Optically detected electron-phonon resonances in hyperbolic Poschl-Teller quantum wells

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In this paper, the author investigate the linear optically detected electrophonon resonance (ODEPR) effect and linewidths in the hyperbolic quantum well with Poschl-eller potential type.

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Nội dung Text: Optically detected electron-phonon resonances in hyperbolic Poschl-Teller quantum wells

OPTICALLY DETECTED ELECTRON - PHONON RESONANCES ¨ IN HYPERBOLIC POSCHL-TELLER QUANTUM WELLS LE DINH 1 TRAN THI NGOC PHAM TUAN VINH 2 1 Hue University of Education, University of Hue 2 Dong Thap University ANH1 , Abstract: The explicit expressions for linearly optical conductivity and absorption power of electromagnetic wave caused by confined electrons in the hyperbolic quantum well is obtained in the case of electron-longitudinal optical phonon scattering. Linearly optically detected electrophonon resonance (ODEPR) effect in a specific GaAs/AlAs hyperbolic quantum well with P¨oschl-Teller potential type is investigated. Conditions for ODEPR are discussed based on the curves expressing the dependence of absorption power on the photon energy. From these curves we obtained ODEPR - linewidths as profiles of the curves. Computational results show that linear ODEPR- linewidths increase with temperature and decrease with well parameters. Keywords: absorption power, quantum well, hyperbolic, P¨oschl Teller potential, ODEPR- linewidths 1 INTRODUCTION In recent years, electron-phonon resonance effect (EPR) and optically detected electron-phonon resonance (ODEPR) is being interested by domestic and international physicists. These studies have contribution in clarifying new properties of electrons under the effect of external fields, therefore provide information of optical properties of semiconductors to the technology of fabricating optoelectronic components. EPR phenomena arise from an electron scattering due to the absorption and emission of phonons when the energy difference of two electric subbands equals the longitudinal-optical (LO) phonon energy. If these processes are accompanied by the absorption or emission of photons, we will have the ODEPR effect. The EPR effect has been started to study since 1972 by Bryskin and Firsov for the nondegenerate Journal of Sciences and Education, Hue Universitys College of Education ISSN 1859-1612, No 01(45)/2018: pp. 15-23 Received: 06/10/2017; Revised: 11/10/2017; Accepted: 23/10/2017 16 LE DINH et al. semiconductor in strong electric field [1]. The study of EPR effects is very important in understanding transport phenomena in the modern quantum devices. For electrons in quantum wells, the investigation of multi-subband transport effects such as the effective mass, the energy levels, and the electron-phonon interaction has received some attentions [2, 3, 4, 5]. There are now a lot of works focusing on this phenomenon in two-dimensional semiconductor [6, 7, 8], and quantum wires [9, 10]. Most studies, however, focus on conventional quantum wells such as square potential quantum wells, parabolic potential quantum wells, etc. Hyperbolic quantum well with P¨oschl-Teller potential type is still new direction with some studies and has attracted some interest in recent years [11, 13, 14, 15]. Tong [12] suggested several applications in semiconductor heterojunction devices and in optical systems. Tong and Kiriushcheva [13] showed that it can be used in the reduction of noise in resonance tunneling devices and other devices. Radovanovic et al. [14] worked several intersubband absorption properties of the potential. Yildirim and Tomak [15] studied some intersubband nonlinear optical properties of the potential. In this paper, we investigate the linear ODEPR effect and linewidths in the hyperbolic quantum well with P¨oschl-eller potential type. First, we derive the analytical expression of linear absorption power. From curves on graphs of absorption power as a function of photon energy, we obtain ODEPR-linewidths as a profiles of curves by using the Profile method presented in one of our previous papers [16]. The dependence of linear linewidths on temperature and the well parameter are discussed. The paper is organized as follows: The model and theoretical framework is described in Section 2, the results and discussions are presented in Section 3, and finally, the conclusions are given in Section 4. 2 MODEL AND THEORY We consider a quantum well with the electron confined potential expressed in the form of [17] V1 − V2 cosh(αz) , (1) V (z) = sinh2 (αz) where V1 , V2 and α are three parameters representing the properties of potential. Solving Schrodinger equation, we obtain the energy spectrum and corresponding electron wave function as follows [18]: α2 ~2 2 2 Eα = = Enz + Ekx ky = kx + ky − [υ − µ − (1 + 2n)]2 , ∗ 2m 8 1 n = 0, 1, 2, ...., (υ − µ − 1) . 2 (2) OPTICALLY DETECTED ELECTRON - PHONON RESONANCES... 17 1 ψα (x, y, z) = p ei(kx x+ky y) Cuδ (1 − u)ε 2 F1 [−n, n + 2 (δ + ε + 1/4) ; 2δ + 1/2; u] , Lx Ly (3) √ p p 1 1 , 8 (V1 + V2 ) + α2 , v = 2α 8 (V1 − V2 ) + α2 , ε = −2E where δ = 41 + µ2 , µ = 2α α 2 αz u = tanh 2 . The coefficient C is expressed in the form of Gamma function: q C = 2αεΓ (n + µ + 1) Γ (n + µ + 2ε + 1) /n!Γ(µ + 1)2 Γ(n + 2ε + 1). Consider an electromagnetic wave with angular frequency ω and amplitude E0 , the linear absorption power delivered to the system is given by E02 Re [σzz (ω)] , 2 here σzz (ω) is the z component of optical conductivity tensor: X fβ − fα σzz (ω) = −e lim+ hziαβ hjz iβα , ∆→0 ~¯ ω − Eβα − Γαβ ω) 0 (¯ αβ P0z (ω) = (4) (5) where e is the electric charge of the electron, hXi ≡ hα|X|βi is the matrix element of operator X, fα(β) is the Fermi-Dirac distribution function of the electron at energy state Eα(β) , αβ + ω ¯ = ω − i∆ (∆ → 0 ); Γ0 (¯ ω ) is called the spectral lineshape function. The matrix element of position operator z is calculated by Z Lz h 0 i CC 0 u2δ (1 − u)2ε 2 F 1 −n , n0 + 2 (δ + ε + 1/4) ; 2δ + 1/2; u hziαβ = δkx0 ,kx δky0 ,ky 0 × 2 F 1 [−n, n + 2 (δ + ε + 1/4) ; 2δ + 1/2; u] zdz = δkx0 ,kx δky0 ,ky I1 . The matrix element of current density operator jz is calculated by Z ∞ ie~ 0 hjz iβα = uδ (1 − u)ε 2 F1 [−n0 , n0 + 2 (δ + ε + 1/4) ; 2δ + 1/2; u] δ 0 δ 0 CC m∗e kx ,kx ky ,ky 0 δ−1 1 1 ε ε δ × δu (1 − u) + u ε (1 − u) − 1 .2 F1 −n, n + 2 δ + ε + ; 2δ + ; u 4 2 (−n) (n + 2δ + 2ε + 1/2) + uδ (1 − u)ε 2δ + 12 1 3 ie~ × 2 F1 −n + 1, n + 1 + 2 δ + ε + ; 2δ + ; u dz = ∗ δkx0 ,kx δky0 ,ky I2 4 2 m Take the real part of σzz (ω), we obtain the general expression of the linear absorption power in z direction 2 X X fk0 x ,k0 y ,n0 − fkx ,ky ,n B0 (ω) e2 ~E0z P0 (ω) = δkx0 ,kx δky0 ,ky I1 I2 . (6) 2m∗e k ,k ,n k0 ,k0 ,n0 (~ω − Eβα )2 + B02 (ω) x y x y 18 LE DINH et al. In the above equation, we denote Eβα = Eβ −Eα = Ek0 x ,k0 y ,n0 −Ekx ,ky ,n . The relaxation rate B0 gets the form as follows: X F01 1 Lx Ly Dm∗e 1 + B0 (ω) = 16π 3 ~2 (fβ − fα ) n00 M01 M02 (k 0 x + M01 )2 (k 0 x − M01 )2 " # 1 1 1 F02 1 × + + + M03 M04 (k 0 x + M03 )2 (k 0 x − M03 )2 (k 0 y + M02 )2 (k 0 y − M02 )2 #! " 1 1 N11 × 2 + 0 0 (k y + M04 ) (k y − M04 )2 F03 1 1 + + M05 M06 (−kx + M05 )2 (kx + M05 )2 " # 1 1 F04 1 1 × + + + M07 M08 (−kx + M07 )2 (kx + M07 )2 (−ky + M06 )2 (ky + M06 )2 " #! ) 1 1 + N31 . (7) × (−ky + M08 )2 (ky + M08 )2 where fα (fβ ) is the Fermi-Dirac distribution function of electron with energy Eα (Eβ ), e2 ~ωLO 1 1 D= − ; 2ε0 χ∞ χ0 21 2m∗e 2 M01 = M03 = kx + 2 (~ω ± ~ωLO − En00 + En ) ; ~ 21 2m∗e 2 M02 = M04 = ky + 2 (~ω ± ~ωLO − En00 + En ) ; ~ 12 2m∗e 02 M05 = M07 = k x − 2 (~ω ± ~ωLO − En0 + En00 ) ; ~ 12 2m∗e 02 M06 = M08 = k y − 2 (~ω ± ~ωLO − En0 + En00 ) ; ~ 2 −1 ~ M01 2 2 F01 = (1 + Nq ) (1 − fα ) 1 + exp θ M02 + En00 − EF ; 2m∗e " −1 # 2 ~ M01 2 M02 2 + En00 − EF − Nq fα 1 − 1 + exp θ 2m∗e 2 −1 ~ M03 2 2 F02 = Nq (1 − fα ) 1 + exp θ M04 + En00 − EF 2m∗e " 2 −1 # ~ M03 2 − (1 + Nq ) fα 1 − 1 + exp θ M04 2 + En00 − EF ; 2m∗e OPTICALLY DETECTED ELECTRON - PHONON RESONANCES... 19 2 −1 # ~ M05 2 F03 = (1 + Nq ) fβ 1 − 1 + exp θ M06 2 + En00 − EF 2m∗e 2 −1 ~ M05 2 2 − Nq (1 − fβ ) 1 + exp θ M06 + En00 − EF ; 2m∗e " 2 −1 # ~ M07 2 F04 = Nq fβ 1 − 1 + exp θ M08 2 + En00 − EF 2m∗e 2 −1 ~ M07 2 2 − (1 + Nq ) (1 − fβ ) 1 + exp θ M08 + En00 − EF ; 2m∗e 2 2π 2 2 1 + 2ε 2 02 1 ; N11 = N21 = C C 2 18 − 30ε + 12ε 1 α 2δ + 2 Lz 2 2π 2 00 2 1 2 2 1 + 2ε N31 = N41 C C 18 − 30ε + 12ε . 1 α2 2δ + 2 Lz " Substituting B0 (ω) into Eq. (6), we obtained the explicit expression of linear absorption power in the quantum well. We found that the analytical result is quite complex. However, physical meaning can be obtained from numerical computation and graphical plotting. 3 NUMERICAL RESULTS AND DISCUSSIONS To clarify the obtained analytical results, we carry out some numerical computations and graphical plotting for a specific AlGaAs/GaAs quantum well. Parameters used are [19, 20]: electrical charge e = 1.6 × 10−19 C, electron effective mass m∗ = 6.097 × 10−32 kg, Planck constant ~ = 1.0544 × 10−34 Js, Boltzmann constant kβ = 1.38066 × 10−23 J/K, permittivity of free space 0 = 13.5, high frequency permeability χ∞ = 10.9, static frequency permeability χ0 = 12.9, LO-phonon energy ~ωLO = 36.25 meV, external field amplitude E0 = 105 V/m. The expression of the linear ODEPR condition is expressed by ~ω ± Eβα ± ~ωLO = 0. (8) When the linear ODEPR conditions are satisfied, in the course of scattering events, electrons in the state |αi could make transition to state |βi by absorbing one photon of energy ~ω, accompanied with the absorption and/or emission of a LO-phonon of energy ~ωLO . Figure 1 describes the dependence of the linear absorption power P0 (ω) on the photon energy at T = 200 K, with parameter α = 2.2 × 108 m−1 . From the figure we can see three resonance peaks describing the different transitions of electrons, satisfying the various resonance conditions: