An Improved 3D Monte Carlo Simulation ofReaction Diffusion Model for Accurate Prediction

on the NBTI Stress/RelaxationSeongwook Choi and Young June Park

School of Electrical Engineering and Computer ScienceSeoul National University

Seoul, KoreaEmail: church7@snu.ac.kr

Chang-Ki Baek and Sooyoung ParkDepartment of Creative IT Excellence Engineering

and Future i-LabPOSTECH

Pohang, Korea

Abstract—We developed the 3D Monte Carlo simulation frame-work for the Reaction Diffusion model of NBTI. For the firsttime, we report that the RD model can predict the experimentalfeatures of NBTI stress/relaxation when the microscopic prop-erties are considered in 3D space such as the capture cross-section, density and reaction energy of Si-H bonds. We expectthat our simulation framework would be effectively used forthe accurate prediction of trap generation/relaxation in generalsemiconductor/insulator systems.

Index Terms—NBTI, Reaction Diffusion model, Monte CarloSimulation, Relaxation Characteristics, Discrete trap, Langevinequation

I. INTRODUCTION

Negative Bias Temperature Instability (NBTI) is one ofthe most serious challenges in the scaled VLSI devices [1].The Reaction Diffusion (RD) model was widely used for theprediction of the interface trap generation under NBTI afterits first proposal in 1977 [2]. However, it has been seriouslycriticized because of its inaccuracy in predicting the fastmeasurement results on relaxation phases [3], [4], making theNBTI model highly controversial until now [4], [5]. Therefore,it is worthwhile to verify the competing models for NBTIin order to rule out such controversies. In this context, weinvestigate recent arguments [5] on the RD model in orderto study the validity of the model by proposing an advancedsimulation framework.

The “failure” of the conventional RD model [4], [6], [7] isdescribed in Fig. 1. In the figure, the universal fitting [6] fromthe RD model is shown with the own measurement data. Theproblem is that the 50% of the interface trap is recovered atthe point where the stress time is same as the relaxation timewhile the actual measurement results show much less recovery.Moreover, the recovery is completed in ∼6 decades accordingto RD model but the actual recovery is slower.

So far, tuning of any parameters of the RD model could notresolve the problems shown in Fig. 1. Recently, the effectivediffusion constant has been speculated in the relaxation phaseto resolve the problem on the RD model [5]. According to thespeculation [5], H should laterally diffuse (hover) in order to

10-4 10-3 10-2 10-1 100 101 102 103 1040.00

0.25

0.50

0.75

1.00

range forNBTI recovery

much less than50% recovery

at trelax=tstN

it/Nit(t

r=0)

trelax/tst

actual recoveryis slower

50% recoveryat trelax=tst

measurement (tst=5000s) measurement (tst=1500s) Analytic RD model

Fig. 1. NBTI relaxation phase where the data is scaled by the universalrelaxation law [6]. The measurement result (symbols) are much slower thanthe simulation result of the analytic RD model (solid line). For measurementdata, the Subthreshold Swing method is used to extract the interface trapdensity. The range for NBTI recovery reported by the literatures are alsoshown.

find the unsaturated Si- bond for recombination as describedin Fig. 2. Since the diffusing trajectory become lager dueto the lateral diffusion, the effective diffusion constant forthe recovery phase becomes smaller resulting in the slowerrecovery. In order to justify this speculation, some authors trieda stochastic simulation in a 2D space using equally spaceddangling bonds but every attempt have been failed to correctthe relaxation behavior [8], [9]. In this work, we developa 3D Monte Carlo Reaction Diffusion (MCRD) frameworkas suggested by the authors [10] and conduct the stochasticNBTI simulation to consider the random nature of the Si/SiO2

interface without spatial discretization. Contrary to the previ-ous work [8], [9], we could observe that the lateral diffusionis naturally included. With our 3D-MCRD framework, wedemonstrate that the RD framework can accurately predictboth NBTI stress/relaxation phases.

SISPAD 2012, September 5-7, 2012, Denver, CO, USA

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185 ISBN 978-0-615-71756-2

interface

σc

SiSiSi

HH

Si

Si atom with dangling bond

oxide

H

H

HH

H

H

hover

interface

trappassivated

interface trap

rnn

H

Fig. 2. Schematic representation of the Si/SiO2 interface. The trajectory of Hbecomes longer than the 1D case because hydrogen have to find unsaturatedSi- bond, making the effective diffusion constant becomes smaller.

II. 3D MCRD MODEL

In order to trace the trajectory of each H particle, we adoptthe Langevin equation [11] which describes the Brownianmotion of the particle in a 3D space as

du

dt= −Γu+A(t) +K/m (1)

where u is a velocity, m is a mass of the particle, A representsthe white noise and K is a local electric field. In the aboveequation, Γ is a dynamical friction which can be written as

Γ =kBT

mD(2)

where D is the diffusion constant of the particle, kB is theBoltzmann constant and T is temperature.

In order to solve (1), Chandrasekhar [11] derived equationswhich can be used for MC simulations. The derivation can besuccessfully adopted for many systems including the mixedparticle MC simulation on the semiconductor problems [12].We follow Chandrasekhar’s derivation with notations in [12]to solve the position and velocity of the hydrogen particle inthe oxide if the initial values of the position (x0) and velocity(u0) at time t0 are given. For 1D case, the position and thevelocity at t = t0 +∆t, are determined as follows [11], [12]:

x = x0 + Γ−1(1− e−Γ∆t)(u0 −K

Γ) +

K

Γ∆t+ σxω1 (3)

u = u0e−Γ∆t +

K

Γ(1− e−Γ∆t) + xσuω1 + σu|xω2 (4)

where

σx = F 1/2, σu = G1/2, σu|x = G1/2(1− H2

GF)1/2 (5)

and F, G and H can be written as follows:

F = αΓ−3(2Γ∆t− 3 + 4e−Γ∆t − e−2Γ∆t) (6)

G = αΓ−1(1− e−2Γ∆t) (7)

H = αΓ−2(1− e−Γ∆t)2 (8)

0.2 0.4 0.6 0.8 1.00.00.2

0.4

0.6

0.8

1.0

1.2

0.00.20.40.60.81.0

final point

z (

a.u

.)

y (a.u.)

x (a.u.)

initial point

Fig. 3. The trajectory of a sample particle in the 3D space, calculated bythe Langevin equation (1). The Brownian motion of the hydrogen particle isobserved.

In the above equations, ω1 and ω2 are Gaussian randomvariables with unit variance so that x(t) and u(t) are deter-mined stochastically. The equations described above can beeasily extended to the 3D case for our 3D MCRD model. Therandom 3D motion of a sample hydrogen particle is shown inFig. 3 implemented by (3) and (4).

At the Si/SiO2 interface, the Si-H bond can be dissociatedor recovered by the following reaction:

Si−Hkf−−⇀↽−−kr

Si·+H (9)

The dissociation reaction probability (kf ) for each Si-H bondduring ∆t is

kf∆t = eH exp(Ef · g(σf )/kBT ) (10)

and the recovery reaction probability (kr) during ∆t is

kr∆t = CHU(σc − rnn) exp(Er · g(σr)/kBT ) (11)

where eH is a dissociation constant, CH is a recovery constant,rnn is the distance between H and unsaturated Si bond, σc isthe capture cross-section of the unsaturated Si bond, U is aunit step function, Ef and Er are activation energies for thereactions and g is the Fermi-derivative function which canrepresent the Gaussian distribution with variance σc and σr.

In summary, the RD equations [2], [5] are solved by theMC method in 3D using (3) and (4) for diffusion; (10) and(11) for reaction at the interface to obtain the interface trapnumber (Nit).

III. SIMULATION RESULT

We conduct the 3D MCRD simulation for NBTI and observethe behaviors of individual random traps as shown in Fig. 4.For the Monte Carlo simulation, the number of particles isfixed to 5000 and the area of device is scaled according to thegiven Si-H density. The hydrogen atoms (H 0) are assumed tobe diffuse out while the hydrogen molecules (H2) are actualdiffusing species [5] [8]. Since the difference of the H and H2model gives a slightly different result in stress phase only [5],

186

0 50 1000

50

100 W

(nm

)

L (nm)

1s

0 50 1000

50

10010s

W (n

m)

L (nm)

0 50 1000

50

100200s

W (n

m)

L (nm)0 50 1000

50

100325s

W (n

m)

L (nm)

Fig. 4. 2D discrete distribution of the interface trap under NBTI stress (1,10, 200s) and relaxation phase (325s).

10-3 10-2 10-1 100 101100

101

102

103

104

10-3 10-2 10-1 100 101 1020.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 10

t1

t0.25

Nit (#

)

time (s)

t0.25

t0.43

Dit0=2.00×1014 cm-2

Dit0=5.00×1013 cm-2

Dit0=1.25×1013 cm-2

Dit0=5.56×1012 cm-2

Dit0=2.00×1012 cm-2

RD model (1D)

(b)(a)

decrease Dit0

N

it / Nit (tr =0)

time (s)

previouswork [8] c=3.5

Fig. 5. 3D MCRD simulation results for the stress phase (a) and the relaxationphase (b) with various Dit0.

the assumption does not distort the recovery behavior whichis our main concern.

In order to demonstrate the correction of the 1D RD model,we investigate the effect of the total density of the Si- bond(Dit0) and the σc in (11) which are expected to mostly affectto the lateral diffusion in terms of the average distance betweenSi- bonds and the free flight time of H, respectively.

The 3D MCRD simulation results on the stress and re-laxation phase are shown in Fig. 5 with various Dit0 (σc

is set to 3.5 A). We assume the homogeneous Si-H bonds(σf = σr = 0eV ) same as the assumption in the conventionalRD model. When Dit0 is 2.00× 1014cm−2, the characteristicreaction limited region (Nit∝tn with n = 1) and diffusionlimited region (n = 0.25) are well reproduced by the 3DMCRD simulation in contrary to the previous attempt [8],[9] (refer Fig. 5(a)). Therefore, both the microscopic and themacroscopic model are consistent in Nit for the NBTI stress

10-3 10-2 10-1 100 101100

101

102

103

104

10-2 10-1 100 101 1020.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

decrease c

Nit (#

)

time (s)

t0.25

t1

decrease c

Nit / N

it (tr =0)

time (s)

c=2.5A c=3.5A c=4.5A c=5.5A RD model (1D) Dit0=5.56×1012 cm-2

(a) (b)

Fig. 6. 3D MCRD simulation results for the stress phase (a) and the relaxationphase (b) with various σc.

phase. Note that n of 0.25 can be coverted to n of 0.16 whichis observed in experiments, when the H2 is considered asdiffusing species. Interestingly, as Dit0 decreases, the recoverybecomes slower as shown in Fig. 5(b). When Dit0 is lessthan 1.25×1013cm−2, the recovery becomes slower than thatpredicted by the 1D RD model. It can be explained in termsof the average distance; as the Dit0 decreases, the averagedistance between Si-H bonding increases. Hence, the hydrogenparticle should fly more to find the unsaturated Si- bond andthus the effective diffusion constant become smaller. In sum,an additional lateral diffusion is enhanced so that the recoverybecomes slower as the Dit0 decreases. Notwithstanding therecovery behavior is in the reasonable range when Dit0

is less than 1.25×1013cm−2, the stress behavior becomesinconsistent (n > 0.25) under such range of Dit0 as shown inthe inset of Fig. 5(a).

The simulation results with various σc are shown in Fig. 6under Dit0 of 5.56×1012cm−2. When σc is smaller than 3.5A,the recovery characteristics enter into the proper range. This isbecause the free flight time of hydrogen becomes longer so thatthe effective diffusion constant is again decreased. However,the reduction of σc further increases n in the stress phasewhich is contrary to the experimental observation. To sumup, the 3D MCRD simulation with the homogeneous Si-Hbond reveals the followings; 1) As the speculation in [5], thelateral diffusion do affect to the recovery characteristics. 2)the recovery speed of the RD model are influenced by themicroscopic parameters such as Dit and σc. 3) Even thoughthe recovery can be corrected, there is no combination of Dit

and σc which satisfy the experimental features of both stressand relaxation phases.

Now, the inhomogeneity in Si-H bonds [13] in (10) and(11) (σf = 0.06eV , σr = 0.07eV ) are considered for the3D MCRD simulation and data for Nit with various Dit0 andσc are shown in Fig. 7 and Fig. 8, respectively. For all theconditions of Dit0 and σc, we found that the slower relaxationfollowed by the appropriate stress behavior (n∼0.25), whichagrees well with the criteria of both stress/relaxation phases

187

10-3 10-2 10-1 100 101

102

103

3x103

10-3 10-2 10-1 100 101 1020.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Nit (#

)

time (s)

t0.25

c=3.5

without diffusion(well-based model

w/ small dispersion)

Nit / N

it (tr =0)

decrease Dit0

time (s)

Dit0=5.00×1013 cm-2

Dit0=1.25×1013 cm-2

Dit0=5.56×1012 cm-2

Dit0=3.13×1012 cm-2

RD model (1D)

(a) (b)

Fig. 7. 3D MCRD simulation results for the stress phase (a) and therelaxation phase (b) with various Dit0. The dispersion of the reaction energyare considered as σf = 0.06eV in (10) and σr = 0.07eV in (11).

10-3 10-2 10-1 100 101

102

103

104

10-3 10-2 10-1 100 101 102 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Nit (#

)

time (s)

t0.25

decrease c

1 10

t0.25

t0.20

Dit0=5.56×1012 cm-2

decrease c

Nit / N

it (tr =0)

time (s)

c=3.1A c=4.1A c=5.1A c=6.1A RD model (1D)

(a) (b)

Fig. 8. 3D MCRD simulation results for the stress phase (a) and the relaxationphase (b) with various σc. The dispersion of the reaction energy are consideredas σf = 0.06eV in (10), σr = 0.07eV in (11).

(refer Fig. 1). This is in contrast to the well-based model [4](an alternative model for NBTI) which cannot reproduce thefeatures on the interface trap with reasonable values [5], [13]for σf and σr as shown in Fig. 7 by the gray line. Hence, itcan be inferred that physics from competing NBTI models (thediffusion from RD model and the dispersion of reaction energyfrom well-based model) should be simultaneously consideredfor an accurate NBTI model covering the stress and relaxationphases.

IV. CONCLUSION

In order to resolve the controversies on the RD model,we developed the 3D MCRD simulation framework which isa microscopic version of the analytic RD model. Since ourMCRD framework can trace the motion and reaction of eachhydrogen particle stochastically, we can verify the recentlyraised speculation on the RD model including the conceptof lateral diffusion [5]. For the first time, we report that theRD model is indeed influenced by the lateral diffusion and

the recovery behavior can be adjusted to fit the experimentalfeatures when Dit0 and σc are considered. However, we cannotadjust the parameters of the MCRD model to explain bothstress and relaxation phases properly, which means that thepresent form of the RD model is insufficient even if the lateraldiffusion is considered. Interestingly, when the dispersion ofreaction energy of Si-H bonds (σf and σr) are considered, bothstress and relaxation phases are accurately predicted underthe MCRD framework. Our work calls more investigations tojustify whether the 3D MCRD model can accurately predictthe numerous data on NBTI including AC/DC, temperature,voltage dependencies, etc. We expect that the 3D MCRDsimulation framework would be a vital tool for exploring thevalidity of the RD based model.

ACKNOWLEDGMENT

This work was supported by the BK 21 program, SamsungElectronics Co. Ltd, National Research Foundation of KoreaGrant funded by the Korean Government (2011-0027321) andthe Center for Integrated Smart Sensors funded by the Ministryof Education, Science and Technology as Global FrontierProject (CISS-2011-0031845).

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