Journal of the Korean Physical Society, Vol 64, No 4, February 2014, pp 572∼578 Negative Absorption Coefficient of a Weak Electromagnetic Wave Caused by Electrons Confined in Rectangular Quantum Wires in the Presence of Laser Radiation Nguyen Quang Bau∗ and Nguyen Thi Thanh Nhan Department of Physics, College of Natural Sciences, Hanoi National University, Hanoi, Vietnam Nguyen Vu Nhan Department of Physics, Academy of Defence Force - Air Force, Hanoi, Vietnam (Received 17 January 2013, in final form 22 October 2013) Analytic expressions for the absorption coefficient (ACF) of a weak electromagnetic wave (EMW) caused by electrons confined in rectangular quantum wires (RQWs) in the presence of laser radiation are calculated using the quantum kinetic equation for electrons in the case of electron-optical phonon scattering The dependence of the ACF of a weak EMW on the intensity E01 and the frequency Ω1 of the external laser radiation, the intensity E02 and the frequency Ω2 of the weak EMW, the temperature T of the system and the size L (Lx and Ly ) of the RQWs is obtained The results are numerically calculated and discussed for GaAs/GaAsAl RQWs The numerical results show that the ACF of a weak EMW in RQWs can have negative values Thus, in the presence of laser radiation, under proper conditions, a weak EMW is increased This is different from the similar problem in bulk semiconductors and from the case without laser radiation PACS numbers: 78.67.Lt, 78.67.-n Keywords: Rectangular quantum wires, Absorption coefficient, Electron-phonon interaction, Laser radiation DOI: 10.3938/jkps.64.572 I INTRODUCTION semiconductor that is located in a weak EMW The influence of laser radiation on the absorption of a weak EMW in normal bulk semiconductors has been investigated [16–19] However, in that problem, the ACF of a weak EMW has only positive values Similar studies on low-dimensional systems, in particular, RQWs, have not been done Therefore, in this paper, we use the quantum kinetic equation for electrons to theoretically calculate the ACF of a weak EMW caused by electrons confined in a RQW in the presence of laser radiation The results are numerically calculated for the specific case of a GaAs/GaAsAl RQW We show that the ACF of a weak EMW in a RQW can have negative values This is different from the similar problem in bulk semiconductors and from the case without laser radiation Thus, for a RQW, in the presence of laser radiation, under proper conditions, the weak EMW is increased The nature of this effect is due to our system being low-dimensional; i.e., the system has a size around the De Broglie wavelength of the carriers We can use this effect as one of the criteria for quantum-wire fabrication technology Quantum wires are one-dimensional semiconductor structures In quantum wires, the motion of electrons is restricted in two dimensions, so they can only flow freely in one dimension Hence, the energy spectrum of the electrons becomes discrete in two dimensions, and a system of electrons in a quantum wire is similar to a onedimensional electron gas The confinement of electrons in one-dimensional systems remarkably affects many of the physical properties of the material, including its optical properties, and those properties are very different from the properties of normal bulk semiconductors [1–5] Among the optical properties, the absorption of electromagnetic waves by matter is very interesting and has been developed in both theory and experiment The linear absorption of a weak electromagnetic wave (EMW) and the nonlinear absorption of a strong EMW in lowdimensional systems have been studied [6–15] Experimentally, measuring the absorption coefficient (ACF) of a strong EMW directly is very difficult, so in an experiment, one usually studies the influence of the strong EMW (laser radiation) on the electrons in a ∗ E-mail: nguyenquangbau54@gmail.com -572- Negative Absorption Coefficient of a Weak Electromagnetic Wave· · · – Nguyen Quang Bau et al II ABSORPTION COEFFICIENT OF A WEAK EMW IN THE PRESENCE OF A LASER RADIATION FIELD IN A RQW where n and (n, =1, 2, 3, ) denote the quantization of the energy spectrum in the x and the y directions, respectively, pz = (0, 0, pz ) is the wave vector of an electron along the wire’s z axis, and m∗ is the effective mass of an electron We consider a field of two EMWs: laser radiation as a strong EMW with an intensity E01 and a frequency Ω1 , and a weak EMW with an intensity E02 and a frequency Ω2 : We consider a wire of GaAs with a rectangular cross section (Lx ×Ly ) and a length Lz , embedded in GaAsAl The carriers (electron gas) are assumed to be confined by infinite potential barriers in the xOy plane and to be free along the wire’s axis (the Oz-axis), where O is the origin The EMW is assumed to be planar and monochromatic, to have a high frequency, and to propagate along the x direction In a RQW, the state and the electron energy spectrum have the forms [20] ⎧ ipz z ⎪ ⎪ ⎨ √2e ψn, ,pz = sin nπx Lx sin Lx Ly Lz ⎪ ⎪ ⎩ E(t) = E01 sin (Ω1 t + ϕ1 ) + E02 sin (Ω2 t) (3) The vector potential of that field of the two EMWs is ≤ x ≤ Lx ≤ y ≤ Ly πy Ly -573- A(t) = otherwise c c E01 cos(Ω1 t + ϕ1 ) + E02 cos(Ω2 t) (4) Ω1 Ω2 (1) εn, (pz ) = 2 pz 2m∗ H= 2 + n, ,pz + n, ,n , ,pz ,q The Hamiltonian of the electron-optical phonon system in the RQW in that field of two EMWs in the second quantization representation can be written as n + L2x Ly π 2m∗ pz − εn, , e Az (t) a+ n, c (2) ,pz an, ,pz ω q b+ q bq + q Cq In, ,n , (q⊥ )a+ n , ,pz +qz an, ,pz (bq + b+ −q ), 2 In, ,n , (q⊥ ) = × n+n − (−1) cos(qx Lx ) 2 (qx Lx ) − 2π (qx Lx ) (n2 + n ) + π (n2 − n ) 32π (qy Ly ) − (−1) (qy Ly ) − 2π (qy Ly ) ( + + 2) cos(qy Ly ) + π4 ( Because the motion of the electrons is confined in the e ω0 1 [7–9], |Cq | = 2ε χ∞ − χ0 , where V and ε0 are 0V q the normalization volume and the electronic constant, and χ0 and χ∞ are the static and the high-frequency dielectric constants, respectively In,l,n ,l (q⊥ ) is the electron form factor (which characterizes the confinement of electrons in a RQW) This form factor can be written as [20] where e is the elemental charge, c is the velocity of light, ωq ≈ ω0 is the frequency of an optical phonon, (n, , pz ) and (n , , pz + qz ) are the electron states before and after scattering, respectively, a+ n, ,pz (an, ,pz ) is the creation (annihilation) operator of an electron, bq+ (bq ) is the creation (annihilation) operator of an phonon for a state having wave vector q = (qx , qy , qz ), and qz = (0, 0, qz ) Cq is the electron - optical phonon interaction constant 32π (qx Lx nn ) (5) − )2 2 (6) xOy plane, we only consider the current density vector -574- Journal of the Korean Physical Society, Vol 64, No 4, February 2014 of electrons along the z direction in the RQW, which has the form jz (t) = e m∗ pz − n, ,pz e Az (t) nn, c ,pz (t) the electrons in a RQW, we use the general quantum equation for the statistical average value of the electron particle number operator (or electron distribution func[16]: tion) nn, ,pz (t) = a+ n, ,pz an, ,pz (7) t The ACF of a weak EMW caused by the confined electrons in the presence of laser radiation in the RQW takes the form [16] 8π α= √ jz (t)E02 sin ωt c χ∞ E02 t i |Cq | |In, ,n , a+ n, ,pz an, ,pz , H t (9) Using the Hamiltonian in Eq (5) and the commutative relations of the creation and the annihilation operators, we obtain the quantum kinetic equation for electrons in the RQW: (8) In order to establish the quantum kinetic equations for ∂nn, ,pz (t) =− ∂t ∂nn, ,pz (t) = ∂t +∞ (q⊥ )| Ju (a1z qz )Js (a1z qz )Jm (a2z qz )Jf (a2z qz ) u,s,m,f =−∞ n , ,q × exp {i {[(s − u)Ω1 + (m − f )Ω2 − iδ] t + (s − u)ϕ1 }} t × dt2 {[nn, ,pz (t2 )Nq − nn , ,pz +qz (t2 )(Nq + 1)] −∞ × exp + [nn, × exp − [nn , × exp − [nn , × exp i [εn , (pz + qz ) − εn, (pz ) − ωq − s Ω1 − m Ω2 + i δ] (t − t2 ) ,pz (t2 )(Nq i + 1) − nn , [εn , (pz + qz ) − εn, (pz ) + ωq − s Ω1 − m Ω2 + i δ] (t − t2 ) ,pz −qz (t2 )Nq i − nn, ,pz (t2 )(Nq + 1) − nn, ,pz (t2 )Nq ] [εn, (pz ) − εn , (pz − qz ) + ωq − s Ω1 − m Ω2 + i δ] (t − t2 ) , where a1z and a2z are the z-components of a1 = eE02 , m∗ Ω22 + 1)] [εn, (pz ) − εn , (pz − qz ) − ωq − s Ω1 − m Ω2 + i δ] (t − t2 ) ,pz −qz (t2 )(Nq i ,pz +qz (t2 )Nq ] eE01 m∗ Ω21 and a2 = respectively Nq is the balanced distribution function of phonons, ϕ1 is the phase difference between the two electromagnetic waves, and Jk (x) is the Bessel function In Eq (10), the quantum numbers n and characterize the quantum wire These indices are not present in the previously-published quantum kinetic equation for (10) the electrons in a similar problem, but in normal bulk semiconductors [17] The first - order tautology approximation method is used to solve this equation [16–19] The initial approximation of nn, ,pz (t) is chosen as ¯ n, ,pz , n0n, ,pz +qz (t2 ) = n ¯ n, ,pz +qz , n0n, ,pz (t2 ) = n nn, ,pz −qz (t2 ) = n ¯ n, ,pz −qz As a result, the expression for the unbalanced electron distribution function nn, ,pz (t) can be obtained: Negative Absorption Coefficient of a Weak Electromagnetic Wave· · · – Nguyen Quang Bau et al nn, ,pz (t) =n ¯ n, ,pz − |Cq | |In, ,n , -575- +∞ (q⊥ )| Js (a1z qz )Jk+s (a1z qz )Jm (a2z qz )Jr+m (a2z qz ) k,s,r,m=−∞ n , ,q n ¯ n , ,pz −qz Nq − n ¯ n, ,pz (Nq + 1) exp {−i {[kΩ1 + rΩ2 + iδ] t + kϕ1 }} × kΩ1 + rΩ2 + iδ εn, (pz ) − εn , (pz − qz ) − ωq − s Ω1 − m Ω2 + i δ n ¯ n, ,pz Nq − n ¯ n, ,pz Nq ¯ n , ,pz +qz (Nq + 1) n ¯ n , ,pz −qz (Nq + 1) − n − + εn, (pz ) − εn , (pz − qz ) + ωq − s Ω1 − m Ω2 + i δ εn , (pz + qz ) − εn, (pz ) − ωq − s Ω1 − m Ω2 + i δ n ¯ n, ,pz (Nq + 1) − n ¯ n , ,pz +qz Nq , (11) − εn , (pz + qz ) − εn, (pz ) + ωq − s Ω1 − m Ω2 + i δ × where n ¯ n, ,pz is the balanced distribution function of electrons, and the quantity δ is an infinitesimal and appears due to the assumption of an adiabatic interaction of the EMW e4 n0 ω0 √ α= 2πε0 c 2πχ∞ m∗ kB T m∗ Ω32 Z1 Z2 Substituting nn, ,pz (t) into the expression for jz (t), we calculate the ACF of the weak EMW by using Eq (8) The resulting ACF of a weak EMW in the presence of laser radiation in a RQW can be written as 1 − χ∞ χ0 +∞ cos α2 IIn, ,n , n, ,n , =1 (H0,1 − H0,−1 ) + (G0,1 − G0,−1 ) + (H−1,1 − H−1,−1 + H1,1 − H1,−1 ) 32 × (D0,1 − D0,−1 ) − 1 (G−1,1 − G−1,−1 + G1,1 − G1,−1 ) + (G−2,1 − G−2,−1 16 64 − +G2,1 − G2,−1 ) , where ξ − 2ks,mT Ds,m = e B K0 |ξs,m | 2kB T ξ − 2ks,mT Hs,m = a21 cos2 α1 e B ξ − 2ks,mT Gs,m = a41 cos4 α1 e 2 π 2m∗ n L2x eE01 , m∗ Ω21 Z1 = εn, = a1 = + B (12) ε − kn,T e B 2 4m∗ ξs,m n=1 B π 2m∗ x +∞ n L2x , Z2 = ξs,m = εn , − εn, + ω0 − s Ω1 − m +∞ ,n , = ε − kn,T + L2y e B ε e B , Nω0 = π2 +∞ − 2m∗ k T L2 e =1 Ω2 , with B y Nω0 , − (Nω0 + 1) − e − (Nω0 + 1) − e ε −ξs,m n , kB T Nω0 , ε −ξs,m n , kB T Nω0 , , ω0 e kB T −1 , s=-2, -1, 0, 1, 2, and m=-1, dqx dqy |In, ,n , (q⊥ )| = [A1 (1 −∞ −∞ (n2 +n ) 5(n2 +n ) , where A1 = L1x π3 + 2π(n −n )2 + 2πn2 n 3π 105 3π 105 B1 = Lx + 16πn2 , B2 = Ly + 16π IIn, −ξs,m n , kB T − kn,T |ξs,m | 2kB T K2 ε |ξs,m | 2kB T K1 π n2 +∞ − 2m∗ k T L2 e 1/ 2 4m∗ ξs,m , εn , = L2y − (Nω0 + 1) − e − δn,n ) + B1 δn,n ] [A2 (1 − δ A2 = α1 is the angle between the vector E01 and the positive direction of the Oz axis, and α2 is the angle between the Ly π + ( 2+ 2) 2π( − )2 , + ) + B2 δ 5( + 2π 2 ) , ], , vector E02 and the positive direction of the Oz axis Equation (12) is the expression for the ACF of a weak -576- Journal of the Korean Physical Society, Vol 64, No 4, February 2014 Fig (Color online) Dependence of α on T Fig (Color online) Dependence of α on Ω1 EMW in the presence of external laser radiation in a RQW As one can see, the ACF of a weak EMW is independent of E02 ; it depends only on E01 , Ω1 , Ω2 , T, Lx , and Ly This expression is different from that in the normal bulk semiconductors [17] From Eq (12), when we set E01 = 0, we will obtain the expression for the ACF of a weak EMW in the absence of laser radiation in a RQW In Section III, we will show clearly that under the influence of laser radiation, the ACF of a weak EMW in a RQW can have negative values III NUMERICAL RESULTS AND DISCUSSION In order to clarify the analytical expression for the ACF of a weak EMW in the presence of laser radiation in a RQW and to show clearly that the ACF can have negative values, in this section, we numerically calculated the ACF for the specific case of a GaAs/GaAsAl RQW The parameters used in the calculations are as follows [8,21]: χ∞ = 10.9, χ0 = 13.1, m = 0.066m0 , m0 being the mass of free electron, n0 = 1023 m−3 , ω0 = 36.25 meV , α1 = π3 , and α2 = π6 Figure describes the dependence of α on the temperature T for five different values of E01 , with Ω1 = 3×1013 Hz, Ω2 = 1013 Hz, Lx = 24 nm, and Ly = 26 nm Figure shows that when the temperature T of the system rises from 20 K to 400 K, the curves have a maximum and a minimum Figure describes the dependence of α on the frequency Ω1 of the laser radiation for three different values of T , with Ω2 = 1013 Hz, Lx = 24 nm, Ly = 26 nm, and E01 = 11 × 105 V /m This figure shows that the curves can have a maximum or no maximum in the investigated interval Figure describes the dependence of α on the frequency Ω2 of the weak EMW for three different values of T , with Ω1 = × 1013 Hz, Lx = 24 nm, Fig (Color online) Dependence of α on Ω2 Ly = 26 nm, and E01 = 15 × 106 V /m From Fig 3, we see that the curves have a maximum (peak) at Ω2 = ω0 and smaller maxima (peaks) at Ω2 = ω0 The frequencies Ω2 of the weak EMW at which ACF has maxima (peaks) not change as the temperature T is varied Figure shows the ACF as a function of the intensity E01 of the laser radiation for three different values of T , with Ω1 = × 1013 Hz, Ω2 = × 1013 Hz, Lx = 24 nm, and Ly = 26 nm From the figure, we see that the curves have a maximum in the investigated interval Figure describes the dependence of α on Lx for three different values of T , with Ω1 = × 1013 Hz, Ω2 = × 1013 Hz, Ly = 26 nm, and E01 = 15 × 106 V /m From this figure, we also see that the curves have many maxima (peaks) These figures show that under influence of laser radiation, under proper conditions, the ACF of a weak EMW in a RQW can have negative values This is different from the similar problem in bulk semiconductors and Negative Absorption Coefficient of a Weak Electromagnetic Wave· · · – Nguyen Quang Bau et al Fig (Color online) Dependence of α on E01 -577- The distance between consecutive energy levels has to be significantly greater than the thermal energy of the kB T , where ε1 and ε2 are two concarriers: ε2 − ε1 secutive energy levels of the electrons in a quantum wire, and kB is the Boltzmann constant If the electron gas is degenerate and has a Fermi energy level ζ, the following condition is needed: ε2 > ζ > ε1 In the opposite case, when ζ ε2 − ε1 , in principle, we can observe quantization effects due to size reduction, but the relative amplitudes are very small In addition, the distance between consecutive energy levels has to be significantly greater than the error in the energy: ε2 − ε1 τ , where τ is the average lifetime of the carrier in a quantum state with a set of determined quantum numbers If a quantum wire satisfies the above conditions [22, 23], when we change parameters such as E01 , Ω1 , Ω2 , T , Lx , and Ly in a proper way, a negative ACF of a weak EMW in the presence of laser radiation will be observed The negative ACF effect is an important characteristic that only low-dimensional systems have Thus, it can be used as one of the criteria to check the fabrication of low-dimensional systems in general and quantum wires in particular If a quantum wire is fabricated successfully, when we change the parameters in a proper way, the ACF of a weak EMW in a quantum wire in the presence of laser radiation will have a negative value; if this effect does not appear, the fabrication has failed IV CONCLUSIONS Fig (Color online) Dependence of α on Lx from the case without laser radiation The main scientific reason leading to the negative ACF of weak EMW in the presence of laser radiation in low-dimensional semiconductors in general and in quantum wires in particular is that the systems are low-dimensional Namely, when the size of the system is reduced down to around the De Broglie wavelength of carriers, the quantum laws markedly appear, leading to new properties of the system appearing, the so-called size effect One of those properties is that the ACF of a weak EMW in the presence of laser radiation can have a negative value; i.e., the weak EMW is increased This property does not appear entirely for bulk semiconductors (not low-dimensional systems); i.e., no increase in the weak EMW occurs in bulk semiconductors, even when the parameters are adjusted [17,18] However, if we want to observe quantum effects, the quantum wires must satisfy the following conditions: In this research, we investigated the negative absorption coefficient of a weak EMW caused by electrons confined in RQWs in the presence of laser radiation We obtained an analytical expression for the ACF of a weak EMW in the presence of laser radiation in a RQW for the case of electron-optical phonon scattering The expression shows that the ACF of a weak EMW is independent of E02 and depends only on E01 , Ω1 , Ω2 , T , Lx , and Ly This expression is different from that in normal bulk semiconductors From this expression, the ACF of a weak EMW in the absence of laser radiation in a RQW can be obtained by setting E01 = The ACF is numerically calculated for the specific case of a GaAs/GaAsAl RQW Computational results show that the dependence of the ACF on various physical factors of the system is complex Figure shows that a resonant peak appears for Ω2 = ω0 and that many smaller resonant peaks appear for Ω2 = ω0 Figure shows that the ACF of a weak EMW has many maxima (peaks) These results show that under the influence of laser radiation, the ACF of a weak EMW in a RQW can have negative values Thus, in the presence of a strong EMW, under proper conditions, a weak EMW is increased This is different from the similar problem in bulk semiconductors and from the case without laser radiation We can use this effect as one of the criteria for quantum-wire -578- Journal of the Korean Physical Society, Vol 64, No 4, February 2014 fabrication technology ACKNOWLEDGMENTS This research was done with financial support from Vietnam NAFOSTED (number 103.01-2011.18) REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] R Tsu and L Esaki, Appl Phys Lett 22, 562 (1973) J S Harris, Jr., J Mod Phys B 4, 1149 (1990) P Zhao, Phys Rev B 49, 13589 (1994) N Nishiguchi, Phys Rev B 52, 5279 (1995) R Dingle, Confined carrier quantum states in ultrathin semiconductor heterostructures (Festkorperprobleme XV, H J Queisser, Pergamon Vieweg, New York, 1975) N Q Bau, N V Nhan and T C Phong, J Korean Phys Soc 42, 647 (2003) N Q Bau and T C Phong, J Phys Soc Jpn 67, 3875 (1998) N Q Bau, N V Nhan and T C Phong, J 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(1983) [23] A Shik, Quantum Wells: Physics and Electronics of Two-dimensional Systems (World Sientific, Singapore, 1998)  ... that the ACF of a weak EMW has many maxima (peaks) These results show that under the in uence of laser radiation, the ACF of a weak EMW in a RQW can have negative values Thus, in the presence of. .. expression, the ACF of a weak EMW in the absence of laser radiation in a RQW can be obtained by setting E01 = The ACF is numerically calculated for the specific case of a GaAs/GaAsAl RQW Computational... radiation The main scientific reason leading to the negative ACF of weak EMW in the presence of laser radiation in low-dimensional semiconductors in general and in quantum wires in particular is that the