Journal of the Korean Physical Society, Vol 61, No 12, December 2012, pp 2026∼2031 Calculations of the Acoustoelectric Current in a Quantum Well by Using a Quantum Kinetic Equation Nguyen Quang Bau, Nguyen Van Hieu and Nguyen Vu Nhan∗ Faculty of Physics, Hanoi University of Science, Vietnam National University, 334-Nguyen Trai, Thanh Xuan, Hanoi, Vietnam (Received 18 June 2012, in final form 18 September 2012) An analytic expression for the acoustoelectric current (AC) j ac induced by electron-external acoustic wave interactions and electron-internal acoustic wave (internal phonons) scattering in a quantum well (QW) is calculated by using the quantum kinetic equation for electrons The physical problem is investigated in the region ql (where q is the acoustic wave number and l is the electrons mean free path) The dependence of the AC j ac on the external acoustic wave frequency ωq , the width of the QW L and the temperature T for a specific QW of AlGaAs/GaAs/AlGaAs is achieved by using a numerical method The computational results show that the dependence of the AC j ac on the temperature T , the external acoustic wave frequency ωq , the width of the QW L is non-monotonic and that the peaks can be attributed to transitions between mini-bands n → n The dependence of the AC j ac on the temperature T and the Fermi energy F is obtained, and a maximum of the AC j ac for F = 0.038 eV and ωq = × 1011 s−1 seen at T = 50 K, which agrees with the experimental results for AlGaAs/GaAs/AlGaAs QWs All these results are compared with those for normal bulk semiconductors and superlattices to show the differences Finally, the quantum theory of the acoustoelectric effect in a quantum well is newly developed PACS numbers: 84.40.Ik, 84.40.Fe Keywords: Quantum well, Quantum acoustoelectric current, Electron-external acoustic wave interaction, Electron-internal phonons scattering, Quantum kinetic equation DOI: 10.3938/jkps.61.2026 I INTRODUCTION When an external acoustic wave is absorbed by a conductor, the transfer of the momentum from the acoustic wave to the conduction electron may give rise to a current that is usually called the acoustoelectric (AE) current j ac 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 ), which, we believe, should find an important place in the acoustoelectronic devices In low-dimensional systems, the energy levels of electrons become discrete and to be different from other dimensionalities [1] Under certain conditions, the decrease in dimensionality of the system for semiconductors can lead to dramatically enhanced nonlinearities [2] Thus the nonlinear properties, especially electrical and optical properties of semiconductor quantum wells (QWs), superlattices (SLs), quantum wires, and quantum dots (QDs) have attracted much attention in the past few years For example, the linear absorption of a weak elec∗ E-mail: nguyenquangbau54@gmail.com; Tel: +84-913-348-020 tromagnetic wave caused by confined electrons in lowdimensional systems has been investigated [3–5] Calculations of the nonlinear absorption coefficients of an intense electromagnetic wave by using the quantum kinetic equation for electrons in bulk semiconductors [6], in quantum wells [7] and in quantum wires [8] have also been reported Also, the AE effect has been studied in detail in bulk semiconductors by using both the Boltzmann classical kinetic equation and the quantum method [9–15] In recent years, the AE effect in low-dimensional structures has been extensively studied experimentally and theoretically So far, however, almost all those works [1624] have been studied theoretically by using the Boltzmann classical kinetic equation method, and are, thus, limited to the case of the electron-external acoustic wave interaction The AE effect in superlattices [16–18], the AE current in one-dimensional channel [19], the AE effect in a finite-length ballistic quantum channel [20], the AC in a ballistic quantum point contact [21], the AC through a quantum wire containing a point impurity, the AC in submicron-separated quantum wires [22,23], and the AE effect in a carbon nanotube [24] have also been studied In addition, the AE effect has been studied experimentally in a SL [25] and in a QW [26] However, -2026- Calculations of the Acoustoelectric Current in a Quantum Well · · · – Nguyen Quang Bau et al the calculation of the AE current j ac in a QW by using the quantum kinetic equation method is still open for study In the present work, we use the quantum kinetic equation method to study the AC current j ac induced by the electron-external acoustic wave interactions and the electron-internal acoustic wave (internal phonons) scattering in a QW The present work is different from previous works [16-24] because 1) the AE current is a result of not only the electron-external acoustic wave interaction but also the electron-internal phonons scattering in the sample, 2) we use the quantum kinetic equation method, 3) we show that the present results can explain the experimental results [26] This paper is organized as follows: In Sec II, we outline the quantum kinetic equation for electrons confined in a QW The analytical expression for the AE current in the case of the electron-external acoustic wave and electron-internal phonon scattering is obtained in Sec III The numerical results and a brief discussions are presented for a specific QW AlGaAs/GaAs/AlGaAs in Sec IV Finally, we present conclusions in Sec V II QUANTUM KINETIC EQUATION FOR ELECTRONS IN A QUANTUM WELL -2027- of the electron is quantized, or the motive direction of electron is limited); electrons are free on the (x-y) plane The motion of an electron is confined a QW and its energy spectrum is quantized into discrete levels in the Oz direction Let us suppose that an external acoustic wave of frequency ωq is propagating along the quantum well axis (Oz) When ωq /η = cs q/η and ql 1, (1) where η is the frequency of the electron collisions, q is the modulus of the external acoustic wave-vector and l is the electrons mean free path The acoustic wave is considered the region ql Under such circumstances, the external acoustic wave can be interpreted as monochromatic phonons having the 3D phonon distribution function N (k), and the acoustic flux can be presented as a δ-function distribution in k-space N (k) = (2π) ωq cs Φδ(k −q), where Φ is the flux density of the external acoustic wave (external phonon) with frequency ωq In the presence of an external acoustic wave with frequency ωq , the Hamiltonian of the electron-external phonon and electron-internal phonon system in a QW in second quantization representation can be written as (we select = 1) We use a simple model for a QW, in which an electron gas is confined by an infinite potential along the Oz direction (along the Oz direction, the energy spectrum H = H0 + He−ph ; H0 = n,p⊥ He−ph = n,p⊥ ,n ,q εn (p⊥ )a+ n,p⊥ an,p⊥ + Dk In,n (kz )a+ n,p⊥ ,n ,k n,p⊥ = a (b n,p⊥ +k⊥ n ,p⊥ k n2 π p2⊥ + 2m 2mL2 + b+ ), −k (3) (4) Here, m is the effective mass of the electron, L is the width of the QW, and p⊥ is the transverse component of the quasi-momentum in the (x-y) plane Un,n (q) is the matrix element of the operator U = exp(iqy − λl z): (−1)n+n exp(−λl L) − (−1)n−n exp(−λl L) − − , (n + n )2 π (n − n )2 π λl L + λl L + λl L λl L where λl = (q − ωq2 /c2l )1/2 is the spatial attenuation factor of the potential part of the displacement field, Cq and (2) k k Cq Un,n (q)a+ n,p⊥ +q⊥ an ,p⊥ cq exp(−iωq t) + where n denotes the quantization of the energy spectrum + in the Oz direction, n = 1, 2, ; a+ n,p⊥ and an,p⊥ (bk and bk ) are the creation and the annihilation operators of the electron (internal phonon), respectively, cq is the annihilation operator of the external phonon | n, p⊥ and | n , p⊥ + k⊥ are electron states before and after scattering The electron energy takes the simple form Un,n (q) = ω k b+ b k , (5) Dk are the electron-external phonon and the electron- -2028- Journal of the Korean Physical Society, Vol 61, No 12, December 2012 internal phonon interaction factors, respectively and take the form In ,n (kz ) = Cq = iΛc2l (ωq3 /2ρ0 ΞS)1/2 , + σl2 + 2σl Ξ = q σl = | Dk |2 = − c2s /c2l + σt2 , 2σt σl −2 σt 1/2 σt = − c2s /c2t , 1/2 (6) ac | Cq |2 | Un,n (q) |2 N (q) [f ( = −π ∂fn,p⊥ ∂t i Here, Λ is the deformation potential constant, cl and ct are the velocities of the longitudinal and the transverse bulk acoustic waves, respectively, cs is the velocity of the acoustic wave, ρ0 is the mass density of the medium, S = Lx Ly is the surface area, and ∂fn,p⊥ ∂t nπ nπ z sin z exp(ikz z) L L (7) dz sin In order to establish the quantum kinetic equations for electrons in a QW, we use the electron distribution function fn,p⊥ = a+ n,p⊥ an,p⊥ t : , Λ2 k 2ρ0 cs L L = [a+ n,p⊥ an,p⊥ , H] t , ac (8) where Ψ t denotes a statistical average value at the moment t; Ψ t = T r(W Ψ) (W is the density matrix operator) Starting from the Hamiltonian, Eqs (2), (3), and (8), and realizing operator algebraic calculations, we obtain the quantum kinetic equation for electrons in a QW: − f( n,p⊥ ) n ,p⊥ +q⊥ )]δ( n ,p⊥ +q⊥ − n,p⊥ − ωq ) n ,q +[f ( × δ( n,p⊥ ) − f( − − n,p⊥ + ωq ) + [f ( n,p⊥ ) n,p⊥ ) − f( n ,p⊥ −q⊥ )]δ( n ,p⊥ −q⊥ | Dk |2 | In,n (kz ) |2 N (k) [f ( n,p⊥ ) − f( n ,p⊥ −q⊥ +π n ,p⊥ +q⊥ )]δ( n ,p⊥ +q⊥ n,p⊥ + ωq ) + [f ( − f( − n ,p⊥ −q⊥ )] n,p⊥ n ,p⊥ +k⊥ )]δ( n ,p⊥ +q⊥ − ωq ) − n,p⊥ + ω q − ωk ) n ,k +[f ( n,p⊥ ) − f( n ,p⊥ −k⊥ )]δ( n ,p⊥ −k⊥ − Equation (9) is fairly general and can be applied for any mechanism of the interaction In the case of vanishing electron-internal phonon interaction, it gives the same results as those obtained Refs 16 – 18 (9) ∂fn,p⊥ /∂t = (∂fn,p⊥ /∂t)ac + (∂fn,p⊥ /∂t)th = 0, (10) where (∂fn,p⊥ /∂t)ac is the rate of change caused by the electron-external acoustic wave and inertial phonons interaction and (∂fn,p⊥ /∂t)th is the rate of change due to the interaction of the electron with thermal phonons, impurities, and one another Substituting Eq (9) into Eq (10) we obtain the basic equation of the problem: The external acoustic wave is assumed to propagate perpendicular to the Oz axis of the QW After a new | Cq |2 | Un,n (q) |2 N (q) [f ( − ω q + ωk ) equilibrium has been established, the distribution function f of the electrons will obey the condition III ACOUSTOELECTRIC CURRENT (∂fn,p⊥ /∂t)th = π n,p⊥ n,p⊥ ) − f( n ,p⊥ +q⊥ )]δ( n ,p⊥ +q⊥ n,p⊥ + ωq ) + [f ( − n,p⊥ − ωq ) n,n q +[f ( × δ( n,p⊥ ) n ,p⊥ −q⊥ − f( − n ,p⊥ +q⊥ )]δ( n ,p⊥ +q⊥ n,p⊥ + ωq ) + [f ( n,p⊥ ) − f( | Dk |2 | In,n (kz ) |2 N (k) [f ( −π − n,p⊥ ) n ,p⊥ −q⊥ )]δ( n ,p⊥ −q⊥ n,p⊥ ) − f( − f( − n ,p⊥ −q⊥ )] n,p⊥ n ,p⊥ +k⊥ )]δ( n ,p⊥ +q⊥ − ωq ) − n,p⊥ + ω q − ωk ) n,n k +[f ( n,p⊥ ) − f( n ,p⊥ −k⊥ )]δ( n ,p⊥ −k⊥ − n,p⊥ − ω q + ωk ) (11) Calculations of the Acoustoelectric Current in a Quantum Well · · · – Nguyen Quang Bau et al (∂fn,p⊥ /∂t)th = −f1 /τp ; τp is the momentum relaxation time Thus, We linearize Eq (11) by replacing f ( n,p⊥ ) with fF ( n,p⊥ ) + f1 , where fF ( n,p⊥ ) is the equilibrium Fermi contribution function As indicated in Ref 27, | Cq |2 | Un,n (q) |2 N (q) [fF ( f1 = −πτ n,p⊥ ) -2029- − fF ( n ,p⊥ +q⊥ )]δ( n ,p⊥ +q⊥ − − ωq ) n,p⊥ n ,q +[fF ( × δ( n,p⊥ ) n ,p⊥ −q⊥ − fF ( − n,p⊥ n ,p⊥ +q⊥ )]δ( n ,p⊥ +q⊥ + ωq ) + [fF ( n,p⊥ ) n,p⊥ − fF ( | Dk |2 | In,n (kz ) |2 N (k) [fF ( −πτ − n,p⊥ ) + ωq ) + [fF ( n,p⊥ ) n ,p⊥ −q⊥ )]δ( n ,p⊥ −q⊥ − f( − fF ( − n ,p⊥ −q⊥ )] n,p⊥ n ,p⊥ +k⊥ )]δ( n ,p⊥ +q⊥ − − ωq ) n,p⊥ + ωq − ωk ) n ,k +[fF ( n,p⊥ ) − fF ( n ,p⊥ −k⊥ )]δ( n ,p⊥ −k⊥ − n,p⊥ − ω q + ωk ) The density of the AE current j ac in the direction of the external acoustic wave vector q is expressed j ac = n 2e (2π)2 vp f1 dp⊥ , (13) | Un,n |2 exp(− j ac = A1 n,n π n2 )(B+ − B− ) + A2 2mL2 kB T where A1 = A2 = C± ρ0 cs exp F kB T (2π)2 e ∧2 τ (2mkB T π)1/2 exp (2π)5 ρ0 cs mωq , F kB T (m∆n,n ± mωk )2 , 2mKB T π2 (n2 − n ) = 2mL2 ∆n,n c= n,n π n2 )(C+ − C− ), 2mL2 kB T (14) IV NUMERICAL RESULTS AND DISCUSSION 1+ b± K5/2 [2(b± c)1/2 ] , 4c mkB T ± ∆n,n ± ωk ∆n,n ± ωk = exp − m∆n,n ± mωk 2kB T b± = | In,n |2 exp(− , + a± where vp is the electron velocity given by vp = ∂ n,p /∂p Substituting Eq (12) into Eq (13) and solving for a nondegenerate electron gas, and taking τp to be constant, we obtain for the AE current Equation (14) is the analytical expression for the AE current in a QW when the momentum relaxation time is a constant In the case of the vanishing electron-internal phonon interaction, this result is the same as that obtained by using the Boltzmann kinetic equation in a QW 2 D± D± exp(− ), mkB T 2mkB T m(ωk − ωq ) m∆n,n q + ± , = q q (m∆n,n ± ωk )2 π 1/2 exp(−2(b± c)1/2 ) = 4c3/2 1/2 × [2c + 2a± (b± c) + a± ] B± = D± (2π)2 eΦ ∧2 τ c4l ωq2 (12) , , 8mkB T (15) Here, F is the Fermi energy, kB is the Boltzmann constant, and Kn (x) is the Bessel function of 2nd order To clarify the results obtained so for, in this section, we consider the AE current This quantity is considered to be a function of the temperature T , the acoustic wave number q, the acoustic intensity Φ, and the parameters of the AlGaAs/GaAs/AlGaAs QW The parameters used in the calculations are σ = 5300 kg m−3 , τ = 10−12 s, m = 0.067 m0 , m0 being the mass of a free electron, Φ = 104 W m−2 , and cl = × 103 ms−1 , ct = 18 × 102 ms−1 , cs = × 102 ms−1 Figure shows the dependence of the AE current on the temperature and the Fermi energy The dependence of the AE current on the temperatures and the Fermi energy are not monotonic have a maximum at T = 50 K, F = 0.038 eV for ωq = × 1011 s−1 This result agrees with the experimental results [26] According to [26] the peak appears at T = 50 K for ωq ≈ × 1011 (s−1 ) However, Refs 26 contains no explanation for -2030- Journal of the Korean Physical Society, Vol 61, No 12, December 2012 Fig Dependence of the j ac current on the temperature T and the Fermi energy (ωq = × 1011 (s−1 ), n = → 2, n = → 2) Fig Dependence of the j ac current on the frequency ωq of the external acoustic wave at different values of the QW’s widths L = 30 nm (solid line), L = 31 nm (dot line), and L = 32 nm (dashed line) Here T = 50 K, F = 0.038 eV, n = → 3, n = → this behavior From our calculation, we conclude that the dominant mechanism for such a behavior is electron confinement in the QW Figure presents the dependence of the AE current on the frequency ωq of the external acoustic wave at different values for the QW’s widths Figure shown some maxima when the condition ωq = ωk ± ∆n,n (n = n ) is satisfied This result is different from the AE current in a bulk semiconductor [9–15], because in a bulk semiconductor, when the q increase, the AE current increases linearly The cause of the difference between the bulk semiconductor and the QW is characteristics of a low-dimensional system, in low-dimensional systems, the energy spectrum of electron is quantized and exists even if the relaxation time τ of the carrier does Fig Dependence of the j ac current on the width of the quantum well at different values of the external acoustic wave frequency ωq = 32 × 1010 s−1 (solid line), ωq = 31 × 1010 s−1 (dot line), and ωq = 30 × 1010 s−1 nm (dashed line) Here T = 50 K, F = 0.038 eV, n = → 3, and n = → Fig Dependence of the j ac current on the width of the quantum well at different values of the temperature T = 50 K (solid line), T = 52 K (dot line), and T = 54 K (dashed line) Here ωq = × 1011 (s−1 ), F = 0.038 eV, n = → 3, and n = → not depend on the carrier energy In Fig 2, there are two peaks This is attributed to the transitions between mini-bands (n → n ), and the two peaks correspond to (n = → n = 2) and (n = → n = 3) transitions or intersubband transitions as the main contribution to j ac When we consider the case n = n Physically, we merely consider transitions within subbands (intrasubband transitions), and from the numerical calculations we obtain j ac = 0, where mean that only the intersubband transition (n = n ) contribute to the j ac In superlattices [16–18], the AC appears even if the intrasubband transitions In addition, the positions of the peaks change sig- Calculations of the Acoustoelectric Current in a Quantum Well · · · – Nguyen Quang Bau et al nificantly when the quantum well’s width increases This can be explained by assuming that only electrons whose momenta comply with the condition ωq = ωk ± ∆n,n (n = n ) contribute considerably to the effect Figure shows the dependence of the j ac on the width of the QW for different values of the external acoustic wave frequency ωq and Fig shows the dependence of the j ac on the width of the QW for differen values of the T The figures show that the dependence of j ac on the width of well is strong and nonlinear The dependence of the j ac on the width of the QW has maximum values (peaks) The figures also show that when we consider different transitions, we obtain peaks at different values of L when the condition ωq = ωk ± ∆n,n (n = n ) is satisfied Moreover, Fig shows that at small QW width, peaks are smaller and that the peaks move to the smaller QW width when the frequency of the acoustic wave increases In contrast, Fig shows that the positions of the maxima nearly are not move as the temperature is varied because the condition ωq = ωk ± ∆n,n (n = n ) not depend on the temperature This means that the condition is determined mainly by the electron’s energy V CONCLUSION In this paper, we have obtained analytical expressions for the j ac in a QW by using the quantum kinetic equation for the distribution function of electrons interacting with an external acoustic wave and internal phonons We have shown the strong nonlinear dependence of j ac on the temperature T, the frequency ωq of the external acoustic wave and the width L of the QW The importance of the present work is the appearance of peaks when the condition ωq = ωk ± ∆n,n (n = n ) is satisfied, the results are complex and different from those obtained in bulk semiconductors [9–15] and the superlattices [16–18] The numerical results obtained for the AlGaAs/GaAs/ AlGaAs QW show that a peak exists at T = 50 K, ωq = × 1011 s−1 and F = 0.038 eV, which fits with the experimental results [26] Our result indicates that the dominant mechanism for such a behavior is electron confinement in the QW and transitions between mini-bands n → n The j ac exists even if the relaxation time τ of the carrier does not depend on the carrier energy, and the results are similar to those for superlattices [16,18] This differs from bulk semiconductors, because in bulk semiconductors [9–15], the AE current vanishes for a constant relaxation time ACKNOWLEDGMENTS This work was completed with financial support from the Vietnam National University, Ha Noi (No QGTD.12.01) and Vietnam NAFOSTED (No 103.012011.18) -2031- REFERENCES [1] Y Zhang, K Suenaga, C Colliex and S Iijima, Science 281, 973 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equation method... considered to be a function of the temperature T , the acoustic wave number q, the acoustic intensity Φ, and the parameters of the AlGaAs/GaAs/AlGaAs QW The parameters used in the calculations are σ =... current in a QW when the momentum relaxation time is a constant In the case of the vanishing electron-internal phonon interaction, this result is the same as that obtained by using the Boltzmann kinetic