Proc Natl Conf Theor Phys 35 (2010), pp 124-134 PARAMETRIC TRANSFORMATION AND PARAMETRIC RESONANCE OF CONFINED ACOUSTIC PHONONS AND CONFINED OPTICAL PHONONS IN QUANTUM WELLS DO MANH HUNG, NGUYEN QUANG BAU Department of Physics, College of Natural Science, Vietnam National University in Hanoi Abstract The parametric transformation and parametric resonance of confined acoustic phonons and confined optical phonons in quantum wells in the presence of an external electromagnetic field are theoretically studied by using a set of quantum kinetic equations for phonons The analytic expression of the parametric transformation coefficient (K1 ) and the threshold amplitude (Eth ) of the field in quantum wells are obtained Unlike the case of unconfined phonons, the formula of K1 and contains a quantum number m characterizing confined phonons Their dependence on the temperature T of the system and the frequency Ω of the electromagnetic field is studied Numerical computations have been performed for GaAs/AlAsAl quantum wells The result have been compared with the case of unconfined phonons which show that confined phonons cause some unusual effects Keyword: Parametric transformation and parametric resonance, quantum well I INTRODUCTION It is well known that in the presence of an external electromagnetic field (EEF), an electron gas becomes non-stationary When the conditions of parametric resonance (PR) are satisfied, parametric interactions and transformations (PIT) 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., energy exchange processes between these excitations will occur [1, 2] The PIT of acoustic and optical phonons has been considered in bulk semiconductors [3 - 5] The physical picture can be described as follows: due to the electron - phonon interaction, propagation of an acoustic phonon with a frequency → ω− q is accompanied by a density wave with the same frequency When an EEF with frequency Ω is presented, a charge density waves (CDW) with a combination frequency → ω− q ± N Ω (N = 1, 2, ) will appear If among the CDW there exists a certain wave having a frequency which coincides, or approximately coincides, with the frequency of the optical → phonon, ν− q , optical phonons will appear These optical phonons cause a CDW with a ∼ → , a certain CDW causes the → → combination frequency of ν− q ± N Ω, and when ν− q ± N Ω = ω− q acoustic phonons mentioned above The PIT can speed up the damping process for one excitation and the amplification process for another excitation There have been a lot of works on the PIT for low dimensional semiconductors in the case of unconfined phonons [6 - 8] However, parametric transformation and parametric resonance of acoustic and optical phonons in quantum wells in the case of confined phonons have not been studied PARAMETRIC TRANSFORMATION AND PARAMETRIC RESONANCE OF 125 yet Therefore, in this paper, we have studied parametric transformation and parametric resonance of acoustic and optical phonons in quantum wells in the case of confined phonons The comparison of the result of confined phonons to one of unconfined phonons shows that confined phonons causes some unusual effects To this clarify, we estimate numerical values for a GaAs/AlAsAl quantum well, and we discuss the conditions under which the parametric resonance occurs II THE PARAMETRIC RESONANCE OF CONFINED ACOUSTIC PHONONS AND CONFINED OPTICAL PHONONS IN QUANTUM WELLS It is well known that the motion of an electron and phonon in a quantum wells is confined and that its 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 - confined acoustic (confined optical) phonon system in a quantum well in the second quantization representation can be written as: H (t) = εn − → n, k ⊥ + → m,− q → − − e→ → + A (t) a+ − k⊥− → a − n, k ⊥ n, k ⊥ c ⊥ ⊥ → m,− q ⊥ ⊥ + → m,− q ⊥ ⊥ → m,− q ⊥ ⊥ ⊥ + → → cm,− νm,− → q + q cm,− q ⊥ + → → bm,− ωm,− → q q bm,− q − → n,n , k ⊥ − → n,n , k ⊥ + m m C− → q In,n a + m m D− → q In,n a − → → n , k ⊥ +− q ⊥ − → → n , k ⊥ +− q ⊥ ⊥ ⊥ → an,− k ⊥ → an,− k ⊥ → + b+ bm,− → q m,−− q ⊥ → + c+ cm,− → q m,−− q ⊥ , ⊥ (1) ⊥ here, n, n’ are denotes the quantization of the energy spectrum in the z direction (n, → − → − − n’ = 1, 2, 3, ), n, k ⊥ and n , k ⊥ + → q ⊥ are electron states before and after → − − scattering, respectively; k ⊥ , → q ⊥ is the in-plane (x, y) wave vector of the electron (phonon); a+ − → a − → , n, k ⊥ n, k ⊥ + → → (b+ , bm,− , cm,− → → q ; cm,− q ) are the creation and the annihilam,− q q ⊥ ⊥ ⊥ ⊥ tion operators of the electron (acoustic phonon; optical phonon), e is the charge of the → − electron, c is the of light velocity, A (t) is the vector potential of an EEF, respectively → − → − → → A (t) = Ωc E cos (Ωt) and ωm,− νm,− is the energy of the confined acoustic (opq q ⊥ ⊥ − → − → tical) phonon, ωm q (νm, q ) is frequency confined acoustic (optical) phonon; m is the ⊥ → − quantum number characterizing confined phonons εn k ⊥ is the energy spectrum of the electron in quantum wells take the simple form [7]: εn → − k⊥ = −2 2→ k⊥ 2m∗ + π n2 2m∗ L2 (2) 126 DO MANH HUNG, NGUYEN QUANG BAU m m where, L is the well width, m∗ is the effective mass C→ D→ is the electron - confined − − q⊥ q⊥ acoustic (electron - confined optical) phonon interaction constant take the form [9] m C→ − q = ⊥ m D→ − q = ξ2 ρυa V mπ → − q 2⊥ + L → e2 νm,− q⊥ V ε0 (3) 1 − χ∞ χ0 (4) → − q 2⊥ + mπ L Here, V, ρ, υa , and ξ are the volume, the crystal density, the acoustic wave velocity, and the deformation potential constant, respectively ε0 is the electronic constant; χ0 and χ∞ are the static and high - frequency dielectric constant, respectively The electron form m (q ), is written as [10] factor In,n z ⊥ L (qz ) = L m In,n Nm (z) sin n πz L nπz dz L sin (5) mπz mπz + η (m + 1) sin (6) L L ( η (m) = if m even and η (m) = if m ext) In order to establish a set of quantum transport equations for confined acoustic and confined optical phonons in quantum wells, we use the general quantum distribution function [11] for the confined acoustic (confined → → optical) phonons, bm,− and cm,− : q q Nm (z) = η (m) cos t ⊥ i t ⊥ ∂ → b − ∂t m, q ⊥ = ∂ → c − ∂t m, q ⊥ = → bm,− q , H (t) ⊥ (7) t and i → cm,− q , H (t) ⊥ (8) t ∧ ∧ ∧ Where ψ t denotes a staticical average value at the moment t and ψ t = T r(W ψ)(W being the density matrix operator) Hamiltonian Eq (1), (7) and (8) and using the commutative relations of the creation and the annihilation operators, we obtain the quantum kinetic equation for confined acoustic (confined optical) phonon in quantum wells: ∂ → b − ∂t m, q ⊥ × fn t → + iωm,− q ⊥ → bm,− q t ⊥ =− ∞ i dt1 e Jν − → n,n , k ⊥ ν,µ=−∞ t → − → − − k⊥ −→ q ⊥ − fn k ⊥ m In,n − → εn k ⊥ −εn − → → k ⊥ −− q ⊥ λ Ω Jµ λ Ω (t1 −t)−iνΩt1 +iµΩt −∞ → × |Cm,− q ⊥| → bm,− q ⊥ t1 + b+ → m,−− q ⊥ m m + C− − → → q D− q t1 ⊥ ⊥ → cm,− q ⊥ t1 + c+ → m,−− q ⊥ t1 (9) PARAMETRIC TRANSFORMATION AND PARAMETRIC RESONANCE OF 127 and ∂ → c − ∂t m, q ⊥ t → + iνm,− q → cm,− q ⊥ t ⊥ 2 m In,n λ Ω Jν − → n,n , k ⊥ ν,µ=−∞ t → − → − − k⊥ −→ q ⊥ − fn k ⊥ × fn ∞ =− − → εn k ⊥ −εn i dt1 e − → → k ⊥ −− q λ Ω Jµ (t1 −t)−iνΩt1 +iµΩt ⊥ −∞ m m × D− − → → q C− q ⊥ → bm,− q t1 ⊥ ⊥ + b+ → m,−− q ⊥ → + |Dm,− q ⊥| t1 → cm,− q ⊥ t1 + c+ → m,−− q ⊥ → − → − k ⊥ is the distribution function of the electron in the state n, k ⊥ ; Jν Where, fn − → − → t1 (10) λ Ω q ⊥ One finds that the final result consists of an infinite is the Bessel function; λ = e EmΩ → → set of coupled equations for the Fourier transformations Bm,− q (ω) of q (ω) and Cm,− ⊥ ⊥ → bm,− q ⊥ and t → cm,− q t ⊥ , respectively For instance, the equations for → Bm,− q (ω) and ⊥ → Cm,− q (ω), can be written as: ⊥ ω− → ωm,− q⊥ → Bm,− q⊥ (ω) = m In,n ⊥ n,n + ∞ 2 m In,n m m C− → − → q q D− ⊥ ⊥ n,n m C− → q → νm,− q⊥ s=−∞ → ωm,− q⊥ − Π0 m, → q ⊥, ω → Bm,− q ⊥ (ω) → ω + ωm,− q⊥ − Πs m, → q ⊥, ω → Cm,− q ⊥ (ω − sΩ) → ω − sΩ + νm,− q⊥ (11) and ω− → νm,− q⊥ → Cm,− q⊥ (ω) = m In,n m D− → q ⊥ n,n + ∞ 2 m In,n m m C− → → q q D−− ⊥ ⊥ n,n → ωm,− q⊥ s=−∞ → νm,− q⊥ − Πs m, → q ⊥, ω → Bm,− q ⊥ (ω − sΩ) → ω − sΩ + ωm,− q⊥ From Eq (11), we have:  (12)  ω − ω m,− → q⊥ − m In,n m C− → q n,n = − Π0 m, → q ⊥, ω → Cm,− q ⊥ (ω) → ω + νm,− q⊥ ⊥ → −  Bm,− → → ωm,− q ⊥ Π0 m, q ⊥ , ω q ⊥ (ω) − Πs m, → q ⊥, ω → Cm,− q ⊥ (ω + sΩ) − → ω + sΩ + ν m, q ⊥ s=−∞ ∞ m m In,n C−− → q n,n ⊥ m D− → q ⊥ → νm,− q⊥ ω+ → ωm,− q⊥ (13) → From Eq (12), after some mathematical transformations, and for ω = ωm,− and s = N, q ⊥ we find the expression 128 DO MANH HUNG, NGUYEN QUANG BAU 2 → → Cm,− q ⊥ ωm,− q ⊥ + NΩ = Im n,n n,n − → m m ω → − C− D→ − → − m,− q ΠN (m, q ⊥ ,ω+N Ω)Bm,→ q q q  m D→ − q⊥ Im n,n n,n  ω+N Ω−ν → m,− q ⊥−  ⊥ ⊥ ⊥ νm,→ − q⊥ ωm,→ − q ⊥ − Π0 m,→ q ⊥ ,ωm,→ − q +N Ω+νm,→ − q ⊥ ⊥ − ωm,→ q ⊥  +N Ω  2ω → q⊥  m,− ⊥ We obtain equations dispersion describe interaction between confined acoustic phonon and confined optical phonon in quantum wells:   2 → − m m ω − ω m,−  → → In,n C− ωm,− → q⊥ − q ⊥ Π0 m, q ⊥ , ω q ⊥ n,n   → × (ω + N Ω)2 − ν m,− q⊥ − 2 m In,n m D− → q ⊥ n,n = m In,n n,n m C− → q ⊥ m D− → q ⊥ ∞ → −  → νm,− q ⊥ Π0 m, q ⊥ , ω + N Ω − − ΠN m, → q ⊥ , ω ΠN m, → q ⊥, ω + NΩ → → ωm,− q ⊥ νm,− q⊥ s=−∞ (14) In Eq (14), the first terms describe the interaction between phonons that belong to the same kind (acoustic - acoustic phonon or optical - optical phonon) while the second terms describe interaction between phonons that belong to different kinds (acoustic - optical → → phonon) We limit our calculation to the case of the first order resonance, ωm,− q ± νm,− q = Ω Because the solution to the general dispersion equation, Eq (14), is complex Here, m we assume that the electron - phonon interactions satisfy the condition In,n m D− → q ⊥ m approximation can be regarded as In,n m C− → q ⊥ m C− → q , ⊥ m D− → q zero, and the ⊥ solution Eq (14) by means of the disturbance, we obtain:   2 → − m m =0 ω − ω m,− → → In,n C− ωm,− → q⊥ − q ⊥ Π0 m, q ⊥ , ω q (15) ⊥ n,n   (ω + N Ω)2 − ν m,− → q⊥ − m In,n m D− → q ⊥ n,n → − =0 → νm,− q ⊥ Π0 m, q ⊥ , ω + N Ω (16) In these limitations, if we write the dispersion relations for confined acoustic and confined − − optical phonons as ωac (m, → q ⊥ ) = ωa + iτa and νoc (m, → q ⊥ ) = ω0 + iτ0 with conditions |ωa | |τa | and |ω0 | |τ0 |, and consider the case of N = 1, we obtain: m m → → → ωa ≈ ωm,− + In,n C− ReΓm,− ωm,− (17) → q q q⊥ q ⊥ ⊥ → ω0 ≈ νm,− + q ⊥ ⊥ n,n m In,n n,n m D− → q ⊥ → → ReΓm,− q ⊥ νm,− q⊥ (18) PARAMETRIC TRANSFORMATION AND PARAMETRIC RESONANCE OF τa = − n,n   → βm∗ εn,n ωm,− q⊥ m /2 f0  e−βεn − eβ exp − → − → − 2 q 2πβ q ⊥ ⊥ n,n   ∗ε 3/ − → βm ν n,n m, q ⊥ m f0  e−βεn − eβ exp − → − → − 2 q⊥ 2πβ q ⊥ m In,n 2 m C− → q ⊥ 129 − ωm,→ q ⊥ (19) τ0 = − m In,n m D− → q ⊥ − νm,→ q ⊥ (20) λ Λ= Ω m In,n m m − → − → C− → → q D− q ReΓm, q ⊥ ωm, q ⊥ ⊥ ⊥ (21) n,n and π2 2n π 2 n2 f0 m∗ exp −β − exp −β ⊥ ⊥ 2πβA 2m∗ L2 2m∗ L2 − 2→ π 2 n − n2 q 2⊥ → A= + + ωm,− q ⊥ 2m∗ L2 2m∗ ∗ 2 f0 m π n π 2 n2 → → = νm,− ReΓm,− exp −β − exp −β q q ⊥ ⊥ 2πβA1 2m∗ L2 2m∗ L2 − 2→ π 2 n − n2 q 2⊥ → + + νm,− A1 = q ⊥ 2m∗ L2 2m∗ − 2→ q⊥ → → εn,n ωm,− − ωm,− q ⊥ = εn − εn − q⊥ 2m∗ − 2→ q⊥ → → εn,n νm,− − νm,− q ⊥ = εn − εn − q⊥ 2m∗ We obtain the resonant acoustic phonon modes → ReΓm,− q ± ω± = ωa + → ωm,− q = (υa ± υ0 ) ∆ (q) − i (τa + τ0 ) ± [(υa ∓ υ0 ) ∆ (q) − i (τa − τ0 )]2 ± Λ2 (22) (23) (24) (25) (26) (27) (28) (±) In Eq (27) the signs (±) in the sub-script of ω± correspond to the signs in front of (±) the root and the sings (±) in the superscript of ω± correspond to the other sign pairs → → These signs depend on the resonance condition νm,− ± ωm,− = N Ω For instance, the q q ⊥ ⊥ (−) existence of a positive imaginary part of ω+ implies a parametric amplification of the confined acoustic phonons ωa and ω0 are the renormalization (by the electron - phonon interaction) frequency of the acoustic phonon and optical phonon; ∆ (q) = q − q0 , the distance to the intersection of dispersion curves, q0 being the wave number for which the resonance is satisfied;υa (υ0 ) is the group velocity of the acoustic (optical) phonon;τa τ0 , are electronic decrease constant of the acoustic and optical phonons, β = 1/kB T , kB is the Boltzmann constant, f0 is the density of electron In such case that λ 1, the maximal resonance, q = qx (qy = qz=0), we obtain: − F = Imω+ = Im ωa + −i (τa − τ0 ) + [− (τa − τ0 )]2 − Λ2 (29) 130 DO MANH HUNG, NGUYEN QUANG BAU The condition for the resonant acoustic phonon modes to have a positive imaginary part F > so > leads to |Λ|2 > 4τa τo , Using this (τa − τ0 )2 + |Λ|2 −i (τa − τ0 ) + condition and Eqs (19) - (21), yields the threshold amplitude for EEF:  1 → → → → ωm,− ImΓm,− νm,− q q q q 2m∗ Ω2  ImΓm,−  ⊥ ⊥ ⊥ ⊥ Eth =   − e → q⊥ → − → ReΓm,− ω q m, q ⊥ → ImΓm,− q ⊥ ⊥ × exp −β → ImΓm,− q ⊥ =− → ωm,− q π 2 n2 2m∗ L2 → νm,− q =− π 2 n2 2m∗ L2 ⊥ 2m∗ π βm∗ A exp − − β 2 → q⊥ → ωm,− q ⊥ exp β ⊥ × exp −β m∗ f0 2π q⊥ sh β 2 → ωm,− q ⊥ (31) βm∗ A21 2m∗ π exp − − β 2 → q⊥ m∗ f0 2π q⊥ → νm,− q ⊥ exp β (30) sh β → νm,− q ⊥ (32) Equation (30) means that parametric amplification of the confined acoustic phonons is achieved when the amplitude of the EEF is higher than some threshold amplitude and easy to come back to the case of unconfined phonons [7] when m → III THE PARAMETRIC TRANSFORMATION OF CONFINED ACOUSTIC PHONONS AND CONFINED OPTICAL PHONONS IN QUANTUM WELLS Parametric transformation of confined acoustic phonons and confined optical phonons in quantum well is determined by the formula: KN = → Cm,− q → νm,− q → Bm,− q → ωm,− q ⊥ ⊥ ⊥ ⊥ → → Cm,− νm,− are determined from Eq (13) Using the parametric resonant conditions q q ⊥ ⊥ → → ωm,− + N Ω ≈ νm,− q q , the parametric transformation coefficient is obtained: ⊥ ⊥ KN = n,n m In,n → − m Dm Π → C− → − → N m, q ⊥ , ωm,− q q q ⊥ ⊥ δ + iγ0 ⊥ (33) PARAMETRIC TRANSFORMATION AND PARAMETRIC RESONANCE OF where, the quantity δ is infinitesimal Consider the case of N = and note |δ| get: Γ K1 = 2γ0 with λ m m m − → − → Γ= In,n C− → → q D− q ReΓm, q ⊥ ωm, q ⊥ Ω ⊥ ⊥ 131 γ0 , we (34) (35) n,n γ0 = − m In,n m D− → q → ImΓm,− q ⊥ ⊥ n,n → νm,− q (36) ⊥ → → → → Where, ReΓm,− ωm,− and ImΓm,− νm,− are determined by the formula (22), q q q q ⊥ ⊥ ⊥ ⊥ and (32) γ0 is the decline electronic constant of the optical phonon Equation (34) means that parametric transformation coefficient of confined acoustic phonons and confined optical phonons in quantum well is achieved when the amplitude of the EEF is higher When m → 0, easy to come back to the case of unconfined phonons, is determined by the formula: Γ∗ (37) K= 2γ ∗ Where, λ − − → → D → q⊥ C → q ⊥ ReΓ− ω− (38) Γ∗ = q q ⊥ ⊥ Ω n,n γ∗ = − → ReΓ− q ⊥ → ImΓ− q ⊥ → ω− q ⊥ → ω− q ⊥ = − D → q⊥ n,n ∗ m f0 2πβA2 2 π2 2n 2m∗ L2 βm∗ A22 m∗ /2 f0 exp − =− √ − 2→ q2 2πβ q⊥ ⊥ A2 = π2 n2 − n2 2m∗ L2 ⊥ ⊥ exp −β + − exp −β exp − − 2→ q 2⊥ 2m∗ (39) → ν− q → ImΓ− q β π A22 2m∗ L2 π 2 n2 2m∗ L2 (40) → − exp β ω− q ⊥ (41) → + ω− q (42) ⊥ IV NUMERICAL RESULTS AND DISCUSSIONS IV.1 In the case parametric resonance In order to clarify the mechanism for parametric resonance of acoustic and optical phonons in the case of confined phonons, we consider a AlAs/GaAsAl quantum well The parameters used in this calculation are as follows [12]: χ∞ = 10.9, χ0 = 12.9, L = 100A0 , m = 0.067m0 , (m0 being the mass of free electron), ν0 = 36.25mev, Ω = × 1014 Hz, ξ = 13.5ev ρ = 5.32g.cm−3 υs = 5370m.s−1 , E0 = 106 v/m, e = 1.60219×10−19 C, = 1.05459 × 10−34 J.s Figure show the dependence of the threshold amplitude Eth on the magnitude of − wave vector → q at temperature T = 72K As shown in, the threshold amplitude reaches 132 DO MANH HUNG, NGUYEN QUANG BAU Fig The dependence of Eth v.cm−1 on the q m−1 with T = 72K Fig The dependence of Eth v.cm−1 on the T with q = 2.8 × 108 m−1 the maximum value when q = 1.2 × 108 m−1 Other cases of unconfined phonon, the curve has a sub-maximal when q = 2.5 × 108 m−1 The cause of this difference is due to the wave vector of phonon quantum chemical confined phonon Because the wave vector of phonon is quantized of the energy in the confined phonon direction Figure (solid line - confined and dot line - unconfined) show the dependence of the threshold amplitude on the temperature T for both the confined phonon and unconfined phonon From the graph shows, at the same temperature, the confined phonons makes the threshold amplitude increases IV.2 In the case parametric transformation In order to clarify the mechanism for the parametric transformation of acoustic and optical phonons in the case of confined phonons, in this section, we will consider quantum wells The parameters used in this calculation are as follow [12]:χ∞ = 10.9, χ0 = 12.9, L = 100A0 , m = 0.067m0 , (m0 being the mass of free electron), ν0 = 36.25mev, Ω = × 1014 Hz, ξ = 13.5ev ρ = 5.32g.cm−3 υs = 5370m.s−1 , E0 = 106 v/m, e = 1.60219 × 10−19 C, = 1.05459 × 10−34 J.s Figure 3, and Figure shows the influence of confined phonon on the changing phenomenon of the parameter between the acoustic phonon and optical phonon Concretely, PARAMETRIC TRANSFORMATION AND PARAMETRIC RESONANCE OF 133 the confinement of phonon makes an increase of the coefficient-changing parameter between the acoustic phonon and optical phonon in quantum well In a same range of temperature T, (in the case confined phonon) the coefficient oscillation around the unit, with the case unconfined phonon the threshold amplitude is very small Because, when phonon is confined, the energy bands of phonon are divided into mini-bands like electrons in potential well Therefore, the probability of occurrence greater resonance conditions In other words, the chance of changing acoustic phonon into optical phonon and vice versa becomes bigger In short, the coefficient of parametric transformation between acoustic phonon and optical when phonon is confined is more stronger than unconfined phonon From all figures above, we can see clearly the effect of confined phonons on the parameter transformation coefficient Namely, the confined phonons increase the phonon transformation coefficient in quantum wells Fig The dependence of K1 on the T (In the case confined phonon) V CONCLUSIONS In this paper, we analytically investigated the possibility of parametric transformation and parametric resonance of confined acoustic phonons and confined optical phonons We obtained a general dispersion equation for parametric amplification and transformation Fig The dependence of K1 on the T (In the case unconfined phonon) 134 DO MANH HUNG, NGUYEN QUANG BAU of phonons However, an analytical solution to the equation can only be obtained within some limitations Using these limitations for simplicity, we obtained dispersions of the resonant confined acoustic phonon and confined optical phonon modes and the threshold amplitude of the field for acoustic phonon parametric amplification and optical phonon parametric amplification Similarly to the mechanism pointed out by several authors for bulk semiconductors, parametric amplification for acoustic phonons in a quantum well can occur under the condition that the amplitude of the external electromagnetic field is higher than some threshold amplitude We have numerically calculated and graphed the threshold amplitude and the parametric coefficient for AlAs/GaAsAl quantum well clearly show the predicted mechanism Parametric amplification for acoustic phonons and optical phonon and the threshold amplitude depend on the physical parameters of the system and are sensitive to the temperature Calculated result shows that the confinement of phonon makes an increase of the coefficient changing parameter between acoustic and optical phonon Based on this idea, we can put forward a capability about changing the functions of low - semiconductors It plays important sense in application especially in material science, electronics In addition, we can manufacture super mini (based nanostructures) and multi - functions (based on devices properties which could be controlled from outside ACKNOWLEDGMENT This research is completed with financial support from the Vietnam - NAFOSTED (N0 103.01.18.09) REFERENCES [1] P Silin, Parametric Action of the High - Power Radiation on Plasma, 1973 National Press on Physics Theory, Literature, Moscow [2] G M Shmelev, N Q Bau, Physical phenomena in semiconductors, 1981 Kishinev [3] E M Epstein, Sov Phys Semicond 10 (1976) 1164 [4] M V Vyazovskii, V A Yakovlev, Sov Phys Semicond 11 (1977) 809 [5] S M Komirenko, K W Kim, A A Dimidenko, V A Kochelap, M A Stroscico, Phys Rev B 62 (2000) 7459; J Appl Phys 90 (2001) 3934 [6] K Ploog, G H Doller, Adv Phys 32 (1983) 285 [7] T C Phong, N Q Bau, J Korea Phys Soc 42 (2003) 647 [8] T C Phong, L V Tung, N Q Bau, J Korea Phys Soc 53 (2008) 1971 [9] N Q Bau, D M Hung, N B Ngoc, J Korea Phys Soc 54 (2009) 765 [10] N Q Bau, D M Hung, L T Hung, Journal of the USA Physics Progress In Electromagnetics Research Letters (2010) 175 [11] N Q Bau, D M Hung, Journal of the USA Physics Progress In Electromagnetics Research B (2010) 39 [12] Y He, Z Yin, M S Zhang, T Lu, Y Zheng, Mat Sci Eng B 75 (2000) 130 Received 10-10-2010 [...]... MANH HUNG, NGUYEN QUANG BAU of phonons However, an analytical solution to the equation can only be obtained within some limitations Using these limitations for simplicity, we obtained dispersions of the resonant confined acoustic phonon and confined optical phonon modes and the threshold amplitude of the field for acoustic phonon parametric amplification and optical phonon parametric amplification Similarly... mechanism Parametric amplification for acoustic phonons and optical phonon and the threshold amplitude depend on the physical parameters of the system and are sensitive to the temperature Calculated result shows that the confinement of phonon makes an increase of the coefficient changing parameter between acoustic and optical phonon Based on this idea, we can put forward a capability about changing the... Similarly to the mechanism pointed out by several authors for bulk semiconductors, parametric amplification for acoustic phonons in a quantum well can occur under the condition that the amplitude of the external electromagnetic field is higher than some threshold amplitude We have numerically calculated and graphed the threshold amplitude and the parametric coefficient for AlAs/GaAsAl quantum well clearly... functions of low - semiconductors It plays important sense in application especially in material science, electronics In addition, we can manufacture super mini (based nanostructures) and multi - functions (based on devices properties which could be controlled from outside ACKNOWLEDGMENT This research is completed with financial support from the Vietnam - NAFOSTED (N0 103.01.18.09) REFERENCES [1] P Silin, Parametric. .. [9] N Q Bau, D M Hung, N B Ngoc, J Korea Phys Soc 54 (2009) 765 [10] N Q Bau, D M Hung, L T Hung, Journal of the USA Physics Progress In Electromagnetics Research Letters (2010) 175 [11] N Q Bau, D M Hung, Journal of the USA Physics Progress In Electromagnetics Research B (2010) 39 [12] Y He, Z Yin, M S Zhang, T Lu, Y Zheng, Mat Sci Eng B 75 (2000) 130 Received 10-10-2010 ... financial support from the Vietnam - NAFOSTED (N0 103.01.18.09) REFERENCES [1] P Silin, Parametric Action of the High - Power Radiation on Plasma, 1973 National Press on Physics Theory, Literature, Moscow [2] G M Shmelev, N Q Bau, Physical phenomena in semiconductors, 1981 Kishinev [3] E M Epstein, Sov Phys Semicond 10 (1976) 1164 [4] M V Vyazovskii, V A Yakovlev, Sov Phys Semicond 11 (1977) 809 [5] ... THE PARAMETRIC TRANSFORMATION OF CONFINED ACOUSTIC PHONONS AND CONFINED OPTICAL PHONONS IN QUANTUM WELLS Parametric transformation of confined acoustic phonons and confined optical phonons in quantum. .. for confined acoustic and confined optical phonons in quantum wells, we use the general quantum distribution function [11] for the confined acoustic (confined → → optical) phonons, bm,− and cm,−... possibility of parametric transformation and parametric resonance of confined acoustic phonons and confined optical phonons We obtained a general dispersion equation for parametric amplification and transformation