theory and experiments on surface l f noise

8/8/2019 Theory and Experiments on Surface l f Noise

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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-19, O. 2, FEBRUARY 1972 27

Devices,vol. ED-8, pp. 370-377, Sept. 1961.itance diode with large igure-of-mer it, IRE Trans. Electron [SI P. Brook and C. S. Whitehead, Hyperabrupt unctions in

[j ] M . Shinoda, Capacitance of the hyperabrupt unction fabri-Au-Si Schottky diodes by on-implantation, Electron. Lett.,

cated with alloy diffusion technique, J . Inst . Electron. Commun. 191 M . P. Lepselter and S. M. Sze, Silicon Schottk y barrie r diodevol. 4, o. 16, pp. 335-337, 1968.

Eng., Jup. (Abstracts), vol. 47, no. 3, pp. 14-15, 1964. with near-ideal I- V characteristics, Bell Syst. Tech. J., vol.

diffused variable capacitance diode, Solid-state Electron., vol. [lo] J . W. Mayer, L. Eriksson, and J. A. Davies, Ion-Implantation in

[7] S. Nakanuma, Silicon variable capacitance diodes with high [ l l ] R. A. Moline, Ion-implanted phosphorus in silicon: Profiles6, no. 1, pp. 1-24, 1960. Semiconductors. Kew Yo rk: Academic Press.

voltage sensitivity by low temperature epitaxial growth, EEE using C-V nalysis, J . Appl . hys., Aug. 1971.Trans. Electron Devices,vol. ED-13, pp. 578-559, July 1966. [12] R. W. Treible and R. A. Moline, unpublishel.

[6] T. Sukegawa, K. Fujikawa, nd J . Nishizawa, Silicon lloy- 47, p. 195, 1968.

Theory and Experiments on Surface l / f Noise

HORNG-SEN FU, MEMBER, IEEE, AND. CHIH-TANGSAH, FELLOW, IEEE

Absfracf-A theoretical ow-frequency oise model orhe

epitaxial-channel surface field-effect structure is presented whererandom modulation of the channel conductance arises from fluctua-tion of charges rapped at the oxide trap states near the Si-SiOzinterface. In this model, charge fluctuation in the oxide traps arisesfrom c,arrier tunneling between the fast interface surface states andthe oxide trap states. A second fluctuation, at higher frequencies,arises from the random thermal emission and capture of electronsand holes at the fast interface states through the thermal r Shockley-Read-Hall process. Different oxide trap densities were introducedinto th e interface egion of t he metal-oxide-silicon ield-effectstructur es using a carefully controlled and reproducible oxygen heattreatment techniqu e. Energy distributions of th e oxide trap densiti esare obtained from capacitance measurements. Humps are observ edbetwelen the lat band and he ons et of str ong surface nversion(lower half of t he bandg ap) in both the noise power and the oxidetrap density versus gate voltage or surface band bending) plots.

Theoretical noise power calculations using th e experimental oxidetrap density profile from he capacitance-voltage data agree very wellwith the experimental noise humps n both magnitudes and finestructures. I t is shown that the frequency spectra of noise dependstrongly on the oxide trap density profile in the oxide. It is uggestedthat the oxide traps a re due to the excess xygen at the SiOz-Si inter-face.

I . NTRODUCTION

H E 1,s noise has been studied by many investi-gators. Early s tudies were primarily on germani-um ilaments whose urfaces were xposed to

different ambients. nlany different models have beenproposed to explain the noise data [1]-[9]. McWhorter[ l o ] proposed -the tunneling model to explain the widerange of frequency spectrum observ ed in Ge filaments,which has been used by many author s to accoun t forthe surface l/ f noise in RlOS field-effect devices [11]-

This work was supported n part by the U. S. Air Force Office ofManuscript received March 23, 1971; revised September 18, 1971.

Scientific Research and he Advanced Research Projec ts Agency.The work is based n part on the doctoral thesis of H. S. F u sub-mitted to th e Gr aduate College of the University of Illinois, Urbana,Ill.

and he Materials Research Laboratory, University of Illinois, Ur-The authors are with the Department of Electrical Engineering

bana, I l l . 61801.

[18]. In adopting McWhorters tunneling model, these

authors assumed that carr iers in conduction or valencebands tunnel directly into the surface states which arelocated a t some energy in the semiconductor bandgapand at some distance away from the hterfac e in thesurface oxide. Electron energy of ab ou t half of th e semi -conductor ga p (-0.5 eV) must be dissipated. I t wasshown byKane [19] that neither he uger- impactmechanism, nor the photon mechanism, nor the multi-photon processes are plausible. Thus, an inter mediat es t a t e is essential.

In hi spaper, a newmodel of carrier unnelingthrough n ntermediate tate is roposed th at isanalogous to the tunneling odel used earlier to explain

th e excess currents observed in gold-doped silicon tun -nel diodes [20]. I n this model, the carriers in conductionor valence bands communicate with he fast surfacestates ocated at he nterf ace hrough he Shockley-Read-Hall (SRH) process. T h e carriers hen unnelinto or out of the oxide traps located a t some distanceaway from the interface elastically. This model is moreplausible than early models since i t is well known thatthe hermal SRH process s very efficient an d ha tcontinuous (in energy) istribut ion of fast urfacestates at he nterface is a common characteristic ofsilicon-silicon-dioxide interface. n t.his model,separate time constant for the tunneling process is then

obta ined, in addition o he Shockley-Read ime con-stant . This is in contrast to the earlier models just re-ferred o where he unneling process is tacitly ab-sorbed nto he Shockley-Read hermal capture atecoefficients.

In getting acorrelation between he surface statesor oxide raps and he l/ f noise, the main difficultyencountered n he previous nvestigat ions lies in thelack of deta ile d da ta on the spatial and energy distribu-tions of the surface state s. A certa in type of s patial orenergy distribution was generally assumed a p r io r i ithe earlier work. R ecently, Sah and Hielscher [ l l ] have


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274 IEEE TRANSACTIONS ON ELECTRON DEVICES,EBRUARY 1972

shown xplicitly th at a linear elationship betwcmsurface state density and he l/ f noise exists. Th.:ydemonstrated he ccurr ence of corre spond ing fi r1.estructu res in both the magnit ude of th e low-frequen.:ynoise and the surface state density as a funct ion of c n -ergy n he band gap. In his paper, a comprehensir.est ud y of th e l/ f noise an d the surf ace stat es or oxi l:etraps will be given. The surface sta te densities a Iddistributions are obtained from capacitance measur e-ment s that are then used to calculate the noise P O M . ~

through he new theory, an d com par ed o he noisiedata. Epitaxial-channel silicon surface field-effect strt.c-tures are used, since his allows us o probe a mu:hwider range of energy in the bandg ap in both n oise anldsurface state measurements. These devices were fab a;-cated n ur aboratory nder arefully control'l8:'dconditions, so tha t the oxide trap and interface statesare reproduced over a wide range of density.

The results will be presented n hree parts. Fir,:,t,a der iva tion of th e noise power from the generatioil-recombination-tunneling inetics ased on the newmodel is summarized. Second, a small-signal equivale I tcircuit is given th at includes the SR H proc ess betwel:ln

the fast surface states and the conduction or valen :ebands and the tunnelin g process between the fast sur-face states and the oxide traps. This circuit is used 113extract he oxide rap densi ty rom he capacitanxdata. Finally, he experimental results on he surfa,:esta te or oxide trap densit y and the l/f noise power a represented nd orrelated o ach other hrough ll'etheory developed.

11. SURFACE OISETHEORYFig. 1 shows he coordinate ystem and he crcljs

sect ion of th e epitaxial n-c han nel urf ace field-effe:ttransistor used in this analysis. With suitable notatic.m

changes, his analysis canalso beapplied o he 3,-channel as well. Fig. 2 shows some definitions of tlueterminologies used here. Surface states which occur 11an infinitesimally hin ayer at th e interface ( x = O , ; J )are known as fast surface states or interface states a r dthey can e xchan ge char ges read ily with th e conductic.111and valence bands. Those states which are located at :icertain distance away from the interface (x = x , y ) a -13known as slow surface states or oxide t raps since thtycannot readily exchange charges directly with the co:1-duction or valence bands. The tr ansi tion processes a r dth e energy band dia gra m of the device are shown 11Fig. 3. These ransition mechanisms arediscussed 1 1more detail in the following sections.

A . Charge Trapping Mechan i sm

A fast surface state with energy level a t ET with.11the energy bandgap will act as a Shockley-Read-H::Ilrecombination center. I t will thermally capture ele':-trons from he conduction band or emit rapped ele:-trons to the conduction band. Similarly, oles will mal..^:transitions between the surface states and the valen1::eband states. A trapped electron n t h e fast nterfa#::(:

DrainSect ion

thin s l a b

Fig. 1. Cross section and coordinate system for theepitaxial n-channel MOS translstor.

f ixed oxidecha rge 4 dra(in

fast surfocesta te

gates t a t e )l ec t rode

source

Fig. 2. Definitions of fast surface states and oxidetraps in an MOS transistor structure.

i h

Metal OxideemiconductorX - I L X '

Fig. 3. Energy band diagram and charge changingprocesses in the epitaxial n-channel MOS transistor.

surface state can also tunnel into an empty oxide traplocated n he oxide a t some distance away from heinterface and vice versa. Consider an empty urfacesta te located a t (x = 0 , y) . It can capture an electronfrom the conduction band at the same position in thechannel process (a )] an d become negatively charged.At the same time, an electron from the n+ source con-ta ct will drift nto he channel o maintain he samenumber of electrons. ince lectrons remajoritycarriers here, he electron drift ime is the dielectricrelaxation ime,whi ch is of the rder of s [21]T he foregoing capt ure process herefore causes a ne tcharge fluctuation a t ( x = 0, y ) . The electrons capturedby the surface states can also be thermally excited tothe conduction band [process (b)] or recombine with ahole captured [process (c) ] from he valence band a t( x= 0 , y) . If the trappe d electro n is thermally excited


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PU AND SAH: SURFACE I / ~ N O I S ~

into the conduction band, t will be swept away rapidlyfrom ( x = 0, y) to (x, y ) by the surface electric field inthe x direction, causing a net charge change t ( x = 0, y).However, if a hole from the valence band s captured byoccupied urface tates, here will be no net chargechange a t (x = 0 , y) since holes cannot communicatewith the external circu.it due to the large barrier fromsurface band bending. The same applies if a hole is

thermally excited rom a surface state process d)].Because he hole pocket is isolat ed from he externalcircuit (there s no hole cur rent ), hole ransitions be-tween he surface states and valence band states willnot nduce ne t charge luctuation n he conductionchannel and therefore will not give low-frequency noise.A trap located in the oxide close o he nterface canexchange charge with the fast surface states or the bandstates a t the nterface hrough unneling. An electroncaptured by a ast surface state will tunne l nto anoxide trap at th e sa me ene rgy evel [process (e)] locateda t a distance from the interface and induce a charge ofq ( x , - % ) / x , a t (x = 0, y), where x. is the t hickness of t he

oxide. Similarly, he everse process process f)] willinduce a net charge of - q ( x , - x ) / x , in the conductionband.

In order to calculate the trapping time constants forthe charge fluctuations n he fast surface states andoxide traps, the following parameters will be defined.

Surface densit ies of electr ons and holes,respectively, at th e nterface x = 0, in num-ber/cm2.Capture robabilities or lectrons ndholes, reslpectively, by a sh eet of interf acestates, in cm2/s.Emission ates or lectrons nd oles,respectively, from he nterface states, nnumber/s.Densit ies of electrons nd holes, espec-tively, rapped in a sheet of inter facestates, in number/cm2.Electron and hole oncentrations in th eoxi de tr aps , in Umber/Cm.(nTo( = nToAx)an d $TO( = p r o a x ) are the sheet densitiesof electron s and holes, respect ively, in theoxide raps ocated rom x to x + A x , innumber/cm2.)Tunneling probabilities or electron un-neling into or out of a trap, in cmz/s.

The rates (number/cm2.s) of processes (a), (b), (c),(d ) , (e), and (f) in Fig. 3 can then be expressed as

275

Th e ne t ra te of change of electron and hole concen-trations n he conduction and vaSence ban ds, due ocapture and emission processes at the fast surf ace state s(neglecting ll ther eneration--recombination ro-cess es) , and the net rate of ch ange of t rapped electron sat the surface states and oxide traps (located from x tox+Ax from he nterface) can be obtained rom herates of these processes. Th ey ar e

dns j N

at- (b) - a) - -

4- -

wherejN is the densi ty of electron drift current due tothe presence of the electric fie ld, resul ting in a sho t oisespectrum.

B . Charge Trappin g TimeCons tan t s

The fluctuating components of ns, Psl nss, and nTOcan be obtained from the foregoing equations by sepa-rating the steady-state components. Let

ns = Ns + 6ns ( 3 4p s = Ps $. sps (3 b)

ns s = Nss + 6nss (34%TO = 1VTO + &%TO (3 d)

j~ = J N + j N (3 4where NE, P B , SS, NTO, nd J N are the sLeady-statequantities and S denoted the fluctuating component.

The t ime d ependence of 6ns, 6 p , ~ ,ass, an d 8nTO canbe obtained by substituting (3a)--(3e) into (2a)-(2d).This involves four time constants whose exact solutionsare tedious to obtain. However, the time constant forthe conduction electron fluctuation 6ns is much shorterthan those of the trapped charges 6p8, Gnss, and 6nTothus, this problem can be simplified by considering theslower ime constants only since we are ntereste d nthe ow-frequency noise. The alge bra is given n Ap-pendix I. The time constant s associate d with 6nss an d6nTo ar e given by

1

c ~ ~ ~ $ ~ sen s -I- P s P s 4- ep s78s = . _ _ _ ( 4 4

I

where Tl = r l exp ( - 8x) , T2= 2 exp ( -28x), and

Here 0 is the attenuating constant [ 2 2 ] of the wave


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216 IEEE TRANSACTIONS ON ELECTRON DEVICES, FEBRUARY 197

functions in the oxide; T ~ , ~ , nd T~ are unknown c m -s tan t s ; m* is the effective mass of electro ns in the ox.itle;1' is thepotential barrier between he ast surlslce

state and the oxide trap ; and , is the Planck's const: nt.I t is important to note that th e transition p robabili iesbetween he oxide rap states and he surface st: 1:escome mainly rom hose surface states with ener!,;iesclose to he rap energy level elastic). Transiti ~ n sbetween an oxide trap state and the states in th e c m-duction or va lence bands are also possible inelast I:),but re not mportant due o arge nergy hange. Fig. 4. Lumped ircuit model for a n MOS transistor.

C. Induced Charge Fluctuation in the Conducting Charinel

The net charge fluctuation per unit area in the c'm-duction band of the conducting channel, Q N ,s th e s jmof the ind uced ch arge du e to the cha rge fluctuation:, inthe surface states, oxide traps, and holes at th e inTa--face. T ha t is,

+ shot noise term [ 5 )where he actor C w / ( C w + C o ) results rom he f ; t l c tthat the induced charges are shared by two capacitcts,Cw (depletion layer capacitance) and C, (oxide capz1c.i-tance), and on ly those cha rges induced n C w ap pe ar iinthe onducting hannel. Th e luctuation of 6 n s isdropped since t has a very short time constant resultingfrom he presence of the surface electri c ield, 1vhic.hsweeps ut all the urface lectrons nto he bCikregion. I t is shown n Appendix I that he domin2ntlow-frequency term in ( 5 ) is the first term which con ICSfrom electron unneling nto or out of th e oxide r:.pstates located a t a region from x t o % + A x . Therefore,the charge fluctuation Q N is given by

where Q N is the electron charge density per unit areain the conducting channel and can be obtained from

D

QN = S, n(x ' )dx 'where D is the dep th of t he conducting channel an dn ( x ' ) is the electron concentra tion in the conducting

channel.To obt ain the oise voltage across the drain electrode,

a small-signal umped model for he field-effect str uc -ture is used [ 2 3 ] . This is shown in Fig. 4. The entirechannel is divided nto hree subsections, namely, hethin lab, he ource ection, and he drain ection.p, an d e,, represent the amplification factor and the gatesignal of the source section, an d ,u d an d egd representthose of the drain section. 8 v d is the open-ci rcuit noisevoltage a t the drain electrode, due to the fluctuation nthe thin s lab a t , and is given by

This fluctuati on is amplified by the drai n sect ion only.T he power spe ctrum of the luctuati on 6 v d , SavdA

where Af is the frequency bandwidth, can be obtained' e x p (-&)G n To ( o ) A x :) from he Wiener-Khintchine heorem.

where 6nTo(0) is the stea dy-s tate valu e of &To.

D. ur f ace No i seThe drain noise current 8 i d ,which is induce d by l.he

charge luctuations n he raps, can be expressed asfollows :

where ,un is the mobi lity of elec tron s and is assumEdto be constant, 2 is the width of the cond ucti ng chan -nel, and A V y s the dc voltage drop across the thin sl l bof the semiconductor located from to y + Ay along I: l econducting channel Fig. 1). Th e voltage luctuationacross the thin slab, A V y , caused by the current luctLl;.-tion 8 i d can therefore be related to AVY an d is given 'l)y

where the relation [ 2 3 ] , [24 ]

was used. w is the angular frequency, NTTo is the con-cent ratio n of oxide tr ap s a t x , n d f t o s the equilibriumFermi-Dirac occupation factor of the traps. The total


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PU AND SAH: SURFACE ~ I ~ N O I S E 277

TABLE IPARAMETERSF THE N-CHANNEL EVICESND OXYGEN EA TTREATMENTOHT) CONDITIONS

Substrate or Epitaxial LayerEpitaxial Gate Effective

Dopingayer Oxidehannel Gate O H TDevicerienta-ensityhicknesshicknessengthidthrea A Tempera-ime

Numberypeion ND(cm-8) D b m ) xo(A) L b m ) Z(pm) (10-8cma) ture(OC) (h )_ _ . _ _

1 n on p+ (111) 0.5X106 3. 2 1188 5 1.2 2542 3 -25 none

- -2 n o n p+ (111) 0.5.2 1320 51.2.2 5 1000 0.53 n on p+ (111) 0.5.2 1216 51.2 2542 3.25 800 1 . 05 n o n p (100) 1 o 4.703 06.5 2547 3.14

(100 )8 nn p (100) 1.0.7 1128 46.5 2547 3.14 600

(1 00)n on p 1 o 3 .5 2045 36. 0 2570 2 73 600 96 .010 n on p ( 100) 1 o 3.42 2080 50.0 2500 3.2 8 600

(100)1 P 5.0 nonepitaxial 1830 36.0 2570 2.6612 P (100) 5.0 - 1770 36.0 2570 2.70

2542-

4a, b, cn p+ (111) 0. 5.2 1218 51 2 2542 3.2 5 600 24.0

6 n on p (100) 1.0.7 1040 46.5 2547 1000 0. 5none

73.14

n on p 1 o 4.71506.5 2547 3.14 80 0 1 .024.0

24.0

Inone

13none

- 2100 36.0 2570 2.9 4 90014 P (100) 5 . 0 - 1970 36 O 2570 3.0 600 42 .Q

3 .0

-

P (100) 5.0

I

voltage luctuation rom all of the rap s n he oxidefor the entire channel is then given by

N T T 0 f t 0 ( 1 - f t o ) T 11 0 4 V Y4 Y

. - _ _( ) 1 + W 2 T T 0 2 dVydz. (13)The fac tor f t o ( l fto) in he ntegrand behaves ike

a 6 function and is peaked at the electron quasi-Fermilevel F n ( y ) (Fig. 3 ) . Therefore, only hose raps withenergy evel coinciding with F n ( y ) contr ibute o henoise. Since F n ( y ) is a function of y along the length ofthe channel [or V y ( y ) ] ,he noise measured at the dra inelectrode, which is biased to V D V , is actually the sumof the contributions from those traps with the energylevels approximately in a range of 0.75 V D V when V D

is small (Aq5,=0.75AV~ for n-t ype silicon with Fermienergy of 0.84 eV above the edge of the val enc e ban d,where & is the surface potential and VC s the effectivegate voltage). Hence the smaller the V D , he narrowerth e energy ange n he gap hat ca n be probed. Atsmall V D , the channel can be considered as uniform,an d Q N , pd, an d AV,/Ay are nde pen den t of V, (y ) .AVy/Ay can then be replaced by V D / . L , here L is th echannel ength, and p d < < l . Equation (13) can hen besimplified to

and the short-circuit noise cur rent is

111. DEVICEFABRICATIONTh e devices used in this st ud y were fabricated in our

laboratory under carefully controlled and reproducible

conditions. A thick layer of oxide was first grown on achemically tched nd egreased ilicon lice. Thewindows or he n+ islands were cu t using photoen-gravi ng echn iques , hen followecl by a phosphorousdiffusion. The oxide was hen stripped off and a hinlayer (about 1000 A) of silicon dioxide was regrown onthe surface in an ultrapure oxygen ambient a t 1000Cand hen withdrawn from he furnace n an ultrapureargon ambient. T o reduce he sodium contaminationan d subsequent drift of th e surface states and oxidetraps, his was followed by aphosphorous getteringcycle. The slice was then broke n nto four pieces andeach was processed eparately. Piece number 1 wasused as the control. In order to control the amount of

oxide charges, pieces 2 , 3, an d 4 were heated n hepure oxygen ambient (which will be called oxygen hea tt rea tment or O H T in this paper) separately a t 1000C,800C, and 600C for 0.5, 1, an d 24 h, respectively.Contact holes for drain and source electrodes were cutby he photoengraving process. This was followed byaluminum evaporation forming the gate electrod e andsource an d drain contacts. To minimize any possiblecontamination, special care was given o maintainingthe cleanlin ess of th e sli ces througho ut the processi ng.

Th e device geometry, doping, and OH T conditionsare given in Table .

IV . EXPERIMENTAL ESULTS N OXIDETRAPA N D SURFACE TATE ENSITIES

The capacitance-dc gate voltage (C- V ) character-istics a t different signal frequencies are used to o bta in :1) the total oxide and surface state charge density, Qo(number/cm2); 2) the ota l (all time constants) oxidetrap and surface state density p(states/cm2 eV); and3) th e oxide rap density, pTT()(states/cm2. eV), as afunction of t he time const ant.


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I-8 -7 .E - 5 - 4 -3 - 2 - 1

-I.0 1

'6 dol's

Fig. 5 . Low-frequency a nd high-frequency capacitance vers 1:3the gate voltage curves for devices (4a), ( l ) , nd (5).

7":6 t

5 [

jl- Argon I__I.. _L-_.,..

-L00 700 900 I100

T"C

Fig. 6. Fixed oxide charge density versus the temperature of oxy;. nheat reatment. 0 (111) n on p+ ; A (100) n on p; w (1 I)n-type argon annealmg); 0 (111) n-type; A (100) n on p.

A . Fixed Oxide Charge Density Qo

Typical C - V curves a t different frequencies for d j"-ferent OHT conditions are shown n Fig. 5. The norm; I-ized (to kT/q ) surface potential U s is also abeled f xdevice unit4(a).These urves how he ncreasi~igsurface state or oxide tra p effect a t low OHT temper I-ture nd low-signal requency. T h e oxide charl!:edensity Qo for each device is obta ined by c ompar ing l .I?high-frequency (150 kHz) C- V curves with the theorel .cal (zero surface states and oxide traps) - L curves. Tl t:results are shown in Fig. 6 and compared with those I 11the Deal 's riangle [25] (solid ine). I t is evident h: I.the highest oxide charge density is obtained for a dt .vice abricated on (111) surfa ce nd oxygen heatt r ea t ed at he lowest temperatur e (600C). O H T z t

1200C yields the owes t number of oxide charge!..Heat treatme nt in an argon ambient a t different teal-peratures does not change he oxide charge densitlisignificantly. Similar esults are obtained or device,;fabricated on a (100) surface, except hat he oxidt:charge densities are much lower.

4

2 x 10j1

IO"I1-

Ib

* rI

I

II

/ a

* I

I i

: ,

0.. b Z

?*- 7 - 6 -5 - 4

vc v o l t s

Fig. 7 . Total density of states versus gate voltage or devicesfabricated on (1 11) surfaces acd oxygen heat treated at 600C.

B . Tota l Dens i ty of States p

The to tal (all time constants) oxide trap and surfacestate density, p(states/ cm*.eV), is obtained using heTerman echnique [26 ] which nvolves he graphicaldifferentiation of the high-frequency C-V curves. Theresults or ypical (111) sample s wit h OHT at 600C(units 4a and 4b) are shown n Fig. 7 which ndicatepeaked urface tates of 5 Xl0 l1 s t a t e s / cm2 .eV tE,+0.3 eV and 2.8X10" states/cm2.eV at E,+0.4 eV.The la rger peak a t .3 eV+E, compares favorably withthat bserved on the ery hin oxide apacitancepreviously reported [ 2 7 ] .

C. O x i d e Tr a p D e n s i t y PTTO

The freque ncy depende nce of the oxide trap density,pTTo(states/cm2 eV), is obtained rom he requencydepende nce of t he C -V curves shown in Fig. 5. pTTo isobtained from the measured C(f) and C(HFE1.50 kHz)using the small-signal quivalent ircuit ivennAppendix 11.

T h e d a t a of pTTo are obtained from

(see Appendix I1 for the derivat ion) where C T O is theequivalent paral lel apaci tance of chargi ng nd dis-charging of t he oxide traps by t unnel ing and s given by

The dat a of C(w) and C(150 kHz) = C( 00) are used toevaluate CTo(w) from (17). CFo(w) is then used to e valu-ate PTTO(W) from 16).


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FU AND SAH: SURFACE l / f NOISE 279

I

'1

Fig. 8. Oxide trap density, or oxide trap capacitance, versus gatevoltage, with frequency as a parameter, fo r device (4a).

Fig. 9. Noise power versus gate voltage for devic e(4a) with frequency as a parameter.

A typical result of p T To ( w ) is shown n Fig. 8 or a(111) device OH T a t 600C (unit 4a) where he otaldens ity of s ta te s p(al1 w ) from Fig. 7 is also shown asdashed ines.Humps re observed between he latband ndhense t of stro ng urfacenversion( U S = % p ) . T h e secondary peak of p ( w ) a t E,+0.4 eVis not vident nhe ~ T T O ( W ) a ta ue ohe decreasing N-resolution a t higherrequencieshensing16). E

10-20

10-21

Iy -z

'v. EXPERIMENTA.L ESULTS N 1/f NOISElo-=

e , A

Th e noise power (short-circuit rms noise current) ofthe devices was measured a t a drain voltage V D of SO

mV, using a Hewlett-Packard 302-A wave analyzer, Ahigh-gain ow-noise reamplifier asesigned orthese measurements. The dra in vol tag e of SO mV cor-responds o about 1.5 kT (a t room emperature) ofenergy pread. The noise measured here herefore is

A . Dependence on Tr a p E n e rgy

Fig. 9 shows the typical narrow-band noise power asfunction of gate voltage V Q rsurface Fermi energyposition E-E, , a t frequencies of 10, 20, 40 , and 100

H z for the (111) device O H T a t 6OO0C (unit 4a). Noisea t higher requencies decreases and becomes difficultto detect due to amplifier oise because of t he low dr ai nvoltage used here. Large humps were observed at th esame surface Fermi evel position, E,+0.4 eV, as hehumps observed on ow-frequency C-I/ plots and hetrap density plots Figs. 7 and 8 ) . Th e extreme uni-formity of the noise power attrib uted to the r eprod uci-

bility of the fabrication echnique on a silicon slice isdemonstrated in Fig. 10 for three devices (units 4a, 4b,and 4c) at two frequencies, 1 0 and 100 Hz.

T h e effects of oxyg en heat reat:ment on the noise

are hown n Fig. 11. OHT increases and hifts henoise peaks toward more negative V G ue to increasingpositive charge Qo. T h e solid curves show the relativemagnitude s of noise h um ps at f = 0 Hz or devicesfabricated on the (111) surface and oxygen heat treateda t different emperatures. I t is obvious that a lowertem per atu re OHT giv es hig her lo\nr-.frequency noise aswell as higher oxide charges Qo.The dashed curves show


8/13

280 IEEE TRANSACTIONSON ELECT RON DEVICES, FEBRUARY 19

I I I I I I I-7 -6 -5 -4 -3 - 2 -I 0

v V O l t S

Fig. 11. Noise hump s versus gate voltag e for devices (device n 1rn-

an d (100) surfaces (dashed lines) and oxygen heat treated a t clif-bers abeled as shown) fabricated on (111) surfaces (solid li 18,:s)

ferentemperatures. @ 600C; A 800C; X 1000C; 0control.

the similar, plots for devices fabricated o n (100) 5;ur-faces. No te ha t he noise evels of th e (100) devicesare lower tha n t hos e of the (111) devices due to a lo-f.ernumbe r of surface states. No significant noise hu m/ ) sobserved for the control sample (unit 5) which also lladno hump in the C-V plot (Fig. 5).

B. FrequencyS@ectrunz

T h e frequency pectra of th e noise were obtailedfor fixed g ate and drain voltag es. Fig. 12(a) shows thelog-log plots of noise power versus frequency for th e(111) device OHT a t 600C (unit 4a). Gate voltageparameters are V c ( E - E E , ) -3.0 (1.0 eV ), -5.6 :1,9.4e17 mai n hump ) and -8. 0 (0.0 eV) V , which corresp md

to the surface accumulation, weak inversion, and strmginversion egions, espectively. T h e slopes of th e dottend o change rom 1.0 to abo ut 2.0 as he su r ac echanges rom accumulation o trong nversion. Theslope a t V O - .6 V (nea r the peak of th e noise ve ':jusV c curve) is about 1.57. Fo r gate voltages betv,c:en-5.0 and 6.0 V, th e slopes luctuate between 1.08to 1.62, as can be understood from Fig. . Slopes gre aterthan 1.0 are at tribute d to t he lar ge spa tial variatiorl ofpTTo in th e oxide. Th ese slopes do not cha nge sigr1,ifi-cantly as the drain voltage of the device is increase1:l u pto nd beyond hannel urrent aturation (5.0 V).This ind icates th at the noise measured beyond currentsaturation is generated mainly n he channel re :ionnear he n+ ource contact. The noise generatelf inthis region is amplified by the whole transitor. Fig.12(b) shows the similar plots or a (100) device Jiith600C O H T (unit 8). The gate voltages here also cor-respond o surface accumulation, weak nversion, m dstrong surface inversion. A slope of 1.35 is observed near

4

dI000

f H z f tJ 2

( a> (b )Fig. 12. Noise power versus frequency a t different surface

potential ranges. (a) Device (4a). (b) Device (8).

t

11000

-20 -I5 -I0 -5 0v vo115

Fig. 13 . Noise humps versus gate voltage a t f = 2 0 H z for devices(9) an d (10). Curve @)-original; Curv e @)-after apositivebias-temperature est; curve (c)-after sequential positive andnegative bias-temperature tests.

the noise peak . This implies th a t devices fabricated o(111) an d (100) surfaces have different p Z r T Odistributionsince the slope of the power spectra (i,v2 versus f)sensitive to p r T o versus x in the oxide.

C. E f e c t of the Bias-Tempera ture Tes t and PhosphorouGettering

Because of the ion drif t or th e slow trap pin g effect,the bias -temperature test will caus e t he C- V curve (andhence t he noise humps) to shift along th e gate voltageaxis. T h e noise profiles of the device are measured be-fore and after electrically stressing the device a t 250Cfor about 10 min under an electric field of l o6 V/cm.The results are shown in Fig. 13 for unit 9 (no phos-phorous ettering) nd nit 10 (with hosphorousgettering 28]-[30]); oth were (100) devices OHTa t 600C. Curve ( a ) is the original noise profile, andcurves ( b ) an d ( c ) are th e profiles after sequential posi-tive and negative bias tests. Very different results areobserved between these two devices. First, the magni-


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FU A N D SAH: SURFACE 1/f NOISE 281

tude of the noise for unit 10 is much lower than that forunit 9 [curves ( a ) ] ,Second, he direction of the shift son the noise curves (as well as C- V curves) are different.For unit 9 (not gettered), a positive gate bias causesthe curve to shi f t to a more negative direction, and anegative gate bias causes the curve to shift to a morepositive direction. For unit 10 (gettered), there is prac-tically no s hift. Third, t he noise is essentially constant(in fact i t decreases slightly for the nongettered unit 9)after drift. The second and third observations indicatethat positive on (sodium [31]) drift toward the inter-face does not affect the noise so that these ions are notthe noise source, and the drift of negative ions, whichcould be the noise sources, might be present since thel/ f noise is reduced slightly after drift (unit 9). T h efirst observation suggests that the PzO6 layer (unit 10)forms a barrier preventing he oxygen rom eachingthe nterface to form he oxide raps during ow-tem-perature OHT. In uni t 10, a negative bias causes hecurve to shift slightly toward a more negative direction.This is known as he slow rappi ng effect [32]-[33].

'This slow tra ppi ng effect was observed only on 6OO0CO H T samples. K O shi ft was observed on (100) controlsamples after bias-temperature est. The device hatshows a larger slow trapping effect at elevated tempera-ture also shows higher noise a t room temperature. Thus,we sugg est that the slow tra ppi ng effect might be origi-nated rom unneling nto or ou t of th e oxide raps,which also give rise to the noise.

Several authors [18], [34] have repo rted he use ofthe bias-temperature stress t o create the surface statesfor tudying he noise characteristics, but he oxidetrap density distribution n energy and position wereno t obtained. Thus, adirect correlation between henoise power and the surface state density could not be

made.

D . Relat ionBetweenFixedOxid e Charge Dens ityandNoise Power

I t was thought that the total oxide charge density oand the oxide trap charge density P T TO (near the inter-face) are proportional Ell]. Hence Qo must be propor-tional to th e l/ f noise since th e noise is proportion al toth e oxide rap density P T T O . In Fig . 14(a) , he noisepower in terms of th e quiva lent noise resistance(referred to he gat e) Rg n s plotted versus Qo for hedevices fabricated on p-type substrates and oxygen heattreated at different emperatures these are nversionn-channel field-effect structures). A straight ine rela-tionship is observed. Fig. 14(b) shows he normalizednoise power plotted versus Qo for epitaxial n-channelM OS field-effect devices fabricated on (1 11) and (100)surfaces and oxygen heat treated a t different tempera-tures. Again a linear dependence is observed. The factth at th e noise power levels are proportiona l to the o xide

I

1 0 4 L -

.I

IO " 10'2 1010 !O" 1012

( 4 (b )

Q, tf/crn'Q, k / c r n 2

Fig. 14. Noise ower versus fixed oxide charge density (at flat band)

different oxygen heat treatments.for devices gev ice numbers abeled as shown) fabricated with

charge densities of the devices with no sodium drifts(a ypical example s unit 10 in Fig. 13) proves very

convincingly that the oxide traps near the 02-Si inter-face and the oxide charges are of the sam e origin (oxy-gen) in these samples.

V I. COMPARISON F NOISE A.ND OX:IDETRAPDENSITY DATA

The experimental esults btained rom hewopreceding ections re ow omparedu hus emon-strating he one-to-one orrespondence of theoxidetra p den sit y and noise power as a funct.ion of th e tra penergy evel or Fermi evel posi-tion a t the nterface.This also serves o demonstrate hat he l/ f low-fre-quency noise ndeed originates rom he oxide raps.

T he d a t a of noise and oxide rap density versussurface Fermi level position or gate voltage are shown inFig. 15(a)-(c). T h e vertical eparations of th e noisecurves a t higher frequencies are somewhat greater thanthose of t he oxide trap den sity curv es. The noise humpis also wider than the width of th e corresponding oxidetrap density curve. These differences come rom th efactor T T O / ( 1 + w ~ T ~ o ~ )hich appears in the integrandof th e noise equation. This difference can be easilyrealized by omparing he requency dependence ofth e ntegr ands of the nd P T T O equations; ha t s,


10/13


6 x

4

1 I

V G " O l l 5 vg "0115

(a > (b') (c )Fig. 15 . Experimental correlations between noise power and ox de

traps for device ( 4 4 . (a) Noise power versus gate voltage. :I))Oxide trap densities versus gate voltage. (c) Total density 'Ifstates versus gate voltage.

The double an d sharp humps n he otal surfazes t a t e densi ty da ta of Fig. lj( c) compared with ! lebroader noise hump indicate that the surface states i tthe main peak of E, +0.3 eV are probably rather f a s tst at es at oom temperature and hence do not contributo muc h of t he ow-fr equen cy noise observed. This i ' jalso consistent with our earlier data for this peak whichhad to be taken a t 130K instead of 300K to increa:r:the time constants to a convenient range [27].

VII. COMPARISON F EXPERIMENTAL NDTHEORETICAL OISE CURVES

T o compute the theoretical noise power of the device

using (1 j), the spatial distribution of the oxide ra 2,concentration N T T ~ ( x ) ust be known. This s evaluate41from he xperimental oxide rap apacitance date,shown in Fig. 8 , which gave P T T O ( O ) and then NTTO(CO)= 4 0 k T p T T ~ ( w ) .ince most of the contri butions o h':

p ~ ~ o ( w )ome from the oxide traps a t OTTO = 1, we haw:NTTO(W)'VNTTO(~/'TTO).ence rom T T O = T O exp (20%'and NTTo(u) NTTO T ~ - , exp ( - 20x) , the position o 'traps can be obtained from the signal frequency w . i i ,value of T~ = s an d f3-1 = 2 A were used n Fig. 1 6Change in T O will shift the relative trap concentratio1along the x axis and a change in 6 will change the slopcof the ra p once ntra tion profile. However, hestchanges will not ffect he mag nit ude of th e noiscpower ignificantly ince x


11/13

F U AN D S A H : SURFACE I/f NOISE 283

states. The trapped charges communicate with conduc-tion or valence bands via the thermal or S R H process.I t i s emonstrated ot hheoretically nd xperi-mentally that charge fluctuations in the oxide traps areresponsible for he surface l/ f noise. The details pre-sented here cover he weak urface nversion ange,since this is where th e large humps appe ar in bo th noisepower nd he xide rap ensity. Th e requency

spe ctr a of t he noise depend trongly on he patialdis tri buti on of the oxide rap dens ity. An ncreasingdensity of the oxide trap as we go deeper into the oxi defrom he nterface s needed o account for he noisepower spectra l/f with n greater han 1.0. I t is sug-gested from the experimental evidence that these oxidetraps arise from excess oxygen a t t h e SiO2-Si interface.

APPENDIX

DERIVATION F TIME CONSTANTS AN D ~ Q NI t w a s pointed ou t in Section 11-A tha t he ime

const ant of 6ns is much shorter t han those of 6 $ ~ ,8n58,and 6 n ~ o .Using (2a)-(2d), an d neglecting an,, we have

Since Io r 1, the reciprocal of the time cons tant for theShockley-Read-Hall ecombination processes a), b),(c), and (d), is much larg er than that of the tunnelingprocess 1y ] we have1 01 1>>Iy . Hence in the inversionrange, we have l o r 1-1 E l> > / P I , 171, 1 1 1 1 .The imeconstants for Gnss and &??TO ca n be approximated by

which are also given as (4a) and (Lib).By using (20) in (5), those terms containing 6nTo(0)

a re A13, Az3, and A M .Terms containing either Gns~(0) rS$s(O) are much smaller than those containing To(O).Hence (5) can be expressed as

X0 - X8Q X = qsnTO(0)

cw

[- EO + y - 1X

cw + co X0 X0

- d P + Y + E - ) a 2 ] . (26)XX 0

If we concentrate only on the ow-frequency com-ponent s, hat is, hose erms containing exp ( X 3 t ) , wehave


12/13


APPENDIX rSMALL-SIGNAL EQUIVALENTIRCUIT F A N MO!:STRUCTURE INCLUDINGHE TUNNELING ROCESI.:

T h e small-signal equivalent circuit which includes theSRH process between the interface states and the con-duction or valence band states and the tunneling rol::essbetween he xide rap tates located ro m x tox + A x ) and he nterface states can be derived rom(2c) and 2d) sing la)-(lf). The etailed mathe-matical derivation s will be presented elsewhere. '!'heeffect of bu lk generation nd ecombination n chedepletion layer is neglected here since i t was pointed outby Nicollian and Goetzberge r [ 3 6 ] th at th is effect 7,QilIcause a constant high conductance in the strong invm-sion range, which was not observed in this study. Fig.18(a) s hows the equivalen t circuit for the traps local.t:din the range of Ax a t x . The circuit elements are gi-c,c:nas follows:

(..9)

(2 )

r

Fig. 18. Small-signal equiv alent circuit s for an MOS structure in-cluding the unneling mechanism. a) With raps located a t xto % + A x . (b) Equivalen t parallel capacitance and conductancefor the en tire MOS structure.

the equivalent parall el capacitance and condu ctance ofthe oxide traps. They are given by

(3 7)

By taking the derivative f CT O ith respect to w andusing (34), (16) can be obtained.

C t, = - ' ~ ~ o f t O ( 1f tD) F/cm32

k T

ACKNOWLEDGMENT

( 3 3 ) The aut hor s would ike to han k M . J. McNutt andP. S. Chung, who have assiste d n he edious noise

( 3 L) measurements.

where W is the width of the depletion layer. C, an dare he hole and lectron apacitances, C t , is t l (2capacitance due to the interface states, G,, an d G,, alt:the hole and lectron ecombination onductanc c:through the SRH process, C to s the capacitance due I ( Ioxide traps, and G t is the conductance representing t b t :tunneling between he nterface states and he oxiclc:traps. u+ an d us are the small-signal potential a t x anlla t the interface normalized to k T / q . u L , nd u t , are t k r csmall ignal omponents of hequasi-Fermi nerglevels of the interface states and the oxide traps no]malized to k T / q . K O s he dielectric constant of t h :silicon dioxide an d eo is the electric permittivity of t h :free space. For distributed oxide traps, the equivalentcircuit is just the parallel combination of the circuit f a reach rap. The equivalent circuit for he entire MOi3structure is shown in Fig. 18(b), where CTO nd GTOarl:

REFERENCEST. E. Firle and H. Winston, . A p p l .Phys. , vol. 26, p. 716, 1955.F. J . Hyde, Proc. Phys. SOL. ondon , vol. B69, p. 242, 1956.

A. Van der Ziel, Physica (Utrecht), vol. 16, p. 359, 1950.H. C. Montgomery, Bell Syst . Tech. J . , vol. 31, p. 950, 1952.

L. Bess, Phvs . Rev. , vol. 91, p. 1569, 1953.F. K. DuPre, Phys . Rev. , vol. 78, p. 615, 1950.

S. R. Morrison, Phys. Rev. ,vol. 99, p. 1904, 1955., Phys.'Rev., vol. 103, p. 72, 1956.

R. L. Petritz, in Semiconductor Surface Physics,R. H. Kingston,

A. L. McWhorter, Sc.D. hesis, Dep. Elec. Eng., Mass. nst.Ed. Philadelp hia: Univ. Pa., Press, 1957, p. 226.

Tech., Cambridge, 1955.C. T. Sah and F. Hielscher, Phys . Rev. Le t t . ,vol. l,?, p. 956, 1966.

and l / f noise in MO S transistors, IEEE Trans.ElectronDe-G. Abowitz, E. Arnold, and E. A;, eventhal, Surface states

vices,vol. ED-14 , pp. 775-777, Nov. 1967.S. Christensson, I . Lundstrom, and C. Svensson, Solid-SlateElectron. , vol. 1 1, p. 797, 1968.I . Flinn, G. Bew, and F. Berz, Solid-State Electron. , vol. 10, p.833, 1967.F. Berz, Solid-Slate Electron. ,vol. 13, p. 631, 1970.R. J . Hawkins, I. R:M Mansour, and G. G. Bloodworth, B r i t .J . A p p l . P h y s . , ser. 2, vol. 2 , p. 1059, 1969.

J . Appl. Phys . , ser. 2 , vol. 2 , p. 1063, 1969.I. R. M. Mansour, R. J . Hawkins, and G. G. Bloodworth, B r i t .

S. T. Hsu, Solid-State Electron. , vol. 13, p. 843, 1970.E. Burstein and S . Lundquist, Tunneling Phenomena in SolidsNew York: Plenu m, 1969.C. T. Sah, Phys . Rev. , vol. 123, p. 1594, 1961.

-


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IEEE TRANSACTIONS ON ELECTRON DEVICES, FEBRUARY 1972 285

[21] --, The equivalent circuit model in solid-state electronics-Par t I : The single energy level defect centers, Proc. IE EE , vol.55, pp. 654-671, Ma y 1967.--

[22] L. Schiff, Quanfum Mechanics, 3rd ed. New York: McGraw-Hill, 1968, ct . 5.

[23] C. T. Sah, The ory of low-frequency generation noise n unc-tion-gate field-effect transistors, Proc. IE EE , vol. 52, pp . 795-

, Solid State Electron., vol. 13, p. 1547, 1970.

814., July 1964...

[24] NI. Lax, Rev. Mod. P hys, ., ol. 32, p. 2 5 , 1960.I251 B. E. Deal, M. Sklar, A. S. Grove, and E. H. Snow, J . Electro-1

chew. Soc., Vol. 114, p. 266, 1967.

[26] L. M . Terman, Solid-State Electron., ol. 5, p. 285, 1962.[27] W. R. unter, D. H. Eaton, and C. T. Sah, Appl. Phys. Lett.,vol. 17. D. 211. 1970.

[28] D . ~ R . err, J. S. Logan, P. J. Burkhardt, and IV. A . Pliskin,, ~~~

IB M J . Res. Develop., vol. 8 , p. 376, 1964.

[29J J. M . Eldridge and P. Balk, Trans. Met. SOC. I M E , vol. 242 ,p. 539, 1968.

[30] P. Balk and J . M . Eldridge, (Phosphate glass stabilization ofFE T devices, Proc. IEEE, vol. 57, pp. 1558-1563, Sept. 1969.

[31] E. H. Snow, A. S. Grove, B. E. Deal, and C. T. Sah, J . Appl .Phys., vol. 36, p. 1664, 1965.

[32] S. R. Hofstein, A n investigation of instability nd harge

Electron Devices, vol. ED-13, pp. 222-237, Feb. 1966.motion n metal-silicon oxide-silicon struct ures, IEEE Trans.

[34] B. L. Jones and R. E. Hurlston, VHF noise due o surface[33] -, Solid-state Electron., vol. 10, 9; 657, 1967.

states in MOS devices, Proc. IEEE (Lett .), vol. 58 , pp. 152-

[35] S. C. Gupta, Transformation and late VariabIe Methods in153, Ja n. 1970.

[36] E. H. Nicollian and A . Goetzberger, Bell. Syst . Tech. J ., vol. 46.Linear System. New York: Wiley, 1965, p. 200.

p. 1055, 1967.

Correspondence

Calculation of Electron Drift Velocity at Si AvalanchePETER C. Y. CHEN

Abstract-In evaluating the electron drift elocity at Si avalanche,the electron distribution is obtained y solving the Boltzmann trans-port eq uation i n the maximum anisotropy truncati on (MAT) approx-imation of Baraff. Improvem ents are made to the MAT procedureso that he symmetrical part of the distribution unction can beobtained in a straightforward manner. The temperature dependenceof electron drift velocity is studied by including both optical phononemission and absorption rather than emission alone. The calc ulatedelectron drift velocity varies from X l o 7 o 2.9 X IO7 cm/s at Si ava-lanche.

En thi s correspondence, th e electron drift velocities a t Si avalancheare calculated. So far, interests in semiconductor avalanche [1]-[5]have been lirnited to ionization rate studies. There has hardly been

any publication related o electron drift velocity [6]. However, recentwork which uses IMPATT diodes as millimeter-wave solid-state sources(see, for example, 171) makes the drift velocity st udy at semicon-ductor avalanche useful for device analysis. This is understandablefrom a device standpoint because the conventional Read-type an-alysis or p+n-n + diodes becomes questionable n th e millimeter-wave range n view of t he fact that the trans it-ti me effect in theavalanche region cannot be neglected. Of course, it is also useful forstudying silicon pvn avalanche transit-time diodes [ 8 ] .

Several heoretical studies on carrier distribution unctions atsemiconductor avalanche have been available in the literature [1]-[4]. n 1962, Baraff [9] published a n elaborate calculation of a dis-tribution function whose validity was irst verified in Si by comparingthe heoretical and experimental values of ioniza tion ate. Later,Baraff [4 ] was able to show that the electron distributions that wereobtained by the use of a simple maximum anisotropy tru nca tio n(MAT) scheme to sclve the general Boltzmann equation agree wellwith the more elaborate numerical calculation. This MAT method

was extended to examine he warm electron distributions by hepresent author and C . A. Lee [ lo] . T he major merit of M AT is th atit can give valid solutions for cases ranging from he high field where

Air Force Systems Command, Rome Air Development Center, Griffiss Air ForceManuscript received August 2 3 , 1971. This work was supported in part by the

Base, N. Y .The author was with the School of Electrical Engineering, Cornel1 University,

Ithaca, N . Y . He is now with the De partment of Electrical Engineering, Universityof Pittsburgh, Pittsburgh, Pa. 15213.

an almost spherically carrier distribution is considered [l ] to therelatively low field case where th e effective pa rt of t he distri bution isspike-shaped [2]. In this correspondence, a n alternative procedureof the MAT method is presented so tha t the olution can be obtainedin a straightforward way. Furthermore, the temperatu re dependenceis studie d by ncluding both emission and absorption in optical phononscattering.

Under the case of a high electric field applied to a homogeneousnondegenerate semiconductor, two fundamental scattering processes,optical phonon emission, and absorption and onization, are con-sidered. A constant mean free path A is assumed for optical phononscattering. The optical phonon is assumed to have an energy E R .There is a threshold energy E { , below which no ionization occ urs, andabove which the ionization mean free path li , is constant. The distri-bution function is expanded in a series of Legendre polynomials, Le.,

where is th e cosine of th e angle between field and momentum , and aspherical parabolic band characterized by e = f i 2 / 2 m is assumed. Fol-lowing th e MAT scheme of Baraff, the first two of the well-knowninfinite Boltzmann hierarchy of coupled-differential equations takethe form:

1 1 cosh ( eHD+ R/2hT)- D W Z I ( E )--3 1 + r [ cosh ( e R / 2 k T )

+ exp ( d ) M n ( E ) - m& ) ( la)1

where

d

d eD = -- > mo(e) = e ~ o ( e ) , MI() E B I ( ~ ) ,

Y = A/Ji, Q = eE l

with

l/l = l / A + l/li.It is noted t hat an exact relation m&+eH) =exp (enD)m&) is usedinstead of th e commonly used Taylo r expansion. The value of Zi isto be regarded as infinite for energy below ei. In th e Case where th e

theory and experiments on surface l f noise

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