Materials Transactions, Vol 56, No (2015) pp 1408 to 1411 Special Issue on Nanostructured Functional Materials and Their Applications © 2015 The Japan Institute of Metals and Materials The Dependence of a Quantum Acoustoelectric Current on Some Qualities in a Cylindrical Quantum Wire with an Infinite Potential GaAs/GaAsAl Nguyen Vu Nhan1,+, Nguyen Van Nghia2,4 and Nguyen Van Hieu2,3 Faculty Faculty Faculty Faculty of of of of Physics, Academy of Defence Force Air Force, Son Tay, Hanoi, Vietnam Physics, Hanoi University of Science, Vietnam National University, 334-Nguyen Trai, Hanoi, Vietnam Physics, Danang University, 459 Ton Duc Thang, Danang, Vietnam Energy, Water Resources University, 175 Tay Son, Hanoi, Vietnam The quantum acoustoelectric (QAE) current is studied by a quantum kinetic equation method and we obtain analytic expression for QAE in a cylindrical quantum wire with an infinite potential (CQWIP) GaAs/GaAsAl The computational results show that the dependence of the QAE current on the radius of CQWIP GaAs/GaAsAl, the Fermi energy ¾F and temperature T is non-monotonic, and the appearance of peak when the h" ðB20 B2 ị n ;N n;N (n 6ẳ n0 and N 6¼ N ) is satisfied Our result indicates that the dominant mechanism for such a behavior is the condition ẵq~ ẳ ẵk~ ỵ 2mR2 electron connement in the CQWIP GaAs/GaAsAl and transitions between mini-bands All these results are compared with those for normal bulk semiconductors and superlattice to show the differences The dependence of a QAE current on some qualities in a CQWIP GaAs/GaAsAl is newly developed [doi:10.2320/matertrans.MA201514] (Received January 22, 2015; Accepted July 1, 2015; Published August 25, 2015) Keywords: cylindrical quantum wire, quantum acoustoelectric current, electron-external acoustic wave interaction, electron-acoustic phonon scattering, quantum kinetic equation Introduction When an acoustic wave propagating in a conductor creates a net drag of electrons and hence an acoustoelectric (AE) current or, if the circuit is disconnected, a acoustoelectric potential difference The study of this effect is crucial because of the complementary role it may play in the understanding of the properties of low-dimensional systems (quantum wells, superlattices, quantum wires+) As we know, low-dimensional structure is the structure in which the charge carriers are not free to move in all three dimensions The motion of electrons is restricted in one dimension (quantum wells, superlattices), or two dimensions (quantum wires), or three dimensions (quantum dots) In lowdimensional systems, the energy levels of electrons become discrete and the physical properties of the electron will be changed dramatically and in which the quantum rules began to take effect Thus, the electron-phonon interaction and scattering rates1) are different from those in bulk semiconductors The linear absorption of a weak electromagnetic wave have been studied in the low-dimensional structure.24) The quantum kinetic equation was used to calculate the nonlinear absorption coefficients of an intense electromagnetic wave in quantum wells5) and in quantum wires.6) Also, study on the effect of AE in the normal bulk semiconductor has received a lot of attention.710) Further, the AE effect was measured experimentally in a submicronseparated quantum wire11) and in a carbon nano-tube.12) However, the calculation of the QAE current in a CQWIP by using the quantum kinetic equation method is unknown Throughout,5,6) the quantum kinetic equation method have been seen as a powerful tool So, in a recent work13) we have used this method to calculate the QAME field in a QW In the present work, we use the quantum kinetic equation method for electron-external acoustic wave interaction and electron+ Corresponding author, E-mail: nvnhan@excite.com acoustic phonon (internal acoustic wave) scattering in the CQWIP GaAs/GaAsAl to study the QAE current The present work is different from previous works710) because: 1) the QAE current is a result of not only the electronexternal acoustic wave interaction but also the electronacoustic phonon scattering in the sample; 2) we use the quantum kinetic equation method; 3) we show that the dependence of QAE current on the Fermi energy ¾F, the temperature T of system and the characteristic parameters of CQWIP GaAs/GaAsAl is nonlinear; 4) we discussed for the CQWIP GaAs/GaAsAl, which is a one-dimensional system (the CQWIP GaAs/GaAsAl) and these results are compared with those for the bulk semiconductor,710) superlattice.14,15) This paper is organized as follows: In Section 2, the QAE current is calculated through the use of the quantum kinetic equation method In section 3, the QAE current is discussed for specific CQWIP GaAs/GaAsAl Finally, we present a discussion of our results in section The Analytical Expression for QAE Current in a CQWIP GaAs/GaAsAl We consider a CQWIP structure of the radius R and length L with an infinite confinement potential Due to the confinement potential, the motion of electrons in the Oz direction is free while the motion in (x-y) plane is quantized into discrete energy levels called subbands Then the eigenfunction of an unperturbed electron in the CQWIP is expressed as pz ẳn;N;~pz ~rị ẳ p expinị exp i z ẳn;N ~rị h" R2 L r < RÞ; ð1Þ here N = 1, 2, 3, + is the radial quantum number; n = 0, «1, «2, + is the azimuth quantum number; R is the radius ~ ẳ 0; 0; pz ị of the CQWIP; L is the length of the CQWIP; p The Dependence of a Quantum Acoustoelectric Current on Some Qualities in a Cylindrical Quantum Wire is the electron’s momentum vector along z-direction; ðBn;N r=Rị ẳn;N ~rị ẳ JJnnỵ1 Bn;N ị is the radial wave function of the electron in the plane Oxy, with Bn,N are the N level root of Bessel function of the order n The electron energy spectrum takes the form ắn;N;~pz ẳ h" p2z h" B2n;N ỵ ; 2m 2mR2 ð2Þ where m is the effective mass of the electron We assume that an external acoustic wave of frequency ½q~ is propagating along the CQWIP axis (Oz) and the acoustic wave will be considered as a packet of coherent phonons with ~ ~ ẳ 2ị ~ ~ ị, the Ô-function distribution in k-space Nkị ẵq~ vs Ôk À q where º is the flux density of the external acoustic wave with frequency ½q~, vs is the speed of the acoustic wave, q is the external acoustic wave number We also consider the external acoustic wave as a packet of coherent phonons Therefore, we have the Hamiltonian describing the interaction of the electron-internal and external phonons system in the CQWIP in the secondary quantization representation can be written as X Hẳ ắn;N;~pz aỵ a pz n;N;~ pz n;N;~ n;N;~ pz X ỵ In;N;n0 ;N Ck~ aỵ0 a 0 p0z bk~ pz ỵk~ n ;N ;~ n ;N ;~ n;N;n0 ;N ;k~ ỵ X k~ ỵ ỵ bỵ ~ ị k h" ẵk~ bỵ b k~ k~ X n;N;n0 ;N ;~q Cq~ Un;N;n0 ;N aỵ a ~ tị; p0z bq~ expiẵ q pz þ~q n0 ;N ;~ n0 ;N ;~ ð3Þ p where Ck~ ẳ k=2vs SLị is the electron-internal phonon interaction factor, μ is the mass density of thepmedium, $ isffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the deformation potential constant, Cq~ ¼ iÃv2l h" ẵ3q~ =2FSị is the electron-external phonon interaction factor, with F ẳ qẵ1 ỵ ã 2l ị=2ã t ị ỵ ã l =ã t 2ị1 ỵ ã 2t ị=2ã t ị, ã l ẳ v2s =v2l ị1=2 , ã t ẳ v2s =v2t ị1=2 , S = ³R2 is the surface area, vl (vt) is the velocity of the longitudinal (transverse) bulk acoustic wave, aỵ (an;N;~pz ) is the creation (annihilation) n;N;~ pz operator of the electron; bỵ (bk~ ) is the creation (annihilation) k~ operator of internal phonon and bq~ is the annihilation ~ is the operator of the external phonon The notation jn; ki ~ electron states before interaction and jn ; k ỵ q~ i is the electron states after interaction Un,N,nA,NA is the matrix element of the operator U = exp(iqy klz): Z expkl Lị R Un;N;n0 ;N ẳ ẳn0 ;N ;~p0 ~rị z R2 L ẳ n;N;~pz ~rị expiq? rịdr; 4ị here kl = (q2 (ẵq/vl)2)1/2 is the spatial attenuation factor of the potential part the displacement field and In,N,nA,NA is the electronic form factor: Z R In;N;n0 ;N ¼ JjnÀn0 j ðq? RÞ¼ Ãn0 ;N ;~p0 ~rịẳn;N;~pz ~rịrdr; 5ị z R with q? is the wave vector in the plane Oxy To set up the quantum kinetic equation for electrons in the presence of an ultrasound, we use equation of motion of statistical average value for electrons ih" @hfn;N;~pz tịit @t ! QAE ẳ hẵaỵ a pz ; Hit ; n;N;~ pz n;N;~ 1409 ð6Þ where the notation hXit is mean the usual thermodynamic average of the operator X and fn;N;~pz tị ẳ haỵ a pz it is the n;N;~ pz n;N;~ particle number operator or the electron distribution function Use the Hamiltonian in the eq (3) replaced into the eq (6) and realizing operator algebraic calculations like in Ref 13), we obtain the solution of the quantum kinetic equation for electrons in CQWIP GaAs/GaAsAl in the form of the function f (t) as follows 2á X ftị ẳ jCk j2 jIn;N;n0 ;N j2 Nk ffn;N;~pz fn0 ;N ;~pz ỵk~ ị h" 0 ~ n ;N ;k Ôắn0 ;N ;~pz ỵk~ ắn;N;~pz h" ẵk~ ị ỵ fn;N;~pz fn0 ;N ;~pz k~ ịÔắn0 ;N ;~pz k~ ắn;N;~pz ỵ h" ẵk~ ịg X jCq j2 jUn;N;n0 ;N j2 Nq fðfn;N;~pz À fn0 ;N ;~pz ỵ~q ị ỵ h" n0 ;N ;~q Ôắn0 ;N ;~pz ỵ~q ắn;N;~pz ỵ h" ẵk~ h" ẵq~ ị fn0 ;N ;~pz ~q fn;N;~pz ị Ôắn0 ;N ;~pz ~q ắn;N;~pz h" ẵk~ ỵ h" ẵq~ ịg; 7ị where is relaxation time of momentum, fn;N;~pz is the electron distribution function, Nq is the particle number external phonon, Nk is the particle number internal phonon and Ô is the Kronecker delta symbol We found that the expression (7) has the same form as the expression obtained in Ref 13), but the quantities of expressions additional indicators specific to quantum wires and they also have the completely different values The density of the QAE current is generally expressed as Z 2e X QAE j vpz fðtÞdpz ; ẳ 8ị 2h" n;N here vpz is the average drift velocity of the moving charges and it is given by vpz ẳ @ắn;N;~pz =@pz Substituting eq (7) into eq (8) and taking ¸ to be constant, we obtain for the density of the QAE current in the CQWIP GaAs/GaAsAl Z 2e¸ X QAE j vpz jCk j2 jIn;N;n0 ;N j2 Nk ¼À h" 0 n;N;n ;N ;k~ ffn;N;~pz fn0 ;N ;~pz ỵk~ ịÔắn0 ;N ;~pz ỵk~ ắn;N;~pz h" ẵk~ ị þ ðfn;N;~pz À fn0 ;N ;~pz Àk~ Þ Â Ôắn0 ;N ;~pz k~ ắn;N;~pz ỵ h" ẵk~ ịgdpz Z eá X ỵ vpz jCq j2 jUn;N;n0 ;N j2 Nq h" n;N;n0 ;N ;~q  ffn;N;~pz fn0 ;N ;~pz ỵ~q ị Ôắn0 ;N ;~pz ỵ~q ắn;N;~pz ỵ h" ẵk~ h" ẵq~ ị fn0 ;N ;~pz ~q fn;N;~pz ị Ôắn0 ;N ;~pz ~q ắn;N;~pz h" ẵk~ ỵ h" ẵq~ ịgdpz : 9ị By carrying out manipulations, we have received analytic expressions for the density of the QAE current in the CQWIP GaAs/GaAsAl as follows: 1410 N V Nhan, N Van Nghia and N Van Hieu 3 eájj2 f0 2m eÂắF 2h" vs mẵq h"  X Âh" 2 B  jIn;N;n0 ;N j exp À 2m n;N n;N;n0 ;N & 2m ỵ 3ỵ eỵ K3 ỵ ị ỵ 3K2 ỵ ị h"  ! ỵ 3K1 ỵ ị ỵ K0 ỵ ị jQAE ẳ ỵ 3À eÀ²À 2m² À h" ¢ 3 K3 ð² ị ỵ 3K2 ị !' ỵ 3K1 ị ỵ K0 ị Fig The dependence of the QAE current on the radius R of the CQWIP GaAs/GaAsAl at different values of the temperature T = 290 K (dot line), T = 295 K (dashed line), T = 300 K (solid line) Here ½q ẳ 1011 sạ1 3 4m ÂắF e ¢ h" μFSvs X ¢h" 2 Bn;N  jUn;N;n0 ;N j2 exp À 2m n;N;n0 ;N ỵ eájj2 v4l ẵ2q f0 feằỵ ằ5=2 ỵ ẵK 52 ằỵ ị ỵ 3K 32 ằỵ ị ỵ 3K 12 ằỵ ị ỵ K 12 ằỵ ị eằ ằ5=2 ẵK 52 ằ ị ỵ 3K 32 ằ ị ỵ 3K 12 ằ ị ỵ K 12 ằ ịg; 10ị h"  h" Bn0 ;N Bn;N ị h" Âẵk here ặ ẳ 2m ặ mẵq , ằặ ẳ ặ ặ , with 2R2  = 1/kBT, kB is the Boltzmann constant, T is the temperature of the system and ¾F is the Fermi energy The eq (10) is the expression of the QAE current in the CQWIP GaAs/GaAsAl The results show the dependence of the QAE current on the temperature of system, the Fermi energy and the radius of the CQWIP GaAs/GaAsAl are nonlinear These results are different from the results of other authors have obtained in the bulk semiconductor,710) superlattice.14,15) The cause of the difference between the bulk semiconductor,710) superlattice14,15) and the CQWIP GaAs/ GaAsAl is characteristics of a one-dimensional system, in one-dimensional systems, the energy spectrum of electron is quantized in two dimensions and exists even if the relaxation time ¸ of the carrier does not depend on the carrier energy Numerical Results and Discussions To clarify the results obtained, in this section, we consider the QAE current in the CQWIP GaAs/GaAsAl This quantity is considered to be a function of the temperature T, the Fermi energy ¾F and the radius R of CQWIP GaAs/GaAsAl The parameters used in the numerical calculations6,13) are as follow: = 10ạ12 s, = 104 W m¹2, μ = 5320 kg m¹3, vl = © 103 m s¹1, vt = 18 © 102 m s¹1, vs = 5370 m s¹1, $ = 13.5 eV, m = 0.067 me (me is the mass of free electron) Figures 1, present the dependence of the QAE current on the radius R of the CQWIP GaAs/GaAsAl at different values for the temperature T and the external acoustic wave frequency ½q~, respectively In Fig 1, there is one peak when the condition ẵq~ ẳ ẵk~ ỵ h" B2n0 ;N B2n;N ị 2mR2 (n 6ẳ n0 and N 6¼ N ) is satisfied The existent peak in the CQWIP Fig The dependence of the QAE current on the radius R of the CQWIP GaAs/GaAsAl at different values of the acoustic wave frequency ẵq ẳ 1011 sạ1 (dot line), ẵq ẳ 1011 sạ1 (dashed line), ẵq ẳ 1011 sạ1 (solid line) Here T = 295 K, ắF ẳ 0:048 eV GaAs/GaAsAl may be due to the transition between minibands (n ! n0 and N ! N ) When we consider the case n = nA and N = NA Physically, we merely consider transitions within sub-bands (intrasubband transitions), and from the numerical calculations we obtain jQAE ¼ 0, where mean that only the intersubband transition (n 6¼ n0 and N 6¼ N ) contribute to the jQAE These results are different from those in the normal bulk semiconductors,710) in the limit of R approximates micrometer-sized, the electron confinement ignore, there does not appear peaks, this result is similar to the results obtained in the normal bulk semiconductors.710) These results are also different from those in superlattice.14,15) Here, the difference is about shape graph and number of peaks In addition, Fig shows that the peaks move to the larger frequency of the radius when the frequency of external acoustic wave ½q~ increases In contrast, Fig shows that the positions of the maxima nearly are not move as the temperature is varied because the condition ½q~ ẳ ẵk~ ỵ h" B2n0 ;N B2n;N ị 2mR2 (n 6¼ n0 and N 6¼ N ) not depend on the temperature Therefore, We can use these conditions to determine the peak position at the different value of the acoustic wave frequency or the parameters of the CQWIP GaAs/GaAsAl This means that the condition is determined mainly by the electron’s energy The Dependence of a Quantum Acoustoelectric Current on Some Qualities in a Cylindrical Quantum Wire 1411 CQWIP GaAs/GaAsAl has a maximum peak at a certain value R = Rm although we change the temperature of system However, if the frequency of acoustic wave varies, the peaks position have a shift The QAE exists even if the relaxation time ¸ of the carrier does not depend on the carrier energy, and the results are similar to those for two-dimensional systems.14,15) This differs from bulk semiconductors, because in bulk semiconductors,710) the QAE current vanishes for a constant relaxation time These results are also different from the results of other authors have superlattice.14,15) So, the dependence of a QAE current on some qualities in a CQWIP GaAs/GaAsAl is newly developed Fig The dependence of the jQAE current on the temperature T and the Fermi energy ¾F Here ẵq ẳ 1011 sạ1 Figure shows the dependence of the QAE current on the temperature and the Fermi energy ¾F The dependence of the QAE current on the temperatures and the Fermi energy are not monotonic have a maximum at T = 295 K, ¾F = 0.044 eV for ẵq ẳ 1011 sạ1 From the results of research on the absorption coefficient of electromagnetic wave in superlattice, quantum well, quantum wire36) was explained by transition between the mini-bands and electron confinement in the low-dimensional structures This is basic to conclude the existent peak in the CQWIP GaAs/GaAsAl may be due to the electron confinement in one-dimensional structures and transition between mini-bands (n ! n0 and N ! N ) Conclusion In this paper, we have theoretically investigated the QAE in the CQWIP GaAs/GaAsAl We found the strong nonlinear dependence of the QAE current on the temperature T, the Fermi energy and the radius of the CQWIP GaAs/GaAsAl The importance of the present work is the appearance of peak when the condition ẵq~ ẳ ẵk~ ỵ h" B2n0 ;N B2n;N ị 2mR2 (n 6¼ n0 and N 6¼ N ) is satisfied Our result indicates that the dominant mechanism for such a behavior is the electron confinement in the CQWIP GaAs/GaAsAl and transitions between minibands The result of the numerical calculation was done for the CQWIP GaAs/GaAsAl This result have shown that the dependence of the QAE current on the radius R of the Acknowledgments This work is completed with financial support from the National Foundation for Science and Technology Development of Vietnam (NAFOSTED) under Grant no 103.012015.22 REFERENCES 1) N Mori and T Ando: Phys Rev B 40 (1989) 61756188 2) R Mickevičius and V Mitin: Phys Rev B 48 (1993) 1719417201 3) N Q Bau, N V Nhan and T C Phong: J Korean Phys Soc 41 (2002) 149154 4) N Q Bau, L Dinh and T C Phong: J Korean Phys Soc 51 (2007) 13251330 5) N Q Bau, D M Hung and N B Ngoc: J Korean Phys Soc 54 (2009) 765770 6) N Q Bau and H D Trien: J Korean Phys Soc 56 (2010) 120127 7) R H Parmenter: Phys Rev B 89 (1953) 990 8) M Rotter, A V Kalameitsev, A O Grovorov, W Ruile and A Wixforth: Phys Rev Lett 82 (1999) 2171 9) E M Epshtein and Y V Gulyaev: Sov Phys Solid State (1967) 288 293 10) Y M Galperin and V D Kagan: Sov Phys Solid State 10 (1968) 20382045 11) J Cunningham, M Pepper and V I Talyanskii: Appl Phys Lett 86 (2005) 152105 12) B Reulet, A Y Kasumov, M Kociak, R Deblock, I I Khodos, Y B Gorbatov, V T Volkov, C Journet and H Bouchiat: Phys Rev Lett 85 (2000) 28292832 13) N Q Bau, N V Hieu and N V Nhan: Superlatt Microstruct 52 (2012) 921930 14) S Y Mensah and F K A Allotey: J Phys Condens Matter 12 (2000) 5225 15) S Y Mensah, F K A Allotey, N G Mensah, H Akrobotu and G Nkrumah: Superlatt Microstruct 37 (2005) 8797 ... energy The Dependence of a Quantum Acoustoelectric Current on Some Qualities in a Cylindrical Quantum Wire 1411 CQWIP GaAs/ GaAsAl has a maximum peak at a certain value R = Rm although we change the. .. result indicates that the dominant mechanism for such a behavior is the electron confinement in the CQWIP GaAs/ GaAsAl and transitions between minibands The result of the numerical calculation was done... the creation (annihilation) k~ operator of internal phonon and bq~ is the annihilation ~ is the operator of the external phonon The notation jn; ki ~ electron states before interaction and jn