On the analysis of random doping induced

fluctuations in ultra small semiconductor devices by

linearization

P. AndreiECE Department, Florida State University

Tallahassee, FL 32310, USApandreigeng.fsu.edu

Abstract-The basic assumptions used in the analysis of randomdoping induced fluctuations through the linearization oftransport equations are carefully considered. It is proved that, inthe framework of the density-gradient model, the terminalcurrents, transconductance, and threshold voltages of MOSFETand SOI devices are more or less linear with respect to themagnitude of variation of the doping fluctuations. This factsupports the hypothesis of the linearization techniques andexplains the good agreement existent in the literature between theresults of statistical analysis of semiconductor devices by usingthe linearization technique and the Monte-Carlo simulations.

Keywords- random dopingfluctuations; linearization.

I. INTRODUCTIONA new technique for the analysis ofrandom doping induced

fluctuations on the intrinsic values of parameters ofsemiconductor devices was proposed in [1-4]. This techniqueproved to be computationally very efficient and reliable for theanalysis of parameters variations in MOSFET devices. Thetechnique is based on linearization of the transport equationswith respect to the fluctuating quantities (doping, electrostaticpotential, electron and hole concentrations, etc) and on theevaluation of the variances of quantities of interest by usingsuperposition. Although the linearization technique has beenapplied extensively to the analysis and evaluation of randomdoping induced fluctuations [5-8], the errors induced by thelinearization have not been analyzed. In this article we carryout a detailed analysis of the linearization errors by verifyingthe assumption of linearity between the device parameters andfluctuations of the doping concentration at different locations.

The main assumption of the linearization technique is thatfluctuations are small and, in the first-order approximation, thevariations of any parameter A of the semiconductor devicedepends linearly on the magnitude of fluctuations of the dopingconcentration at different locations:

A= SA(r)D(r)dr' (1)

where Q represents the semiconductor region, D (r) denotesthe fluctuations of the doping concentration at location r, andSA (r) is some "weight" function, called the sensitivityfunction of parameter A [1,3,6]. In (1) D(r) should be

regarded as a random field equal to D (r) - (D (r)), where

(D (r)) is the average value of the doping concentration atlocation r. The variance of parameter A is:

2= I LSA (r)SA (r')ACF[D(r),D(r')]drdr', (2)

where ACF [D (r), D (r')] is the autocorrelation function ofthe doping concentration. If the doping concentrations atdifferent locations are independent Poisson random variables,the variance of A can be shown to be [2]:

2 = f[SA(r)]2 D(r)dr.Q

(3)

It is apparent from the previous equations that the accuracy inthe calculation of CA by using (2) [or (3) in the case of Poissondistributions] depends on wheatear fluctuations A can bewritten as a linear combination of D5(r) [ see (1)]. In order tocheck if this assumption holds for normal semiconductordevices, we carry out simulation experiments in which we varythe doping concentration at different locations inside the deviceand measure the variation of intrinsic parameters of the device(e.g. threshold voltage, small-signal parameters, etc.). If therelation between the doping concentration and the parametersof the device is linear, the linearization technique can be safelyused to compute the variances of device parameters. If thisrelationship is not linear in the range of variation of the dopingconcentration, the linearization technique might overestimateor underestimate CA .

This work is supported by the Army High Performance ComputingResearch Center under the auspices ofthe Department ofthe Army, ARLunder Cooperative Agreement number DAAD19-01-2-0014.

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II. LINEARIZATION TECHNIQUEIt is apparent from the previous section that the variance of

some parameter of the semiconductor device can be computedif the sensitivity function of that parameter is known.Therefore, in this section we will describe the procedures forthe computation of the sensitivity functions for some of themost important parameters of the semiconductor device:terminal currents, threshold voltages, and transconductance. Inorder to make the linearization technique suitable for numericalimplementation on standard semiconductor device simulators,we present it in compact matrix form.

Let us write the discretized transport equations in vectorform [2]:

F(X,D,VG)=O, (4)

of the doping concentration. By solving (5) and (6) for thefluctuations of the terminal current we get:

(7)laO= -L_a FD+(ID )H DS

where gt is the transpose of column vector go that can befound by solving the following linear system of equations:

FXOga = IX' (8)

where FX denotes the transpose of matrix F'X. By comparing(1) and (7), it can be inferred that the sensitivity function of theterminal current is given at each mesh point by:

where F is a nonlinear vector function of the unknown "state"vector X, doping concentration vector D, and gate voltageV,. In the simulations presented in this work the transportmodel is the density-gradient model [8], and the componentsstate vector consist of the mesh point values of electrostaticpotential, electron and hole concentrations and quasi-Fermipotentials. Doping concentration vector consists of the meshpoint values of the doping concentration in the device, and canbe written as a sum of its average value (D) and fluctuationsD: D = (D) + D. Doping fluctuations induce fluctuations ofthe state variable: X = (X)+X, which can be computed byusing transport equations (4).

A. Fluctuations of Terminal CurrrentsLet Ia denote the current through terminal a of the

semiconductor device. To compute the sensitivity function ofIa it is convenient to write terminal currents as explicitfunctions of the state vector and doping concentration:Ia = Ia (X, D). The fluctuations of terminal currents Ia canbe found by linearizing this equation with respect to thefluctuating quantities:

SI = CL-( g FD ) + (IDa)i A ViA (9)

where AV, is the volume of the mesh cell i . The variances ofterminal currents can be evaluated by using (3), which, in

discretized form, reads: cI = g'aF-(ID) ] (D) ALj

where (D)i is the average doping concentration at location i

B. Fluctuations of Threshold VoltageThe technique for the calculation of the sensitivity function

of the threshold voltage in MOS device can be developedsimilarly to the technique for the calculations of SI .Fluctuations of the threshold voltage should be understood asfluctuations of the gate voltage VT = VG when the drain currentis constant (fluctuations are zero): 'Drain = 0. Thus, one canwrite the following system of equations:

FXX + FDD+F V =0

( X ) ( VD )

(5)

where I,' and I,' are the derivatives of jA with respect to thestate variable and doping concentration, while superscript tdenotes the transpose of the given vector. Since VG is constant,linearization of transport equations (4) gives:

where FV are the derivatives of the transport equations withrespect to VG . By solving this system we obtain the followingequation for the fluctuations of the threshold voltage:

V 9gDrain FD + (ID)a(1))VT gt.F *D, (11)

(10)

FxX + FDD=O,

where FX and FD are the derivatives of F with respect toXand D, respectively. All derivatives are computed at the givendc bias point and by assuming constant (non-fluctuating) values

where vector gDrain can be computed from (8) with a = Drain.

The sensitivity function of VT is given at each mesh point

by SV = S/(gD F ), where SI' are the mesh point

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(6)

= ( Xa )' a)t bla 'i. + (ID

SISPAD 2006 257

values of the sensitivity function of the drain current. As we

will show in the next section, product gt * FV is equal to thenegative value of the transconductance of the device (gm),which gives the following simple relation between thesuperposition coefficients of VT and 'Drain SV = -S/gmS

C. Fluctuations ofthe Transconductance

Transconductance can be computed as g. = ajIiDraIVG =

XDrai XG, where x is the solution of FxXG+FV =0. BY~G G

using the last two equations we obtain

gm=-raFn( ) FV =-gt Fv . Fluctuations of the dopingconcentration induce fluctuations of the transconductance. Thesensitivity function of the transconductance can be shown to beequal to the components of vector ftF^D where is thesolution of the following system of equations [9]:

FIt t = Bt IDrainXX~ gDrain -1X X G (12)Figure 1. Fully depleted, double-gate SOI (a) and MOSFET (b) devices used

in this study. Doping concentration al locations (1), (2), (3), and (4) are

changed and device parameters are computed.

and matrix B can be approximated by using:

B = FX(X+EXa>Fx (X)£

(13) -8._;

where £ is a small parameter and ID"aI" denote the Hessianmatrix of the drain current. In simulations, - can be chosenbetween 10-3 and 10-9. By computing matrix B using (13), weavoid the evaluation of second order derivatives of thetransport equations, which would require cumbersomeimplementation procedures for in standard device simulators.

III. ANALYSIS OF LINEARIZATION ERRORSAs mentioned in the introduction, the accuracy in the

computation of variances of parameters of semiconductordevices by using the linearization technique depends onwhether transport equations are linear with respect to dopingfluctuations. To test the validity of this assumption we considervarious semiconductor devices and carry out simulationexperiments in which we vary the doping concentration atdifferent points and compute the intrinsic parameters of thedevice (e.g. terminal currents, threshold voltage, small-signalparameters, etc.) as a function of the doping concentrations atthese points. We consider the following three devices withsimplified architectures: (a) a fully depleted, double-gate SOIdevice with the thickness of the semiconductor 10 nm, theoxide thickness 2 nm, and doping concentration of 1018 cm3 inthe channel [see Fig. 1(a)]; (b) a MOSFET device withimproved architecture (retrograde well and halo implantation)with oxide thickness of 3 nm, doping concentration in thechannel between 1016 CM3 and 10l8 CM3 [see Fig. 1(b)]; (c) a

p-njunction diode with the doping concentration at the cathode101 cm-3 built on a p-type substrate doped with 1016 cm-3.

-2x10l lxlO u lxlO 2x10

Doping concentration (cm3)

-2x10l lxlO lxlO 2x10 3x10

Doping concentration (cm3)

Figure 2. Dependence of the drain current and transconductance on the dopingconcentration at mesh points denoted by (1), (2), and (3) in Fig. l(a).

0o07

0.06

ff 005

- 0 04

0.03

_ 0 02

0 01

0.00

-0.01°0 lxio' 2x10

Doping concentration (cm-3)

5,0

0

7C,0

U)

H-r

-0.2610

-0.2613

-0.2616

-0.2619

-0.2622

-0.2625

5

4

3

2

-1

lxilo1 2x10 3x10

Doping concentration (cm-3)

0 1x10i 2x1018 3x1018

Doping concentration (cm-3)

Figure 3. Dependence of the drain current, transconductance, and threhsoldvoltage on the doping concentration at mesh points denoted by (1), (2), (3),

and (4) in Fig. 1(b).

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Location (3)

I -E Location (4)

K, A_

MOSFET Device

'I 7~--Fz 'x -x

Location (1) ° -l Location (2)- Location (3) >

Location (4)

SISPAD 2006 258

ci)00

ci)I

0

c.)

C)

1.095- )- in the depletion region

1.094 - in the n-type semiconductorin the p-type semiconductor

1.093 -

1.092 -

1.091

1.090

1.089-

1.08818-1.0x 0 -5.Ox101 0.0 5.Ox1 01 1 .0)x1o18

Doping concentration (cm 3)Figure 4. Current through a p-n junction diode as a function of the doping

concentration at different locations.

poly silicon oxideSource - Drain

-2E 10 1 |

220-X

0> 30640

70o -1280 -1490 30 60

Along channel direction (nm)

Fig. 5: Sensitivity function of VT of the MOSFET device represented in Fig.l(b). Blue areas represent regions highly sensitive to random doping fluctua-tions, while red areas represent regions less sensitive to doping fluctuations.

-,

EC:

16

14

12

10

8

6

4

2

Drain voltage (V)Figure 6. Drain current (continuous lines) and standard deviations of the drain

current (vertical bars) for the MOSFET device represented in Fig. 1(b).

The locations where the doping concentrations were variedare represented by numbers from (1) to (4) in Figs. 1(a)-(b).The results of the simulations are represented in Figs. 2, 3, and4 for the double gate SOI device, MOSFET, and p-n junctiondiode, respectively. There is a remarkable linear dependence ofthe parameters of the SOI and MOSFET on the doping

concentrations, which strongly supports the basic assumptionsof our linearization technique. This linearity accounts for thegood agreement that exists in the literature between thelinearization technique and the statistical methods [1,10] in thecase ofMOSFET devices. In the case of the p-n junction diodethe current characteristics are not linear for a large range ofvariation of the doping concentration. However, if thefluctuations of the doping concentration are small, thedependence of the terminal current on the dopingconcentrations remains linear. To test the accuracy of thelinearization technique in the case of p-n junctions we havegenerated 200 diodes with different doping concentrations butthe same average values of the doping and computed theterminal currents for given forward potentials across the diode.The statistical (Monte Carlo) techniques give a standarddeviation of the terminal currents of 0.26,uA, while thelinearization technique gives 0.31 ,uA; the average value of thedrain current is 1 uA in both cases. The relatively goodagreement between the two techniques suggests that thelinearization of transport equations is still a fairly goodapproximation for p-n junction diodes. Finally, the sensitivityfunction of the threshold voltage of the MOSFET device ispresented in Fig. 5. The drain current and the standarddeviations of the drain current for the DG-SOI devices arerepresented in Fig. 6.

In conclusion, the linearization technique provides apowerful and accurate tool for the analysis of random dopinginduced fluctuations in semiconductor devices. A remarkablylinear dependence was observed between the parameters of theMOSFET and SOI devices and doping fluctuations, whichsupports the assumptions of the linearization technique.

REFERENCES

[1] I.D. Mayergoyz and P. Andrei, "Statistical analysis of semiconductordevices," J. Appl. Phys., vol. 90, pp. 3019-3029, 2001.

[2] P. Andrei and I.D. Mayergoyz, "Quantum mechanical effects on randomoxide thickness and random doping induced fluctuations in ultrasmallsemiconductor devices ," J. Appl. Phys., vol. 94, pp. 7163-7172, 2003.

[3] P. Andrei and I.D. Mayergoyz, "Analysis of fluctuations in ultra-smallsemiconductor devices ," SISPAD 2003 Proceedings, pp. 95-98, 2003.

[4] A. Wettstein, 0. Penzin, E. Lyumkis, and W. Fichner, SISPAD 2004Proceedings, pp. 91.

[5] P. Andrei and I.D. Mayergoyz, "Analysis of fluctuations insemiconductor devices through self-consistent Poisson-Schrodingercomputations ," J. Appl. Phys., vol. 96, 2071-2079, 2004.

[6] P. Andrei and I.D. Mayergoyz, "Sensitivity of frequency characteristicsof semiconductor devices to random doping fluctuations," Solid-StateElectron., vol. 48, pp. 133-141, 2004.

[7] DESSIS Technical Manual, Release 9.0, ISE Integrated SystemsEngineering, 2003.

[8] M. G. Ancona and G. J. lafrate, "Quantum correction to the equation ofstate of an electron gas in a semiconductor," Phys. Rev. B, vol. 39, pp.9536-9540, 1989.

[9] I.D. Mayergoyz and P. Andrei, "Random doping fluctuations of small-signal parameters in nanoscale semiconductor devices," Int. J. HighSpeed Electronics and Systems, vol. 12, pp. 345-356, 2002.

[10] A. Asenov, A. R. Brown, J.H. Davies, S. Kaya, and S. Slavcheva,"Simulation of Intrinsic Parameter Fluctuations in Decananometer andNanometer-Scale MOSFETs ," IEEE Trans. Electron Devices, vol. 50,pp. 1837-1852, 2003.

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