RAPID COMMUNICATIONS

PHYSICAL REVIEW B, VOLUME 63, 201306~R!

Quasiperiodic current and strange attractors in ac-driven superlattices

David Sanchez,1 Gloria Platero,1 and Luis L. Bonilla2,*1Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain

2Escuela Polite´cnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Legane´s, Spain~Received 5 December 2000; revised manuscript received 28 February 2001; published 2 May 2001!

Intriguing routes to chaos have been experimentally observed in semiconductor superlattices driven by an acfield. In this work, we analyze theoretically the time-dependent current in ac-driven superlattices. Experimentshave shown that the current is quasiperiodic and its Poincare´ maps are distorted and twisted for certain valuesof the ac intensity, at fixed dc voltage. This marks the appearance of very complex attractors and routes tochaos as the ac signal amplitude increases. We show that discrete well-to-well motions of domain walls duringspiky high-frequency self-oscillations of the current play a crucial role in the origin of the experimentallyobserved distorted maps.

DOI: 10.1103/PhysRevB.63.201306 PACS number~s!: 73.50.Fq, 73.40.Gk, 73.40.2c

an

nso

hto

suthicai-tiethse

toecallreeto

-e

ase

hey-rsorb

in

hesfo

ede

SLrei-asetder

ese

llytheto

eac

ledargens.no-ja-

delse-

opi-icro-

ower

rgeo are-velsesf-

enticred

nm

Spatiotemporal chaos has been analyzed in fluidchemical systems1 as well as in solid-state systems.2,3 Differ-ent routes to chaos, as the system is driven by a frequeincommensurate with its natural frequency, start from quaperiodic attractors and end in chaos via frequency lockingdirectly. Interesting theoretical and experimental researcof this topic have been realized in semiconducsuperlattices.4–9

Nonlinear dynamics of weakly coupled semiconductorperlattices~SLs! driven by dc and ac biases has beenresearch topic of many experimental and theoretworks.10–17 The nonlinearity manifests itself in many physcal situations as, for instance, in their transport properunder finite dc or ac bias. Under appropriate dc voltage,current through the SL displays natural oscillations cauby creation, motion, and annihilation of domain walls~DW!in the SL.14 The observed frequencies range from kHzGHz. Superimposed on a smooth current oscillation, thappear faster current spikes whose frequencies are typiin the GHz regime.14 Each spike reflects the well-to-wemotion of DWs causing the self-oscillation, and therefotheir frequency is related to the characteristic tunneling timNumerical simulations show that DW may correspondcharge monopoles or dipoles,15,16although experimental evidence suggests that charge monopoles are responsiblself-oscillations observed in the available samples.14 acdriven SLs display a rich dynamical behavior including qusiperiodic and chaotic oscillations with nontrivial phaspace structure.4,6–9

Many studies fix the frequency of the ac drive as tgolden mean number (11A5)/2'1.618 times the frequencof the natural oscillations~i.e., the frequency ratio is an irrational number hard to approximate by rational numbe!,which is convenient to obtain complex dynamical behaviThen the system presents rich power spectrum, complexfurcation diagrams, and different routes to chaos includquasiperiodicity, frequency locking, etc.4,6,7,9 First return orPoincare´ maps~PM! are used to analyze unambiguously tunderlying attractors.7 PMs usually consist of smooth loopfor quasiperiodic attractors, and of a set of discrete pointsfrequency locking. More exotic PM resembling distortdouble loops in the quasiperiodic case have been experim

0163-1829/2001/63~20!/201306~4!/$20.00 63 2013

d

cyi-r

esr

-el

sed

relly

.

for

-

.i-

g

r

n-

tally observed in the middle of the second plateau of aI –V characteristic~at the onset of the second plateau, PM asmooth and not distorted!.6,7 These PM correspond to quasperiodic nonchaotic strange attractors having a fractal phspace structure.9,18,19 The origin of distorted maps was nounderstood at the time of their observation, although disorand sample imperfections were invoked.6 Luo et al.7 showedthat a combination of signals with different frequenci~whose origin was not ascertained! was needed to reproducexperimentally observed distorted double layer PM.

The aim of the present work is to establish theoreticathat high-frequency current spikes play a relevant role inoccurrence of exotic PM. In turn, current spikes are duethe well-to-well motion of DW during each period of thself-oscillations. Thus distorted PM reflect DW motion indriven SLs.

Theoretical studies of nonlinear effects in weakly coupSL are based upon discrete versions of Poisson and chcontinuity equations plus appropriate boundary conditioEarly models were discrete drift models with a phenomelogical fitting of sequential tunneling current between adcent wells and boundary conditions~see Refs. 14, 15, andreferences therein; more complicated rate equation mowith tunneling currents derived from simple quantum mchanical considerations were also used15!. More recently, thesequential tunneling current has been obtained microsccally, and the electrostatics at the contacts has been mscopically modeled and included self-consistently.12,13,16De-pending on SL configuration and contact doping, we knthat undriven self-oscillations are due to recycling of eithmonopole DWs or dipole waves.16 In the latter case, a dipolewave consists of two DWs, one corresponding to a chaaccumulation layer and another one corresponding tdepletion layer. Between these DWs there is a high-fieldgion encompassing several wells, and the dipole wave trathrough the whole SL. Recycling of dipole waves causcurrent self-oscillations similar to the well-known Gunn efect in bulk semiconductors.

In this paper, we have analyzed the tunneling currthrough a dc1ac biased SL by means of the microscopmodel described in detail in Refs. 12,13. We have considea 50-period SL consisting of 13.3 nm GaAs wells and 2.7

©2001 The American Physical Society06-1

lls

lf-ee-

enu

e

thera

ien

soat

u

snete

ety

Sc

h

o

eceobt

ir

a-

ens

isentho-

ionfhis

ural

re-yer

d

t

her

RAPID COMMUNICATIONS

DAVID SANCHEZ, GLORIA PLATERO, AND LUIS L. BONILLA PHYSICAL REVIEW B 63 201306~R!

AlAs barriers, as described in Ref. 14. Doping in the weand in the contacts areNw5231010 cm22 and Nc5231018 cm23, respectively. With these doping values, seoscillations are due to recycling of monopole DWs. Wchoose not to analyze the original 9/4 sample where theperiments were performed.6 In this sample, the second plateau of theI –V characteristics ends at a resonant peak duG2X tunneling. In the simpler SL configuration chosehere, theX plays no role at the first and second plateaus. OSL is subject to an applied voltage,V(t)5Vac(t)1Vdc ,whereVac(t)5Vac sin(2pfact). The ac frequencyf ac equalsthe golden mean times the natural frequency. Since theergy associated withf ac ~in the MHz range! is very smallcompared to the typical energy scales of the system~both theenergy difference between different well subbands andlevel broadening due to scattering, are of the order of sevmeV!, we can assume that the ac bias modifies adiabaticthe potential profile of the SL.

We evaluate microscopically the current across barrby means of the transfer Hamiltonian model. Electroelectron interaction is accounted for by solving the Poisequation through the SL including the leads, with appropriboundary conditions.12,13,16 The condition that all voltagedrops across the different regions of the nanostructure madd up to the applied bias becomes:

V~ t !5(i 50

N

Vi1(i 51

N

Vwi1D11D21EF

e. ~1!

HereVi , Vwi , D1 , D2, andEF denote the potential dropat the i th barrier and well, the energy drops at emitter acollector contacts, and the collector Fermi energy, resptively. Tunneling and displacement currents are relathrough the Ampere’s law for the total current densityJ5J(t):

J5Ji ,i 111e

d

dVi

dt. ~2!

Here Ji ,i 11 , d and e are the tunneling current through thi th barrier, its thickness, and the GaAs static permittivirespectively.

Figure 1 shows the time-averaged current through theversusVdc , the time evolution of the current at a fixed dbias, its Fourier spectrum, and its PM forVdc54.2 V andVac519 mV. This value of the dc bias corresponds to tonset of the second plateau of the time-averagedI –V char-acteristic curve@Fig. 1~a!#. The current trace of Fig. 1~b! isquasiperiodic and does not present observable superimphigh-frequency oscillations~spikes!. The natural oscillationnear the onset of the plateau is caused by monopole rcling very close to the collector contact. Thus the DW donot move over many wells and no current spikes areserved. The power spectrum in Fig. 1~c! contains peaks athe fundamental frequencyf 0'39 MHz, the frequency ofthe applied ac fieldf ac , the combination of both and thehigher harmonics. The PM depicted in Fig. 1~d! is a smoothloop with a double layer structure indicating nontrivial qusiperiodic oscillations. Let us now fix the dc bias atVdc

20130

x-

to

r

n-

eallly

rs-ne

st

dc-d

,

L

e

sed

y-s-

55.1 V, in the middle of the second plateau. The undrivself-oscillation is caused by recycling of monopole DWwhich periodically run through part~approximately 35%) ofthe SL and disappear at the collector. Similar dynamicsobserved with ac-driven bias. The corresponding currtrace may show high-frequency spikes depending on the csen initial field profile. A flat initial field profile gives a spikycurrent trace until the latter settles to the stable oscillat~without appreciable current spikes!; see the earlier part oFig. 2~a!. The power spectrum and PM corresponding to tcase differ markedly from those in Figs. 1~c! and 1~d!. Theharmonic content of the power spectrum in Fig. 2~b! ~spikespresent! is greater than that in Fig. 1~spikes absent!. Addi-tional peaks correspond to higher harmonics of the natfrequency ~i.e., the low fundamental frequency,f 0'33 MHz; the spikes have frequenciesf s'566 MHz), theapplied frequency and combinations thereof. The corsponding PM presents a distorted loop with a double la

FIG. 1. ~a! Time-averagedI (V) curve for a 50-period 13.3-nmGaAs/2.7-nm AlAs SL with Nw5231010 cm22 and Nc5231018 cm23. ~b! I (t) for Vdc54.2 V, f 0539 MHz,Vac519 mV.Spikes are not resolved.~c! Power spectrum showing barely formehigher harmonics of the main frequency.~d! Poincare´ map, con-structed by plotting the current at the (n11)st ac period versus thaat thenth period. The times are taken at the minima ofVac(t).

FIG. 2. ~a! I (t) for Vdc55.1 V @see Fig. 1~a!#, f 0533 MHz,Vac519 mV. Spikes are present in the transient regime~see inset!.~b! Power spectrum displaying several higher harmonics off 0. ~c!Poincare´ map showing a somewhat distorted loop due to the higharmonics in~b!.

6-2

tas,n

te

cedintfot

e-ddulnaesnconthe

ysco

th

eon

sly

dc

ur-tsdingn inof

her

Mctalctal

-hey-aloucture

on

d

RAPID COMMUNICATIONS

QUASIPERIODIC CURRENT AND STRANGE . . . PHYSICAL REVIEW B63 201306~R!

structure which shows a strong similarity with experimenresults;6,7 see Fig. 2~c!. From these numerical observationwe conclude that distorted PM are linked to spiky curretraces, even if such traces change to smooth ones aftransient.20

In Ref. 7, higher harmonics observed in real-time traof the current are held responsible for the occurrence oftorted PM in the quasiperiodic case. Further experimeevidence is found in the frequency-locked state. Supportthis claim lies on numerical studies of power spectrum daThe PM plot of a signal containing both the natural frquency and the ac field oscillations is a perfect ellipse. Aing a new signal with the second harmonic frequency resin a PM with a twisted loop. Adding more and more harmoics may reproduce reasonably well the experimental mThus any nonsinusoidal natural periodic signal in the prence of a weak ac potential of incommensurate frequeseemingly produces a distorted PM. The corresponding Frier spectrum presents peaks at the main natural frequeand its higher harmonics. To observe self-crossings infirst return map, peaks at higher harmonics in the powspectrum should be large enough for the current traceacquire at least one additional maximum. What is the phcal origin of these additional peaks? We argue that theyrespond to current spikes.

Figure 3~a! shows the current through a 50-well 13.3 nm2.7 nm SL driven by both a dc voltage bias~5.5 V! and an acbias having 2 mV amplitude and frequency equal togolden mean ratio times the natural frequency~about 20MHz!. Doping in the wells and in the contacts areNw5231010 cm22 and Nc5231016 cm23, respectively ~lowcontact doping is needed to obtain sharper monopoles!. Thecurrent trace deviates from a sine due to the presencspikes at low current values, which results in higher harmics in the frequency spectrum@Fig. 3~b!#. The first returnmap is strongly distorted@see Fig. 3~c!#: its twisted arm ex-tends from about 49mA to 55 mA, exactly the region cov-ered by the spikes in Fig. 3~a!. Let us see that current spikeultimately cause the twist of the PM loop. If we slightincrease contact doping,Nc5431016 cm23, both the dis-tortion in the current PM and the spikes vanish@Fig. 4~a!#.Although the power spectrum shows higher harmonics@Fig.

FIG. 3. ~a! I (t) ~full line! for Vdc55.5 V and Vac

52 mV, Nw5231010 cm22, Nc5231016 cm23. Fitting to asine function~dashed line! is depicted for comparison.~b! Powerspectrum.~c! Poincare´ map.

20130

l

tr a

ss-alr

a.

-ts-p.-y

u-cyer

toi-r-

/

e

of-

4~b!#, the corresponding first-return map@Fig. 4~c!# issmooth without self-crossings.

To reinforce the previous conclusions, we change thevoltage toVdc51.5 V ~middle of the first plateau, 4 MHznatural frequency! and Vac52 mV, so that undriven self-oscillations are caused by dipole waves, yielding spikier crent traces.16 Figure 5~a! depicts the current trace and iinset shows superimposed current spikes. The corresponPM presents three well-defined distorted loops, as seeFig. 5~c!. Since loops in the PM are due to a combinationstrong enough signals of different frequencies,7 the greaterstrength of the high-frequency spikes gives rise to higharmonic content and additional loop structure@Fig. 5~b!#.

As mentioned above, the double-layer structure of Pindicates nonchaotic quasiperiodic attractors having fraphase space structures. We have calculated the multifradimensions of these attractors,Dq , for three cases corresponding to Figs. 1, 2, and 5; see Fig. 6. In all cases, tcorrespond to strange attractors whoseDq presents the kneelike structure typical of chaotic attractors with multifractdimensions.9 Albeit they are not chaotic, it is important tstress that strangeness means fractal phase space strand does not involve necessarily sensitive dependence

FIG. 4. ~a! I (t) for Vdc55.5 V and Vac52 mV, Nw5231010 cm22, Nc5431016 cm23. ~b! Power spectrum.~c! Poin-caremap.

FIG. 5. ~a! I (t) for Vdc51.5 V, f 054 MHz, Vac52 mV,Nw5231010 cm22, Nc5231016 cm23. Spikes are superimposeon the current throughout the signal~see inset!. ~b! Power spectrum.Higher harmonics off 0 have finite amplitude.~c! Poincare´ map.

6-3

v

eni

il-he

ll.

cur-ncyec-thea

oleral

lethe

thenichers inac

dy-bet-

s-

the-of

RAPID COMMUNICATIONS

DAVID SANCHEZ, GLORIA PLATERO, AND LUIS L. BONILLA PHYSICAL REVIEW B 63 201306~R!

initial conditions~as for chaotic attractors!. This distinctionis well established both theoretically,18,19 andexperimentally.21

In summary, we have explained recent experimental edence showing complex Poincare´ maps in the quasiperiodicregime.7 We have analyzed theoretically the time-dependcurrent through an ac driven SL and have characterizedtriguing Poincare´ maps corresponding to quasiperiodic osclations. The strange attractors which define them have t

FIG. 6. Multifractal dimension. Labels~1!, ~2!, and ~3! corre-spond to the attractors in 1~c!, 2~c!, and 5~c!, respectively.

20130

i-

tn-

ir

physical origin in the complex dynamics of the domain waWe have shown that distorted loops in the Poincare´ maps arerelated to the presence of high-frequency spikes in therent traces. Their frequencies combine with the ac frequeand the low natural frequency to yield a richer power sptrum. Since spikes are associated to extended motion ofDW, distorted Poincare´ maps are observed at the middle ofI –V plateau and not at its beginning, where the monopmoves over too small a part of the SL. The case of natuself-oscillations due to dipole recycling is different: dipoDW are generated at the emitter contact and move overwhole SL. Then spikes are more prominent than inmonopole case, yielding current traces with higher harmocontent and distorted PM. Thus our model shows that higfrequency current spikes are responsible for new featurethe time traces of the electronic current when driven by anpotential. Hopefully our results may help understandingnamical properties of doped semiconductor superlatticester.

We thank H. Grahn and O. Bulashenko for fruitful discusions. This work has been supported by the DGES~Spain!Grants No. PB98-0142-C04-01 and No. PB96-0875, byEuropean Union TMR Contracts No. ERB FMBX-CT970157 and No. FMRX-CT98-0180 and by the CommunityMadrid, Project No. 07N/0026/1998.

r

ent

*Also at Unidad Asociada al Instituto de Ciencia de Material~CSIC!, Spain.

1M. C. Crosset al., Rev. Mod. Phys.65, 851 ~1993!.2G. A. Held et al., Phys. Rev. Lett.55, 887 ~1985!.3A. M. Kahn et al., Phys. Rev. Lett.68, 369 ~1992!.4O. M. Bulashenkoet al., Phys. Rev. B53, 10 008~1996!.5K. N. Alekseevet al., Phys. Rev. B54, 10 625~1996!.6K. J. Luo et al., Phys. Rev. Lett.81, 1290~1998!.7K. J. Luo et al., Phys. Rev. B58, 12 613~1998!.8J. C. Cao and X. L. Lei, Phys. Rev. B60, 1871~1999!.9O. M. Bulashenkoet al., Phys. Rev. B60, 5694~1999!.

10H. T. Grahnet al., Phys. Rev. Lett.67, 1618~1991!.11H. T. Grahnet al., Jpn. J. Appl. Phys., Part 134, 4526~1995!.12R. Aguadoet al., Phys. Rev. B55, R16 053~1997!.

es13L. L. Bonilla et al., Phys. Rev. B62, 2786~2000!.14J. Kastrupet al., Phys. Rev. B55, 2476~1997!.15A. Wacker, inTheory and Transport Properties of Semiconducto

Nanostructures, edited by E. Scho¨ll ~Chapman and Hall, NewYork, 1998!, Chap. 10.

16D. Sanchezet al., Phys. Rev. B60, 4489~1999!.17H. Steueret al., Appl. Phys. Lett.76, 2059~2000!.18E. Ott, Chaos in Dynamical Systems~Cambridge University

Press, New York, 1993!, Sec. 6.4.19J. W. Shuai and K. W. Wong, Phys. Rev. E57, 5332~1998!.20As expected, the frequency of the spikes is a multiple of th

natural one, thus reinforcing the harmonic content of the curretrace.

21W. L. Ditto et al., Phys. Rev. Lett.65, 533 ~1990!.

6-4