boundary condition for the modeling open-circuited devices

VLS1 DESIGN1998, Vol. 8, Nos. (1-4), pp. 567-572Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach SciencePublishers imprint.

Printed in India.

Boundary Condition for the Modeling of Open-circuitedDevices in Non-equilibriumJOSEPH W. PARKS JR. and KEVIN F. BRENNAN*

School of Electrical and Computer Engineering, Georgia Instituteof Technology, Atlanta, GA 30332-0250

A boundary condition specifically designed to model open-circuited devices in amacroscopic device simulator is introduced. Other simulation techniques have relied onan external circuit model to regulate the current flow out of a contact thus allowing thepotential to remain the controlled variable at the boundary. The limitations of thesemethods become apparent when modeling open-circuited devices with an exceptionallysmall or zero output current. In this case, using a standard ohmic-type Dirichletboundary condition would not yield satisfactory results and attaching the device to anarbitrarily large load resistance is physically and numerically unacceptable. Thisproposed condition is a true current controlled boundary where the external current isthe specified parameter rather than the potential. Using this model, the external currentis disseminated into electron and hole components relative to their respectiveconcentration densities at the contact. This model also allows for the inclusion oftrapped interface charge and a finite surface recombination velocity at the contact.An example of the use of this boundary condition is performed by modeling a silicon

avalanche photodiode operating in the flux integrating mode for use in an imagingsystem. In this example, the device is biased in steady-state to just below the breakdownvoltage and then open-circuited. The recovery of the isolated photodiode back to itsequilibrium condition is then determined by the generation lifetime of the material, thequantity of signal and background radiation incident upon the device, and the impactionization rates.

Keywords." Current boundary condition, macroscopic simulation, charge storage, avalanchephotodiodes

1. INTRODUCTION

Many macroscopic device simulators provide for a

connection to an external circuit primarily through

either ohmic or Schottky contacts. In these cases,the system variables are constructed such that theelectrostatic potential is the independent para-meter describing the boundary condition, and

*Corresponding author. Email: [email protected].

567

568 J.W. PARKS JR. AND K. F. BRENNAN

other values such as the current flowing throughthe contact are calculated in a post-processing step[1, 2]. These boundary conditions are adequate forsituations where the device can be approximatedas a voltage controlled current source; however,instances exist where this may not be the case.Most notably this occurs when the energy storedwithin the internal capacitance of a device becomesthe dominant controlling factor for the predictionof the potential across the device. In these cases,the use of a current controlled boundary is moreapplicable.A situation requiring a current controlled

contact exists when simulating the recovery toequilibrium of an open circuited device. Such acondition occurs when examining the performanceof an avalanche photodiode configured in thecharge storage mode for imaging applications[3, 4]. In this configuration, the sensor is periodi-cally reset by applying a large reverse bias to thedevice. Following the reset, the device is open-circuited and begins to equilibrate through carriergeneration. If the device is used as a photodetec-tor, light shining upon the structure generatesexcess charge which is then stored within thedepletion region of the device. At the end of theintegration cycle, the quantity of stored chargewithin the device is used as an indication of thetotal illumination upon the pixel [5].The present work introduces a current con-

trolled boundary condition which is capable ofsimulating an electrically isolated photodiode.With this model, a steady-state voltage is appliedbetween the ohmic contacts using the traditionalDirichlet boundary condition. The device is thenisolated from the external circuit. The voltageacross the device is no longer fixed invalidating theuse of a voltage controlled boundary condition.Alternatively, a current controlled boundary con-dition must be employed to describe the devicewhich includes effects such as surface recombina-tion, trapped surface charge, and leakage currentfrom the read-out electronics. Application of thismodel to a generic APD is also presented.

2. BOUNDARY CONDITION

A current controlled boundary condition isrequired to accurately model the reverse recoverywithin a photodiode. When the diode is discon-nected from the external circuit or driving bias, thevoltage at the cathode floats. The metal contactcan no longer be treated in its usual manner as aninfinite source of carriers and the assumption thatthe electron and hole concentrations are dictatedby their equilibrium values may not be valid.Moreover, the voltage across the device is deter-mined solely by the distribution of charge and theinternal capacitance of the diode and not by theexternal circuit. Simulation techniques whichoperate by allowing the potential to remain theindependent variable, while acceptable for model-ing switching characteristics, are inappropriate forcases with either very low or zero external current.

Attempting to model the open-circuited devicewith the standard ohmic condition and a very largeshunt load resistance is unacceptable on physicalas well as numerical grounds.The primary concern with developing the current

controlled model lies in the apportioning of theelectron and hole fluxes from the external current atthe contact. It is assumed here that the total currentis divided into the electron and hole currentsrelative to their respective densities at the contact.Thus, the relation between the currents is given by,

n.jp_ p__p__.jn=qRsurf (1)n+p n+p

where J, and Jp are the electron and hole currentsleaving the contact as indicated in the illustrationof the discretized control volume as shown inFigure [6]. The carrier concentrations are takento be those of the neighboring control volumewithin the device. Rsurf is the recombination rateowing to surface states at the contact [7] using thestandard relation:

SSp(np n2i 6(x) (2)Rsurr Sn(n + nl + Sp(p + pl

BOUNDARY CONDITION MODELS 569

FIGURE Illustration of the sample control volume at thesurface of the simulation domain. Current continuity requiresthe conservation of flux within the control volume. Here, thecurrent boundary is entered directly into the formulation wherethe sum of the electron and hole currents balance that of theexternal current.

where Sn and Sp are electron and hole recombina-tion velocities. Additionally, Eqs. (3) and (4) belowrequire that the carrier currents sum to theexternal current and that the band bending at thecontact is proportional to the trapped interfacecharge.

Jn -- Jp Jext (3)

-n Oint (4)

Combining Eqs. (1) and (3) yields expressionssetting the values of the current into a contact.

Jp qRsurf + p"Jext (5)n+p

Jp -qRsurf 4-" Jext (6)n+p

These currents are used directly within the controlvolume formulation of the discretized continuityequations [8]. Furthermore, partial derivatives ofthe current expressions with respect to the systemvariables are analytically obtained thus leading toincreased convergence of the non-linear system.A two-dimensional analysis of the photodiode is

necessary to study the full carrier transport ofeffects such as premature edge breakdown andpixel crosstalk; however, this study is concerned

solely with the examination of the charge storagecapabilities of the APD. Therefore, a one-dimen-sional analysis is satisfactory. This greatly simpli-fies the model by allowing the external currentdensity, Jext, to be directly proportional to thecross sectional area of the device contact and thetotal current. Multi-dimensional extensions of thisboundary can be produced by either imposing anadditional constraint upon how the output cur-rents sum over the range of the contact such asequipartitioning or by solving Gauss’ law describ-ing the current flow through the metal comprisingthe contact [9].To study the charge storage of the photodiodes,

the current boundary condition is incorporatedinto a drift-diffusion simulator which self-consis-tently solves Poisson’s equation and the current

continuity equations [10]. Since the smallestdimensions of the device structures are on theorder of microns, it is believed that hot carriertransport can be neglected in these devices andthat a full hydrodynamic simulation is unneces-sary. Within this model, the generation-recombi-nation rates include terms for SRH, Auger, andradiative recombination, impact ionization, andwavelength dependent photoillumination. Addi-tionally, the standard field dependent mobility andimpact ionization models for silicon are incorpo-rated into the simulator [11]. By using a completelynumerical model, many of the non-linear attri-butes of the carrier transport of a photodiode incharge storage mode can be included without theuse of many of the simplifying approximationstypically employed [3, 4].

3. EXAMINATION OF CHARGE STORAGEIN PHOTODIODES

To demonstrate the utility of the current con-trolled boundary condition, the reverse recovery ofan avalanche photodiode designed for night-skyimaging is examined. Here the device, shown inFigure 2, is a reach-through avalanche photodiode50 gm in length. During the initial reset period, a


2.0m

1.5txm

46.51.tm

0.1

25.0m

n+ le18

p 2e16

x le12

p+ le20

FIGURE 2 Doping profile and geometry of reach-throughavalanche photodiode used to study the current boundarycondition.

large electric field is established in the p-typemultiplication region and a modest drift field existsin the intrinsic region. The illumination spectrumused in this study is that of a cloudless, moonless,night sky with a total illumination of 5.4x1016photons per cm2 over a range of wavelengths from0.3 to 1.1 microns [12]. The total internal quantumefficiency for this spectrum is approximately 55%.

If the photodetector was continuously biased,the photo-generated carriers would be sweptacross the depletion region, collected at the devicecontacts, and interact directly with the externalcircuit. In the charge storage mode of operation,the connection to the external circuit is removed.In this case, the internally stored electric field stillcauses the carriers to be swept across the depletionregion where they may impact ionize. However,the charge is stored near the edges of the depletionregion as opposed to interacting with the externalcircuit. This accumulated charge counteracts thedepleted charge from the reset and tends to drivethe reverse recovery. Additional factors whichpromote the recovery of the diode include theleakage current inherent in the readout electronics,

Jext, the dark current due primarily to thermalgeneration of carriers from SRH centers, andsurface recombination.

Figure 3 shows the dynamics of the chargestorage within the APD for various photoillumi-nation intensities. For this case, intensities rangingfrom 0.05 to 500 times the nominal night skyspectrum are considered (2.7x 1015- 2.7x 1019photons per cm2 per second). The initial bias onthe device is established to produce an initialsteady-state gain of fifteen. The first part of thevoltage recovery is dominated by the influence ofholes filling the intrinsic region resulting in a lineardecay of internal bias with only a very smalldegradation in gain. Once the drift region has beenfilled, the charge is stored at the edge of the highfield multiplication region giving rise to theexpected parabolic voltage recovery. A non-lineardecrease in the internal gain is observed as the highfield region recovers. Under extreme illuminationconditions, the diode completely recovers to itsequilibrium level and begins to become forwardbiased. Recombination processes prevent thediode from continued storage of charge once thediode becomes sufficiently forward biased.

i 50

4o

"- 2010 0X’’"...........................0 ""-i0 2 4 6 8 10

time (msec)

FIGURE 3 Calculated reverse bias recovery of the diodeshown in Figure 2 as a function of integration time. Thedifferent curves represent separate values of photoillumina-tion.

BOUNDARY CONDITION MODELS 571

o IE+6

" ..,..,--’’’’""’*’’" ..,..,..,...o..,..,..o" o.,.,,.o.,,....,,.

IE+20.0001 0.001 0.01 0.1 10

time (msec)

FIGURE 4 Calculated stored electron charge of the deviceshown in Figure 2 as a function of integration time for variousillumination intensities.

Typically, the actual stored charge rather thanthe device bias is quantified at the end of theintegration period to determine the level ofphotoillumination [5]. Figure 4 illustrates the totalelectron charge stored within the diode as afunction of the integration time. As expected, thefirst part of the integration gives rise to a linearcollection of charge with a constant gain. As theamount of charge increases, the gain begins tosaturate with a square root dependence [10, 13].Once the device has recovered to the point whereinternal gain becomes negligible, the linear recov-ery is again observed. As the diode becomessignificantly forward biased, as in an open-circuited solar cell, the charge saturates.

4. CONCLUSIONS

A current controlled boundary condition applic-able to macroscopic device simulators is presented.This boundary type establishes the systems vari-ables such that the external current becomes theindependent quantity at the contact, thus allowingthe contact potential to be self-consistently setwithin the simulation. As an example of this

model, a reach-through avalanche photodiode,operating in the charge storage mode, is examined.Here it is observed that the model allows one tocapture the full transient recovery of the devicefrom the initial reset point all the way through theforward bias condition in the case of extremephotoillumination.

RefeYences

[1] Iwamuro, N. and Tagami, S. (1993). "Stable Calculationfor Power Device Simulation with Inductive Load Circuit:Application of an Integral Method", Microelec. Jour., 24,139-145.

[2] Kausel, W., Nylander, J. O., Nanz, G., Selberherr, S. andPoetzl, H. (1990). "BAMBI- A Transient 2D-MESFETModel with General Boundary Conditions IncludingSchottky and Current Controlled Contacts", Microelec.Jour., 21(5), 5-21.

[3] Weckler, G. (1967). "Operation of p-n Junction Photo-detectors in a Photon Flux Integrating Mode", IEEE J.Solid State Cir., SC-2(3), 65-73.

[4] Komobuchi, H. and Ando, T. (1990). "A Novel High-Gain Image Sensor Cell Based on Si p-n APD in ChargeStorage Mode Operation", IEEE Trans. Elec. Dev., 37(8),1861-1868.

[5] Mendis, S., Kemeny, S. E. and Fossum, E. R. "CMOSActive Pixel Image Sensor", IEEE Trans. Elec. Dev., 41(3),452-453.

[6] Basore, P. A. and Rover, D. T. (1985). "PC-1D Installa-tion Manual and User’s Guide", Iowa State UniversityResearch Foundation.

[7] Green, M. A. (1982). Solar Cells, Prentice-Hall, Engle-wood Cliffs, NJ, 1982.

[8] Pantankar, S. V. (1980). Numerical Heat Transfer andFluid Flow, Hemisphere Publishing Corp., New York.

[9] Gough, P. A., Johnson, M. K., Higgins, S. A., Slatter,J. A. G. and Whight, K. R. (1987). "Two DimensionalSimulations of Power Devices with Circuit BoundaryConditions", NASCODE V, 213- 8.

[10] Parks, J. W., Smith, A. W., Brennan, K. F. and Tarof,L. E., "Theoretical Study of Device Sensitivity and GainSaturation of Separate Absorption, Grading, Charge, andMultiplication InP/InGaAs Avalanche Photodiodes",IEEE Trans. Elec. Dev., 43(12), 2113-2121, Dec. 1996.

[11] Selberherr, S. (1984). Analysis and Simulation of Semi-conductor Devices, Springer-Verlag, Wein, New York.

[12] Engstrom, R. W. and Rodgers, R. L., "Camera Tubes forNight Vision", Optical Spectra, 26-30, 438-442, Feb.1971.

[13] Webb, P. P., Mclntyre, R. J. and Conradi, J. (1974)."Properties of Avalanche Photodiodes", RCA Review, 35,234-278.

Authors’ Biographies

Joseph W. Parks Jr. was born in Oak Ridge, TNon May 21, 1970. He received his B.S. degree in


electrical engineering from the University ofTennessee, Knoxville in 1992 and his M.S.E.E.from the Georgia Institute of Technology, Atlanta,GA, in 1993. He is presently working on his Ph.D.degree also in electrical engineering at the GeorgiaInstitute of Technology. His research work in-volves the numerical modeling of semiconductordevices with emphasis in the drift-diffusion andhydrodynamic simulation of photodetectors andavalanche photodiodes.

Kevin F. Brennan received the B.S. degree inphysics from the Massachusetts Institute ofTechnology, Cambridge, in 1978, and the M.S.degree in physics and Ph.D. degree in electrical

engineering from the University of Illinois, Urba-na-Champaign, in 1984.He is currently Institute Fellow and Professor,

School of Electrical and Computer Engineering,Georgia Institute of Technology, Atlanta. Hiscurrent research interests include the physics andmodeling of semiconductor devices. Of particularinterest are the physics and modeling of avalanchephotodiodes, confined state ionization devices,high field effects in semiconductors, photoconduc-tors, and high speed transistors.

Dr. Brennan was the recipient of a PresidentialYoung Investigator Award through the NationalScience Foundation.

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