A comprehensive analytical and numerical model of polysilicon emitter contacts in bipolar transistors

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL.

ED-31, NO. 6 , JUNE 1984

773

A Comprehensive Analytical and Numerical Model of Polysilicon Emitter Contacts in Bipolar Transistors ZHIPING YU,

STUDENT MEMBER, IEEE,

BRUNO R I C C ~AND , ROBERT W. DUTTON,

FELLOW, IEEE

to be responsible for the increased current gain of these devices. These two models have been used, both analytically [4] and numerically [ 5 ] , [ 6 ] , to study the I-V characteristics of the poly-emittertransistor. Fossum [4] proposed to characterize the minority-carrier behavior in polysilicon ,with an effective mobility and a lifetime constant, thus resulting in a reduced effectivesurface recombination velocity as viewed fromthe monocrystallineemitter.Eltoukhy[6] applied thetunneling current expression derived by Mandurah for grain boundaries [7] to the polysilicon%licon interface and incorporated it in numerical analysis to model the effect of the interface on I-V characteristics. Qualitative agreement with published data are achieved in the efforts mentioned above. However, two major issues, the expression for minority-carrier mobilityin the polysilicon and the majority-carrier tunneling through the interface are not well formulated. Previous analysis has used one overall expression forthe minority-carrier mobilityin polysilicon, and adopted itdirectly I. INTRODUCTION from the expression developed for themajority-carrier mobility HE USE OF heavily doped polysilicon as the diffusion in polysiliconby Baccarani [8] . Thismobilityreduction is source ofemittersand/oremittercontactsof bipolar assumed to be due to potential barrier at the grain boundaries. transistorshasfound wide application in IC structureswith Realizing that potential barriers for the majority carriers are shallow junctions due to several superior properties. Among not necessarily those for the minoritycarriers, especially when these the most significant feature is the dramatic current gain they are caused by trapping of the majority carriers, the inincrease-higher than that of the conventional transistor by as adequecyofthe use of the same mobility expression for much as a factor 7-10 [2], [3]. minority carriers is obvious. Moreover, inthe analytical Various modelshave been used to explain I-Vcharacteristics, approach [4] , the effect of the polysilicon-silicon interface inparticularthecommon-emittercurrent gain increase, of was not includedin the calculation of the effectivesurface polysilicon emitter(poly-emitter)transistors.Thereduction recombination velocity, thus the use of this parameter in the of the base current, which consists mainly of the recombina- device analysis is largely limited. tion current of minority carriers injected into the emitters in The formula of the tunneling current for the majority carriers most bipolar devices, is atfirst attributed to the possibly re- used in [6] is less revealing as regarding to the effect of the duced bandgap narrowing in polysilicon [ 2 ] . This hypothesis interface on the majority-carrier current. Moreover, its validawas later considered to be lacking of experimentalsupport tion needs to be justified because the original formula derived [3] , Two other mechanisms-tunneling of carriers through an by Mandurah [7] is based on the Boltzmann statistics, whereas oxide-likeinsulatinglayerat the polysilicon-silicon interface forthe heavily doped polysilicon emitterthe Fermi-Dirac (tunnelemittermodel) [3] and theretardedtransport of statisticsapparently applies to the majority carriers. Anew, injectedminority carriersdue to the lower mobilityinthe complete formula for the tunneling current of the degenerate polysilicon region (two-region model) [ 11 -have been proposed carriers is thus needed. In this paper, we verify and extend Fossum’s concept of the Manuscript received May 12, 1983; revised December 20,1983. This work was initially supportedby an industrial grant from FairchildSemi- effective surface recombination velocity to include the effect conductor and by ongoing support~fromTexas Instruments, Inc. of the interface on the minority carriers at low or intermediate Z. Yu and R. W. Dutton are with the Integrated Circuits Laboratory, biases. This is done byfirstcharacterizing grains and grain Stanford University, Stanford, CA 94305. , and then using an B. Ricc6 is with Istituto di Elettrotecnica ed Elettronica, Universita boundaries with different mobilities [9] di Padova, Padova, Italy. approximateform of thetunnelingcurrent expression for Abstract-A comprehensive model-both analytical and numerical-is proposed as a tool to analyze heavily doped emitters of transistors with polycrystallinesilicon(polysilicon)contacts.Thegrainsandgrain boundaries of polysilicon,theinterfacialoxide-likelayerbetween polycrystalline and monocrystalline silicon are lumped respectively into “boxes” in which the drift minority current component is neglected. on the whole is exThe mobility reduction of carriers in polysilicon plicitly attributed to the additional scattering dueto the lattice disorder in the grain boundaries and the carrier tunneling through the interface. The effect of the poly-contacts on transistors can be modeled as a reduced surface recombination velocity for minority carriers in combination with a series emitter resistance for majority carriers. Furthermore, by characterizing the monocrystalline emitter with an effective recombination velocity, the effect of the polysilicon layer on the currentgain can be analyzed analytically. Computer simulation is used to verify the assumptions of the model formulation. Using published data [ 1J , the analytical and numerical approaches are compared and i t is shown that for thesedevicesa uniquecombination of physicalparameters are needed for the model to fit the data.

T

001 8-9383/84/0600-0773$Ol .OO 0 1984 IEEE

IEEE TRAPJSACTIONS ELECTRON ED-31, DEVICES, VOL. ON

114

minority carriers. After applying the diffusion equation (neglecting the drift component of the minority-carrier curren ,), a set of linear equations of excess minority-carrier concentlations can be obtained, whichleads totheconceptof tE effective recombination velocity (hereafter referred to as EIP‘J) for minority carriers viewed from the monocrystalline emitt1:r. The effect of the monocrystalline emitter on the base currerit at low or intermediate bias range can also be characterized )y an effective recombination velocity at the emitter edge of the emitter-basespace-charge region through solving a first-orc;,er differential equationfor generalized effective rec0mbinati.m velocity S(x) [ 101 with the effectivesurface recombinatim velocity of the polysilicon layer as the initial value. Due to the nonlinear I-V characteristics oftheinterfacetunneling currentat high biases, numerical methodsmust be used 1.0 obtain the complete device I- V characteristics. The simulai e:d results using the interface properties (thickness, barrier heights) as fittingparameters is in good agreement with publisked data [ l ] . The paper is organized as follows. InSection 11, a“bcx” modelfora slice ofsemiconductor or amorphousmate .ial (grain boundary) is proposed; in Section I11 the formulas u jl:d to describetunneling through an insulatinglayerare deriil:d and it is shown that the box approximation can be exteniilsd to include the minority-carrier tunneling at low or intermed: bias. A quasi-analytical model (ingeneral, the differential equation on S(x) has to be solved numerically) for the monocrystalline emitter is proposed in Section IV. The analytical formulation of the current gain increase based on the abe’ve model is presented in Section V , and an example of the polyemitter transistor is analyzed both analytically and numerically in Section VI, the two results are compared. Finally con:lusions are drawn in Section VII. 11.BOX

ANALYSIS OF POLYSILICON GRAINS AND GRAIN BOUNDARIES

It is well known that polysilicon consists of monocrystalline grains and disordered grain boundaries [7], [ l l ] , [ 121 . For heavily doped polysilicon emitter transistors, the built-in dectric field inside the grains due to the potential barrier catlsed by the trapping of majority carriers [ l l ] can be consid :re‘ negligible, then one can neglect the drift current compo:lent in solving thecontinuityequationforminority carriers. By applying the diffusion equation to each grain, the currenx at the ends of the grain can be easily expressed as the linear function of the excess minority-carrier concentrations at the same grain ends (see Fig. l(a))

NO. 6 , JUNE 1984

x - l o (a)

cb)

Fig. 1. Schematic diagram of the box model for the grain (a), and for the grain boundary (b).

p 1 , p 2 are the excess minority-carrierconcentrations at the edges of the grain itself (excluding the grain boundary), D p ,L p are the hole diffusion constant and diffusion length inside the grains, respectively, which are assumed to have the same values as in monocrystalline silicon, and dg is the grain size (not including the grain boundary). In the above equations, carrier recombination inside of grains has been taken into consideration, At grain boundaries, additional scattering occurs due to the lattice distortion and extra recombination takes place because of the trapping states located at the boundary. The common model for the grain boundary is to neglect its thickness and ate assume the carrier concentration unchanged across the boundary. That is, the recombination is assumed to occur in an infinitely thin layer [ 5 ] , [6] . The overall mobility reduction in polysilicon is then attributed to the potential barrier of the space-charge region abouttheboundary. As pointedout in the previous section, such model is not suitable for explaining the mgbility reduction of minority carriers. Here we use the mobility model suggested by Kim [9J , that the mobility in the grain boundary is different from that in the grain. Then the overall mobility reduction is explicitly attributed to the lower mobility b,& in the grain boundary with finite thickness. As to the recombination due to the energy states in the bandgap at the grain boundary, one can either define an average lifetime whichmay vary with the carrier concentration at the grain boundary [SI,or assume that the recombination only occurs at the interface between the grain boundary and grains using an interface recombination velocity S g b . In view of the easy usage of the interface recombination velocity,in this paper we use the latter t o model recombination at the grain boundaries. Thus the grain boundary is modeled as having finite thickness from the perspective of carrier mobility and being infinitesimally thin for the purpose of computing recombination. The use of the model parameters P g b , S g b , is shown below. This treatment again results in the linear relationship of the current densityand excess carrier concentrationatthe grain-grain boundary interface (see Fig. l(b)).

where with

YU e t al. : POLYSILICON EMITTER CONTACTS

115 INTERFACE

where S is thewidthofthe grain boundaryregion,and is the hole mobility in the grain boundary, which is called the extended state mobility and takes the value of &b = 2 cm2/V s in [9]. Equation (3a) may be rewritten as

n+ POLYSILICON

SILICON

I

j g b = qTgb(P2 - P 3 )

(44

with EMITTER

Later, wewillsee the minority-carrier tunneling through the silicon-polysilicon interface can also be expressed in the same form as (‘la), while Tgb is replaced by the tunneling coefficient provided- that the applied voltage drop across the interface is small compared to the thermalvoltage kT/q. The interface recombination velocity Sgb at the edge of the grain boundary is here defined as

(The grain boundary in Fig. l(b) is assumed to be contiguous to the grain in Fig. l(a) from left). Ignoring the dependence of the recombination at the trapping stateson Fermi-levels [SI leads to a simple expression for s g b [4]

Fig. 2. Schematic diagram of polysilicon contact to the emitter of the bipolar transistor.

series of boxes of grains and grain boundaries from one-dimensional point of view. Let us now suppose there are N grains with the same grain size within the polysiliconlayer along the current path, and that the ERV at the metal contact to the polysilicon is , S (Fig. 2).

j ’(i) = j(x

+ dg)

with x i = (i - I ) d , d = dg + S , the entire length of the grain, andi= l;*-,N. From ( 1 ) and (7), we obtain a tridiagonal system of a 2 N - 1 linear equations.Theunknowns are p’(l), p(i), p’(i)(i= 2 , * * ,N), i.e., the excess minority-carrier concantration at the edges of the grain regions, each of which has a linear dependence on p ( l ) , the injected carrier concentration at the polysilicon side of the polysilicon-silicon interface. Thusp’(1) can be expressed as

-

where the density of trapping states at the grain boundary N,, is split into halves at grain boundary edges, c p is the capture cross section of holes and uth the thermal velocity. Substituting (4a) into (5) results in

(7a)

P ’ ( 1 ) = CP(1)

where C represents a constant having the form of of

C=

(sa) (8b) Equations (1) and (7) show the same linear relationship between the terminal currents and carrier concentrations of the boxes in Fig. 1 , though in Fig. l(a) the grain region is shown with a general, hyperbolic dependence of p ( x ) , whereas Fig. l(b) shows the grain boundarywith linear p(x) dependence. This dependence makes the extension of the concept of ERV possible since the current at any grain and grain boundary is proportional to the excess carrier concentration at that point. In fact, later we will show that as long as the electric field and physical parameters D,, 7, are functions of the dopingprofile only (low-level injection assumption), this linear relationship always holdsforminority carriers. Themodelthus derived from the above approximations are hereafter named with the term “box model” to indicate that only terminal currents and carrier concentrations are concerned. Also, theterm is the analogy ofthe physicalentities-polysiliconconsisting ofa

with Seffthe ERV at the end of the first grain region. The injected current expression is then obtained by substituting the above equations into (1 a) j(1) = 4 [a,p(l) - bgp ’(I)] = qS,,p(l) with

S ~= ’ag =

v,

b; ag + Seff

JG + Seff

v,

~ 0 t (dg/Lg) h ~ 0 t (dg/Lg) h + Seff

where V, = D,/L, may be called the diffusion 3velocity. The linear relationship between the current and concentration at the same point makes it possible to define an effective recombination velocity S p f to characterize the effect of the polysiliconlayer on the behavior of the injected minority carriersin the emitter. That is, the effect of the polysilicon contact without considering the interface between poly- and

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IEEE T b ANSACTIONS ON ELECTRON DEVICES, VOL. ED-31, NO. 6, J U N E 1984

monocrystalline silicon is to transform ERV from SM to Ypt. I If Spr is smaller than S, the blocking effect of the polysilicon due to the low p g b is dominant and an increase of the curl ent gain then occurs. Otherwise, the extra recombination at g x i n boundaries prevails over the mobility reduction and no adyrantage can be taken from the polysilicon layer as far as the derice electric performance is concerned. It is a straightforward matter to solve for Spt knowing the / VARYING POLY THICKNESS, physical structure of a polysilicon layer-even withnonunifo.rm dg:230A grain sizes. The geometric parameters involved are the nurrber of grains, grain size and grain boundary width. The physkal 1 2 3 L 5 6 7 8 9 GRAIN NUMBER N parameters are theminority-carriermobility andlifetime in(ERV) asa side of grains, p g , T ~and , p g b , N,, at grain boundaries as de- Fig. 3. The polysiliconeffectiverecombinationvelocity function of geometric and physical parametersof polysilicon. fined above. For example upper dash line in Fig. 3 shows when the recombination effect when N = 1 (one grain) (due t o large capture cross section c p ) exceeds the mobility reductioneffect,the increase of the grain number can only deteriorate the characteristics of the transistor through increasing Spr. This prediction is different from previousresults (Fig. 3 of [SI), where the increase of the number of grains when N = 2 (two grains and one grain boundary) always results in the decrease of the ERV. As for the majority carriers, the effect of polysilicon layer is takenintoaccount by the effective mobility averaged over grains and grain boundaries as expressed in [4] . with 111. TUNNELINGMODEL OF THE -2 POLYSILICON-BULK INTERFACE Due to the process of depositing polysilicon on the emitter window of the substrate, an insulating layer is inevitably introActually, it is not necessary to solve the whole set of linear duced between the polysilicon and silicon. Thisinsulating equations simultaneously in order to obtain the value of .;,D,. layer is oxide-like in nature. Because of its larger bandgap the By repeated application of (10) from the metal contact, using insulating layer forms apotential barrier for both electrons corresponding values for S, ag, b, at each box being ev;~luand holes. ated,one canreadily obtain Spt. Noticethatthebox here A rectangular potential barrier is assumed for the resulting refers to both grain and grain boundaries. It should be poinkd interface. In a way similar to themodeling of grain boundaries, out that a similar method has also previously been used by the recombination attheinterface is lumped equally on its Dunkley [13] . Our method differs in that an one-way propatwo sides. Carriers pass through the interface bytunneling. gating method from the surface for calculating the ERV (:an Thermionic emission can be included in the tunneling mechabe used as a result of using the surface recombination velocity nism by assuming the tunneling probability to be unity when at the electrode contact as the boundary condition. The rethe carrier energy in the current direction exceeds the barrier sults provide a simpler calculation method with aclear phys. cal basis for determining parameters. The electric field is neglec ted height. However, when the barrier height, measured from the in the previous model formulation, though adoping depend :nt band edge in the nondegenerate case or from the Fermi-level electric field can be incorporated in the model as indicated by in the degenerate case, is larger than several kT, the thermionic contribution to the currentis normally negligible. Dunkley [ 131 for the case of constant field. The formulas of the tunneling currents through arectangular The effect of the polysilicon layer on the ERV Spr is shown potential barrier are derived in the Appendix, using the W.K.B. in Fig. 3 based on the computation of (10) and (1 1) as we1 as approximation and with an approach similar to that used in the further terms with grain number N larger than 2 . Two [14]. The results areshown below, and the formula for the competingfactors change Spl in different ways. Theextra majority carriers is obtainedunderthe assumption that the recombination occurring at the grain boundaries increases :he Fermi-level for electrons is much higher than the conduction totalrecombinationcurrent,thus resultingin higher whereas the smaller mobility at the grain boundaries decreases band edge. ,!;i,t;

the slope of the injected minority-carrierdistribution in h e region adjoining the polysilicon layer, i.e., the monocrystalline emitter [ 1] , resulting in the reduced S p t , When the lattereff :ct dominates, the increase of the number of grains, either with fixed grain size dg (solid line in Fig. 3) orfixed total polysilic on layer thickness L , (lower dash line) will reduce Spt. While ..he

Ill

YU et al. : POLYSILICON EMITTER CONTACTS POLYSILICON INTERFACE SILICON

(12a); whereas in the case of the minority carriers, the tunneling current is largely causedby the difference of the carrier concentrations, corrected by the band-edge displacement due t o the voltage drop across the barrier, on the both sides of the barrier as is shown in bracketed terms of (12b). It is perhaps worth noting that (12b) is essentially the same as (3), ( 5 ) in [6] except that (12b) shows more cleqrly the relationship between the tunneling current and the carrier concentrations on both sides of the barrier. Equation (12a), however, is different from (3) of [6] in that the carrier concentration is assumed the same across the interface. Sincethe polysilicon-silicon interface is in series with the emitter-base junction, at intermediate bias or below, the main part of the applied voltage falls across the p-n junction so that usually V