulyav. Sov. Phys. Solid Sa, ol 9, No, 88 (967) [3] K.B. olygo and Z.. Uriskii, Zh ksr. or. Fiz 3, 99 (956) [] M.. Kaganov, Sh.. Mvlyu and.m.suslov, Sov. Phys., J.E..P. 5,, 89 (98) [3] M. Pomranz, Phys. Rv., 3, 38, (964) [] A.. Margulis and.a. Margulis, J.Phys. Condns. Mar 6, 639, (994) [3] S.Y. Mnsah, F.K.A. Alloy, S.K. Agyaong and N.G. Mnsah Sulais and Mirosru (9) 453 (97). 8

9 [33] B.K. Ridly Quanum Pross in Smionduors Cha 3, Clarndom Prss, Oford 9 (999). [34] M. Pomranz, Pro, EEE 53, 9 (966). [35] G.. O Clok and M.. uffy, Al. Phys.. 3, 55 (973). [36] K. Shimada,. Soa and K. Suzuki, J. Al. Phys. 84, 495, (998). [37] B.K. Ridly, B.E. Fouz and.f. Easman, Phys. Rv. B 6, 686 (). 9