JEFE IKANSACTIONS ON ELECrRON DEVICES, VOL 15. NO 3. MARCH 1988 325

Inverse-Narrow-Width Effects and Small-Geometry MOSFET Threshold Voltage Model

Abstract-An analytical threshold voltage model is developed based on the results from our three-dimensional MOSFET simulator, MI- CROMOS. This model is derived by salving Poisson’s equation ana- lytically and is used to predict the threshold voltage of MOSFET’s with fully recessed oxide isolation (the trench structure). Coupling was ob- served between the short-channel effect and the inverse-narrow-width effect. This coupling results from the mutual modulation of the deple- tion depth and is used to extend the analytical inverse narrow-width model to small-geometry devices.

This model is compared with experimental data obtained from the literature as well as with our three-dimensional simulator. Satisfactory agreement for channel lengths down to 1.5 pm and channel widths down to 1 wm has been obtained.

I. INTRODUCTION HE EVER-INCREASING packing density of T VLSI /ULSI chips requires ever-advancing innova-

tive techniques and novel structures for future processing. The fully recessed oxide isolation for individual MOS- FET’s seems to be a very attractive candidate at present for highest packing densities [ 11. However, the existing inverse narrow-width effect [2] of the device fabricated with this technology creates serious variations of subthreshold current and threshold voltage. As the device is scaled down, the inverse narrow-width effect increases the subthreshold current and reduces the threshold volt- age. However, in device design it is necessary that the quantities be accurately predictable. A first attempt at this prediction was carried out by Akers [3] using a geomet- rical approximation and a conformal mapping technique to form a model for this effect. However, the assumption of constant electrostatic potential along the thick-oxide sidewall from the surface to the bottom creates errors in the predictions, especially when the width is scaled down to 1 ,urn or less. Actually, the potential varies quadrati-

Manuscript received Septemher 16, 1986; revised October 30, 1987. This work was supported in part by the Semiconductor Research Corpo- ration.

K . K . Hsueh was with the Center for Solid-state Electronics Research, Arizona State University, Tempe, AZ 85287. He is now with Gould Inc., Semiconductor Division. Pocatello, ID 83201,

J . J . Sanchez was with the Center for Solid-state Electronics Research, Arizona Statc University, Tempe, A 2 85287. He is now with Intel Cor- poration, CTMG Division, Chandler, AZ 85224.

T. A . DeMassa and L. A. Akers are with the Center for Solid-State Electronics Research, Arizona State University. Tempe, A Z 85287.

lEEE Log Number 87 I88 1 1 .

cally along this sidewall and this variation can be signif- icant. Hong and Cheng [4] used the same problem-solv- ing technique as Akers [3] while also including the quadratic variation of the sidewall potential to predict the threshold voltage. However, their choice of boundary conditions as well as the short-channel model incorpo- rated [5], lead to predictions of the threshold voltage that do not closely correspond with simulation and experimen- tal data. Hence, in the present analysis the variation of potential along the thick-oxide sidewall is accounted for by using a formula developed from our three-dimensional MOSFET simulator (MICROMOS [6]). Additionally, the enhanced surface potential at the edge of the channel width (due to the edge fringing electric field shown in Fig. 1) is also obtained from our 3-D simulator. These formulas are then used as boundary conditions to solve Poisson’s equa- tion in two dimensions (width and depth), utilizing a tech- nique similar to that of Ratnakumar and Meindl [7].

In the first part of our analysis, the two-dimensional Poisson’s equation is solved considering only the narrow- width effect. However, after comparison of our results with those from the simulator, it was evident that short- channel effects (as discussed in [7]) modified the results and must be included with the narrow-width effect. In fact, it is a coupling of both of these effects that gives rise to a mutual modulation of the depletion depth. Hence, we ex- tended our first model to include short-channel effects.

In the next section, we begin by obtaining a model that includes only the inverse narrow-width effect. In Section 111, a short-channel model is briefly discussed. Following this, the results are extended to obtain the small-geometry model. The results and validity of these models are then discussed in Section V . Important conclusions and com- parisons are then presented.

The model for threshold voltage is developed directly from the basic expression for the n-channel case given by

(1 ) QA Cb,

v, = vF8 + 24F f -

where VFB is the flat-band voltage, 2+F is the potential required to invert the surface, Chx is the oxide capacitance per unit area, QA is the depletion charge per unit area, and the backgate bias V,, is included in QA. From Gauss’ law ( E N = Q ; / C , ~ ) , we can eliminate (3; in terms of the normal

0018-9383/88/0300-0325$01 .OO 0 1988 IEEE

IEEE TRANSACTIONS ON ELECTRON DEVICES. VOL. 35, NO. 3, MARCH 1988

' VG

000 '-

Fig. 1 . Surface potential of a MOSFET with fully recessed isolation ox- ide. L = 2 pm, W = 1.0 pm, V, = 0.1 V, NB = 4 x 10'' ~ m - ~ , and to, = 1000 A.

electric field yieiding

The analyses in the next two sections involve the deter- mination of EN in terms of the depletion width taking into account namw-width and short-channel length MOS

II. INVERSE-NARROW-WIDTH EFFECT MODEL In the inverse-narrow-width model, a very long channel

is assumed 90 t;trat the x-dependence can be neglected. The key factor to the SUQC~SS of this model is the definition of the rela€ioaship between the suiface potential at the edge of the channel width, esT in Fig. 2(b), and the surface potential at the middle of the channel width, *sM in Fig. 2(b). Sugino et J. [ 1 J demonstrated that is about 13 percent higher than eSM with a 90" recessed angle. This is partially true because, in their 2-D simulation, they only solved Poisson's equation without also solving the conti- nuity equation. For the fully recessed oxide isolation (trench structure), even when the device is under moder- ate inversion, the edge area along the thick sidewall is already in the strong-inversion regime because of the en- hanced field at the edge, as shown in Fig. 3. Therefore, the surface potential at the thick sidewall is somewhat pinned at a particular value as is illustrated in Fig. 4. For the weak-inversion case, there is ti sibi

StNCtUleS.

I - 0 W

(b) Fig. 2. (a) Width cross section of a fully recessed oxide isolation MOS-

FET. (b) The solid line is the assumed variation of the surface potential at threshold and is an approximation to the true $s indicated by the dashed line.

Fig. 3. Vertical component of electric field at surface. Device is with NB = 1 x 1 0 ' ~ ~ m - ~ , V, = 2 V, V, = 0.1 V, L = 1 pm, R = 1 pm, to, = 500 A , and toxn = 7000 A .

between qsT and eSM. As the gate bias is increased to the onset of strong inversion, the difference is reduced ap- proximately to tr, = 0.026 V at room temperature. For the strong-inversion case, the difference is small.

HSUEH er a / . MOSFET THRESHOLD VOLTAGE MODEL

l'O1

0.55

0.50-

0.V5- > .

0 . w -

- I

- . r n . .- . =1 0.35- * . 0 . n.

0.30-

0.25-

0.20-

0.15-

0. 10-

0.05-

321

o . o o L , , , , , , .- , ~

0.00 0.05 0.10 0 . 1 5 0.20 0.25 0.90 0 .35 0.VO 0.V5

Depth ( urn )

(C)

Fig. 4. The variation of potential as a function of y (depth position). Curve Cis the potential curve at the middle of the channel width; C' is the potential curve along the thick sidewall. (a) Heavy inversion case. Since both qSM and 'Psr are at strong inversion values, they are very close. Device is with f,,, = 200 A , VG - VFB = 1.24 V. (b) On-set of strong inversion. qsr can be approximated by (8). Device is with f,,, = 200 A , VG - VFB = I V . (c) Weak inversion case. 'Ps,can be approximated by (8). Device is with f,,, = 600 A, Vc - VFB = 1.24 V. All devices are with N , = IO' ' cm-', L = I O pn, W = 3 pm, and t,,,, = 6000 A .

328 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 3. MARCH 1988

Note that, in Fig. 3, there is an enhanced edge field at the thick sidewall. However, this field attenuates drasti-

,cally over a distance approximated by [ estox dw/EOx which will be apparent later, and is an important param- eter for the understanding of the derivation of the rela- tionship between qST and *sM. The constants eS and E,,

are the permittivity of the semiconductor and silicon- dioxide, -respectively, dw = [ ~ E , * S / ~ N B ] ~ / ~ is the deple- tion depth under the gate, NE is the substrate doping con- centration, and at threshold $s = VBG + 2+ where VB, is the back gate bias.

If we neglect the inversion carrier density caused by the enhanced edge field along the thick sidewall, Le., similar to Sugino et al. [l], the 2-D Poisson's equation under the gate, as is illustrated in Fig. 2(a), can be written as

a2* a2* - + 7 = - p / E s ay2 az

where p = -qNB, y is in the depth direction, and z is in the width direction. The vertical electric field at the sur- face Esy can be approximated by [7]

(3)

Therefore

dEsy Esy

dY d W ( 3 4 - = --

where Vh = VG - VFB. Substituting (3a) into (2), we have

(4) d2*s - ( W Z ) - $ 0 )

dz2 x'o where

*O = vG - vFB - 4NBtoxdW/Eox ( 5 4

and

xo = ( € s t o x d W / E o x ~ 1 ' 2 . (5b)

Solving (4) with the following boundary conditions pro- vided from Fig. 2 as

z = 0, *s = *s* ( 6 4

z = WO, *s = *ST (6b)

and

yields

where tox is the thin gate oxide thickness and Wo is a fitting parameter taken to be 300 A (a value providing reason- able results). As suggested earlier, (7) indicates that the field attenuates over a distance x. Equation (7) provides the 13-percent incmase of *ST over *sM as in [l].

If the strongly inverted electrons at the edge area are included in the calculations, however, the difference be- tween ?PsT and e~~ is less than 13 pe&ent because of the

pinning property of qST at the edge. For a device in strong inversion, the surface potential can vary between 2af + VBG and 2af + nuT + VBG [9], where @f is the Fermi potential and vT = k T / q , n is an integer taken as a max- imum of 6, and VBG is the back bias to the device. Then for this case, an empirically fitted formula for the rela- tionship between *ST and *sM is

( 8 ) kT 4

*ST = - exp (Wolxo) + *SM.

As stated previously, qsM can be in the range from 2af + VBG to 2af + nuT + VBG. A conventionally used value for 9sM at threshold is taken as 2af + VBG, and d o = [ 2 ~ , ( 2$f + vBG)/q&]1/2. Equation (7) is valid for a device op- erating under very weak inversion, while (8) is valid for moderate- to strong-inversion cases. Both equations are further justified by the simulation data from MICROMOS [6] as listed in Table I. Note that, at strong inversion, the difference between *ST and *sM is very close to up

The surface potential of (7) or (8) is now combined with the two-dimensional form of Poisson's equation, given by

(9) a 2 q + a2* ay2 az2 - - P I E S - - =

where y Is in the depth direction and z is in the width direction. This equation is written for the depleted region ABCD in Fig. 2(a) and can be solved subject to (7) or (8).

To obtain closed form solutions, the following simpli- fying assumptions (with brief justification) are made:

1) Complete depletion inside ABCD; this is reasonable considering the entire area ABCD, even though our 3-D simulator does indicate that when the middle region is moderately inverted, the edge regions are already heavily inverted.

2) Uniform substrate doping; this is justifiable, even if the substrate surface is ion implanted because an average doping concentration could then be used.

3) Constant surface potential; this assumption is again reasofiable because the difference between 'J"ST and 9sM is smal!as shown in Fig. 4(c). Also, Wo in (7) and (8) is 300 A and hence quite small compared to present channel widths (as small aSO.5 pm). The assumed surface poten- tial is, therefore, the solid line as shown in Fig. 2(b).

The boundary conditions used are

where F is a fitting factor to account for the edge electric field effects and varies between 1.12 and 1.5 depending upon the substrate doping. The recommended values of the fitting factor F are shown in Table II. Additionally,

HSUEH r / ( I / . : MOSFET THRESHOLD VOLTAGE MODEL

v$c

0.1

2

329

0 .2 0 .5 2

0.533 0.711 0.840

0.564 0.729 0.842

and Vas given in (loa) yields the overall 2-D solution as

m

1 sinh

TABLE 1 S I M ~ I A I IO^ R k s u i TS, DFVICF W I T H L = 1 pm, We= 1 pm, A N D N , = 1

x 10" cm-' A N D I , , , = 500 A

Surface P o t e n t i a l a t the Edge of t he Width Y S T

+t----t 0.1 0.756 0.818 0.869

( 2 n + 1 ) a ( z - w ) 2d,

- sinh Surface Potential of the Middle

of t he Width ISM

( 2 n + 1). +

1 2

+ - Ky2 - Kd,.y + 'PsM TABLE I1 R I C O M M F N D F D V A L U E S OF T H E F I T T I N G FACTOR F I N (9)

S u b s t r a t e Doping/

where N B 1 F

- ( - , ) , I

( 2 n + 1)2*2F

where V , = \ksr - \EsM and W is the device width. The electric field in the y-direction is obtained by differentiat- ing (12) with respect to y yielding

2 1 x 1 0 l 6 1 1.5

d,, is the fixed depletion depth away from the edges in the width direction as illustrated in Fig. 2(a).

To solve Poisson's equation, we set K = q N B / c s (where p = - q N R ) and let

* ( z , y) = .( y ) + V(Z, Y ) ' ( l o a )

( 2 n + 1 ) a y

2dW

24%.

Q, cos = Kd,,. - c

n = ~ ( 2 n + ~ ) T W sinh

Substituting (loa) into (2) yields two separated equations as follows:

( 2 n + 1 ) a z 2dW

sinh d 2u q = K

2

( ( 2 n + 1)aF)'

1 sinh (2n + 1 ) a ( 2 - w) 2d,.

(2n + 1)aF2

and

a2v a2v K ay2

2 + - = o .

The solution of (lob) is next obtained using boundary conditions from (9a) and (9b) as follows: ( 2 n + 1 ) w F

-

8Kd,,. (2n + 112a2

+ + V I ) - KY To obtain the solution of (IOc), the same technique is

used as specified in [ 6 ] . Combining the solutions for U To obtain an expression for d,,., we use the condition

330 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 3, MARCH 1988

that

+(;, d,) = 0

and solve for d,. This will yield the minimum depletion depth (at the middle of the width direction). Substituting z = W / 2 and y = d, into (12) and rearranging yields

KdZ, - 2 i s M - - n = O (@ ,

4 d W cosh

+ (2n + 1 ) .

(14) Because the series converges rapidly with n, we retain only the ri = 0 term and solve for the quantity d t as fol- lows:

3 2 ~ q s ( F +; 2 - 1)

F2K'(32 - r3 cash B) d i =

2*SM

r3 b s h B

+ K(1 -

where

B = rW/4dw.

Note, for large W, d , = do at threshold. This is easily seen from (14) because when W >> d,, the series van- ishes yielding d t = 2esM/K or d, = (24+sM/EsNB)1/2, which is 4 at threshold. For smaller values of W, d , be- comes greater than do. The interpretation of this is that the effects of the increased electric field at the edges of the width are equivalent to reducing the substrate doping resulting in a larger depletion depth.

Having determined d,, the normal electric field at the surface can be determined from (1 3) as a function of d,. Thus, at the surface

E N E E ( z , Y ) I z = w / 2 (15) y = o

and therefore

1 ' - 8qST m

EN(d,) = Kd, - C n=O cosh (212 + 1)aw d,

4 d W

2

((2n + 1 ) r F ) Z n + l)rF2

( 16) Again, because of the rapid convergence of the series, retaining only the first term in (16) yields

EN(d,) = Kd, - cosh B

(17) This equation can be substituted directly into the thresh- old voltage equation (la) yielding

(18) ES VT = VFB + 24F + - & ( d , ) . G x

Using (18), the narrow-width effect on VT is determined. Since d, 2 do for the case of a fully recessed structure,

the threshold voltage will reduce as Wdecreases (for fixed applied bias voltages). This indicates the inverse-narrow- width effect.

111. SHORT-CHANNEL EFFECT MODEL For a short-channel MOS device, we dso assume a very

large W value. Many excellent 2-D shat-channel models have been presented [1]-[3], [7], [lo]. Among them, a rather easy and explicit 2-D short-channel model with boundary conditions shown in Fig. 5 was developed by Ratnakumar and Meindl E71 where their potential expres- sion is given as

m ' sin (2n + l)n-y/2dl i ( x , Y ) = 20 si& (2n + 1 ) z L / u L

(sinh (2n + 1 ) T ( L - x ) / 2 d L )

- (sinh (2n + 1 ) ax/UL)

m u m ui. : MOSFET THRESHOLD VOLTAGE MODEL

S o u r c e

331

. . . . . . . . . . 1‘ *L

D r a i n - - - - - -JL - _ - - H=, 82-

0 L (b)

Fig. 5 . (a) Channel length cross section of MOSFET. (b) Boundary con- ditions and assumptions used in short-channel model [ 5 ] .

where V, = qss - q,,, \kss = qB, - VBG, qso = 24F - VBG, V, is the drain voltage and VB, is the built in poten- tial. Additionally, we denote dL as the depletion depth taking into account short-channel effects. As in the pre- vious section, the normal electrical field will be derived directly from (19). The position along the channel at which the minimum depletion depth occurs, x,, can be obtained from the condition aEN (x, O)/dx = 0 and an approximate value for small V, was obtained as [7]

L / 2 - ( 2 d L / a ) log [ I v D / ( 6 B ~ - v B G ) ] .

( 2 0 ) Our approach is to determine d, in the same manner as

in the previous section by setting \E (x,, d L ) = 0. Hence, substituting into (19) and retaining only the first term in the series, an expression for d t is obtained as

( 1 - K , ) d i = L

+ -t

2 l K ; + . sinh wL/2dL

(sinh 7r(L - X n l ) / ( 2 d L ) T

sinh TX,,, /( 2dL)

4VD __ sinh 7rxfn/ ( 2 4 ) I (21 ’ ) 7r

where

K 1 = ( 3 2 / ( ~ * sinh 7rL / (2dL) ) <

* (sinh R ( L - x m ) / ( 2 d L ) - sinh n x , / ( 2 d L ) ) . 1 Note that, in this short-channel case, similar arguments

to the inverse-narrow-width case can be made. In partic- ular, as L is reduced, dL > do, and this will cause a re- duction in v,.

Having determined dL, the normal electric field at the surface can be determined (as in the previous section) as a function of dL. Hence

m I

E N ( d L ) = KdL - ? sinh ( 2 n + 1 ) 7rL/(2dL)

(2n + 1) T ( L - x,) - sinh 2dL

+ (2(v2d: VD) + T 2 ( 2 n + 1)’

9KdL

( 2 2 ) >- sinh ( 2 n + 1 ) TX, - 2dL

Equation (22) is the normal field to be substituted into the threshold voltage equation resulting in

Again, this expression indicates that V , is reduced as L is reduced.

IV . SMALL-GEOMETRY EFFECTS Expressions (16) and (22) represent the narrow-width

and short-channel normal electric field, respectively. Fur- thermore, (14) and (18) show the modulation of the de- pletion depth ( d , or d L ) representing the narrow-width and short-channel cases, respectively. As both the chan- nel width and length are scaled down, there is also a cou- pling effect causing further reduction in threshold voltage. Thus, an analysis to account for the inverse-narrow-width effect, the short-channel effect, and the coupling between these two is necessary. An accurate analytical model solv- ing the 3-D Poisson’s equation with boundary conditions in all six planes might be the best approach [ 131, though quite complicated. In our 3-D study [6] of the small-ge- ometry effect of a MOSFET, we have found that there is a mutual modulation of the depletion depth ( d o ) under- neath the gate as shown in Fig. 6. Consider, for extmple, devices with NB = 1 x 10l6 cm-3 and to, = 500 A (but of various sizes). Further, define d,., as the depletion depth (measured from the surface to the position with an elec-

, , trostatic potential of 0.05 V), where i depends on device

332 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 3, MARCH 1988

0.61 -

0 .53 -

0 . U -

10

LENGTH 1U)Il

LEGEND: P S I - 0.05 ---_.-- 0.20 ----- 0.35 --- 0.51 - -0 .66 - 0.81 -0.96

(a)

70

0.76 I

LENGTH lUMl

0.05 _ _ _ _ _ _ _ 0.20 ----- 0.35 --- 0.51 0.6.6 - 0.81 - 0.96

LEGENO: P S I - -- e).

10

Fig. 6 . Mutual modulation of the depletion-depth underneath the gate: (a) large device, d,, = 0.258 pm; (b) short-channel device, d,* = 0.256 pm; (c) narrow-width device, d,, = 0.262 pm; (d) small-geometry device, d, , = 0.275 pm.

HSUEH ef ul MOSFET THRESHOLD VOLTAGE MODEL 333

V'H3 5 .-I

L=5 urn

j II_' i 4

L 6 . l !

4 : from MICROMOS

.__7 -*__ _-- L 5A , , , , , , . J

N i ~ l n

Fig. 7 . Inverse narrow-width effects: devicg with N , = 4 X 10" cm-' and t,>, = 1000 A .

size as indicated below, and Ay is the difference of d,.; and the large-geometry value d,, , i .e., Ay = d,, - dy , . Then

1) for the large geometry device with L = 5 pm and W - 5 pm, d!, = 0.258 pm;

2) for the short-channel device with L = 1 pm and W = 5 pm, dX2 = 0.265 pm, and Ayz = 0.007 pm;

3) for the narrow-width device with L = 5 pm and W - 1 pm, d!3 = 0.262 pm, and Ay3 = 0.004 pm; and

4) for the small-geometry device with L = 1 pm and W = 1 pm, dJ4 = 0.275 pm, and Ay4 = 0.017 pm.

Obviously, Ay4 # Ay, + Ay3. Therefore, there is a mu- tual modulation of do and the small-geometry effect is not the sum of the short- and the narrow-channel effects. An additional coupling effect should exist for a small-geom- etry device. Using this concept, we have found that the following procedure also leads to satisfactory prediction of threshold voltage variations.

First, we determine which effect (inverse-narrow-width or short-channel) is more dominant by comparing the val- ues of d,,, and dL. Then, knowing which effect dominates, we calculate the depletion width for the other case.

To aid in our description, consider the case where short- channel effects are dominant (meaning that V , is reduced more by the short-channel effects and dl > dw); then we calculate dl , from (14) or (14a) with do replaced by d,.. Following this, d;, (prime to denote that this is a modified

-

-

value of d L ) is calculated by replacing do in (21) by dw. Next, the normal field EN ( d t ) is obtained from (22) and the threshold voltage is calculated from

where d t in this equation has been modified by dw. We interpret the threshold voltage V+, as accounting for short- channel effects and its coupling effects, but not the in- verse-narrow-width effect. This is because we explicitly have only d i in (23). The treatment provided here on the minimum depletion depth is analogous to the effect of a change in substrate doping on do in the long-channel case.

In order to include the narrow-width contribution, we estimate this by the change in V , due to the reduction in W as

where Vrwb is the threshold voltage given by

and Vrw is obtained from ( 1 8) as

vrn = V F B f 24/. + E\) (d:, ) (27) CO,

with d)', determined from (14a). The final expression for

334 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 3, MARCH 1988

VTH 0.7-

0.6-

0.5-

E :

F “‘“7

Y i I

e . p 0 .3-

i i ; E

0.2-

0.1-

0 . 0 7

- : This model

0 : Hong and Cheng’s model [41

o O ~ O ~ O O O

0

0

0

I . . . . . . . . . I . . . . . . . . . I . . . . . . . . ‘ r 0 I 2 3 u

the threshold short-channel minating is thus

of a small-geometry device with

VT = Vk - AV,. (28) If the inverse-narrow-width effect is dominant, d, >

dL, a similar procedure is used with the result that

VT = V b - AVn. (29) Note that, in applying (24) to (28), special cases may

occur. For instance, when W >> d , and the short-channel effects are dominant, (14a) gives us d, = dL and for this case

Similarly, for the case where inverse-narrow-width ef- fects are dominant, L >> d,, we have

V. RESULTS AND DISCUSSIONS Fig. 7 shows the calculated threshold voltage values as

a function of channel width with VD = 0.1 V. Noticeable inverse-narrow-width effects were observed when the channel width is approximately 2 pm or less. This cal- culation was compared wjth the results obtained from our three-dimensional MOSFET simulator and good agree- ment was observed. A more detailed comparison will be presented in [ 141. The influence of the back gate bias is shown in Fig. 8. Similar trends were observed between our model and that of Hong and Cheng [4]. A simple jus- tification of this point can be made by using (1) (with Q; = qNBdo), which results in a VT = 0.2065 V for a long-

1E.P.E TRANSACTIONS ON ELECTRON DEVICES. VOL. 35. NO. 3, MARCH 1988 335

d /j_j : Shigyo & D a n g ' s m o d e l [Z]

0 : M I C R O M O S

- : T h i s m o d e l

2 I 2 3 4

H l O T H ' L'fl 1

Fig. 9 . Threshold voltage versus channel width: channel length varies froi? 1.5 to 4 pni by 0.5 pn. N , = 4 x 10" c n i ~ I , I , , , = 1000 A . r,,r,, = 8000 A .

channel device with N H = 1 x cmp3, t,,, = 1.50 A , and W = 4 pm. Our model predicted a V , = 0.2054 V , while [4] gave a value of 0.1934 V.

Fig. 9 shows the small-geometry effects for devices ob- tained from the literature [2] with the fo!lowing structure data: N , = 4 X IO" cmp3, f o x = 1000 A , and the thick- oxide thickness = 8000 A . Note that for L greater than 3 pm, the inverse-narrow-width effect dominates. But, when the channel length is reduced to 2.5 pm or less, a larger slope is observed and apparent small-geometry effects appear. This is due to the coupling effects of the narrow-width and the short-channel length. If the channel length is further reduced, a stronger short-channel effect and its couplings with the inverse-narrow-width effect causes a much more pronounced reduction in V,. The tri- angles indicated for the L = 2 pm case were obtained from the work of Shigyo and Dang [2]. Very close agree- ment is observed for a similar device, thus providing as- surance of the validity of our model. However, diver- gence was observed for the channel length less than 1.5 pm. This is due to the overestimation of the short-channel effect because the short-channel model [7] assumes an in- finite junction depth Y,. This overestimation is illustrated in Fig. I O . Very good prediction was observed for the short-channel model until the length was reduced to

around 1 .5 pm. Additionally, such divergence arises since the device parameters chosen accentuate the short-chaa- ne1 effect, i .e. , N B = 4 x 10" cmp3 and t,,, = 1000 A . In more realistic device designs based on proper scaling currently used in industry, such a combination of param- eters is not chosen. A more realistic design is then taken for these cases, as shown in Fig. 11, with devices using L = 1 pm, W = 1 pm, and N B = 1 x 10l6 cm-3. Rea- sonable agreement was observed for oxi ie thicknesses curreptly used in production (t,, < 400 A ). For t,, > 400 A the large deviations attributed to the Ratnakumar model as suggested by Fig. 10 become dominant. As one gradually reduces the oxide thickness, the control of the lateral field is reduced thus minimizing the importance of finite junction depths with cylindrical shapes. The reduc- tion in oxide thickness results in improved agreement be- tween the model and MICROMOS down to channel lengths of 1.5 pm. Agreement is expected for shorter channel lengths if the oxide thickness t,,, is further scaled or the doping concentration increased. The agreement, however, cannot be found by using Hong and Cheng's model [4] (e .g . , V , 0.5570 V for L = 1 pm, W = 1 pm, and f,,, = 1000 A device; V , = 0.2875 V for L = 1 pm, W = 4 pm, and t,,, = 500 A case).

In this model, deviations are more strongly evident in

336 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 3, MARCH 1988

I

- : Ratnakumar's Model 161

A : MICROMOS

0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 1 2 3 u LENGTH i pn 1

Fig. 10. Threshold voltage predictions of the short-channel model. Device with NB = 4 X IO" fox = loo0 A , and W = 10 p m .

the submicrometer case. Such deviations were attributed to rj, which is assumed to be infinite in this model. Inclu- sion of rj is possible; however, this requires the cylindri- cal solution of Poisson's equation and results in a lengthy and complicated expression for the potential [ 141. The ad- vantage of the approach presented here is that the equa- tions are more tractable and easier to solve. For channel lengths in submicrometer cases, a sophisticated short- channel model [l 11 can be incorporated with the inverse- narrow-width model suggested here, whieh would im- prove the small-geometry model,

We interpret Figs. 6 and 9 as showing that short-chan- ne1 effects are more dominant, while the inverse-narrow- width effect provides a modulation influence. We believe our model predicts both of these effects quite well. This model can also be extended to the isoplanar or the LO- COS devices with moderate changes (the value qsT and proper choice of the value F ).

VI. CQNCLUSION

We have studied the inverse-narrow-width effects of a fully recessed oxide isolated MOSFET. An analytical

model for this effect was proposed, which implicitly shows the narrowwidth effects depend exponentially on the channel width and nearly a linear dependence on the po- tential rise at the edge of the channel width. An approach has been suggested to eliminate this rise by providing a 60' recessed oxide isdatian technology, but this tech- nique has proven to be impractical [ 13. Another choice is to apply a heavier implant to the edge region dong the thick-oxide sidewall [ 161, but the stronger electrig field in this area may cause increased impact ionization and even- tually degrade the current drive capability [ 171.

The small-geometry effects arise from the mutual mod- ulation of the depletion depth by the short-channel effect and the inverse-narrow-width affect. The dominant effect can be obtained by comparing the before-modulated de- pletion depths. The coupling effects are then included by the mutual modulation of the depletion depth separately. The overall reduction of the thteshold voltage is therefore the sum of the modulated reduution due to the short-chan- ne1 effects and the modulated reduction due to the inverse- narrow-width effects.

The validity of this model has been checked using our

337

0 : MILROYOS - : This model

3 c

U 1 3 T H ' L ' M I

Fig. 1 1 . The variations of threshold voltage due to the change of oxide thickness, L = 1 pm. W = 1 pin. N , = I X 10"' cni '. and r , , , , , = 8000 A

three-dimensional simulator MICROMOS and addition- ally by data reported in the literature. The results suggest that this model is a satisfactory candidate for CAD appli- cations.

A c K NOW L t i x M E N T

The author wishes to thank the reviewers for the valu- able suggestions to this work.

REFERENCES M . Sugino. L. A . Akers, a n d J . M. Ford. "Optimum p-channel iso- lation structure for CMOS." IEEE 7 i . t r r i . s . Elrc~tro/i Dc,i.ic~c,s, v o l . ED- 31. n o . I?. pp. 1823-1829. 1984. N Shigyo and K . L . M . Dang. "Anal)sis of inverse narrowchannel ellect h:ised on ;I three-diiiien\ional \imulator." in P ~ o c . . S x r i i p VLSI

L. A . Akers, "The in \e rw narrow-width ellect," I i , iceLrtr . . k o l . EDL-7. n o . 7. pp. 419-421. 1986. ti. M. Hong and Y . C . Chcng. "An anal)tical model for the inb'erse- narrowgate ellect of a n ie ta l~oxidc~se~i i~conductor field-effect tran- sistor." J . AppI . PIixs.. vol. 61. no. 6. pp. 2387-2392. Mar. IS. 1987. L. I). Y:Iu. "A \imple thcoi-y to predict the threshold voltage of short- channel lGF!zT'\. ' . S o / i t / - S / t i / c L / e ~ ~ / r t m . . v( i I . 17. pp. 1059-1063. 1974. I.. A . ALers and ti. 1.. Hweh. "4 thi-ce-dimen\ioiial MOSFET \ i n - uliltor." S/AM Tcd i . Ah.\rr.. p . ? ? A . 1985.

T~,c~/lrlo/. . pp. 53-54. 1982.

171 K. N . Rntnakumar and J . D. Meindl. "Short-channel MOST thresh- old voltage model," lEEE J . So/it/-Srriic Circ.irirc, vol. SC-17, n o . 5 , pp. 937-947, Oct. 1982.

181 F. Hsu. R . Muller. C . H u , and P. Ko . "A simplc punch-through model for short-channel MOSE'b'I"s." IEEE 7 r t r m . E/rcrro/ i D P i, ims, vol. ED-30. n o . 10. pp. 1354- 13.59. 1983.

[9] Y . Tsividis, "Problema in precision modeling of the MOS transistor for analog application,'' /EEE Trtrri.5. Co/?i/,rcfc,r-Aidri/ Des igr i . vol. CAD-3, n o . I . pp. 72-81. 1984. D. R . Poolc and D. L . Kwong. "Two-dimensional analytical mod- eling of threshold voltage of short-channel MOSFET'\." lEEE E l e ~ trmi D r i s i c ~ ~ Le// . . v o l . EDL-S. n o . I I . p . 443-445. 1984. S . K . Mehta and R . Muralidharan. "Analytical model f o r threshold voltage of kin-implanted ahon-channel IGFET's." I /roil Dc3ricrs. v o l . ED-33. no. 7. pp. 1073-1074, 1986. J . D. Kcndall and A . R . Boothro)d. "A two-diniensional analytical threshold voltage model for MOSFET's with arbitrarily doped sub- strates," IEEE Elc~rrori Dc4c.c Lcrr.. vol . EDL-7, n o . 7, pp. 401- 403. 1986.

o t t . a n d J . D. Meindl. "Perf'oriiiance limits o f CMOS ULSI." Tru/i.\. 6lrcrroti Dci , i c~ , s . v o l . ED-32. n o . 2.

J . Sanchez. K . Hsueh. T . DcMa model for small-gcometr MOSF

' ' Anal 4 si LI of a n ii nonial ou s s ti bt h res hol d d oxide MOSFET uring ii three-diinension 7'rtrri.c. k / m r o r t / I C ~ I ~ ~ C ~ C . \ . v o l . ED-32. n o .

2. pp. 441-445. Feh 1,985 S. Sa\vud;i. Y . Matwinoto. S . Sliinorahi. and 0. Oiowa . "Eflccts 01' field boi-on dow on wb\tratc c ~ i r i w i i ~ i n n i i r iou cliiinncl LDD MOS- FETs," in IEDM T c ~ h . Dig . , pp. 778-781. 1984.

t 338 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35. NO. 3 . MARCH 1988

Kelvin Kuey-Lung Hsueh (S’83-M’86) was born in Tainan, Taiwan, Republic of China. He re- ceived the B.S.E.E. degree from the Tatung In- stitute of Technology, Taipei, the M.E. degree in automatic control from Feng-Chia University. Taichung, and the M.S.E.E. and Ph.D. degrees from Arizona State University. Tempe, in 1983 and 1987, respectively.

He worked with the Tatung Company and the Tatung Institute of Technology as an Engineer/ Researcher/Lecturer from 1974 to 1981 after

Thomas A. DeMassa (S’60-M’66-SM’83) was born in Detroit. MI, in 1937 He received the B S.E E degree, the M.S E E degree, the M S degree in physics, and the Ph D. degree from the University of Michigan in 1960, 1961, 1963, and 1966, respectively.

He has been a member of the faculty in the De- partment of Electrical and Computer Engineering at Arizona State University since 1966 He has published numerou\ technical articles in the area of solid state electronic\ and is uresentlv com-

completing the compulsory military service in Taiwan. He came to Arizona State University in 1981 where he worked as a Graduate Research Asso- ciate with the Center for Solid-State Electronics Research until 1987. Cur- rently, he is with the Gould Inc. Semiconductor Division in Pocatello, ID, where he is a Design Engineer. His research interests include analog MOS circuit design, device modeling, multidimensional simulation, and the physics of small-geometry MOSFET’s.

pleting a textbook Electronic Devices, Circuirs, Sysrerns and Insrrurnenrs. Dr. DeMassa is a member of Tau Beta Pi, Eta Kappa Nu, and Sigma

Xi. He is a past chairman of the Phoenix IEEE Waves and Devices Chapter and past officer in the IEEE Phoenix Section. One of his proudest achieve- ments was receiving Tau Beta Pi’s Teacher of the Year Award at ASU in 1981.

*

*

Lex A. Akers (S’68-M’75-SM’83) was born in Washington, DC, on May 7 , 1950 He received the B S.E.E. degree in 1971, the M.S E E de- gree in 1973, and the Ph.D. degree in 1975, all from Texas Tech University

He joined the Southwest Research Institute after graduation as a Research Engineer where he worked on instrumentation design and mathemat- ical modeling In 1976, he joined the Department of Electrical Engineering, University of Ne- braska, as an Assistant Professor He was active

in the area of multidimensional modeling of MOSFET’s. In 1980, he moved to the Department of Electrlcal and Computer Engineering, Arizona State University, as an Associate Professor He was promoted to Professor in

1985 He is currently working on neural networks and submicrometer CMOS development. He has also worked at the California Institute of Technology Jet Propulsion Laboratory, Pasadena, as a NASA Fellow, the Microelectronics Research and Development Center at Rockwell Interna- tional, the Advanced Products Research and Development Laboratory at Motorola, and the Semiconductor Research and Development Laboratory at Motorola He has spent a semester teaching or doing research at the University College, Cork, Ireland, and the University of Edinburgh, Scot- land

Dr Akers is a member of Sigma Xi, Tau Beta Pi, Eta Kappa N u . and I \ a past Chairman of the Phoenix IEEE Electron Devices Chapter

Jutian J. Sanchez (S’81-M’86) was born on June 19, 1956 in Belen, NM He received the B S E E and B.S. degrees in applied mathematics from the University of Maryland, College Park, in 1979, and the M S and Ph D degrees in electrical en- gineenng from Arlzona State University, Tempe, in 1986 and 1987, respectively

From 1980 to 1981, he was with Motorola G E G., where he worked on surface acoustic wave devices From 1981 to 1987, he attended Arizona State University where he was involved

in the research of small-geometry MOSFET’s In addition, he 5erved a< d consultant to the CTMG division of Intel from 1983 to 1987 In 1987, he joined the Intel Corporation in Chandler, AZ