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 a
new 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.Comment: 4 pages,3 Post-Script figure

A. M. Kahn

C. Maes

F. J. Higuera

H. Kroemer

H. Le Person

I. R. Cantalapiedra

J. B. Gunn

J. Kastrup

J. Rubinstein

L. L. Bonilla

M. C. Cross

M. J. Bergmann

M. P. Shaw

V. L. Bonch-Bruevich

English

arXiv

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.Peer Reviewe

Bonilla, Luis L.

Rodríguez Cantalapiedra, Inma

UPCommons

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

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.Peer ReviewedPostprint (published version

UPCommons. Portal del coneixement obert de la UPC

PHYSICAL REVIEW E SEPTEMBER 1997VOLUME 56, NUMBER 3Universality of the Gunn effect: Self-sustained oscillations mediated by solitary wavesL. L. BonillaEscuela Polite´cnica Superior, Universidad Carlos III de Madrid, Butarque 15, 28911 Legane´s, SpainI. R. CantalapiedraDepartament 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 purelyresistive circuit mediated by solitary wave dynamics of the electric field on an attached appropriate semicon-ductor. By means of an asymptotic analysis, it is argued that Gunn-like behavior occurs in specific classes ofmodel 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.EkIn semiconductors where the local current density as afunction of the local electric field is N shaped, the Gunneffect is a ubiquitous phenomenon @1–5#. The Gunn effect@6# consists of time-periodic oscillations of the electric cur-rent flowing through an external purely resistive circuit at-tached to a semiconductor sample subject to dc voltage bias.The current oscillations correspond to the generation, one-dimensional motion, and annihilation of solitary waves of theelectric field inside the semiconductor. Besides this, the on-set of the Gunn effect can be quite interesting, as the currentmay display intermittency accompanied by spatiotemporalstructures of the electric field inside the semiconductor @7#.Recently the onset of the Gunn instability was analyzed bysingular perturbation methods which provide the governingamplitude equation for long semiconductors @8#. Gunn-likephenomena may also explain the experimentally observedself-sustained oscillations of the current in doped weaklycoupled superlattices @9# whose dominant transport mecha-nism is resonant tunneling between adjacent quantum wells@10#. In these cases, the oscillations are due to recycling ofelectric-field wave fronts ~charge monopoles! instead of soli-tary waves @10#. The difference in the type of waves may betracked to the boundary condition at the injecting contact@11,12#. Gunn-like phenomena have also been numericallyobserved in a driven diffusive lattice-gas model of hoppingconductivity @13#.A natural question that comes to mind in relation withthese phenomena concerns their universality: Given that theGunn instability appears in widely different semiconductorsystems and models, what are the features a given model hasto have in order to present the Gunn instability? Notice thatthe Gunn effect is in principle a nonequilibrium phenomenonwhich may happen far from any bifurcation points. Thus thequestion of its universality may not be related to linearizationabout fixed points of a renormalization transformation. Nev-ertheless an asymptotic analysis allows us to understanddeeply the Gunn effect and to try to give a precise meaningto the notion of universality far from equilibrium. This papertries to give an answer to the universality question, and italso puts the Gunn instability into perspective by comparingit to phenomena occurring in other pattern forming systems@14#.561063-651X/97/56~3!/3628~5!/$10.00From the study of the Gunn instability in semiconductormodels, we can extract the following common features thatseem to be necessary for its occurrence.~1! The model should be able to support solitary wavesmoving in a privileged direction on a large enough spatialsupport.~2! It should include an integral ~over space! constraint.~3! It should have appropriate boundary conditions ~Di-richlet, Neumann, mixed, etc.! which render unstable the sta-tionary solutions for certain values of the integral constraint.We shall illustrate these points by constructing a simplemodel that displays the Gunn instability:]u]t1K]u]x5]2u]x21J2g~u !, ~1!1L E0Ludx5f . ~2!In these equations the unknowns are u(x ,t) and J(t), witht.0 and 0,x,L; g(u) is a function having a local maxi-mum gM5g(uM) followed by a local minimumgm5g(um) for u.0 (0,uM,um), while K and f are non-negative parameters. Equations ~1! and ~2! are to be solvedwith an appropriate initial condition for u(x ,0)>0 and Di-richlet boundary conditions:u~0,t !5u~L ,t !5rJ~ t !,gmum,1r,gMuM. ~3!In semiconductor models u , J , and fL correspond to theelectric field, total current density and dc voltage bias, re-spectively. Boundary conditions ~3! correspond to Ohm’slaw relating the electric field and the current at the injectingand receiving contacts. ~We assume that both contacts haveidentical resistivity r.0 for simplicity.! Other boundaryconditions ~fixed u , mixed boundary conditions! do notqualitatively change the character of the solutions @1,11#.The model represented by Eqs. ~1! and ~2! with K50 andzero-flux boundary conditions instead of Eq. ~3! is known asthe constrained Cahn-Allen equation, and it was recently in-3628 © 1997 The American Physical Society56 3629UNIVERSALITY OF THE GUNN EFFECT: SELF- . . .troduced by Rubinstein and Sternberg as a nonlocal reaction-diffusion model of nucleation akin to the mass-conservingfourth-order Cahn-Hilliard equation @15,16#. Equation ~1!with a fixed constant J and K50 is the well-known bistableFisher-Kolmogorov-Petrovskiıˇ-Piskunov ~FKPP! equation,which includes among its possible solutions a variety of trav-eling fronts and pulses ~solitary waves! moving on an infiniteone-dimensional spatial support @17,14#. The pulses of theFKPP equation are unstable solutions: they either shrink orexpand when an infinitesimal disturbance is added @17#. Theglobal integral constraint ~2! and Dirichlet boundary condi-tions ~3! convert the FKPP equation into a model very simi-lar to the typical semiconductor ones: the constrained Cahn-Allen equation. This model does not present the Gunninstability if K50 because the x$2x symmetry implies nopreferred direction of motion for traveling waves. A largeenough nonzero convective term K.0 breaks the x$2xsymmetry, and it favors waves moving from left to right. Theresulting model satisfies conditions ~1!–~3! above, and it dis-plays the Gunn effect; see Fig. 1. It may be observed that thepresent model is also related to Kroemer’s model of theGunn effect in n type GaAs @2#: we just change the convec-tion coefficient to a constant K in Ampe`re’s law and set thediffusivity equal to 1 in the dimensionless Kroemer’s modelstudied in Ref. @12#. These changes exclude the straightfor-ward extension of our previous asymptotic analysis, as wecannot use the shock waves and particular solutions specificFIG. 1. ~a! The function J(t) for g(u)5100(u20.2)(u20.4)(u20.6). Parameter values are K52, L5100, r51.5, andf50.32. ~b! The corresponding profiles of u(x ,t) evaluated at thetimes marked in part ~a! of this figure.of Kroemer’s model to describe the Gunn effect @12#.To understand these results, we shall assume thate51/L!1. Then it is convenient to rewrite Eqs. ~1! and ~2!in terms of the ‘‘slow’’ variables s5et and y5ex . The re-sult ise]u]s1eK]u]y 2e2 ]2u]y2 5J2g~u !, ~4!E01u dy5f . ~5!In the limit e!0 the solutions of this system are piecewiseconstant: on most of the y interval u is equal to one oranother of the zeros of g(u)2J , separated by transition lay-ers that connect them. At y50 and 1 there are boundarylayers ~quasistationary most of the time!, which we will callinjecting and receiving layers, respectively. Let us assumethat uM,f,um , and denote by u1(J),u2(J),u3(J) thethree zeros of g(u)2J . Let the initial profile u(y ,0) satisfy-ing Eq. ~5! be a square bump u5u3(J) forY 1(0),y,Y 2(0) and u5u1(J) elsewhere, plus terms of or-der e , as in the time marked by Eq. ~1! in Fig. 1~b!. Locatedat y5Y 1 and y5Y 2 , Y 1,Y 2 , there are sharp wavefronts ofwidth O(e) connecting u5u1(J) and u5u3(J). This initialprofile will naturally evolve into the Gunn effect as timegoes on ~see below!. The initial value of J follows from Eq.~5!:f5u1~J !1@u3~J !2u1~J !#~Y 22Y 1!1O~e!. ~6!The boundary layers and the fronts connecting u1(J) andu3(J) are built from trajectories of the phase plane:dudj 5v ,dvdj 5mv1g~u !2J , ~7!where j5e21@y2Y i(s)# , c5dY i /ds , and m5K2c . Theboundary layers are separatrices connecting the vertical lineu5rJ in the phase plane (u ,v) to the saddles (u1,0) or(u3,0) for c50: u(x)!ui(J) as x!` and u(x)!ui(J) as(x2L)!2` ~i51 and 3! are the matching conditions. Foreach fixed value of J between gm and gM we can find aunique value c1(J) such that u(2`)5u1(J) andu(`)5u3(J) @corresponding to a heteroclinic orbit connect-ing (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 heteroclinicorbit connecting (u3,0) to (u1,0) with v,0#. The functionsc6(J) are depicted in Fig. 2. They intersect when J5J*given byJ*51u32u1Eu1u3g~u !du , c65K . ~8!Starting at s50, the fronts Y i(s) move with speedsdY 1ds 5c1~J !,dY 2ds 5c2~J !, ~9!3630 56L. L. BONILLA AND I. R. CANTALAPIEDRAwhereas their positions are related to the bias f through Eq.~6!. We find an equation for J by differentiating Eq. ~6! andthen inserting Eq. ~9! into the result:dJds 5A~J !@c1~J !2c2~J !# , ~10!A5~u32u1!2f2u1g381u32fg18.0, ~11!where gi8(J)[g8(ui), and we used that gui(J)5J implies]ui /]J51/g8(ui). This is a simple equation for J , demon-strating that J tends to J* exponentially fast. Notice that thisis a very simple explanation of the well-known observationthat a pulse detached from the boundaries moves at constantspeed and J , given by the equal area rule ~8!, @1#.After a certain time, the wave front Y 2 reaches 1, and wehave a new stage governed by Eq. ~6! with Y 251 and Y 1given by Eq. ~9!. The equation for J becomesdJ/ds5Ac1.0, and its solution increases @compare J and uat time ~2! in Fig. 1# until it surpasses the value Jc such thatu2(J)5rJ . ~At Jc , @]u/]x#x50 changes sign and the qua-sistationary injecting layer becomes unstable! @12#. Let s1 bethe earliest time at which J5Jc . After s5s1 , the profile ofu changes within the boundary layer at y50: this injectinglayer becomes unstable, and it sheds a new wave during afast stage described by the time scale t5(s2s1)/e . To findwhat happens next we need to perform a more complicatedanalysis keeping O(e) terms in the outer ~bulk! expansion ofu and J , and just the leading-order term in all inner expan-sions ~boundary layers and wave fronts!. This calculation hasbeen performed in detail for a semiconductor model @18#. Itcan be shown that the sheddingFIG. 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).of a new wave from the injecting layer is governed by thefollowing 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)5rJ(t;e), with J(t;e)5Jc1eJ (1)(t),J ~1 !~t !5h8~t!1ah~t!2gE2`te2b~t2t !h~ t !dt , ~12!h~t!5~u32u1!c1~t2t0!2E0`@u~x ,t!2u1#dx ~13!~in this equation all functions of J are calculated at J5Jc ;t0 is a constant and a, b, and g are positive parameters! @18#,and the following matching condition on an appropriateoverlap domain: u(x ,t)2u0x;J(s)!1, as t!2` ,s!s12 . Here u0x;J(s) is the quasistationary injectinglayer solution of Eq. ~7! with m5K such thatu00;J(s)5rJ(s) and u0`;J(s)5u1(J(s)) for s,s1 ,J(s1)5Jc . The function h(t) is the area lost due to themotion of the old front during the time t minus the instanta-neous excess area under the injecting layer.The solution of the previous semi-infinite problem revealsthe formation, growth, and motion of a new pulse in theinjecting layer, driven by h(t) through the effective excesscurrent ~12!. This process ends when the new pulse isbounded by two well-formed wave fronts ~detached from theinjecting layer! which are located at Y 3 and Y 4 , Y 3,Y 4 @seethe u profile at time ~3! in Fig. 1~b!, in which Y 3 and Y 4 havealready moved from their initial positions O(e ln e) at thebeginning of this stage#. It may be seen that the injectinglayer becomes unstable and sheds a new wave when its widthreaches a critical size Dy5O(e ln e) @18#.If f is large enough, we have a stage where the old wavefront located at Y 1,1 coexists with the newly formed pulsebounded by the two wave fronts located at Y 3 and Y 4 :f5u1~J !1@u3~J !2u1~J !#~12Y 11Y 42Y 3!1O~e!.~14!Differentiating this equation and using that Y 1 and Y 3 movewith speed c1 whereas Y 4 moves with speed c2 , we obtaindJ/ds5A(2c12c2). Starting from Jc , J decreases furtherto J† @the zero of (2c12c2)# if 2c1(Jc),c2(Jc) ~thestable case with Jc.J† in Fig. 2!. After the old wave reachesy51, we again obtain Eqs. ~6!–~10! and recover the initialsituation. Thus a full period of the Gunn oscillation is de-scribed; see Fig. 1. On the other hand, if 2c1(Jc).c2(Jc)(J*,Jc,J†), J increases after the formation of the newpulse, and it is possible for the injecting layer to shed morewaves into the bulk, as shown by the numerical simulationsof Fig. 3. How many waves are shed depends both on thevalue of Jc ~and therefore on the injecting resistivity r! andon the length L . A rough estimation would give(n11)c1(Jc)5nc2(Jc) as e!0 for the number n of shedwaves. This shedding mechanism seems to have the effect ofbreaking the spatial coherence of the sample, which maylead to complex spatiotemporal phenomena ~intermittencieswith a varying number of pulses present in the sample atdifferent times!. The unstable case will be further analyzed inthe near future.56 3631UNIVERSALITY OF THE GUNN EFFECT: SELF- . . .In conclusion, we have investigated what are the mainfeatures that a given model should have in order to presentthe Gunn effect. These features are demonstrated by studyinga simple model by means of a general asymptotic analysiscorroborated by direct numerical simulations. As a result theGunn effect is reduced to solving a sequence of very simpleproblems ~one equation for J each time! plus a canonicalproblem for shedding new pulses. Our asymptotic analysisexplains qualitatively and quantitatively the formation, mo-tion and annihilation of pulses in the Gunn effect. This worksheds light on several puzzling aspects of the Gunn oscilla-tions ~see the chapter on open problems in Ref. @19#!: ~i!Why do pulses move with the well-known equal-area-rulevelocity at constant J when they are far from the contacts@the corresponding current is a stable equilibrium of Eq.FIG. 3. ~a! Density plot for u(x ,t) with r54, f50.3 andg(u) as in Fig. 1 ~lighter color means larger u). Here multipleshedding of pulses occurs at the injecting layer: two pulses areformed during each period. The second shed reaches and overtakesthe first one. ~b! The corresponding profile of J(t).~10!#? ~ii! How does the wave speed change when it arrivesto the receiving contact? ~iii! How are new waves created atthe injecting contact? In addition, we have described an in-stability mechanism consisting of multiple pulse sheddingduring each oscillation of J , which appears for appropriatevalues of the boundary parameters at the injecting contact.Similar work has been performed in diverse semiconductormodels: Gunn oscillations in ultrapure closely compensatedp-type Ge @18#, Kroemer’s model of Gunn oscillations inbulk n-type GaAs @20#, and slow oscillations in semi-insulating GaAs @21#. A modification of the asymptoticmethod presented here describes the charge monopole recy-cling responsible for the self-oscillations in n-doped weaklycoupled superlattices @22#. Irrespective of the physicalmechanism responsible for the existence of the wave frontand pulses, our asymptotic method describes the Gunn oscil-lations in these models. The model presented here perhapsillustrates in the simplest way what the method consists of:~i! find the equations and boundary conditions which charac-terize the shape of the wave fronts and their speed as func-tions of the current density J . ~ii! Derive the equations whichdetermine J as a function of the slow time scale dependingon the number of wave fronts present in the sample. Thefield profile follows adiabatically the evolution of J . ~iii!Add the semiinfinite problems responsible for wave sheddingat the contacts. The solution and matching of these problemsyields an approximation of the Gunn effect in the givenmodel. Of course, solving some of these steps may be initself a rather complicated technical problem for particularmodels requiring special asymptotics @21#.This work was supported by the DGICYT Grant No.PB94-0375, and by the EC Human Capital and Mobility Pro-gram 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, ThePhysics of Instabilities in Solid State Electron Devices ~Ple-num, 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. Bergmann, S. W. Teitsworth, L. L. Bonilla, and I. R.Cantalapiedra, Phys. Rev. B 53, 1327 ~1996!.@4# H. Le Person, C. Minot, L. Boni, J. F. Palmier, and F. Mollot,Appl. Phys. Lett. 60, 2397 ~1992!.@5# V. A. Samuilov, in Nonlinear Dynamics and Pattern Forma-tion in Semiconductors and Devices, edited by F.-J. Niedernos-theide ~Springer-Verlag, Berlin, 1995!, p. 220.@6# J. B. Gunn, Solid State Commun. 1, 88 ~1963!.@7# A. M. Kahn, D. J. Mar, and R. M. Westervelt, Phys. Rev. B43, 9740 ~1991!; 45, 8342 ~1992!.@8# L. L. Bonilla and F. J. Higuera, SIAM ~Soc. Ind. Appl. Math.!J. Appl. Math. 55, 1625 ~1995!.@9# J. Kastrup, H. T. Grahn, R. Hey, K. Ploog, L. L. Bonilla, M.Kindelan, M. Moscoso, A. Wacker, and J. Gala´n, Phys. Rev. B55, 2476 ~1997!.@10# L. L. Bonilla, in Nonlinear Dynamics and Pattern Formationin Semiconductors and Devices ~Ref. @5#!, p. 1.@11# H. Kroemer, IEEE Trans. Electron Devices ED-15, 819~1968!.@12# F. J. Higuera and L. L. Bonilla, Physica D 57, 161 ~1992!.@13# C. 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 flowwhich conserves mass and is local, the constrained Cahn-Allenequation is a simple nonlocal gradient flow that conservesmass. See a discussion and comparison of the two models inRef. @15#. A nice analysis of the front motion in the one-dimensional 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 Biom-3632 56L. L. BONILLA AND I. R. CANTALAPIEDRAathematics Vol. 19 ~Springer-Verlag, New York, 1990!, Chap.11; P. C. Fife, Mathematical Aspects of Reacting and DiffusingSystems, Lecture Notes in Biomathematics Vol. 28 ~Springer-Verlag, New York, 1979!.@18# L. L. Bonilla, P. J. Hernando, M. A. Herrero, M. Kindelan, andJ. J. L. Vela´zquez, Physica D ~to be published!.@19# V. L. Bonch-Bruevich, I. P. Zvyagin, and A. G. Mironov, Do-main Electrical Instabilities in Semiconductors ~ConsultantsBureau, 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 pub-lished!.

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

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

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