The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector

The Hamiltonian of electrons interacting with interface phonons in three-barrier resonant tunneling structure is established using the first principles within the models of effective mass and polarization continuum. Using the Green's functions method, the temperature shifts and decay rates of o...

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Date:	2014
Main Authors:	Tkach, M.V., Seti, Ju.O., Grynyshyn, Y.B., Voitsekhivska, O.M.
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Language:	English
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Cite this:	The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector / M.V. Tkach, Ju.O. Seti, Y.B. Grynyshyn, O.M. Voitsekhivska // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23704:1-10. — Бібліогр.: 17 назв. — англ.

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spelling	irk-123456789-1534852019-06-15T01:26:17Z The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector Tkach, M.V. Seti, Ju.O. Grynyshyn, Y.B. Voitsekhivska, O.M. The Hamiltonian of electrons interacting with interface phonons in three-barrier resonant tunneling structure is established using the first principles within the models of effective mass and polarization continuum. Using the Green's functions method, the temperature shifts and decay rates of operating electron states are calculated depending on geometric design of three-barrier nano-structure GaAs/AlxGa₁₋xAs which is an active region of quantum cascade detector. It is established that independently of the temperature, the energy of quantum transition during the process of electromagnetic field absorption is a nonlinear weakly varying function of the position of the inner barrier with respect to the outer barriers of the structure. З перших принципiв у моделi ефективних мас та поляризацiйного континууму встановлено гамiльтонiан системи електронiв взаємодiючих з iнтерфейсними фононами у трибар’єрнiй резонансно-тунельнiй структурi. Методом функцiй Грiна розраховано температурнi змiщення й загасання найнижчих (робочих) електронних станiв у залежностi вiд геометричної конфiгурацiї наносистеми GaAs/ AlxGa₁₋xAs як активної зони квантового каскадного детектора. Встановлено, що незалежно вiд температури системи енергiя квантового переходу в процесах поглинання електромагнiтного поля є нелiнiйною слабозмiнною функцiєю вiд положення внутрiшнього вiдносно зовнiшнiх бар’єра наносистеми. 2014 Article The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector / M.V. Tkach, Ju.O. Seti, Y.B. Grynyshyn, O.M. Voitsekhivska // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23704:1-10. — Бібліогр.: 17 назв. — англ. 1607-324X arXiv:1407.2431 DOI:10.5488/CMP.17.23704 PACS: 78.67.De, 63.20.Kr, 72.10.Di http://dspace.nbuv.gov.ua/handle/123456789/153485 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The Hamiltonian of electrons interacting with interface phonons in three-barrier resonant tunneling structure is established using the first principles within the models of effective mass and polarization continuum. Using the Green's functions method, the temperature shifts and decay rates of operating electron states are calculated depending on geometric design of three-barrier nano-structure GaAs/AlxGa₁₋xAs which is an active region of quantum cascade detector. It is established that independently of the temperature, the energy of quantum transition during the process of electromagnetic field absorption is a nonlinear weakly varying function of the position of the inner barrier with respect to the outer barriers of the structure.
format	Article
author	Tkach, M.V. Seti, Ju.O. Grynyshyn, Y.B. Voitsekhivska, O.M.
spellingShingle	Tkach, M.V. Seti, Ju.O. Grynyshyn, Y.B. Voitsekhivska, O.M. The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector Condensed Matter Physics
author_facet	Tkach, M.V. Seti, Ju.O. Grynyshyn, Y.B. Voitsekhivska, O.M.
author_sort	Tkach, M.V.
title	The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector
title_short	The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector
title_full	The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector
title_fullStr	The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector
title_full_unstemmed	The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector
title_sort	effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/153485
citation_txt	The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector / M.V. Tkach, Ju.O. Seti, Y.B. Grynyshyn, O.M. Voitsekhivska // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23704:1-10. — Бібліогр.: 17 назв. — англ.
series	Condensed Matter Physics
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fulltext	Condensed Matter Physics, 2014, Vol. 17, No 2, 23704: 1–10 DOI: 10.5488/CMP.17.23704 http://www.icmp.lviv.ua/journal The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector M.V. Tkach∗, Ju.O. Seti, Y.B. Grynyshyn, O.M. Voitsekhivska Chernivtsi National University, 2 Kotsubynsky St., 58012 Chernivtsi, Ukraine Received April 16, 2014, in final form May 17, 2014 The Hamiltonian of electrons interacting with interface phonons in three-barrier resonant tunneling structure is established using the first principles within the models of effective mass and polarization continuum. Using the Green’s functions method, the temperature shifts and decay rates of operating electron states are calculated depending on geometric design of three-barrier nano-structure GaAs/AlxGa1−xAs which is an active region ofquantum cascade detector. It is established that independently of the temperature, the energy of quantum transition during the process of electromagnetic field absorption is a nonlinear weakly varying function of the position of the inner barrier with respect to the outer barriers of the structure. Key words: resonant tunneling nano-structure, interface phonons, quantum cascade detector PACS: 78.67.De, 63.20.Kr, 72.10.Di 1. Introduction Quantum cascade detectors (QCD) have been investigated for over a decade but a growing attention to these devices is still observed [1–4]. From practical point of view, such an interest is caused by the unique characteristics of QCD. Occupying the whole infrared and terahertz frequency range of electromagnetic waves, these devices can operate in a wide range of temperatures (from cryogenic to room ones) and so on. From theoretical point of view, the interest is caused by the fact that the basis of QCD functional ele- ments are the open quasi-two-dimensional resonant tunneling structures (RTS). The physical properties of electronic transport in these structures are not still clear enough. The theory of spectral parameters and dynamic conductivity of electrons in open RTS is well devel- oped [5–7] without taking into account the electron-phonon interaction. The obtained results well cor- relate with the experimental data [3]. However, from physical considerations it is clear that the effect of phonons can be neglected only at cryogenic temperatures when the average occupation numbers of phonon states are small and electron-phonon binding is weak. Studying the nano-structures with strong binding or at high temperatures (modern QCDs operate at room temperatures [3]), one should consider the electron-phonon interaction. The theory of electron-phonon interaction in spherical, cylindrical and plane closed single [8, 9] and multi-shell nano-heterosystems [10–15] has been developed for a long time using the models of effective mass and dielectric continuum. It was established that, contrary to the massive three-dimensional sys- tems, the so-called interface phonons (I-phonons) exist in low-dimensional nano-heterosystems (besides the confined polarization phonons). The effect of I-phonons increases when the thickness of nano-layers decreases. The effect of phonons on the transport properties of electronic current through RTS was mainly in- vestigated for the two- and three-barrier nano-structures [5–7]. Studying the probabilities of quantum ∗E-mail: ktf@chnu.edu.ua ©M.V. Tkach, Ju.O. Seti, Y.B. Grynyshyn, O.M. Voitsekhivska, 2014 23704-1 http://dx.doi.org/10.5488/CMP.17.23704 http://www.icmp.lviv.ua/journal M.V. Tkach et al. transitions using the Fermi golden rule, it is enough to use the Hamiltonian of electron-phonon interac- tion in the representation of the second quantization over the phonon variables only, obtained by Mori and Ando [8] for a double heterostructure. In this study, we investigate the electron spectrum renormalized due to I-phonons in three-barrier RTS which is an active region of QCD. The Hamiltonian of electron-I-phonon system is obtained in the representation of occupation numbers over all variables. It is further used in the method of temperature Green’s functions in order to study the shifts and decay rates of two lower electron states - the operating states of QCD active region. This makes it possible to study in detail the effect of various mechanisms of electron-I-phonon interaction on the parameters of two operating electron states depending on the design of three-barrier RTS at cryogenic and room temperatures. 2. Hamiltonian and Fourier image of Green’s function of the system of electrons interacting with interface polarization phonons in a three- barrier nano-structure The theory of spectral parameters (resonance energies and decay rates) and dynamic conductivity of electrons in three-barrier RTSwithout taking into account the electron-phonon interactionwas developed in detail in [5]. It was established that when the widths of nano-structure outer barriers were bigger than 3–4 nm, the resonance energies in open and closed models were almost the same and the resonance widths were two-three orders smaller than the energies. Considering the widths of outer barriers of a three-barrier RTS (the active bands of experimental QCD) as rather big (3–6 nm) [6], we develop the theory of electron-I-phonon interaction using the model of closed three-barrier RTS (figure 1) with fixed effective masses m (z) = {mw (II, IV); mb (I, III, V)} and rectangular potential energy profile neglecting the small decay rate U (z) = { 0, 0 É z É a1 (II), a1 +b É z É a1 +b +a2 (IV), U , −∞É z É 0 (I), a1 É z É a1 +b (III), a1 +b +a2 É z É∞ (V). (1) Z a 2 a 1 b 2 bb 1 Z 4 Z 3 Z 2 Z 1 Z 0 =0 VIVII IIII U U(z) Figure 1. Potential energy profile of closed three- barrier RTS (solid line). The boundaries of outer bar- riers with the widths (b1; b2) for the correspondingopen system (dashed lines). Expressing the electron wave function in the form ΨE~k (~r ) = 1p S ei~k~ρΨE (z) , (2) where ~k and ~ρ are quasi-momentum and radius- vector of electron in the plane xO y and S is the square of the main region in this plane. For z-th component of this function, we obtain the Schrodinger equation{ −ħ2 2 d dz 1 m (z) d dz +U (z) } ΨE (z) = EΨE (z) . (3) The complete electron energy in the region un- der the barrier (E ÉU ) consists of two terms En~k = En + ħ2k2 2m∗ n . (4) Here, ħ2k2/(2m∗ n) is the kinetic energy of elec- tron moving in the plane perpendicular to Oz axis. It is determined, as in [16], by the effective mass correlated over the RTS 1 m∗ n = ∞∫ −∞ \|Ψn (z) \|2 dz m (z) , (5) 23704-2 The effect of interface phonons on operating electron states. . . whereΨn (z) wave functions are the solutions of one-dimensional stationary equation (3) Ψn (z) =  ∑ j=2,4 Ψ j n (z) = ∑ j=2,4 ( A j n coskn z +B j n sinkn z ) ,∑ j=1,3,5 Ψ j n (z) = ∑ j=1,3,5 ( A j neχn z +B j ne−χn z ) . (6) Here, kn =ħ−1 √ 2mwEn , χn =ħ−1 √ 2mb \|U −En \| = √ 2mbUħ−2 −k2 nmb/mw . (7) The discrete energy spectrum En and coefficients A j n , B j n are fixed by fitting conditions Ψ j n(z) ∣∣ z=z j =Ψ j+1,n(z) ∣∣ z=z j , 1 m j ∂Ψ j n ∂z ∣∣∣ z=z j = 1 m j+1 ∂Ψ j+1,n ∂z ∣∣∣ z=z j , j = 1,2,3,4 (8) together with the condition that the wave function vanishes at z →±∞ (B1n = A5n = 0) and the normality condition ∞∫ −∞ Ψ∗ n(z)Ψn′ (z)dz = δnn′ . (9) In the region above the barrier (E ÊU ), the energy of electron longitudinal movement is continuous. Thus, introducing the longitudinal quasi-momentum kz , it is written as Ekz = ħ2k2 z /(2mw). Finally, the complete energy has the form: Ekz~k = Ekz + ħ2k2 2mw . (10) Now, the solution of equation (3) for the wave functionΨkz (z) becomes the expression (6) with kn → kz , χn → iχ . The fitting conditions (8) are valid at n → kz , and the normality condition, similarly to thede Broglie wave, is written as follows: L/2∫ −L/2 ∣∣Ψkz (z) ∣∣2 dz = 1, (11) where Ψkz (z) function satisfies the periodic condition Ψkz (−L/2) =Ψkz (L/2) at a big span of the main region having the length L. All coefficients (A j kz , B j kz ) are defined from these conditions and the wavefunctionΨkz (z) is obtained. Introducing the generalized index ñ = { n, EÉU kz , EÊU }, for the compact analytics, we perform a transition to the representation of the second quantization using the quantized wave function _ Ψ (~r ) =∑ ñ~k Ψñ~k (~ρ, z) _ a ñ~k= ∑ ~k [∑ n Ψn(~ρ, z)an~k + ∑ kz Ψkz (~ρ, z)akz~k ] (12) and obtain the Hamiltonian of uncoupling electrons in the representation of their occupation numbers _ He= ∑ ñ,~k Eñ~k a+ ñ~k añ~k (13) with the electron spectrum Eñ~k , creation (a+ ñ~k ) and annihilation (añ~k ) Fermi operators of electron states,satisfying the anti-commutative relationships. It is well known [8–10] that in the dielectric continuum model, the phonon spectra and potential of polarization field Φ(~r ) are obtained from the following equation ε j (ω)∇2Φ(~r ) = 0, (14) 23704-3 M.V. Tkach et al. where ε j (ω) is dielectric constant of j -th layer of a nano-structure composed of two materials ε j (ω) = ε j∞ ω2 −ω2 L j ω2 −ω2 T j . (15) Here, ε j∞ is high-frequency dielectric constant, ωL j , ωT j are the frequencies of longitudinal (L) andtransversal (T ) phonons of the bulk material creating j -th layer of nano-structure. The spectrum and potential of polarization field of interface phonons is obtained, according to [9], from the equation (14) if ∇2Φ(~r ) = 0. (16) Thus, the solution of this equation is the potential Φ(~r ) =∑ j ,~q C (q)ϕ j (q, z)ei~q~ρ , (17) where ~q , ~ρ are two-dimensional vectors and functions ϕ j (q, z) =α j e−qz +β j eqz , j = 1, . . . ,5 (18) satisfy the system of equations obtained from the fitting conditions for the intensity and induction of polarization field  ϕ j (q, z j ) =ϕ j+1(q, z j ), ε j (ω) ∂ϕ j (q, z) ∂z ∣∣∣ z=z j = ε j+1(ω) ∂ϕ j+1(q, z) ∂z ∣∣∣ z=z j , j = 1,2,3,4 (19) and considering that at z →±∞ ϕ1(q, z) ∣∣ z→−∞ =ϕ5(q, z) ∣∣ z→∞ = 0. (20) Within the transfer-matrix method [11], the coefficients α j , β j and the potential of polarization field Φ(~r ) are obtained from the system of equations (19), (20). The condition of nontrivial solution determines the dispersion equation 5∏ j=1  [ 1+ ε1(Ω) ε0(Ω) ] [ 1− ε1(Ω) ε0(Ω) ] e−2qz j−1[ 1− ε1(Ω) ε0(Ω) ] e2qz j−1 [ 1+ ε1(Ω) ε0(Ω) ] = ( 1 0 0 1 ) . (21) Here, ε0(Ω) and ε1(Ω) are the dielectric constants in the wells and barriers, respectively. Its solutions Ωλ~q = ħωλ~q define the energy spectrum of interface phonons. For the nondegeneratedcase, the number of phonon modes (λ) is equal to the twice number of all interfaces between nano- structure layers. Quantizing the polarization field using the known quantum mechanics method [9], we obtain the Hamiltonian of interface phonons _ H I= ∑ λ,~q Ωλ~q ( b+ λ~q bλ~q + 1 2 ) , λ= 1, . . . ,8 (22) and the Hamiltonian of electron-I-phonon interaction _ He-I=−eΦ(~ρ, z) =− ∑ λ,~q , j eCλ(q)ϕ j (λ, q, z)ei~q~ρ ( bλ~q +b+ λ,−~q ) (23) in coordinate representation over the electron variables (~ρ, z) and in the representation of occupation numbers over the phonon variables with operators b+ λ~q , bλ~q , satisfying commutative relationships andwith the known coefficients Cλ(q) and functions ϕ j (λ, q, z). 23704-4 The effect of interface phonons on operating electron states. . . Performing the transition to the representation of electron occupation numbers in (23) using the quantized wave function (12), we obtain the Hamiltonian of electron-I-phonon interaction in the rep- resentation of the second quantization over all variables of the system _ He-I= ∑ ñ1,ñ,~k λ,~q Fñ1ñ(λ,~q)a+ ñ1~k+~q a ñ~k ( bλ~q +b+ λ,−~q ) , (24) where binding functions Fñ1ñ(λ,~q) =− √ 8πΩabRy L2qN (λ, q) 5∑ j=1 z j∫ z j−1 Ψ∗ ñ1 j (z)Ψñ j (z) [ α j (Ωλq )e−qz +β j (Ωλq )eqz] dz (25) contain the normality coefficient N (λ, q) = 5∑ j=1 ω(q=0) λ=1 ∂ε j (ω) ∂ω ∣∣∣ ω=ωλ(q) [ β2 j (Ωλ~q ) ( e2qz j −e2qz j−1 )−α2 j ( Ωλ~q )( e−2qz j −e−2qz j−1 )] . (26) Here, Ry= e2/2ab, Ω=ħω(q=0) λ=1 , ab is Bohr radius. Integral in (25) is analytically calculated but we do notpresent it due to its sophisticated form. We should note that at q → 0, the binding function Fñ1ñ(λ,~q) ∼ q−1/2 and, thus, a further integration in MO is not divergent. The obtained Hamiltonian of electron-I-phonon system in three-barrier RTS H = He+HI+He-I (27) allows us to calculate the Fourier-image of electron Green’s function in the quasi-stationary part of the spectrum according to the rules of Feynman-Pines diagram technique [9], at the finite temperature, when Dyson equation is valid Gn(~k,ħω) = [ ħω−En~k −Mn(ħω,~k) ]−1 (28) with mass operator (MO) Mn(ħω,~k) calculated (due to the weak electron-I-phonon binding) in one-pho- non approximation (η→±0) Mn(ħω,~k) = ∑ ñ1,λ,~q F∗ nñ1 (λ,~q)Fñ1n(λ,~q) [ 1+νλ~q ħω−Eñ1 (~k +~q)−Ωλ~q + iη + νλ~q ħω−Eñ1 (~k +~q)+Ωλ~q + iη ] , (29) where νλ~q = [ eΩλ~q /kT −1 ]−1 is the average number of I-phonons occupation numbers. Further, using this MO, we study the contributions of different mechanisms of electron-I-phonon in- teraction into renormalized spectral parameters (energy shifts (∆n) and decay rates (Γn)) of n-th electronstate. In experiments [1–3], the electrons move perpendicularly to the planes of three-barrier RTS, thus in MO (25) we put~k = 0 and neglect the frequency dependence of MO in the vicinity of En energies takinginto account aweak electron phonon binding (further proven by numeric calculations). To distinguish the role of different mechanisms of electron-I-phonon interaction, one should extract the real and imaginary part in MO Mn(~k = 0, ħω= En) = ReMn(~k = 0,ħω= En)+ iImMn(~k = 0,ħω= En) =∆n − iΓn/2, (30) as well as the terms describing the partial contributions of I-phonons due to interaction with electrons from different states ∆n =∆nn +∆nd+∆nc , Γn = Γnn +Γnd+Γnc , (31) where ∆nn is the partial shift of the n-th state due to the intra-level interaction within I-phonons ∆nn = ( L 2π )2 ∑ λ,± P ∫ ∫ d2~q(En −En~q ∓Ωλ~q )−1 ∣∣∣F (λ,~q) nn ∣∣∣2 ( νλ~q + 1 2 ± 1 2 ) , (32) 23704-5 M.V. Tkach et al. Γnn is the decay rate of the n-th state due to the intra-level interaction within I-phonons Γnn = L2 2π ∑ λ,± ∫ ∫ d2~q [ δ(En −En~q ∓Ωλ~q ) ]∣∣∣F (λ,~q) nn ∣∣∣2 ( νλ~q + 1 2 ± 1 2 ) , (33) ∆ n ( cd) and Γ n ( cd) are the partial shifts and decay rates of the n-th state due to the interaction within I- phonons with all (except the n-th) states of quasi-discrete (d) spectrum (∆nd = ∑ n1,n∆nn1 , Γnd =∑ n1,n Γnn1 ) and with the states of continuum (c) (∆nc =∑ kz ∆nkz , Γnc =∑ kz Γnkz ), ∆ n ( cd)− iΓ n ( cd) = ( L 2π )2 ∑ ( kz n1,n ) ∑ λ,± ∫ ∫ d2~qF∗ n (n1 kz )(λ,~q)F n (n1 kz )(λ,~q) ( νλ~q + 1 2 ± 1 2 ) × [ P ( En −E n (n1 kz )∓Ωλ~q )−1 −2πiδ ( En −E n (n1 kz )∓Ωλ~q )] . (34) SymbolP in formulas (32), (34) means that the respective integrals are taken as Cauchy principal values. Using the developed theory, we numerically calculated the energies renormalized due to phonons (Ẽn = En +∆n) and decay rates (Γn) of electron states in quasi-discrete spectrum at the fixed physical andgeometrical parameters of three-barrier RTS. 3. Parameters of electron spectrum as functions of temperature and de- sign of three-barrier RTS (GaAs/AlxGa1−xAs) The complete and partial shifts and decay rates of electron spectrum in three-barrier RTS were cal- culated for GaAs/AlxGa1−xAs nano-structure, being the active element of the experimentally investigatedQCD [2, 3, 17]. The physical parameters are presented in table 1. Table 1. Physical parameters of nanostructures. ε∞ ħωL,meV ħωT, meV me /m U , meV GaAs 10.89 36.25 33.29 0.067 Al0.15Ga0.85As 10.48 35.31 33.19 0.079 120 Al0.45Ga0.55As 9.66 33.66 32.77 0.104 320 In figure 2, the electron spectrum as the function of the position of the inner barrier with respect to the outer ones is presented at different Al concentrations: x = 0.15 and low potential barrierU = 120meV (a); x = 0.45 and high potential barrierU = 320meV (b). The thicknesses of the inner barriers (b = 1.13 nm) and the sum of both well widths (a = a1 +a2 = 13.9 nm) are the same for the both structures. The figure proves that independently of Al concentration, the quasi-discrete energy levels (En) qualitatively simi-larly depend on the position of the inner barrier (fixed by the width of input well (a1)): the energies Enare the symmetric functions with respect to the average position of the inner barrier in the common well (a1 = a2 = a/2) with n maxima. Two nearest operating levels (E1, E2) have one maximum at a1 = a/2 and two maxima— at a1 = a/4, 3a/4, respectively. The anti-crossing is observed at a1 = a/2, where the distance between E2 and E1 is minimal, due to the presence of two wells in a three-barrier RTS.In figure 3, the energy spectra of interface phonons (Ωλ~q )are presented for the both three-barrierRTS (a), (b). The spectra contain two groups having four modes of energies of a weak dispersion. The high-energy group is placed between the energies of longitudinal phonons (ΩL1 ,ΩL2 ) and the low-energygroup— between the energies of transversal phonons (ΩT1 , ΩT2 ) of the respective layers. In each group,the pair of modes with a higher energy has a negative dispersion while with lower energy — positive dispersion. The position of the inner barrier with respect to the outer ones weakly effects the magnitude 23704-6 The effect of interface phonons on operating electron states. . . 0 4 8 12 0 50 100 E n , m e V a 1 , nm a x=0.15 E 3 E 2 E 1 0 4 8 12 0 50 100 150 200 250 300 E 1 E 3 E 2 E 4 E n , m e V a 1 , nm b x=0.45 Figure 2. Energy spectrum of electron (En ) noninteracting with phonons as a function of the inner barrierposition (a1) at x = 0.15 andU = 120meV (a) and x = 0.45 andU = 320meV (b); a = 13.9 nm, b = 1.13 nm. of the dispersion: Ωλ~q varies at 2–3 % at big q and the energies become almost the same at a small q . An increase of Al concentration in the barriers does not qualitatively vary the dispersion of all phonon modes but increases its magnitude at a small quasi-momentum. In order to study the effect of electron-I-phonon interaction on the magnitude of the electromagnetic field energy absorbed by a nano-structure ( Ẽ21 = Ẽ2− Ẽ1) arising due to the quantum transition from thefirst (\|1〉) into the second (\|2〉) quasi-stationary state, we calculated the complete and partial shifts (∆n , ∆nd, ∆nc) and decay rates (Γn , Γnd, Γnc) of two operating states (n = 1,2). The results are presented in fig- ure 4 at T = 0 and 300 K for a nano-structure (a) because Al concentration causes their small quantitative changes for a nano-structure (b). Figure 4 (a), (b) proves that at cryogenic temperatures (formally at T = 0 K), the both operating states (\|1〉 and \|2〉) shift into the region of smaller energies (∆1, ∆2 < 0) independently of three-barrier RTS design due to electron-I-phonon interaction. The magnitudes of complete shifts nonlinearly depend on the position of the inner barrier (a1) and are of the same order. The complete shifts are mainly producedby intra-level interactions with partial shifts ∆11, ∆22. The shifts ∆1d, ∆2d are produced by inter-levelinteraction due to all states of a quasi-discrete spectrum and are smaller than ∆11, ∆22. Only at somemagnitudes of a1, the partial shifts ∆22 and ∆2d have correlating magnitudes. The partial shifts (∆1c, ∆2c)are caused by the interaction with the continuum states and are smaller or much smaller than the others. The shifts are negative because at T = 0 K only virtual phonons exist in the system. The decay rate of electron spectrum (Γn) is regulated by the energy conservation law which, as it isclear from MO (23) at T = 0 K, is determined by δ-function δ(En −En1 −Ωλ~q −ħ2q2/2m). Consequently, 0,0 0,1 0,2 0,3 0,4 0,5 33 34 35 36 a x=0.15 T 1 T 2 L 1 L 2 q , m eV q, /a 0,0 0,1 0,2 0,3 0,4 0,5 33 34 35 36 b x=0.45 T 1 T 2 L 1 L 2 q , m eV q, /a Figure 3. Energy spectra (Ωλ~q ) of interface phonons uncoupling with electrons as a function of quasi-momentum (q) at x = 0.15 and U = 120 meV (a) and x = 0.45 and U = 320 meV (b); a = 13.9 nm, b = 1.13 nm; ΩL1 , ΩL2 are the energies of longitudinal phonons, ΩT1 , ΩT2 are the energies of transversalphonons in the wells (1) and barriers (2). 23704-7 M.V. Tkach et al. when for the n-th state the condition n É n1 fulfills, the difference of energies becomes Enn1 = En −En1 É 0; thus, δ(−\|Enn1 \|−Ωλ~q−ħ2q2/2m) = 0 and, since, ΓnÉn1 = 0. As far as En < Ekz , the same reason brings to Γnkz = 0. Physically, it means that the intra-level and inter-level interaction between electrons from lower states and electrons from higher states of a discrete (d) and a continuum (c) spectrum due to virtual I- phonons (T = 0 K) occurs without decay. When the electrons from higher (n) states interact with virtual I-phonons through the lower (n1) states, at n > n1, the difference of the energies Enn1 = En −En1 > 0, thus at Enn1 <Ωλ~q , δ(Enn1 −Ω−ħ2q2/2m) = 0 and Γnn1 = 0 while at Enn1 >Ωλ~q , δ(Enn1 −Ω−ħ2q2/2m) , 0 0,0 0,5 1,0 1,5 0 4 8 12 25 30 35 40 45 50 E 21 E 21 < q E 21 > q E 21 > q q E 2 1 , m eV a 1 , nm (c) T = 0 K 2 = 21 2 , m eV 0 4 8 12 0,0 0,4 0,8 1,2 1,6 2,0 (f) T = 300 K a 1 , nm , m eV -0,05 -0,04 -0,03 0 4 8 12 -4 -3 -2 -1 0 0 4 8 12 (b) 2d a 1 , nm , 2 n ' , m eV C 2 , m eV a 1 , nm -0,07 -0,06 -0,05 -0,04 0 4 8 12 -4 -3 -2 -1 0 0 4 8 12 (e) , T 2 n ' , m eV T 2d a 1 , nm T C 2 , m eV a 1 , nm -0,04 0,00 0 4 8 12 -3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0 4 8 12 (a) 1d a 1 , nm , 1 n ' , m eV C 1 , m eV a 1 , nm -0,04 0,00 0 4 8 12 -3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0 4 8 12 (d) T 1d a 1 , nm , T 1 n ' , m eV a 1 , nm T C 1 , m eV Figure 4. Electron energy shifts and decay rates as a function of the inner barrier position at x = 0.15 and T = 0 K (a), (b), (c) and T = 300 K (d), (e), (f); a = 13.9 nm, b = 1.13 nm. In figures (d), (e), (f), ∆T1 , ∆T2 and ΓT1 , ΓT2 are the total shifts and decay rates of (\|1〉)and (\|2〉) states. ∆T11, ∆T22 and ΓT11, ΓT22 are the partialshifts and decay rates caused by intra-level interaction due to I-phonons; ∆T1d, ∆T2d and ΓT1d, ΓT2d are thepartial shifts and decay rates caused by inter-level interaction with the states of discrete spectrum due to I-phonons; ∆T1c, ∆T2c and ΓT1c, ΓT2c are the partial shifts and decay rates caused by inter-level interactionwith the states of continuum due to I-phonons. In figures (a), (b), (c), the same magnitudes calculated at T = 0 K are presented without index T. 23704-8 The effect of interface phonons on operating electron states. . . and Γnn1 , 0. In the last case, only the inter-level interaction due to phonons causes the finite decay of higher states at cryogenic temperatures (T = 0 K). Figure 4 c presents thementioned reasons determining the decay rates of electron states at T = 0 K. In the figure one can see the decay rate (Γ2 = Γ21) of the energy of quantum transition E21 = E2−E1 detectedat the absorption of electromagnetic wave, as a function of design of three-barrier RTS (a1) and the stripe,where all modes of I-phonon energies (Ωλ~q ) are located. The figure proves that the decay rate of the firststate (\|1〉) is absent (Γ1 = 0) and, as far as Γ22 = 0, the decay rate of the second state (\|2〉) is caused only by the inter-level interaction due to phonons (Γ2 = Γ21). Thus, Γ2 = 0 in the range 4.75 É a1 É 9.15 where E21 ÉΩλq and Γ2 , 0 in the ranges 0 É a1 É 4.75 and 9.15 É a1 É 13.9 where E21 >Ωλq .At the finite temperature, the average occupation number of phonon states is not equal to zero (νλ~q , 0). Therefore, as it is clear from MO (23), the spectral parameters of both operating states are produced not only by virtual (as at T = 0 K) but also by real I-phonons, both in the processes of their creation, described by the first term of MO proportional to (1+νλ~q ) and in the processes of their annihilation,described by the second term of MO proportional to νλ~q .In figure 4 (d), (e), (f), the complete and partial shifts (∆Tn , ∆Tnd, ∆Tnc) and decay rates (ΓTn , ΓTnd, ΓTnc) ofthe operating electron states (n = 1,2) are presented at T = 300 K. It is clear that the temperature weakly changes the shapes and magnitudes of both states shifts depending on the position of the inner barrier (a1). The decay rates (ΓT1 , ΓT2) are produced by partial contributions of intra-level (ΓT11, ΓT22) and inter-level(ΓT12, ΓT21) interactions with I-phonons in the processes of their creation and annihilation. 0,0 0,5 1,0 1,5 2,0 2,5 0 4 8 12 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0 4 8 12 a 1 , nm TT , m e V a 1 , nm , m e V Figure 5. A complete shift (∆T) and the decay rate(ΓT) as a function of the inner barrier position (a1)at T = 300 K; a = 13.9 nm, b = 1.13 nm. In figure 5, a complete shift (∆T =∆T2 −∆T1) anddecay rate (ΓT = ΓT1+ΓT2) of energy (Ẽ21 = E21+∆T)in the process of electromagnetic wave absorp- tion as functions of the inner barrier position (a1)is presented at T = 300 K. The both functions are strongly nonlinear while the respective mag- nitudes are not big. We should note that sharp minima at the curves of all shifts and respective maxima at the curves of the decay rates (figures 4, 5) are mainly caused by a bigger contribution of inter-level in- teraction between the second and third level in those two configurations of a three-barrier RTS where the anticrossing arises between them. According to physical considerations, the de- cay increases and the magnitude of the detected energy weakly decreases when the temperature increases, which qualitatively correlates with the experimental results [17]. It is clear that one should consider in the model the confined polarization and acoustic phonons in order to quantitatively compare the theoretical and experimental data. This rather complicated and so- phisticated work will be done in further investigations based on the approach proposed in this paper. 4. Main results and conclusions From the first principles (without any fitting parameters), we obtained the Hamiltonian of electron- I-phonon system in the representation of the second quantization over all variables for a three-barrier RTS. The renormalized electron spectrum was calculated at cryogenic and room temperatures using the thermo-dynamical Green’s functions method. For GaAs/AlxGa1−xAs nano-structure, we studied in detailthe effect of various mechanisms of electron-phonon interaction (intra-level and inter-level with quasi- discrete and continuum spectrum) at the formation of energy shifts and decay rates of electron states depending on the geometric design of three-barrier RTS. The energies of electron states, the energy of electromagnetic field absorbed by a three-barrier RTS which is an active element of QCD, at quantum transition between the first and the second states, the 23704-9 M.V. Tkach et al. shifts of the energies and decay rates due to the electron-I-phonon interaction are not big over the magni- tude, while nonlinear functions depend on the position of the inner barrier between the outer barriers. It is proven that independently of the geometric design of a three-barrier RTS, an increase of tempera- ture causes bigger decay rates of both operating electron states and their shift into the low-energy region. The shift of the first level is smaller than that of the second one. Thus, being detected by QCD, the energy of electromagnetic field absorbed at a quantum transition decreases, which qualitatively correlates with the experiment. References 1. Luo H., Liu H.C., Song C.Y., Wasilewski Z.R., Appl. Phys. Lett., 2005, 86, 231103; doi:10.1063/1.1947377. 2. Hofstetter D., Graf M., Aellen T., Faist J., Appl. Phys. Lett., 2006, 89, 061119; doi:10.1063/1.2269408. 3. 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B, 2001, 15, No. 27, 3539; doi:10.1142/S0217979201007804. 12. Yan Z.W., Ban S.L., Liang X.X., Int. J. Mod. Phys. B, 2003, 17, No. 31, 6085; doi:10.1142/S0217979203023653. 13. Wu B.H., Cao J.C., Xio G.Q., Lio H.C., Eur. Phys. J. B, 2003, 33, 9; doi:10.1140/epjb/e2003-00135-2. 14. Huang W.D., Wei S.Y., Ren Y.J., Wang Y.H., Int. J. Mod. Phys. B, 2007, 21, No. 25, 4407; doi:10.1142/S0217979207037843. 15. Jiraushek C., Kubis T., Appl. Phys. Rev., 2014, 1, 011307; doi:10.1063/1.4863665. 16. Gao X., Botez D., Knezevic I., J. Appl. Phys., 2007, 101, 063101; doi:10.1063/1.2711153. 17. Gendron L., Carras M., Huynh A., Ortiz V., Appl. Phys. Lett., 2004, 85, No. 14, 2824; doi:10.1063/1.1781731. Вплив iнтерфейсних фононiв на електроннi робочi стани трибар’єрної резонансно-тунельної наноструктури як активної зони квантового каскадного детектора М.В.Ткач,Ю.О.Сетi,Ю.Б. Гринишин, О.М.Войцехiвська Чернiвецький нацiональний унiверситет iм.Ю. Федьковича, вул. Коцюбинського, 2, 58012 Чернiвцi, Україна З перших принципiв у моделi ефективних мас та поляризацiйного континууму встановлено гамiльтонi- ан системи електронiв взаємодiючих з iнтерфейсними фононами у трибар’єрнiй резонансно-тунельнiй структурi.Методом функцiй Грiна розраховано температурнi змiщення й загасання найнижчих (робочих) електронних станiв у залежностi вiд геометричної конфiгурацiї наносистеми GaAs/ AlxGa1−xAs як актив-ної зони квантового каскадного детектора. Встановлено,що незалежно вiд температури системи енергiя квантового переходу в процесах поглинання електромагнiтного поля є нелiнiйною слабозмiнною фун- кцiєю вiд положення внутрiшнього вiдносно зовнiшнiх бар’єра наносистеми. Ключовi слова: резонансно-тунельна наноструктура, iнтерфейснi фонони, квантовий каскадний детектор 23704-10 http://dx.doi.org/10.1063/1.1947377 http://dx.doi.org/10.1063/1.2269408 http://dx.doi.org/10.1109/JQE.2009.2017929 http://dx.doi.org/10.1063/1.3462300 http://dx.doi.org/10.1134/S1063782611030195 http://dx.doi.org/10.1103/PhysRevB.40.6175 http://dx.doi.org/10.1007/s100510050357 http://dx.doi.org/10.1142/S0217979201007804 http://dx.doi.org/10.1142/S0217979203023653 http://dx.doi.org/10.1140/epjb/e2003-00135-2 http://dx.doi.org/10.1142/S0217979207037843 http://dx.doi.org/10.1063/1.4863665 http://dx.doi.org/10.1063/1.2711153 http://dx.doi.org/10.1063/1.1781731 Introduction Hamiltonian and Fourier image of Green's function of the system of electrons interacting with interface polarization phonons in a three-barrier nano-structure Parameters of electron spectrum as functions of temperature and design of three-barrier RTS (GaAs/AlxGa1-xAs) Main results and conclusions

The effect of interface phonons on operating electron states in three-barrier resonant tunneling structure as an active region of quantum cascade detector

Institution

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