A relationship between statistical time to breakdown distributions and pre-breakdownnegative differential resistance at nanometric scaleR. Foissac, S. Blonkowski, M. Kogelschatz, and P. Delcroix

Citation: Journal of Applied Physics 116, 024505 (2014); doi: 10.1063/1.4888183 View online: http://dx.doi.org/10.1063/1.4888183 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analysis and modeling of resistive switching statistics J. Appl. Phys. 111, 074508 (2012); 10.1063/1.3699369 Nanoscale probing of dielectric breakdown at SiO 2 / 3 C-SiC interfaces J. Appl. Phys. 109, 013707 (2011); 10.1063/1.3525806 Prediction of breakdown in ultrathin SiO 2 films with fractal distribution of traps J. Appl. Phys. 106, 063708 (2009); 10.1063/1.3212986 Statistics of electrical breakdown field in Hf O 2 and Si O 2 films from millimeter to nanometer length scales Appl. Phys. Lett. 91, 242905 (2007); 10.1063/1.2822420 Photo-enhanced negative differential resistance and photo-accelerated time-dependent dielectric breakdown inthin nitride-oxide dielectric film Appl. Phys. Lett. 78, 3241 (2001); 10.1063/1.1373409

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A relationship between statistical time to breakdown distributionsand pre-breakdown negative differential resistance at nanometric scale

R. Foissac,1,2 S. Blonkowski,1 M. Kogelschatz,2 and P. Delcroix1

1STMicroelectronics, 850 rue Jean Monnet, 38926 Crolles Cedex, France2Univ. Grenoble Alpes, LTM, F-38000 Grenoble, France and CNRS, LTM, F-38000 Grenoble, France

(Received 23 May 2014; accepted 27 June 2014; published online 11 July 2014)

Using an ultra-high vacuum Conductive atomic force microscopy (C-AFM) current voltage,

pre-breakdown negative differential resistance (NDR) characteristics are measured together with

the time dependent dielectric breakdown (TDDB) distributions of Si/SiON (1.4 and 2.6 nm thick).

Those experimental characteristics are systematically compared. The NDR effect is modelled by a

conductive filament growth. It is showed that the Weibull TDDB statistic distribution scale factor

is proportional to the growth rate of an individual filament and then has the same dependence on

the electric field. The proportionality factor is a power law of the ratio between the surfaces of the

CAFM tip and the filament’s top. Moreover, it was found that, for the high fields used in those

experiments, the TDDB acceleration factor as the growth rate characteristic is proportional to the

Zener tunnelling probability. Those observations are discussed in the framework of possible break-

down or forming mechanism. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4888183]

I. INTRODUCTION

Dielectric breakdown is known to be a key issue in the

reliability of microelectronic devices and understanding its

behaviour is crucial to get reliable MOS structures and to

extrapolate their lifetime. Dielectric breakdown is also a key

issue for oxides based Resistive Random Access Memories

because it constitutes the first step in forming process.

Dielectric breakdown models are generally based on the for-

mation of a conductive percolation path through the oxide by

the generation of randomly distributed defects.1,2 These mod-

els reproduce satisfactorily the experimental time dependant

dielectric breakdown (TDDB) distributions observed when a

constant voltage stress (CVS) is applied. It has been shown,

for example, that the shape factor of the TDDB Weibull dis-

tribution is proportional to the gate oxide thickness.3 The

shape factor dependence on thickness is not the same if one

considers that the defects are uniformly distributed through

the oxide thickness or in a non-uniform way.4 In the latter

case, we will, maybe abusively, speak of invasive percola-

tion. It is also well known that when the voltage stress is

stopped before the dielectric breakdown occurs, a stress

induced leakage current (SILC) appears.5 SILC has been also

extensively studied and modelled considering “classical” per-

colation6,7 as well as invasive percolation8 to cite some exam-

ples among the wide literature on the subject.

Conductive atomic force microscopy (C-AFM) under

ultra-high vacuum (UHV) is a well-adapted tool to study

TDDB9,10 and SILC7,11 characteristics. In particular, it has

been experimentally evidenced that the acceleration factor

and Weibull shape factor of TDDB distributions measured

with C-AFM are the same than their counterparts measured

on standard devices.9 Moreover, thanks to the reduced tun-

nelling current due to a small tip/sample contact area it has

been possible to observe a pre-breakdown negative differ-

ential resistance (NDR)11 effect when the voltage sweep

direction is reversed just before reaching the dielectric

Breakdown voltage (Vbd). The backward current-voltage

characteristic corresponds to the SILC in this case. Since

the NDR characteristic accounts dynamically for what

appends before breakdown and for the SILC, it is legitimate

to look for a relationship between TDDB statistical distribu-

tion and NDR characteristics. This is the purpose of this

paper.

With this end in mind, we first compare the parameters

extracted from ramped current-voltage measurements lead-

ing to the NDR effect to the parameters extracted from

TDDB statistical distributions. Practically, I-V characteris-

tics will be recorded at different ramp speeds and TDDB sta-

tistical distributions will be recorded at different CVS by

C-AFM under UHV. This will be done on p-Si/SiON sam-

ples with two SiON layer thicknesses, 1.4 nm and 2.6 nm.

The parameters of the current voltage characteristic will be

extracted via a modified version of a model presented in

Ref. 11. Those parameters will be then compared to the

Weibull shape and scale parameters of the TDDB distribu-

tion and, in particular, to the voltage acceleration factor.

Those extracted values will be discussed and gathered con-

sistently and a possible breakdown mechanism based on

invasive percolation will be discussed.

II. EXPERIMENTAL DETAILS

In the present study, C-AFM electrical tests were carried

out on a 1.4 nm and a 2.6 nm SiON layer on Silicon substrate

(5.07� 1015 cm�3, p-type).

The nitrided silicon oxide layers have been prepared in

the following way: after a HF-SC1 clean, a thin SiON inter-

facial layer was formed by oxidation of the silicon substrate

with an 800 �C rapid thermal oxidation (RTO), followed by

Inductively Coupled Plasma (ICP) nitration and a post nitra-

tion anneal at 1000 �C.

C-AFM measurements were performed at room temper-

ature with an Omicron AFM/scanning tunnelling microscopy

0021-8979/2014/116(2)/024505/7/$30.00 VC 2014 AIP Publishing LLC116, 024505-1

JOURNAL OF APPLIED PHYSICS 116, 024505 (2014)

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system under UHV (<10�9 Torr) with conductive diamond

tips (B doped). The AFM tip was used as a top electrode (tip

area �10 nm2), and a negative voltage was applied to the

substrate. By convention, all of the current voltage character-

istics are presented according to a positive gate voltage

(inversion). The current was recorded using a Keithley 6430

equipped with a sub-femtoampere sourcemeter. Electrical

contact between the sample and the stage was assured with

indium solder, and all of the samples were outgassed at

150 �C for 3 h at 4� 10�8 Torr. IV and I(t) measurements

were done in contact mode AFM (normal force¼ 20 nN),

while respectively applying a ramp voltage stress (RVS) or a

CVS to the oxide layer. All the measurements were recorded

on fresh oxide at different positions on the sample.

III. EXPERIMENTAL RESULTS

A. Conduction current measurements

Typical Current Voltage (IV) characteristics obtained on

fresh 1.4 nm and 2.6 nm oxide layers are shown in Fig. 1. For

both thicknesses, the conduction current follows on more

than two decades a Fowler-Nordheim (F-N) tunnelling

mechanism, consistently with previous observations.12 For

parameter fitting, the following expression of the Fowler

Nordheim tunnelling current is used:12

IFN ¼ Stipq3

16p2�h/b

m0

meff

� �V � Vfb

Tox

� �2

� exp�4

ffiffiffiffiffiffiffiffiffiffi2meff

p3q�h

� /32

bTox

V

!; (1)

where Stip the tip/sample contact area, �h is the reduced

Planck’s constant, q the elementary charge, V the tip voltage,

Tox the oxide thickness, /b the barrier height at the Si/SiON

interface, m0 the electron mass, and meff the effective elec-

tron mass in the oxide layer. According to Refs. 11 and 13,

Stip has been taken equal to 5 nm2 and /b was taken equal to

3 eV. In Fig. 1, the conduction current through the 2.6 nm

thick layer has been reproduced with Eq. (1) using an

effective mass equal to 0.5m0. The conduction current

through the 1.4 nm has been fitted with meff¼ 0.9m0.

Thickness-dependent effective masses of the electron have

already been reported in Refs. 14–16.

B. Time dependant dielectrics breakdown experiments

The dielectric breakdown is known to be a highly local-

ized phenomenon;5 the application of nano-scaled stress

using the reduced contact area between the AFM tip and the

sample surface enables to study the intrinsic breakdown

characteristics of the oxide. The C-AFM has already been

validated by Refs. 9 and 13 for statistical dielectric break-

down studies on thin gate oxides. In this paper, TDDB distri-

butions have been measured by constant voltage stress on

the 1.4 nm and on the 2.6 nm SiON layers. When a CVS is

applied to the oxide layer, a current time characteristic I(t) is

recorded as shown in Fig. 2. The dielectric breakdown

occurs at the time to breakdown (Tbd) which corresponds to

the time for a sharp increase of the current (Hard

Breakdown). As it is shown in Fig. 2, this time can be deter-

mined using a current threshold value above typically 60 nA

and a threshold value of 100 nA was practically used. For

each distribution, a minimum of 60 I(t) characteristics have

been recorded. Time to breakdown (Tbd) cumulative failure

distributions P(Tbd) has been obtained by C-AFM measure-

ments on both thicknesses at two different electric fields

(2.03 V/nm for the 2.6 nm and 2.64 V/nm for the 1.4 nm).

Those distributions are displayed in Fig. 3 in a Weibull scale

(i.e., ln(-ln(1-P(Tbd))) versus ln(Tbd) and follows a linear

shape in this scale. As usually observed,2–4 the TDDB distri-

bution’s shape parameters increase with the thickness of the

oxide layer. Notice that voltage stresses and consequently

electric fields are much higher than those used under stand-

ard test conditions (micrometer and submicrometer scale).

The fact that the thinner oxide can undergo a higher electric

field with longer Tbd has already been observed10,13 and is

explainable by extreme value statistics (see, for example,

Ref. 17). Thus, this TDDB study considers phenomenon

occurring at electric fields of about 20 MV/cm, which is a

high value. In Figs. 4 and 5 are displayed the Tbd cumulative

failure distributions obtained on the 1.4 nm thick SiON layer

FIG. 1. Tunneling current measured on the 2.6 nm and the 1.4 nm SiON

layer with C-AFM. The current has been fitted with a Fowler Nordheim tun-

neling current with Stip¼ 10 nm2 and meff¼ 0.5m0 for the 2.6 nm layer and

0.9m0 for the 1.4 nm layer.

FIG. 2. Typical I-t obtained on p-Si/SiON 1.4 nm by C-AFM at 3.8 V. The

current threshold for hard breakdown detection is 100 nA.

024505-2 Foissac et al. J. Appl. Phys. 116, 024505 (2014)

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for four voltages and on 2.6 nm thick SiON layer for three

voltages, respectively. For both thicknesses, the TDDB dis-

tributions present a Weibull behaviour, consistent with previ-

ous studies.4

C. Negative differential resistance experiments

Thanks to the reduced tip/sample contact area, C-AFM

measurements enable us to detect low pre-degradation cur-

rents occurring at high electric fields that could not be

observed on micrometric devices. In this part, we focus on

the pre-breakdown current appearing during current-voltage

measurements close to the breakdown voltage as described

in Sec. III B. For this purpose, the applied staircase voltage

is ramped up until a maximum voltage (Vmax) is reached and

then it is ramped down. The ramp is completely described by

its voltage step height (DV), its step duration (Dt), and the

maximum voltage Vmax. According to previous C-AFM

studies,5 if the voltage ramp is not reversed, the IV curve

exhibits a sharp dielectric breakdown followed by a decreas-

ing post breakdown current when the ramp direction is

reversed. In this study, following Ref. 11, the Vmax values

were chosen to reverse the ramp direction just before the

dielectric breakdown occurs. This leads to NDR characteris-

tic obtained on SiON 1.4 nm by C-AFM and displayed in

Fig. 6(a). This characteristic is compared to a simulated FN

FIG. 4. TBBD distributions obtained on SiON 1.4 nm by C-AFM at 3.6 V,

3.7 V, 3.8 V, and 3.9 V. The distributions have been reproduced with

Monte Carlo simulations on 150 devices with A¼ 112.2 V/nm and

s¼ 6.9� 10�15 s.

FIG. 5. TBBD distributions obtained on SiON 2.6 nm by C-AFM at 5.2 V,

5.3 V, and 5.4 V. The distributions have been reproduced with Monte Carlo

simulations on 150 devices with A¼ 84 V/nm and s¼ 6.9� 10�15 s.

FIG. 6. (a) I(V) characteristic on SiON 1.4 nm obtained by C-AFM with

Dt¼ 60 ms (w) reproduced with filament current of equation (Eq. (9))

(dashed line) and tunneling current on SiON 1.4 nm obtained under a

0.04 lm2 electrode (plain line). (b) I(V) characteristic on SiON 1.4 nm

obtained by C-AFM with Dt¼ 60 ms (w) and 1600 ms (D). The negative

differential resistance has been reproduced using equation (Eq. (9)) with

meff¼ 0.9 m0 and Af¼ 111.9 V/nm and 110 V/nm.

FIG. 3. TBBD distributions obtained by CVS on SiON 2.6 nm at 2.03 V/nm

and on 1.4 nm SiON at 2.64 V/nm by C-AFM.

024505-3 Foissac et al. J. Appl. Phys. 116, 024505 (2014)

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tunnelling current on the same sample but with a 0.04 lm2

electrode, which is the typical surface of a standard device (§

Eq. (1)). It is clear from Fig. 6(a) that the tunnelling current

magnitude precludes the observation of the aforementioned

NDR effect. Thanks to the small tip surface of the AFM, the

NDR characteristics current value is above the tunnelling

current and makes the observation possible.

Typical IV curves obtained by this method on a 1.4 nm

thick layer and a 2.6 nm thick layer are shown in Figs. 6(b)

and 7 in a linear scale. It can be seen that during the first

steps of the down ramping, the current still increases until

reaching a maximum value (Imax) though the voltage is

decreasing (negative differential resistance effect). Then, the

current decreases with decreasing voltage. This behaviour

was reproducible at different positions. In Fig. 6(b) are

reported the IV characteristics for two voltage ramps applied

to the 1.4 nm layer with the same Vmax (3.7 V) and the same

DV¼ 0.05 V but two different Dt: 60 ms and 1600 ms.

Fig. 6(b) clearly shows the influence of the ramp speed on

the current. When Dt is increased, the NDR effect is

increased leading to a higher Imax. This negative differential

resistance behaviour is also observed on a 2.6 nm SiON layer

(Fig. 7). The voltage ramps in Fig. 7 have two different Vmax

(5 V and 5.3 V) and two different ramp speeds (0.6 V/s and

2 V/s). For these two characteristics, DV was equal to 0.1 V

and only Dt was varying. By decreasing the ramp speed, the

NDR effect is amplified even if Vmax is decreased.

IV. DISCUSSION

A. Time dependant dielectric breakdown distributions

The experimental TBBD distributions displayed in Figs.

4 and 5 have been fitted classically with a Weibull cumula-

tive distribution

PðtÞ ¼ 1� expð�ðt=sÞbNÞ; (2)

where b is the shape parameter, s the scale parameter, and N,

the area scaling parameter. In a Weibull scale, the

cumulative failure distribution W(t)¼ ln(�ln(1 – P)) is a lin-

ear function of ln(t) with slope b

WðtÞ ¼ b lnðtÞ � b lnðsÞ þ lnðNÞ: (3)

The extracted slopes for both oxides show that this parameter

is thickness dependant5 with b1.4 nm¼ 0.96 and b2.6 nm¼ 1.2

(these values are consistent with the values found in

Ref. 18).

Considering the very high field in our experiments

(�20 MV/cm), we choose a 1/E acceleration factor field de-

pendence like the anode hole injection (AHI) model.19

Hence, s can be expressed as

s ¼ s0 expðA=EÞ; (4)

where s0 is a pre exponential factor, A the acceleration fac-

tor, and E the applied electric field. The 1.4 nm SiON layer’s

distributions have been fitted simultaneously using

A¼ 112.2 V/nm and s0¼ 6.9� 10�15 s which is a reasonable

order of magnitude. The fits are provided in Fig. 4. By fitting

the three distributions obtained on the 2.6 nm SiON layer,

the following parameters could be extracted for this thick-

ness: A¼ 84 V/nm, s0¼ 6.9� 10�15 s, and N¼ 39.8. We

have displayed in Figs. 4 and 5 the distributions obtained

from Monte Carlo simulation of 150 devices with the above

values of the parameters. Note that the A parameter of the

thinner layer is much larger than the one of the thicker layer.

The parameters used for these fits are gathered in Table I.

B. Negative differential resistance modelling

Pre-breakdown NDR observations at nanoscale have

been reported on a 2.6 nm thick SiON layer.11 It has been

shown that a filament growth mechanism is the most appro-

priate to reproduce the NDR characteristics obtained by

C-AFM. Such a filament field assisted nucleation has been

used to describe switching in phase change memory devi-

ces20 and more recently dielectric breakdown.21 Here, we

will use a similar approach. Let us consider that the inva-

sive percolation consists of a number of growing filaments

(which can be considered as a linear succession of gener-

ated defects) starting at one of the interfaces. One can

assume for example (this point is not necessary) that the

growth rate is proportional to the distance between the top

of the filament and the opposite electrode. This is the ana-

logue to the non-transformed volume in a growth process.

The filament length X(t) obeys then to the differential

equation

dX

dt¼ 1

sfðTox � XÞ; (5a)

FIG. 7. I(V) characteristic on SiON 2.6 nm obtained by C-AFM with

Dt¼ 60 ms (w) and 1600 ms (D). The negative differential resistance curves

have been reproduced using equation (Eq. (9)) with meff¼ 0.5mf and

Af¼ 82.9 V/nm and 83.25 V/nm.

TABLE I. Extracted parameters from the fit of the TDDB distributions

obtained on p-Si/SiON layers (1.4 nm and 2.6 nm thick) using Eqs. (3)

and (4).

A (V/nm) s0 (s) N

SiON 2.6 nm 84.0 6.9� 10�15 39.8

SiON 1.4 nm 112.2 6.9� 10�15 31.8

024505-4 Foissac et al. J. Appl. Phys. 116, 024505 (2014)

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where the growth rate 1sf

depends on the electric field

sf ðtÞ ¼ sf 0 expðAf=EðtÞÞ: (5b)

Notice that this expression is very similar to the electric field

dependence of the scale parameter s of the TDDB distribu-

tion (Eq. (4)).

Since in the case of a ramped voltage stress the electric

field is a function of time, the filament length from the for-

mer differential equation is given by

XðtÞ ¼ Tox 1� exp �ðt

0

t0=sf ðt0Þdt0� �� �

: (6)

The electric field E(t) between the tip and the substrate is

directly related to the staircase ramped voltage V(t) intro-

duced in Sec. III C

E tð Þ ¼ V tð ÞTox

¼ DV

Tox

XVmax=2DV

j¼0

H t� jDtð Þ �XVmax=DV

j¼Vmax=2DV

H t� jDtð Þ

8<:

9=;;

(7)

where H(t) is the Heaviside function.

The filament length X(t) can be deduced using Eqs. (6)

and (7). We will now assume following Ref. 11 that elec-

trons can tunnel from the top of the filament to the opposite

electrode. The transmission probability will depend on the

electric field between the top of the filament and the opposite

electrode named Efil(t) given by

Efil tð Þ ¼ V tð ÞTox � X tð Þð Þ ¼

E tð Þ1� X tð Þ=Tox

� � : (8)

The total current through the dielectric layer is then assumed

to be the sum of the tunnel current from the tip/sample con-

tact area through the whole layer thickness and the one from

the filament cross section through the residual layer thick-

ness. The sum of these two contributions is given by

I ¼ Stipq3

16p2�h/b

me

mef f

� �E tð Þð Þ2

� exp�4

ffiffiffiffiffiffiffiffiffiffiffi2mef f

p/

32

b

3q�hE tð Þ : 1� 1� qV tð Þ/b

� �� �" #

þ Sf ilq3

16p2�h/b

me

mef f

� �Ef il tð Þ� �2

� exp�4

ffiffiffiffiffiffiffiffiffiffiffi2mef f

p/

32

b

3q�hEf il tð Þ : 1� 1� qV tð Þ/b

� �� �" #: (9)

Equations (6) to (9) allow computing the current during the

voltage ramp up and down. The experimental NDR charac-

teristics from Figs. 6 and 7 can be fitted by solving Eqs.

(6)–(8) and combining them with Eq. (9) to obtain the cur-

rent I(t) as a function of the applied voltage V(t). The effec-

tive masses and barrier height are given by the current

voltage characteristics fitted values from Sec. III A and equal

to 0.5m0 and 0.9m0 and /b was taken equal to 3 eV, respec-

tively. We have extracted Stip¼ 5 nm2, and a filament diame-

ter of 0.4 nm (Sfil¼ 0.12 nm2) consistent with previous

assessments from Sec. III A. As it is shown in Figures 6 and

7, the model describes satisfactorily the experimental charac-

teristics. From the fits of the IV (NDR) characteristics of the

two voltage sweeps, one can extract the Af and s0f parame-

ters (Figs. 6 and 7). In the case of the 2.6 nm SiON layer

(Fig. 7), the parameter Af was evaluated equal to 82.9 V/nm

and 83.8 V/nm for the ramp with the 60 ms step duration and

the ramp with the 1600 ms step duration, respectively. As the

two ramps correspond to two different experiments, the

small variations of Af are not surprising and these values

will be discussed in Sec. IV C. The NDR characteristics

obtained on the 1.4 nm layer and reproduced with Eq. (9) are

given in Fig. 6(a). For this thinner layer, the Af parameter’s

value was found equal to 111.9 V/nm for the Dt¼ 60 ms

ramp and 110 V/nm for the Dt¼ 1600 ms ramp, which is

much larger than the Af values obtained on the 2.6 nm layer.

For both thicknesses, the characteristic lifetime sf0 was found

to be 1.2� 10�16 s. All the values extracted from the fit of

the NDR effect have been reported in Table II.

According to equations (X(t) and E(t)), the degradation

growth is governed by the electric field as a function of time.

For voltages close to Vmax, the filament starts to grow lead-

ing to current increase. The filament continues its growth

even few steps after the voltage ramp direction is reversed.

The electric field at the end of the filament Efil(t) increases

with the filament length and thus continues to increase while

the applied voltage is ramped down. During the steps where

Efil(t) is still increasing, the tunnelling current between the

tip and the apex of the filament increases in turn. It is this fil-

ament growth persistence during the ramping down that

explains the observed NDR. The final filament length will

depend on the chosen Vmax. The higher Vmax is, the longer

the filament will be and the NDR will reach a higher Imax.

The filament growth is also enhanced with increasing Dt

because the applied field is maintained at high values for lon-

ger durations. This explains that the NDR is observed for

lower voltages when the time step is increased.

C. TDDB NDR (IV) comparison

According to Tables I and II, we can notice that the Af

parameters extracted from the two ramps for the two thick-

nesses are remarkably close to the respective A parameters

extracted from TDDB distributions for the same thickness.

This similarity can be understood from the fact that the time

to breakdown corresponds to the time taken by the longest

filament to cross the gate dielectric. TDDB distributions

allow extracting a mean value of the acceleration factor,

TABLE II. Extracted parameters from the fit of the NDR characteristics

obtained on p-Si/SiON layers (1.4 nm and 2.6 nm thick) for different ramp

speed using Eq. (9).

Af (60 ms) (V/nm) Af (1600 ms) (V/nm) sf0 (s)

SiON 2.6 nm 82.9 83.8 1.2� 10�16

SiON 1.4 nm 111.9 110.0 1.2� 10�16

024505-5 Foissac et al. J. Appl. Phys. 116, 024505 (2014)

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whereas a single NDR is a particular realization of the fila-

ment growth event. Focusing on the statistical description of

the breakdown, we can extract the mean time to failure of

the Weibull distribution, which is

hTbdi ¼ s� N�1bC 1þ 1

b

� �: (10)

In Eq. (10), C is the Gamma function.

On the other hand, one can consider that the breakdown

will occur when the filament has almost crossed the layer.

Thus, the filament length is given by: Xfil¼Tox� d, where dhas the order of magnitude of an atomic size. The time taken

by the filament to cross the gate stack, tcross, is then deter-

mined for the case of CVS by solving Eq. (6)

tcross ¼ sf � lnTox

d

� �: (11)

Because of the exponential dependence of the tunnelling cur-

rent between the top of the filament and the opposite elec-

trode (see Eq. (9)), the fit of the IV (NDR) characteristics

corresponds likely to the longest filament, which takes the

least time to grow through the SiON layer. It is then reasona-

ble to compare the two aforementioned quantities tcross and

hTbdi. Eliminating the term exp(A/E) in both Eqs. (10)

and (11) gives

sf 0 � s0 �N�

1bC 1þ 1

b

� �

lnTox

d

� �0BBB@

1CCCA: (12)

Considering the thicknesses of the SiON samples (1.4 and

2.6 nm), we can roughly estimate the ratio Tox/d between 4

and 10. Taking into account the experimental values of b, we

can then approximate the logarithm of this ratio by 1, which

leads to

sf 0 � s0 � N�1b

� �: (13)

Equation (13) allows to estimate sf0 by taking into account s0

and N from Table I and to compare the result with sf0 from

Table II (1.2� 10�16). From TDDB distributions (Table I),

we can calculate s0 N�1/b¼ 3.2� 10�16 s for the 2.6 nm layer

and 1.6� 10�16 s for the 1.4 nm layer. This comparison con-

firms the consistency of our assumptions. In the Weibull dis-

tribution, the scaling parameter N is the number of random

variable of a parent distribution whose asymptotic tail behaves

as a power law (see, for example, Ref. 17). In our filamentary

approach, one can naturally identify N with the ratio between

the C-AFM tip and filament cross section.

In Sec. IV B, the tip/sample contact area was found

equal to 5 nm2 and the filament diameter was taken equal to

0.4 nm (filament cross section Sfil¼ 0.12 nm2), which gives

N¼ 39.5. This value is in full agreement with the N values

extracted from statistical Weibull distributions fits in Table I.

This comparison shows also the consistency of our approach,

and the scaling power law (Eqs. (12) and (13)) constitutes

one of the main results of this study. Comparison of the

values of Af extracted from the filament growth during the

NDR effect (Table II) and A, the voltage acceleration factor

extracted from the TDDB distributions (Table I) shows that

the values are the same. Moreover, Af and A appear to be

thickness dependant. The A values are larger than the field

parameter AFN ¼4ffiffiffiffiffiffiffiffi2mef f

p/

32b

3q�h in the Fowler-Nordheim current

that presents the same functional dependence than the volt-

age acceleration factor of the filament growth rate (see

Eq. (1)). AFN varies from 25 to 33 V/nm as A is about 80 to

110 V/nm according to thickness. Then, the Fowler

Nordheim mechanism cannot explain our experiments. In

Ref. 11, the A value was temperature dependant with a tem-

perature activation much larger (�4 eV) than experimental

(�0.6 eV) and was not dependant on thickness. If one, how-

ever, invokes a holes generation mechanism, it is interesting

to assume that Zener effect (band to band tunnelling) may

arise. The Zener tunnelling probability is proportional to

exp(AZener/E), where AZener given by Eq. (14) presents the

same functional dependence as AFN,22

AZener ¼p2

hq

ffiffiffiffiffiffiffiffiffiffiffi2mef f

pEGapð Þ

32: (14)

In Eq. (14), EGap is the gate dielectric band gap energy. For

a 2.6 nm SiON layer, with EGap¼ 6 eV (Ref. 23)

and meff¼ 0.5m0, the Zener coefficient is found equal to

83.5 V/nm and for a 1.4 nm SiON layer with meff¼ 0.9m0 the

Zener coefficient is found equal to 112 V/nm. The AZener val-

ues are clearly in excellent accordance with the experimental

values. The thickness dependence of AZener is due to the

thickness-dependence of the electron effective masses.14–16

From those comparisons, one can propose a breakdown

mechanism for the high field used in those experiments:

holes are generated in the gate dielectric by Zener band to

band tunnelling and are driven to the bottom Si electrode by

the effect of the electric field. Generated holes can be trapped

preferentially at the end of a filament, which is at the same

potential than the cathode. Hole capture at the end of the fila-

ment results in a dangling bond Si-O, which may be replaced

by a Si-Si bond, thus increasing the length of the filament as

it is illustrated in Fig. 8. Then, the growth rate depends on

the number of holes created upstream from the end of the fil-

ament the filament. The growth rate is then proportional to

distance between the filament tip and the anode (see

Figure 8) and one retrieves the growth equation from part B:dXdt ¼ 1

sfðTox � XÞ used here. The driving force of the growth

mechanism can be assumed to near to near holes captures.

(This can be thought as a near to near trap generation.)

For statistical point of view, the time to breakdown cor-

responds to the time taken by the first filament that crosses

the dielectric film. The distribution of this particular event

is the distribution of the shortest time tcross among N fila-

ment tcross times. The shortest time tcross among N is

described by an extreme (smallest) value distribution. If, as

it was aforementioned, the tcross times statistical distribution

asymptotic tail behaves like a power law, this extreme value

distribution is the Weibull statistics.24 It is worth noting

that the electric field acceleration factor, A, depends on the

024505-6 Foissac et al. J. Appl. Phys. 116, 024505 (2014)

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dielectric thickness via the effective mass thickness de-

pendence (in the range of thicknesses used in our experi-

ments). For practical lifetime prediction reasons, it is often

simpler to use a thickness independent acceleration factor.

Doing this, the acceleration can neither be interpretable by

a field effect nor by a voltage effect which may lead to

errors in lifetime predictions.

V. CONCLUSION

Using an UHV C-AFM, we have measured current volt-

age, pre-breakdown negative differential resistance charac-

teristics together with the TDDB distributions for two

dielectric thicknesses. It was found that all those experimen-

tal characteristics can be consistently explained within the

framework of a conductive filaments growth that can be

thought as an invasive percolation mechanism. The Weibull

characteristic corresponds to the distribution of the first fila-

ment that shunts the dielectric film. The Weibull distribution

scale factor is proportional to the characteristic growth rate

of an individual filament. The proportionality factor is a scal-

ing power law of the number of filaments with the exponent

1/b (b being the Weibull shape parameter). This scaling law

can be used at standard device area. Moreover, it was found

that, for the high fields used in our experiments, the accelera-

tion factor is proportional to the Zener tunnelling probability.

According to those facts, one can infer that holes generated

by Zener effect may act as the filament growth driving force.

A complete statistical model, in particular accounting for the

Weibull shape factor, is feasible but is beyond the scope of

this article. This will be detailed in a further work.

ACKNOWLEDGMENTS

This work was partly supported by the French

RENATECH network.

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FIG. 8. Schematic representation filament growth induced by holes genera-

tion and capture. Holes generated by Zener band to band tunneling in the

region located between x and Tox are captured at the top of the filament.

This capture corresponds to a Si-O bond breaking. Once the Si-O bond is

broken, it can be replaced with a Si-Si bond (black bold circles). The repeti-

tion of this process constitutes the filament growth driving force.

024505-7 Foissac et al. J. Appl. Phys. 116, 024505 (2014)

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