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A NEW ANALYTICAL AND
STATISTICAL-ORIENTED APPROACH FOR THE TWO-DIMENSIONAL ANALYSIS OF
SHORT-CHANNEL MOSFET’s
M. Conti, C. Turchetti, G. Masetti
To cite this version:
M. Conti, C. Turchetti, G. Masetti. A NEW ANALYTICAL AND STATISTICAL-ORIENTED APPROACH FOR THE TWO-DIMENSIONAL ANALYSIS OF SHORT-CHANNEL MOSFET’s.
Journal de Physique Colloques, 1988, 49 (C4), pp.C4-253-C4-256. �10.1051/jphyscol:1988452�. �jpa-
00227950�
JOURNAL DE PHYSIQUE
Colloque C4, suppl6ment au n09, Tome 49, septembre 1988
A NEW ANALYTICAL AND STATISTICAL-ORIENTED APPROACH FOR THE TWO-DIMENSIONAL ANALYSIS OF SHORT-CHANNEL MOSFET's
M. CONTI, C. TURCHETTI and G. MASETTI'
Depfz. of Electronics University of Ancona, v. Brecce Bianche, I-60131 Ancona, Italy
"DEIS University of Bologna, v. le Risorgimento 2 , I-40136 Bologna.
Italy
ABSTRACT
An approximated analytical solution of Poisson's equation for the short-channel MOSFET operating in the subthreshold regime is presented. It is shown that the proposed approach predicts a dependence of the threshold voltage on process parameters and drain and substrate voltages in very good agreement with two-dimensional analysis and with available experimental data. Finally, the method of this work, which permits to gain a factor of about 103 in CPU time with respect to numerical modeling for threshold predictions, seems particularly suited for statistical modeling.
1. INTRODUCTION
In order to improve circuit performances and to reduce chip size, devices used in today MOS VLSl circuits have very small channel lengths andlor channel widths. Besides, when device dimensions are scaled down, second-order phenomena, such as drain induced barrier-lowering and influence of the curvature of the source and drain junctions [I-21, become essential in establishing the electrical behaviour of the MOSFET. As a consequence, the device performances should be determined, in principle, by the adoption of two- ( or even three- ) dimensional device simulators (see, f.i., 13-61 ).
However, these simulators are very time consuming and, thus, very cost effective when multiple analyses, such as those required for statistical modeling, must be performed. On the other hand and as well known: i) in order to develop designs with the aid of circuit simulators, circuit designers require very simple (even if strongly approximated) analytical device models, while ii) in most cases, device designers, to get a general view of the way a process is going on or to evaluate the spread existing between the characteristics of devices processed with the same or with different runs, are interested in the knowledge of only a few device parameters of the MOSFET (such as threshold voltage and gain factor).
The purpose of this paper is to derive, through the weighted-residual method [8], an approximated analytical solution of the two dimensional Poisson's equation for a short-channel MOSFET operating in the subthreshold region.
In particular, it is shown that the proposed approach is able to predict a distribution of the surface potential along the channel close to that achievable with the two-dimensional simulator HFIELDS [6]. Besides, this approach can also satisfactorily predict the dependence of the threshold voltage in short-channel devices for a wide range of values of both device parameters (such as oxide thickness, junction depth and substrate doping) and bias voltages.
Finally,it is shown that with the analysis of this work threshold voltage predictions can be achieved with a reduction of a factor of about 103 in CPU time with respect to standard numerical analysers 161; this result enables to qualify the approach as a cheap method to predict the sensitivity of threshold voltage on fabrication tolerances in short-channel MOSFETs.
2. MODEL FORMULATION
For a device operating in the subthreshold regime, by neglecting carrier concentrations Poisson's equation can simply be expressed as
q Na
+ -
in the semiconductora 2 0 ( x . y ) + = ES
a
x2 aP
0 ( 1 )in the oxide
where 4 is the electrostatic potential, N, is the bulk doping and eS the permittivity of silicon.
Eqs. 1
,
once defined the proper boundary conditions, can be solved by means of standard numerical procedures [3-61. However, some approximate mathematical methods that reduce partial differential equations to a set of ordinary differential equations are available and more easily solved. One of the most powerful methods is the so-called weighted residual approach [81. Following this technique we can seek for an approximate solution ~ ' ( x , Y )Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988452
C4-25 4 JOURNAL
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PHYSIQUEto ( 1 ) in the linear form
+ +
= a j + j ( x S y )j = l
where ej(x,y) are independent arbitrary functions to be properly chosen and aj are coefficients to be determined.
The errors, or residuals ( Rsi and Ro, ), introduced by approximation ( 2 ) can thus be defined for eq. ( 1 ) as in the silicon
i n the silicon-dioxide ( 3b )
Usually, to reduce the residuals, a suitable number of integrals of the error, weighted in different ways through some weighted parameters Wk, are required to be zero, that is
where
r
is the boundary of the domainn
in which equations (1) have to be solved,while Rr and Rn are the residuals evaluated alongr
and insiden,
respectively.In this work, in order to get a relatively simple and low-time consuming approach, we choose to fit the electrostatic potential with a simple polynomial in x and y with independent coefficients, i.e.
Morever, always in the light to achieve a simple analytical formulation, we made a choice of the relationship (2') where, by imposing the boundary conditions, all the coefficients result as functions of only one of theme. This results in a strong simplification of the analysis which reduces the set of equations (4) for the residuals to one equation only. As a consequence, the evaluation of $' results dependent on the evaluation of one parameter ajk only.
Now, due to the very simple formulation (2') for $' , the application of the weigthed-residual method gives a simple linear equation for the minimum value of the surface potential:
$smin = a VGS .-
P
( 5 )where a and
p
are numbers which depend on device geometry, device physical parameters ( Na, tox, VFB ) and on bias voltages.3. SURFACE POTENTIAL AND THRESHOLD BEHAVlOUR
An example of the behaviour achievable for the surface potential along the channel of the device $s(y) by using the approach presented in this work is shown in Fig. 1 . Device parameters were L=1.5 pm, Na= 10'6 cm-3, Nd=
1018 cm-3, to,= 25 nm, xj= 0.25 pm, VBS= -2 V, VDS= 1 V, VGS
-
VFB= 1.2 V. In the same figure, for comparison, the behaviour of the surface potential achieved with the two-dimensional analyzer HFIELDS 161 is shown as dotted line. As can be seen, the agreement between the two behaviours is resonably good and, in particular, the values found for the minumum value $,min of the surface potential are pratically coincident. As far as the CPU time is concerned, for the results of Fig. 1 we found that our approach enables a reduction in computer time of factor 1: 1000 with respect to the calculations required from HFIELDS. This factor represents also an average value for the CPU time ratio for all the other comparisons we made by varying device parameters in the following ranges: L=0.6 1 10 pm, Na= 4 150
*
10'5 cm-3, b x = 10 1 70 nm, xj= 0.15 I 0.45 pm, VgS= 0 1 - 4 V, VDS= 0.1 1 5 V.4. THRESHOLD VOLTAGE PREDICTIONS
In MOS technology a parameter extremely useful to "conventionally" define the conduction state of a single device and to identify process variations throughout a wafer and among chips built on different wafers with the same run (or different runs) is the threshold voltage. For the sake of simplicity and a) to take into account that the carrier injection in the channel of the MOSFET almost exponentially depends on the variations of the barrier-height at the source-channel junction with respect to its value at equilibrium and b) to account for the strong dependence of such a barrier-height on device parameters and bias voltages in short-channel devices, we defined the threshold voltage VTH as the gate-source voltage which gives a certain value $* for the surface potential Osmin
.
SO doing we get:Fig2 shows, as continuous lines, the threshold-voltage value, calculated from (6) and using @* = 2 $F ( $F being the Fermi-voltage), as a function of channel length for several values of the body bias. Device parameters were: Na=
1016 ~ m - ~ , b x = 25 nm, xj= 0.25 pm, VDS= 5 V. The results achieved with CADDET
[A
are shown as dots in the figure. As can be seen, the agreement between comparison is very good. Similar conclusions can be drawn as far as the dependence on oxide thickness, substrate doping and drain voltage; a comparison between the results predicted from the model of this work and the numerical simulations is shown for three values of the oxide thickness in Fig. 3.Finally, a comparison between the results predicted from our model and experimental data
[q
is shown in Fig. 4.As can be seen, for the examined channel-length and oxide-thickness range, the proposed model satisfactorily predicts the threshold behaviour.
5. STATISTICAL ANALYSIS OF THRESHOLD VOLTAGE
Due to the very low CPU time required, the approach presented in this work seems particuraly suited for statistical analysis of the threshold voltage in short-channel MOSFETs. In fact one can assume that threshold voltage given by eq.(6) is a random function of some process variables
VTH = VTH ( L,
tax.
Na, xj ) ( 7 )Then, by considering that the process variables can be characterized with a certain probability distribution, one can easily derive the distribution probability of threshold voltage. In particular we assumed a gaussian distribution for all the above variables and calculated the distribution p(VTH) of the probability density associated to the threshold voltage, the mean value and the root-mean square error of the threshold.
Fig. 5a shows, for three values of channel length, the p(VTH) distribution achieved by considering a constant mean-square error in channel length of 0.1 pm. The other device parameters were considered constant and equal to N,= 1016 cm-3, tOx=25 nm, xj= 0.25 pm, VDS= 5 V, VBS= -2 V. As can be seen, as the channel length decreases the p(VTH) distribution becomes more broadened with a more pronounced asymmetry toward lower values of threshold.
These conclusions are fully confirmed from the results shown in Fig. 5b, where the mean value and the value of the root-mean square error of the threshold voltage are reported against channel length. In particular, for the process considered, one can derive an increase from 0.07 % to 3.5 % in the ratio between the root-mean square error and the mean value of the threshold voltage when the channel length is decreased from 5 ym to 1 ym.
Similar conclusions can also be drawn by considering a gaussian distribution for the oxide thichness, the substrate doping and the junction depth.
6. SUMMARY AND CONCLUSIONS
In this work the weighted-residual method was utilized to find an approximated analytical solution of the two dimensional Poisson's equation in short-channel MOSFETs. In particular, it was shown that the proposed method:
i ) is able to predict a distribution of the surface potential along the channel very close to that achievable with two-dimensional simulators;
i i ) can satisfactorily predict the dependence of the threshold voltage in short channel devices for a wide range of variation of process parameters ( such as channel length, oxide thickness, substrate doping and junction depth ) and bias voltages;
i i i ) for threshold evaluation it requires a CPU time lower of about three order of magnitude with respect to two-dimensional simulators.
Finally, due to property iii). above, the analysis was extended to find the probability distribution of the threshold voltage induced by variation in process parameters. In particular, results were found for gaussian distributions of channel length, oxide thickness, substrate doping and junction depth.
ACKNOWLEDGEMENTS
This work has been supported by the Consiglio Nazionale delle Ricerche under the Project: "Materiali e Componenti per I'Elettronica a Stato Solido". Support from SGS-THOMSON Microelectronics is also gratefully acknowledged.
Thanks are also due to the numerical analysis group of the University of Bologna for the support in the usage of HFIELDS code.
REFERENCES
[ 1 ] R.R.Troutman, IEEE Trans. Electron Devices, vol. ED-26, pp.461, 1979.
[ 2 ] B.Eitan and D.Frohman-Bentchkowsky,lEEE Trans. Electron Devices, vol. ED-29, pp.254, 1982.
[ 3 ] A.Husain and S.G.Chaimberlain,lEEE Trans. Electron Devices, vol. ED-29, pp.631, 1982.
[ 4 ] W.I.Engl et al, Proc. IEEE, vol. 71, pp. 10, 1983.
[ 51 SSelberherr, "Analysis and simulation of semiconductor devices", Ed. Springer, Wien 1984.
[ 6 ] G.Baccarani et at, Proc. of the Fourth International Conference on the Numerical Analysis of Semiconductor Devices and Integrated Circuirts, J.J.H.Miller Ed., Dublin: Boole Press, pp. 3, 1985.
[ 7 ] T.Toyabe and S.Asai, IEEE Trans. Electron Devices, vol. ED-26, pp. 453,1979.
[ 8 ] B.A.Finlayson, "The Method of Weighted Residuals and Variation Principles", New York: Academic, 1972.
JOURNAL
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PHYSIQUE0. .2 .La .6 .8 1.
Y / ~ e f f
Fig. 1
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