DSpace at VNU: Acoustomagnetoelectric effect in a superlattice

VNII Journal o f Science, Mathematics - Physics 25 (2009) 131-136 Acoustomagnetoelectric effect in a superlattice Nguyen Quang Bau‘, Nguyen Van Nguyen I'hi rhanh Iluycn', Nguyen Dinh Nani‘, Tran Cong Phong'^ ^Faculty o f Physics College o f Science VNU 334 N^uvcn Traĩ, Thanh Xuan Hanoi Meĩnưm ^ỈỈIÍC University o f Education, 32 La Lot Hue, Vietnam Received 10 July 2009 Absiract The acoustomagnetoclcciric (AME) eíĩcct in a superiattice (SL) is investigated for an acoustic wave whose wavelength A '2ĩĩ/(Ị is smaller than the mean free path / of the electrons and hypersound in the region ql ^ (where q is the acoustic wave number) The anaiyticai expression for the AME current j is calculated in the case of relaxation time of momentum r is constant approximation The result indicates that the existense of in a SL may be due to the finite gap band , and ihe periodicity of the electron spectrum along the SL axis Numerical calculations have been done and the result is analysed for the GaAs/AlAs SL All the rcsLilts are compared with the normal bulk semiconductors (both theory and experiment) in the weak niamieiic field rcíỊÌon to show the difference I I n tr o d u c tio n It is well kiunvíì that, when an acoustic wave propaqatcs imough a conductor, it is accompanicd by a transfer o f energy and m om entum to the conducting clcctrons This gives rise to what is callcd the acoustoelcclric cíĩcct The study o f acoustoclccric cfTcct in the bulk sem iconductor has rcccivcd a !oi o f attcntion[ 1-4] Recently, Mcnsali has investigated this ciTect in a supcrlattice [5] and there has been a izrowinu: interest in observinii this cfTcct in mesoscopic structures [6 -8 ] However, in the prcscncc o f Ihc magnetic field the acoustic wave propagating in the conductor can produce another ellect called the acoustom agnctoelcctric (A M E ) elTect T he A M E effcct is creating an A M E current (if the sample is short circuited in the Hall direction), or an A M E field (if the sample is open) when a sample placed in a inamiclic field H carrics an acoustic wave propagating in a direction perpendicular 10 / / I he A M E eíTcct w as first fbrcsecd liicorelically by G rinbcrg and Kramer [9] for bipolar sem icon­ ductors and was obser'vcd experimenlally in bismuth by Yamada [10] In past times, there are more and more interests in studyinu and discovering this elTccl, such as in a monopolar semiconductor [11], and in a Kane seniiconduclor [ 12] in this specim en they observed that the A M E efTect occurs mainly bccausc o f the dcperidcnce o f the electron relaxation time r on the energy and when T = constant, Corresponding author nguyenvanhicudn a gmail.com 131 N Q Ban cl UỈ 132 I'NU Journal of Saencv Mcỉlhematics - Phystcs 25 (2()()9) Ỉ3 Ị-Ị3 the eíTect vanishes Like the classical magnetic field, the clTect also exists ill the case o f a quaiiiti/cd magnetic field [Icccntly, D Margulis and A Margulis [13,14] have studied the quantum acoustom agnctoclectric (Ọ A M E ) eilcct due to Rayleigh sound waves The A M E eíTcct is similar to the liall eíTect in the bulk semiconductor, where, the s o u n d flux Ỉ plays the electric currcnt J role The esscnce o f the A M F efi'cct is due to the existeiisc o f partial current generated bv the difTcrcnt cneruy groups o f electrons, when the total acoustoelcctric (lo A ■ - i + U-Vr -f ^ ' 2A s i n ( ^ ) ^ - ỂỈ (9 ) in The inequality (9) is condition for acoustic wave q to the A M E cỉĩect exists, riiereforc vvc have obtained the expression o f the A M I: curent ■AMF UJ^ r l-( 27ĨV,U\ỵơ Ư2 ( 10 ) 2A sin(^) the Hq.(lO) is the A M E current in SL for the case degenerate electron gas, that is onlv obtained if the condition o f inequality (9) is satisfied We can see that the dependence o f the A M E current o n the frequencv Ljfj is nonlinear Numerical results and discussions The parameters used in the calculations are as follow [ 12,13,16]: A — O le V , d — 100 m = 0.0G7rrio, nio being the mass o f free electron, H = 2.10^ = 10 * i r m ” ^, Vs = 5370 m “ ^ The result in Eq.{10) can be written in terms o f the acoustoelectric currcnt 11 the A M E currcnt depends on the magnetic field /7, the quantity Q r s e rv in g as a measure o f the is equal to i i r This result is quite interesting as a sim ilar magnetic strength T he ratio o f j ratio calculated for the case o f the Ọ A M E due to the Rayleigh sound wave was o f that order [15j In their ease, Q r :$> (quantized magnetic field) and the sample was a bulk material, bulk sem iconductor 112,13) cC 3.5 X X 10' q{1/cm) 10 X 10 Fig The dependence of AME current on the q for the case of the bulk semiconductor q(1/cm ) X 10 Fig The dependence of AME current on the q for the case of the superlattice It is plausible that m echanism responsible for the existence o f the A M E effect in a SL may be due to the finite band gap and the periodicity o f the electron spectrum along the z axis and not the N.Q Ban et (iL / VNLỈ Journal o f Science, Mathematics - Physics 25 (2009) Ỉ3Ĩ-Ĩ36 135 dcpenidcncc o f r on f ^7 112,15) I'hc calculation was done on the basis o f r= c o n stan t and according to [11] the A M E cfTect should be zero However, for > A w hen the SL behaves as a bulk monopolar semifconductor with the parabolic law o f dispersion, j —> as expected for T = constant [11] 'I his is readily dcduccd from the conservation laws T he non-linear dependence o f param eters A and d and the frequency ujfj and particularly the strong spatial dispersion o f on the SL once ag ain can only be attributed to the finite band gap and periodicity o f the energy spectrum o f electron alo n g the z axis Figure show s the dependence o f the A M E current on the acoustic w ave num ber in the case o f the b'ulk sem iconductor n-InSb[ 14] It can be see from figure I , W hen the q rises up, the A M E current incre:ascs linear and value o f A M E current is verv' small, approxim ation 10“ ^^ ?7?/l However, in figure when w e liave investigated for the case o f superlaltice, there appear distinct maxima and the \ a l u c o f ilic A M E current is larucr than that o f the A M E current in the case o f bulk semiconductor nInSb T he cause o f the diflercnce between the bulk sem iconductor and the superlattice, because o f the low -dinicnsional svstems characteristic, meanly, in the low-dim ensional systems the energy spectrum o f electron is quantized, like this, the A M E effect have been appeared in SL for the case degenerates elect runs tjas, and note that it exists even if the relaxation tim e r o f the carrier independs on the carrier energy atid has a strong spatial dispersion In the limit case at iUfj — 10^^ s " ^ , and H ™ 2.10^ A n i~ ^ the A M E currcnl is obtained the value about 10” '* r n A c r n “ ^, this value fits with the experimental result in [ 13] C onclusion In this paper, wc have obtained analytical expressions for the A M E current in a SL for the case o f th e iiet»enerate eieciron izas T he stroriíỉ dependences o f on the frequency n o f the mĩnetic íìcld acoustic wave, the SL parameters A and (I arc m iniband h a lf width and the period o f ll; c SIv, r e s p e c t iv e ly , r i i e r e su lt s h o w s that it e x i s t s e v e n i f th e r e la x a t io n t i m e T o f th e carrier d o e s not d c p c iid on the c a rric r energy and has a strong spatial d isp e rsio n , w hich result is diil'crcnt com pared t(' \hosc obtained i:i bulk s e m i c o n d u c t o r [ 11,12], acc ord in g to [ 11J in the e as e T — c o n sia n i the cí ĩcc t only exists if the electron gas is non-degenerate, if the electron gas is degcnarate, the ciTcct docs not appear However, our result indicates that in (SL) the A M E eíTect exists both the non-degenerate and the degenerate electron gas when the relaxation time T o f the carrier does not depend on the carricr energy In addition, our analysis shows that I*ie result has value, which is smaller than its in [5,12,15,17] and increases linearly ÍỈ o f the magnetic field This result is sim ilar to the semiconductor and the sijpcrlalticc for the case o f the non-degenerate electron gas in the weak magnetic field region [12,14,15] Unlike the semiconductor, in the SL the A M E current is non-linear with the acoustic wave (Ỵ Especially, in the limit case at ijfỊ ~ 10^3 5-1^ and / / = 10^ / h n ^ the A M E current is obtained the value about 10“ * ĩì ìA c m " ^ , this value fits with the experimental result in [13] 'I'he numerical result obtained for a G aAs/AIAs SL show s that the A M E effect exists when the q o f the acoustic w ave com plies with specific conditions (9) which condition dependens on the frequency o f the acoustic w ave Lc/g- Fermi energy, the mass o f electrons, the m iniband h alf width A and the period of the SI d N am el\ to have A M E c u r r e n t , the acoustic phonons energy are high enough and satisfied in the some in te n a ! to impact much m om entum to the conduction electrons N.Q Bau ei al / VNU Journal o f Science, Maihematics - Physics 25 (2009) I3Ỉ~Ỉ36 136 A ck n o w led g m e n ts This work is completed with financial support from the N A F O S T E D (103.01.18.09) and QG-09.02 References [1] R.ÍỈ Parmcnler P hys Rev 89 (1973) 990 [2] M Roller, A.v Kalamcit, A o Grovorov, Phys Rev Lett 82 (1999) 2171 [3] [4] [5] [6] P.E Lippens, M Lannoo, J.F Pouliqucn, J A p p i P hys 66 (1989) 1209 v v Afonin, Yu Gal'prin, Sem iconductor 27 (1993) 61 S,Y Mensah, F.K.A Allotey, S.K Adjepong Phy's (1994) 6783 J.M Shillon, VI Talvanskii, M Pepper, D Ritchie, C ondens M a tter (1996) 531 [7] F.A Maa, Y Galperin, Phys Rev B 56 (1997) 4028 [8] M.J Hoskins, H Morko, B.J ỉỉunsinger, AppỊ Phỵs Lett [9] N I 41 (1982) 332 Kramer, o v P hys D okl (1965) 552 [10] T Yamada / P hys Soc Japan, 20 (1965) 1424 [11] E.M Epshtein, YU.V Gulyaev Sov Phys Solid s state (1967) 28 [12] N.Q Anh, N.Q Bau N v Huong J Fhys VN, (1990) 12 [13] A.D M a r g u l is '/ P hys (1994) 6139 [14] K Mincichi T Shoji, J P hys Soc Japan, 30 (1970) [15] S.Y Mcnsah, F.K.A Allotcy, s Adjcpong y Phys (1996) 1235 [16] M.I Kaganov, 5ơu Phys-JETP, 51 (1967) 189 [17] G.M Shmelev, E.M Epshticn, N.Q Anh S o viet Phys St Sol II (1981) 3472  ... obtained ax T = constant !l is also nonlinear dependent on the SL parameters Num erical calculations are carried out a specific G aAs/AlAs SL to clarify our results The papíỉr is organized as... solving a] .(5) we shall consider a situation w hereby the sound is propagating along the SL axis (0/) , the m agnetic field II is parallel to the (ox) axis and the A M E current appears parallel... runs tjas, and note that it exists even if the relaxation tim e r o f the carrier independs on the carrier energy atid has a strong spatial dispersion In the limit case at iUfj — 10^^ s " ^ , and