Universality of the Gunn effect: self-sustained oscillations mediated by solitary waves

Universality of the Gunn effect: Self-sustained oscillations mediated by solitary waves L. L. Bonilla Escuela Polite ´cnica Superior, Universidad Carlos III de Madrid, Butarque 15, 28911 Legane ´s, Spain I. R. Cantalapiedra Departament de Fı ´sica Aplicada, Universitat Polite ´cnica de Catalunya, Gregorio Maran ˜ o ´n 44, 08028 Barcelona, Spain ~Received 11 July 1996; revised manuscript received 5 November 1996! The Gunn effect consists of time-periodic oscillations of the current flowing through an external purely resistive circuit mediated by solitary wave dynamics of the electric field on an attached appropriate semiconductor. By means of an asymptotic analysis, it is argued that Gunn-like behavior occurs in specific classes of model equations. As an illustration, an example related to the constrained Cahn-Allen equation is analyzed. @S1063-651X~97!15508-8# PACS number~s!: 03.40.Kf, 05.60.1w, 07.50.Ek In semiconductors where the local current density as a function of the local electric field is Nshaped, the Gunn effect is a ubiquitous phenomenon @1–5#. The Gunn effect @6#consists of time-periodic oscillations of the electric current flowing through an external purely resistive circuit attached to a semiconductor sample subject to dc voltage bias. The current oscillations correspond to the generation, onedimensional motion, and annihilation of solitary waves of the electric field inside the semiconductor. Besides this, the onset of the Gunn effect can be quite interesting, as the current may display intermittency accompanied by spatiotemporal structures of the electric field inside the semiconductor @7#. Recently the onset of the Gunn instability was analyzed by singular perturbation methods which provide the governing amplitude equation for long semiconductors @8#. Gunn-like phenomena may also explain the experimentally observed self-sustained oscillations of the current in doped weakly coupled superlattices @9#whose dominant transport mechanism is resonant tunneling between adjacent quantum wells @10#. In these cases, the oscillations are due to recycling of electric-field wave fronts ~charge monopoles!instead of solitary waves @10#. The difference in the type of waves may be tracked to the boundary condition at the injecting contact @11,12#. Gunn-like phenomena have also been numerically observed in a driven diffusive lattice-gas model of hopping conductivity @13#. A natural question that comes to mind in relation with these phenomena concerns their universality: Given that the Gunn instability appears in widely different semiconductor systems and models, what are the features a given model has to have in order to present the Gunn instability? Notice that the Gunn effect is in principle a nonequilibrium phenomenon which may happen far from any bifurcation points. Thus the question of its universality may not be related to linearization about fixed points of a renormalization transformation. Nevertheless an asymptotic analysis allows us to understand deeply the Gunn effect and to try to give a precise meaning to the notion of universality far from equilibrium. This paper tries to give an answer to the universality question, and it also puts the Gunn instability into perspective by comparing it to phenomena occurring in other pattern forming systems @14#. From the study of the Gunn instability in semiconductor models, we can extract the following common features that seem to be necessary for its occurrence. ~1!The model should be able to support solitary waves moving in a privileged direction on a large enough spatial support. ~2!It should include an integral ~over space!constraint. ~3!It should have appropriate boundary conditions ~Dirichlet, Neumann, mixed, etc.!which render unstable the stationary solutions for certain values of the integral constraint. We shall illustrate these points by constructing a simple model that displays the Gunn instability: ] u ] t1K ] u ] x5 ] 2u ] x21J2g~u!,~1! 1 L E 0 Ludx5 f .~2! In these equations the unknowns are u(x,t) and J(t), with t.0 and 0,x,L;g(u) is a function having a local maximum gM5g(uM) followed by a local minimum gm5g(um) for u.0(0,u M ,u m ), while Kand f are nonnegative parameters. Equations ~1!and ~2!are to be solved with an appropriate initial condition for u(x,0)>0 and Dirichlet boundary conditions: u~0,t!5u~L,t!5 r J~t!,gm um ,1 r ,gM uM.~3! In semiconductor models u,J, and f Lcorrespond to the electric field, total current density and dc voltage bias, respectively. Boundary conditions ~3!correspond to Ohm’s law relating the electric field and the current at the injecting and receiving contacts. ~We assume that both contacts have identical resistivity r .0 for simplicity.!Other boundary conditions ~fixed u, mixed boundary conditions!do not qualitatively change the character of the solutions @1,11#. The model represented by Eqs. ~1!and ~2!with K50 and zero-flux boundary conditions instead of Eq. ~3!is known as the constrained Cahn-Allen equation, and it was recently inPHYSICAL REVIEW E SEPTEMBER 1997VOLUME 56, NUMBER 3 56 1063-651X/97/56~3!/3628~5!/$10.00 3628 © 1997 The American Physical Society troduced by Rubinstein and Sternberg as a nonlocal reactiondiffusion model of nucleation akin to the mass-conserving fourth-order Cahn-Hilliard equation @15,16#. Equation ~1! with a fixed constant Jand K50 is the well-known bistable Fisher-Kolmogorov-Petrovskiı ˇ-Piskunov ~FKPP!equation, which includes among its possible solutions a variety of traveling fronts and pulses ~solitary waves!moving on an infinite one-dimensional spatial support @17,14#. The pulses of the FKPP equation are unstable solutions: they either shrink or expand when an infinitesimal disturbance is added @17#. The global integral constraint ~2!and Dirichlet boundary conditions ~3!convert the FKPP equation into a model very similar to the typical semiconductor ones: the constrained CahnAllen equation. This model does not present the Gunn instability if K50 because the x↔2xsymmetry implies no preferred direction of motion for traveling waves. A large enough nonzero convective term K.0 breaks the x↔2x symmetry, and it favors waves moving from left to right. The resulting model satisfies conditions ~1!–~3!above, and it displays the Gunn effect; see Fig. 1. It may be observed that the present model is also related to Kroemer’s model of the Gunn effect in ntype GaAs @2#: we just change the convection coefficient to a constant Kin Ampe `re’s law and set the diffusivity equal to 1 in the dimensionless Kroemer’s model studied in Ref. @12#. These changes exclude the straightforward extension of our previous asymptotic analysis, as we cannot use the shock waves and particular solutions specific of Kroemer’s model to describe the Gunn effect @12#. To understand these results, we shall assume that e 51/L!1. Then it is convenient to rewrite Eqs. ~1!and ~2! in terms of the ‘‘slow’’ variables s5 e tand y5 e x. The result is e ] u ] s1 e K ] u ] y2 e 2 ] 2u ] y25J2g~u!,~4! E 0 1udy5 f .~5! In the limit e →0 the solutions of this system are piecewise constant: on most of the yinterval uis equal to one or another of the zeros of g(u)2J, separated by transition layers that connect them. At y50 and 1 there are boundary layers ~quasistationary most of the time!, which we will call injecting and receiving layers, respectively. Let us assume that uM, f ,um, and denote by u1(J),u2(J),u3(J) the three zeros of g(u)2J. Let the initial profile u(y,0) satisfying Eq. ~5!be a square bump u5u3(J) for Y1(0),y,Y2(0) andu5u1(J) elsewhere, plus terms of order e , as in the time marked by Eq. ~1!in Fig. 1~b!. Located at y5Y1and y5Y2,Y1,Y2, there are sharp wavefronts of width O( e ) connecting u5u1(J) and u5u3(J). This initial profile will naturally evolve into the Gunn effect as time goes on ~see below!. The initial value of Jfollows from Eq. ~5!: f 5u1~J!1@u3~J!2u1~J!#~Y22Y1!1O~ e !.~6! The boundary layers and the fronts connecting u1(J) and u3(J) are built from trajectories of the phase plane: du d j 5v,dv d j 5 m v1g~u!2J,~7! where j 5 e 21@y2Yi(s)#,c5dYi/ds, and m 5K2c. The boundary layers are separatrices connecting the vertical line u5 r Jin the phase plane (u,v) to the saddles (u1,0) or (u3,0) for c50: u(x)→ui(J)asx→`and u(x)→ui(J)as (x2L)→2`~i51 and 3!are the matching conditions. For each fixed value of Jbetween gmand gMwe can find a unique value c1(J) such that u(2`)5u1(J) and u(`)5u3(J)@corresponding to a heteroclinic orbit connecting (u1,0) to (u3,0) with v.0#and a unique value c2(J) such that u(2`)5u3(J) and u(`)5u1(J)@a heteroclinic orbit connecting (u3,0) to (u1,0) with v,0#. The functions c6(J) are depicted in Fig. 2. They intersect when J5J* given by J*51 u32u1 E u1 u3g~u!du,c65K.~8! Starting at s50, the fronts Yi(s) move with speeds dY1 ds 5c1~J!,dY2 ds 5c2~J!,~9! FIG. 1. ~a!The function J(t) for g(u)5100(u20.2)(u 20.4)(u20.6). Parameter values are K52, L5100, r 51.5, and f 50.32. ~b!The corresponding profiles of u(x,t) evaluated at the times marked in part ~a!of this figure. 56 3629UNIVERSALITY OF THE GUNN EFFECT: SELF-... whereas their positions are related to the bias f through Eq. ~6!. We find an equation for Jby differentiating Eq. ~6!and then inserting Eq. ~9!into the result: dJ ds5A~J!@c1~J!2c2~J!#,~10! A5~u32u1!2 f 2u1 g3 81u32 f g1 8 .0, ~11! where gi 8(J)[g8(ui), and we used that g„ui(J)…5Jimplies ] ui/ ] J51/g8(ui). This is a simple equation for J, demonstrating that Jtends to J*exponentially fast. Notice that this is a very simple explanation of the well-known observation that a pulse detached from the boundaries moves at constant speed and J, given by the equal area rule ~8!,@1#. After a certain time, the wave front Y2reaches 1, and we have a new stage governed by Eq. ~6!with Y251 and Y1 given by Eq. ~9!. The equation for Jbecomes dJ/ds5Ac1.0, and its solution increases @compare Jand u at time ~2!in Fig. 1#until it surpasses the value Jcsuch that u2(J)5 r J.~At Jc,@ ] u/ ] x#x50changes sign and the quasistationary injecting layer becomes unstable!@12#. Let s1be the earliest time at which J5Jc. After s5s1, the profile of uchanges within the boundary layer at y50: this injecting layer becomes unstable, and it sheds a new wave during a fast stage described by the time scale t 5(s2s1)/ e . To find what happens next we need to perform a more complicated analysis keeping O( e ) terms in the outer ~bulk!expansion of uand J, and just the leading-order term in all inner expansions ~boundary layers and wave fronts!. This calculation has been performed in detail for a semiconductor model @18#.It can be shown that the shedding of a new wave from the injecting layer is governed by the following semi-infinite problem for x.0, 2`, t ,`: u(x, t )~far from the old wave dying at y51!solves Eq. ~1! and u(0, t )5 r J( t ; e ), with J( t ; e )5Jc1 e J(1)( t ), J~1!~ t !5h8~ t !1 a h~ t !2 g E 2` t e2 b ~ t 2t!h~t!dt,~12! h~ t !5~u32u1!c1~ t 2 t 0!2 E 0 ` @u~x, t !2u1#dx ~13! ~in this equation all functions of Jare calculated at J5Jc; t 0is a constant and a , b , and g are positive parameters!@18#, and the following matching condition on an appropriate overlap domain: u(x, t )2u0„x;J(s)…!1, as t →2`, s→s12. Here u0„x;J(s)…is the quasistationary injecting layer solution of Eq. ~7!with m 5Ksuch that u0„0;J(s)…5 r J(s) and u0„`;J(s)…5u1(J(s)) for s,s1, J(s1)5Jc. The function h( t ) is the area lost due to the motion of the old front during the time t minus the instantaneous excess area under the injecting layer. The solution of the previous semi-infinite problem reveals the formation, growth, and motion of a new pulse in the injecting layer, driven by h( t ) through the effective excess current ~12!. This process ends when the new pulse is bounded by two well-formed wave fronts ~detached from the injecting layer!which are located at Y3and Y4,Y3,Y4@see the uprofile at time ~3!in Fig. 1~b!, in which Y3and Y4have already moved from their initial positions O( e ln e )atthe beginning of this stage#. It may be seen that the injecting layer becomes unstable and sheds a new wave when its width reaches a critical size Dy5O( e ln e )@18#. If f is large enough, we have a stage where the old wave front located at Y1,1 coexists with the newly formed pulse bounded by the two wave fronts located at Y3and Y4: f 5u1~J!1@u3~J!2u1~J!#~12Y11Y42Y3!1O~ e !. ~14! Differentiating this equation and using that Y1and Y3move with speed c1whereas Y4moves with speed c2, we obtain dJ/ds5A(2c12c2). Starting from Jc,Jdecreases further to J†@the zero of (2c12c2)#if 2c1(Jc),c2(Jc)~the stable case with Jc.J†in Fig. 2!. After the old wave reaches y51, we again obtain Eqs. ~6!–~10!and recover the initial situation. Thus a full period of the Gunn oscillation is described; see Fig. 1. On the other hand, if 2c1(Jc).c2(Jc) (J*,Jc,J†), Jincreases after the formation of the new pulse, and it is possible for the injecting layer to shed more waves into the bulk, as shown by the numerical simulations of Fig. 3. How many waves are shed depends both on the value of Jc~and therefore on the injecting resistivity r !and on the length L. A rough estimation would give (n11)c1(Jc)5nc2(Jc)as e →0 for the number nof shed waves. This shedding mechanism seems to have the effect of breaking the spatial coherence of the sample, which may lead to complex spatiotemporal phenomena ~intermittencies with a varying number of pulses present in the sample at different times!. The unstable case will be further analyzed in the near future. FIG. 2. The functions c6(J) for K52, and g(u) as in Fig. 1. We marked the value J5J†for which 2c1(J)5c2(J). 3630 56 L. L. BONILLA AND I. R. CANTALAPIEDRA In conclusion, we have investigated what are the main features that a given model should have in order to present the Gunn effect. These features are demonstrated by studying a simple model by means of a general asymptotic analysis corroborated by direct numerical simulations. As a result the Gunn effect is reduced to solving a sequence of very simple problems ~one equation for Jeach time!plus a canonical problem for shedding new pulses. Our asymptotic analysis explains qualitatively and quantitatively the formation, motion and annihilation of pulses in the Gunn effect. This work sheds light on several puzzling aspects of the Gunn oscillations ~see the chapter on open problems in Ref. @19#!:~i! Why do pulses move with the well-known equal-area-rule velocity at constant Jwhen they are far from the contacts @the corresponding current is a stable equilibrium of Eq. ~10!#?~ii!How does the wave speed change when it arrives to the receiving contact? ~iii!How are new waves created at the injecting contact? In addition, we have described an instability mechanism consisting of multiple pulse shedding during each oscillation of J, which appears for appropriate values of the boundary parameters at the injecting contact. Similar work has been performed in diverse semiconductor models: Gunn oscillations in ultrapure closely compensated p-type Ge @18#, Kroemer’s model of Gunn oscillations in bulk n-type GaAs @20#, and slow oscillations in semiinsulating GaAs @21#. A modification of the asymptotic method presented here describes the charge monopole recycling responsible for the self-oscillations in n-doped weakly coupled superlattices @22#. Irrespective of the physical mechanism responsible for the existence of the wave front and pulses, our asymptotic method describes the Gunn oscillations in these models. The model presented here perhaps illustrates in the simplest way what the method consists of: ~i!find the equations and boundary conditions which characterize the shape of the wave fronts and their speed as functions of the current density J.~ii!Derive the equations which determine Jas a function of the slow time scale depending on the number of wave fronts present in the sample. The field profile follows adiabatically the evolution of J.~iii! Add the semiinfinite problems responsible for wave shedding at the contacts. The solution and matching of these problems yields an approximation of the Gunn effect in the given model. Of course, solving some of these steps may be in itself a rather complicated technical problem for particular models requiring special asymptotics @21#. This work was supported by the DGICYT Grant No. PB94-0375, and by the EC Human Capital and Mobility Program Contract No. ERBCHRXCT930413. We thank M. J. Bergmann, P. J. Hernando, M. A. Herrero, F. J. Higuera, M. Kindelan, M. Moscoso, S. W. Teitsworth, J. J. L. Vela ´zquez, and S. Venakides for fruitful discussions. @1#M. P. Shaw, V. V. Mitin, E. Scho ¨ll, and H. L. Grubin, The Physics of Instabilities in Solid State Electron Devices ~Plenum, New York, 1992!. @2#H. Kroemer, in Topics in Solid State and Quantum Electronics, edited by W. D. Hershberger ~Wiley, New York, 1972!,p.20. @3#M. J. 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Maes and W. Vanderpoorten, Phys. Rev. B 53, 12 889 ~1996!. @14#M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 ~1993!. @15#J. Rubinstein and P. Sternberg, IMA J. Appl. Math. 48, 249 ~1992!. @16#While the Cahn-Hilliard equation is the simplest gradient flow which conserves mass and is local, the constrained Cahn-Allen equation is a simple nonlocal gradient flow that conserves mass. See a discussion and comparison of the two models in Ref. @15#. A nice analysis of the front motion in the onedimensional Cahn-Allen model can be found in the paper: L. G. Reyna and M. J. Ward, Eur. J. Appl. Math. 5, 495 ~1994!. @17#J. D. Murray, Mathematical Biology, Lecture Notes in BiomFIG. 3. ~a!Density plot for u(x,t) with r 54, f 50.3 and g(u) as in Fig. 1 ~lighter color means larger u). Here multiple shedding of pulses occurs at the injecting layer: two pulses are formed during each period. The second shed reaches and overtakes the first one. ~b!The corresponding profile of J(t). 56 3631UNIVERSALITY OF THE GUNN EFFECT: SELF-... athematics Vol. 19 ~Springer-Verlag, New York, 1990!, Chap. 11; P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics Vol. 28 ~SpringerVerlag, New York, 1979!. @18#L. L. Bonilla, P. J. Hernando, M. A. Herrero, M. Kindelan, and J. J. L. Vela ´zquez, Physica D ~to be published!. @19#V. L. Bonch-Bruevich, I. P. Zvyagin, and A. G. Mironov, Domain Electrical Instabilities in Semiconductors ~Consultants Bureau, New York, 1975!. @20#L. L. Bonilla, I. R. Cantalapiedra, G. Gomila, and J. M. Rubı ´, Phys. Rev. E 56, 1500 ~1997!. See also G. Gomila, J. M. Rubı ´, I. R. Cantalapiedra, and L. L. Bonilla, ibid. 56, 1490 ~1997!. @21#L. L. Bonilla, P. J. Hernando, and M. Kindelan ~unpublished!. @22#L. L. Bonilla, M. Kindelan, M. Moscoso, and S. Venakides, SIAM ~Soc. Ind. Appl. Math.!J. Appl. Math. ~to be published!. 3632 56 L. L. BONILLA AND I. R. CANTALAPIEDRA