, ! DSlHQC GrAHANQr QU0C VIE.TNAM NATIONAL UNIVERSITY, HANOI t s Su NB 6' 68 ?l r; rnF e' KHO ' J E J U R N AtLr F ffitr l E , I P," { - Physics24,No lS (2008)240-243 VNU Journalof Science,Mathematics The parametricresonanceof confinedacousticphononsand confinedopticalphononsin quantumwells NguyenThi Mai Nhien,Le 'Ihai Hung*, Do Manh Hung,NguyenQuangBau Faculty of Physics, Collelie of Science, WU, 334 Nguyen Trai, Hanoi, Vietnant Received 17 November 2008 Abstract The parametric resonance of confined acoustic and confined optical phonons in quantum wells by using a set of quantum kinetic equations for phonons is studied The analytical expression of the tfueshold amplitude E6 of the field in quantum wells is obtained The dependenceof it on the temperature T of the s;ystemand the frequecy O of the external electromagneticfield is sfudied Numerical compuLtations and graphs are performed for GaAsGaAsAl quantum well The results are compared wilh the case of unconfined phonons Introduction field (EEF), an electrongas As we known that in the presenceof an extemalelectromagnetic becomesnon-stationary.When the conditionsof parametricresonanceare satisfied,parametnc ( PIT) of somekinds of excitationssuchasphonon-phonon, plasmoninteractionsandtransformations plasmon-phonon will arise, the exchange energy kind such as: different of excitations plasmon,or of processbetweentheseexcitationswill occur U-zl.Ttre PIT of acousticand opticalphononshasbeen in bulk semiconductors considered [3-4] There have been a lot of works on the PI'f for low dimentional semiconductors15-10] However, the parametricresonanceof acousticand optical phononsin quantumwell in the caseof confined phononshave not sfudied yet Therefore,in this paper, we continue to study the the parametricresonanceof acousticand opticalphononsin quantumwell in the presenceof EEF in the caseof confinedphonons.This resulthasbeencomparedwith the caseof unconfinedphonons We usea simplemodelfor a quantumwell, in which a two-dimentionalelectron,phonongasis well alongthe z directionand electronsand confinedby the infinity potentialV(z) of a rectanguLlar phononsare free on the x-y plane.A laserfield E ==E, sin(Q/) inadiatesthe samplein a direction as a which is normalto the x-y plane,its polarizationis arlongthe x axis and its strengthis expressed and optical phononsinteraction If the ele,;tron-acoustic vector potential A0=lE".os(fU) f) potentialis used,the Hamiltonianfor the systemof the electronand the acousticand opticalphonons in caseof confinedphononsis writtenas: (1) H = H " * H n r n + H " p n + H " - n p t+ H r - o p n In order to establisha set o1'quantumkinetic equationsfor acousticand optical phonons, usethe generalquantumdistnbutionfunctionsfor (U^,r^l,and (c^.u^),*h * (t4), a.notesa stastical Conesponding I I 82@yahoo'com'\/n E-mail:hunglethaiI9 author 240 I ; I l I N-I.M Nhien et al / WU Journal of Science,Mathematics- Physics24, No IS e\Aq phonons, theconfined average at thernoment(W), =rrl*i),(,| t t 240-243 241 ir rt densirymatrixoperator), Hamintonianin Eqs(1) and realizingoperator algebraiccalculatioru The parametric resonanceof confined acousticphononsand confined optical phononsin quantum wells usingFourier rransforma tion,(v^.a,) ,=+-Tr^,u,(ap-'^aat, vmdt(rd:)\r^u,),{^ (2) _ _ € _ @ We obtain the general dispersion equation for the parametric interaction and tranformation betweenconfined acousticphonons and optical phononsin compositional surperlatice: - rr)"1,.Ne), - r,,, - :,,1' (m, a+n,") (o -,,^,u, Q,, |ZV lryu.lIq I o^,i.n I =#}Vr",l'|ry,l'lri,l'a^,4,e^4,n*(*,4,,r)rt*(m,Q',at+NO) (3) Here: = (*)r., (* n,(*,d,,,ul +,e), Zt )r,u,(ot (4) !r (a+rc)= (s) ?b^W ) -',,V,- q,)- nfi - tto+ We assumethat electron- phononinteractionsthe condition: o Vnl^lq,l'1ry,]' (6) If we write the dispersionrelationof acousticandopticalphononas: to,(*,Q ) = o)oI ir o, ,o,(*,4,) = 0o + ir o (7) ,o: -!ZV:,,1'lri,l' *r;,u(,^,,,\ (8) (-^,r,\ ',u =- lZV * r,,u u * ' ' i,^,1'ln;,1' h (e) We obtaintheresonance acousticphononmode: I * ,ot= ,, *;L(u, t ,,)n(q)- i(r, + r")+ withA:#zvl,,,|'t{,DflRer^,u(r^,u,\ I (10) (ll) (11)thesigns(t) in thesubcriptof atl correspond ln equation to theslgns(t) in thefrontof to the othersignpairs.The signsdependon theroot andthe sig"s (t) in subcriptof atl correspond condition: theresonant 242 N.T.M- Nhien et ol / VNU Journr'il of Science,Mathematics- Physics24, No lS (2005) 240-243 ^r, 4!)' r^l ^.r,(,,,,,,- [R"f'''a-, @^JT- For instance,the existenceof ir positiveimaginarypart of atf impliesa parametricamplication of the acousticphonon.In suchcasethat L