CONDUCTIVITY OF THREE-BARRIER RESONANCE TUNNEL STRUCTURE

Transcription

1 CONDENSED MATTER CONDUCTIVITY OF THREE-BARRIER RESONANCE TUNNEL STRUCTURE M. V. TKACH, YU. O. SETI, O. M. VOITSEKHIVSKA, G. G. ZEGRYA Yu. Fed kovich Chernovtsy National University, 58 Chernovtsy, Ukraine, A. F. Ioffe Physical-Technical Institute of Russian Academy of Science, 94 St. Petersburg, Russia, Received May 4, Within the model of rectangular potentials and different effective electron masses it is developed a theory and performed a quantum-mechanical calculation of dynamical conductivity for the electrons interacting with electromagnetic field in open three-barrier resonance tunnel structure with arbitrary outer barriers. For the experimentally investigated structure with In.53 Ga.47 As-wells and In.5 Al.48 As-barriers, it is shown that there exist the optimal geometrical configurations determined by the position of the inner barrier respectively the outer ones, providing the optimal operation of nano-device as a separate detector or an active element of such a detector in desired frequency range. Key words: resonance tunnel structure, dynamical conductivity, detector.. INTRODUCTION The experimental studying of the quantum cascade lasers (QCL) [-3] operating in the actual tera Hertz electromagnetic wave range has been activated after the creation of first nano-lasers [4, 5] using the idea of quantum transitions between electronic states [6]. Later, the quantum cascade detectors (QCD) well detected in the infrared and tera Hertz ranges were created [7, 8]. Recently, the technical characteristics of both type devices were experimentally improved [9 ] due to their different design. The specific new properties of these quantum devices find application in many fields of engineering, medicine, biology, etc. The separate cascade of QCL or QCD consists of an active element, being an open two-, three-, or four-barrier resonance tunnel structure (RTS), and an injector or extractor, respectively. Therefore, the operating characteristics of QCL and QCD essentially depend on the physical properties of their RTSs determined by the active conductivity of electrons and their interaction with the electromagnetic field. Rom. Journ. Phys., Vol. 57, Nos. 3 4, P. 6 69, Bucharest,

2 Conductivity of three-barrier resonance tunnel structure 6 The quantum transitions between different quasi-stationary states are accompanied either by emission (QCL) or absorption (QCD) of the quanta of electromagnetic field. Herein, there arises the applied interest to develope the theory of electrons active conductivity for the open nano-rts. The main problem for the theory is to solve the complete Schrödinger equation for the electronic current flowing through the RTS taking into account the interactions with electromagnetic field, between electrons, with phonons and other dissipative subsystems. It is a mathematically complicated task because in the most of experiments [ ], as a rule, the energy spectrum, oscillator strengths of quantum transitions, and electron-phonon-interaction relaxation times, etc. are theoretically estimated at the base of the wave functions of electrons stationary states obtained from the stationary Schrödinger equation for the closed systems. The calculation of dynamical conductivity within this approach is impossible at all. In refs. [ 5], the theory of dynamical electrons conductivity for the open RTS with different number of wells and barriers is developed within the more consistent quantum-mechanical model using the analytical solution of complete Schrödinger equation with open bounder conditions. Avoiding the mathematical complications, the authors are using a simplified model of constant effective electron mass over all the nano-system layers and δ -like approximation for the rectangular potential barriers and are researching the influence of various interactions (electron-electron or electron-phonon) at the operating of QCL both in ballistic and non-ballistic regimes. In refs. [6 8] it is proven that the above mentioned model heavily overestimated the magnitudes of resonance energies compared to a more realistic model. In paper [9] it is developed the theory of dynamical conductivity for an open three-barrier RTS as a separate nano-detector or as an active element of QCD, in the frames of realistic model of different effective electron masses in various elements of RTS and rectangular potential wells and barriers. For the nano-system with In.53 Ga.47 As-wells and In.5 Al.48 As-barriers, that was often experimentally researched [,4,5,7,9,], it was shown that there existed three RTS geometrical configurations at which nano-detector operated in most optimal regime in the tera Hertz frequency range. The numeric calculations and detail analysis was performed for the three-barrier RTS with equal outer barriers. In this paper, we are going to study the more general case of optimal design for the RTS with arbitrary widths of outer (input and output) barriers and compare the results to the system with equal ones.. COMPLETE SCHRÖDINGER EQUATION. DYNAMICAL CONDUCTIVITY FOR THREE-BARRIER RTS Open three-barrier RTS with the geometrical parameters presented in Fig. is studied in the Cartesian coordinate system. The small differences of the lattice constants for the wells and barriers allow researching this system within the model of effective masses and rectangular potentials

3 6 M. V. Tkach et al. 3 m,, reg.,, 4, 6 mz ( ) = ( ) = U z. () m, U, reg.,3, 5 We consider the electronic beam with energy Е and concentration of uncoupling electrons n, falling at the three-barrier RTS from the left side perpendicularly its planes. Fig. Geometry and potential energy scheme for a three-barrier RTS. According to quantum mechanics, the RTS conductivity is determined by the density of electronic current through the system, in its turn defined by the wave function of an electron interacting with a periodical in time electromagnetic field. The general theory of solving of complete Schrödinger equation Ψ(,) zt i = ( H H(,) z t ) Ψ(,) z t () t where H = U( z) zm z z ( ) is the electron Hamiltonian in a stationary case, iωt iωt ( ) 5 ( 5) ( ) H(,) z t =eє zθ z ( z z) θ z z e e (4) is the Hamiltonian of electron interaction with time-dependent electromagnetic field of the frequency ω and amplitude Є of the electrical field intensity is presented in details in paper [9]. Thus, in this paper we are going to mention the main results only. Within the small signal approximation, when the amplitude of electromagnetic field intensity is considered as small, the solution of equation () is written in the form ω i t i( ω ω) t i( ωω) t Ψ ( z, t) =Ψ ( z) e Ψ ( z) e Ψ ( z) e ( ω = E ). (5) (3)

4 4 Conductivity of three-barrier resonance tunnel structure 63 Ψ z (formula (), ref. [9]) is a solution of the stationary Schrödinger equation and the second term presents the first-order correction in one-mode approximation with Ψ ( z) functions obtained from the equation Here, the function ( ) ( H ( )) ( z) eє z ( z) z5 z ( z z5) ( z) ω ω Ψ θ ( ) θ Ψ = (6) Solutions of heterogeneous equations (6) are super positions of the functions where ( z) equations, and ( z) ( ) ( ) ( ) Ψ z =Ψ z Φ z. (7) Ψ (formula (4), ref. [9]) are the solutions of homogeneous Φ (formula (6), ref. [9]) are the partial solutions of heterogeneous equations (6). The continuity conditions of wave functions and respective densities of currents at all RTS interfaces p ( ) Ψ ( ) ( p) ( ) ( p) ( p ) dψ z d z Ψ ( zp) =Ψ ( zp) ; =. (8) m dz m dz () z= z () p z= zp bring to the system of twelve heterogeneous equations from which all twelve () (6) ( p) ( p) unknown coefficients B, A, B, A ( p= 5) are determined. Finally, Ψ ( z) functions and first-order corrections are found and the complete wave function Ψ ( zt, ) becomes known. The density of current of uncoupling between each other electrons (with concentration n ) is fixed by the expression ie n * * jzt (,) = (,) zt (,) zt (,) zt (,) zt mz ( ) Ψ Ψ Ψ Ψ z z. (9) Assuming the small sizes of RTS comparing to the electromagnetic wave length, the density of guided current is further calculated in quasi-classic approximation which determines a real part of dynamical conductivity n 5 ( ) ( ) () (6) () (6) σω ( ) =σ ( ω ) σ ( ω ) = ω k B A k B A. () zmє Here σ ( ω), σ ( ω ) are the partial components of conductivity, arising due to the electronic currents interacting with the electromagnetic field and flowing ahead ( σ ) and backward ( σ ) the RTS, respectively the primary direction of the electronic beam falling into RTS.

5 64 M. V. Tkach et al OPTIMAL GEOMETRICAL DESIGN OF RTS WITH ARBITRARY INPUT AND OUTPUT BARRIERS OPERATING AS DETECTOR It is well known that energetic and frequency detector characteristics are mainly determined by the properties of RTS dynamical conductivity (σ), depending, in its turn, on the spectral parameters (resonance energies Е n and widths Г n ) of quasi-stationary electron states defined by geometrical and material parameters of RTS. The three-barrier RTS under study is characterized by five independent geometrical parameters: the widths of inner ( ), input ( ) and output ( ) outer barriers and the widths of input ( b ) and output ( b ) wells. The numeric calculations of spectral parameters and conductivity versus all geometrical parameters of RTS were performed for the experimentally researched [,, 4, 7, 9, ] plane three-barrier nano-rts (Fig. ), composed of In.53 Ga.47 Aswells and In.5 Al.48 As-barriers with the physical parameters: m =.46 me, m =.89 me, a =.5868 nm, a =.5867 nm, and U = 56 mev, well satisfied the theory conditions. The rather small concentration of electrons 6-3 ( n = cm ) allows to neglect the electron-electron interaction. 4 E n [mev] 3 (a) n=3 - ω III n= ω I ω II n= -4 b I b II b III b ln Г n 4 E n [mev] 3 ( b) -4 n= b n=3 n= ln Г n - Fig. Dependences of resonance energies (Е n ) and logarithms of resonance widths (ln Г n ) for the first three quasi-stationary states (n=,,3) on input well width (b ). The dependences of resonance energies (Е n ) and logarithms of resonance widths (ln Г n ) for the first three quasi-stationary states (n=,,3) on the position of inner barrier between the outer ones ( b b, b= b b - common width of both wells) for the RTS with narrow inner barrier ( <, = 3.6 nm, =.4 nm, = 3 nm, b=.8nm) and broad one ( >, = 3.6 nm, = 7. nm, = 3 nm, b=.8nm) are shown in Fig. a,b respectively. It is to be

6 6 Conductivity of three-barrier resonance tunnel structure 65 mentioned that for the convenient comparison, instead of the magnitudes of resonance widths ( Г n ) and conductivities ( σ n=,3,4, σ ) their logarithmic values are presented (in the units Г = mev, σ = S/cm ). When the position of inner barrier respectively outer ones changes, there appear either anti crossings of the resonance energies (Fig. a, narrow inner barrier) or a collapse of the resonance energies (Fig. b, broad inner barrier) with a proper smooth (Fig. a) or abrupt (Fig. b) inter-section of the resonance energies. An increases of input well width ( b ) does not qualitatively change the properties of RTS spectral parameters according to the physical reasons and the numeric calculations. Herein, the resonance energies (Е n ) of all quasi-stationary states are decreasing, complying the quadratic law and the resonance widths (Г n ) - the exponential law. Further, it is performed the calculation of maximum values of conductivity σ n=,3,4, σ and, respectively, ln σ n=,3,4, lnσ depending on the input well width ( b ) for the RTS with equal and different widths of outer barriers (Fig. 3a-f). ln σ (a) (d) ln σ ln σ 4 b I = b b ln σ n=, 3, 4, ln σ - = =.4 nm ln σ n=, 3, 4, ln σ - =.4 nm; =3.6 nm ln σ ln σ - ln σ n=, 3, 4, ln σ (b) b II 5, 5,6 6, b b III 8, 8,4 8,6 b ln σ n=, 3, 4, ln σ ln σ n=, 3, 4, ln σ - - =3.6 nm; =.4 nm (e) b

7 66 M. V. Tkach et al. 7 - ln σ n=, 3, 4, ln σ (c) - =. nm; =4.8 nm b - ln σ n=, 3, 4, ln σ - =4.8 nm; =. nm (f) b Fig. 3a-f - Evolution of ln σn=,3,4 and ln σ, ln σ as functions of the input well width. From the figure one can see that the maximum values of dynamical conductivity logarithms ( ln σ n=,3,4) formed by all quantum transitions from the first quasi-stationary state to other ones are the complicated non linear functions, depending on the position of the inner barrier in common potential well. These functions, qualitatively similar, are essentially quantitatively different. Analyzing this fact and some general properties of ln σ n=,3,4 as functions of b one can determine the optimal configurations (OC) of RTS operating as a separate detector or as an active QCD element in the desired range of frequencies. Considering the transition from the first quasi-stationary state to the second one, it is necessary to place the inner barrier between the outer ones in such a way that two evident conditions are fulfilled:. The transitions from the first quasi-stationary state to any other, except the second one, must not disguise the detected frequency ω, i.e. it is necessary that ln σ > ln σ n= 3,4,..... The density of electronic current through the RTS from the second quasistationary state is to be maximally possible, i.e. a partial term σ produced by the output current has to predominate the term σ produced by the input one. According to the position of inner barrier respectively the outer ones, these conditions are either fulfilled or not. Therefore, in order to define the optimal geometric design of RTS operating as detector, it is necessary to study the evolution of dynamical conductivity depending on the arbitrary width of input ( ) and output ( ) barriers. Fig. 3 presents a typical example of evolution of ln σ n=,3,4 and ln σ, ln σ as functions of the input well width ( b ). For the convenient comparison, it is also shown the behaviour of conductivity for the RTS with equal widths of input

8 8 Conductivity of three-barrier resonance tunnel structure 67 and output barrier (Fig. 3a and Fig. 3b the same case but in details) and for the RTS with different widths of outer barriers at the condition = const = 6nm (Fig. 3c-f). Analyzing the conductivity properties for RTS with equal outer barriers (Fig. 3a,b) one can see that there are three advantageous (non-tinted) and three disadvantageous (tinted) ranges for the detector optimal work. Since, there are three optimal geometrical configurations determined by the position of inner barrier respectively the outer ones when a maximum conductivity ( σ ) is reached in the forward direction and all not favourable factors ( σ, σ 3, σ 4 ) become simultaneously minimal. The first optimal configuration (I OC) corresponds to the RTS with the inner barrier close to the input one (in the limit case b, the three-barrier RTS transforms into a two-barrier one with the input barrier width: and output one: ). The second optimal configuration (II OC) is a nearly symmetrical RTS ( b b/), and the third one (III OC) is an asymmetrical RTS ( b 3 b/4). Each optimal configuration has its advantages and disadvantages. Analyzing the spectral parameters, fig. a, one can see that the I and III OC are characterized I III by nearly equal resonance energies ( E = 4. mev, E = 58. mev ), as well as nearly equal energies of the detected field ( ω I = E I I E =. mev, III III III ω = E E = 67. mev ). Comparing to the I and III OC, the II one has a II bigger resonance energy ( E = 98.3 mev ), but on the other hand, it allows to II detect much smaller electromagnetic field energy ( ω = 3. mev ). As far as the RTS with arbitrary ratio of input and output barriers widths ( δ= / ) is concerned, the pattern of evolution of conductivity logarithms and their components during various quantum transitions is more various (Fig. 3c f). It appears that the number of OCs can change from zero to three depending on the values δ and. Figs. 3c f present the dependences ln σ n=,3,4 and ln σ on b for the RTSs with constant sum of outer barrier widths ( = 6 nm ) and four characteristic ratios of input and output barrier widths ( δ= / 4; /3; 3/; 4 ). From the figure one can see that if the output barrier width considerably exceeds the input one ( >> ), the system has not OC and is ineffective as detector (Fig. 3c, δ= / 4 ). Herein, the conductivity is formed by a prevalent reverse current ( σ >>σ ). When the input barrier width is less than output one ( < ) but a total width of the input and output ones is more than that of the output barrier

9 68 M. V. Tkach et al. 9 ( > ), the system has I OC, and II and III ones are degenerated into a single common OC at which the internal barrier is located in the third quarter of the common well (Fig. 3d, δ=/3). When the input barrier width exceeds, remaining commensurate with the output barrier width ( ), the system possesses either all three OCs (I, II, III) when a little bit exceeds or only the II and III ones because σ has a very small magnitude in the I OC and therefore, it is practically ineffective (Fig. 3e, δ=3/). When the input barrier width considerably exceeds the output one ( >> ), see Fig. 3f, δ=4, the conductivity possesses an OC at which the inner barrier can be located in the whole fourth quarter of a common well, since it is formed by a prevalent forward current ( σ >> σ ). As to the configuration at which the inner barrier is located in the second quarter of common well, the value σ is so small that the corresponding RTS is completely ineffective as a detector although the conditions: σ >>σ, σ >>σ = 3,4 are fulfilled. n 4. CONCLUSIONS In the paper it is established the fact that depending on the position of inner potential barrier respectively the outer ones, the three-barrier RTS has up to three geometrical configurations with the optimal conditions for operating either as a separate detector or an QCD active element. In the optimal configurations, due to the quantum transitions between the first and second quasi-stationary states, the forward electronic current essentially exceeds either reverse one or all the currents arising due to the transitions into all other quasi-stationary states. When the electromagnetic fields with the frequencies exceeding ten tera Hertz are detected, it is reasonable to use either two-barrier RTS with wider input barrier stable to negligible size deviations or three-barrier RTS with the inner barrier, located at the distance of a well quarter from a thinner output barrier comparing to the input one. The advantage of optimal three-barrier RTS configuration is that its active conductivity several ten times exceeds the two-barrier one. However, the disadvantage is its great sensibility to the position of inner barrier because a negligible (several angstroms) change in its position leads to the sharp conductivity decrease. Further, we are going to study the dynamical conductivity dependence on RTS output barrier width to reveal that one, ensuring the detector operating in the

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