Modified charge-control model for MOS transistors

So/~d.Sfer

Ekrmnrr.

1978. Vol 21. pp. 593-S94.

Pergamon Press

Printed in Grcal Britain

NOTES MODIFIED CHARGE-CONTROL MODEL FOR MOS TRANSISTORS (Receioed

4 March

1977; in reoised form

Several attempts have been made to show that a considerable improvement may be obtained in the prediction of the dynamic properties of bipolar transistors by extending the charge-control concept through inclusion of the time delays in the relations between the controlling steady-state charge and the terminal voltages and currents[l-31. In MOS transistors also, the device current does not follow the voltages at its electrodes instantaneously[4]. This is particularly true at the high frequencies when the duration of the transient processes (determined mainly by stray and load capacitances) becomes comparable to the inherent time lag of the transistor. Based on an analogy to the bipolar transistorsIS], the intrinsic time lag associated with the MOS transistors during switching[4] is derived here. Further following Winkel’s approach[l] for bipolar transistors, this really important effect of delay time is incorporated in the corresponding charge-control equations for MOS transistors161 wherefrom the modified charge control model is constructed. The enhancement achieved in model characteristics is explicitly demonstrated through application to a practical device. The phenomenon of charge variation in the channel of an MOS transistor during switching, determining the current stabilization time, can be arbitrarily divided into two physical transient processes: the formation of a depletion layer after the removal of majority carriers from the surface (distribution of the gate voltage between the oxide and depleted layer); the formation of a channel because of carrier injection through the drain and source electrodes (voltage redistribution). The duration of the first process, caused by relaxation effects (as an example the time constant of which amounts to IO-‘* sets for Si with a resistivity of 1 ohm-cm) can be ignored as compared to the duration of the second process and we can assume that the depletion region is formed immediately. The process of channel formation by charge build up is examined below. It is well established that for the MOS transistor the diffusion and generation recombination components of the channel current are much less than the drift component and their effect is hence ignored in what follows. Also for the sake of simplicity, as a first approximation. the dependence of the delay time on substrate doping level and mobility variation with drift field is neglected. On application of a step function input A V,(r) = vg eJy’ to the gate of the MOS transistor, operating in the saturatton region (when charge is injected from source terminal only), the process of charge storage and dissipation in its channel can be pictorially represented as in Fig. I. In a time interval 8r the average current across the channel is the shaded charge averaged over distance d-viz. 26d . Q(O)/Sd. multiplied by its mean velocity, which is the distance its centre of gravity moves, 4d/7, divided by 6r (see Appendix). Thus

13 June 1977)

d

Fig. 1. Process of change in channel charge distribution. where V(x). the gate-channel voltage distribution, is the same in form during transient as that in the steady state (from quasi static approximation), and p is the effective mobility of current carriers. Since V(d) = V, (1 - x/dP2, from (1) and (2) we obtain

St= & The delay time is evidently channel length L so that

G

.&d.

(3)

the time taken for d to reach the

8L2 =--0387, 2llLVo

(4)

where 7, = L’/gV, is of the order of the steady-state channel transit time of the carriers. The expression in (4) for the intrinsic lag of MOS transistor is the same as derived in 141from solution of the continuity equation obtained through an analysis at high frequencies. Incorporating this. the chage control eqns (32) of 161 revert to (small signal sinusoidal variation has been assumed)

A[,(,)=, I

26d

d+8d

.,W’=dBQ(I) dr

4d A,,,(,,=/~.e’%;~. I

where QtOl= G, t V_o- VT) = G, V,. Co, is the oxide capacitance. VA is the d.c. gate bias w.r.t. source. V, is the threshold voltage of the device and Vo = VA - V,. Also from independent considerations, the output current is given by the channel charge per unit length multiplied by the velocity of current carriers. i.e.

(5)

and

ALSO

from 161. AQ(r) = Q, Q =

(2) 293

e’“‘. where (t/W&V,

@ I + jwt4/l?)r,~

(6)

594

Notes over here introduces an excess phase shift in the forward transadmittance. The significant improvement resulting in the Huang’s model161 characteristics is clearly illustrated in Fig. 3 through application to an actual device DNC-20 for which experimental measurements have been made by Das[7]. To summarize, the charge control concept for MOS transistors has been reformulated by incorproating the propagation effects to overcome the existing shortcomings. The simple and straight forward extension of the charge control principle brings it to a level of accuracy that is adequate for most practical purposes (in particular the fast response circuits).

Fig. 2. The extended

charge control model in the pinchoff region.

for MOS

transistors

Acknowledgements-The authors are grateful to Prof. S. C. Dutta Roy for helpful suggestions. The financial support from the Council of Scientific and Industrial Research. India is gratefully acknowledged. Depanmenr of Electrical Engineering Indian Insfiture of Technology New De/hi-l 10029. India

UMESH KUMAR A. B. BHAITACHARYYA

REFERENCES I. 1. te Winkel. IEEE Trans. Electron Dec. ED20. 389 (1973). 3 Deu. ED-8. _. A. N. Baker and W. G. May, IRE Trans. Electron 152 (1%1). Radio Engg. and Electron. Phys. 8, 1021 3. V. I. Shveykin. (1963). 4. E. R. Karakhanyan, Yu. R. Nosov and V. A. Shilin. Radio Engg. Electron. Phys. 17.668 (1972). Chap. 5 and 6. Pergamon 5. J. J. Sparkes, Junction Transistors, Press, Oxford (1966). 6. J. S. T. Huang, IEEE Trans. EIecrron Dev. ED16. 775 (1%9). 7. M. B. Das, IEEE Trans. Electron &LA ED16. 1049 (1%9).

The channel current can be derived from charge storage considerations as below. Assume that at an instant of time t after the application of step voltage to the gate, the channel has been formed through carrier injection from source upto a distance d. In an infinitesimal time interval &, the incremental change in charge stored (shaded area in Fig. I) is given by

=; Fig. 3. Experimental actual device: (-o-)

and model values for Experimental; (-----) (-x--x-_) Huang model.

&,(l,y,,) Extended

for an model:

Hence using (5) and (6), the new expression for the particularly affected short-circuit admittance parameter y2, is

(7) The corresponding modified small-signal equivalent circuit for MOS transistor in the pinch off region is shown in Fig. 2. It is evident from (7) that the delay time considered and emphasized

Q(0). 6d.

(Al)

Therefore, the increase in charge stored per unit length in the small time interval 81 is 2Q(O). Sd/3d. The distance traversed by this charge during this time can be determined from consideration of the movement of centre of gravity (c.g.) at which the whole charge in motion may be assumed to be concentrated. The c.g. of the area below the curve (I -x/d)“’ lies at the intersection of the curves (1 -2x/d)“* and (I - $d)“‘/2. i.e. at a distance 4d/7 from L = 0. Hence the velocity of the incremental charge is (4d/7)St and the average channel current at time I is