Modeling of Discrete Dopant Effects on Threshold

Voltage Shift by Random Telegraph Signal

Ken'ichiro Sonoda, Kiyoshi Ishikawa, Takahisa Eimori, and Osamu TsuchiyaProduction and Technology Unit, Renesas Technology Corp., Mizuhara 4-1, Itami, Hyogo 664-8641, Japan

Email: sonoda.kenichiro@renesas.com, Telephone: +81-72-784-7325, Fax: +81-72-780-2693

Abstract- This paper discusses the discrete channel dopanteffects on the threshold voltage shift by random telegraph signal(RTS) in MOSFETs. Appropriate grid spacing to incorporatediscrete dopant effects in three dimensional device simulationis addressed to obtain consistent results with continuum limit.Considering discrete dopant effects, the threshold voltage shiftof MOSFETs by RTS follows the log-normal distribution, whilethe threshold voltage itself follows the normal distribution. Ananalytical model for the distribution of the threshold voltage shiftis also presented. The threshold voltage shift by RTS will becomea serious concern in 50nm flash memories and beyond.

I. INTRODUCTIONThe random telegraph signal (RTS)[1] caused by trapping

of a single carrier at the Si/SiO2 interface will becoming aserious issue for MOSFETs with sub 100nm dimensions asa source of, not only low-frequency noise in analog circuits,but also functional error in digital logic circuits and memories.Flash memories, for example, are susceptible to RTS[2] owingto thick gate dielectrics.

The threshold voltage shift AVth which is related to the cur-rent change AId by RTS as -AId/gm is sometimes anoma-lously larger than the value estimated by the conventionalformula q/CoxWeffLeff[3], where gm is the transconductance,q is the elementary charge, Co, is the capacitance of the gatedielectric per unit area, and Weff and Leff are the effectivechannel width and the effective channel length, respectively.A probable explanation is that the strategically located traps inthe inhomogeneous channel influence the magnitude and thespreading of the RTS amplitudes[1][3][4].

In this paper, the threshold voltage shift by RTS is ana-lyzed using three dimensional device simulation consideringinhomogeneous carrier concentration in the channel causedby random discrete dopant[5]. An analytical model for thedistribution of the threshold voltage shift is also presented.

II. MODELING

Before simulation of the threshold voltage of MOSFETsis done, a grid spacing dependence of conduction currentis investigated to assess consistency between continuous anddiscrete cases. The conduction current in p-type silicon withthe average acceptor concentration NA is simulated. Uniformrectangular grid with spacing d and constant mobility are usedthroughout this work. In the discrete case, discrete randomvariables following the Poisson distribution with the averageNAd3 are used for the number of dopant atoms associatedwith each nodes.

Majority carrier current in the discrete case decreases as thegrid spacing decreases as shown in Fig. 1, because the depthof the potential wells which originate in ionized dopant atomsbecomes deeper and more majority carriers are trapped inthe potential wells[7]. Minority carrier current shows oppositedependence on the grid spacing because of the mass actionlaw. If the rectangular region with constant charge densityis represented by a sphere of the radius rimp with the samevolume of a grid as shown in Fig. 2, the depth of the potentialwells is given by EB = (3/2)(q/47wErimp). The radius rimp isobtained by solving (4/3)7rr p = d3.The depth EB of the potential wells is regarded as the

ionization energy of dopant atoms. The simulation result inthe continuous case using the energy EB given by the aboveformula for the ionization energy EBO agrees with the averagevalue of the simulation results in the discrete case as shownin Fig. 1. In order to obtain consistent simulation resultsbetween continuous and discrete cases, the width d of therectangular region in the discrete case should be determined sothat the energy EB is equal to the ionization energy of dopantatoms in the continuous case. When B is used as acceptorsin silicon, the width d = 6.6nm should be used to matchthe ionization energy with the measured value 45meV[8]. Iffiner grid spacing is necessary to express spacial variation, thedopant atom is expanded so that the energy EB is equal to theionization energy.

III. RESULTS AND DISCUSSION

Discrete dopant effects on the threshold voltage ofnMOSFETs and its shift by RTS are analyzed using threedimensional device simulation. The threshold voltage shift isdefined as the difference between threshold voltages with andwithout an electron captured in an interface trap located at thecenter of the channel as shown in Fig. 3.The threshold voltage Vth in the discrete case fluctuates

around the value in the continuous case as shown in Fig. 4,and follows the normal distribution as shown in Fig. 5 (a).The standard deviation is well described by the analyticalmodel[9]. The threshold voltage shift AVth in the discretecase also fluctuates around the value in the continuous caseand follows the log-normal distribution as shown in Fig. 5 (b).An analytical model for the fluctuation of AVth is derived

to explain the origin of the log-normal distribution. Moderateinversion is assumed because threshold voltage is discussedin the following. Using the total number of carriers in the

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channel N, the drain current of MOSFETs is expressed asId = q,uNVds/L'ff[10], where ,u is the carrier mobility andVds is the drain-source voltage. Suppose the channel is dividedinto small cells[6] with the area At = 7rr , in which theconductivity is affected by trapping of a single carrier asshown in Fig. 6. The local threshold voltage Vthi of thecell i is defined so that the number of carriers in the cellis expressed as Ni = Noexp(q(Vgs -Vthi)/nkT), whereVg, is the gate-source voltage, n(= 1 + Cdep/Cox) is thesubthreshold factor, Cdep is the depletion layer capacitanceper unit area under the channel, k is the Boltzmann constant,T is the temperature, and NO is a constant. By trapping ofa single carrier at the trapping site in the cell j, the localthreshold voltage of the cell increases by AVthj q/CoxAt,and the current change is AId = (q,uVdSNO/L2ff) exp(q(Vgs-Vthj)/nkT)(exp(-qAVthj /nkT) - 1). The correspond-ing threshold voltage shift is AVth = -AId/gm g(q/CoxWeffLeff)Ntotal/ Ei exp(q(Vth -Vthi)/nkT), whereNtotal _ WeffLeff/At is the total number of the cells. Thelogarithm of the AVth is, therefore,

log(A Vth) log ( Cox WeffLeff) qkT(hj -Vthave ), (1)

where Vthave is the average of the local threshold voltage.The derived model Eq. (1) suggests that the threshold volt-

age shift AVth becomes larger than the conventional formulaqlCOxWeffLeff if the local threshold voltage Vthj of the cellthat has a trapping site is lower than the average local thresholdvoltage, and vise versa. The strategically located trap at the sitethat has lower local threshold voltage and higher local carrierconcentration enlarges the threshold voltage shift by RTS.The model Eq. (1) also suggests that AVth obeys not normalbut log-normal distribution if the local threshold voltage Vthjfluctuates in normal distribution. The standard deviation ofthe local threshold voltage fluctuation is related to the surfacepotential fluctuation as =Vth= noo,,. The standard deviationof log(AVth) is, therefore, expressed as

q(1og(AVth) kT S (2)

The power spectrum of the surface potential fluctuation byrandom discrete dopant is calculated using the approximateGreen's function for the electrostatic potential in the depletionlayer[lI]. The standard deviation of the surface potential isobtained by integrating the power spectrum as

(70sNA ( q N\ fmaxi Cqw

d 3

4'T KCOX+FS) ]qminQ q2d (3)

47FQt\\Eo ESq J

CoV 47TQs cox + csi

x arctan (qmax -arctan (qmin)

where NA is the dopant concentration, w is the depletion layerwidth, and Q, = (C0, + Cdep)/(E(ox + Esi ). The minimumwavenumber, qmin, is determined by the longer of the channel

width and the channel length as shown in Fig. 7. The max-imum wavenumber, qmax, is determined by the diameter ofthe area in which the conductivity is affected by trapping ofa single carrier as shown in Fig. 6. The term 1 e-qw wasreplaced with a constant co,, in order to integrate the powerspectrum analytically. The constant co,, is treated as a fittingparameter and set to 0.2 in this work.

The proposed analytical model defined by Eqs. (2), (4) welldescribes not only the numerical simulation results as shownin Fig. 5 (b), but also the measured results as shown in Fig. 8.Substrate doping concentration dependence of (71og(AVth) isalso expressed by the proposed model. Note that (71og(VAth)is proportional to N)1/2 as shown in Fig. 9 (b), while (Vth isproportional to NA/4[9] as shown in Fig. 9 (a).

Finally, an impact of the threshold voltage shift on flashmemories and SRAMs is estimated by the proposed model asshown in Fig. 10. The coupling coefficient of flash memoriesis assumed to be acg = 0.6. As the dimension shrinks, both50 percentile and 99.9 percentile increase in proportional to1/WeffLeff. For 50nm-flash, 99.9 percentile exceeds 100mV,which suggests that the threshold voltage shift by RTS shouldbe considered in flash memory design because the marginfor the threshold voltage is only several hundreds mV inmulti-level-cell flash[2][12]. In contrast, 99.9 percentile for50nm-SRAM is far below lOmV and little attention would benecessary for SRAMs.

IV. CONCLUSIONThe discrete dopant effects on the threshold voltage shift of

MOSFETs by RTS have been discussed. Considering discretedopant effects, the threshold voltage shift by RTS follows thelog-normal distribution. An analytical model for the distri-bution of the threshold voltage shift has been presented andconfirmed by using numerical simulation results and measuredresults. The proposed model predicts that the threshold voltageshift by RTS should be considered in the design of flashmemories of 50nm and beyond.

ACKNOWLEDGMENTThe authors would like to thank Shun'ichi Narumi, Yoshi-

hiro Ikeda, Makoto Ogasawara, Shiro Kamohara, YutakaOkuyama, Hideaki Kurata, Yoshitaka Sasago, and HitoshiKume for their technical supports and discussions.

REFERENCES

[1] K. S. Ralls et al., Phys. Rev. Lett., vol.52, p. 228, 1984.[2] H. Kurata et al., Symposium on VLSI Circuits, 13.3, 2006.[3] M. J. Uren etal., Appl. Phys. Lett., vol.47, p. 1195, 1985.[4] L. K. J. Vandamme etal., Solid-State Electron., vol. 42, p. 901, 1998.[5] A. Asenov et al., IEEE Tarns. Electron Devices, vol. 50, p. 839, 2003.[6] R. W. Keyes, IEEE J. Solid-State Circuits, vol. SSC-10, p. 245, 1975.[7] N. Sano et al., IEDM Tech. Dig., p. 275, 2000.[8] S. M. Sze, Physics of Semiconductor Devices, second edition, John Wiley

& Sons, 1981.[9] P. A. Stolk et al., IEEE Trans. Electron Devices, vol. 45, p. 1960, 1998.[10] M-. H. Tsai et al., IEEE Electron Device Lett., vol. 15, p. 504, 1994.[11] G. Slavcheva et al., JAppl. Phys., vol. 91, p. 4326, 2002.[12] S. Lee et al., ISSCC Dig. Tech. Papers, p. 52, 2004.

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Lx=Ly=Lz=0.05pm, Na=5el7cm 3, V=O.O1V

0.01 0.02 0.03d (gim)

Fig. 1. Majority (upper) and minority (lower) carrier current flowing througha p-type silicon cube as a function of grid spacing d using three dimensionaldevice simulation. Two electrodes are placed on the opposite sides of the cube.EBO is the ionization energy used in the calculation of incomplete ionizationof dopants. rimp is a function of d. The band degeneracy factor is set to 1.The averages and the ±3o-deviations of 30 distinct samples are shown forthe discrete dopant case.

A _x-- ,/

x

(a)

A

x

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(b)Fig. 2. Charge distribution and corresponding potential distribution. (a) rect-angular grid, (b) sphere approximation ((4/3)-FO p = d3).

imp

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Fig. 4. Simulated Vg-Id characteristics of MOSFETs. 30 distinct samplesare used for the discrete dopant case.

99.9999999.999999.99999.9999.9

999590

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1051

0.10.01

0.0010.0001

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Weff/Leff=50nm/45nm, tox=9nm, Na=5el 7cm Vd=0.05V

0-2 1o-0

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3.0

2.0

1.0

0.0

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(b)Fig. 5. The Cumulative distribution function of the simulated (a) thresholdvoltage Vth, and (b) threshold voltage shift AVth by RTS for 30 distinctsamples. A trap is located at the center of the channel. The parameter co, =

0.2 is used in the analytical model for AVth.

1-4244-0404-5/06/$20.00 2006 IEEE

:f= ~~~~~~ -------------

---CONTINUOUS (EBU=ueV)- CONTINUOUS (EBO=0.045eV)

CONTINUOUS (EBO=(3/2)(q/4icFRimp))0 DISCRETE (EBO0=OeV)

3e-06

2e-06

1 e-06

Oe+001 e-20

8e-21

6e-21

a,

4e-21

2e-21

Oe+OO0.00

- CONTINUOU(-E--BO-0eV)

CONTINUOUS (EBO0=OeV)----CONTINUOUS (EB0=0.045eV)

--- CONTINUOUS (EBO=(3/2)(q/4icFRimp))0 DISCRETE (EBO0=OeV)

CONTINUOUS (R-=6.661 V, a=0.OOOV)0 - - O DISCRETE (Ri=0.659V, a=0.065V)--- ANALYT. MODEL

II

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1

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SISPAD 2006 117

energy

IIII

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mobile carriers

100

z 10

q

rtrapped electron

-2

10Fig. 6. Schematic diagram of the mobile carrier concentration around thetrapped electron.

100GATE

2*rc

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SOR DRAIN.~~Lf g

, ,... ,.w. ..........

Fig. 7. Schematic diagram of the minimum and the maximum wavenumbersof the surface potential which affects the local threshold voltage fluctuation.

< 101c,)0

1 015

Weff/Leff=50nm/45nm, tox=9nm, Vd=0.05V

O DEV. SIM. (CONFI. LEVEL=99%)---- ANALYT. MODEL

I'll~~~~~~, -'_- I

,,-(5-)-" . _ _~~~10 16 1 01

Na (cm )

(a)

10 1C

O DEV. SIM. (CONFI. LEVEL=99%)-- ANALYT. MODEL

'Ii_I -

I -I

I-

0 16 10107Na (cm )

Weff/Leff=50nm/45nm, tox=9nm, Vd=0.05V

10 1 0

Weff/Leff=24nm/50nm, tox=9nm, Na=5el7cm~~~~~~~~~~~i .. .....

10 10o

(b)Fig. 9. The simulated standard deviation of (a) the threshold voltage Vth, and(b) the logarithm of the threshold voltage shift AVth by RTS as a function ofsubstrate doping concentration. Error bars indicate 99% confidence intervals.

100

3.0

1o-02.0

1.0 10-2

1OuAVth (V)

Fig. 8. Measured and simulated threshold voltage shift. Poisson distributionis assumed for the number of traps at the interface with the averageNt (kTlq)Weff Leff, where Nt is the trap density per unit area and unitenergy.

101 102Weff=Leff (nm)

Fig. 10. The 50 and 99.9 percentiles of the simulated threshold voltage shiftcaused by single electron RTS.

1-4244-0404-5/06/$20.00 0 2006 IEEE

99.99

99.9 k

99 h

L-C) 95 F

90 F

80

70

(O MEAS. (CONFI. LEVEL=99%)---- SIM. (Nt=5e10cm-2eV-')

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SISPAD 2006 118