O N E LECTRON D EVICES, V OL. E D-26, NO. 4 , A PRIL 1979
A nalytical Models of Threshold Voltage and
B reakdown Voltage of Short-Channel
MOSFET‘s Derived from
Ana l ysis
TORU TOYABE, MEMBER,
I EEE, A ND
voltage of short-channel MOSFET’s are derived from the combination of analytical consideration and two-dimensional numerical analysis. An approximateanalytical solution forthesurfacepotential
is used to
derive the threshold voltage, in contrast with the charge conservation approac:hwhich has been usuallytaken.It
is shown that the surface
potentid depends exponentially on the distance from the drain, and thiscauses thethreshold voltage to decreaseexponentiallywithdecreasing: channel length. The analytical dependence of threshold voltage on device dimensions, doping, and operating conditions is verified by accurate two-dimensional calculations, and the accuracy of the model is attained by slight modification.
The breakdown voltage of a short-channel n-MOSFET is lowered by a posi1:ive feedback
effect of excess
twodimensional analysis of thismechanism,a
simpleexpression of the
breakdown voltage is derived.
Using: this model, the scaling down ofMOSFET‘sis discussed. The simple models of thresholdandbreakdownvoltage
MOSFET’s arehelpfulbothforcircuit-orientedanalysisandprocess diagnosis where statistical use of the model is often needed.
I . I NTRODUCTION
EMAND for larger scale integration of MOS circuits has
urged miniaturization of MOSFET’s with the support of
the progress of the lithographic technique. When the channel length of MOSFET’s is of the order of 1 pm, the threshold
voltage decreases with the decrease in channel length [ 11, and
Manuscript received August 11, 1978;revised October 14, 1978. The authors are with the Central Research Laboratory, Hitachi, Ltd., Kokubunji, Tokyo 185, Japan.
SHOJIRO ASAI, MEMBER,
breakdown voltage also decreases in the case of an n-channel device . Theseshort-channeleffects pose manyproblems
in the device design in the development of modern MOS LSI.
The threshold voltage of a short-channel MOSFET has been
analyzed by Lee  and Yau  using a charge-conservation
condition including a drain and source depletion layer charge. They used simple geometrical division of the depletion layer in the substrate to derive the expressions of threshold voltage. On the other hand, two-dimensional nurnerical analysis has
MOSFET’s [ l ] , [ 5], 161 and
breakdown voltage [ 2 ] . This
approach gives accurate prediction of these quantities.
However, two-dimensional analysis is not quite adequate for
theapplication to circuit analysis or statisticalmodeling in process diagnosis because it is complex and takes much computation time. In this paper, threshold and breakdown voltage models are proposed in an analytically closed form as a function of structural parameters and operating conditions. Twodimensional analysis is fully utilized to derive themodels.
Implications of the present models are discussed in relation to the scaling down of MOSFET’s.
11. T HRESHOLD VOLTAGE
A . Automatic Threshold Voltage Calculatbn Method in
When the channel length of MOSFET’s becomes smaller, the
V , begins to decrease
due to twothe
0018-9383/79/0400-0453$00.75 @ 1979 IEEE
dimensional field effect of the drain junction. Thisbehavior cannot be explained by a one-dimensional model. Therefo x , a c omputer program for numerical analysis is developec to
in MOSFET's. This
called CADDET,' solves Poisson's equation and the cur .ent
continuity equation for electrons (an n-channel device is considered) as a setof difference equations....