Patterns in bistable resonant-tunneling structures

PHYSICAL REVIEW B VOLUME 56, NUMBER 20 Patterns in bistable resonant-tunneling structures 15 NOVEMBER 1997-II B. A. Glavin an V. A. Kochelap Institute of Semiconuctor Physics, Ukrainian Acaemy of Sciences, Pr. Nauki 45, Kiev 252028, Ukraine V. V. Mitin Electrical an Computer Engineering Department, Wayne State University, Detroit, Michigan 48202 Receive 20 November 1996 We report a theoretical investigation of the phenomenon of the formation of patterns transverse to the tunneling current in resonant-tunneling ouble-barrier heterostructures. Such patterns arise in heterostructures with an intrinsic bistability of the current-voltage characteristic. The patterns are characterize by a nonuniform istribution of resonant electrons in the quantum-well layer an, consequently, a nonuniform tunneling current ensity through the heterostructure. Patterns exist for coherent an for sequential mechanisms of the resonant tunneling. Possible types of stationary patterns epen on the applie voltage, an can be controlle by conitions on the eges of the heterostructure. In fact, the patterns are two or three imensional in character, since the nonuniform electron istributions inuce a complex configuration of the electrostatic potential in barrier regions. In aition to stationary patterns, moving patterns are consiere. They escribe the switching of the heterostructure from one uniform state to another. S0163-18299705544-6 I. INTRODUCTION Resonant tunneling through a ouble-barrier heterostructure was observe two ecaes ago, 1 but this phenomenon still attracts consierable interest because of its funamental character an increasing number of prospective applications microwave oscillations, 2 circuit applications, 3,4 cascae lasers, 5,6 etc. A resonant quasiboun state is forme in a quantum well between two barriers. These barriers separate the well from electroes heavy ope emitter an collector regions. The energy of the quasiboun state, measure with respect to the bottom of the quantum well, 0, is usually chosen to be above the Fermi level E F of electroes when no voltage is applie. The applie bias shifts the energy of quasiboun state below E F, an the electric current passes mostly through the quasiboun state. The current increases with the bias until the energy of the quasiboun state lowers below the bottom E 0 of the emitter ban. At this region the current falls to a low value, an negative ifferential resistance occurs. Different particular mechanisms can be responsible for the resonant tunneling coherent, 7 sequential, 8 phonon-assiste mechanisms, 9,10 etc.. However, when the quasiboun state is in resonance with the emitter electron states, there is a finite ensity of electrons, i.e., a built-up charge, in the quantum well. This built-up charge etermines the voltage istribution across the heterostructure, an consierably affects the current-voltage characteristic. The other important effect inuce by the built-up charge is the intrinsic bistability of the system uner consieration. For some range of biases at a fixe bias, two stable states exist. One state is characterize by the large built-up charge, resonanttunneling conitions, an a large current; the other one correspons to resonance breaking, a lowering of the quasiboun state below the bottom of the emitter ban, an a low built-up charge an current. The possibility of electrical instability ue to an accumulation of the charge in the well was pointe out by Ricco an Azbel. 11 Subsequently, the intrinsic bistability was observe by a number of experimental groups 12 14 for various oublebarrier structures. Calculations an moeling of the bistability were one by various authors. 15 17 In these papers, the tunneling was consiere one imensional, an the transport through the ouble-barrier heterostructure was suppose to be epenent on only one coorinate, perpenicular to the barriers. Actually, most oublebarrier resonant tunneling structures are layere ones, an the tunneling electron can move not only across the layers vertical transport, but also along these layers horizontal, or lateral, transport. Because the applie voltage is uniformly istribute over the highly conuctive emitter an collector regions, a one-imensional picture of the electron transport through barriers an quantum-well layers is sufficient for regimes with a single state note, however, that transverse patterns in this case are possible in the structures with special conitions on their bounaries 18. However, for the case of bistability, ifferent states of the nonequilibrium system can coexist. This leas to nonuniform in the plane of the layers istributions of the tunneling current, an built-up charge an potential energy. That is, uner a bistable tunneling regime one can expect spontaneous formation of transverse patterns. There is a well-known analogy between light waves in the Fabry-Perot interferometer an the electron waves in oublebarrier structures. The analogy qualitatively illustrates the resonant behavior of electron transmission through the structures. The analogy can be extene to the bistability regimes. A meium with optical nonlinearity, embee insie a resonator, gives rise to optical bistability or multistability. 19 Uner these conitions, ifferent stationary an moving transversal patterns are realize. 19,20 In the case of tunneling, nonlinearity in the wave equation appears ue to electrostatic interaction. Similarly to a nonlinear interferometer, one can imagine various transverse patterns for tunneling in oublebarrier structures. In contrast to the case of a nonlinear inter- 0163-1829/97/5620/1334614/$10.00 56 13 346 1997 The American Physical Society

56 PATTERNS IN BISTABLE RESONANT-TUNNELING... 13 347 ferometer, the patterns of resonant tunneling can appear for both types of tunneling, coherent an incoherent. One of the goals of this paper is to emonstrate the possibility of these patterns. There are a variety of self-consistent patterns known in soli-state physics: stationary an moving Gunn omains, an current filaments at N- an S-shape current-voltage characteristics, respectively; 21 transverse omains in semiconuctors with equivalent valleys; 22,23 stationary an wave patterns uner resonatorless optical bistability; 24,25 etc. These an other known examples are relate to nonlinear classical electron transport, while, for the case analyze in this paper, at least in the vertical irection, the transport has a quantum character. Since horizontal electron transfer is the main process etermining the transverse patterns, let us consier it qualitatively. This transfer can be epicte as follows. The electron is injecte from the emitter to the well in general with a finite horizontal component of the momentum p or velocity v p/m* m* is the effective mass. The velocity epens on the position of the quasiboun-state energy with respect to the Fermi energy E F in the emitter: when the energy quasiboun level moves from E F through the bottom of the emitter ban E 0, the velocity changes from zero to the Fermi velocity v F 2E F /m*. For estimates, one can say that v an v F have the same orer of magnitue. We can introuce a characteristic time for horizontal transfer: a time of tunneling escape from the well es. The characteristic istance of the horizontal transfer is L ch v es. One can expect that the scale of the patterns in question is of the orer of L ch. For sharp resonant level from the uncertainty relation for the quasiboun state we can write 0 es. Combining this inequality with the fact, that E F, 0, an the kinetic energy of the horizontal motion m*v 2 /2 are of the same orer of magnitue, for the in-plane wave vector k we fin kl ch p L ch m*v2 es 0 es 1. The latter estimate shows that horizontal transfer can be consiere as classical. Base on this conclusion, we evelop a theory of the patterns, assuming that the vertical transport is quantum an the horizontal transfer is classical. From the same uncertainty conition we can euce that the characteristic scale L ch greatly excees the well with. We assume L ch is much larger than the thickness of the whole structure: L ch. The paper is organize as follows. In Sec. II the moel an basic equations necessary for an investigation of the patterns are given. In Sec. III we show the existence of bistability for uniform tunneling within the propose moel. An analysis of the patterns for the limiting case where a local approach to electron transport is applicable, is one in Sec. IV. A more general consieration base on the kinetic equation is presente in Sec. V. Section VI summarizes the main results of the paper. The erivation of some necessary equations, an methos of their simplification, are presente in Appenixes A D. 1 FIG. 1. The scheme an energy-ban iagram of the resonanttunneling structure. II. MODEL AND BASIC EQUATIONS Since the problem of the transverse patterns requires at least a two-imensional spatial analysis, we use a simple moel, showing the main features of the bistability an the patterns. We eal with the moel of a resonant-tunneling heterostructure, schematically shown in Fig. 1. The structure is treate as a system of three parts, weakly couple by tunneling: emitter (E), quantum well QW, an collector (C). The electroes E an C are usually heavy ope semiconuctors, an are suppose to be ieal conuctors with zero screening length. The energy height of the barriers B 1, an B 2 is V, an their thicknesses are B1 an B2 respectively. Charge accumulation in the well causes a change of the potential profile in the whole structure. It alters the position of the quasiboun state with respect to the bottom of the energy ban of the emitter an, in general, with respect to the bottom of the quantum well. We isregar the latter effect, an consier that the built-up charge shifts the well bottom an the quasiboun level equally. Such a case correspons to the very thin quantum well, where the built-up charge can be accounte for as an infinitely thin sheet. The thinner the well is with respect to B1 an B2, the better our moel escribes the real structure. Introucing the area concentration n of electrons in the well, for the assumption iscusse above we can write the Poisson equation in the form 4e2 znr where (r,z) is the electrostatic potential energy for electrons, rx,y, is the ielectric constant, an e is the elementary electric charge. The coorinate system is shown in Fig. 1. The bounary conitions at the electroes are 2 r,z B1 0, r,z B2, 3 where is the external voltage bias in energy units. Uner the conitions of weak coupling between emitter, quantum well, an collector, for the electron istribution

13 348 B. A. GLAVIN, V. A. KOCHELAP, AND V. V. MITIN 56 function in the well f (r,p,t) one can erive the Boltzmannlike kinetic equation see Appenix A, Eq. A5 f t p m* f f r r p Gr,t,p f es I f, where p is two-imensional momentum, (r)(r,z0) is the electrostatic potential energy in the well, G(r,t),p is the local rate of tunneling from the emitter to the well, es is the tunneling escape time, an I f is the collision integral for the electrons insie the well. As we state in Sec. I, lateral transport of resonant electrons is classical, which is reflecte in the semiclassical character of Eq. 4. One can see that Eq. 4 is a conventional Boltzmann equation with two aitional terms on the right-han sie: G an f / es. The first of these escribes tunnel injection of electrons from the emitter to the quantum well, an the secon one escribes the tunnel escape of electrons from the quantum well to the electroes. The classical character of the lateral transport is reflecte once more in the local character of tunneling injection an escape terms: G an es are functions of at fixe r. They are expresse through the tunneling probabilities an the Fermi istribution of electrons in the emitter see Appenix A. If these functions are known, Eqs. 2 an 4, along with the efinition of concentration 4 nr,t p f r,p,t, 5 compose the system of couple nonlinear equations. Besies the bounary conitions at the electroes Eq. 3, aitional bounary conitions have to be impose in the x,y plane. We can consierably simplify the system using inequality 1. Then the ifferential equation 2 with bounary conitions of Eq. 3 can be presente in the integral form of Eq. B1 as an equation for : B1 rkrrnr. 6 The kernel function K(r) is calculate in Appenix B. From Eqs. B4 an B5 one can see that K(r) has a maximum at r0, an ecays almost exponentially with characteristical length of the orer of. Since the spatial scale of the function n(r) isl ch, using Eq. 1 we can approximate rkrrnrnr rkrr. As a result, we obtain the solution of the electrostatic problem, B1 r 4e2 B1 B2 nr. Let us iscuss the bounary conitions in the x,y plane. For a heterostructure with infinite horizontal imensions, we require finite magnitues of solutions at x,y. For restricte horizontal imensions, ifferent kins of effects etermine the bounary conitions. The simplest is a straightforwar scattering of electrons at the eges of the quantum 7 well. Such an ege scattering can be consiere analogously to the case of thin metal layers, etc. 26,27 For two-imensional electrons limiting cases of iffusive an specular bounary scattering were stuie in Refs. 26 an 27. Also, parameters of the system at the eges thicknesses of the barriers an the well, etc. can be ifferent from those ones in the bulk. If these changes are localize in a region that is small with respect to L ch, they also can be inclue in the bounary conitions. III. BISTABILITY UNDER UNIFORM TUNNELING Let us show that the moel formulate above allows bistable vertical transport regimes with uniform tunneling in the x,y plane. In such a case the r an t epenences are absent, an, from the kinetic equation 4, one can fin the areal electron concentration n es g. Since the left-han sie of Eq. 8 is a function of, we obtain two algebraic equations 7 an 8 for two variables n an. It is convenient to rewrite this system as Ln 4e 2 B1 B2 B1 R. 9 For the particular heterostructure the latter equation has one controlling parameter: the external bias. The right-han sie is a linear function of an. The left one is more complicate function, having, generally, a superlinear epenence in the bias range, where the quasiboun level crosses the bottom of the emitter ban. This epenence can generate more than one solution of Eq. 9. We calculate the functions es () an g() for a heterostructure with parameters structure I V1 ev, m* 0.067m 0, 5.8 nm, B1 2 nm, 0 0.1 ev, 11.5, an a scattering broaening of the quasi-boun-state of 0.054 mev. In Fig. 2 the left- an right-han sies are shown for E F 56 mev an zero temperature. Cases a c correspon to ifferent biases. The epenences n() can be unerstoo as follows. Because the energy 0 excees the Fermi level E F at high, the injection of the carriers into the well is small, an the built-up concentration is low. If is negative an ecreases, the quasiboun state is shifte own an the concentration increases. At high negative the resonant state escens below the emitter ban bottom, the concentration sharply rops own in accorance with qualitative consieration. This results in a superlinear epenence n() see Figs. 2a 2c. This epenence changes only weakly with the total bias. This means that the main control parameter epenence comes from the right-han sie of Eq. 9. For voltage biases l, only one solution with a high electron concentration exists. At l 0.285 ev, the secon solution with a low concentration appears Fig. 2a. In the range l h 0.318 ev three solutions exist, an are well separate Fig. 2b. Two of them are stable; they correspon to the bistable regime of tunneling. At h two high-ensity solutions coalescence Fig. 2c, an isappear at h. The electron current through the heterostructure is shown in Fig. 3. In the bistability range the 8

56 PATTERNS IN BISTABLE RESONANT-TUNNELING... 13 349 FIG. 3. Z-shape current-voltage characteristic of structure I. 2 4e 2 B1 B2 m*, 11 where the tunneling times ew an cw are calculate at E F, sc is the scattering time see Appenix C, where we introuce sc. In this case the imensionless range of bistability q( h l )/E F can be evaluate as a function of k an. Note that can be expresse through the effective Bohr raius a B in the material of the system: a B 2 /4 B1 B2. In Fig. 4 the epenence q(k) is shown at 1.7. Accoring to Fig. 4, bistability exists at any k. But the bistability is more evelope for asymmetric heterostructures, for which ew sc cw / cw sc. This fact explains the epenence of the bistability on the temperature, which is usually observe in experiments: at higher temperatures sc ecreases, an this washes out the asymmetry of tunneling in the system. Note that a large bistability range is possible even for completely incoherent resonant tunneling, when sc cw, but ew sc IV. PATTERNS IN THE LOCAL APPROACH FOR THE HORIZONTAL TRANSFER FIG. 2. Self-consistent solutions of the bistability problem uner uniform tunneling for structure I: the right an left sies of Eq. 9 are shown separately. a 0.29 ev. b 0.305 ev. The otte line shows the epenence n 0 (), in the limit of zero broaening of the resonant level. c 0.32 ev. In Appenix C it is shown that sufficiently smooth transverse istributions of the electrons can be escribe by the iffusionlike ifferential equation C13, combine with the current-voltage characteristic has a Z-type shape; the high an low currents can be realize at the same voltage bias. If one assumes the resonant level to be infinitely sharp the corresponing n() epenence is shown in Fig. 2b, it is possible to analyze the epenence of the bistability range on the parameters of the heterostructure. For this purpose we introuce two imensionless parameters cw sc k ew cw sc, 10 FIG. 4. Depenence of the bistability range q( h l )/E F on the coefficient of asymmetry of the barriers k for 1.7.

13 350 B. A. GLAVIN, V. A. KOCHELAP, AND V. V. MITIN 56 solution of electrostatical problem of Eq. 7. To avoi cumbersome formulas, we introuce functions D 4e2 B1 B2 D, 12 R 4e2 B1 B2 g 1 es B1. 13 Then, for we obtain the equation with the nonlinear iffusivity an the source-rain term: t r D r R. 14 In Eq. 14 we neglect by the ifference between n 0 ()/t an n/t because the ifference has a higher orer of magnitue for smooth patterns. At the eges of the quantum well, rr e, we shoul impose bounary conitions. In the local approach these conitions can be written in the form of conitions on the horizontal flux at the eges: j n Sn e n, rr e. 15 Here j n is normal component of current C10. These bounary conitions take into account the fact that, near the eges for istances much less than L ch, the injection an escape rates can iffer from those in the bulk of the well layer. In general, the parameters S an n e epen on r. In the terms of the functions introuce in Eqs. 12 an 13, the bounary conitions for are D r S e, n e 4e2 B1 B2 n e, rr e. 16 For transverse patterns with a characteristical length L st, Eq. 14 an bounary conitions of Eq. 16 are vali for L st v F es. 17 Below we analyze this requirement. We restrict ourselves to a consieration of oneimensional patterns epening on y only. In this case for the stationary patterns we fin y D y R. The first an secon integrals of Eq. 18 are 0 y D2 1 DR, 0 D yy 0, 2 0DR 18 19 20 respectively. Here 0 an y 0 are two integration constants. The simplest way to classify possible types of implicit solutions of Eqs. 19 an 20 is to employ phase portrait FIG. 5. Z-shape current-voltage characteristic of structure II. analysis. It is easy to see that the uniform solutions of Eq. 9, stuie in Sec. III, correspon to zeros of R() an, consequently, to the singular points of the phase plane ( /y). In the bistable range of the bias, l, h, there are three or two for l, h singular points. As we shall see below, the local approach is vali for all within the bistability range only in the case of a weak bistability. Here we eal with a structure structure II with values of barrier thicknesses an scattering broaening ifferent from those use in Sec. III: B1 2 nm, B2 3 nm, an the scattering broaening 0.35 mev. The current-voltage characteristic of this structure is shown in Fig. 5. In Fig. 6 the phase portraits of Eq. 18 are shown for structure II for external bias within the bistable range. In fact, the phase portraits represent possible solutions of Eq. 18 on the (/y) plane. For all cases the two singular points the left l an the right r are sales, while the mile one m is the center; s an s label the separatrixes. Cases a c iffer in the behavior of the separatrixes. For case a one of the separatrixes (s) originates from the right sale, an finishes in the same sale forming the close trajectory. Case b is very special one, where two separatrixes, s an s connect the sales. Case c is similar to case a, but the close separatrix originates from the left sale. For all these cases the separatrixes isolate the region of the plane with close trajectories. These close trajectories an separatrixes correspon to solutions which are finite in space even for y. They escribe patterns in heterostructures with infinitely large transverse imensions. The close trajectories other then separatrixes give perioical patterns. The separatrixes correspon to the aperioical patterns: soliton type a, antisoliton type c, an kinklike b. Spatial epenences of the electron ensity, corresponing to the aperioical patterns, are shown in Fig. 7. The kinklike pattern occurs at unique bias c, which can be obtaine from the first integral 19: rd,c R, c 0. 21 l Note that Eq. 21 is the analog of the rule of equal areas, which is vali for many nonequilibrium patterns of ifferent origin. 21 25

56 PATTERNS IN BISTABLE RESONANT-TUNNELING... 13 351 FIG. 6. Solutions of Eq. 18 on the phase plane (/y) for structure II at ifferent voltages: a 0.244 ev c, b 0.2475 ev c, an c 0.246 ev c. The patterns shown in Fig. 7 can be interprete in terms of the electron current. Actually, solitonlike an antisolitonlike patterns correspon to aitional negative an positive built-up charges localize in finite omains of the quantumwell layer. The greater local electron concentration is ue to the larger tunneling injection of electrons into the well, an, therefore, to larger local electric current. Thus the solitonlike patterns of Fig. 7a the antisolitonlike patterns of Fig. 7c means a local increase ecrease in the electric current through the heterostructure, i.e., high low current strip layer. The kinklike pattern of Fig. 7b can be thought of as FIG. 7. Electron-ensity profiles in units 10 15 m 2 for the patterns in structure II corresponing to phase portraits a, b, an c in Fig. 6. a transient region between two uniform states with low an high built-up charges an currents. From the abovementione facts, can see that such a coexistence of the two states is possible only at certain bias c. Let us estimate the valiity of the local approach. From Eq. 18 one can estimate the length scale of the patterns as L st D es 1/2 where D an es are average values, an, 22

13 352 B. A. GLAVIN, V. A. KOCHELAP, AND V. V. MITIN 56 FIG. 8. The epenence of the characteristical length L ch on the potential energy in the quantum well for structure II. 4e2 B1 B2 L R. Here L an R are average values of the left an right sies of Eq. 9, an is a characteristical variation of within the pattern. As can be seen, D es L ch an the conition of valiity of the local approach is 1. This conition can be realize in two cases: a for all voltages within the bistability range if the latter is small; an b for voltages corresponing to the eges of bistability for the arbitrary range l, h : nearby l, it is vali for soliton types of solutions, nearby h, for antisoliton types. For the numerical examples of Fig. 7, the scale of the patterns is about 500 1000 nm. The epenence of L ch on for structure II is shown in Fig. 8. As can be seen, in this case the conition for valiity of the local approach is satisfie. A. Solutions for finite transversal imensions For a heterostructure with finite transverse imension, L y /2yL y /2 one shoul impose the bounary conitions that follow from Eq. 16: D y S e, y L y 2 23 FIG. 9. a Bounary conition curves an possible phase trajectories not in scale for structure II with finite transverse imensions. b Possible istributions of the electron ensity in units 10 15 m 2. Calculations are performe for structure II with horizontal imension L y 1030 nm. The ege conitions are symmetric an correspon to bounary conition curves (1) in a. c The same for asymmetrical ege conitions, which correspon to bounary conitions curves (1) an (2) in a. Bounary conitions can control possible patterns. In orer to emonstrate this, let us combine Eqs. 23 with the phase portraits illustrate in Fig. 9a. For simplicity, we assume that parameters S () an e are inepenent of. The bounary conition curves (1) are given for S () () 1 S 1 10 4 m/s an (Sn e ) () 1 (Sn e ) () 1 1.510 20 m 1 s 1. The physical meaning of these constants is the following. If these bounary conitions are a result of some generation an recombination in the region near the sies of the structure with size, for example, l s 10 nm, then S10 4 m/s correspons to the surface recombination time s l s /S1 ps, an the surface generation rate Sn e 1.510 20 m 1 s 1 correspons to prouction in this region of surface ensity n s Sn e s /l s 10 16 m 3 for the time s. The value of the external voltage is the same as for the phase portrait in Fig. 6a. The trajectories, satisfying the bounary conitions, have to start at the curve an en on the proper curve. The irection of motion along the trajectory is etermine by the conition that at positive /y the value increases uring this motion, an vice versa. For given transversal imensions one must select the trajectories for which

56 PATTERNS IN BISTABLE RESONANT-TUNNELING... 13 353 1 L y y. 24 The latter etermines the integration constant in Eq. 19. For large enough L y an the symmetric bounary conitions (1) an (1) a number of ifferent trajectories exist. For L y 1030 nm five of them are shown on the phase portrait in Fig. 9a they are marke s1, s2, s3, s4, an s5. Corresponing coorinate epenences are epicte in Fig. 9b. Applying such an analysis, one can see that, if at least one of the ege generation rates (Sn e ) () is large, so that the proper bounary conition curve oes not intersect the separatrix s, the patterns exist only at c.at c only single state with high is to be realize. Curve a1 in Fig. 9a is shown for such a case for the parameters S (2) 3.5 10 4 m/s, (Sn e ) (2) 4.510 20 m 1 s 1 the proper bounary conition curve (2) is presente in Fig. 9a. This result means also that such a bounary conition conceals the low current branch of current-voltage characteristic of the whole evice in the range l, c. Only one type of phase trajectory, a1, which correspons to the profile with high electron an current ensities, shown in Fig. 9c, can exist. Analogously, one can show that if there is aitional electron rain recombination on the bounaries, the high current branch is conceale in the range c, h. Strongly asymmetric bounary conitions, combining the abovementione ones at yl y /2, conceal the multivalueness. For the current-voltage characteristic this means the realization of a high current branch at c, an a low current branch at c. At a critical bias c the currentvoltage characteristic has a vertical portion. Obviously, each point of this vertical portion correspons to the kinklike pattern, whose position etermines the value of the current. Varying S () an (Sn e ), one can provie a number of ifferent patterns, incluing perioical ones. As a result, we can conclue that the bounary conitions rastically affect patterns an allow the manipulation of the current-voltage characteristic. B. Nonstationary solutions Above, we consiere stationary patterns. In general, Eq. 14 allows ifferent nonstationary solutions. The simplest of these are solutions in the form of autowaves, (y vt), where v is the velocity of such a wave. For these autowave processes the aitional term v(/y) appears in Eq. 18. The usual analysis of the phase plane for the latter equation shows that, for a fixe bias within the bistability range, there is a single velocity v(), for which a solution in the form of a moving kink can exist. The solution can be thought of as a front, switching the system from one uniform state to another. At c the velocity v is zero, an we obtain a stationary kinklike solution, iscusse above. In Fig. 10a the calculate v() is shown for structure II. The positive velocity correspons to the autowave, switching the system from a high to a low current state. FIG. 10. a Velocity of switching wave in units v F as a function of the voltage bias for structure II. b Depenences of switching wave velocity in units v F on imensionless voltage parameter ( l )/( h l ) for structure I. Soli line results of calculations within the steplike moel; ashe line results of variational calculations. work beyon the local approach if the resonant-tunneling structure emonstrates a wie range of bistability. For these cases one must analyze the kinetic equation 4 an algebraic equation 7. Intheapproximation the collision integral can be approximate as I f f 1 / sc, where f 1 is asymmetrical part of istribution function f 1 (y,p)f 1 (y,p), an sc is the scattering time. We are going to consier the case of ballistic electron horizontal transfer, which takes place if sc es. As shown in Sec. III, the wie bistability range can be realize if ew is smaller then sc an cw. This proves that, in structures with wie bistability, horizontal electron transfer can be ballistic. In this case the kinetic equation is f t p m* f y f y p G f. es 25 Here p labels the y component of momentum p. One can solve Eq. 25 in terms of the characteristic curves V. PATTERNS UNDER BALLISTIC REGIMES OF HORIZONTAL TRANSFER Let us consier the range of parameters for which the local approach is not applicable. In particular, one shoul pp 0 2 2m*y 0 ypp 0,y 0,y, 26 where p 0 is the momentum of the electron, injecte into the well at the point yy 0. The general solution of the kinetic equation has the form

13 354 B. A. GLAVIN, V. A. KOCHELAP, AND V. V. MITIN 56 y f y,p m* Pp,y,y Mp,y,yGPp,y,y,y, 27 where the kernel M(p,y,y) epens on the particular shape of the potential (y) an the bounary conitions. Results of calculations of M are presente in Appenix D. Using Eqs. 27, 5, an 7, one can obtain the following integral equation for : B1 4e2 B1 B2 m* p y P MG. 28 This nonlinear integral equation takes place of partial ifferential equation 4 an relationship 7. Nonlinear integral equation 28 is too complex to be solve analytically. Moreover, there is no general approach for a numerical solution. We analyze the problem of patterns, introucing two simplifications into Eq. 25. First, we assume the steplike character of G an es as functions of : G0, 1 2m* h F p2, es l, h, 0 0 0 0, 0 0 0 0, 29 30 where F is the Fermi function see Appenix A. This means that we assume that electron tunnel injection from the emitter to the well EW takes place only if the resonant level lies above the bottom of emitter ban an the rate of injection in this case oes not epen on the position of the resonant level. Secon, in Eq. 25, we neglect by a term, proportional to the force i.e., we consier the horizontal transfer as free motion of electrons. These assumptions are vali if a variation of within the pattern is smaller then E F : uner this conition i the population of states in the emitter, which are in resonance with a quasiboun state, is almost constant within the pattern; an ii the kinetic energy of electrons is large with respect to their potential energy. Within the escribe moel the only parameters of the potential profile which affect the solution of the kinetic equation are the positions of the bounaries of the injection region, where 0 0 i.e., where tunnel injection from the emitter exists. This allows us to solve the problem of patterns. Qualitatively, we obtaine the same results as for the case of weak bistability in the local approach. The critical voltage at which the kinklike pattern exists correspons to the center of the bistability region: c ( l h )/2. At l c a solitonlike pattern is possible, while at c h an antisolitonlike pattern can be realize. At T 0 the with of soliton L s the with of the injection region is etermine by the equation 2 l h l 1F L s v F h, 31 an the with of antisoliton L a with of the region with no injection is etermine by the equation where 2 l h l 1F L a v F l, Fx 4 0 1 1z 2 exp x zz. 32 Autowave patterns exist as well. Their spee v is etermine by the equation l 1 1 h l 2 v v F 1 v2 1/2 v F 2 arcsin v v F. 33 In Fig. 10c the epenence of v/v F on the voltage parameter ( l )/( h l ) is shown by the soli line. In orer to obtain results without the above-mentione assumptions, we applie the following variational proceure for the solution of self-consistent integral equation 28. Let us introuce the functional J yl 34 where B1 4e2 L B1 B2 m* p y P MG. 35 Functional J equals zero for the exact solution of Eq. 28. For a particular solution we can choose some probe functions pr (y,c i ), where c i are variational parameters. These parameters are etermine by the conition of minimization of J(c i ). Using this metho, we analyze all three types of basic solutions: soliton, antisoliton, an kinklike. Here we present the switching kinklike autowaves in the case of ballistic electron transfer. As in this case we eal with a nonstationary kinetic equation one must introuce an aitional shift m*v in the momentum epenence of G in Eq. 27. Applying ifferent probe functions, we foun that the best fit correspons to an arctanlike spatial epenence of : pr 1 2 l r 1 r l arctany, 36 where l an h are the potential energies in the well, corresponing to the low an high charge ensity uniform states. This kin of probe function has two parameters: an the switching velocity v. The epenence of v/v F on the imensionless voltage ( l )/E F for structure I is shown in Fig. 9b. As iscusse in Sec. IV B, the positive velocity means switching from a high-current state. One can see that the latter result is in an agreement with approximation of Eq. 33. In the case of the wie range of bistability an ballistic horizontal electron transfer the spatial scale of the patterns is of the orer of L ch. The strong epenence of the tunneling injection rate on the position of the resonant level an the ballistic motion of electrons in the quantum well lea to a consierable spatial broaening of the collector current en-

56 PATTERNS IN BISTABLE RESONANT-TUNNELING... 13 355 FIG. 11. The spatial epenence of the current for the solitonlike pattern. In the upper part the current fiel is epicte. In the lower part the emitter curve 1, collector curve 2, multiplie by factor 5, an two-imensional lateral curve 3 currents are presente. Calculations are one for structure I. The voltage bias correspons to the conition ( l )/( h l )0.3. sity with respect to the emitter current ensity. In Fig. 11, spatial epenences of emitter, collector, an twoimensional lateral current ensities are shown for the solitonlike pattern in structure I at ( l )/( h l )0.3. In the upper part of Fig. 11 the current fiel in the structure is shown the spacing between the current lines is proportional to the value of current ensity. The current fiel in the quantum-well layer is presente conitionally. In the lower part of Fig. 11 all three currents are plotte. One can see consierable current leakage over the quantum well. VI. DISCUSSION Different layere heterostructures with the resonanttunneling mechanism of the carrier transport frequently emonstrate a bistable behavior of the tunneling current. The physical reason for the intrinsic bistability is the ynamic builup of the electric charge, which leas to two possible positions of the quasiboun state in the quantum well: low an high currents at the same voltage bias. In general, the electric charge can buil up in the quantum well between the barriers, in the emitter spacer notch before the barriers, in the barrier layers for heterostructures of type II, etc. As a result a variety of current-voltage characteristics with the intrinsic bistability is observe: a Z type, 12 14 an S type, 28 more complex butterflylike, 29 etc. The bistable effects attract attention, are well unerstoo, an are escribe by ifferent one-imensional theories. In the framework of such one-imensional approaches only carrier motion perpenicular to heterojunctions is taken into account, an parallel lateral carrier transfer is isregare. Meanwhile, lateral carrier transfer exists in layere structures an is of importance uner bistable effects. The lateral transfer brings about the simultaneous coexistence of ifferent possible states of the system, i.e., the formation of selfsustaine spatially nonuniform istributions of the built-up charge, resonant current, etc. The formation of patterns is well known in macroscopic soli-state physics, but bistable resonant tunneling systems provie an example with the quantum character of the carrier motion at least in one perpenicular irection. In this paper we evelope an approach which allows us to consier the tunneling in a ouble-barrier structure, to take into account self-consistently the nonuniform built-up charge an lateral carrier transfer. We obtaine that the patterns in question are characterize by a lateral scale exceeing the thickness of the structure in the perpenicular irection consierably. The scale of the patterns L ch is etermine by the time of electron escape from the well an the Fermi velocity of electrons in the emitter (L ch v F es ). This is a result of the ballistic or quasiballistic character of electron horizontal transfer. The shape of the patterns epens on the applie bias, an can be of soliton, antisoliton, an kinklike forms. For heterostructures with finite imensions of the layers, the patterns can be more complicate. Conitions on the eges of the heterostructure shoul be involve in consieration. The approach evelope allowe us to consier ifferent ege conitions. We showe that the number of patterns an their properties are strongly influence by these conitions. In particular, these ege conitions can cancel some branches of the current-voltage characteristic corresponing to low or high current through the entire structure. In this context, we point out that, espite a vast number of papers on resonant tunneling, only a few pai attention to the ege effects on the tunneling current. 18,30 Our analysis showe that lateral transfer has a large characteristic scale, an the ecrease in lateral imensions of resonant structures shoul certainly lea to size effects in the resonant tunneling. It is worth mentioning that the effect of horizontal electron leakage can be important even if the structure oes not possess bistable behavior. 18,30 Besies stationary patterns, mobile patterns have been foun. In particular, we escribe patterns which prouce a switching of the heterostructure from one uniform current state to the another. The velocity of such switching waves epens on voltage an is of the orer of the Fermi velocity v F. Let us briefly iscuss the problem of stability of stationary patterns. This is an important question since it etermines the possibility of pattern observation in the experiments. It requires special investigation. Here we restrict ourselves to a short iscussion of this problem in the case of weak bistability patterns escribe by Eq. 14. Equations of this type often appear in ifferent problems of self-organization see, for example, Ref. 21. Stability problem of such patterns against small perturbations can be formulate mathematically in terms of the Sturm-Liouville eigenvalue problem. For infinite imensions the result is that soliton an antisoliton patterns are unstable, while a kinklike pattern is stable. If the system uner consieration has some imperfections or efects, all three basic solutions can be stabilize by pinning on efects. Uniform solutions are stable against small perturbations. However, the soliton an antisoliton patterns are those which inspire the propagation of switching waves, escribe in Sec. IV B. That is, uniform high an low current states are unstable with respect to strong perturbations. That is, the low high current uniform state at l c ( c h )

13 356 B. A. GLAVIN, V. A. KOCHELAP, AND V. V. MITIN 56 can be switche to the high low current uniform state by means of strong enough perturbation of built-in charge, localize in a finite spatial region. For finite lateral imensions of the structure, the situation is more complicate, an stability strongly epens on the bounary conitions at the eges of the heterostructure an the lateral imensions. In particular, for fixe values of the electron ensity at the eges of the quantum-well layer, patterns can be stable. 21 It is worth aing that the stability of the patterns can also epen on the properties of the external circuit in which the resonant-tunneling ioe is inclue. In conclusion, we stuie the effect of the pattern formation in ouble-barrier resonant-tunneling heterostructures with an intrinsic bistability of the current-voltage characteristic. The effect consierably involves the lateral carrier transport, an exists for both coherent an sequential mechanisms of the resonant tunneling. The patterns are characterize by an alternative position of the resonant level in the quantum well, a nonuniform istribution of resonant electrons in the quantum-well layer, an a nonuniform tunneling current ensity through the heterostructure. ACKNOWLEDGMENTS The authors woul like to thank Dr. J. Schulman for reaing the manuscript an iscussions, an Dr. F. Vasko an Dr. V. Sheka for iscussions. This work was supporte by U.S. ARO an by the Ukrainian State Committee for Science an Technology Grant No. 2.2/49. APPENDIX A: EQUATION FOR HORIZONTAL ELECTRON TRANSFER In electroes, electrons can be characterize by the z projection of the momentum p z or corresponing energy of vertical motion an lateral momentum pp x,p y. The istribution functions are suppose to be the Fermi functions F(EE F ): f e p,p z F p2 p z 2 2m* E F, f c p,p z F p2 2 p z 2m* E F. A1 For classical motion along the well, one can introuce the istribution function f (r,p,t), which epens on twoimensional vectors, rx,y an p. We assume that the transversal coorinate epenence of patterns is so smooth that the tunneling can be accounte as strictly vertical process along z at fixe r. Then, in terms of tunneling transitions between the emitter, quantum well, an collector EW, CW, the total erivative of f (r,p,t) can be written as 31 f t ew W p,p z ;p, f e p,p z f r,p 0 p,p z cw W p,p z ;p, f c p,p z f r,pi f, 0 p,p z A2 where the first an secon terms on the right-han sie are the rates of tunneling between the emitter an the well an between the well an collector, respectively. The probabilities of tunneling W (ew) an W (cw) generally epen on the electron scattering. The last term in Eq. A2 escribes changes in the istribution function ue to scattering of electrons uring their quasiclassical motion along the well. If we neglect broaening of the quasiboun level an assume the conservation of the electron lateral momentum upon tunneling, it is possible to write ew W p,p z ;p, w ew 0 0 p,p pz,p 0, p 0 2m* 0, A3 cw W p,p z ;p, w cw 0 0 p,p pz,p 1, p 1 2m* 0, A4 where w ew an w cw are the probabilities of one imensional tunneling through the emitter an collector barriers for the electron energies 0 an 0, respectively. For Eq. A3 the signs correspon to the transitions E W. For Eq. A4 the signs correspon to the transitions W C. The coefficients w ew an w cw o not epen on the momenta p, but they are, in general, functions of the potential profile of biase heterostructure; is the electrostatic potential energy in the well. In orer to calculate w ew an w cw, we efine the energy of the resonant level as follows. Since in our moel we assume a narrow quantum well, the position of the resonant level with respect to the well bottom is, mainly, etermine by the with an epth of the well. Finite thicknesses of the barriers cause small corrections to this value. Let 0 () be the position of the level if we neglect finite transmission coefficients of the barrier. Then, the true resonant energy as a function of the bias can be written as 0 0 1 0,, 2 A4 where V () 0, an,,1 12 V e 2 12 Ve 2, 2 3 2m* 1/2 B1 2 V 3 V 3, A4 2 3 2m* 1/2 B2 2 V 3 V 3. For w (ew) an w (cw), we obtain w cw 8 w ew 8 V 0V 0 e 2 0, V 0 V 0 e 2 0.

56 PATTERNS IN BISTABLE RESONANT-TUNNELING... 13 357 Assuming a strong voltage bias, we can neglect tunneling from the collector to the well an rewrite Eq. A2: f t 1 0 ew F Gp, f es I f. p2 2m* 0E F f es I f A5 Here is the Heavisie function, an we introuce the local rate of tunneling injection of the electrons into the well layer: Gp, 1 ew 0 F p2 2m* 0E F, A6 an ew 1/w (ew), cw 1/w (cw), an es 1/(w (ew) w (cw) ). Formulas A3 A6 are vali for the limit of zero with of the quasiboun level. Broaening of this level can be important for the voltage bias, aligning the level an the bottom of the emitter ban. Two processes lea to the broaening: finite transmission of the barriers an scattering in the quantum well. We assume ensity of states, associate with the level in the form, 0 1 0 2 2, A6 where is energy of vertical motion, an is the broaening of the level. Two above-mentione processes contribute to the broaening: (1/ es 1/ sc )/2, where sc is the scattering time for the electrons in the well. Equation (A6) is vali for weak broaening: 0 see Ref. 32. Taking the broaening into account, one can moify the kinetic equation A5 as follows. In Eq. A6, for the injection rate one shoul substitute 0 0 0,0. A6 Of course, in this case one shoul put 0 E F in the argument of the Fermi function in Eq. A6. This moification of G(p,) is important for biases near the ege of the bistability region for any broaening, an it shoul be inclue into consieration at all biases if the bistable region is narrow. Writing the total erivative f/t in Eq. A5 in an explicit form, we obtain the basic kinetic equation 4. APPENDIX B: ONE-DIMENSIONAL APPROXIMATION FOR ELECTROSTATIC ENERGY The Poisson equation (4e 2 /)n(x,y,z) uner bounary conitions 3 has the solution x,y,z z 4e2 xyzgxx,y y,z,znx,y,z, B1 where G is the Green function with the bounary conitions G z0 G z 0. To satisfy these conitions, we present G as a series: Gxx,yy,z,z k For F k, one can fin kz B1 sin F k xx,yy,z. B2 F k x,y,z 1 kz sin B1 K 0 2 k x2 y, B3 where K 0 is the McDonal function. For the electrostatic energy in the well, (x,y, B1 )(x,y), we obtain Eq. 6 with the kernel function Krr 4e2 k 2 k B 1 sin K 0 k rr. B4 For the patterns, epening on the one transversal coorinate, say y, only the integral on y remains in Eq. 6. In this case the kernel function is Kyy 4e2 k sin 2 k B 1 e kyy/. B5 From Eqs. B4 an B5, it follows that the kernel function exponentially ecays for the argument, exceeing the thickness of the structure. For the smooth epenences n(x,y) this proves the one-imensional consieration of the electrostatic problem employe in Eq. 7. APPENDIX C: LOCAL APPROACH FOR HORIZONTAL TRANSFER Integrating kinetic equation 4 over p, one can easily obtain the balance equation for horizontal transport: n t iv gr,t n, es where we introuce the two-imensional electron flux an total injection rate p g 0 ew p F C1 p f r,p C2 m* p2 2m* 0E F. C3 From the theory of electron transport it is well known that an equation in form of Eq. C1 can be significantly simplifie for the case where a local approximation is applicable i.e., current C2 can be expresse through the concentration n, the potential, an their erivatives. For this the length an time scales of the problem shoul be sufficiently greater than the relaxation length an time of the electron momentum see, for example, Ref. 33. Such a hierarchy of the characteristic scales provies an almost symmetric istribution function in the momentum space. The analysis given in Sec. III showe that, for bistable regimes, momentum relaxation is not the fastest process, an can be completely absent. This

13 358 B. A. GLAVIN, V. A. KOCHELAP, AND V. V. MITIN 56 means, that electron istribution relaxation occurs mainly as a result of tunneling exchange between the quantum well an the electroes. This exchange can lea to an almost symmetric istribution function even if the momentum relaxation is negligible. The stanar approach cannot be use in the case in question. We introuce the local approximation in another way. Supposing that the potential (r,t) is given, we fin the concentration n(r,t) through an its erivatives on r an t. Then, this result an relationship 7 will compose the self-consistent system of equations which will escribe the patterns in the local approximation. In orer to erive the equation for n n,(/r),( 2 /r 2 ),(/t),... in the local approximation, it is convenient to present the istribution function as f (r,p,t) f () (r,p,t) f () (r,p,t), where f () (r,p,t) f () (r,p,t) is symmetric, while f () (r,p,t) f () (r,p,t) is an asymmetric function of p. Since the injection rate G is symmetric, we obtain the couple equations f p t m* f t p m* f r f r p p f f G, p es f p f es f sc, C4 C5 where we suppose elastic scattering insie the well, an write the collision integral in the approach with momentum inepenent scattering time sc. Then we assume a smoothness of the patterns an a graient expansion of all functions. To trace the erivation we formally introuce two imensionless parameters 1 an 2 by the following replacements: (/t) 1 (/t), (/r) 2 (/r) in the final formulas we set 1 2 1. Then all functions can be presente as expansions in series with respect to 1 an 2 : f f 00 1 f 10 2 2 f 02, f 2 f 01 1 2 f 11 3 2 f 03, nn 0 1 n 10 2 2 n 02, 2 01 1 2 11 2 3 03, C6 Thus the terms proportional to 1 s 1 2 s 2 contain the s 1 th power of the time erivative an the s 2 th power of the graient; from Eqs. C4 an C5 we easily fin the lowest approximations. f 00 r,p,t es Gr,p, f 01 r,p,t eff p m* n 0 es g, 01 effn 0 m* 2 effn 0 p, f r m* r D r, C7 C8 C9 C10 where 1/ eff 1/ es 1/ sc, an is average electron kinetic energy: FIG. 12. Moel potential profile illustrating three possible types of trajectories of ballistic electrons injecte in the quantum well. p p2 2m* f 00 C11 Now we can write the corrections of high orers for n: n 0 n 10 es n 0 t, n 02 es iv 01. C12 We restrict ourselves to the three-term approximation of n given by Eq. C6. Then we fin n in the form n, 2 r 2, t n 0 es n 0 t es iv D r, C13 where we set 1 2 1. Combining formula C13 an relationship 7, one can obtain the equation for (r,t). In the above erivation we neglect the terms with erivatives of the higher orer. This is vali only for the patterns with smooth time an coorinate epenences see criteria 17. APPENDIX D: KERNELS FOR INTEGRAL EQUATION 27 The moel fragment of potential (y) in Fig. 12 illustrates three possible cases for injecte electrons. If an electron is injecte with lateral motion energy, exceeing a maximum of (y)p 2 /(2m*)(y 0 )max (y), no turning points exist, an Mp 0,y 0,yyy 0 e hp 0,y 0,y, p 0 0, D1 where hp 0,y 0,y y0 y y es ypp 0,y 0,y. If p 0 0, one must replace y y 0 on the right-han sie of Eq. D1. If there is a single turning point y 2 *y 0, one can obtain, for p 0 0,