Self-sustained current oscillations in weakly-coupled superlattices are
studied by means of a self-consistent microscopic model of sequential tunneling
including boundary conditions naturally. Well-to-well hopping and recycling of
charge monopole domain walls produce current spikes (high frequency modulation)
superimposed on the oscillation. For highly doped injecting contacts, the
self-oscillations are due to dynamics of monopoles. As the contact doping
decreases, a lower-frequency oscillatory mode due to recycling and motion of
charge dipoles is predicted. For low contact doping, this mode dominates and
monopole oscillations disappear. At intermediate doping, both oscillation modes
coexist as stable solutions and hysteresis between them is possible.Comment: 4 pages, 4 figure

Aguado, Ramon

Bonilla, Luis L.

Moscoso, Miguel

Platero, Gloria

Sanchez, David

English

arXiv

Self-sustained current oscillations in weakly coupled superlattices are studied by means of a self-consistent microscopic model of sequential tunneling, naturally including boundary conditions. Well-to-well hopping and recycling of charge monopole domain walls produce current spikes - high-frequency modulation - superimposed on the oscillation. For highly doped injecting contacts, the self-oscillations are due to the dynamics of monopoles. As the contact doping decreases, a lower-frequency oscillatory mode due to recycling and motion of charge dipoles is predicted. For low contact doping, this mode dominates and monopole oscillations disappear. At intermediate doping, both oscillation modes coexist as stable solutions and hysteresis between them is possible. ©1999 The American Physical Society.We thank Rosa Lo´pez and Professor Holger Grahn for helpful discussions. This work has been supported by the DGES ~Spain!, Grant Nos. PB97-0088 and PB96-0875, by the European Union TMR Contract Nos. ERB FMBX-CT97- 0157 and FMRX-CT98-0180, and by the Community of Madrid, Project No. 07N/0026/1998.Peer Reviewe

Sánchez, David

Moscoso, M.

Aguado, Ramón

Digital.CSIC

PHYSICAL REVIEW B 15 AUGUST 1999-IVOLUME 60, NUMBER 7Current self-oscillations, spikes, and crossover between charge monopole and dipole wavesin semiconductor superlatticesDavid Sa´nchezInstituto de Ciencia de Materiales (CSIC), Cantoblanco, 28049 Madrid, SpainMiguel Moscoso and Luis L. BonillaEscuela Polite´cnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad 20, 28911 Legane´s, Spainand Unidad Asociada al Instituto de Ciencia de Materiales (CSIC), Cantoblanco, 28049 Madrid, SpainGloria Platero and Ramo´n AguadoInstituto de Ciencia de Materiales (CSIC), Cantoblanco, 28049 Madrid, Spain~Received 22 March 1999!Self-sustained current oscillations in weakly coupled superlattices are studied by means of a self-consistentmicroscopic model of sequential tunneling, naturally including boundary conditions. Well-to-well hopping andrecycling of charge monopole domain walls produce current spikes—high-frequency modulation—superimposed on the oscillation. For highly doped injecting contacts, the self-oscillations are due to thedynamics of monopoles. As the contact doping decreases, a lower-frequency oscillatory mode due to recyclingand motion of charge dipoles is predicted. For low contact doping, this mode dominates and monopoleoscillations disappear. At intermediate doping, both oscillation modes coexist as stable solutions and hysteresisbetween them is possible. @S0163-1829~99!06031-2#I. INTRODUCTIONSolid-state electronic devices presenting negative differ-ential conductance, such as resonant tunneling diodes, Gunndiodes, or Josephson junctions,1 are nonlinear dynamical sys-tems with many degrees of freedom. They display typicalnonlinear phenomena such as multistability, oscillations, pat-tern formation, or bifurcation to chaos. In particular, verticaltransport in weakly coupled semiconductor-doped superlat-tices ~SL’s! has been shown to exhibit electric-field domainformation,2–4 multistability,5 self-sustained currentoscillations,6–8 and driven and undriven chaos.9–11 Stationaryelectric-field domains appear in voltage-biased SL’s if thedoping is large enough. When the carrier density is below acritical value, self-sustained oscillations of the current mayappear. They are due to the dynamics of the domain wall~which is a charge monopole accumulation layer or, briefly, amonopole! separating the electric-field domains. This domainwall moves through the structure and is periodically re-cycled. The frequencies of the corresponding oscillation de-pend on the applied bias and range from the kHz to the GHzregime. Self-oscillations persist even at room temperature,which makes these devices promising candidates for micro-wave generation.7 Theoretical and experimental work onthese systems has gone hand in hand. Thus the paramountrole of monopole dynamics has been demonstrated by theoryand experiments. Monopole motion and recycling can be ex-perimentally shown by counting the spikes—high-frequencymodulation—superimposed on one period of the current self-oscillations: current spikes correspond to well-to-well hop-ping of a domain wall through the SL. In typical experimentsthe number of spikes per oscillation period is clearly lessthan the number of SL wells.7,12 It is known that monopolesare nucleated well inside the SL ~Refs. 7 and 8! so that thePRB 600163-1829/99/60~7!/4489~4!/$15.00number of spikes tells over which part of the SL they move.In this paper we study the nonlinear dynamics of SL’s.We have extended the model proposed in Ref. 14 for thestationary case in order to include the dynamics, i.e., the timedependence of the electronic current. From our model weobtain self-sustained oscillations of the current and currentspikes reflecting the motion of the domain wall as observedexperimentally. Furthermore, when contact doping is dimin-ished, we predict a crossover from monopole to dipole self-oscillations resembling those in the Gunn effect.13 Indeed,our results show that there is an intermediate range of contactdoping and a certain interval of external dc voltage for whichmonopole and dipole self-oscillations with different frequen-cies are both stable. Hysteretic phenomena then exist.II. MODEL AND SUPERLATTICE SAMPLE.We analyze the tunneling current through the SL bymeans of the transfer Hamiltonian. The dynamics is consid-ered in the model through Ampe`re’s law for the total currentdensity J5J(t):J5Ji21,i1eddVidt . ~1!Here Ji21,i is the tunneling current through the ith barrier ofthickness d,Ji ,i1152e\kBTp2m*(j51nmax E g@~e2eri1 !21g2#g@~e2eri11j !21g2#3Ti11~e!lnF 11e (ev i2e)kBT11e(ev i112e)kBTG de , ~2!4489 ©1999 The American Physical Society4490 PRB 60BRIEF REPORTSwhere erij is the j th resonant state of the ith well (nmax is thenumber of subbands participating in the transport! and Ti(e)is the transmission through the ith barrier. Scattering istreated phenomenologically by considering the spectral func-tions of the wells as Lorentzians (g is the halfwidth!. Thelast term in ~1! is the displacement current at the ith barrierwhere the potential drop is Vi and e is the static permittivity.There are N11 equations ~1! for the current from the emitterto the first well, the current from the i well to the i11 well,and from the N well to the collector.We include the Coulomb interaction in a mean-field ap-proximation by means of discrete Poisson equations relatingthe potential drops in wells (N unknowns!, barriers (N11unknowns!, and contacts ~two unknowns!. The boundaryconditions at the contacts contain four equations describingthe lengths of the depletion and accumulation layers as wellas the charge density at the leads ~four unknowns! ~see Refs.14 and 15 for a detailed discussion of the electrostaticmodel!. The other unknowns are the Fermi energies in thewells (N unknowns! and the total current. The final set of3N18 equations and unknowns is closed by two equationsof conservation of charge and voltage. The current dependsexplicitly on the Fermi levels and potential drops ~throughthe resonant level positions! and implicitly through thetransmissions, which depend on the local electrostaticdistribution.We have studied a 13.3-nm GaAs/2.7-nm AlAs SL at zerotemperature consisting of 50 wells and 51 barriers, as de-scribed in Ref. 12. Doping in the wells and in the contactsare Nw5231010 cm22 and Nc5231016 cm23, respec-tively.III. MONOPOLE-MEDIATED SELF-OSCILLATIONSOF THE CURRENTFigure 1~a! depicts the current as a function of time for adc bias voltage of 5.5 V on the second plateau of the SL I-Vcharacteristic curve. J(t) oscillates periodically at 20 MHz.Between each two peaks of J(t), we observe 18 additionalspikes. The electric-field profile is plotted in Fig. 1~b! at thefour different times of one oscillation period marked in Fig.1~a!. There are two domains of almost constant electric fieldseparated by a moving domain wall of ~monopole! chargeFIG. 1. ~a! Self-sustained oscillations of the total currentthrough the SL due to monopole recycling and motion. Bias is 5.5V and emitter doping, Nc5231016 cm23. ~b! Electric-field profilesat the times marked in ~a! during one period of the current oscilla-tion.accumulation ~which is extended over a few wells!. Mono-pole recycling and motion occur on a limited region of theSL ~between the 30th and the 50th well! and accompany thecurrent oscillation.7,8 Well-to-well hopping of the domainwall is reflected by the current spikes until it reaches the 46thwell which is close to the collector. Then the strong influenceof the contact causes that no additional spikes appear. In-stead the current rises sharply triggering the formation of anew monopole closer to the emitter contact but well insidethe SL; see Figs. 1~a! and 1~b!. The number of wells tra-versed by the domain wall ~almost! coincides with the num-ber of spikes per oscillation period, a feature not found inprevious models. Figure 1~b! shows the recycling of a mono-pole: between times ~1! and ~3! there is a single monopolepropagating towards the collector; at ~4! a new monopole isgenerated at the middle of the structure and the old one col-lapses at the collector.IV. CURRENT SPIKESWhat is remarkable in Fig. 1~a! ~as compared to previousstudies! are the spikes superimposed near the minima of thecurrent oscillations. Such spikes have been observed experi-mentally and attributed to well-to-well hopping of the do-main wall.12,16 They are a cornerstone in interpreting the ex-perimental results and in fact support the theoretical pictureof monopole recycling in part ~about 40%! of the SL duringself-oscillations. The identification between the number ofspikes and of wells traversed by the monopole rests on volt-age turn-on measurements supported by numerical simula-tions of simple models during early stages of stationary do-main formation.16 These models do not predict spikessuperimposed on current self-oscillations due to monopolemotion.4,7,17 To predict large spikes, a time delay in the tun-neling current12 or random doping in the wells18 have to beadded. Unlike these models, ours reproduces and explainsspikes naturally, thereby supporting their use to interpret ex-perimental results.Figure 2~a! depicts a zoom of the spikes in Fig. 1~a!. Theyhave a frequency of about 500 MHz and an amplitude of 2.5mA. Figure 2~b! shows the charge density profile at fourdifferent times of a current spike marked in Fig. 2~a!. Noticethat the electron density in Fig. 2~b! is larger than the welldoping at only three wells ~40, 41, and 42! during the timesrecorded in Fig. 2~a!. The maximum of electron densitymoves from well 40 to well 41 during this time interval sothat ~i! tunneling through the 41st barrier ~between wells 40and 41! dominates when the total current density is increas-ing, whereas ~ii! tunneling through barriers 41 and 42 is im-portant when J(t) decreases. The contributions of tunnelingand displacement currents to J(t) in Eq. ~1! are depicted inFigs. 2~c! and 2~d!.More generally, the spikes reflect the two-stage hoppingmotion—fast time scale—of the domain wall: at time ~1!~minimum of the current!, the charge accumulates mainly atthe i-th well. As time elapses, electrons tunnel from this wellto the next one, the (i11)st, where most of the charge islocated at time ~3! ~maximum of the current!. This corre-sponds to a hop of the monopole. As the monopole moves, itleaves a lower potential drop on its wake. The reason is thatthe electrostatic field at the (i11)st well and barrier becomePRB 60 4491BRIEF REPORTSabruptly flat between times ~1! and ~3!, as they pass from thehigh to the low field domain. This means that a negativedisplacement current has its peak at the (i11)st barrier, nearthe wells where most of the charge is. Between times ~1! and~3!, the tunneling current is maximal where the displacementcurrent is minimal and the total current increases. After that,some charge flows to the next well @time ~4!# but both, tun-neling and displacement currents, are smaller than previ-ously. This occurs because the potential drop at barrier (i12) ~in the high-field domain! is larger than at barrier (i11). Then there is a smaller overlap between the resonantlevels of nearby wells—the tunneling current decreases—and the displacement current and, eventually, J(t) decreases.This stage lasts until well i is drained, and most of the chargeis concentrated at wells (i11) ~the local maximum ofcharge! and (i12) ~slightly smaller charge!. Then the nextcurrent spike starts.V. DIPOLE SELF-OSCILLATIONS OF THE CURRENTAn advantage of our present model over other discreteones4,17,19 is our microscopic modeling of boundary condi-tions at the contact regions. Thus we can study what happenswhen contact doping is changed. The result is that there ap-pear dipole-mediated self-oscillations as the emitter dopingis lowered below a certain value. There is a range of voltagesfor which dipole and monopole oscillations coexist as stablesolutions. This range changes for different plateaus. Whenthe emitter doping is further lowered, only the dipole self-oscillations remain. Figure 3 presents data in the crossoverrange ~below Nc54.131016 cm23 and above Nc51.731016 cm23 for the second plateau!, for the same sample,doping and bias as in Figs. 1 and 2. Except for the presenceof spikes of the current, dipole recycling and motion in SL’sare similar to those observed in models of the Gunn effect inbulk GaAs.13 These self-oscillations have not been observedso far in experiments due to the high values of the contactFIG. 2. ~a! Zoom of Fig. 1 showing the spikes of the current. ~b!Electron density profiles ~in units of the doping at the wells!, ~c!tunneling current, and ~d! displacement current within the mono-pole at the times marked in ~a!.doping adopted in all the present experimental settings. No-tice that current spikes appear differently than in the mono-pole case, Fig. 1~a!. The main difference is that now there aremany more current spikes, 36, for the dipoles recycle at theemitter and traverse the whole SL. See Figs. 3~b! and 3~c!.Charge transfer and balance between tunneling and displace-ment current during a spike are similar to those occurring inmonopole oscillations. For a simpler model4,8 the velocity ofa charge accumulation layer ~belonging to a monopole or adipole! has been shown to approximately obey an equal arearule. Then monopole and dipole velocities are similar but amonopole traverses a smaller part of the SL than a dipoledoes. Therefore dipole oscillations have a lower frequencythan monopole ones. Our results agree with the following:the frequency of the dipole oscillations discussed above isabout 8 MHz, 40% of the frequency of monopole oscilla-tions.Dipole self-oscillations have also been predicted to occurin weakly coupled SL’s as the result of assuming a linearcurrent-field relation at the injecting contact on a simplermodel.17,20 Since such an ad hoc boundary condition has noclear relation to contact doping, no crossover between differ-ent oscillation types could appear in that work.VI. MULTISTABILITYMonopole and dipole waves coexist in both the first andthe second plateaus. The time-averaged current as a functionof dc voltage in the first plateau ~whose crossover range isbelow Nc52.131016 cm23 and above Nc51.531016 cm23)has been plotted in Fig. 4. Notice that the average current ofdipole oscillations is lower than that of monopole oscilla-tions. Let us start at a bias of 0.5 V ~for which the stationarystate is stable! and adiabatically increase the voltage. Theresult is that we go smoothly from the stationary state to thefast monopole self-oscillation at about 1.3 V. This branch ofoscillatory states eventually disappears at about 2.6 V. If wenow adiabatically lower the bias, we reach a slow dipoleself-oscillation at about 2.4 V. There is a small hysteresisloop between dipole oscillations and the stationary state be-tween 2.4 V and 2.6 V: the former may start as a subcriticalHopf bifurcation. At about 0.8 V the dipole oscillationdisappears and we are back at the stable stationary state.We therefore find the hysteresis loops marked by arrows inFig. 4.FIG. 3. ~a! Dipole-mediated self-oscillations of the current at 5.5V for Nc5231016 cm23. ~b! Detail of the current spikes. ~c!Electric-field profiles at the times marked in ~a!.4492 PRB 60BRIEF REPORTSVII. CONCLUSIONIn conclusion, we have dealt with self-sustained oscilla-tions of the current in SL’s whose main mechanism is se-quential tunneling. Depending on contact doping, these os-cillations may be due to recycling and the motion of twodifferent charge density waves: monopoles and dipoles. Ex-perimentally, only the monopole oscillations have been ob-served, for the contacts doping is usually set to values thatare too high. The dipolelike oscillations could be observedconstructing samples with lower doping at the contacts. Infact, as the doping of the contacts is reduced, we predictFIG. 4. I-V characteristics at the first plateau for both sweepdirections showing bistability between self-oscillations mediated bymonopole and by dipole dynamics. Notice the hysteresis cycle.current oscillations due to dipole charge waves. The cross-over between both types of self-oscillations occurs at inter-mediate emitter doping values for which stable monopoleand dipole oscillations coexist. Then the diagram of averagecurrent versus voltage is multivaluated, presenting hysteresiscycles and multistability between monopole and dipole os-cillations ~and between oscillatory and stationary states!. Thetime-resolved current in the oscillatory modes presents anumber of sharp spikes. They occur because well-to-wellhopping of charge accumulation layers occurs in two stages:during the stage where the current rises, charge is mainlytransferred through a single barrier. The charge is transferredthrough two adjacent barriers at the stage in which the cur-rent decreases. All these properties form the basis for pos-sible applications of SL’s working as multifrequency oscil-lators in a wide range of frequencies. A quantitativedescription of such multifrequency oscillators requires thecalculation of typical output power characteristics and noiselevels. This is the purpose of a future work.ACKNOWLEDGMENTSWe thank Rosa Lo´pez and Professor Holger Grahn forhelpful discussions. This work has been supported by theDGES ~Spain!, Grant Nos. PB97-0088 and PB96-0875, bythe European Union TMR Contract Nos. ERB FMBX-CT97-0157 and FMRX-CT98-0180, and by the Community ofMadrid, Project No. 07N/0026/1998.1 M. P. Shaw, V. V. Mitin, E. Scho¨ll, and H. L. Grubin, The Phys-ics of Instabilities in Solid State Electron Devices ~PlenumPress, New York, 1992!.2 K. K. Choi et al., Phys. Rev. B 35, 4172 ~1987!.3 H. T. Grahn et al., Phys. Rev. Lett. 67, 1618 ~1991!.4 L. L. Bonilla et al., Phys. Rev. B 50, 8644 ~1994!.5 J. Kastrup et al., Appl. Phys. Lett. 65, 1808 ~1994!.6 R. Merlin et al., in Proceedings of the 22nd International Con-ference on the Physics of Semiconductors, edited by D. J. Lock-wood ~World Scientific, Singapore, 1995!, p. 1039.7 J. Kastrup et al., Phys. Rev. B 55, 2476 ~1997!.8 L. L. Bonilla et al., SIAM ~Soc. Ind. Appl. Math.! J. Appl. Math.57, 1588 ~1997!.9 O. M. Bulashenko and L. L. Bonilla, Phys. Rev. B 52, 7849~1995!.10 Y. Zhang et al., Phys. Rev. Lett. 77, 3001 ~1996!.11 K. J. Luo et al., Phys. Rev. Lett. 81, 1290 ~1998!.12 J. W. Kantelhardt et al., Phys. Status Solidi B 204, 500 ~1997!.13 F. J. Higuera and L. L. Bonilla, Physica D 57, 161 ~1992!.14 R. Aguado et al., Phys. Rev. B 55, R16 053 ~1997!.15 J. In˜arrea et al., Europhys. Lett. 33, 477 ~1996!.16 J. Kastrup et al., Phys. Rev. B 53, 1502 ~1996!.17 A. Wacker, in Theory and Transport Properties of SemiconductorNanostructures, edited by E. Scho¨ll ~Chapman and Hall, NewYork, 1998!, Chap. 10.18 F. Prengel et al., in Proceedings of the 23rd International Con-ference on the Physics of Semiconductors, edited by M. Schef-fler and R. Zimmermann ~World Scientific, Singapore, 1996!, p.1667.19 F. Prengel, A. Wacker, and E. Scho¨ll, Phys. Rev. B 50, 1705~1994!.20 L. L. Bonilla, E. Scho¨ll, and A. Wacker ~unpublished!.

Current self-oscillations, spikes, and crossover between charge monopole and dipole waves in semiconductor superlattices

Current self-oscillations, spikes and crossover between charge monopole
  and dipole waves in semiconductor superlattices

Current self-oscillations, spikes and crossover between charge monopole and dipole waves in semiconductor superlattices

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