Frequency-Domain-Based Carrier Transport Model for a Lateral p-i-n photodiode

Transcription

1 Frequency-Doman-Based Carrer Transport Model for a ateral p--n photodode Kohkch Konno Osamu Matsushma Kyohto Hara Gaku Suzuk Dondee Navarro and Mtko Mura-Mattausch Graduate School of Advanced Scences of Matter Hroshma Unversty Japan Phone: Fax: E-mal: kohkch@hroshma-u.ac.jp. Introducton Recently optcal nterconnecton has been studed wth ntense vgor to acheve fast swtchng operaton n USI []. In order to desgn optoelectronc ntegrated crcuts (OEICs) models descrbng the electronc and optcal characterstcs of optoelectronc devces for crcut smulaton are necessary. We focus on a p--n photodode as an optoelectronc devce. Although the vertcal photodode has been studed by many authors [2] work on the lateral type as shown n Fg. (a) s stll mssng. In ths paper we present a formulaton of carrer transport n a lateral p--n photodode and develop a model applcable for crcut smulaton of OEICs. Computatonal accuracy of the model s comparable wth the 2-dmensonal devce smulator MEDICI [3]. Snce the model equatons are solved n the frequency doman t s applcable for harmonc balance smulaton [4]. 2. ateral p--n Photodode Structure and Formulaton for Carrer Transport We consder a smplfed structure of a lateral p--n photodode shown n Fg. (b). In the case of the lateral p--n photodode the drecton of electrc feld whch s along the x axs s orthogonal to the travelng drecton of the ncdent lght whch s along the y axs. We adopt the followng assumptons n formulatng the devce equatons: () homogeneous rradaton only on the -regon () deep - and -regon compared to the ncdent lght penetraton depth () constant electrc feld E = (E x E y ) = (E ) n the -regon (v) neglgble change of electrc feld E due to the ncdent pulse. The frst assumpton allows us to focus only on the -regon. The second assumpton s ntroduced to realze homogeneous electrc feld n the -regon. Ths s mportant to acheve hgh-speed operaton of the lateral p--n photodode because the transport of carrers s governed by the electrc feld exstng n the regon. The thrd assumpton s valdated under the second assumpton and under moderately hgh reverse appled bas. Snce under normal operatng condton the lght ntensty s below the level where the output current shape s dstorted by screenng effect assocated wth hgh llumnaton the last assumpton s also justfed. The equatons governng the transent carrer transport s gven by nxyt ( ) Jxn ( x y t) = Gn ( x y t) t q t pxyt ( ) Jxp ( x y t) = Gp ( x y t) t q t where n and p are the carrer number densty of electrons and holes q s the elementary charge J xn and J xp are the x components of current densty vectors for electrons and holes and G n and G p are the generaton rate of carrers. Here we neglected recombnaton terms because photogenerated carrers quckly evacuate to the or regon by the hgh electrc feld. The current densty s descrbed domnantly by the drft component J xn ( xyt ) = qµ nnxyte ( ) J ( xyt ) = qµ pxyte ( ) xp p where µ n and µ p are the moblty of electrons and holes respectvely. Furthermore the generaton rate of carrers s wrtten as α y Gnp ( x y t) = αe φ( t) where α s the absorpton coeffcent and φ s the effectve photon flux. The shape of the nput ncdent lght determnes the functonal form of φ (t). We focus on the non-statonary descrpton of carrer transport n the p--n photodode. For ths purpose we expand tme dependent varables va Fourer expanson as t f( x y t) = f ( x y) e where f denotes n p φ J xn or J xp. Under the boundary condton n ( y ) p ( y = ) = we can obtan the soluton for current as I = qµ E W e φ e where W s the devce wdth. µ E t 3. Modelng Results and Dscusson To demonstrate the photodode transent response usng the developed model we consder a snusodal nput wth a sngle frequency and constant optcal ampltude φ. In our calculatons = 2µm and V PN = 5V. The output photocurrent s gven by the real part of I. For an nput of GHz the photocurrent response exhbts no transt delay as shown n Fg.2. At GHz whch s n the devce cut-off regon the photocurrent shows phase shft and reduced ampltude as shown n Fg.2. Both are

2 due to the fact that carrers can no longer respond to very fast swtchng nput. The -dependence of Re[I ] for fxed φ exhbts the frequency response characterstcs of the photodode. The cut-off frequency f T s derved from the value of Re[I ] / Re [I =] at 3dB. Usng the developed model two methods for enhancng the frequency response are recognzed. One method s by ncreasng the appled bas whch ncreases f T as shown n Fg.3(a). The next s by reducng the -regon length of the photodode. Fgure 3(b) shows ncreasng f T as s reduced for two dfferent appled bases. Steep ncrease n f T s observed for below µm n the deal case. Smlar results can be obtaned for any lght ntensty lower than the crtcal value from whch screenng effect occurs. Next we valdate the accuracy of our model wth measured transent photocurrent of a fabrcated S lateral p--n photodode. The devce structure s shown n Fg.(a) wth -regon length =2µm. For our calculaton we adopt the spectral method usng the Fast Fourer Transform n obtanng the tme-doman model response. The model correctly reproduces measured photocurrent response for a Gaussan lght pulse (λ ~ 532nm) wth ~ ps full wdth at half maxmum and appled bas V PN = 7V as shown n Fg.4(a). In ths calculaton we used the same values of devce dmenson as those of the fabrcated devce. The dfference n the end regon can be attrbuted to carrers generated n the and/or regon due to dffracton effects along the permeter of the Al openng. Computatonal accuracy of the model as compared wth MEDICI s shown n Fg.4(b). 4. Concluson We have developed an analytcal model for carrer transport n a lateral p--n photodode. The frequency response of the photodode s enhanced by reducng the -regon length and applyng hgher reverse bas. The model acheves excellent reproducton of measured photocurrent down to ~ ps pulse wdth nput at hgh reverse bas. At low appled bas the y-dependence of E becomes sgnfcant and thus should be ncluded n the modelng to acheve more accurate results. The developed model s approprate for crcut smulaton of OEICs. References [] See e.g..c. Kmerlng Appl. Surf. Sc (2). [2] K. Konno O. Matsushma D. Navarro and M. Mura-Mattausch J. Appl. Phys (24) and references theren. [3] MEDICI User s Manual Synopsys (23). [4] K. S. Kundert and A. Sangovann-Vncentell IEEE Trans. CAD-5 52 (986). Fg.2 Photodode response for (a) GHz and (b) GHz nput. Fg.3 (a) Photoresponse as a functon of. ( b) Cutoff frequency as a functon of. Fg. (a) Fabrcated lateral p--n photodode. (b) Smplfed Structure of a lateral p--n photodode for a model. Fg.4 Normalzed photocurrent calculated usng the spectral method. (a) Calculated transt delay agrees wth expermental result. (b) Computatonal accuracy s comparable wth the two-dmensonal devce smulator MEDICI.

3 FREQUENCY-DOMAIN-BASED CARRIER TRANSPORT MODE FOR P-I-N PHOTODIODES Towards Crcut Smulaton for Optoelectronc Integrated Crcuts K. Konno O. Matsushma K. Hara G. Suzuk D. Navarro and M. Mura-Mattausch Graduate School of Advanced Scences of Matter Hroshma Unversty BACKGROUND "Interconnecton Bottleneck" Contnuous shrnkng of devce dmensons brngs about hgher devce cut-off frequency. Sgnal propagaton delay due to conventonal nterconnects overwhelms transstor gate delay and hnders fast swtchng operaton. Optcal Wavegude ED SO2 S Substrate Necessty of Optcal Interconnecton Photodetector Optoelectronc Integrated Crcut (OEIC) PURPOSE In order to fully utlze optcal nterconnects n crcuts models descrbng the optcal and electronc characterstcs of optoelectronc devces are necessary. We present a frequency-doman-based formulaton of carrer transport n p--n photododes whch are used as photodetectors n optcal nterconnects. We am at developng a photodode model sutable for crcut smulaton of OEICs. PROCEDURE FOR TIME-DOMAIN SIMUATION. Expandng nput optcal sgnals nto Fourer modes wth sngle frequency by usng Fast Fourer Transform (FFT). 2. Dervng the soluton of output current for each mode labeled by. 3. Summng the Fourer modes of current to construct the fnal output current n real space. Photododes Optcal sgnals FFT Electrc sgnals FFT Analytc soluton n Fourer space......

4 BASIC EQUATIONS FOR DESCRIBING CARRIER TRANSPORT Contnuty equaton: Current densty equaton: n(tx) J t n (tx) = G n (tx) R n (tx) q p(tx) + J t p (tx) = G p (tx) R p (tx) q J n (tx) = qµ n[n(tx) E(tx) + J p (tx) = qµ p[p(tx) E(tx) + Equaton for the electrc feld: E(tx) q ε Dn µn Dp µp [N D + (x) N A (x)] n(tx)] p(tx)] VERTICA P-I-N PHOTODIODE Assumptons:. Homogenety n all devce parameters and n radaton ntensty perpendcular to the lght radaton 2. Constant electrc feld E n the -regon 3. Neglgble potental drop n the and regon 4. Shallow regon wth respect to penetraton depth 5. No change of the electrc feld due to the ncdent lght pulse x ght pulse hole Eg Ev Ec electron V PN E Generaton rate: Fourer Expanson Method Recombnaton rate: G p (xt) = p(xt) p τ p G np (xt) = α φ(t) e αx = α (Σ φ e t ) e αx ( n(xt) p(xt) J n (xt) J p (xt) ) = Σ ( n (x) p (x) J n (x) J n (x) ) e t Soluton: qαµ n E qα p αµ n E ( τ p ) /2 + α p J total (t) = Σ [ (e α e /µ n E ) e a ] φ e t Normalzed Current Densty Model 2D smulator MEDICI Statonary approxmaton 2 Current Smulaton 2 4 Tme [ps] Ths fgure shows the photocurrent usng our model as compared wth those obtaned by a conventonal 2D devce smulator MEDICI and by a statonary approxmaton: e α Jtotal(t) = q ( ) φ (t). + α p [See S. M. Sze (98)] S vertcal p--n photodode depth of and -regon:. µm. µm mpurty concentraton n and -regon: 2 cm 3 2 cm 3 5 cm 3 Gaussan lght pulse wdth: σ ps wavelength: λ 532nm

5 Assumptons: ATERA P-I-N PHOTODIODE. Homogeneous rradaton only for -regon 2. Deep and regon wth respect to penetraton depth 3. Constant electrc feld E x n the x-drecton of the -regon 4. No change of the electrc feld due to the ncdent lght pulse Al Soluton: W ght pulse Eg Fourer Expanson Method W dz I n () = Jn(xyt)dy = Σ [ ( e / µ ne x )] Current Smulaton y hole E x electron x= x= q µ n E x W φ e t Fabrcated Devce and Measurement ght Pulse D p+ n+ p+ p-s () = 2µm D =.5µm ARC OCOS 2 cm 3 2 cm 3 substrate 5 cm 3 Synchronzed Sgnal Input Voltage PD Trgger Sgnal 47pF 47Ω sgnal 47pF Pulse aser p-s DUT Beam Spltter Trgger lne 47Ω V PN Osclloscope (Rn=5Ω) TRG sgnal Wavelength 532nm CH Ec Ev Pulse Wdth(FWHM) ~ps Normalzed Current Measurement Model Input Pulse Normalzed Current V PN = 7V Measurement MEDICI Tme [ns] Ths fgure shows the photocurrent usng our model as compared wth those obtaned by a conventonal 2D devce smulator MEDICI and by a statonary approxmaton: J total(t) = q W φ (t). SUMMARY Tme [ns] Analytc soluton of p--n photodode current n Fourer space has been derved. Smulaton usng spectral method has been successfully performed to construct photocurrent n spte of a sgnfcant reducton n calculaton tme. Our model s applcable to crcut smulaton of OEICs.