Nonresonant Transparency Channels of a Two-Barrier Nanosystem in an Electromagnetic Field with an Arbitrary Strength

Transcription

1 ISSN , JETP Letters, 2012, Vol. 95, No. 5, pp Pleiades Publishing, Inc., Original Russian Text N.V. Tkach, Yu.A. Seti, 2012, published in Pis ma v Zhurnal Eksperimental noi i Teoreticheskoi Fiziki, 2012, Vol. 95, No. 5, pp Nonresonant Transparency Channels of a Two-Barrier Nanosystem in an Electromagnetic Field with an Arbitrary Strength N. V. Tkach* and Yu. A. Seti Chernovtsy National University, Chernovtsy, Ukraine * ktf@chnu.edu.ua Received January 11, 2012; in final form, January 30, 2012 The theory of the quasistationary spectrum of the system of electrons interacting with a high-frequency electromagnetic field with an arbitrary strength in a two-barrier resonant tunneling nanostructure has been proposed on the basis of the exact solution of the total one-dimensional Schrödinger equation. It has been shown that nonresonant transparency channels in the planar two-barrier resonant tunneling structure appear owing to the superposition of a certain quasistationary state with first field resonance harmonics of both nearest states; this superposition is due to the interaction of electrons with the electromagnetic field. DOI: /S INTRODUCTION Interest in study of resonant tunneling nanosystems is stimulated both by their known application in quantum cascade lasers, detectors, and other nanodevices [1 5] and by new properties and phenomena interesting for fundamental physics. As is known, the theory of the transport of electron fluxes through resonant tunneling structures was developed in detail in the small-signal approximation [6 9]. The dependences of the properties of ballistic electron transport through resonant tunneling nanosystems on the strength of the electric component of the electromagnetic field were analyzed in [10 12]. In particular, using the previously developed perturbative method, Pashkovskii [13] analyzed the features of the electron transport through the two-barrier resonant tunneling structure near resonance levels with allowance for the strong interaction with the electromagnetic field. It was revealed that, as the amplitude of the field, a resonance level is first broadened and then split, forming two transparent channels. The physical reason for splitting was not discussed. The aim of this work is to study the ballistic electron transport through the two-barrier resonant tunneling structure in an electromagnetic field with an arbitrary strength on the basis of the exact solution of the total Schrödinger equation. It is shown that nonresonant transparency channels in the two-barrier resonant tunneling structure appear owing to the superposition of a certain quasistationary state with first field resonance harmonics of both nearest states; this superposition is due to the interaction of electrons with the electromagnetic field. ANALYTICAL AND NUMERICAL CALCULATIONS We consider a planar symmetric two-barrier nanostructure in a uniform high-frequency electromagnetic field (t) = 2 cos(ωt) with arbitrary strength and frequency ω. For certainty, we assume that the monoenergetic electron flux with the energy E is incident on the two-barrier resonant tunneling structure from the left (along the z axis) perpendicularly to its layer. Neglecting scattering inside the resonant tunneling structure and assuming that the motion of the electron is one-dimensional, we write the time-dependent Schrödinger equation for the system under study in the form i Ψ ( z, t) = t ( e )2 sin 2 ( ωt) mω U( z) + H 2mz 2 int Ψ( z, t). (1) Here, e is the elementary charge and m is the effective mass of the electron. The Hamiltonian of the system contains the kinetic energy of the electron (first term); its potential energy in the two-barrier resonant tunneling structure in the typical δ-barrier approximation [11 13], U( z) = UΔ[ δ( z) + δ( z ], (2) where U and Δ are the height and width of the potential barriers, respectively, and a is the width of the potential well; the potential energy of the interaction of the electron with the electromagnetic field = e () t { z[ θ( z) θ( z ] + aθ( z }, (3) H int 271

2 272 TKACH, SETI and the potential energy of the field (the fourth term). The Hamiltonian in Eq. (1) for the system of electrons interacting with the high-frequency field in the resonant tunneling structure is complete. It includes the energy of interaction described by a term linear in the field strength Δ (the exact solution with which is known from [12]) and the energy of the field itself described by a term quadratic in the strength. Both these terms of the Hamiltonian make the same contributions quadratic in to the exact wavefunction. Equation (1) in the inner region (0 z of the two-barrier resonant tunneling structure has two exact linearly independent solutions: i - Et which describe the forward and backward waves with the quasimomentum k 0 = 1 2mE. The solutions are obviously simplified beyond the resonant tunneling structure. For this reason, the exact wavefunction of the system as a linear combination of both solutions (4) with allowance for the expansion of all periodic functions in Fourier series taking into account superpositions over all field harmonics can be represented in the form where ψ ± ( z, t) = exp ± ik 0 z 2e z sin( ωt) ω 2e cos( ωt) mω 2 (4) 2( e ) mω 1 sin( 2ωt) t 2 2ωt, Ψ( E, ω, z, t) = ψ + ( E+ pω, ω, z, t ) + ψ ( E+ pω, ω, z, t), p = = e i - ( E+ pω)t ψ ± ( E+ pω, ω, z, t) sin( 2παβ 2 ) π n 1, n 2 = + f( ± k p, ω, z, t)e ik p z ( 1) n 1 j n2 ( αβ 2 ) n 1 + 2αβ 2 ± + i ( n 1 + 2n 2 )ωt f( ± k p, ω, z, t) A ± ± 0p θ( z) A 2παβ 2 = + sin( ) 1p π n 3, n 4, n 5, n 6 = i 2n 3 ± n j n 3 + 2αβ 2 n4 αβ 2 j n6 2β z ā - i n e 3 + 2n 4 + n 5 + n 6 ( )ωt ( )j n5 ( 4αβak p ) [ θ( z) θ( z ], (5) (6) (7) and j n is a Bessel function of the nth order. Here, we introduce the convenient compact notation 2 2 k Ω ω; U a e a; α a U = = = ; β = ----; a 2mΩ Ω (8) k p = 1 2m[ E+ pω]; k a = a 1, with the obvious physical meaning: α is the kinetic energy of the electron with the quasimomentum k a and β is the potential energy of the interaction of the electron with the field of strength divided by the field energy Ω. ± All unknown expansion coefficients ( A ( 0, 1, 2)p ) are unambiguously determined by the conditions of the continuity of the wavefunctions and fluxes of their densities at the interfaces between media (z = 0, at an arbitrary time t, Ψ( E, ω, +η, t) = Ψ( E, ω, η, t) ( η 0); ---Ψ ( E, ω, z, z = +η ---Ψ ( E, ω, z, z = η = 1 ā - UΔ iβsin( ωt) Ψ( E, ω, 0, t); αωa (9) Ψ( E, ω, a + η, t) = Ψ( E, ω, a η, t); ---Ψ ( E, ω, z, z = a+ η ---Ψ ( E, ω, z, z = a η UΔ = Ψ ( E, ω, a, t), αωa 2 and, therefore, separately for each harmonic p. Taking into account the monochromaticity of the electron flux incident on the two-barrier resonant tunneling structure through the main channel (p = 0), i.e., the absence of any other fluxes incident on the system + through all other channels (p 0), A 0p 0 = A 2p = 0. System (9) contains an infinite number of equations owing to an infinite number of harmonics. Real calculations can be performed with necessary quite large numbers of positive (N + ) and negative (N ) harmonics, taking into account that the latter number is limited by the number of so-called open channels determined by the obvious condition N [E/Ω]. According to [14], the transparency coefficient of the nanostructure can be determined in terms of the calculated forward (J + ) and backward (J ) electron flux densities at the input (z = 0) and output (z = of the resonant tunneling structure as the sum of partial terms: ± + A 2p n 3 = j n3 ( 2β)e in 3 ωt θ( z, D( Eω, ) = N + p = N J + ( E+ pω, ω, J + ( E, ω, 0) (10)

3 NONRESONANT TRANSPARENCY CHANNELS 273 Fig. 1. Transparency channel D versus E near the energy E 2 for various field strengths U a and for ( Ω = Ω 21 = 77.4 mev and (b) Ω = Ω 32 = mev. Expression (10) for the transparency coefficient of the two-barrier resonant tunneling structure makes it possible to determine the main and an arbitrary number of satellite quasistationary states of the system of electrons interacting with the high-frequency field of arbitrary strength and frequency ω. According to the presented theory, the interaction of the electron with the high-frequency field in the two-barrier resonant tunneling structure should lead not only to the renormalization of purely quasistationary electronic states but also to the appearance of satellite quasistationary states corresponding to all possible field harmonics. In view of this circumstance, it should be expected that, if a certain satellite quasistationary state of one electronic state is in resonance with another electronic state, mixed quasistationary states will appear and be manifested in the form of the corresponding maxima of the transparency coefficient. The transparency coefficient was calculated for the two-barrier resonant tunneling structure In 0.52 Al 0.48 As/In 0.53 Ga 0.47 As with the parameters m = 0.043m e, where m e is the mass of the electron in vacuum; U = 516 mev; a = 18 nm; and Δ = 9 nm typical of experimentally studied nanoheterostructures [2, 5]. Figure 1 shows the transparency coefficient D as a function of the energy of the electron E near the resonance energy E 2 of the second quasistationary state at two fixed energies of the field ( Ω 21 = E 2 E 1 and (b) Ω 32 = E 3 E 2 corresponding to the differences between the resonance energies for several field strengths ( or U a = e. As can be seen in the inset in Fig. 1b, an increase in the field strength is accompanied by the deformation of the initial (at U a = 0) Lorentzian shape of the coefficient D(E): first, the maximum of D expands and decreases gradually and, then, two maxima appear and the distance between them increases. Two maxima of D in Fig. 1a correspond to the resonance energies of two quasistationary states, which are superpositions of the second main quasistationary state (or the zeroth harmonic) with the first positive satellite (first harmonic) from the first main quasistationary state. Since the resonance energies corresponding to both maxima of D are only lower ( E 2; 1 ( +1) ) or only higher ( E 2; 1 ( +1) ) than E 2, it is reasonable to introduce the proposed obvious notation. Similarly, two maxima of D in Fig. 1b correspond to the resonance energies of quasistationary states, which are superpositions of the second main quasistationary state with the first negative satellite with the field energy Ω 32 from the third main quasistationary state. The energies corresponding to peaks of D(E) are also only lower ( E 2; 3( 1) ) or only higher ( E 2; 3( 1) ) than E 2. As can be seen in Fig. 1b, the complex quasistationary state that is a superposition of the main and satellite states is quite well separated for field strengths at which E 2; 3( 1) E 2; 3( 1) Γ 2, where Γ 2 is the resonance width of the second quasistationary state in the absence of a field. According to Fig. 2, the dependences of the resonance energies of complex states on the field energy Ω near Ω 21 (Fig. 2 and Ω 32 (Fig. 2b) have the character

4 274 TKACH, SETI 2; 3 ( 1) ( 1) Ω ( 1) ( 1) Fig. 2. Energies E 2;1 ( +1) and E 2; 3( 1) of the complex quasistationary states of the electron near E 2, as well as the transparency coefficient D, versus the field energy Ω for the field strengths U a = (1) 0, (2) 10, and (3) 20 mev near ( Ω 21 and (b) Ω 32. Fig. 3. Transparency channel D versus E near the energy E 8 for various field strengths U a and for ( Ω = Ω 78 = mev and (b) Ω = Ω 98 = mev. of anticrossings whose sizes increase with the field strength U a. At energies E 2; 1 ( +1) and E 2; 3( 1) near the resonance energies of the field Ω 21 and Ω 32, complex states appear and create two channels in the two-barrier resonant tunneling structure with approximately or exactly the same transparency D( ) E 2; 1 ( +1) E 2; 1 ( +1) E 2; 1 ( +1) D( E 2; 1 ( +1) ) and D( E 2; 3( 1) ) D( E 2; 3( 1) ). With an increase in the strength U a, the field energies at which both nonresonant channels have the same transparencies, D( ) = D( ) and D( ) = D( E 2; 3( 1) ), as well as transparencies themselves, decrease. Near the energy E 2, a pair of channels with E 2; 3( 1)

5 NONRESONANT TRANSPARENCY CHANNELS 275 tion of the energy E at various field strengths U a near the eighth quasistationary state is shown. E 9 Ω According to Fig. 3, the transparency coefficient loses a Lorentzian shape with an increase in the field strength. Then, a low-energy maximum E 8; 7(+1), and, then, a high-energy maximum E 8; 7 ( +1) appear near the central maximum E 8; 7 ( +1). They are formed by the superposition of the eighth quasistationary state, the first positive harmonic of the seventh quasistationary state, and the first negative harmonic of the ninth quasistationary state, respectively. As can be seen in Fig. 4a, near the energy E 8, a set of quasistationary states whose energies E 8; 7 ( +1), E 8; 7(+1),, and E 8; 7 ( +1) depending on Ω form triple anticrossing degenerate at the Ω values at which the corresponding maximum of D disappears (Fig. 4b). In view of this circumstance, two or three nonresonant channels with different transparencies can be implemented near resonance energies. CONCLUSIONS Fig. 4. ( Energies E 8; 7 ( +1) and E 8; 7(+1), of the complex quasistationary states near E 8, as (b) well as the transparency coefficient D, versus the field energy Ω for the field strengths U a = (1) 0, (2) 100, and (3) 150 mev. the same transparencies which is formed by the superposition of the second main quasistationary state and the first positive harmonic of the first main quasistationary state (Fig. 2 is more transparent than a pair formed by the superposition of the second main quasistationary state and the first negative harmonic of the third main quasistationary state (Fig. 2b). The formation of complex quasistationary states and the corresponding transparency channels of twobarrier resonant tunneling structures owing to the interaction of electrons in high excited states with the high-frequency field differs from the formation described above because the resonance widths of quasistationary states (without the field) are comparable with each other and with the difference between the resonance energies of neighboring quasistationary states. For this reason, both harmonics from the neighboring lower and upper quasistationary states, rather than one of them, are near the energy of the nth quasistationary electronic state in the field energy scale (Ω). This situation is exemplified in Fig. 3, where the evolution of the transparency coefficient as a func- To summarize, we have shown that the interaction of electrons with the two-barrier resonant tunneling structure with the high-frequency field of the resonance energies Ω n, n ± 1 = E n E n ± 1 forms complex quasistationary states, which are superpositions of the nth quasistationary state with the positive harmonic of the (n 1)th quasistationary state and the negative harmonic of the (n + 1)th quasistationary state. For this reason, a pair of nonresonant channels with a quite high but not complete transparency appears near the resonance energies of low quasistationary states in the two-barrier resonant tunneling structure at field frequencies ω 1 E n E n ± 1, and two (at ω 1 E or three (at ω 1 n E n ± 1 E n E n ± 1 ) channels appear near the resonance energies of high quasistationary states. Since nonresonant transparency channels of the resonant tunneling structures in quite strong electromagnetic fields can significantly affect the working parameters of devices based on these structures (quantum cascade lasers, detectors, etc.), the existence of these channels should be taken into account when studying physical processes occurring in such nanosystems. The proposed method for the Fourier expansion of the exact solution of the total Schrödinger equation can be applied, in particular, to calculate the active conductivity of multilayer resonant tunneling structures in electromagnetic fields with arbitrary strength.

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