VNU J O U R N A L O F SCIENCE, Nat Sci., t.xv r fil - 1999 T H E Q U A N T U M THEORY OF A M PLIFIC A TIO N OF S O U N D (A C O U STIC PH O N O N S) B Y L A SE R W AVE IN N O N - D EG EN ER A TE SE M IC O N D U C T O R N guyen Q uang B au, N guyen Vu N h an and C h h o u m m N avy FHciiIty o f Physics - College o f Natural Sciences - V N U A b s t r a c t Based on the quantum, transport eq u a tio n for the electron - p h o n o n system o f s e m i c o n d u c t o r s , the a m p lific a tio n o f s o u n d (a c o u s tic p h o n o n s ) by L a s e r w a v e w ith m u lH p h o to n a b s o r p t io n p ro ce ss IS th e o r etica lly stu died T h e a n a ly tic e x p r e s s io n s f o r the co efficien t o f a m p lific a tio n o f s o u n d (a c o u s tic p h o n o n s ) a n d th e c o n d i t io n s o f aTTi.plificat.ion of so u n d (acou stic ph on on s) in n on -degenerate se m ic o n d u c to rs an d in the ca se w ith s u m m a t i o n o v e r all va lu es w ith m { n - ^ ~ a fr e q u e n c y o f L a s e r w a v e i J are obtained The difference of the coefficient of amphfication o f s o u n d ( a c o u s tic p h o n o n s ) a n d the c o n d itio n s o f a m p lific a tto n o f s o u n d ( a c o u s tic p h o n o n s ) m the case w ith m u ltip h o to n a b s o r p tio n from, the case w i th m o n o p h o t o n absorjition IS discussed I INTRODUCTION The tlieoiy of amplification of sound (acoustic phonons) by Laser wave ÊoSin(ÍÌỂ) (ÍỈ - the frequency of Laser wave) in semiconductors has been studied [l 3] In [1 2] the physics problem was restricted for degenerate semiconductors in the case of monophoton ahsorptioii Tho rpsnlts of works [1,^] iìulir-o that tho ahnoiption coofficicnt of aouiid (acoustic phoiions) can bo negative in some regions of values of acoustic wave vector ỹ That is t he absorption coefficient of sound (acoustic phonons) changes into the coefficient of aniplificat.ion of sound (acoustic phonons) In [3] the analytic expressions for the absoiption roefhciont of sound (acoustic phonons) have been obtained for the case of non(legpneiate semiconductors with multiphotoii absorption process, but in restricted values: A exp (13) we have a ( ^ ) and if (14) A{u>ạ) < exp [ - ^ ) A { - U J ) , we have a(if)(0 and which has the fonii a { q ) = - Ỉ I>0 ( ni \ 1/2 ) f'xp ps \ k T J \ exp I exp ( - ^ ) Ả(-U>ạ) - A(Wự-)| in 2qH-T V2 m 'V I ;-0 J! (15 It moans that we have again the coefficient of amplification of sound (acoustic phonons) From the analytic fonimla (12) we make analyze the absorption coefficient of sound in Ihuil Lơậ ^ / ỉ t t , X ^ AT, ujq’ ^ AT aiiei k \ ị T q ^ / m tln u tlio iumiiiu nt of the Bessel function is bigger than , the summation only value j ~ and lo(^) ^ c^y/2nz Om result for tho absorption coofficient of sound ill this limit cases is tho saiiK^ losiilt of 3].Not that if the condition of amplification of sound (14) is satisfied then tho absorption coefficient of sound changes into the coefficient of amplification of sound We ninnorically evaluate and plot thf* analytic formula (7) for the casr of Iiioiioplioton absorption (Fig.l) and (12) for the caso of inultiphoton absoiption (F'ig.2) in the sanii' condition Thò absorption coefficient of sound (acoustic phonons) is plotted as a fuiK tion of tho froqiu'ury of Laser wave (Q) and of the froquency of souml (acoustic phonon ) (ujy-) From the graphics we can see - For the nionophoton absorption : when UJ^/Q > we have the absorption coi'fficient of sound (acoustic phonons) a{q) > - Ill the same condition for the case of multiphoton absorption wo have again tho coefficient of amplification of sound (acoustic phonons) a{q) < Note that, the (lependonce of the coefficients of amplification of soinitl (acoustic phonons) on the frequency of Laser wave (Í2) and on the frequency of souiul (acoustic phonons) (iu’ỹ) is nonlinear and complicated The Q u an tu m Theory o f Amplification o f Sound Mto® OMEGA.ấ)dft Fụj 7:Tho c oofficient of amplification of sound (acoustic phonons) ill the case of inonophoton absoipfion K- axis = (\{(])~ axis : o m e g a - axis = - axis : O M E G A - axis = ÍÌ - axis X 10 OMÊOA-w»s a /'/Ự : TIh^ coi'fficifuit of amplifii'ation of soiiiul (acoustic phouons) in th(‘ casi^ of Iiiult iphoioM ahsoi ptioii S3- axis ^ HX I S I OlllGgii - iixis — i j j q - axis ; O M E G A - axis = Q - axis.; V CONCLUSION In the conclusion, we want to (‘Iiiphasizo that : The analytic expiossions for the' condition uJệ/íì /2ni- and UJ^ kT A > k T th(‘ absorption coefficient of sound with multiphot.on absorption process (12) return to the rosult of [3 In the case of monophoton absorption (Section 3) : our results are different from results of [1,2] The reavSon of difference is th at results of [1,2] for the case of dogenorate semiconductors, but our results for the case of non-dogeiierate seinicoiuliK tors In the multiphoton absorption (Section 4) : the expressions for the coiulition and the coefficient of amplification of sound (acoustic phonons) (14), (15) show that tlie different dependencies of that ones ill comparison with results of [1,2 and Section The reason of difference is th at results of 1,2] and section for the case of monophoton absorption, but the expressions (14) and (15) for the case of multiphoton absorption A c k n o w le d g m e n t: We would like to thank the niomber of group of Theoretical Solid s ta te Physics, Faculty of Physics , College of Natural Science , Vietnam National Univpisit v for their discussions and the National Project on Basic Science K T 04 foi financial support REFEREN CES 1] E.M Epstein Radio physics, 18(1975) 785 2] E.M Epstein Lett JET P, 13(1971) 511 Nguyen Hong Son, G.M Shmelev and E.M Epstein Izt) VUZOB S SS R J Physics 5(1984)19 41 D.K Ferry and Carlo Jacoboni Qĩiarứuin transport in semiconductors New YorkLondon 1992 5] G.M Shmelev and Nguyen Quaiig Bail Physical phenomena IĨÌ sennconductors Kishinev (1981)12 6] Nguyen Quang Bau and Nguyen Van Huong J Science of HSU, Physics, 3(1990) 7J L Sholimal lunnel effects in semiconductors and appb call OTIS Moscow 1974 TAP CHI KHOA HOC ĐHQGHN KHTN t.x v , vPl - 1999 LÝ THUY ET LƯỢNG T Ử VÉ SỰ GIA TANG SÓNG ẢM (PH ONO ÀM) BỜI SĨNG LASER TRONG BAN DAN KHỊNG SUỴ BIEN N g u y ễ n Q u a n g B u , N g u y ễ n V ũ N h â ĩi, C h h o u m m N a v y Khoa Vật ìý - Đại học K H Tự nhiên, ĐH QG Hà Nội Trịn cư sờ phương tiìuli động lượng tiV hệ điện tiV - phonon c ua hán (lan, ỉighi(Mi cứu lý thuyết gia tầng sóng ảm (phonon âni) bờ sóng Laser có kể đốn q trình hấ]^ thụ nhieu photon Thu biểu thức giải tích cho hệ số gia táng sóng ảni (phoiioii âiiì) đieii kiện gia tăng sóng âm (phonon ảin) trư n g h ạp có kơ đến tíiih toii^ theo tất đại lượng chứa fíì{Q, — tần số sóng Laser, / — 0, ±1; ± ; ) Thảo luận VP khác hệ số gia tăng sóng âm (phonon âm) trư n g hợp hấp thụ nhieii photon so với trường hợp hấp thụ photon ... that, the (lependonce of the coefficients of amplification of soinitl (acoustic phonons) on the frequency of Laser wave (Í2) and on the frequency of souiul (acoustic phonons) (iu’ỹ) is nonlinear... It uii-aiis tliat W(' have the coofficient of amplification of sound (acoustic phonoiis) IV AMPLIFICATION OF SOUND (ACOUSTIC PHONONS) IN NON- DEGENERATE SEMINCONDUCTORS IN MULTIPHOTON ABSORPTION... nonlinear and complicated The Q u an tu m Theory o f Amplification o f Sound Mto® OMEGA.ấ)dft Fụj 7:Tho c oofficient of amplification of sound (acoustic phonons) ill the case of inonophoton absoipfion