This article was downloaded by: [University of Tennessee At Martin] On: 06 October 2014, At: 06:19 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Integrated Ferroelectrics: An International Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ginf20 The Dependence of the Magnetoresistance in Quantum Wells with Parabolic Potential on Some Quantities under the Influence of Electromagnetic Wave a a b Nguyen Dinh Nam , Do Tuan Long & Nguyen Vu Nhan a Department of Physics, College of Natural Science, Viet Nam National University, Ha Noi, Viet Nam b Department of Physics, Academy of Defence force-Air force, Ha Noi, Viet Nam Published online: 23 May 2014 To cite this article: Nguyen Dinh Nam, Do Tuan Long & Nguyen Vu Nhan (2014) The Dependence of the Magnetoresistance in Quantum Wells with Parabolic Potential on Some Quantities under the Influence of Electromagnetic Wave, Integrated Ferroelectrics: An International Journal, 155:1, 45-51, DOI: 10.1080/10584587.2014.905122 To link to this article: http://dx.doi.org/10.1080/10584587.2014.905122 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, 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NAM,1,* DO TUAN LONG,1 AND NGUYEN VU NHAN2 Department of Physics, College of Natural Science, Viet Nam National University, Ha Noi, Viet Nam Department of Physics, Academy of Defence force-Air force, Ha Noi, Viet Nam The magnetoresistance is one of the important properties of semiconductors Starting from the hamiltonian for the electron-acoustic phonon system, we obtained the expression for the electron distribution function and especially the expression for the magnetoresistance in quantum wells with parabolic potential (QWPP) under the influence of electromagnetic wave (EMW) in the presence of magnetic field We estimated numerical values and graphed for a GaAs/GaAsAl quantum well to see the nonlinear dependence of the magnetoresistance on the temperature of the system T, the amplitude E0 and the frequency of the electromagnetic waves, the magnetic field B, the parameters of the quantum well and the momentum relaxation time τ clearly Keywords Dependence of magnetoresistance I Introduction In the past few years, there have been many scientific works related to the properties of the low-dimensional systems such as the optical, magnetic and electrical properties [1–9] These results show us that there are some differences between the low-semiconductor and the bulk semiconductor that the previous work studied The magnetoresistance is also interested However, it has not been resolved in the quantum wells with parabolic potential under the influence of electromagnetic wave The calculation of the magnetoresistance in the QWPP in the presence of magnetic field under the influence of EMW is done by using the quantum kinetic equation method that brings the high accuracy and the high efficiency [1–4] Comparing the results in this case with in the case of the bulk semiconductors, we also see some differences II The Magnetoresistance in Quantum Wells with Parabolic Potential under the Influence of Electromagnetic Wave in the Presence of Magnetic Field It is well known that in the quantum wells, the motion of electrons is restricted in one dimension, so that they can flow freely in two dimensions [1, 2, 7] We consider a quantum Received July 23, 2013; in final form January 12, 2014 ∗ Corresponding author E-mail: dinhnamt2@yahoo.com 45 46 N D Nam et al well with parabolic potential of GaAs embedded in AlAs It subjected to a crossed electric − → − → field E1 = (0, 0, E1 )and magnetic field B = (0, B, 0) In the presence of an EMW with − → − → electric field vector E = E0 sin t (where E0 and are the amplitude and the frequency of the EMW, respectively), the Hamiltonian of the electron-acoustic phonon system in the above-mentioned QWPP in the second quantization presentation can be written as: H = e − → − → A (t) a + − εN kx − ω− → → → →+ →a − q q +b− q b− c N, kx N, kx − → − → q N, kx Downloaded by [University of Tennessee At Martin] at 06:19 06 October 2014 + − →− → q N,N , kx , + → CN,N (− + b+− q )a + − → → b− → − → → + aN,− q −q kx N , kx + qx → φ(− q )a + − →, → − →a − N, kx + qx N, kx − → q (1) + ) and b− where N is the Landau level index (N = 0,1 ,2 ), a + − → → and a − → , (b− → q q N, kx N, kx − → are the creation and the annihilation operators of the electron (phonon), |N, kx > and − → − qx > are electron states before and after scattering, ω− |N , kx + → → q is the energy of an − → acoustic phonon; CN,N ( q ) = C− → → q is the electron-phonon interacq IN,N (qz ), where C− → tion constant and IN,N (qz ) is the electron form factor [4], φ − q is the scalar potential − → − → of a crossed electric field E1 , A (t) is the vector potential of an external electromagnetic − → − → wave A (t) = eE0 sin( t)/ If the confinement potential is assumed to take the form V (z) = mω0 (z − z0 )2 /2, then the single-particle wave function and its energy are given by [1, 2]: → ψ(− r )= εN (kx ) = ωp N + →− → i− e k⊥ r ψ(kx , z), 2π + 2m∗ 2 kx (2) kx ωc + eE1 ωp − (3) , Here, m∗ and e are the effective mass and charge of conduction electron, respectively, k⊥ = (kx , ky ) is its wave vector in the (x,y) plan, z0 = ( kx ωc + eE1 ) /mωp2 , ωp2 = ω02 +ωc2 , ω0 and ωc are the confinement and the cyclotron frequencies, respectively, and ψm (z − z0 ) = Hm (z − z0 ) exp −(z − z0 )2 /2 , (4) with Hm (z)being the Hermite polynomial of mth order From the quantum kinetic equation for electron in single scattering time approximation and the electron distribution function, using the Hamiltonian in the Eq 1, we find: ∂f − → N, kx + ∂t − → e E1 − → − → + ωc kx , h ∂f − → 2π N, kx − → = ∂ kx N − → ,q → CN,N (− q) 2N− → q +1 +∞ Jl2 (αqx ) × f l=∞ − → − → δ εN (kx + qx ) − εN (kx ) − l → − fN,− k N , kx + qx x (5) Dependence of Magnetoresistance in QWPP 47 Downloaded by [University of Tennessee At Martin] at 06:19 06 October 2014 − → − → − → where kx = (kx , 0, 0), h = B /B is the unit vector in the direction of magnetic field, f − → is an unknown distribution function perturbed due to the external fields, Jl (x) is the N, kx th l - order Bessel function of argument x and N− → q is the time-independent component of the electron distribution function For simplicity, we limit the problem to case of l = −1, 0, If we multiply both sides − → of the Eq by (e/m∗ ) kx δ(ε − εN (kx )), carry out the summation over N and kx and use J02 (αqx ) ≈ − (αqx )2 /2, we obtain: − → R (ε) − → − → − → − → + ωc h , R (ε) = Q (ε) + S (ε), τ (ε) (6) − → R (ε) = (7) where 2π e − → S (ε) = − m∗ e − → kx f − → δ (ε − εN (kx )) , ∗ N, kx − →m N, kx − → q CN,N (q) (2N− → q + 1)(αqx ) N , − → f − → − → − → kx → − fN,− k N , kx + qx x N, kx × 2δ εN (kx + qx ) − εN (kx ) − δ εN (kx + qx ) − εN (kx ) − −δ εN (kx + qx ) − εN (kx ) + e − → Q (ε) = − ∗ m − → N, kx → − → − kx F , , (8) ∂f − → N, kx − → δ (ε − εN (kx )), ∂ kx (9) with ε − εN (kx ) − → − → ∇T (10) F = eE1 − ∇εF − T − → − → − → Finding R (ε) in term of Q (ε), S (ε)and through some computation steps, we obtain the expression for conductivity tensor: σim = τ (εF ) τ (εF ) e c0 δik +d0 d1 δik −ωc τ (εF ) εikl hl + ωc2 τ (εF ) hi hk ∗ 2 m 1+ωc τ (εF ) 1+ωc2 τ (εF ) + d0 d2 ) τ (εF − + ωc2 τ (εF − ) δik − ωc τ (εF − ) εikl hl + ωc2 τ (εF − ) hi hk + d0 d3 ) τ (εF + + ωc2 τ (εF + ) δik − ωc τ (εF + ) εikl hl + ωc2 τ (εF + ) hi hk × δkm − ωc τ (εF ) εkmn hn + ωc2 τ (εF ) hk hm } (11) where c0 = N eLx π 0θ ( ), (12) 48 N D Nam et al eLx ξ kB T e2 E02 eE1 ωc I (qz ), 4π m∗ ηυ ω4 ω02 N,N d0 = N,N d1 = ( +3 )θ ( )θ ( 1) −2 N,N − ( +3 )θ ( )θ ( Downloaded by [University of Tennessee At Martin] at 06:19 06 October 2014 d2 = √1 d3 = √1 = ≈ +3 )θ ( )θ ( ), )θ ( 2) + √0 3) − 1 ( (13) eE1 ωc ω02 2m∗ ωp2 ω2 = = = = − − 4 − 5 )θ ( ), (15) θ( )θ ( ), (16) 2m∗ ωp3 N + ω2 ω2 2m∗ ωp2 ω2 2m∗ ωp2 ω2 2m∗ ωp2 ω2 − e2 E12 − 2m∗ ωp2 εF 2 ω2 2m∗ ωp2 2m∗ ωp2 (14) θ( εF − ωp N + = θ( (17) , εF − ωp N + (18) , εF + − ωp N + , εF − − ωp N + , εF − − ωp N + , εF + − ωp N + (19) (20) where Lx ξ, η, υ, kB , T , εF are the x-directional normalization lengths, the deformation potential constant, the density, the acoustic velocity, the Boltzmann constant, the temperature of system and the Fermi energy, respectively In this work, we consider the case of electron-acoustic phonon scattering and the − → presence of electric field E1 Comparing with the case of electron-optical phonon scattering Dependence of Magnetoresistance in QWPP 49 − → and no electric field E1 that was studied previously [10], we see some differences in the expression of conductivity tensor and also in the expression of magnetoresistance The magnetoresistance is given by the formula: ρ σzz (H ) σzz (0) −1 = 2 (H ) ρ σzz (H ) + σxz (21) Using the Eq 11, we obtain the explicit formula of the magnetoresistance as following: Downloaded by [University of Tennessee At Martin] at 06:19 06 October 2014 ρ = ρ N,N + e τ (εF ) d0 d1 τ (εF ) c0 + − ωc2 τ (εF )h2 ∗ m + τ (εF ) + ωc2 τ (εF ) d0 d2 τ (εF − ) − ωc2 τ (εF )τ (εF − 2 + ωc τ (εF − ) × − ωc2 τ (εF )τ (εF + )h2 × N,N × 2m∗ εF − ω0 N + × ⎧⎧ ⎨⎨ ⎩⎩ N,N )h2 + e2 τ (εF )Lx + τ (εF ) m∗ π θ εF − ω0 N + d0 d2 τ (εF − ) × − ωc2 τ (εF )τ (εF − + ωc2 τ (εF − ) + d0 d3 τ (εF + ) − ωc2 τ (εF )τ (εF + + ωc2 τ (εF + ) N,N + e τ (εF ) d0 d1 τ (εF ) c0 + − ωc2 τ (εF )h2 ∗ m + τ (εF ) + ωc2 τ (εF ) + + ) d0 d3 τ (εF + 2 + ωc τ (εF + ) )h2 )h2 e τ (εF ) d0 d1 τ (εF )2ωc τ (εF )h2 ω τ (ε )h + c c F m∗ + ωc2 τ (εF ) + ωc2 τ (εF ) ) d0 d2 τ (εF − × ωc [τ (εF ) + τ (εF − + ωc2 τ (εF − ) ) d0 d3 τ (εF + ωc [τ (εF ) + τ (εF + + + ωc2 τ (εF + ) )] h2 )] h −1 −1 (22) Eq 22 is the analytical expression of the magnetoresistance in the QWPP It shows the dependence of the magnetoresistance on the external fields, including the EMW In the next section, we will give a deeper insight into this dependence by carrying out a numerical evaluation In Eq 22, we can see that the formula of the magnetoresistance is easy to come back to the case of bulk semiconductor when ω0 reaches to zero [11, 12] III Numerical Results and Discussion For the numerical evaluation, we consider the model of a quantum well of GaAs/GaAsAl with the following parameters: εF = 50 meV , kB = 1.3807 × 10−23 J K −1 , υ = 5220 m/s Downloaded by [University of Tennessee At Martin] at 06:19 06 October 2014 50 N D Nam et al Figure The dependence of the magnetoresistance on the temperature and m∗ = 0.0067m0 with m0 is the mass of a free electron For the sake of simplicity, we also choose N = 0, N = 1, τ = 10−12 s [1, 2] Figure shows the magnetoresistance as a function of the temperature The value of the magnetoresistance increases sharply when the temperature is low, after that it decreases steadily With the different values of the electric field E1 , we get the resonant peaks at the different points of temperature Figure shows us the dependence of the magnetoresistance on the amplitude of the EMW The higher the amplitude of the EMW is, the faster the magnetoresistance grows up The line of the dependence of the magnetoresistance on the amplitude E0 of EMW also changes when we change the value of the frequency of the EMW We see that there are some differences in the dependence of the magnetoresistance on the temperature and the amplitude from the case of electron-optical phonon scattering [10] We also get the same graphs as in the case of bulk semiconductor [11, 12] when the confinement frequency ω0 reaches to zero Figure The dependence of the magnetoresistance on the amplitude of the EMW Dependence of Magnetoresistance in QWPP 51 IV Conclusions Downloaded by [University of Tennessee At Martin] at 06:19 06 October 2014 In this paper, we obtain the analytical expression of the magnetoresistance in QWPP under the influence of EMW in the presence of magnetic field We see that the magnetoresistance in this case depends on some quantities such as: the magnetic field B, the temperature T, the parameters of QWPP, the momentum relaxation time τ , the amplitude E0 and the frequency of EMW Estimating numerical values and graph for a GaAs/GaAsAl quantum well to see this dependence clearly Looking at the graph, we see that the magnetoresistance gets the negative values and the dependence of the magnetoresistance on the temperature, the amplitude and the frequency of the EMW are nonlinear When ω0 reaches to zero, we obtain the results as the case of bulk semiconductor that was studied [11, 12] Funding This research is completed with financial support from Vietnam NAFOSTED (103.012011.18) and TN13-04 References T C Phong and N Q Bau, Parametric resonance or acoustic and optical phonons in a quantum well J Korean Phys Soc 42, 647–651 (2003) N Q Bau, L T Hung, and N D Nam, The nonlinear apsorption coefficient of strong electromagnetic waves by confined electrons in quantum wells under the influences of confined phonons JEWA 13, 1751–1761 (2010) N Q Bau, L Dinh, and T C Phong, Absorption Coefficient of Weak Electromagnetic Waves Caused by Confined Electrons in Quantum Wires J Korean Phys Soc 51, 1325–1330 (2007) N Q Bau and H D Trien, The nonlinear absorption coefficient of strong 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magnetoresistance of semiconductors Sov Phys Semicond 18, 739–741 (1976) 12 V L Manlevich and E M Epshtein, Photostimulated kinetic effects in semiconductors J Sov Phys 19, 230–237 (1976) ... electromagnetic wave The calculation of the magnetoresistance in the QWPP in the presence of magnetic field under the influence of EMW is done by using the quantum kinetic equation method that brings the. .. us the dependence of the magnetoresistance on the amplitude of the EMW The higher the amplitude of the EMW is, the faster the magnetoresistance grows up The line of the dependence of the magnetoresistance. .. Parabolic Potential under the Influence of Electromagnetic Wave in the Presence of Magnetic Field It is well known that in the quantum wells, the motion of electrons is restricted in one dimension,