Dynamic behaviors in coupled self-electro-optic effect devicesY. Ohkawa, T. Yamamoto, T. Nagaya, and S. Nara Citation: Applied Physics Letters 86, 111107 (2005); doi: 10.1063/1.1875760 View online: http://dx.doi.org/10.1063/1.1875760 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/86/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The structure measurement of micro-electro-mechanical system devices by the optical feedback tomographytechnology Appl. Phys. Lett. 102, 221902 (2013); 10.1063/1.4807283 Carrier relaxation dynamics and steady-state charge distributions in coupled InGaN ∕ GaN multiple and singlequantum wells J. Appl. Phys. 101, 093515 (2007); 10.1063/1.2727437 Theory of delayed optical feedback in lasers AIP Conf. Proc. 548, 87 (2000); 10.1063/1.1337760 Analysis of switching dynamics of asymmetric Fabry–Perot symmetric self-electro-optic effect devices withextremely shallow quantum wells J. Appl. Phys. 82, 1936 (1997); 10.1063/1.366002 Dynamic optical switching of symmetric selfelectrooptic effect devices Appl. Phys. Lett. 59, 2631 (1991); 10.1063/1.105920

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Dynamic behaviors in coupled self-electro-optic effect devicesY. Ohkawa, T. Yamamoto, T. Nagaya, and S. Naraa!

Department of Electrical & Electronic Engineering, Faculty of Engineering, Okayama University,700-8530 Okayama, Japan

sReceived 19 August 2004; accepted 12 January 2005; published online 10 March 2005d

Dynamic behaviors of two coupled bistable elements optically connected in a series are theoreticallypredicted and discussed based on a phenomenological model with respect to photocarrier densities,where each bistable element is a self-electro-optic effect devicesSEEDd. Feedbackfrom one of thetwo SEEDs to the power of an incident light beam was introduced, which resulted in various typesof oscillatory solutions and bifurcation structures depending on incident light power and choices ofparameter values included in our model. ©2005 American Institute of Physics.fDOI: 10.1063/1.1875760g

Self-electro-optic effect devicessSEEDsd have beenstudied intensively because they are candidates for funda-mental device element in realizing optical computers thatcould be massively parallel processors. On the other hand,recent significant developments in biological sciences in-cluding brain research have produced other viewpoints aboutsuch devices, that is,9neuromorphic or biomorphic devices9that mimic the excellent functionalities of biological and/orneural systems. One can find, for instance, Chua’s cellularneural network,1 Aihara’s chaotic neurla network,2 the neu-romorphic devices developed by the Meed group,3 or theexample of Nothmore.4 In this letter, we propose9dynamicSEED9 as a potential active element in such neuromorphichardware devices.

Figure 1 shows the SEED elements. A typical singleSEED consists of inversely biasedp-i-n semiconductor lay-ers. Since SEED was proposed by Milleret al.,5,6 a numberof variations have been extensively investigated; Tokudaetal.7–11 in particular, with two serially connected SEEDsfFig.2 sleftdg, showed complex multi-stable responses to input op-tical power that can be quite useful when one considers thatreversible switching between binary states can be accom-plished with “only positive intensity pulses.”11,12 On theother hand, when realizing neuromorphic devices, it is cru-cially important to obtain dynamic behaviors that are tempo-rally exciting such as neuron firing in biological brains,which will be obtained by introducing an idea to seriallyconnected SEEDs, shown below.

Now, let us briefly illustrate a SEED model shown inFig. 1 sleftd, that gives rate equations for photocarrier densityn in a single SEED

dn

dt= −

n

t+

aPinV0

sv − j0d2 + sV0/2d2 . s1d

The first term on the right side denotes carrier relaxation,where withint the carrier is depleted from the intrinsic re-gion dominated by the electric field applied across the layer.The second term expresses the photocarrier productionbrought from dissociated excitons generated by light absorp-tion in the intrinsic region consisting of a quantum well. Weassume the absorption spectrum of an exciton has a Lorent-

adAuthor to whom correspondence should be addressed; electronic mail:nara@elec.okayama-u.ac.jp

FIG. 1. sad Illustration of a single SEED.sbd Typical stationary solution ofdn/dt=0 that gives bistable behavior of SEED.scd An example of experi-mental results obtained by Tokudaet al. ssee Ref. 11d.

FIG. 2. Illustration of serially connected SEEDs without feedbacksleftd andwith feedbacksrightd; in the latter, light input power by LASER is designedto bePin=P0

in+PsV2d , PsV2d=AV2, V2=R2I2, andI2=hn2.

APPLIED PHYSICS LETTERS86, 111107s2005d

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zian form, wherePin is the power of incident light,v is lightfrequency,j0 is the central frequency of the exciton absorp-tion spectrum, andV0 is the half width of the absorptionspectrum corresponding to the inverse of the lifetime of theexcitons. We introduce quantum confined Stark effectsQCSEd by j0=v0−bVin, wherev0 is the central peak fre-quency of the exciton absorption spectrum in the absence ofexternally applied electric field across the quantum wellsiregiond, whereb is a parameter representing the rate of spec-tral redshift due to QCSE via applied voltageVin. AppliedvoltageVin can be represented asVin=V0− IR, whereVin isthe voltage across the quantum well,V0 is the sum of anexternally applied source voltage and the built-in potential ofthe p-n junction formed in the device, the current,I andR,the resistive load, respectively. We assume that currentI isrepresented asI =hn, whereh is a constant parameter. Weemployed a constanth because, for a SEED with a reversebiasedp-n junction, the current is nearly independent of thevoltage across the junction. Finally, we obtainedj0=v0−bsV0−Rhnd that gives nonlinear terms with respect ton inEq. s1d. When the system is stationary, the time derivativeshould vanish in Eq.s1d. By employing appropriate valuesfor the parameters included in the equationssee Table Id, wecan reproduce the experimental results shown in Fig. 1sbdthat indicate typical bistable behavior. For comparison, anexample of the experimental results is shown in Fig. 1scd.

Extending the model to serially connected SEEDs isquite staightforward as shown in Fig. 2sleftd that indicatescomplicated multi-stable behaviorsRef. 9d. The essential ori-gin of these behaviors is based on the fact that the input lightenters the lower SEED after it has been absorbed partly bythe upper SEED with bistable behavior, which is introducedby replacing incident optical powerPin with Pin−m1n1,wheren1 is the photocarrier density in the upper SEED.

Furthermore, to obtain dynamic behaviors, in this letter,we added a simple feedback to incident light power depend-ing on voltageV2 of load registerR2 in the serially connected

SEEDs shown in Fig. 2srightd, so thatPin was replaced withPin=P0

in+AV2, whereA is a feedback gain constant, andV2=R2I2, andI2=h2n2, respectively. This feedback is designedto have positive feedback effects on the terms correspondingto photocarrier generation in the rate equation. The finalforms of the rate equations are

dn1

dt= −

n1

t1+

a1sP0in + Ah2R2n2dV01

hv − v01 + b1sV01 − h1R1n1dj2 + sV01/2d2 ,

s2d

dn2

dt= −

n2

t2+

a2sP0in + Ah2R2n2 − m1n1dV02

hv − v02 + b2sV02 − h2R2n2dj2 + sV02/2d2 ,

s3d

where we used suffixes “1”supper SEEDd and “2” slowerSEEDd for all variables and parameters. This feedback effectdrastically changed the behaviors of solutions depending onthe choice of parameter values. First, let us show the station-ary solutions of both equations forn1 and n2 to light inputpowerP0

in sFig. 3d.It is quite peculiar that a region has no stable solutions.

In unstable regions, we can find periodic oscillatory solutionsas shown in Fig. 4. Note that Hopf bifurcation occurs at thepoint connecting stable and unstable solutions

According to choices of appropriate values of param-eters, we can obtain many types of oscillatory solutions andbifurcation structures as well. For instance, Figs. 5 and 6show that the period of limit cycle becomes longer andlonger with approaching the bifurcation points from largeinput powersU→SNd until finally the limit cycle ceases toexist associated with the appearance of stable fixed pointsand one saddle point represented by SN in the figure, whichis called saddle-node bifurcation on a limit cycle.

Furthermore, we found a few cases in which other dy-namic behaviors occur between H and SNsFig. 5d. AfterHopf bifurcation occurred at H, as the input light increases

FIG. 3. Stationary solutions of upperSEED n1 sleftd and lower SEEDn2

srightd as a function of input lightpower P0

in. Dotted line and the graylines in the figures mean that solutionsare not stable but unstable or saddle.Two Hopf bifurcation points are con-tact points between stable solutionsand unstable solutions.

Parameters Values

vfeVg 1.5385v0feVg 1.543bfmeV/Vg 3.086V0fVg 2.74RfMVg 20V0fmeVg 2.1bV0fmeVg 8.46bhRf310−11 meV m3g 1.11taV0f310−9 eV s/m3g 0.37

FIG. 4. Damped oscillation, oscillatory solutions of upper SEEDn1 andlower SEEDn2 for input light power,P0

in=0.118, 0.120, and 0.135, near oneof the two Hopf bifurcation points in Fig. 3.

111107-2 Ohkawa et al. Appl. Phys. Lett. 86, 111107 ~2005!

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from the small power, the limit cycle amplitude increases,and before reaching saddle-node bifurcation point SN, thelimit cycle suddenly vanishes, and only one stable point re-mained. At saddle-node bifurcation point SN, a limit cycleagain appears.

This type of bifurcation is quite rare in usual nonlineardynamic behaviors of system with two degrees of freedom.Detailed analysis and discussion about the obtained results ofthis topic will be given in our subsequent articles.

The authors would like to thank Dr. Peter Davis for valu-able comments. This work has been supported partly by a

Grant-in-Aid for the Promotion of Science No. 16500131from the Japan Society for the Promotion of Science.

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FIG. 5. Stationary solutions of upperSEED n1 sleftd and lower SEEDn2

srightd as a function of input lightpowerP0

in. Dotted lines and gray linesmean that solutions are not stable butunstable or saddle. Note that param-eter values are different from Fig. 3.

FIG. 6. Oscillatory solutions of upper SEEDn1 and lower SEEDn2 forinput light power,P0

in=0.171, 0.223, and 0.258. Note that periods becomelonger as they approach both saddle-node bifurcation points.

111107-3 Ohkawa et al. Appl. Phys. Lett. 86, 111107 ~2005!

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