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VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 425-429Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Three-dimensional Statistical Modeling of the Effectsof the Random Distribution of Dopants

in Deep Sub-micron nMOSFETsE. AMIRANTEa, G. IANNACCONEb’* and B. PELLEGRINIb

aLehrstuhlffir Technische Elektronik, Technische Universitt Mfinchen, 80290 Mfinchen, Germany;bDipartimento di Ingegneria dell’Informazione, Universith degli studi di Pisa, Via Diotisalvi 2, 1-56126 Pisa, Italy

We have performed a three-dimensional statistical simulation of the threshold voltagedistribution of deep submicron nMOSFETs, as a function of gate length, dopingdensity, oxide thickness, based on a multigrid non-linear Poisson solver. We compareour results with statistical simulations presented in the literature, and show thatessentially only the vertical distribution of dopants has an effect on the standarddeviation of the threshold voltage.

Keywords: Statistical modeling; Dopant fluctuations; Deep submicron MOSFETs

1. INTRODUCTION

According to the International Technology Road-map for Semiconductors, 1999 edition [1], themaximum allowed standard deviation avr of thethreshold voltage of MOSFETs should scaleproportionately to the supply voltage Vad. Thiswould imply for the present year a value of

3rvr=50mV (minimum gate length 120nm),for 2001 3rvr=42mV (minimum gate length100nm), for 2004 3rvr=33mV (minimum gatelength 70nm). On the other hand, since thedispersion of Vr is mainly due to randomfluctuations of the dopant distribution in thechannel, crvr would tend to increase withincreasing doping levels and decreasing channel

area, associated with downscaling of CMOStechnology.

It is therefore important to fully understandhow the details of device parameters and of thedoping profile affect the dispersion of Vr and ofthe off-state current. In this paper, we address thisproblem by means of three-dimensional simula-tions on statistically meaningful ensembles ofdevices (200 devices for extracting only the averageand the standard deviation, 1000 devices forobtaining the detailed distribution of Vr).A few attempts at addressing the effects of the

random dopant distribution from a quantitativepoint of view have recently appeared in theliterature [2-5]. However, only Refs. [4] and [5]present a systematic statistical analysis on a three

Corresponding author. Tel.: + 39 050 568677, Fax: + 39 050 568522, e-mail: [email protected]

425

426 E. AMIRANTE et al.

dimensional grid, limited to threshold voltagedispersion. Given the large concentrations ofdopants and carriers in deep submicron MOS-FETs, our model includes Fermi-Dirac statistics ofelectrons and holes, and incomplete ionizationof dopants. In addition, it includes the effects ofa polysilicon gate- recently treated in detail in[6]- and dopant fluctuations in the polysilicongate, source and drain.

2. MODEL AND RESULTS

A Monte Carlo procedure is used to generate thediscrete distribution of dopants in the active areafor each considered device. Let us call NAi and Vithe nominal acceptor doping and the volume ofthe Voronoi cell associated to the i-th gridpoint,respectively (we consider nMOSFETs). The i-thcell will contain on the average (Mi)-NAiVidopants. Given the nature of the ion implantationprocess, the actual number of acceptors in the cell/i is randomly chosen from a Poisson distributionwith average NmV. Then the actual doping of thei-th cell is ai li/Vi. The mesh is chosen in sucha way that in the channel region NAiV << 1, sothat, in practice, at most one dopant atom iscontained in each cell.The non-linear Poisson equation is then solved

with a multi-grid method in the three-dimensionaldomain, considering grounded drain and source,and a voltage Vg applied to the gate electrode:

-q[p(r) n(r)N + (r) (r)],

whereN+ andN are the concentrations of ionizeddonors and acceptors, respectively, p and n are thesemiclassical hole and electron densities computedwith the Fermi-Dirac distribution [6], b is theelectrostatic potential.Once the carrier densities are known, we solve

the continuity equation for a very small drain-source voltage Va, in order to compute theconductance of the channel. We actually assume

that carrier densities at inversion and quasi-inversion do not change to first order in Vas, sothat we obtain a simplified continuity equationwith only the drift term, like in Ref. [4]. Themobility is considered constant in the channel.From the curve of the channel conductance as

a function of the gate voltage, we obtain thethreshold voltage V, as the intercept with thehorizontal axis of the line extrapolating the curvein the inversion region. We have verified that theslope of such a line for a given gate voltage isindependent of the detailed doping distribution:therefore, for each device, we can simply computethe channel conductance for a single value of Vgsufficiently larger than V and, once the slope ofthe line is known, directly extract the thresholdvoltage.

First, we consider a n-MOSFET with effectivechannel length L and width W of 100nm, gateoxide thickness tox- 3 nm, uniform acceptordoping NA--1018 cm -3 in the substrate, n + polygate (NDg--1020cm-3), drain and source junc-tions with a depth of 50nm and donor densityNz) 1019 cm- 3.

In Figure the conduction band edge in thechannel, on the plane at the interface betweensilicon and silicon oxide, is plotted for three

x (nm)

Vo=O V

V=0.2 V

V=0.6 V

O0

y (nm)

FIGURE Conduction band edge in the channel, on theplane at the interface between silicon and silicon oxide, for threedifferent gate voltage: Vg =0, 0.2, 0.6V (surfaces are translatedvertically to avoid overlap). Current flows along the x-axis,drain and source junctions are located at x --0 and x- 100 nm,respectively.

DEEP SUBMICRON nMOSFET’s 427

different gate voltages: Vg-O, 0.2 and 0.6V.Current flows along the x-axis, and drain andsource junctions are located at x-0 andx- 100 nm, respectively. The peaks are associatedto the presence of individual impurities, with largerpeaks due to shallower impurities, and smoothbumps due to deeper regions of higher impurityconcentration.The threshold voltage of such a device, com-

puted in the assumption of uniform doping, is

VTnom--0.498 V. In Figure 2, the threshold vol-tage frequency distribution for a sample of 1000n-MOSFETs is shown: the average threshold vol-tage is (Vr) 0.486 V, slightly smaller than VTnom,and the standard deviation is crvr- 19mV.To understand the influence of the position of

dopants on Vr, in Figure 3 we plot the correlationcoefficient between Vr and the total number ofdopants contained in the substrate region underthe gate, for 0 < z < d (z is the vertical direction,and z- 0 corresponds to the Si-SiO2 interface). Ascan be seen, the correlation coefficient is close to0.9, for d between 14 and 28 nm; then, it decreasesmonotonically for larger values of d. These resultsconfirm those found, on a smaller ensemble ofdevices, in Ref. [8].

In Figure 3 we also plot the correlationcoefficient between Vr and the number of impur-ities contained in a slice with d< z < d / 4nm(circles). The correlation coefficient is maximum

FIGURE 2 Threshold voltage frequency distribution andcorresponding normal distribution for a sample of 1000nMOSFETs with W- L-- 100 nm, tox-- 3 nm, NA 10TM cm -3

FIGURE 3 Correlation coefficient between VT and the totalnumber of dopants contained in the active region at a depthsmaller than d (squares), and correlation coefficient between VTand number of impurities contained in a slice at between d andd+ 4 nm (circles).

for d=0 and decreases with increasing d: itpractically goes to zero for d larger than thedepletion width Wd of the device computed withthe full depletion approximation (in our case

Wd= 34.7 nm). This means that only impuritiesin the depletion region can influence the thresholdvoltage, and that shallow impurities are the mostrelevant for the value of Vv.

Based on an analytical derivation of thestandard deviation of the threshold voltage [3],we have computed, for each device of the sample,the term M, defined as

(z)M-- dz dy dx - UA X y z

(2)

M takes into account the total number of dopantsin the depletion region, associating a larger weightto shallower impurities. In Figure 4 we show thescatter plot of M versus VT for each of the 1000devices in the sample. As can be seen, thecorrelation is extremely good, with a correlationcoefficient r 0.964. It is worthy noticing that Mdoes not take into account the distribution ofdopants in the x and y directions, since in (2) wesimply perform an integration over x and y. Acorrelation coefficient so close to means that the

428 E. AMIRANTE et al.

1400

1300

1200

11o0

10oo

900

800

0.964

0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58

Threshold Voltage [V]

FIGURE 4 Scatter plot ofM defined in (2) versus Vr for eachof the 1000 devices in the sample, and least mean square fittingline. The correlation coefficient is r--0.964.

dispersion of Vr depends in practice only on thevertical distribution of dopants (let us point outthat the same conclusion cannot be drawn for theaverage of Vr). As a consequence, it should bepossible to obtain information on the dispersion ofVr with one-dimensional simulations, where onlythe vertical direction is taken into account. In sucha way, the computational burden of statisticalsimulations could be significantly reduced. Inaddition, one could easily include in the simulationthe effects of quantum confinement in the channel,that are prohibitive for a 3D statistical simulation,but have a significant effect on the thresholdvoltage of deep submicron MOSFETs.

Results from semiclassical and quantum me-chanical statistical simulations in one dimensionwill be shown elsewhere. Here, we can anticipatethat the standard deviation obtained from 1Dsemiclassical simulations is in very good agreementwith the results from 3D simulations shown here,while the downward shift of the threshold voltagecannot be reproduced in 1D, since it is probablydue to the random occurrence of low impedancepaths in the channel between drain and source,which is an inherently 3D phenomenon. On theother hand, the threshold voltage resulting fromquantum-mechanical simulations is on averagesignificantly larger than the semiclassical value(130mV for the device structure mentioned

above), while its standard deviation is only slightly(less than 10%) larger than that obtained with thesemiclassical model. This means that the actualrelative dispersion of Vr is smaller than whatwould appear from a semiclassical simulation.We have repeated our three-dimensional simu-

lations varying impurity concentration NA from5 1017 to 1.5 1018cm -3, the gate length andwidth from 50 to 130 nm, and considering an oxidethickness of 2 and 3nm. Since the standarddeviation of the error on (Vr) and rvr are

r/x/ and rvr/x/-, respectively, where N isthe number of devices per sample, we have usedsmaller samples of 200 devices, obtaining rvr withan accuracy of 5%.

In all cases we have consistently found adownward shift of the average threshold voltagewith respect to VTnom: however, such a shift hasbeen always smaller than 20 mV, with an error barof -2 mV, at least factor of two smaller than thatobtained in Ref. [4] with comparable parameters.The difference could be due, at least partially, tothe different definition of threshold voltage used.Again, with respect to Refs. [4] and [5], we find abetter agreement with results obtained fromexisting analytical models and simpler 1D simula-tions, which take into account only dopingfluctuations in the vertical direction [3, 9].

Regarding the dependence of rvr on the dopingdensity, in Ref. [5] Crvr is found to be proportionalto NA4, while from analitycal models [3,9] itshould be proportional to NA25. In Figure 5 weplot the logarithm of Crvr versus the logarithm ofthe doping density both for the device structuredescribed above and for one with tox= 2 nm andother parameters unchanged. The slopes of theleast mean squart fitting lines, i.e., the exponent ofNA, are 0.239 and 0.289, respectively, confirmingthe predictions of analytical models and the resultsof Ref. [10]. The dependence of rvr on theeffective gate length L is in good agreement withresults in the literature: data in Figure 6 areobtained with NA 10-18 cm-3, tox 3 nm, andW=L, and are compared with the formuladerived analytically in [3].

DEEP SUBMICRON nMOSFET’s 429

1.4

1.3

1.2

1.1

17.6

3 n rn

slope 0.289

. .o .2

Log (N.x 1 cm a)FIGURE 5 Logarithm of av. versus the logarithm of thedoping density NA for a device with L-- W= 100 nm, and twodifferent values of tox: 3 nm (squares) and 2 nm (circles). Alsoplotted are the least mean square best fitting lines with relatedslopes.

45

40

35>’ 30E

25

> 20

15,o5

040 60 80 100 120 140

Effective gate length (rim)

FIGURE 6 avr as a function of the effective gatelengthL= W, for tox--3nm and NA=101gcm -3 obtained fromsimulations (squares) and from the analytical formula ofRef. [31.

In conclusion, we have performed a three-dimensional statistical simulation of the thresh-old voltage distribution of deep submicronnMOSFETs, based on a multigrid non-linear

Poisson solver. We have shown that the standarddeviation of Vr essentially depends only on thevertical distribution of dopants, and that thedistribution of dopants in the horizontal planehas an effect only on the average downward shiftof Vr. Compared with existing results fromstatistical simulations [4, 5], we find some differ-ences that require further analysis.

References

[1] Internation Technology Roadmap for Semiconductors-1999 edition, Semiconductor Industry Association (SanJose, USA, 1999).

[2] Wong, H. S. and Taur, Y. (1993). "Three-dimensional"atomistic" simulation of discrete random dopant dis-tribution effects in sub-0.1 gm MOSFETs", IEDM Tech.Dig., pp. 705- 708.

[3] Stolk, P. A., Widdershoven, F. P. and Klaassen, D. B. M.(1998). "Modeling statistical dopant fluctuations inMOS transistors", IEEE Trans. Electron Devices, 45,1960.

[4] Asenov, A. (1998). "Random Dopant Induced ThresholdVoltage Lowering and Fluctuations in Sub-0.1gmMOSFET’s: A 3-D "Atomistic" Simulation Study", IEEETrans. Electron Devices, 45, 2505.

[5] Asenov, A. and Saini, S. (1999). "Suppression of ran-dom dopant-induced threshold voltage fluctuations insub-0.1gm MOSFETs with epitaxial and b-dopedchannels", IEEE Trans. Electron Devices, 46, 1718.

[6] Asenov, A. and Saini, S. (2000). "Polysilicon gateenhancement of the random dopant induced thresholdvoltage fluctuations in sub-100nm MOSFETs withultrathin gate oxide", IEEE Trans. Electron Devices,47, 805.

[7] Sze, S. (1981). Physics ofsemiconductor devices, New York:Wiley and Sons, 2nd edn.

[8] Vasileska, D., Gross, W. J. and Ferry, D. K. (1998)."Modeling of deep-submicrometer MOSFETs: randomimpurity effects, threshold voltage shifts and gate capaci-tance attenuation", Proc. of IWCE6 (IEEE, Piscataway),p. 259.

[9] Taur, Y. and Ning, T. H. (1998). Fundamentals ofmodernVLSI devices, Cambridge: Cambridge University Press.

[10] Vasileska, D., Gross, W. J. and Ferry, D. K. (2000)."Monte Carlo particle-based simulations of deep sub-micron n-MOSFETs with real space treatment ofelectron-electron interaction and electron-impuritytreatment", Superlattices and Microstructures, 27, 148.

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