AP04_3web.vp 1 Introduction The terahertz frequency band is recently usually consid- ered as the interval 300 GHz – 3 THz that corresponds to the submillimeter wavelength range between 1 mm and 100 �m or to photon energy within the range 1.2 – 12.4 meV. Despite great scientific interest the terahertz frequency range remains one of the least tapped regions of the electromagnetic spec- trum. Below 300 GHz we cross into the millimeter-wave bands. Beyond 3 THz is more or less unclaimed territory: the border between far-infrared and submillimeter radiation is still rather blurry. Recent rapid progress in nanoelectronics and high fre- quency technologies necessitates that heterojunctions, super- lattices, low-dimensional semiconductor structures, quantum wells and barriers are today standard building blocks of modern electronic devices, which find their application in the field of microwave and submillimeter technology or in photonics. The existence of quantum wells and barriers re- sults in the quantum-based mechanism of electron transport, thermionic emission across the barrier and the tunnelling (thermionic-field-emission) through the barrier. These effects should be treated by means of appropriate methods of quan- tum physics. Although the frequency of 1 THz appears to be very high, this is only an appearance. The frequency 1.8 GHz is at present in general use in mobile telephones. It is clear that 1.8 GHz cannot be equal to the transient frequency fT of tran- sistors in the integrated circuits of mobile telephones. The frequency 1.8 GHz should be even lower than the frequency f�, which is defined by the 3 dB drop of the current gain – this means that the frequency fT should be of the order (100 � 300) GHz. Even higher frequency bands are used in radar systems. The resonant tunneling diode (and similar structures with resonant tunneling) are recently typical de- vices for the terahertz fequency band [1]. Moreover, nearly the same theoretical approach that is used for investigat- ing of the terahertz frequency band can also be applied if the interaction of near-infrared radiation with photonic structures is studied. Quantum cascade lasers may provide terahertz bandwith for communications [2]. A typical situation in electronics is that a dc-bias together with a small ac-signal are applied to the structure. The calcu- lation of potential barrier transmittance with dc-bias only is a classical and well-known problem of quantum mechanics [3]. The application of a high-frequency signal to the barrier has been studied only in the last decade. A general method of solution is described in [4, 5] and developed in [6, 7]. However, papers concerning high-frequency phenomena are usually devoted to resonant tunnelling diodes (RTD) and dif- ferent types of potential barriers are rarely investigated, e.g. in [8] (the high-frequency potential step with zero steady state potential). The aim of this paper is to present results achieved in a theoretical investigation of the high-frequency electron transport across the rectangular, triangular, trapezoidal or parabolic potential barriers that are most frequently used in various nanoelectronic or photonic structures. 2 Steady-state transmittance of a potential barrier The term “steady-state” means that only dc-bias is applied to the potential barrier. Although we have mentioned above that the calculation of steady-state barrier transmittance is a classical problem of quantum physics, we briefly summarise and generalise these results. We will consider a rectangular, triangular, trapezoidal or parabolic potential barrier, see Fig. 1. We assume that there is a significant voltage drop only at the barrier region, i.e. out- side the voltage barrier there is no electric field and the elec- trons can be described as free. The barrier height Umax, the barrier width xB, and in fact the whole barrier profile, i.e. the potential energy U(x), depend on the external applied bias. The formulae for the potential energy U(x) are given in Fig. 1. Consider that the electrons are incident on the barrier from the left. In this case, in region A we have incident elec- trons described by the wave function e ik x0 (k0 is the wave vector related to the electron kinetic energy by E k m� �2 0 2 2 ) and electrons reflected from the barrier with wave function r e ik x0 0 � (r0 is the reflection amplitude). The electrons inside the barrier region B are described by some specific wave function that is dependent on the profile (shape) of the potential barrier. In region C (after the barrier) are trans- mitted only electrons with wave function t e iq x0 0 (t0 is the transmission amplitude) with the wave vector q0 satisfying 52 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 3/2004 Nanoelectronic Device Structures at Terahertz Frequency M. Horák Potential barriers of different types (rectangular, triangle, parabolic) with a dc-bias and a small ac-signal in the THz-frequency band are investigated in this paper. The height of the potential barrier is modulated by the high frequency signal. If electrons penetrate through the barrier they can emit or absorb usually one or even more energy quanta ��, thus the electron wave function behind the barrier is a superposition of different harmonics exp( ).�in t� The time-dependent Schrödinger equation is solved to obtain the reflection and transmission amplitudes and the barrier transmittance corresponding to the harmonics. The electronic current density is calculated according to the Tsu-Esaki formula. If the harmonics of the electron current density are known, the complex admittance and other electrical parameters of the structure can be found. Keywords: Nanoelectronic, terahertz, potential barrier, transmittance, Schrödinger equation. E q m� ��B � 2 0 2 2 . Thus the electron wave functions in re- gions A, B, C are: � � � A B C ( ) , ( ) ( ) ( ), ( ) x e r e x a f x b g x x t e ik x ik x� � � � � �0 0 0 0 0 0 iq x0 . (1) Let us turn our attention to the functions f x( ), g x( ) in Eq. (1). These functions contain the information on the potential barrier. In general the wave function �B( )x inside the barrier region is the eigenfunction of the correspond- ing hamiltonian, i.e. it is the solution of the stationary Schrödinger equation Hdc B B� �� E where Hdc � � � � 2 2 22m d dx U x( ) (2) For a rectangular barrier we obtain f x e g x e p m E Uip x ip x( ) , ( ) , ( )max� � � � �0 0 0 2 � (3) where p0 is real for E U� max (this corresponds to the electron emission over the barrier) and p0 is imaginary, p i0 0� � for E U� max (electron tunneling through the barrier). The re- sults for the potential barriers in Fig. 1 are summarized in Table 1. The electron wave functions for any type of barrier obey the standard boundary conditions at the interfaces x xB� � , x � 0 (to simplify the problem equal electron effective mass m is considered throughout the structure): � � � � � � � A B B B A B B B B C B ( ) ( ), ( ) ( ), ( ) ( ), ( � � � � � � � � � � x x x x 0 0 0 0) ( ) .� ��C (4) Substituting the wave functions we obtain a system of four linear equations for the unknown coefficients r t a b0 0 0 0, , , in (1). As the system is sufficiently simple, it can be solved ana- lytically. If the wave functions are known, the single electron quantum mechanical current densities of incident and trans- mitted electrons can be calculate according to the well-known formulae © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 53 Acta Polytechnica Vol. 44 No. 3/2004 max.)( UconstxU == ÷ ÷ ø ö ç ç è æ += Bx x UxU 1)( max ÷ ÷ ø ö ç ç è æ +-= B dc x x eVUxU 1)( max 2 max 1)( ÷ ÷ ø ö ç ç è æ += Bx x UxU -xB 0 x Umax E A B C F B rectangular barrier triangular barrier trapezoidal barrier parabolic barrier x F B -xB 0 Umax E A B C F B -xB 0 x Umax E A B C -xB 0 x Umax E A B C F B eVdc Fig. 1: Different types of potential barrier; E is the energy of the incident electron, U(x) is the potential energy in the barrier region B potential barrier (see Fig. 1) functions f(x), g(x) (see Eq. 1) parameters rectangular f x e ip x( ) � 0 , g x e ip x( ) � � 0 p m E U0 2� �( )max � triangular f x Ai( ) ( )� � , g x Bi( ) ( )� � Airy functions � �� � � � � � � � � � � �x x E UB 1 max , � � � � � � � � 2 2 1 3 mU x max � B trapezoidal f x Ai( ) ( )� � , g x Bi( ) ( )� � Airy functions � � � �( )x � �� � � 2 2 1 3 m U x ( )max �B B� , � � � � x e UB B B � �max parabolic f x U u( ) ( , )� � � , g x V u( ) ( , )� � � parabolic cylinder functions � �� �( )x xB � � � � � 8 2 2 1 4 mU x max � B , u E x m U � � � � B � 2 1 2 max Table 1: Electron wave functions in the barrier region j e im x x j e im inc trans � � � � � � � � 2 2 � � � � � A A A A C * * , * * . � � � C C C x x � � � � (5) The steady state barrier transmittance T Edc( ) is a function of electron energy E and it is defined as the ratio j jtrans inc . We introduce the following short notation: A f x B g x A f x B g x C f x x � � � � � � � � � � � � � � ( ), ( ), ( ), ( ), ( ), 0 0 0 0 B D g x x C f x x D g x x k q q � � � � � � � � � � � � ( ), ( ), ( ) , , / B B B 1 2 0 � � � � � �/ / for the rectangular / triangular / trapeziodal / parabolic barrier (6) The transmission amplitude t0 defined in (1) is then given by � �t e B i B C i C A i A D i Dikx0 2 1 2 1 12 � � � � � � � � � �� � � B ( )( ) ( )( ) (7) and the transmittance reads T E q k tdc( ) � 0 0 0 2. (8) Let us consider the N-Al1�xGaxAs / p �-GaAs abrupt hetero- junction with the following parameters: aluminium mole fraction 0.35, donor concentration in N-region 5×1017 cm�3, acceptor concentration in p�- region 1×1019 cm�3, the depletion layer extends in the N-region and its width is xn nm� 65 for zero bias, the heterojunction built-in voltage is Vbi �18. V. The electron effective mass is considered to be the same throughout the structure and equal to the effective mass of an electron in GaAs, thus m � 0 067. m el . The energy Umax is related to the built-in voltage Vbi and to the external applied voltage as U e V Vamax ( )� �bi . The conduction band profile for various forward bias and the barrier transmittance are shown in Fig. 2. 3 Electron wave function in barrier region with high frequency modulation We will now consider the case if the potential barrier is modulated by a high frequency signal V tac cos( )� where the angular frequency w � �( . )01 10 THz and the amplitude Vac is small and constant; such modulation is called homogeneous. The more general and more complicated case of non-homo- geneous modulation V x tac( ) cos( )� is not considered in this paper. The electron wave function �B( , )x t inside the barrier region is the solution of the time-dependent Schrödinger equation with the hamiltonian Hdc+Hac where Hdc according to (2) represents the barrier profile (including the dc bias) and Hac stands for the high frequency modulation i t m x U x e V � � � � � B dc ac B dc ac ac H H H H � � � � � � ( ) , ( ) , co 2 22 s( )�t (9) It can be immediately proved that the wave function is � � � �B ac B( , ) exp exp sin( )x t i E t i eV t� �� � � � � � � � � ( )x (10) where �B( )x is the solution of the stationary Schrödinger equation (2). We can see that the problem of describing electron wave functions in a uniform sinusoidally oscillating potential (9) is identical to the problem of frequency modula- tion in telecommunications or in signal theory. The wave function (10) can be considered as the frequency modulated wave with carrier frequency w E0 � � , see Fig. 3. We apply the Bessel function expansion to the second term of (10) exp sin( ) exp( )�� � � � � � � �i eV t J eV ip tp p ac ac � �� � � � ��� �� � . (11) This expansion enables us to consider the wave func- tion (10) as the superposition of harmonics exp( )�ip t� , p � � �0 1 2, , ,�, see Fig. 4. Thus, passing the barrier region, the electron is able to absorb or emit one or more energy 54 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 3/2004 V0 = 0 0 < V0 < Vbi V0 = Vbi EC(x) 70 60 50 40 30 20 10 0 barrier width [nm] 1.4 1.0 0.6 0.2 barrier height [eV] barrier transmittance Tdc electron energy E / kT external voltage: (left to right) V0 = 1.8 V = Vbi 1.7 V 1.3 V 1.1 V 0.8 V 0 10 20 30 40 50 60 1.0 0.8 0.6 0.4 0.2 Fig. 2: The parabolic potential barrier at the abrupt Np�-heterojunction for various forward bias Va and the corresponding steady-state transmittance Tdc(E) quantum p��. Its energy can be E p p� � � �� �� , , , ,0 1 2 , the � sign corresponds to the absorption/emission of energy quantum; p � 0 means no emission or absorption. As the electron energy can be E p p� � � �� �� , , , ,0 1 2 , the full electron wave function in regions A, B, C (see Fig. 1) should be the superposition of waves corresponding to these values of energy � � A ( , ) exp exp( ) exp( ) exp( x t i E t ik x r in tn � � � � � � � � � � � 0 � � � � � � � ���� �� � ik xn n ) , � � � � B ( , ) exp exp( ) ( ) ( ) x t i E t in t a f x b g x Js s n � � � � � � � � � � � � s sn eVac �� � � � � � � � �� � � � ����� �� ��� �� �� , (12) � �C( , ) exp exp( ) exp( )x t i E t t in t iq xn n n � �� � � � � ��� �� � � , E n k m E n p mn n� � � �� � � �� � � 2 2 2 22 2, �B . The function �A is the superposition of the incident wave and the reflected waves with the reflectance amplitudes rn, positive and negative values of n correspond to the absorp- tion and emission of energy quanta. The function �C is the superposition of transmitted waves with the transmittance amplitudes tn. The function �B describes the electron mo- tion across the barrier region (both the emission and the tunnelling). The boundary conditions (4) should be now applied to the wave functions (12). Evaluating these relations and equating the terms at harmonics exp( )�in t� we obtain a system of linear equations for the unknown coefficients an, bn, rn, tn, n � � �0 1 2, , ,� . To calculate all these coefficients it would be necessary to solve an infinite set of linear algebraic equations. It is clear that the probability of emission or absorption of en- ergy n�� decreases with increasing number n, thus the system could be terminated at some finite value of the indices n, s in (12). The series expansion in (11) that results in the double summation in (12) is well known in the theory of frequency modulated signals in telecommunications and we can apply the result of signal theory: in the series expansion (11) it is suf- ficient to consider only the terms n N� �0, ,� , where N is ap- proximately equal to eVac ��. If the high frequency signal is small it is sufficient to consider only N � 1 or N � 2, i.e. the generation of the first or the second harmonics or, in other words, the absorption or emission of one energy quantum �� or two energy quanta 2��. If the energy of the incident electron is E, the energy of the reflected or transmitted electron could be E (unchanged, no absorption or emission), E � ��, E � 2�� (absorption of one or two quanta), E � ��, E � 2�� (emission of one or two quanta). For © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 55 Acta Polytechnica Vol. 44 No. 3/2004 (E�h )t (E�h )t Fig. 3: The real part of the electron wave function according to Eq. (10) for a rectangular barrier (height 200 meV, width 20 nm), inci- dent electron energy 50 meV, microwave signal frequency 1.2 THz E + 2 hw … r2 E + hw … r1 E … r0 E - hw … r-1 E - 2 hw … r-2 t2 … E + 2 hw t1 … E + hw t0 … E t -1 … E - hw t -2 … E - 2 hw reflected waves transmitted waves incident electron wave barrier regionEC1 EC2 high- frequency modulation E Fig. 4: High-frequency modulation of a potential barrier; E is the incident electron energy, �� stands for the high-frequency quantum Re exp( sin( ))� �� � �� i eV tac �� � Re exp( )� �� � �� i E t � Re exp( ) exp( sin( ))� � � �� � �� i E t i eV t � � ac � � N � 1 (or 2) we obtain from Eq. (21) the system of 12 (or 20) linear equations for 12 (or 20) unknown coefficients; in gen- eral 8N � 4 linear equations for 8N � 4 unknown coefficients. Such system can be solved analytically in principle, but in practice numerical solution is used. For the purpose of illustrating the above sketched theory it is useful to obtain some analytical results. We will consider the rectangular potential barrier in Fig. 1. If the amplitude of high frequency signal Vac is small and the absorption or emission of only one quantum �� is considered the trans- mission amplitudes t�1 (absorption, the electron energy in region C is E � ��) and t�1 (emission, the electron energy in region C is E � ��) t eV t k p p x i p k q p � � � � � � � � � � � 1 0 0 0 0 0 0 0 0 2 ac B �� ( ) cos( ) � � � � � � � sin( ) ( ) cos( ) sin( p x k p p x i p k q p 0 0 0 1 0 0 0 0 B B p x� � � � � � � � � � � � � � � 1 B) . (13) The transmission amplitude t0 in (13) is given by the gen- eral formula (7), k0, q0, p0 are the electron wave vectors defined in (1) and (3) and p m E U� � � �1 2 ( )max� �� . Sim- ilarly as in (3) the quantities p0, p�1 are real for the electron emission over the barrier, and imaginary, thus p i0 0� � , p i� ��1 1� , for electron tunneling through the barrier. It can be seen in Fig. 3 that the modules t�1 exhibit a strong resonant character at electron energy that corresponds to the barrier height. 4 High frequency barrier transmittance If the transmission and reflection amplitudes are known, the wave functions (12) can be substituted to the general formulae (5) and the single electron quantum mechanical current densities of incident and transmitted electrons can be calculated. The high frequency barrier transmittance is de- fined as the ratio j jtrans inc and can be found for each harmonic. If we adopt the approximation k kn � 0 , q qn � 0 (as the electron energy is high compared with ��) and restrict the calculation to the first harmonics, i.e. to the absorption or emission of one energy quantum, we obtain j e k m j e q m t t t t t t e inc trans i t � � � � �� � � � 0 0 0 0 0 1 1 0 , ( ) (* * * �� �t t t t ei t0 1 1 0* * ) .� � � (14) We can see that jtrans in (14) includes the dc component proportional to t t0 0 * (it is related to those electrons that pass the barrier region without absorption or emission of energy) and the ac component exp( )�i t� related to electrons that emit or absorb one energy quantum in the barrier region. It is clear that the transmittance of the dc component is again given by (8), and it is not affected by the high frequency modulation. As usual in electronics, we use the goniometric functions sin( )�t , cos( )�t in (14) instead of complex functions exp( )�i t� and denote �� � ��arg( ) * *t t t t0 1 1 0 ; the ac component then reads �j e q m t t t t t t t trans ( ) * * * cos cos( )� � �� � # � � � � � 2 0 0 1 1 0 1 0 1 � �t t t1 0 1* sin sin( )� �# (15) and the corresponding transmittances are T q k t t t t T q k t t t t C S 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 2 2 � � � � � � * * * * cos , si � n .�1 (16) More generally, if the N-quantum approximation is con- sidered, the transmitted single electron quantum mechanical current density reads j j j e q m T e T trans trans n n N trans n n in t n � � � � � ( ) ( ) , ( � � � 0 0� * )e in t� (17) and the transmittances Tn in (17) for N � 3 are given by 56 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 3/2004 0.12 0.10 0.08 0.06 0.04 0.02 0.25 0.50 0.75 1.00 1.25 1.50 1.75 electron energy/barrier height transmission amplitude |t -1|, |t +1| |t -1| |t +1| p p/2 0 -p/2 -p 0.25 0.50 0.75 1.25 1.50 1.75 electron energy/barrier height transmission amplitude Arg( t -1), Arg( t +1) Arg( t +1) Arg( t -1) Fig. 5: Transmission amplitudes t�1, t�1 according to Eq. (13) for a rectangular barrier T t t t t t t t T t t t t 0 3 2 2 2 1 2 0 2 1 2 2 2 3 2 1 3 2 2 � � � � � � � � � � � � � � � � * * 1 1 0 0 1 1 2 2 3 2 3 1 2 0 1 1 � � � � � � � � � � � � t t t t t t t t T t t t t t t * * * * * * * � � � � � �� � � � t t t t T t t t t t t t t 0 2 1 3 3 3 0 2 1 1 2 0 3 * * * * * * (18) 5 Electrical parameters of quantum structure If the high frequency transmittances are known the elec- tric current density J n( )� for each harmonic can be calculated by means of the well-known Tsu-Esaki formula [1, 4]. We denote as f ( )� the Fermi-Dirac function integrated over the parallel-to-interface wave vector components f E E k T E E C F C F ( ) ln exp exp ( ) � � � � � � � �� � � � � � � 1 1 1 1 2 2 B �� � � � � � � � � � � � � � � eV k T dc B (19) with the dimensionless energy � � E k TB . The high frequency electric current harmonics can be written in the following way J A n t B n t A em k T q k n n n n ( ) cos( ) sin( ) ( ) � � � � � � � � B 2 2 2 3 0 02 2 � ( ) ( ) cos ( ) ( ) ( ) ( � � � � � � � � � T f B em k T q k n n n d B 0 2 2 2 3 0 02 2 � $ � � ) ( ) sin ( ) ( )T fn n� � � � �d 0 � $ (20) Observe that the origin of the higher order harmonics is related to the quantum character of electron transport in the barrier region rather than to the nonlinearity of current- -voltage or capacitance-voltage characteristics. Thus, their existence is an intrinsic property of the quantum structure. As our aim was to obtain the electrical parameters of the quantum structures, the relations (20) represent in fact the final result of the calculation. Using these formulae it is pos- sible to find, e.g., the module of the higher order current harmonics and their phase shift with respect to the modulat- ing signal (9) or the complex admittance and its real and imaginary part. All these quantities can be investigated as functions of the potential barrier profile (it is included in the barrier transmittance Tn ( )� ), dc bias (included in Tn ( )� and in f ( )� ) or the angular frequency of the high frequency modulat- ing signal (included again in Tn ( )� ). The real and imaginary part of the complex admittance of the rectangular barrier for the first three harmonics as a function of frequency is shown in Fig. 6. The slope of the imaginary part of the admit- tance � �Im ( )y C� �� implies that the capacitance is frequency independent. 6 Conclusions The theory related to the transmittance of different types of potential barriers with dc bias and small high frequency ac signal in the terahertz frequency band was presented in this paper. We have followed the way from the hamiltonian and the time dependent Schrödinger equation to the electric cur- rent densities and complex admittance that can be measured in experiments. At such a high frequency the following effects could play an important role: the electron inside the barrier region can emit or absorb one or even more energy quanta �� where � is the signal angular frequency. The electron wave function outside the barrier and consequently the electric current is a superposition of different harmonics exp( )�in t� . As we know from classical electronics, the generation of higher-order harmonics is due to the non-linearity of the current-voltage or capacitance-voltage characteristics, and it occurs only if the amplitude of the signal is sufficiently large. The origin of the higher-order harmonics at potential barri- ers is different: it is caused by the emission or absorption of one or more energy quantum and occurs even for a small signal; thus their generation is an intrinsic property of the sin- gle-barrier structure. The high frequency quantum effect on potential barriers represents an additional conductivity chan- nel and contributes with a small parallel admittance to the electronic parameters of the structure. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 57 Acta Polytechnica Vol. 44 No. 3/2004 Im ( y / ynorm ) w (THz) 0.1 1.0 10.0 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 10 -10 1 st harmonics 3 rd harmonics 2 nd harmonics 10 -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 0.1 1.0 10.0 Re ( y / ynorm ) w (THz) 1 st harmonics 2 nd harmonics 3 rd harmonics Fig. 6: The real and imaginary part of the complex admittance as functions of the modulation signal angular frequency for a rectangular potential barrier of height 300 meV and width 16 nm. The admittance is normalized by the quantity: y emk T e k TB Bnorm � � � �( ( ) ( ) . 2 2 2 3 112 272 10� � ��1m�2 7 Acknowledgments This research has been supported by the Czech Minis- try of Education in the framework of Research Plan MSM 262200022 MIKROSYT Microelectronic Systems and Technologies. References [1] Roblin P., Rohdin H.: High-speed heterostructure devices: from device concepts to circuit modeling. Cambridge Univer- sity Press, 2002. [2] Shore K. A.: “QC lasers may provide THz bandwith for communications”. Laser Focus World, June 2002, p. 85–91. [3] Bransden B. H., Loachaim C. J.: Introduction to quantum mechanics. Addison-Wesley Longman Ltd., 1989. [4] Coon D. D., Liu H. C.: “Time-dependent quantum-well and finite-superlattice tunnelling”. Journal Appl. Phys., Vol. 58 (1985), p. 2230–2235. [5] Liu H. C.: “Analytical model of high-frequency resonant tunnelling: The first order ac current response”. Phys. Rev. B, Vol. 43 (1991), p. 12538–12548. [6] Truscott W. S.: “Wave functions in the presence of a time-dependent field: Exact solutions and their applica- tion to tunnelling”. Phys. Rev. Lett. Vol. 70 (1993), p. 1900–1903. [7] Fernando Ch. L., Frensley W. R.: “Intrinsic high- -frequency characteristics of tunneling heterostructure devices”. Phys. Rev. B, Vol. 52 (1995), p. 5092–5103. [8] Tkachenko O. A., Tkachenko V. A., Baksheyev D. G.: “Multiple-quantum resonant reflection of ballistic elec- trons from a high-frequency potential step”. Phys. Rev. B, Vol. 53 (1996), p. 4672–4675. RNDr. Michal Horák, CSc. e-mail: horakm@feec.vutbr.cz Department of Microelectronics Brno University of Technology Faculty of Electrical Engineering and Communication Údolní 53 602 00 Brno, Czech Republic 58 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 3/2004