DSpace at VNU: THE EFFECT OF CONFINED PHONONS ON THE ABSORPTION COEFFICIENT OF A WEAK ELECTROMAGNETIC WAVE BY CONFINED ELECTRONS IN DOPED SUPERLATTICES
HỘI NGHỊ VẬT LÝ CHẤT RẮN TOÀN QUỐC LẦN THỨ - Vũng Tàu 12-14/11/2007 THE EFFECT OF CONFINED PHONONS ON THE ABSORPTION COEFFICIENT OF A WEAK ELECTROMAGNETIC WAVE BY CONFINED ELECTRONS IN DOPED SUPERLATTICES Luong Van Tung, Le Thai Hung, Nguyen Quang Bau Department of Physics, Ha Noi National University 334 Nguyen Trai, Thanh Xuan, Ha Noi E-mail: lthung@vnu.edu.vn or hunglethai191182@yahoo.com.vn ABSTRACT The effect of confined phonons on the absorption coefficient of a weak electromagnetic wave by confined electrons in doped surperlattices is theoretically studied by using the Kubo-Mori method In comparison with the case of unconfined phonons, different dependence of the absorption coefficient on the electromagnetic wave frequency (ω), the doping concentration (nD), the number (N) of period, the temperature (T) of the system is obtained The analytic expressions are numerically evaluated, plotted, and disscussed for a specific doping of the n-GaAs/p-GaAs superlattice The results show that confined phonons cause some unusual effects There are two resonant peaks of the absorption coeffient and one of the mass operator The mass operator’s values are larger and the absorption coeffient’s values are smaller than they are in the case of unconfined phonons Keywords: Doped superlatices, Absorption coefficient, Conductivity tensor, Confined phonons INTRODUCTION Recently, there are more and more interests in studying and discovering the behavior of low-dimensional system, in particular two-dimensional systems, such as semiconductor superlattices, quantum wells and doped superlattices (DSLs) The confinement of electrons in low-dimensional systems considerably enhances the electron mobility and leads to unusual behaviors under external stimuli Many papers have appeared dealing with these behaviors, for examples, electron-phonon interaction and scattering rates [1-3] and dc electrical conductivity [4, 5] The problems of the absorption coefficient for a weak electromagnetic wave (EMW) in semiconductor superlattices [6, 7], in quantum wells [8] and in doped superlatices [9] have also been investigated and resulted In this paper, we study the high-frequency conductivity tensor and the absorption coefficient of a EMW due to confined electrons in a DSL with the influence of confined phonons The electron-optical phonon scattering mechanism is assumed to be 512 dominant We shall asume that the EMW has a high frequency Then, we estimate numerical values for a specific doping of the n-GaAs/p-GaAs superlattice to clarify our results Using Kubo’s formula for the conductivity tensor [10] and Mori’s projection operator method [11] in the second-order approximation of the interaction, we obtain the following formula for the components of the conductivity tensor [7, 9, 12, 13]: i σ µν (ω ) = lim( J µ , J ν ) δ − i(ω + η ) + ( J µ , J ν ) −1 δ →0 h iω t − δ t [U , J µ ] , [U , J ν ]I dt ∫0 e +∞ × with here, ( ) −1 (1) hη = [J µ , Jν ] (J µ , Jν ) −1 [U , Jν ]I is the operator (2) [U , Jν ] in the interaction picture, and U is the energy of the electron-phonon interaction The averaging of the operators in Eqs.(1-2) is implemented with the non-interaction Hamiltonian of the electron-phonon system The structure of DSL also modifies the dispersion relation of optical phonons, which leads to interface modes and confined modes [2] However, the contribution from these two modes can be approximated well by calculations with bulk phonons [3] Thus, in this paper, we will deal with bulk confined phonons (2-dimensional) and consider compensated n-p DSL with equal thicknesses dn = dp=d/2 of the n-doping and p-doping layer and equal constant doping concentrations nD = nA in the respective layers THE ABSORPTION COEFFICIENT WITH CASE OF CONFINED PHONONS It is well known that the motion of an electron is confined in each layer of the DSL and that it’s energy spectrum is quantized into discrete levels In this paper, we assume that the quantization direction is the z direction The Hamiltonian of the electron-optical HỘI NGHỊ VẬT LÝ CHẤT RẮN TOÀN QUỐC LẦN THỨ - Vũng Tàu 12-14/11/2007 phonon system in a DSL in the second quantization representation can be written as: H = H0 +U H0 = ∑E k ⊥ ,n ∑ U= (3) k ⊥ ,n a + k ⊥ ,n ∑C I a m ,q ⊥ k ⊥ , n , n ' ∑ hω b ak ⊥ , n + m + q nn ' k ⊥ + q ⊥ ,n ' q⊥ ,m + q ⊥ ,m q ⊥ ,m ( ) (5) ) where rj is the radius vector of jth electron Using the Kubo-Mori’s method, we obtain the following formula for the transverse component of the high-frequency conductivity tensor: where and after scattering, (k ⊥ , q ⊥ ) is the in-plane (x, y) wave vector of the electron (phonon), ak+⊥ , n , ak ⊥ , n + ( bq⊥ ,m , bq⊥ ,m ) are the creation and the annihilation operators of the electron (phonon), respectively, and hω0 is the energy of the optical phonon The electron εk energy ⊥ ,n in doped superlatices takes the simple σ xx (ω ) = γ [− iω + G (ω )]−1 γ = ( J x , J x ), ⊥ ,n h2 = εn + + k⊥ 2m −1 Knowing the high-frequency conductivity tensor, the absorption coefficient can be found by using the common relation: with α xx (ω ) = (6) 1/ 4πe nD n + ε = h 2 κ 0m (7) Here, m and e are the effective mass and the charge of the electron, respectively, κ is the Cq = 2πe hω V 1 − χ χ ∞ mπ q 2⊥ + d 4π Re σ xx (ω ) cρ (13) Here, ρ is the refraction index and c is the light velocity Since the EMW has a high frequency, using Eqs (3-14) and noting that in compensated n-p DSLs, the bare ionized impurities make the main contribution to the superlattice potential, we obtain: α xx (ω ) = electronic constant, nD is the doping concentration and C q is the electron-phonon interaction potential In the case of the electron-optical phonon interaction and confined phonons, we assume that the quantization direction is the z direction, C q [1] is: (11) i + ∞ G (ω ) = lim γ 0−1 ∫ e iωt −δt ([U , J x ], [U , J x ]I )dt (12) δ →0 h form [14]: εk (10) j where H0 is the non-interaction Hamiltonian of the electron-phonon system, n (n = 1, 2, 3, ) denotes the quantization of the energy spectrum in the z direction, (n, k ⊥ ) and n, k ⊥ + q ⊥ are electron states before ( H t1 = −e∑ (r j E )cos(ωt )e δt (4) b ak ⊥ ,n bq ⊥ ,m + bq+⊥ ,m eigenfunction for a single potential well [15] The interaction of the system, which is described by Eqs.(3-5) with a EMW E(t) = E0cos(ωt), is determined by the Hamiltonian: γ G (ω ) 4π 20 , cρ G (ω ) + ω (14) where V / 3e β µ − 2ε γ0 = e (cosh(βε ) + coth(βε ) − 1) (15) 4πβh G (ω ) = G + (ω ) + G − (ω ) (8) (16) 1 − χ∞ χo 1 eβhω −1 hωo × No + ± eβµ 2 hω G ± (ω ) = where V is the normalization volume, m=1, 2, …, N and N is the number of period of DSLs, χ and χ ∞ are the static and the high-frequency dielectric constant, respectively, and [3, 5]: ×∑ I V 1/ e γo 4πβh m nn ' e −1 1 − β n + ε 2 β e2 (λ ± − λ ± ) (17) N d I nnm ' = ∑ ∫ eiqz z Φ n ( z − md )Φ n ' ( z − md )dz (9) m=1 where d is the period of DSL, Φ n ( z − md ) is the with λ± = (n'−n )ε ± hωo − hω (18) No is the equilibrium distribution of optical phonons, µ is the chemical potential The signs (±) in the 513 HỘI NGHỊ VẬT LÝ CHẤT RẮN TOÀN QUỐC LẦN THỨ - Vũng Tàu 12-14/11/2007 superscript of G (ω ) and in the lower-script of the function λ± correspond to the signs (±) in Eq.(17) and Eq.(18) The upper sign (+) corresponds to phonon absorption and the lower sign (−) corresponds to a phonon emission in the absorption process From Eqs.(11,14), we can easily see that G (ω ) plays the role of the well-known mass operator of the electron in the Born approximation in the case of the absence of a magnetic field NUMERICAL CALCULATION AND DISCUSSION ± In order to clarify the different behaviors of a quasi-two-dimensional electron gas confined in a DSL with respect to a bulk electron gas, in this section, we numerically evaluate the analytic formulae in Eqs (14-18) for a compensated n-GaAs/p-GaAs DSL The characteristic parameters of the GaAs layer of the DSL 17 are χ ∞ = 10 , χ = 12 , nD= 10 cm−3, d = 2dn= 2dp= 80 nm, µ = 0.01 meV, m = 0.067m0, and hωo = 36.1 meV, (m0 is the mass of free electron) The system is assumed to be at room temperature T = 293 K Then, we compare with the results in the case of unconfined phonons [9] value of G (ω ) is about 120s-1 in case of unconfined phonons [9] but is 850s-1 in case of the confined phonons at the same the value of N It means that the confined phonons make the life-span of an electron is so much shorter This figure exactly shows that there is a resonant peak of G (ω ) when hω ≈ 50( meV ) Fig.2 Dependence of the absorption coefficient of EMW (cm−1) on the frequency ω of EMW and the period number N with the case of confined phonons, n=n’=11 Figure (2) shows the absorption coefficient α xx (ω ) as a function of the frequency ω of the EMW and the period number N, with case of confined phonons This figure shows the resonant regions in the absorption spectra of the absorption coefficient α xx (ω ) as the EWM is high frequency ωτ >> 1, the resonant regions of the absorption coefficient appear when the values of G(ω) are greater There is one more resonant peak of α xx (ω ) and the values of α xx (ω ) are much smaller than those with the case of unconfined phonons [9] Namely, the resonant peak’s value of α xx (ω ) approximates 0.85 cm-1 in case of Fig.1 Dependence of the operator G (ω ) (1/s) on the frequency ω of EMW and the period number N with the case of confined phonons, n=n’=11 Figure (1) shows the mass operater G (ω ) as a function of the frequency ω of the EMW and the period number N, in case of confined phonons It is seen that G (ω ) depends very strongly on the frequency of the EMW, they are greater when the frequency of EMW increases This figure shows that there is a resonant peak of G (ω ) in the region of the values of N from N=5 to N=20 on the number of periods axis but the values of G (ω ) with case of confined phonons are much larger than their values with the case of unconfined phonons [9] Namely, the resonant peak’s 514 confined phonons but 65 cm-1 in case of unconfined phonons [9] In the figure (3-4), the graphs show that the absorption coefficient of EMW as a function of temperature T, the doping concentration nD Their values are much smaller than those in the case of uncofined phonons [9] Namely, the value of α xx (ω ) is about 60 (cm-1) in case of unconfined phonons [9] but is 0.15 (cm-1) in the case of confined phonons at T=120K; the value of α xx (ω ) is about 63 (cm-1) in case of unconfined phonons [9] but is 2.3 (cm-1) in the case of confined phonons at nD=0.1.1016 In figure (3), the graph shows that in the case of confined phonons did not appear a resonant peak of the absorption coefficient of EMW but in the case of unconfined phonons did [9] HỘI NGHỊ VẬT LÝ CHẤT RẮN TOÀN QUỐC LẦN THỨ - Vũng Tàu 12-14/11/2007 electrons in doped superlattices It’s found that the confined phonons have made the values of the operator G (ω ) increase and the values of the absorption coefficient α xx (ω ) of EMW decrease very much In other words, the confined phonons cause the life-span of an electrons is shorter and the absorption coefficient’s values are much smaller Fig.3 Dependence of the absorption coefficient of EMW (cm−1) on the Temperature T with the case of confined phonons, n=n’=11 Fig.4 Dependence of the absorption coefficient of EMW (cm−1) on the doping concentration nD with case of confined phonon, n=n’=11 CONCLUSIONS In this paper, we have presented out the analytical formulae for the transverse components of the absorption coefficient of a weak EMW due to free carriers in a DSL for the case of confined phonon, Eqs (14)-(18) by using Kubo-Mori method The numerical evaluations of these formulae for compensated n-p doped superlattices show that the confinement of phonons in the doping superlattices not only leads to different dependences of the high-frequency conductivity tensor and the absorption coefficient on the EMW frequency ω, the temperature of system T, doping concentration nD, the number of periods N in comparison with normal semiconductors [6,7] and quantum wells [8] but also creates many significant differences in the absorption coefficient of a EMW from the case of unconfined phonons in a DSL [9] All results show that the confined phonon influences very powerfully on the absorption coefficient of a weak electromagnetic wave by confined Acknowledgments This work is completed with financial support from the program of Basic research in Natural Science, 405906 References [1] N Mori and T Ando, Phys Rev B 40, (1989), 6175 [2] H Rucker, E Molinari and P Lugli, Phys Rev B 45, (1992), 6747 [3] J Pozela and V Juciene, Sov Phys.Tech Semicond 29, (1995), 459 [4] P Vasilopoulos, M Charbonneau and C M Van Vliet, Phys Rev B 35, (1987), 1334 [5] A Suzuki, Phys Rev B 45, (1992), 6731 [6] V V Pavlovich and E M Epshtein, Sov Phys Stat.19, (1977), 1760 [7] G M Shmelev, I A Chaikovskii and Nguyen Quang Bau, Sov Phys Tech Semicond 12, (1978), 1932 [8] Nguyen Quang Bau and Tran Cong Phong, J Phys Soc Jpn 67, (1998), 3875 [9] Nguyen Quang Bau, Nguyen Vu Nhan and Tran Cong Phong, J.Kor.Phys.Soc 42, No.1, (2002), 149-154 [10] R Kubo, J Phys Soc Jpn 12, (1957), 570 [11] H Mori, Prog Theor Phys 34, (1965), 399 [12] G M Shmelev, Nguyen Quang Bau and Nguyen Hong Son, Sov Phys Tech Semicond 15, (1981), 1999 [13] Nguyen Quang Bau, Chhoumm Navy and G M Shmelev, Proceedings of 17th Congress of the Inter Comm for Optics, Daejon; SPIE 2778, (1996), 844 [14] K Ploog and G H Dohler, Adv Phys 32, (1983), 285 [15] T M Rynne and H N Spector, Phys.Chem Sol 42, (1980) 121 515 ... in the case of confined phonons at nD=0.1.1016 In figure (3), the graph shows that in the case of confined phonons did not appear a resonant peak of the absorption coefficient of EMW but in the. .. with the case of unconfined phonons [9] Namely, the resonant peak’s 514 confined phonons but 65 cm-1 in case of unconfined phonons [9] In the figure (3-4), the graphs show that the absorption coefficient. .. K Then, we compare with the results in the case of unconfined phonons [9] value of G (ω ) is about 120s-1 in case of unconfined phonons [9] but is 850s-1 in case of the confined phonons at the