Báo cáo " The dependence of the parametric transformation coefficient of acoustic and optical phonons in doped superlattices on concentration of impurities"

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The parametric transformation of acoustic and optical phonons in doped superlattices is theoretically studied by using a set of quantum kinetic equations for the phonons. The analytic expression of parametric transformation coefficient of acoustic and optical phonons in doped superlattices is obtained, that depends non-linear on the concentration of impurities. Numerical computations of theoretical results and graph are performed for GaAs:Si/GaAs:Be doped superlattices

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VNU Journal of Science, Mathematics - Physics 25 (2009) 123-128 The dependence of the parametric transformation coefficient of acoustic and optical phonons in doped superlattices on concentration of impurities Hoang Dinh Trien*, Nguyen Vu Nhan, Do Manh Hung, Nguyen Quang Bau Faculty of Physics, College of Science, VNU 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Received 30 July 2008; received in revised form 15 August 2009 Abstract. The parametric transformation of acoustic and optical phonons in doped superlattices is theoretically studied by using a set of quantum kinetic equations for the phonons. The analytic expression of parametric transformation coefficient of acoustic and optical phonons in doped superlattices is obtained, that depends non-linear on the concentration of impurities. Numerical computations of theoretical results and graph are performed for GaAs:Si/GaAs:Be doped superlattices. Keywords: parametric transformation, doped superlattices. 1. Introduction It is well known that in presence of an external electromagnetic field, an electron gas becomes non-stationary. When the conditions of parametric resonance are satisfied, parametric resonance and transformation (PRT) of same kinds of excitations such as phonon-phonon, plasmon-plasmon, or of different kinds of excitations, such as plasmon-phonon will arise; i.e., the energy exchange process between these excitations will occur [1-9]. The physical picture can be described as follows: due to the electron-phonon interaction, propagation of an acoustic phonon with a frequency ωq accompanied by r a density wave with the same frequency Ω . When an external electromagnetic field with frequency is presented, a charge density waves (CDW) with a combination frequency υq ± l Ω (l=1,2,3,4…) will r appear. If among the CDW there exits a certain wave having a frequency which coincides, or approximately coincides, with the frequency of optical phonon, υq , optical phonons will appear. r These optical phonons cause a CDW with a combination frequency of υq ± l Ω , and when r υ q ± lΩ ≅ ω q , a certain CDW causes the acoustic phonons mentioned above. The PRT can speed up r r the damping process for one excitation and the amplification process for another excitation. Recently, there have been several studies on parametric excitation in quantum approximation. The parametric ______ * Corresponding author. Tel.: 0913005279 Email: hoangtrien@gmail.com 123
124 H.D. Trien et al. / VNU Journal of Science, Mathematics - Physics 25 (2009) 123-128 interactions and transformation of acoustic and optical phonons has been considered in bulk semiconductors [1-5], for low-dimensional semiconductors (doped superlattices, quantum wells, quantum wires), the dependence of the parametric transformation coefficient of acoustic and optical phonons on temperature T and frequency Ω is has been also studied [6-9]. In order to improve the PRT theoretics for low-dimensional semiconductors, we, in the paper, examine dependence of the parametric transformation coefficient of acoustic and optical phonons in doped superlattices on concentration of impurities. 2. M odel and quantum kinetic equation for phonons We use model for doped superlattice with electron gas is confined by the superlattice potential r r along the z direction and electrons are free on the xy plane. If a laser field E (t ) = E0 sin(Ωt ) irradiates the sample in direction which is along the z axis, the electromagnetic field of laser wave will polarize r cr E0 cos(Ωt ) parallels the x axis and y axis, and its strength is expressed as a vector potential A(t ) = Ω (c is the light velocity; Ω is EMW frequency; E0 is electric field intensity). The Hamiltonian of the electron-acoustic phonon-optical phonon system in doped superlattice can be written as (in this paper, we select h =1): H (t ) = ∑ε α (t ) aα aα + ∑ωq bq+ bq + ∑υq cq cq + + + rrr rrr r r α q q r+ r+ + ∑∑Cq I ' ( q )aα aα (bq + b− q )∑∑Dq I + + ( q )aα aα (cq + c− q ) (1) r r r r r r n , n' n ,n r r q αα '′ q αα '′ r er + where ε α (t ) = ε n ( k⊥ + A(t )) is energy spectrum of an electron in external electromagnetic field, aα , c r (aα ) is the creation (annihilation) operator of an electron for state | n, k⊥ 〉 , bq+ , bq ( cq , cq ) is the + r r r r r creation operator and annihilation operator of an acoustic (optical) phonon for state have wave vector q . The electron-acoustic and optical phonon interaction coefficients take the forms[10] e2ν q qξ 2 r 1 1 −) 2 2 | Cq r| = ; | Dq r| = ( (2) 2 ρν sV 2V χ q χ ∞ χ 0 2 here V, ρ , ν s , and ξ are the volume, the density, the acoustic velocity and the deformation potential constant, respectively. χ is the electronic constant, χ ∞ , χ 0 are the static and high-frequency r dielectric constants, respectively. The electron form factor, I ' (q ) is written as [11]: nn s0 d r (q ) = ∑ ∫eiqd Φ n ( z − jd )Φ ' ( z − jd )dz I (3) nn'′ n j =1 0 here, Φ n ( z ) is the eigenfunction for a single potential well, and s0 is the number of doped superlattices periods, d is the period. Energy spectrum of electron in doped superlattic [12]: r2 r 4π e2 nD 1 k⊥ 1 ε n (k ⊥ ) = ε n + εn = ( ) 2 (n + ) ; (4) χm 2m 2
125 H.D. Trien et al. / VNU Journal of Science, Mathematics - Physics 25 (2009) 123-128 here, nD is concentration of impurities, m and e are the effective mass and the charge of the electron, respectively and en are the energy levels of an individual well. In order to establish a set of quantum kinetic equations for acoustic and optical phonons, we use equation of motion of statistical average value for phonons ∂ ∂ i < bq >t = 〈[bq , H (t )]〉 t ; i < cq > t = 〈[cq , H (t )]〉 t (5) r r r r ∂t ∂t where 〈 X 〉 t means the usual thermodynamic average of operator X. Using Hamiltonian in Eq.(1) and realizing operator algebraic calculations, we obtain a set of coupled quantum kinetic equations for phonons. The equation for the acoustic phonons can be formulated as: rr r ∂ λ λ ∞ 〈bq 〉 t + iωq 〈bq 〉 t = − ∑ ∑Jν ( ) J µ ( )[ f n ( k⊥ − q ) − f ' (k⊥ )] × r r r ∂t Ω Ω n r nn' k −∞ ⊥ t × ∫ dt1{| Cq I + + |2 (〈bq 〉 t + 〈b−q 〉 t ) + C−q Dq I 2 ' (〈 cq 〉 t + 〈 c− q 〉 t )} × r r r rr r r nn' nn −∞ r rr ×exp{i[ε ' (k⊥ ) − ε n ( k⊥ − q )](t1 − t ) − iνΩt1 + i µΩt} (6) n A similar equation for the optical phonons can be obtained in which 〈 cq 〉 t , 〈 bq 〉 t , υq , ωq , r r r r Cq , Dq are replaced by 〈bq 〉 t , 〈 cq 〉 t , ωq , υq , Dq , Cq , respectively. r r r r r r r r r r λ In Eq.(6), f n (k⊥ ) is the distribution function of electrons in the state | n, k⊥ 〉 , J µ ( ) is the Ω rr eE0 q Bessel function, and λ = mΩ 3. The parametric transformation coefficient of acoustic and optical phonon in doped superlattices In order to establish the parametric transformation coefficient of acoustic and optical phonon, we use standard Fourier transform techniques for statistical average value of phonon operators: 〈bq 〉 t , r + + 〈 b− q 〉 t , 〈 cq 〉 t , 〈 c− q 〉 t . The Fourier transforms take the form : r r r ∞ ∞ 1 Ψ q (ω ) = ∫ 〈Ψ q 〉 t eiωt dt; ∫Ψ (ω )e−iωt d ω 〈Ψ q 〉 t = (7) r r r r 2π q −∞ −∞ One finds that the final result consists of coupled equations for the Fourier transformations Cq (ω ) r and Bq (ω ) of 〈 cq 〉 t and 〈 bq 〉 t . r r r For instance, the equation for Cq (ω ) can be written as: r Cq (ω − lΩ) ∞ r r (ω − υq )Cq (ω ) = 2∑ ∑ | I |2 Dq2υq Pl ( q , ω ) + r r rr ω − lΩ + υ nn' r nn' l = −∞ q
126 H.D. Trien et al. / VNU Journal of Science, Mathematics - Physics 25 (2009) 123-128 Bq (ω − l Ω) ∞ r r +2∑ ∑ | I |2 D− q Cq ωq Pl (q ,ω ) (8) rrr ω − l Ω + ωq nn' r nn' l = −∞ In the similar equation for Bq (ω ) , functions such as Cq (ω ) , Cq (ω − l Ω) , Bq (ω − lΩ) , υq , ωq , r r r r r r Cq , Dq are replaced by Bq (ω ) , Bq (ω − lΩ) , Cq (ω − lΩ) , ωq , υq , Dq , C q , respectively. r r r r r r r r r In Eq.(8), we have: λ λ ∞ r ∑J Pl ( q , ω ) = ( ) J µ +l ( )Γ q (ω + µΩ) (9) r µ Ω Ω µ = −∞ r rr [ f ' (k ⊥ ) − f n ( k⊥ − q )] Γq (ω + µΩ) = ∑ r r n (10) r k⊥ ε ' (k ⊥ ) − ε n ( k ⊥ ) − (ω + µΩ) − iδ r n where, the quantity δ is infinitesimal and appears due to the assumption of an adiabatic interaction of the electromagnetic wave (EMW). In Eq.(8), the first term on the right-hand side is significant just in case l=0. If not, it will contribute more than second order of electron-phonon interaction constant. Therefore, we have Cq (ω − l Ω) r r (ω − υq )Cq (ω ) = 2∑ | I ' |2 | Dq |2 υq Pl (q ,ω ) r r r r ω − l Ω + υq nn r nn' Bq (ω − lΩ) ∞ r r +2∑ ∑ | I |2 Dq Cq ωq Pl ( q , ω ) (11) rrr ω − l Ω + ωq nn' r nn' l = −∞ Transforming Eq.(11) and using the parametric resonant condition ωq + mΩ ≅ υq , the parametric r r transformation coefficient is obtained r ∑| I |2 D−q Cq P−l (q ,ω ) rr C (υ ) nn' r r nn' q q = = Kl (12) r δ − i∑ | I B (ω ) |2 | Dq |2 ImP0 ( q ,υq ) r r r r q q nn' nn' r Consider the case of l = 1 ; and assign γ 0 = ∑ nn' | I |2 | Dq |2 ImP0 ( q ,υq ) r r nn' Note that δ = γ 0 , we have r ∑|I |2 D− q Cq P−1 ( q , ω ) rr nn' nn' K1 = (13) iγ 0 Using Bessel function, Fermi-Dirac distribution function for electron and energy spectrum of electron in Eq.(4), we have Γ K1 =| | (14) 2γ 0 λ Γ = ∑ | I ' |2 Dq Cq ReΓ q (ωq ) Where (15) rr r r Ω nn' nn γ0 = ∑| I |2 | Dq |2 ImΓq (υq ) (16) r r r nn' nn'
127 H.D. Trien et al. / VNU Journal of Science, Mathematics - Physics 25 (2009) 123-128 4π e 2 nD 1/2 4π e2 nD 1/2 ' f0m ReΓ q (ωq ) = [exp[− β ( ) ( n + 1/2)] −exp[−β ( ) (n + 1/2)]] (17) r r 2πβ A1 χm χm βυ r βυ r f0 m 2mπ 4π e2 nD 1/2 mA22 ImΓq (υq ) = exp(−β r 2 )exp[−β ( ) (n + 1/2)] ×exp ( q ) sinh( q ) (18) r r r 2π q β χm 2q 2 2 r r 4π e 2 nD 1/ 2 4π e 2 nD 1/2 q2 q2 +( ) ( n − n ) + ωq +( ) ( n − n' ) + υq ' r; A = (19) A1 = r χm χm 2 2m 2m In Eqs.(17) and (18), β = 1/kbT ( kb is Boltzmann constant), f 0 is the electron density in doped superlattices. K1 is analytic expression of parametric transformation coefficient of acoustic and optical phonon in doped superlattices when the parametric resonant condition ωq ± lΩ ≅ υq is satisfied r r 4. Numerical results and discussions In order to clarify the mechanism for parametric transformation coefficient of acoustic and optical phonons in doped superlattices, in this section we perform numerical computations and graph for GaAs:Si/GaAs:Be doped superlattices. The parameters used in the calculation [6,7] ξ = 13.5eV , ρ = 5.32 gcm−3 , ν s = 5378ms −1 , χ ∞ = 10.9 , χ 0 = 12.9 , d = 10 nm , m = 0.066m0 , m0 being the mass of free electron, hω0 = 36.25meV , E0 = 106 V/m ( E0 is electric field intensity), kb = 1.3807 × 10−23 j/K , f 0 = 1023 m −3 , e = 1.60219 ×10−19 C , h = 1.05459 × 10−34 j.s , T=300K, q = 3.2 × 107 m−1 . K1 on nD . Fig. 1. Dependence of the Fig 1 shows the parametric transformation coefficient K1 as a function of concentration of impurities nD . It is seen that the parametric transformation coefficient of acoustic and optical phonons in doped superlattices depends non-linearly on concentration of impurities nD . Especially, when concentration of impurities nD tend toward zero, value of the parametric transformation coefficient of acoustic and optical phonons in doped superlattices will turn back nearly equal that in bulk
128 H.D. Trien et al. / VNU Journal of Science, Mathematics - Physics 25 (2009) 123-128 semiconductors [5] ( K1 ; 0.35 ), when concentration of impurities raises, the parametric transformation coefficient also to increase. This can be explained as follows: when concentration of impurities nD tend toward zero, the doped superlattices as a normal bulk semi- conductors. When concentration of impurities raises large enough, the doped superlattices is as low-dimensional semiconductor. 5. Conclusions In this paper, we obtain analytic expression of the parametric transformation coefficient of acoustic and optical phonons in doped superlattices in presence of an external electromagnetic field K1 , Eqs.(14)-(19). It is seen that K1 depends on concentration of impurities. Numerical computations and graph are performed for GaAs:Si/GaAs:Be doped superlattices, fig 1. The results is seen non- linear dependence of parametric transformation coefficient on concentration of impurities. when concentration of impurities nD tend toward zero, value of the parametric transformation coefficient of acoustic and optical phonons in doped superlattices will turn back nearly equal that in bulk semiconductors [5] ( K1 ; 0.35 ), when concentration of impurities raises, the parametric transformation coefficient also to increase. This can be explained as follows: when concentration of impurities nD tend toward zero, the doped superlattice as a normal bulk semiconductors. When concentration of impurities raises large enough, the doped superlattice is a low-dimensional semiconductor. Acknowledgments. This work is completed with financial support from NAFOSTED and QG.09.02 References [1] G.M. Shmelev, N.Q. Bau, Physical phenomena in simiconductors, Kishinev, 1981. [2] V.P. Silin, Parametric Action of the High-Power Radiation on Plasma (National Press on Physics Theory). Literature, Moscow, 1973. [3] M.V. Vyazovskii, V.A. Yakovlev, Sov. Phys. Semicond. 11 (1977) 809. [4] E.M. Epshtein, Sov. Phys. Semicond. 10 (1976) 1164. [5] G.M. Shmelev, N.Q. Bau, V.H. Anh, Communiccation of the Joint Institute for Nuclear, Dubna, 600 (1981) 17. [6] L.V. Tung, T.C. Phong, P.T. Vinh, N.Q. Bau, J. Science (VNU), 20 (No3AP) (2004) 146. [7] N.Q. Bau, T.C. Phong, J.Kor. Phys. Soc. 42 (2003) 647. [8] N.V. Diep, N.T. Huong, N.Q. Bau, J.Scienc (VNU), 20 (No3AP) (2004) 41. [9] T.C. Phong, L. Dinh, N.Q. Bau, D.Q. Vuong, J. Kor. Phys. Soc. 49 (2006) 2367. [10] N. Mori, T.Ando. Phys. Rev. B 40, (1989) 12. [11] V.V. Pavlovic, E.M. Epstein, Sol. Stat. Phys. 19 (1977) 1760. [12] T.C. Phong, L.V. Tung, N.Q. Bau, Comm.in phys, Supplement (2004) 70.