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 hyper
Trang 1VNII 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 N ani‘, 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 ^ 1 (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 t r o d u c t i o n
It is well kiunvíì that, w hen an acoustic wave propaqatcs im ough 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 m agnetic field the acoustic wave propagating in the conductor can produce another ellect called the acoustom agnctoelcctric (A M E ) elTect T h e A M E effcct is creating an A M E current (if the sam ple is short circuited in the Hall direction), or an A M E field (if the sam ple 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 Kram er [9] for bipolar sem icon ductors and was obser'vcd experim enlally 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 sem iconductor [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 tim e r on the energy and when T = constant,
Corresponding author nguyenvanhicudn a gmail.com
Trang 2the 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 q u an tu m acoustom ag- nctoclectric (Ọ A M E ) eilcct due to Rayleigh sound waves
The A M E eíTcct is sim ilar to the liall eíTect in the bulk semiconductor, w here, 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 d ue to the existeiisc o f partial
current generated bv the difTcrcnt cneruy groups o f electrons, when the total acoustoelcctric (lo<nizitu- dinal) current in specim en is equal to zero W hen this happens, the energy d ependence o f the electron mom entum relaxition tim e will cause average mobilities o f the electrons in the partial current, in ccn- eral, to differ, if an external m agnetic field is perpendicular to the direction o f tlie sound flux, th e Hall currents generated by these groups will not, in general, com pensate one another, and a non-zero AMI:
efFect will result T h e A M B eíTcct problem in bulk sem iconductors for the case ql » 1 (w h e re q is the acoustic wave num bc, Ỉ is the mean fee path) has been investigated [11,12,15] T h e A M E e íĩe c t in
a superlattice still, however, opens for studying, in this paper, we exam ine this efi'ect in a supcrlattice for the case o f electron relaxation time is not dependent on the energy Futherm orc, w e think that the research o f this eiiect m ay help us to understand the properties o f SL material It will be seen that,
due to the anisotropic nature o f the dispersion law, the A M E efiect is obtained ax T = constant !l
is also nonlinear dependent on the SL parameters N um erical calculations are carried out a specific
G aA s/A lA s SL to clarify our results
The papíỉr is organized as follows In section 2 vve outline the lheor>' and conditions n e c c ss a r\ to solve the problem In section 3 we discuss the results and in section 4 w e have som e conclusion
2 A cou stoinagn ctoelcctric current
Following the method developed in [6 ] we calculate the A M E current in a SL T h e acoustic
wave will be considered as a hypcrsound in the region ql ^ 1 and then treated as m onochrom atic
p h o n o t i s ( t r e q u e n c y 1 h e p r o b l e m w i l l b e s u l v c t i Iti t h e q u a s i - c l a s s c a i e a s e , i.e 2 A I '
(2 A is the w idth o f the m iniband, T is the electron relaxation time) T h e magnetic field will also be
considered classically, i.e ÍÌ < ư, lìíì k'fiT (ly is the frcqucncy o f electron collisions, Q is the cyclotron frequency), and weak, thus limiting ourselves to the linear approxim ation o f I I
The density o f the acoustoelectric currcnt in the presence O Í 'magnetic field can be w ritte n n the form [15]
1) where
UffVs I
-f I Ị / ( ^ n , p + g ) “ f{^ n,p) ~ ^n, p ~ ( 2 )
Here $ is the sound flux, Vs is the velocity sound, f { e n p ) , f n p the distribution function and
the energy o f the electron, respectively, n denotes quantization o f the energy spectrum , is the
matrix element o f the electron-phonon interaction and ĩpi is the root o f the kinetic equation given b y [8]
Trang 3N.Q Ban ct id / VNU Journal o f Science, hiliihematics - Physics 25 (2009) I3!~I36 133
ilc rc V', is the electron velocity and *{ } = ( O f / O e ) H V ị; { O f/ d c , ) T he operator Wj; is the collision operator describinii rclcxation o f the non-equilibrium distribution o f electron, and Wj;
is assum ed to be lic m iilia n [16| In ihc V -approxim ation co n s ta n t’, the collision operator has form 11}; " 1 / T We shall seek the solution o f Hq.(3) as
(4) substituting Eq.(4) into Eq.(3) and solving by the method o f iteration, w e get for the zero and the first
approximatioỉì Inscĩliniĩ into E q ( l) and taking into account the fact that I G ợ ịr 1^—I G ;7 ;7 we obtain for llic acoiistoclcctric current the expression
Jz AỈC
V; ( / 7 4 (]) t - l \ { p ) T \ ỗ { e n , p + Ẹ - e n , p - ^ q ) d ^ P
X~T 9 I I I i> rQ.p I / (^rỉ ;7+(ĩ) ~
27ĩ^ìììCU.\Ỵl\>ị J ‘
{ \ ' { p + q ) X / / ) , - X / 7 ) , Ị ỏ ( e „ , , 7 + g - € n , p - u J , j ) d ^ } )
rh e nialrix cleincnl o f the clcctroii-phonon interaction for qd i {d is the period o f the SL) is
mvcii as Ị p i”' A is ihc dcibnnation potential constant and Ơ is the density o f the
S L
ỉn solving ỉa].(5) w e 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 to the
(oy) axis U nder such orientation llic first term in Eq.(5) is responsible for the acoustoelcctric current and solution is found in [6 ] riie sccond term is llie A M E current and is expressed as
\\n
-V c(/^( <f) - v;(p)j(5(e„ ^ + , ^ - €n,ịĩ - í^,ĩ)đ^P,
where i i = v l l / i ì ì ( \
i lie distribution function f { ( n p ) o f dcíỊcncratc electrons gas is given bv
/ ( / , v = i ' , ’
(G)
(7)
w h e n f/.' is Fecnii enerey T he en em y spectrum ^7 o f electron in the SL is given using the usual
Iio t a t o ii b_\ 117|
ỉerc /7, and ])^ arc the transverse and loniiitudinal (relative to the SL axis) com ponents o f the
q uas-nioincnluin, respectively; A,, is llic h air width o f the n t h allow ed m iniband, m is the cfFective
mass o f electron
vVe assume that cleclrons arc confined to the lowest conduction m iniband (n - 1) and omit the
mini:)anci indicates T his is to say that the field does not induce transitions betw een the filled and
Trang 4131 N.Q Bưu cĩ lìỉ / VNU Journal o f Science Mathematics - Physics 25 (2009) Ỉ3 Ỉ-Ỉ3 6
empty m inibands, llius the A n can be written the A.
Substituting nc].(7) and (8 ) into Eq.(6 ) we obtain for the A M Í: current satisfying the coriditiion
^ 2 ' 2 A s i n ( ^ ) ^
-Í/.' > A
T he 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
2 A s i n ( ^ )
(10)
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 d e p en d e n ce o f the A M E current o n the
frequencv Ljfj is nonlinear.
3 Num erical results and discussions
T he param eters 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^ 4 <I> = 10 * i r
m ” ^, Vs = 5370 m 5 “ ^.
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
magnetic strength T h e ratio o f j is equal to i i r T h is result is quite interesting as a sim ilar
ratio calculated for the case o f the Ọ A M E due to the Rayleigh sound w ave was o f that order [15j In
their ease, Q r :$> 1 (quantized magnetic field) and the sam ple was a bulk material, bulk sem iconductor
112,13)
X 10
q{1/cm)
X 10
q (1 /cm ) X 10
Fig 1 The dependence of AME current on Fig 2 The dependence of AME current on
the q for the case of the bulk semiconductor 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
Trang 5dcpenidcncc o f r on f ^7 112,15) I'hc calculation was done on the basis o f r= c o n s ta n 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 w ith the parabolic law o f dispersion, j —> 0 as expected for T = c o n sta n t [11] 'I his is readily dcduccd from the conservation laws T h e non-linear d ependence o f on the SL
param eters A and d and the frequency ujfj and particularly the strong spatial dispersion o f once
a g a in can only be attributed to the finite band gap and periodicity o f the energy spectrum o f electron
a lo n g the z axis
Figure 1 show s the dep en d e n ce 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 1 0 “ ^^ ?7?/l However, in figure 2 when w e liave investigated for the case o f superlaltice, there appear distinct m axim a 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 sem iconductor n- InSb T h e cause o f the diflercnce betw een the bulk sem iconductor and the superlattice, because o f the low -dinicnsional svstem s 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 i f 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 1 0 ” '* r n A c r n “ ^, this value fits with the experimental
result in [ 13]
4 C o n c lu sio n
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 h e stroriíỉ dependences o f on the frequency n o f the maíĩnetic íìcld acoustic w ave, 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 i v e l y , r i i e r e s u lt s h o w s that it e x i s t s e v e n i f t h e r e l a x a t i o n t i m e T o f t h e carrier d o e s
not d c p c iid on th e c a rric r e n e rg y an d has a strong spatial d isp e rs io n , w hich result is diil'crcnt com pared
t(' \hosc obtai ned i:i bulk s e m i c o n d u c t o r [ 11, 12], a c c or d in g to [ 11J in the e a s e T — c o n sia n i the c í ĩc c 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, w hich 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 w eak 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 / / = 2 10^ / h n ^ the A M E current is obtained the value about 10“ * ĩ ì ì A c m " ^ , this value fits with the experimental result in [13].
'I'he numerical result obtained for a G aA s/A IA s 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) w hich condition dependens on the frequency
o f the acoustic w ave Lc/g- Fermi energy, the mass o f electrons, the m iniband h a lf width A and the period
o f the SI d N am e l\ 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 im pact much m om entum to the conduction electrons
Trang 6A c k n o w le d g m e n ts This work is com pleted with financial support from the N A F O S T E D (103.01.18.09)
and QG-09.02.
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