Cyclotron resonance via two-Photon absorption process in cylindrical quantum wire

CYCLOTRON RESONANCE VIA TWO-PHOTON ABSORPTION PROCESS IN CYLINDRICAL QUANTUM WIRE LE DINH - LE TRUNG DUNG College of Education - Hue University HUYNH VINH PHUC Dong Thap University Abstract: Cyclotron resonance (CR) in cylindrical quantum wire (CQW) is investigated via the two-photon absorption process. Expressions for absorp- tion power are obtained when electrons are scattered by acoustic phonons. The CR peaks in the absorption spectrum due to transitions between Lan- dau levels in the case of one- and two-photon absorption process are indi- cated. From curves on graphs of the absorption power as a function of photon frequency, we obtain CR-linewidth as profiles of the curves by using profile method. The temperature and magnetic field dependence of the absorption power and CR-linewidth are considered. The results are compared with those in the case of one-photon absorption process. The calculations demonstrate that two-photon absorption process is sufficient strong to be detected in CR. Keywords: Cyclotron resonance, two-photon absortion process, cylindrical quantum wire 1. INTRODUCTION Cyclotron resonance is known as a good tool for investigating the electron-phonon interaction in the presence of a magnetic field. This effect has been studied both theoretically and experimentally in bulk semiconductors [1, 2, 3], quantum wells [4, 5, 6, 7], quantum wires [8, 9], quantum dots [10, 11] and graphene [12]. In these works, the main CR peak occurs at ω = ωc, where ω and ωc are the frequencies of the electromagnetic wave and cyclotron, respectively. In these studies, CR has been examined under the one-photon absorption process. However, the study of CR via the two-photon absorption process has not been found. Journal of Sciences and Education, Hue University’s College of Education ISSN 1859-1612, No. 3(31)/2014: pp. 5-13 6 LE DINH et al. Multi-photon absorption process occurs when two or more photons are simultane- ously absorbed by the material [13]. Recently, the interest in the development of materials with multi-photon absorption process has increased, depending on their potential application in different fields of science. Multi-photon absorption pro- cess has been studying on the fields of laser scan confocal microscopy [14], optical power limiting [15], multi-photon microscopy [16, 17], micro-fabrication [18], upcon- version lasing [19], photodynamic therapy [20], and multi-photon luminescence [21]. In particular, Boyd and co-workers [21] have demonstrated that the multi-photon luminescence is more sensitive to the local fields than the single-photon lumines- cence. Therefore, the study on multi-photon process is important for understanding in details the transient response of semiconductors excited by electromagnetic field. In an our previous paper [22], we suggested a method for obtaining the explicit ex- pression of absorption power in the presence of magnetic field with multi-photon ab- sorption process. In the case of single-photon absorption process, our result reduces to that presented in the paper of Bhat et al. [6], which is given by the second-order Born golden rule approximation. The result presented in the paper is quite gener- ally, and can be applied for studying CR in different kinds of semiconductors. The expression of the absorption power in different kinds of semiconductors is regulated by the transition matrix element for the electron-phonon interaction, as well as en- ergy spectrum of electrons. The present work is devoted to the study of CR via two-photon processes in CQW. Expression of absorption power is obtained using the perturbation approach. From graphs of the absorption power as a function of photon frequency, we obtained CR-linewidth as profiles of the curves by using profile method [23]. We investigate the dependence of the absorption power and CR-linewidth on the temperature and magnetic field. The present work is quite different from the previous ones because the two-photon absorption process is taken into account. The paper is organized as follows. In Section 2, the theoretical framework used in cal- culations and analytical results is presented. Discussion of the obtained results are given in Section 3. Finally, the conclusions of the paper appear in the fourth section. 2. MODEL OF A CQW AND ANALYTIC EXPRESSION FOR ABSORPTION POWER IN A CQW In the presence of a strong magnetic field parallel to the wire axis, the wave function and energy of electron inside a cylindrical quantum wire of radius R and length Lz, CYCLOTRON RESONANCE VIA TWO-PHOTON ABSORPTION PROCESS... 7 with infinite potential are given by [24] ψ(r, ϕ, z) = 1√ 2piL eimϕeikzzφN,m(r), (1) EN,m,kz = ~2k2z 2m∗ + ~ωc [ N + m 2 + 1 2 + |m| 2 ] , (2) the radial wave function for a circular cross section in Eq. (1) is φN,m(r) = √ (N + |m|)! N ! 1 ac|m|!e −ξ/2ξ|m|/2 × 1F1(−N, |m|+ 1; ξ), (3) where N = 0, 1, 2, . . . and m = 0,±1,±2, . . . denote the Landau-level index and the azimuthal quantum number, respectively; m∗ is the effective mass of electron; ac = (~/m∗ωc)1/2 is the cyclotron radius; 1F1(−N, |m| + 1; ξ) is the general form of the confluent hypergeometric function, ξ = r2/(2a2c). The absorption power can be calculated by relating it to the transition probability for the absorption of photons [22] P (ω) = F 20 √ ε 8pi ∑ i Wifi, (4) where F0 is the intensity radiation; ε is the dielectric constant of the medium; fi is the electron distribution function; and Wi is the transition probability. The sum is taken over all the initial states i of the electron. The transition probability for absorbing photon with simultaneously absorbing and/or emitting phonon W∓i is given by the perturbation approach W∓i = 2pi ~ ∑ f,~q |〈f |He−p|i〉|2 +∞∑ `=−∞ 1 (`!)2 (a0q⊥ 2 )2` δ(Ef − Ei ∓ ~ω~q − `~ω), (5) where 〈f |He−p|i〉 is the transition matrix element for the electron-phonon interac- tion; a0 = (eF0)/[m ∗(ω2 − ω2c )]; Ei and Ef are the initial and final energy state of electrons, respectively; and ~q = (qz, q⊥). The sum is taken over all the final state f of the system. In Eq.(5), the index ` shows the number of photons to be absorbed. Using the wave function given by Eq.(1), the matrix elements for the electron- acoustic phonon interaction in CQW can be written as [8, 24, 25] |〈f |He−p|i〉|2 = |V (~q)|2(N~q + 1/2∓ 1/2)δk′z ,kz±qz |Mm,n,m′,n′(q⊥)|2|JNN ′(a2cq2⊥/2)|2, (6) 8 LE DINH et al. where N~q the distribution function of the phonon in the mode ~q; the sign (−/+) refers to absorption/emission of a phonon with energy E~q = ~ω~q, and |V (~q)|2 = ~ξ 2 2ρu0V0 q; Mm,n,m′,n′(q⊥) = 2 R ∫ R 0 J|m−m′|(q⊥R)ψ∗m′,n′(r)ψm,n(r)rdr, ψm,n(r) = 1 Jm+1(Amn) Jm ( Amn r R ) , ; |JNN ′(x)|2 = N ! N ′! e−xxN ′−N [ LN ′−N N (x) ]2 , N ≤ N ′, where ξ is the deformation potential constant, ρ is the density of the material, u0 is the velocity of sound, V0 is the volume of the system, Amn is the n-th zero of Bessel function of the m-order, Jm(Amn) = 0. The function L M N (x) is the associated Laguerre polynomials. The sum over ~q can be transformed into the integral. By considering the transitions between the states in the extreme electric quantum limit (m = m′ = 0, n = n′ = 1), and making the same approximation as in Ref. [26], (i.e. we take ~2/(2m∗)(q2z − 2kzqz) = 0) in delta functions, then inserting Eq. (6) into Eq. (5), we obtain the following expression for the transition probability W∓i = V0RLz (2pi)2~a2c ∑ N ′ +∞∑ `=−∞ 1 (`!)2 (a0q⊥ 2 )2` ∫ ∞ 0 q⊥dq⊥ ∫ +∞ −∞ dqz|V (~q)|2(N~q + 1/2∓ 1/2) × |M0,1,0,1(q⊥)|2|JNN ′(a2cq2⊥/2)|2δ [(N ′ −N) ~ωc ∓ ~ω~q − `~ω] , (7) We restrict ourselves to the non-degenerate electron gas. In this case the distribution function fi = fN,0,0 ≡ fN in Eq. (4) can be written as follows fN = 2pi2nea 2 cR D e−EN/kBT , D = ∑ N e−EN/kBT , (8) where ne is the electron concentration, kB is the Boltzmann constant, and T is the temperature. The absorption power is calculated by Eq.(4), with the transition probability is shown in Eq.(7) has many contributions from absorption processes of `-photons. We limit ourselves in considering the process of absorbing two photons (` = 1, 2). Inserting Eqs.(7) and (8) into Eq.(4) and making a straightforward calculation of integral over q⊥ and qz, we obtain the following expression of the absorption power in CQW for electron-acoustic phonon interaction P (ω) = A(ω,ωc)kBTξ 2 2ρu20V0a 4 cLz ∑ N,N ′ e−EN/kBT [ (N +N ′ + 1)δ(Z1) + a20 8a2c K(N,N ′, 2)δ(Z2) ] (9) where we have denoted A(ω, ωc) = F 20 √ εneV 3 0 a 2 0 8(2pi)4~RDa2c , Z` = (N ′ −N)~ωc − `~ω, (` = 1, 2), (10) CYCLOTRON RESONANCE VIA TWO-PHOTON ABSORPTION PROCESS... 9 and K(N,N ′, 2) = N ! N ′! ∫ ∞ 0 xN ′−N+2e−x [ LN ′−N N (x) ]2 dx. (11) Here, we have used [27] (N~q + 1/2±1/2)|V (~q)|2 = kBTξ 2 2ρu20V0 and neglected ~ω~q in the arguments of delta functions in compared with ~ωc. Following the collision broadening model, we replace the delta functions in Eq. (9) by Lorentzian of the width Γ, namely δ(x) = (~Γ/pi)/(x2 + (~Γ)2), where Γ2 = kBTξ 2 (2pi~)28ρu20a2cLz . (12) These analytical results are quite clear. However, physical conclusions can be drawn from graphical representations and numerical results, obtained from adequate com- putational methods. 3. NUMERICAL RESULTS AND DISCUSSIONS To clarify the obtained results we numerically calculate the absorption power for a specific CQW made up of GaAs/AlAs. The parameters used in our calculations are as follow [6, 24, 25] ξ = 13.5 eV, ρ = 5.32 g cm−3, u0 = 5378 m s−1, m∗ = 0.067me, me being the mass of free electron, ne = 10 23 m−3, and F0 = 4.0 × 105 V/m 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ِΩc P Ha rb . u n its L Figure 1: Absorption power is shown as a func- tion of ω/ωc at B = 7 T, R = 16 nm, and T = 27 K. Figure 1 shows the dependence of the absorption power on ω/ωc at B = 7 T, R = 16 nm and T = 27 K. The main cyclotron resonance peak at ω = ωc comes from the term (ω − ωc)−2 in a20 contained in (5). The other resonance peaks oc- cur at `ω = pωc, with ` = 1, 2, p = 1, 2, 3, . . .. These peaks describe the cyclotron resonance, which show the fact that an electron transition between Landau levels occurs with the absorption of one-photon (two-photon) with frequencies ω = pωc (pωc/2). Two peaks occur at ω/ωc = 0.5 and 1.5 are due to two-photon absorption process with p = 1 and 3. The peak corresponds to p = 2 is not obviously visible, because its 10 LE DINH et al. position virtually coincide with the main peaks, and its value is much smaller than that of the main peak. The intensity of peak at ω = 3ωc is about 14.0% of that 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ِΩc P Ha rb . u n its L Figure 2: Absorption power is shown as a function of ω/ωc for different values of temperature. The solid, dashed, and dot- ted curves correspond to T = 10 K, 27 K, and 50 K. Here R = 16 nm, and B = 7 T. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ِΩc P Ha rb . u n its L Figure 3: Absorption power is shown as a function of ω/ωc for different values of magnetic field. The solid, dashed, and dotted curves correspond to B = 4 T, 7 T, and 10 T. Here R = 16 nm, and T = 27 K. at ω = 2ωc; the intensity of peak at 2ω = 3ωc is about 15.2% of that at 2ω = ωc. Beside, the peak value in the two-photon process at p = 3 (2ω = 3ωc) is about 112.1% of that in the one-photon process (ω = 3ωc), and this rate is about 17.1% of that at ω = 2ωc. Therefore, the two-photon absorption process is sufficiently strong to be detected in CR. From figures 2 and 3, we can see that absorption power increases with the tempera- ture and magnetic field. This is caused by the absorption power which is proportional to T and a−6c in Eq. (9). As the magnetic field increases, the cyclotron orbit ac de- creases, leading to an increase in the absorption power. We have also studied the dependence of CR-linewidth on the temperature and mag- netic field. It is known that linewidth is often defined in terms of the full-width at half-maximum (FWHM) of the optical field power spectrum. Using profile method [23], we obtain the temperature dependence of the CR-linewidth, as shown in Fig. 4. The figure shows that CR-linewidth increases with temperature. The results are in good agreement with the results of previous papers [1, 2, 3, 8]. As the temperature increases, probability of electron-phonon scattering increases, leading to a general increase in the temperature of CR-linewidth. In addition, the linewidth in the case CYCLOTRON RESONANCE VIA TWO-PHOTON ABSORPTION PROCESS... 11 ã ã ã ã ã ã ã ã ã ã 㠟 Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ 0 20 40 60 80 100 0.0 0.5 1.0 1.5 2.0 Temperature HKL C R - L in ew id th Hm e V L Figure 4: Dependence of the CR- linewidth on the temperature T . The filled and empty correspond to one- photon and two-photon absorption pro- cess, respectively. Here, B = 7 T, and R = 16 nm. Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ ã ã ã ã ã ã ã ã ã ã 5 10 15 20 0.5 1.0 1.5 2.0 B HTL C R - L in ew id th Hm e V L Figure 5: Dependence of the CR- linewidth on the magnetic field B. The filled and empty correspond to one- photon and two-photon absorption pro- cess, respectively. Here, T = 27 K, and R = 16 nm. of two-photon absorption process (empty curves) is smaller than that in the case of one-photon absorption process (filled curves). Hence, the one-photon process makes dominant contribution to the total process. Figure 5 describes the magnetic field dependence of CR-linewidth at T = 27 K and R = 16 nm. From the figure, we can see that CR-linewidth increases with magnetic field B. This result is in good agreement with the results of previous papers [2, 3, 4, 8]. This can be explained that as B increases, the cyclotron radius ac reduces, the confinement of electron increases, the probability of electron-phonon scattering increases, so that the linewidth rises. Beside, we can see from the figure that CR- linewidth varies as √ B, consistent with previous result [4]. 4. CONCLUSIONS In the present paper, by considering the multiphoton process, we have studied CR in CQW. Results are presented for a specific GaAs/AlAs CQW. The absorption spectrum satisfy the condition `~ω = p~ωc. Although being smaller than the one- photon absorption process, the two-photon process is strong enough to be detected in CR. Using profile method, we obtain CR-linewidth as profile of the curves. The values of linewidth are found to increase with the increment of the temperature and magnetic field. Besides, the probability of two-photon absorption process is always smaller than that of one-photon absorption process. The present work can include 12 LE DINH et al. the results obtained in the other works in the case of one-photon absorption process. Especially, these results are much more useful to deal with multiphoton process problems. Finally, we hope that the results above could be used as a guideline for explaining the experimental data which may be obtained in the future. REFERENCES [1] Y. J. Cho, S. D. Choi, Phys. Rev. B 49 (1994) 14301. [2] J. Y. Sug, S. G. Jo, J. Kim, J. H. Lee, S. D. Choi, Phys. Rev. B 64 (2001) 235210. [3] H. Kobori, T. Ohyama, E. Otsuka, J. Phys. Soc. Jpn, 59 (1990) 2141. [4] M. P. Chaubey, C. M. Van Vliet, Phys. Rev. B 34 (1986) 3932. [5] B. Tanatar, M. Singh, Phys. Rev. B 42 (1990) 3077. [6] J. S. Bhat, S. S. Kubakaddi, B. G. Mulimani, J. Appl. Phys. 70 (1991) 2216. [7] R. Masutomi, A. Sekine, K. Sasaki, K. Sawano, Y. Shiraki, T. Okamoto, Physica E 42 (2010) 1184. [8] T. C. Phong, L. T. T. Phuong, H. V. Phuc, Superlattices Microstruct. 52 (2012) 16. [9] Re. Betancourt-Riera, Ri. Betancourt-Riera, J. M. Nieto Jalil, R. Riera, Physica B 410 (2013) 126. [10] K. Takehana, Y. Imanaka, T. Takamasu, M. Henini, Physica E 42 (2010) 915. [11] Shi-Hua Chen, Physica E 43 (2011) 1007. [12] K. C. Chuang, R. S. Deacon, R. J. Nicholas, K. S. Novoselov, A. K. Geim, Phil. Trans. R. Soc. A 66 (2008) 366. [13] Farhad H. M. Faisal, Theory of multiphoton processes, Plenum Press, New York, 1987. [14] P. N. Prasad, J. D. Bhawalkar, J. Swiatkiewicz, P. C. Chang, S. J. Pan, A. Smith, J. K. Samarabu, B. A. Reinhardt, Polym. Commun. 38 (1997) 4551. [15] G. S. He, L. Yuan, N. Cheng, J. D. Bhawalkar, P. N. Prasad, L. L. Brott, S. J. Clarson, B. A. Reinhardt, J. Opt. Soc. Am. B 14 (1997) 1079. [16] W. Denk, J. H. Strickler, W. W. Webb, Science 248 (1990) 73. [17] Jianyong Tang, Ronald N. Germain, Meng Cui, Proc. Natl. Acad. Sci. USA 93 (2012) 10763. [18] S. Kawata, H. B. Sun, T. Tanaka, K. Takada, Nature 412 (2001) 697. CYCLOTRON RESONANCE VIA TWO-PHOTON ABSORPTION PROCESS... 13 [19] S.L. Oliveira, D.S. Correa, L. Misoguti, C. J. L. Constantino, R. F. Aroca, S. C. Zilio, C. R. Mendonca, Adv. Mater. 17 (2005) 1890. [20] J. D. Bhawalkar, G. S. He, P. N. Prasad, Rep. Prog. Phys. 60 (1997) 689. [21] G. T. Boyd, Z. H. Yu, Y. R. Shen, Phys. Rev. B 33 (1986) 7923. [22] H. V. Phuc, L. Dinh, T. C. Phong, Superlattices Microstruct. 59 (2013) 77. [23] T. C. Phong, H. V. Phuc, Mod. Phys. Lett. B 25 (2011) 1003. [24] S. V. Branis, G. Lee, K. K. Bajaj, Phys. Rev. B 47 (1993) 1316. [25] N.A. Zakhleniuk, C.R. Bennett, N.C. Constantinou, B.K. Ridley, M. Babiker, Phys. Rev. B 54 (1996) 17838. [26] P. Vasilopoulos, P. Warmenbol, F.M. Peeters, J.T. Devreese, Phys. Rev. B 40 (1989) 1810. [27] M. P. Chaubey, C. M. Van Vliet, Phys. Rev. B 33 (1986) 5617. Assoc. Prof. Dr. LE DINH Department of Physics, College of Education - Hue University LE TRUNG DUNG, MSc. Department of Physics, College of Education - Hue University Dr. HUYNH VINH PHUC Department of Physics, Dong Thap University