908 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-7.9, NO. 8 , AUGUST 1-97?

Investigations on the Effect of Drift-Field-Dependent Mobility

on MOST Characteristics-Part II: Qg Field Dependent

Abstracf-The MOST V- I static characteristics were investigated in Part I 111, taking the mobility p only as variable with the drift field. In this part, the analysis is extended to include the simultaneous variations of the mobility p and the bulk charge QB along the channel. From the mobility point of view, the analysis follows that of Part I with the two approaches of piecewise and continuous mobility varia- tions. However, in this case the resulting equations are only solvable numerically, and the computed results are plotted. The results show that effects due to the additional variation in QB along the channel are in the same sense and enhance those due to variation in mobility alone. Also, higher doping and greater oxide thickness show them- selves in a way as though the mobility were approaching the limit of velocity saturation.

The overall effects show more reduction in current levels as well as output and mutual conductances, and increased tendency of ear- lier saturation. As far as these effects are concerned, the results favor a MOST of small oxide thickness and long channel. However since a long channel impairs the gain and high-frequency charac. teristics, the results given in this paper help to show how a corn promise may be obtained.

I. INTRODUCTION

N PART I of this investigation [I 1, a detailed analy- sis of MOST V-I static characteristics was given, taking only the effect of mobility variation with th:

drift field into consideration. The analysis of Par t I , i l l

addition to being approximately applicable for high., resistivity substrates, serves also to clarify the trend of analysis followed in this part.

The bulk charge Q B is always a field-dependent parameter and the effect of its variation on the " I S ' : ' characteristics becomes more pronounced in cases o f moderate- and low-resistivity substrates. The effect 'of variation in either Q B alone [2], [ 3 ] or the mobilit.; alone [l], [4], [SI has been considered before.

I n this paper, the simultaneous dependence of the mobility p and the bulk charge QB on the drift field i s taken into consideration. From the mobility point clf view, the analysis follows that of Part I with the two approaches of piecewise and continuous mobility vari:.- tions.

Substituting for QB and p their field-dependent rekL-

E. A. Talkhan and A. I. Barboor are with the Department ,f Manuscript received October 26, 1971; revised January 18, 1972.

I. R. &I. Mansour is with the U.A.R. Atomic Energy Resear1:h Electronics and Communications, Cairo University, Giza, U.A.R.

Laboratories, Cairo, U.A.R.

' 7

4 Fig. 1. Sectional diagram of an n-channel MOST.

tions in the general current equation, the results become so complicated that i t is impractical to look for explicit relations for the drain current or any of the small-signal parameters. Numerical methods were therefore resorted to, using computer type ICL-1905, to obtain plots of the V-I static characteristics, output, and mutual con- ductances.

The derived equations show nonlinear dependence of the transistor parameters on oxide thickness x,, the parameter a€,, as well as on the substrate doping N A . Because of this, i t is not generally possible to normalize the results as was done in Part I .

Although it is feasible to study the effects a t different oxide thicknesses and different values of the parameter a€,, the investigation in this paper is carried for typical values.

Fig. 1 shows a sectional diagram of an n-channel MOST of channel length a and width b. Qss shown in the figure represents the silicon-silicon oxide interface charge and is considered constant. On the other hand, the bulk charge Q B due to the depleted region below the semiconductor surface increases towards the drain.

The drain current ID is given by

where

TALKHAN et ai. : QB FIELD DEPENDENT

and the other symbols are the same as used in Part I. At this point the analysis is continued by substituting (2) for Qn in (1) using for p either one of the formulas used in part I.

11. PIECEWISE MOBILITY VARIATION In this case, p follows the relation

Since our present aim is to predict the effect of simultaneous variation in Q B and p, we shall carry the analysis only for the case of n= 2 that represents in- verse square-law dependence of mobility. This yields three possible situa.tions in the channel depending on the applied voltages. The analysis for each situation proceeds as follows.

A . p Field Dependent Throughout the Channel Combining (1)-(3), the drain current ID comes out as

follows:

Integrating the above equation, we get

where

Equation ( 5 ) is valid up to pinchoff as simply defined by Qn( V,,) = 0 at the drain. The drain saturation voltage VDs comes out from (2) as

and the drain saturation current is obtained by sub- stituting V D = VDS in ( 5 ) . This gives

I D S 2 =' P 2 ( V D S ) - PZ(0). (8)

The range of validity of ( 5 ) and (8) is determined by the condition

where ES is the drift field at the source. At saturation, this condition of validity is represented by the relation

909

I

20

16

12

8

4

0

13 , N*/CM3

14 15 I6 10 IO 10 10

t

1012 d 3 1 0 ' ~ Id6 N,/CM~

( b ) Fig. 2 . Dependence of minimum gate voltage: Volm on substrate

doping NA. (a) At different oxide thicknesses ( a c = 1.0 V). (b) At different channel lengths (xo=2000 b).

where V0lm is the minimum gate voltage above which (8) is valid, and is obtained from (7 ) , (8), and (10) replacing V G ~ by V Q ~ ~ . Computed values of VGln are plotted in Fig. 2 against substrate doping with oxide thickness x. and acc as parameters. As shown in Fig. 2, V Q ~ ~ increases with doping starting with the value of 3 aEc a t high-resistivity substrates. This increase is more rapid at larger oxide thicknesses and longer channels. This demonstrates that for certain substrate doping, the variation of mobility is less pronounced in MOST's of large oxide thicknesses and long channels.

T o get the locus of validity below pinchoff (Val> V Q ~ ~ ) , we substitute ( 5 ) in (9) to get

PS(VD2) = PZ(0) + [paec(VGl - .c"B)]' (11)

a t VDZ, the drain current 102 is given by

Equations (1 1) and (12) determine the required locus plotted as curve I1 in Fig. 4, the area above which rep- resents the region of field-dependent mobility through-

IDS = b(a€c)(VGlm - V B ) (10) out the channel.

910 IEER TRANSACTIONS ON ELECTRON DEVICES, AUGUST 1972

1.0

0.8

0.6

0.4

0.2

0

6

5

' I ; I 4

' I

$ 3

2 I/

I t t

0 0 0.2 0.4 0.6 0.8 1.0

x,/a

(a) (b) Fig. 3. Independence of critical distance xo, critical chann':I voltage VE, drain current ID, and effective gate voltage Val

along the channel. N A = 1.9X IW5, x, =2000 A, and as=l.O V.

B. p Field Dependent in a Part of the Channel rD " I n this case the drift field e reaches the critical value at a point in the channel distant xc from the source l 6 XI - I

given by kJ= 8 V .-.

/ 14 -

x0 Pl(VC> - Pl(0) .e? - = (13: ,+/ a ID '/ VGl = IO v ,

I 2 - Y / r

where V, is the channel voltage at related to 11, through the equation /'

/

aec

C O

ID = 8 __ Qn(Vc) O v c

and the function PI( V ) is given by

(VG1- VI2 4 v i 6 V

Pl(V> = - 8[ + - V B $ f p (1 f -> ] * 3 2 4 f P

Equation (13) is only valid before pinchoff and fcr 4 v 0 <x , /a < 1. The interdependence of x O , V0, drain C U I ' -

rent, and gate voltage along the channel is shown in Fig. 3. Below the curves of VQI = 6 in Fig. 3(a) and ( t ) the transistor always operates under the condition of p variable through only part of the channel. ~b~~~ Fig. 4. V-I characteristics of MOST with piecewise mobility depen-

dence on the drift field, showing the loci of constant mobility (I), curves, depending on biasing, the transistor may 01,- and field-dependent mobility (11). N~=1,9x1016, x0=2000 A, erate under the condition of variable p throughout tlle v p and K = 2 . channel (xc /a = 0), p variable through part of the cha~x- ne1 ( O < x ~ / a < or constant mobility althrough tile corresponding current 101 are obtainable through the channel (x, /a = 1). I t is also to be noticed that for co I- following two simultaneous equations: s tant Vel, x. moves towards the source as the dra.in current increases; whereas at constant drain current, xc PaEc moves towards the source as the gate voltage is reduced. CO

I I I 0 1 2 3 4 5 6 7 VD v

101 = __ Qn(VD1) (17)

The drain current Io in this case is given by and 102 = IDIP1(Vc) - P,(O)] + P,(VD) - P2(Vc). (16) ID1 = Pl(VD1) - PI(()). (18)

Equation (16) is valid for V0 > V D 1 where VOl and t r ~ e These equations give the locus plotted in Fig. 4 as curve

TALKHAN et d. : QB FIELD DEPENDENT 91 1

0 I 2 3 4 5 v D

(a) (b) Fig. 5 . Dependence of normalized output and mutual conductances on drain voltage with gate voltage as parameter.

Loci I and I I of Fig. 7 are also shown. N A = 1.9 X 10l6, x. = 2000 A, and ace = 1 .O V.

I representing the boundary between the region of constant mobility through part of the channel and the region of constant mobility throughout the channel. The drain saturation current in this case is obtained by substituting VD = Vjpsl 10 =IDs, and V,= Vc8 in equa- tions (2), ( 7 ) , (14), and (16) and solving for 10s.

C. Region of Constant Mobility Throughout the Channel This lies to the left of locus I where the current is

given by the formula

I D = Pl(VD) - PI(0) (19)

and the drain saturation current, which in this case oc- curs a t very low gate voltages, is obtained by replacing VD by VDS in the above equation.

With the guide of loci I and 11, and using the above results, we get the family of V D - 1 0 curves shown in Fig. 4 in which the voltage is normalized to ae, and the current to /3(aec) 2 1 which correspond to the chosen values of x0 and aEc. These curves are drawn for a doping N A = 1.9 X 1OI6 chosen so as to correspond to Valm = 6 V as shown in Fig. 2(a). I t should be noticed that p, being a function of po, depends on doping [6] and this must be considered in conjunction with the curves given in this paper whenever absoYute values are required. However, variation in /3 within the practical range of doping used in RIIOST's is small.

D. Small-Signal Parameters Differentiating ( 5 ) , (16), and (19), which give the

drain current in the three regions, the small-signal parameters gd and gm come out as follows.

Region I : P CO

g d l = - Qn(v~) (20)

gm, = PVD. (21)

Region II: P"4Qn2(VD)

g d 2 = C2[2ID + Pl(0) - Pl(V,)] (22)

(23) B ( I D V C + 2(U€C)[Pl(VD) - Pl(V0)Il

gmz = ---. 210 + PdO) -- P O C )

Region 111:

P " 4 2 n 2 ( V D )

P(aac)[Pl(VD) - Pl(Q)I

gds = (24) 2c021D

gm, = -~ (25) I D

The small-signal parameters as given by the above equations are computed and plotted in Fig. 5 against V D with V G as parameter. This figure shows substantial reduction in both gd and g , due to the simultaneous variation in mobility and Q B . 'The reduction in this case is more than the corresponding reduction when Q B

was neglected. In Fig. 5 the loci I and I1 corresponding to those of Fig. 4 are also shown. Earlier tendency of saturation shows up in consistence with Fig. 4.

E. Effect of Doping As clear from the above analysis, doping has ap-

preciable influence on the transistor behavior in many respects as follows.

912 IEEE TRANSACTIONS ON ELECTRON DEVICES~ AUGUST 1972

\ \

. . ' - .,. . \ " 1 ' . . \ . . . , I - . .' d,.' . \' ..' ' .\ \ \ 0 I I I. . I I

... \ . 0 0.2 0.4 0.6 0.8 1.0

X,/a

Fig. 6 . Effect of substrate doping on the relation between critical distance x. and drain current ID. x,=2000 A, a€,= 1.0 V, and Vo1=5.0 V.

From (13), considering the effect of doping on xc, we get the set of curves shown in Fig. 6 for V G 1 = 5 a€,. Referring to this figure, as doping increases the range of the drain current during which the mobility is constant throughout the channel decreases, while the minimum value of x. (corresponding to saturation) moves towards the drain.

From ( l l ) , (12 ) , (17), and (18), the set of curves for loci I and I1 are plotted in Fig. 7(a) with doping as parameter. The saturation current at V G l , increase2 with doping for constant k ( =n /n - l ) , an effect which is similar to that when increasing k with Q B assumed constant [I 1.

Curves of Fig. 7(b) and (c) relate the drain saturation current to drain saturation voltage and effective gate voltage a t different dopings, respectively. The three sets of curves are useful in predicting the V-1 character. istics a t certain doping.

Fig. 8(a) and (b) shows, the variation of gd and g,, against I/= at constant V G 1 and different dopings From Fig. S(a), i t is seen that while high doping haf, the same sense of effect as high k , there is an additiona complete shift of the curves to the left towards the con. stant-mobility region as doping is increased. Fig. 8(b: shows reduction in g, with increased doping at constanl: P G ~ , an effect which is similar to increasing k excep;: that the percentage reduction increases with doping i r . contradiction to its decrease with k [l]. In general g,, is less at higher doping for the same drain and gate voltages.

111. CONTINUOUS MOBILITY VARIATION In this case, the mobility follows the relation

where a is 1 for holes and 2 for electrons [s] . Substi- tuting (26) in (1) before integration, the electric field at any point in the channel comes out as follo~s:

The limit of validity for the above analysis is defined by the condition that the denominator in (27) equal zero. This gives

Equation (28) represents a locus dividing the V-1 plane into two Regions 1 and 2 as shown in Fig. 9.

A. Region 1

From (27), the drain current is given by

For the two cases of a = 1 and CY= 2, we proceed as follows:

TALRHAN et d. : @E FIELD DEPENDENT

8 -

7 -

6 -

5 -

4 -

3 -

2 -

I -

9

913

14 2 x 1 0 4 v

\ high-resistivity 3 V substrate

(b) (C) Fig. 7. Effect of substrate doping. (a) On loci I and 11. (b) On locus of conventional pinchoff ( Q n = O ) .

(c) On transfer characteristics. x, = 2000 d and = 1 V.

914 IEEE TRANSACTIONS ON ELECTRON DEVICES, AUGUST 1972

9,"

0 0 I 2 3 4 5 0 1 2 3 4 5

'D " 'D ( a ) (b)

Fig. 8. Effect of substrate doping on tb.c variation of output and mutual conductances with drain voltage. x. = 2000 A, uec = 1.0 V, and Vu, = 5 V.

7 F'

I 0 I 2 3 4 5 6

I 7

"D ' Fig. 9. V-I characteristics o f MOST with continuous mobility

variation with the drift fie13 ((~=1). N~=1.9XlO's, x,=2000 A, and a s = 1 V.

1) CY= 1: Integrating (29), we get and

Equation (30) is plotted in Fig. 9. On this figure tkle

parison assuming IDs =ID'. in Fig. 11 (a) and (b) against V D for constant VGI.

follow as

locus of conventional pinchoff is also shown for coni- g d and gm as given by the Preceding equations are Plotted

Differentiating (30) , the small-signal paramete:.s 2) a= 2 : Following the preceding steps, the drain current in this case becomes

TALKHAN et al. : QB FIELD DEPEXDENT 915

'D

Fig. 10. V-I characteristics of MOST with continuous mobility variation with the drift field (a = 2 ) . N A = 1.9X1016, x, =ZOO0 d, and ae, = 1 V.

9, n 9," b - - 1.2

t'\% 1.0 -

0.8 - 0

c c

0 - 0

- -7-

1 2 0 I 2

"0 'D

I

( a ) ( b ) Fig. 11. Effect of parameter cy. (a) On the variation of the output conductance with drain voltage.

(b) On the variation of mutual conductance with drain voltage. Na=1.9x1016, ~ , = 2 0 0 0 A, ae, = 1 V, and V Q ~ = 5 V.

Equation (33) has been evaluated numerically and plotted in Fig. 10 -with I/cl as parameter. The small- signal parameters are obtained by differentiating (33) making use of the :mathematical identity. For

Y = f(4 = JoZCCz, Y) dz

dY d x , Y) _ - .- dx

1 - sox dz

The resulting expressions for g d and g, contained

aY

integration terms that are only solvable numerically. Samples are plotted in Fig. 11 with those of 01 = 1 for comparison.

B. Region 2 In this case, the discussion mentioned in Section

IV-B of Par t I applies, and again rigorous solution re- quires two-dimensional analysis.

IV. CONCLUSION In this part of the investigation, we have extended

the analysis to study the effects of the simultaneous

936 IEEE TRAKSACTIONS ON ELECTRON DEVICES, AUGUST 1972

variations of the mobility p and the bulk charge Q B o ~ l MOST characteristics. Since the equations derived i n this case were only solvable numerically, they were processed on a computer to obtain the results plotted in the given figures.

The simultaneous variation in Q B along the channe , which is more pronounced a t higher substrate doping, affects the MOST characteristics in the same sense as

sidered by several authors [7] , [SI, predicted reductions in current levels as well as mutual and output con- ductances. Different approaches were followed using one or the other of the two mobility equations. Similar analysis to that presented in this work would give ex- haustively the characteristics of JFET’s since there are no surface effects, and all the apparent parameters can be taken into consideration.

the effects of variation in mobility alone. Also, higher doping and greater oxide thickness affect the char- acteristics in the same direction as increasing k(n-+ l ) [l] 1. R. NI. Mansour, E. A. Talkhan, and A. I. Barboor, “Investi-

gations on the effect of drift-field-depefdent mobility on MOST in (3) for the mobility. characteristics-Part I : QB constant, IEEE Trans. Electron

I t is interesting to notice that the overall effects of [2] A. S. Grove, Physics and Technology of Semiconductor Devices. Devices, this issue, pp. 899-907.

either the variation in the mobility alone or the simul. New Yo&: Wiley, 1967. taneous variations of mobility and Q B , compared to thl: [3] C. T. Sah and H. C. Pao, “The effects of fixed bulk charge on the

characteristics of metal-oxide-semiconductor transistors,” IEEE simple constant-mobility theory, are mainly earlie- Trans. Electron Devices, v01. ED-13, pp. 393-409, Apr. 1966.

tendency of saturation before Pinchoff and reduction ill on insulated gate field-effect transistors, Solid-State Electron., [4] G. F. Neumark, “Theory of the influence ,pf hot electrons effects

current levels, output, and mutual conductances at thc: ~ 0 1 . 10, PP. 169-175, 1967. same biasing voltages. [5] A. Bar-Lev and S. Margalit, “Changes of mobility along a deple-

tion type MOST channel,” Solid-State Electrolz., vol. 13, pp. 1541- 1546, 1970.

REFERENCES

As far as preceding effects are ‘Onsidered, a tran. [6] D. M. Caughey and R. E. Thomas, (‘Carrier mobilities in silicon sistor of small oxide thickness and long channel is fa. empirically related to doping and field,” Proc, IEEE (Lett.), vored. However, although small oxide thickness is also vO1. 5 5 1 PP. 2192-21933 Dee. i967.

[7] P. David and J . L. Pautrat, Effet de la dependence mobilitCs- an advantage for high gain, long channels are in contra. champ dectrique sur Ies caracteristiques du transistor a effet diction with gain and high-frequency design requ i re . de champs a jonctions,” Solid-State Electron., vol. 11, no. 9, pp.

ments. A compromise must be sought, and the analysi!; [8] D. L. “I and H. Yanai, “Current-voltage characteristics of the junction-gate field-effect transistor with field-dependent mo-

The effect of mobility variations on JFET’s, as con bility,” IEEE Trans. Electron Devices, vol. ED-17, pp 577-586, Aug. 1970.

893-901, 1968.

in this paper helps to show how this may be done.