General framework about defect creation at
the
interface
Cite as: J. Appl. Phys. 105, 114513 (2009); https://doi.org/10.1063/1.3133096
Submitted: 07 January 2009 • Accepted: 11 April 2009 • Published Online: 09 June 2009
C. Guerin, V. Huard and A. Bravaix
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J. Appl. Phys. 105, 114513 (2009); https://doi.org/10.1063/1.3133096
© 2009 American Institute of Physics.
105, 114513
三种Si-H键的打破方式
JOURNAL OF APPLIED PHYSICS 105, 114513 共2009兲
General framework about defect creation at the Si/ SiO2 interface
在Si/SiO2系统中研究，有HK时候呢？
C. Guerin,1,2,a兲 V. Huard,1 and A. Bravaix2
1
ST Microelectronics, Crolles, 850 rue Jean Monnet, 38926 Crolles, France
2
IM2NP-ISEN, UMR CNRS 6137,Maison des Technologies, place G. Pompidou, 83000 Toulon, France
共Received 7 January 2009; accepted 11 April 2009; published online 9 June 2009兲
This paper presents a theoretical framework about interface state creation rate from Si–H bonds at
the Si/ SiO2 interface. It includes three main ways of bond breaking. In the first case, the bond can
be broken, thanks to the bond ground state rising with an electrical field. In two other cases, incident
carriers will play the main role either if there are very energetic or very numerous but less energetic.
This concept allows one to physically model the reliability of metal oxide semiconductor field effect
transistors, and particularly negative bias temperature instability permanent part, and channel hot
carrier to cold carrier damage. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3133096兴
I. INTRODUCTION
The existence of silicon dioxide plays a key role in the
everlasting Si-based semiconductor industry. Since the mid1980s, the continuous scaling down of the dielectric thickness reaches dimensions in actual nodes close to atomic level
共a 1.5-nm-thick dielectric is only 5 atom thick兲. In addition,
at the Si/ SiO2 interface, both electronic and structural properties are changed. As a result, silicon dangling bond 共SDB兲
defects are intrinsically present resulting in electrically active
interface traps 共Nit兲, also called Pb centers.1 In order to minimize the effect of the electrically active states due to SDBtype defects, silicon devices are exposed to hydrogen
through thermal annealing. The goal is that the H atoms will
passivate these interfacial defects by forming Si–H bonds.
Most of reliability degradation modes occurring in the
front-end part of the process are thought to be related to the
dissociation of the Si–H bonds, such as negative bias temperature instability 共NBTI兲, hot carrier 共HC兲, and timedependent dielectric breakdown 共TDDB兲 generally explained
by hydrogen release at Si/ SiO2 interface.2–4 Though all these
degradation modes are related to the same initial SDB-type
defect, many physical explanations have been proposed to
explain the experimentally observed time-dependent hydrogen depassivation, ranging from direct electronic excitation
共DEE兲, vibrational ladder climbing to diffusion-limited
mechanisms. We report here a general framework to explain
hydrogen depassivation at the Si/ SiO2 interface based on the
last published results on Si–H bond local environment. This
work will explain several reported results in literature including various degradation modes focusing particularly here
on our new HC data obtained in the state-of-the art complementary metal oxide semiconductor 共CMOS兲 technology
nodes 共40 nm兲 using medium to ultrathin gate oxides 共Tox
= 1.3– 5 nm兲.
II. ELECTRONIC AND VIBRATIONAL MODE
PROPERTIES OF SI–H BOND
Understanding the Si–H bond properties at the atomic
level is required to control and fabricate MOS devices with
a兲
Electronic mail: chloeguerin@hotmail.com.
0021-8979/2009/105共11兲/114513/12/$25.00
both high performances and reliability. Since some of these
properties are seldom experimentally accessed, simulations
such as first-principles molecular dynamics method are powerful tools for investigating and understanding atomic-level
phenomena. The aim of this paragraph is to propose a short
review concerning the different bond states under carrierinduced excitation, including both recent simulated and experimental results.
The hydrogen atom transfer is viewed as a potential barrier crossing problem. The crossing of such an EB barrier by
thermal activation has been invoked in explaining the motion
of individual atom with the tip of a scanning tunneling microscope 共STM兲.5 In the case of STM desorptions, the underlying microscopic mechanism linked to EB reduction has
been related to vibrational excitation by inelastic scattering
of incident carriers,6 i.e., a vibrational excitation of the
substrate-adsorbate bond. A key ingredient in this context is
the anharmonic coupling between the adsorbate and the incident carrier via an adsorbate-induced resonance state.7 This
resonance is usually modeled by a Lorentzian density of
states 共DOS兲 ␳a, centered at the energy ␧a interacting two
continua of carrier levels representing the incident carriers
and the substrate7–9 共Fig. 1兲. In a MOS transistor case, the
incident carriers are channel carriers 共electrons for n-MOS or
holes for p-MOS兲, tunneling gate carriers or substrate carriers. When carriers are electrons, this resonance concerns the
Si–H 6␴* state.9,10 When they are holes, it is the 5␴ state.8
The precise shape of the DOS is not Lorentzian but it has
been shown to range from 1.5 to 7 eV.8,9 The energy acquired by carriers is represented by a shift between the Fermi
levels of the two continua 共Fig. 1兲. This carrier energy can be
transmitted to the bond by inelastic scattering through the
resonance. Then the energy is dissipated by bond localized
phonon modes, i.e., adsorbate vibrational modes. The two
main Si–H bond vibrational modes are known to be stretching and bending modes11 共Fig. 1兲. Stretching mode corresponds to a path where the adsorbate is moved away from
silicon toward an interstitial site. Bending vibrational mode
corresponds to a path where the adsorbate rotates around Si
toward a neighboring bond center site.12,13 Each mode is
characterized by a bond breaking energy EB, a vibrational
mode energy ប␻, and a relaxation time ␶e. These three pa-
105, 114513-1
© 2009 American Institute of Physics
114513-2
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
Substrate
Adsorbate Incident carriers
Bending
H
Stretching
E
εa
ρa
TABLE II. Si–H bond stretching or bending vibrational mode wave numbers from several references. Availabe vibrational mode lifetimes and
transversal/lateral and optical/acoustical 共T/L and O/A兲 phonon wave numbers are also mentioned in bracket.
FIG. 1. 共Color online兲 Schematic model of inelastic electron scattering via
an adsorbate induced resonance at an energy ␧a with a DOS ␳a near the
Fermi level of the substrate. The energy transfer from carriers through resonance excites bending or stretching bond vibrational modes. After Refs. 7
and 8.
rameters allow a full determination of the adsorbate characterization since the bond breaking is a trade-off between the
energy brought by bond excitation and the relaxation process
linked to ␶e.
A. Bond breaking energy
Several teams studied theoretical Si–H bond dissociation
energy, thanks to calculations based on density-functional
theory. The first step of their work is to model atom cells
including a dangling bond passivated by an hydrogen. Then,
the H atom is moved away from the Si and the required
additional energy is monitored. Results from several references have been reported in Table I for stretching and bending pathways.11–14 It is worth noticing that some of the calculations are done for a Si–H bond in silicon bulk, whereas
others concern Si–H at the Si/ SiO2 interface. However, all
the data related to one mode are coherent. In consequence,
we set the stretching path energy at 2.5 eV and the bending
path energy at 1.5 eV.
Reference
Site
Stretching
共eV兲
Bending
共eV兲
12
13
Bulk Si
Bulk Si
2.5
⬃3.6
14
11
共111兲Si/ SiO2 interface
共111兲Si/ SiO2 interface
3.3
⬍3 共SiO2兲
2.2 共Si兲
1.5
1.75 共path II兲
1.9 共path III兲
1.9
1.5
Phonon
共cm−1兲
2100
共0.8 ns兲
1838–2145
共1.6– 295 ps兲
BC 1998
共7.8 ps兲
HV¯VH 2072
共295 ps兲
650
640 共8 ps兲
TO 463
52
21
15, 17, and 19
Simulation
Simulation
Silicium
20
Silicium
817
共12.2 ps, 5 K兲
53
Si/ SiO2
592
54
Si/ SiO2
2083
TA⬃ 150
LA⬃ 340
LA 357
TA 150
LO 517
TO 460
TO 468
LO 435
626
B. Vibrational mode energy
The presence of light atoms such as hydrogen in a crystalline solid gives rise to localized vibrational modes with
wave numbers 共i.e., energy兲 above the phonon bands of the
solid.15 The infrared absorption spectroscopy 共IRAS兲 allows
measuring the wave number of a Si–H bond excitation mode
by detecting absorption line for each mode.15 Wave number
values from several studies are regrouped in Table II. The
reported values for a given mode are rather identical in spite
of various result sources 共simulation or experiment measurement done on different H-related defects兲. As for the energy
modes, we set the stretching wave number at 2083 cm−1 and
the bending wave number at 610 cm−1. Knowing these wave
number values, the energy of the vibrational mode ប␻ can be
deduced from the relation 1 eV⬅ 8.065 5 ⫻ 103 cm−1.16 It
leads to ប␻s = 0.25 eV for stretching mode and ប␻b
= 0.075 eV for the bending mode.
C. Bond vibrational mode lifetime
A second term can be extracted from an absorption line:
the lifetime of the associated vibrational mode ␶e. In fact, ␶e
is related to the full width at half maximum ⌫ by
⌫=
TABLE I. Si–H bond dissociation energy following stretching or bending
mode from several references. Si–H bonds are placed either in bulk Si or at
the Si/ SiO2 interface.
Bending
共cm−1兲
Source
εF+ energy
εF
Stretching
共cm−1兲
Reference
1
.
2␲c␶e
共1兲
Nevertheless, due to the presence of inhomogeneous
broadening, ⌫ value can be underestimated. A more precise
value of ␶e can be extracted after IRAS by the transient
bleaching spectroscopy technique. In the following it consists of the decrease in the transmission coefficient from the
wave number line measured by IRAS.15 The lifetime values
have been also noted in Table II when given by the reference.
Measurements have been done between 5 and 10 K. Contrary to the wave number, the vibrational mode lifetime is
extremely dependent on the defect structure. Since previous
work17 argued that the Si–H bond of the divacancy binding
114513-3
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
two H atoms 共HV · VH共110兲兲 resembles that of the
H-passivated Pb centers at the Si/ SiO2 interface, we set ␶e at
295 ps for the stretching mode and at 10 ps for the bending
mode.
Finally, we need to extend the knowledge on ␶e from
cryogenic toward high temperatures. A high wave number
excited mode decays into a path of vibrational modes of the
lattice, i.e., phonons, whose wave numbers are smaller than
the wave number of the excited mode. That is why we also
reported the phonon wave numbers in Table II for transversal, longitudinal, acoustic, and optical 共T, L, A, O兲 modes.
The decay of a vibrational mode by a decay channel constituted by several phonons is governed by the following
equation:18
we =
=
1
= 2␲ 兺 兩Gi兩2 f i
␶e
i
with fi
exp共ប␻/kBT兲 − 1
Ni
兿 j=1关exp共ប␻ j/kBT兲 −
1兴
,
共2兲
where the total energy decay rate we is the sum of the rates of
all the decay channels. Gi is the coupling strength of the
channel, f i is its temperature dependence, ␻ is the frequency
of the bond vibrational mode, ␻ j is the phonon frequency,
and Ni is the total number of phonon required for decay.
Furthermore, we know that the wave number of the bond
vibrational mode is equal to the sum of the phonon wave
numbers and that channels with low Ni are likely to dominate
the decay.15
The decay channel of the stretching mode has been modeled by Lüpke19 with five accepting modes with frequency
兵521, 521, 343, 343, 343其. Concerning the bending mode, the
same study has only been carried out by Sun et al.20 but only
on the 817 cm−1 mode, whereas experimental measurements
at the Si/ SiO2 interface show that the Si–H bending mode is
around 610 cm−1. However, the author also showed that the
vibrational lifetimes follow a universal frequency-gap law,
i.e., that the decay time increases exponentially with increasing decay order 共Ni兲. From this law, we suggest that the
bending mode only needs a two orders of magnitude decay, it
can be related to the sum of transversal optical 共TO兲
共460 cm−1兲 and transversal acoustic 共TA兲 共150 cm−1兲 phonon
modes.
In this paragraph, we pointed out values we chose for the
three characteristic parameters of the Si–H bond for the
stretching and bending modes. Our choices have been supported by a wide bibliographic study concerning both simulations and experimental results. We will show that these
values are useful for our general Si–H bond breaking model,
in agreement with the experimental results.
III. A GENERAL MODEL FOR SI–H BOND BREAKING
When the SDB is passivated, the system is in the ground
state. The atom transfer corresponds to the emission of the H
atom to a mobile transport state. This transfer occurs in two
steps: first, the barrier EB needs to be reduced, and then the H
atom can overcome this barrier to the neighboring transport
state via thermal emission.21 We already mentioned in the
Thermal emission
EB
Carrier-induced
excitation
Transport state
Ground state
Thermal emission
Oxide field
E!
EB
Transport state
Ground state
FIG. 2. 共Color online兲 Schematic of the energy levels involved during a
bond breaking. If the barrier EB is reduced either artificially by excitation of
the bonding electron 共top兲 or by the oxide field 共bottom兲, H can be emitted
to a mobile transport state.
previous paragraph a way of reducing EB artificially, thanks
to carrier-induced excitation. This excitation allows carriers
to move on a higher energetic level and then to be easily
emitted. By the second way, EB is really reduced due to the
application of an electric field which has a direct effect on
the ground state.22,23 These two scenarios are schematically
sketched in Fig. 2 and detailed in the two next subsections
since they offer basic explanations for several MOS field
effect transistor 共MOSFET兲 degradation modes. Part A is devoted to the electric field EB reduction, whereas part B deals
with carrier-induced excitation.
A. NBTI: Oxide field driven NIT creation
Our recently proposed NBTI model24 considers two
components in the degradation: a permanent part due to
共fast兲 interface traps and a recoverable one related to 共slow兲
or border traps. It is based on reaction-limited 共RL兲 model
and hole trapping mechanism into the gate insulator. Data
have demonstrated for the first time without ambiguity that
Nit does not get passivated 共even after 1 week of recovery兲
and that the induced Nit is the main contributor to the permanent part. Besides, data also provide additional proof that
NBTI transient effects are uncorrelated to the Nit creation
and that the recoverable part is linked to oxide hole traps
introduced by process steps as nitridation but not to generated traps during NBTI stress.24
As the topic of this study is Nit creation, we will only
focus on the permanent part. The plot of Nit variation under
NBTI stress as a function of stress time presents several time
dependences.25–27 The first part, which follows a linear time
dynamic, is related to RL mechanism. Nit creation is then
expressed by
dNit
= kF共N0 − Nit兲,
dt
共3兲
where kF is the forward reaction rate and N0 is the initial
Si–H bond number. Considering short stress time, Nit is negligible compared to N0, then Nit is linearly generated with
stress time. The slope is equal to kFN0. Knowing N0, it is
possible to experimentally determine the forward reaction
rate kF. It was observed on various oxide interface qualities
114513-4
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
Distribution of dissociation
energies g(E B)
0.08
H
0.06
Eox
Oxide
Si-H bond length d=1.48Å
Si
Bulk
0.04
Electronegativity: Si=1.8, H=2.1
Local field Eloc=(1+χ/3)Eox
EBm=EBm0-d.Z.q.cos²θ
0.02
Eox1
Eox2<Eox1
Z.q=0.2q
0
0.6
0.8
1
*Eloc
FIG. 5. Bending mode related dissociation path allows calculating the Si–H
dipole moment.
1.2
Dissociation Energy EB (eV)
FIG. 3. Fermi-derivative distribution of dissociation energies for two oxide
fields.
B. Channel hot carrier and channel cold carrier:
Energy driven NIT creation
1. Modeling
共i.e., number of available bonds兲 that the forward reaction
rate kF is dispersed over several decades; it is expressed as24
冉
kF = kF0 exp
冊
− EB共Eox兲
.
kT
双井模型
共4兲
This experimental observation is in contradiction with the
reaction-diffusion 共RD兲 model which considers, as main assumption, that kF is constant.28 This kF dispersion related to
EB dispersion is modeled according to a Fermi-derivative
function, where ␴ is the distribution dispersion 关Eq. 共5兲兴
共Fig. 3兲,
1
g共EB, ␴兲 =
␴
冉
exp
EBm − EB
␴
冋 冉
冊
EBm − EB
1 + exp
␴
冊册
2
共5兲
.
The median dissociation energy EBm for a given oxide field is
shown to follow a linear relationship with oxide field 共Fig.
4兲. The median dissociation energy in the absence of oxide
field is thus estimated to be equal to 1.5 eV, which is in close
agreement with the estimated value of the Si–H bending
pathway. Our physical model takes into account the existence of an electrostatic dipole Zq in between Si and H. It
assumes that the Si–H bond is bended by the vertical oxide
field due to the dipole, which, in turn, reduces EB 共Ref. 11兲
共Fig. 5兲. The slope of the oxide field linear dependence is
directly related to the Si–H bond dipole moment Zq. The
dipole moment value is estimated about Zq = 0.2q with our
set of experiments, in close agreement with theoretical values found in literature.29
a): DEE
σ*
Pexc
Si
EBmo = 1.5 eV
1.6
Pcou
b): SVE *
σ
Pexc
Si
Si H
Pemi
N
H
ћω
EB
Si H
σ
EES
1.8
H
σ
!a): EES
2
Mean dissociation energy EBm (eV)
In our physical framework, the Nit creation related to
carrier-induced excitation can be divided into two main
groups depending on the energy of the incident carrier: either
it has enough energy to excite the adsorbate resonance state
or it does not. Each main group can also be divided in two
other subgroups; it leads to have four Si–H bond breaking
ways 共Fig. 6兲.
Let first introduce group 1, linked to high energetic carriers. With sufficient energy, incident carriers might excite
共i.e., have an excitation probability Pexc兲 the resonance. In
spite of a fast excitation decay, a fraction of the atoms can
survive on the excited resonance state long enough to induce
a direct bond breaking. This first desorption pathway is
called DEE7,30 关case 1共a兲兴. The Si–H bond breaking rate Rb is
defined for DEE by the product of the carrier energy distribution function 共CEDF兲 共i.e., the carrier number at a given
energy兲 by the excitation probability of these carriers. The
CEDF integral over the energy is macroscopically linked to
the current I, i.e., in relation to the channel drain current Ids
for HC mode, gate current Igs for TDDB, tip current for
STM, giving
Pexc
Si
1.4
Pcou
σ*
Pemi
H
EB
Si H
σ
1.2
p = -0.056
1
!b): MVE
Pemi
σ*
0.8
0.4
0.2
σ
EBm = EB m0 - p.Eox
0
0
5
Oxide field (MV/cm)
Pexc
Si
电场越高，势垒越低，Kf越大
0.6
10
FIG. 4. Linear dependence of median dissociation energies with oxide field.
H
ћω
EB
Si H
I： Id if HC， Ig if TDDB
FIG. 6. 共Color online兲 Schematics of the four Si–H bond breaking modes
related to carrier-induced excitation: DEE, SVE, assisted or not by EES, and
MVE. Pexc, Pcou, and Pemi are the probabilities of resonance excitation of the
Nth level occupation induced by coupling and of thermal emission from the
Nth level, respectively.
114513-5
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
RbDEE：Bond breaking rate for DEE
RbDEE = IPexc .
I： Id if HC， Ig if TDDB
Pexec：Excitation Probability
共6兲
If the resonance decays from the resonance excited state
before the bond breaking occurs, the energy stored in the
resonance state is transferred via an anharmonic coupling to
low-energy modes associated with a particular motion of the
adsorbate, in competition with fast vibrational relaxation by
bulk phonons emission. The anharmonic coupling is described by a coupling probability Pcou between the resonance
state and the localized vibrational modes. Assuming that only
the highest vibrational level, named N, yields to hydrogen
desorption, the probability Pcou relates to the occupation
probability of the last level N induced by the decay of the
resonance state. This case is related to a multistep vibrational
excitation in a single jump 共or single carrier兲 also called
single vibrational excitation 共SVE兲 共Ref. 31兲 关case 1共b兲兴. In
the case of SVE, the variation in the occupation density of
the Nth level 共nN兲 with stress time t is defined as a competition between the probability of the ground state 共n0兲 excitation by anharmonic coupling and the decay to the N − 1 vibrational level. Since the last vibrational level is part of the
continuum of scattering states just above the barrier, Si–H
bond is further described as Si*H* and the nN variation with
stress time t is given by
Pemi ⬇ ␭t = t␯ exp共− Eemi/kBT兲.
Finally, the Nit creation relates to the probability to emit H to
transport states from the latest excited level H*, which can be
written as
⌬Nit = 关H*兴␭t.
共12兲
Combining Eqs. 共8兲 and 共12兲 gives the total number of intern0是整个的Nit的数量
face traps ⌬Nit,
⌬Nit =
冑
n0␭IPexcPcou 0.5
t .
we
共7兲
注意这时候n=0.5
共13兲
At this point, it is important to notice that the time dynamics is expected to be a power law with an exponent of
0.5. This equation stands as long as generated Nit is negligible compared to the whole population of Si–H bond n0. At
longer stress times and/or larger degradation, saturation is
expected with progressive decrease in n0. From Eqs. 共13兲, the
Si–H bond breaking rate RbSVE can be directly related to the
device lifetime ␶, defined as the time to reach a given degradation criterion, i.e., a given amount of interface traps
⌬Nit,crit with
⌬Nit,crit = 共RbSVE␶兲0.5
with
dnN
= IPexcPcoun0 − we关Si*兴关H*兴,
dt
共11兲
n0␭IPexcPcou
.
RbSVE ⬀
we
共14兲
Lamba是一个常数，见Eqn（9）
共15兲
我们相当于把Pexec*Pcou 用Vg和Vds来表述了
where we defined in part II as the decay rate between two
vibrational levels. As the typical time range of interface traps
creation is relatively long 共seconds or greater兲 compared to
the bond relaxation time 共picoseconds兲, we consider that the
ensemble of bonds reaches a steady state distribution very
quickly. In this case
dnN
IPexcPcoun0
= 0 → 关Si*兴 = ⌬Nit =
.
dt
we关H*兴
共8兲
In the last level, the bond is almost dissociated but it can
still easily be reformed by excitation decay. The H atom will
be dissociated from the Si only if it is transmitted to transport
states. The probability of H emission Pemi from the Nth level
exists, while the excitation on the Nth level is sustained. The
excited state can emit the H with a probability ␭ per unit
time, where ␭ corresponds to a thermal activated emission of
H over a small barrier 共Eemi兲 from the excited state, with an
attempt frequency ␯,21
␭ = ␯ exp共− Eemi/kBT兲.
共9兲
Then,
Pemi = 1 − exp共− ␭t兲.
共10兲
Here, Nit creation is experimentally observed to be a monotonous phenomenon spreading over large time scale. That is
why we consider that stress time t is much shorter than the
emission time ␭−1, such that t␭ Ⰶ 1, which allows to rewrite
Eqs. 共9兲 and 共10兲 as
After defining Rb for two high energetic carrier cases,
Si–H breaking rate with lower energetic carrier is detailed in
group 2. Several authors32–34 applying Monte Carlo to
n-MOSFETs have predicted that electrons heated by
electron-electron scattering 共EES兲 should induce a second
knee in the electron CEDF. This EES phenomenon dominates the high CEDF energy tail and extend it beyond qVds
共with Vds as the drain-source voltage drop兲 until 2qVds. In
fact, this interaction results in an exchange of energy between two electrons, thus promoting one of them toward
higher energy 共encircled electron in Fig. 6兲. This process
may allow this promoted electron to have enough energy to
excite the resonance whereas it was not possible without
EES. Finally, this case 2共a兲 is equivalent to case 1共b兲 except
for the fact that the CEDF is no longer proportional to I but
I2,35 giving
RbEES ⬀
n0␭I2 PexcPcou
.
we
共16兲
The last case 关group 2共b兲兴 is dominant when the carrier
energy is not high enough to excite the resonance even after
EES. The vibrational modes can be directly excited by multiple incident carriers with low energy. Each carrier contributes to the multiple-step vibrational ladder climbing until
reaching EB where each ladder level is separated to ប␻. This
phenomenon is called multiple vibrational excitation 共MVE兲.
It is a competing process since it requires that the excited
vibrational state lifetime ␶e is not too short compared to the
average time between subsequent carrier scattering events.
As a consequence, the MVE degradation is supposedly
strongly correlated to current I 共i.e. the number of electrons
114513-6
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
“hitting” the bond per second兲. H desorption from H passivated Si surfaces STM has been related to MVE due to a
strong dependence on current and voltage.10,30 A model that
has been proven successful in explaining the STM data is the
truncated harmonic oscillator model.10,36–38 It defines the excitation rate Rexc as the excitation to the highest bound vibrational state N. Considering that each vibrational state has the
same energy, N is defined as
N=
EB
.
ប␻
共17兲
dn0
= P dn 1 − P un 0 ,
dt
共18兲
dnn
= Pd共nn+1 − nn兲 − Pu共nn − nn−1兲.
dt
共19兲
If we consider a steady state distribution,
冉 冊
dnn
Pu
nn
= 0 → Pn = =
dt
n0
Pd
n
.
共20兲
For the last N level, N + 1 level does not exist and Eq. 共19兲
gives
dnN
= Pu PN−1 − Pd关Si*兴关H*兴 = PunN−1 − Pd⌬Nit关H*兴
dt
共21兲
and from Eqs. 共12兲, 共20兲, and 共21兲, the total Nit creation is
defined as
冑 冉 冊
n 0␭
Pu
Pd
N
t0.5 .
共22兲
From Eq. 共22兲, the Si–H bond breaking rate RbMVE can be
directly related to the device lifetime ␶, defined as the time to
reach a given degradation criterion
⌬Nit,crit = 共RbMVE␶兲0.5
with
RbMVE ⬀ n0␭
冉 冊
Pu
Pd
共23兲
N
.
共24兲
There are two terms in each Pu and Pd probabilities:39 one
term is purely dependent on the excitation/decay of vibrational modes induced by the lattice 共described by the decay
rate we兲 and the other one is related to the stimulation and
emission of vibrational modes by incoming electrons, which
could either excite 共in Pu兲 or de-excite 共in Pd兲 a phonon
mode 共supposed with a same rate兲 which is equal to
SMVE共Ids / q兲, giving
Pu = we exp共− ប␻/kBT兲 + SMVE共Ids/q兲,
共25兲
共26兲
The SMVE function has been named the “inelastic tunneling
probability” in the case of an STM study,36 i.e., the scattering
rate of the incident carrier with a vibrational mode. Since a
vibrational mode is similar to a localized phonon mode, a
linear electron-phonon coupling can be used to describe
SMVE.8 This electron-phonon interaction is available in literature since widely used by Monte Carlo simulations.32,40 Finally, the breaking rate RbMVE can thus be defined as
RbMVE
Here Pn is the probability of occupation of the nth level,
whereas Pu and Pd are the probabilities up and down 关up for
the excitation from the nth to the 共n + 1兲th level or down for
the decay from the nth to the 共n − 1兲th level兴. The variation in
occupation density nn of each level is defined as Eq. 共18兲 for
the first level and Eq. 共19兲 for level 2 to level N,
⌬Nit =
Pd = we + SMVE共Ids/q兲.
= n 0␯
冤
SMVE
冉 冊
冉 冊
− ប␻
Ids
+ we exp
q
k BT
Ids
SMVE
+ we
q
冉 冊
冥
EB/ប␻
冉 冊
exp
− Eemi
.
k BT
共27兲
It is worth noticing that at low energy 共 ⬍ 4 eV兲, it has been
calculated that the main contribution to RbMVE is due to an
event where one electron gives only one energetic quanta ប␻
to the bond, in comparison with events where several ប␻
could be given at higher energies.8,9
In this paragraph, we have proposed a general theoretical
framework considering the various pathways to excite a
Si–H bond by carriers. We have derived defects generation
rate for each configuration which are all related to intrinsic
properties of the bonds. In parallel, a thorough bibliographic
study has been led to reach an updated status on the values of
these intrinsic properties based on both latest simulations and
experimental results.
2. Experimental validation
The aim of this paragraph is to validate our model of Nit
creation after carrier-induced excitation on experimental results. The defect generation rate has already been shown to
be independent on the way the carriers acquire energy, by
combining data from channel hot electron 共CHE兲, Fowler–
Nordheim tunneling, direct tunneling, and substrate hot
electron.41 Since hot carrier degradation represents a severe
challenge to control in advanced CMOS technologies and
allows in a more flexible way to change independently carrier number or energy, we have made the choice to use hot
carrier stress experiments to investigate the validity of our
energy-driven theory. The Nit creation rate Rb is obtained by
measuring the MOS device lifetime 共␶兲 for fixed parameter
degradation and by taking it inverse 共1 / ␶兲. The criterion is
10% of reverse saturated drain current reduction 共⌬Idsat / Idsato
Rev兲 throughout the paper or 1 k⍀ ␮m of absolute change in
inverse transconductance 共W / gmmax-lin兲 when specified. Because high gate voltage 共Vgs 艌 Vds兲 can also induce electron
trapping in medium to thick gate insulators,42,43 we focus on
ultrathin 共1.7 nm兲 to thin gate oxides 共3 nm兲 with a waiting
time before measurements 共tw = 60 s兲 in order to stay mostly
sensitive to 共fast兲 interface traps.44 Moreover, low noise frequency measurements do not evidence 共border兲 oxide traps
creation in our samples tested under the same HC stress conditions, which points out that ⌬Nit represents the main con-
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
Lifetime τ (a.u.)
4
1.0E+04
10
8
1.E+08
10
0.9V
1V
1.1V
1.21V
1.4V
1.6V
1.8V
2V
1/(τ x Ids), 1/(τ x Ids²) (a.u)
Vds =
6
1.0E+06
10
2
1.0E+02
10
L=0.04µm
0.8V<Vgs <2.8V
0
1.0E+00
10
0
1+
冑
2共Vg − Vtat兲
1+
mEcL
,
3
4
5
6
Edom (eV)
共28兲
where Vo is a halo-based potential, Ec is the critical field for
velocity saturation, and m is a coefficient related to the body
effect. When Vgs increases at a fixed Vds, Ids and Vdsat increase which leads to a VEFF decrease. That is why Fig. 7 can
be roughly divided in two parts depending on the energy
impact: the high and low energy ranges.
a. Channel hot carrier (CHC) modes (SVE and EES) and
STM data. Concerning the high energy range where the reso-
nance can be excited by a single carrier 共SVE and EES兲, the
ratios Rb / Ids and Rb / Ids2 are related to the product PexcPcou
according to Eqs. 共15兲 and 共16兲
RbSVE
⬀ PexcPcou
Ids
Thin oxide, SVE mode
Thin oxide, EES mode
Thick oxide, SVE mode
Thick oxide, EES mode
Nit generation rate Sit
2
with Vdat
2共Vg − Vtat兲/m
2
10
1.E+02
0
tribution in the damage mechanism. By studying the HC
degradation in our 65 nm node devices covering, a wide Vgs
stress range per Vds stress, we have evidenced three
regimes45 similar to a previous team.35 However, we showed
that the third part of their EES-based model cannot explain
our experimental dataset.45 At this time, we proposed MVE
phenomenon as another explanation than EES for the device
lifetime decrease at high Vgs. In this paper, we report data
collected on 45 nm node device using a wider range of Vgs
and Vds stressing voltages than in previous studies.35,45 It
allows reaching higher Ids values 共Fig. 7兲. Figure 7 shows
that the Vds influence 共i.e., carrier energy兲 is important a low
Ids, whereas it is much reduced at higher Ids. This behavior
change can also be explained in terms of energy variation.
The defect generation rate Rb is dominant at a given energy
Edom which can be determined by considering an effective
potential VEFF from the drain to channel pinch-off point,46
=
4
1.E+04
10
0.002
FIG. 7. 共Color online兲 Evidence of two energy ranges: Vds dependence is
strong at moderate Ids 共SVE and EES兲, whereas it is reduced at higher Ids
共MVE兲 共n-MOS, Tox = 1.7 nm兲. The degradation criterion is 10% of
共⌬Idsat / Idsato Rev兲.
VEFF = V0 + Vds − Vdat
6
1.E+06
10
10
1.E+00
0.001
Ids/W(A/µm)
共29兲
FIG. 8. SVE and EES modes: the whole dataset 共for both thin 共1.7 nm兲 and
thick 共5 nm兲 oxides兲 follows a single generation rate function Sit. Equation
共32兲 is adjusted to experimental data 共line兲.
This product is rebuilt experimentally by plotting
1 / 共␶Ids兲 and 1 / 共␶Ids2兲 versus their respective Edom, considering that Edom is close to the energy of the knee 共i.e., Edom
= qVEFF兲 for SVE, whereas it is around 2qVEFF for EES.33
Our experimental dataset shown in Fig. 8 has been adjusted
by a single empirical law similar to the one used to describe
the impact ionization,40,45
PexcPcou ⬀ B共E − EB兲 pit .
共31兲
Here, Pit is around 11 and EB = 1.5 eV. It is worth noticing that EB is the same than that found with NBTI data, and
if its value is modified with the same ␴ dispersion than the
one obtained with NBTI results, it is still possible to fit the
data as well. This desorption yield equation, which is proportional to the Nit number generated per carrier, allows describing STM data as well.10,30,38 The STM yield shows a threshold at a sample bias around 6 V 共Fig. 9兲. This has been
explained as the threshold between DEE at high voltage and
MVE at lower voltage because of the high drain and voltage
dependence of the lowest voltage part. Figure 9 shows that
the yield dependence in the low voltage range is perfectly
modeled by our PexcPcou function 关Eq. 共31兲兴 using the same
power law and Nit threshold energy as for a HC degraded
transistor 共11 and 1.5 eV, respectively兲. The first conclusion
is that this voltage and current dependent part can be explained by SVE and/or EES. Thus, MVE is not reached on
this figure because the energy is too high even at 4 V and the
threshold happens between DEE and SVE. The second point
concerns the DEE part. According to Eq. 共6兲, the yield
Desorption yield (atoms/electron)
114513-7
-5
1.E-05
10
-6
1.E-06
10
-7
1.E-07
10
-8
1.E-08
10
-9
1.E-09
10
STM data
Sit
-10
1.E-10
10
and
3
4
5
6
7
8
9
10
Sample voltage (V)
RbEES
⬀ PexcPcou .
Ids2
共30兲
FIG. 9. The low voltage part of STM desorption yield for hydrogen at
300 K is model by our Sit function. STM data are from Ref. 38.
114513-8
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
8
1.E+08
10
6
1.E+06
10
Vds =1.6V
-40°C
0.8V<Vgs <2.8V
25°C
τ (a.u.)
1.E+06
10
4
1.E+04
10
Lifetime
Lifetime τ (a.u.)
6
2
1.E+02
10
0
1.E+04
10
T
2
1.E+02
10
W/L=10/0.04
Vds =1.6V
W/L=10/0.06
W/L=10/0.05
W/L=10/0.045
W/L=10/0.04
W/L=10/0.035
0
1.E+00
10
125°C
4
0.6V<Vgs <2.4V
0
10
1.E+00
0.001
Ids/W (A/µm)
0
0.002
0.001
Ids/W (A/µm)
0.002
FIG. 10. 共Color online兲 Evidence of a pure single Ids dependence in MVE
mode independent on Vgs and L in contrast with SVE and EES mode behavior 共n-MOS, Tox = 1.7 nm兲.
FIG. 12. 共Color online兲 Device lifetime decrease with increased temperature
in MVE mode 共n-MOS, Tox = 1.7 nm兲.
RDEE / I is proportional to Pexc which means that Pexc is independent of the carrier energy 共Fig. 9兲. Finally, the energy
dependence of PexcPcou is only due to Pcou. To draw a parallel with SMVE, this occupation probability of the last level N
after coupling 共Pcou兲 can also be defined as Sit, the scattering
rate of the incident carrier directly with the last level to be
able to create a Nit,
The L effect on Ibs / Ids is still visible even at high Ids. It
means that energy impact does not disappear totally at high
Ids and that degradation increase should be due to a different
mechanism. Figure 12 also points out another nonordinary
behavior: the high current degradation is activated with an
increased temperature. It is not the case for classical CHC
mode as seen on thicker 32 Å oxide in Fig. 13: at low Ids,
HC degradation is reduced when temperature increases,
whereas it is the contrary at high Ids. T effects and high
current mode distinction 共Fig. 14兲 are also similar on p-MOS
excepted that CCC mode is dominant at smaller Ids than in
n-MOS, as holes gain less energy than electrons. We checked
that p-MOS data de not interfere with NBTI degradation
since they have been collected at 25 ° C and for a maximum
兩Vgs兩 ⬍ 兩Vds兩 with a waiting time before measurements 共tw
= 60 s兲. Therefore we are cautious with respect to the possible influence of NBTI damage which is known to be relative to a recoverable part attributed to the balance between
hole trapping 共E⬘ center兲 and discharging from the tunneling
distance of the substrate 共and gate兲.24,26,44
To go more deeply into CCC mode modeling, we collected data with Vgs much larger than Vds 共Vgs 艌 Vds + 1 V兲.
The Idlin degradation is extracted after 20 000 s of stress instead of the lifetime extraction for a given criterion 共as 10%兲
which could be approximate for small Vds where the degradation is smaller than the criterion. In terms of bond breaking
rate equation 关Eq. 共27兲兴, the beginning of the MVE range
corresponds to Sit共Ids / e兲 ⬇ we exp共−ប␻ / kBT兲. All parameters
Sit ⬀ B共E − EB兲 pit .
共32兲
This Sit function is valid when one single carrier interacts with a bond; it merges both SVE and EES mode data,
generally named CHC modes.
b. CCC mode (MVE). In the highest Ids range in Fig. 7,
carrier energy is typically even lower. Nevertheless, the degradation is worse than in higher energy range. In analogy
with CHC mode, this mode is named channel cold carriers
共CCC兲 mode. Experimental HC lifetimes ␶ acquired for one
Vds but several channel lengths 共L兲 show large channel length
dependence at low Ids, whereas this dependence is almost
gone at higher Ids 共Fig. 10兲. To check if it is not due to a
specific energy behavior, Ibs / Ids is followed for the same
stress conditions than in Fig. 10 共Fig. 11兲. The ratio Ibs / Ids
gives an energy indicator since we have40,46
Ibs
⬇ Sii共qVEFF兲 ⬀ 共E − ␸ii兲 pii ,
Ids
共33兲
where Sii is the scattering rate for impact ionization, ␸ii is the
impact ionization energy threshold 共 ⬇ 1.1 eV兲, and pii ⬇ 4.5.
-5
1.E-05
10
W/L=10/0.16; 25°C
W/L=10/0.12; 25°C
W/L=10/0.16; 125°C
W/L=10/0.12; 125°C
Vds =1.6V
3
1.E+03
10
Lifetime τ (a.u.)
0.8V<Vgs <2.8V
Ibs/Ids
4
1.E+04
10
T
2
1.E+02
10
-6
10
1.E-06
1
1.E+01
10
W/L=10/0.06
W/L=10/0.05
W/L=10/0.045
W/L=10/0.04
W/L=10/0.035
-7
1.E-07
10
0
V ds =3V
1.E+00
10
0
0.001
Ids/W (A/µm)
0.002
FIG. 11. 共Color online兲 Impact ionization ratio for the same conditions as in
Fig. 10 showing that L effect on Ibs / Ids persists at high Ids 共n-MOS, Tox
= 1.7 nm兲.
T
0.5V<V gs <3.5V
0
0.0005
0.001
Ids/W (A/µm)
0.0015
FIG. 13. 共Color online兲 Classical HC thermal activation in n-MOS Tox
= 3.2 nm for high energetic carrier related modes 共SVE and EES兲 but same
thermal activation than found in MVE mode in n-MOS, 1.7 nm 共Fig. 12兲
共lines are guides for the eyes兲.
114513-9
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
Bond breaking rate R bMVE (s )
pMOS, 25°C
11
1.E+11
10
-1
Vds =
7
1.E+07
Lifetime
τ (a.u.)
10
-1.21V
-1.4V
-1.6V
-1.8V
-2V
5
1.E+05
10
3
1.E+03
10
W/L=10/0.04
0.6V<|Vgs |<2V
1
1.E+01
10
0
6
1.E+06
10
1
1.E+01
10
-4
1.E-04
10
-9
1.E-09
10
experimental data
bending model
stretching model
-14
1.E-14
10
0.0005
-7
1.E-07
10
0.001
-5
1.E-05
10
|Ids/W| (A/µm)
-1
1.E-01
10
共34兲
When we substitute it in Eq. 共27兲, the model is in good
agreement with the data 共Fig. 17兲. In this figure, we also
added the same data get at two others temperatures, showing
an increase in the bond breaking rate RB with a temperature
increase. It allows determining the thermal activation barrier
Eemi as 0.26 eV since it is the only unknown parameter. Even
if we cannot claim rather H is emitted toward Si or SiO2, this
value is in the same range than previous results on H diffusion 共0.2– 0.56 eV兲.47–50
Finally, a simplified equation of RbMVE 关Eq. 共27兲兴 is rewritten including the temperature effect as the term
we exp共−ប␻ / kBT兲 is neglected compared to SMVE共Ids / e兲
while this last term is negligible compared to we finally leading to
TABLE III. Chosen parameters for Si–H bond at the Si/ SiO2 interface from
Tables I and II.
Parameters
Stretching
Bending
EB 共eV兲
q␻ 共eV兲
we 共 = 1 / ␶e兲 共ps−1兲
Phonon decay path
2.5
0.25 共or 2072 cm−1兲
1 / 295 共5 K兲
2L0 + 3LA
共2 ⫻ 521+ 3 ⫻ 343 cm−1兲
1.5
0.075 共or 610 cm−1兲
1 / 10 共5 K兲
TO+ TA
共460+ 150 cm−1兲
1
⬀ RbMVE
␶MVE
冋
冉 冊册
⬀ 共Vds − ប␻兲0.5
Ids
W
冉 冊
EB/ប␻
exp
− Eemi
.
k BT
共35兲
That can be simplified as ប␻ = 75 meVⰆ qVds giving for a
given temperature
1
␶MVE
冋 冉 冊册
0.5
⬀ RbMVE ⬀ Vds
Ids
W
EB/ប␻
共36兲
.
The validity of this relationship is presented 共Fig. 18兲
where experimental data obtained from device lifetime extraction under MVE mode 共␶MVE兲 are plotted as a function of
the drain current 共Ids / W兲 showing a weak Vds dependence
共left兲, while this effect disappears when one plots the product
EB/2ប␻
␶MVEVds
on the ordinate scale 共right兲. The experimental
data give an exponent EB / ប␻ = 17.5 corresponding to EB
= 1.4 eV and ប␻ = 80 meV in good agreement with the theoretical values obtained in the range of 1.4 eV⬍ EB ⬍ 1.6 eV
and 75 meV⬍ ប␻ ⬍ 80 meV, i.e., with exponents ranging
between 21 and 17.5.
c. Complete Hot Carriers to Cold Carriers modeling. A
simple equation can also be associated to the two other HC
modes. Because Sit function has been related to
共Ibs / Ids兲m,35,46 Eqs. 共29兲 and 共30兲 give
1
␶SVE
⬀ Ids
-1 -1
in Eq. 共27兲 have been defined for both bending and stretching
modes in the first section and summed up in Table III. Figure
15 shows that experimental data are in agreement with bending mode whereas stretching mode cannot relevantly explain
our results. In this figure, the electron-phonon scattering rate
SbMVE has been set at 10−5, which explains the difference
between the slopes of the data and the model. In fact, the
SMVE depends slightly on the electron energy 共 ⬃ E0.5兲 as
shown by Monte Carlo simulations,40 i.e., on the apply voltage qVds 共Fig. 16兲. Hence, SMVE is obtained by a similar
expression than Sit and Sii 关Eqs. 共32兲 and 共33兲兴 giving
FIG. 15. Comparison between experimental data 共25 ° C兲 and bond breaking
rate model 关Eq. 共27兲兴 using stretching and bending vibrational mode parameters of Table III 共SMVE is kept constant兲.
Electron-phonon scattering SMVE (e s )
FIG. 14. 共Color online兲 Evidence of a single Ids dependence for MVE mode
in p-MOS with the use of a waiting time 共tw = 60 s兲 in order to remain
mostly on the permanent damage 共Nit兲 for the sake of comparison to n-MOS
共Fig. 7兲 with Tox = 1.7 nm.
SMVE ⬀ 共Vds − ប␻兲0.5 .
-3
1.E-03
10
Ids/W (A/µm)
冉 冊
Ibs
Ids
m
共37兲
,
-4
1.E-04
10
-5
1.E-05
10
-6
1.E-06
10
-7
1.E-07
10
0
1
2
3
4
5
Electron energy (eV)
FIG. 16. Electron-phonon scattering rate SMVE simulated by Ref. 40 using
Monte Carlo.
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
6
1.E+06
10
TABLE IV. Summary of experimental parameter extractions from Eq. 共39兲
obtained for the best fits of n-MOS and p-MOS devices from the previous
figures 共Figs. 18 and 19兲.
25C
75C
125C
-1
Bond breaking rate R bMVE (s )
114513-10
4
1.E+04
10
2
1.E+02
10
0
1.E+00
10
n-MOS
p-MOS
W/L = 10/0.04
0.6V<Vds <1.4V
0.0015
0.002
0.0025
Ids/W (A/µm)
FIG. 17. 共Color online兲 Comparison between experimental data at three
temperatures and bond breaking rate model 关Eq. 共27兲兴 using bending vibrational mode parameters of Table III and SMVE of Fig. 16.
␶EES
2
⬀ Ids
冉 冊
Ibs
Ids
m
共38兲
.
As a consequence, a full modeling of cold to hot carrier
degradation is determined by device lifetime extraction 共␶兲
for all the stressing conditions by assuming that all three
degradation modes compete in parallel, which gives a general equation as 1 / ␶ = 兺i1 / ␶i,
␶=
KSVE
冉 冊冉 冊
Ids
a1
W
Ibs
1
m
+ KEES
Ids
a1
a2
a3
Eemi
共eV兲
2.7
1.3
1
1
2.5
3
17.5
12
0.26
0.26
1.8V<Vgs <2.4V
-2
1.E-02
10
0.001
1
m
Device
冉 冊冉 冊
Ids
W
a2
Ibs
Ids
m
a3/2
+ KMVEVds
冉 冊 冉 冊
Ids
W
a3
− Eemi
exp
,
k BT
共39兲
where KSVE, KEES, and KMVE are three constants determined
in their respective dominant mode. We have validated this
modeling for various gate-oxide thickness 共Tox = 5, 3.2, and
1.7 nm兲 under a large set of voltages conditions 共Vgs and
Vds兲, temperatures and device geometries 共W / L兲, both on
n-MOS and p-MOS. Table IV summarizes all the parameters
extracted from the previous figures using Eq. 共39兲 for both
n-MOS and p-MOS. The complete modeling explains also
the change in the damage behavior through the gate-length
dependence 关Fig. 19共a兲兴 where at small Ids it originates from
the Ids and Ibs gate-length dependence, while at higher Ids the
lifetime becomes only dependent on Ids magnitude on not on
L anymore. The influence of temperature is explained by the
temperature dependence of each degradation modes Figure
19共b兲 in ultrathin gate-oxide 共1.7 nm兲 n-MOS. Under EES
corresponding to mode 2, temperature effect is included into
the intrinsic temperature dependence of both currents Ids and
Ibs 共so Ibs / Id兲 while entering into MVE regime 共mode 3兲; the
activation energy is Eemi = 0.26 eV according to a large set of
experimental data and previous results on H diffusion.47–50
Equation 共39兲 leads to the definition of an age function
calculated by the same way than proposed in Ref. 51,
8
1.E+08
10
Vds =1.6V
Vds =
6
1.E+06
Lifetime
τ
(a.u.)
10
4
1.E+04
10
0.9V
1V
1.1V
1.21V
1.4V
1.6V
1.8V
a
Lifetime τ (a.u.)
0.8V<Vgs <2.8V
2
1.E+02
10
0
1.E+00
10
6
1.E+06
10
4
1.E+04
10
2
1.E+02
10
W/L=10/0.06
W/L=10/0.05
W/L=10/0.045
W/L=10/0.04
W/L=10/0.035
0
1.E+00
10
0
0.001
0
0.002
0.001
Ids/W (A/µm)
0.002
a
V ds =
6
1.E+06
10
4
1.E+04
10
2
1.E+02
10
b
0.9V
1V
1.1V
1.21V
1.4V
1.6V
1.8V
W/L=10/0.04
Vds =1.6V
6
1.E+06
10
Lifetime τ (a.u.)
τ x VdsEB/(2ħω) (a.u.)
Ids/W (A/µm)
0.6V<Vgs <2.4V
4
1.E+04
10
T
- 40°C
2
1.E+02
10
slope = -17.5
0
25°C
CHC mode
CCC mode
complete model
0
1.E+00
10
0
0.001
0.002
Ids/W (A/µm)
FIG. 18. 共Color online兲 共a兲 A weak Vds effect is observed on the experimental dependence of the MVE mode extracted from Fig. 7 which is explained
by the simplified relationship of RbMVE 关Eq. 共36兲兴. 共b兲 Once device lifetime
EB/2ប␻
is plotted as ␶MVE. Vds
as a function of Ids / W, a single and universal
degradation dependence is observed independent of the gate length in 40 nm
n-MOS devices 共VDD = 1.1 V, Tox = 1.7 nm, T = 25 ° C兲.
125°C
10
1.E+00
0
0.001
Ids/W (A/µm)
0.002
b
FIG. 19. 共Color online兲 共a兲 Comparison between the complete modeling
共lines兲 and the experimental data of Fig. 10 共symbols兲 showing a clear
change in the gate-length dependence between mode 2 共EES兲 and mode 3
共MVE兲. 共b兲 Comparison between experimental data of Fig. 13 and the complete modeling obtained for three temperatures in L = 40 nm n-MOS with
Tox = 1.7 nm.
114513-11
J. Appl. Phys. 105, 114513 共2009兲
Guerin, Huard, and Bravaix
∆(1/gmmax) ()
1000
8
0.5
t
nMOS, Tox = 32nm, 25°C
100
10
1
1,E-04
10-4
1,E-02
10-2
1,E+00
100
1,E+02
102
Age
FIG. 20. Degradation of the peak transconductance ⌬共1 / gm兲 in linear mode
as a function of the age modeling with Eq. 共40兲 and plotted for 30 different
stress conditions 共distributed in SVE, EES, and MVE modes兲 showing a
single power law trend 共L = 0.15 ␮m, Tox = 3.2 nm, and T ° = 25 ° C兲.
Age =
冋 冉 冊冉 冊
冉 冊冉 冊
冉 冊 冉 冊册
Ids
t
= t KSVE
␶
W
a3/2
+ KMVEVds
Ids
W
a1
Ibs
Ids
a3
exp
m
+ KEES
Ids
W
− Eemi
k BT
.
a2
Ibs
Ids
m
共40兲
The Age function allows merging degradations of the three
modes on the same figure 共Fig. 20兲. This figure confirms that
the time dependence follows a power law with an exponent
of 0.5, for all stressing conditions.
IV. CONCLUSIONS
Interface state creation at Si/ SiO2 has been related to
several mechanisms depending on the stress conditions and
the carrier energy. NBTI data are explained by a reduction in
the bond energy with the oxide field. This barrier is equal to
1.5 eV without oxide field. The same threshold energy is
found under hot carrier damage, whereas the underlying
mechanism is totally different. HC mode is related to the
excitation of bond vibrational modes induced by carriers. If a
carrier has enough energy to excite the adsorbate resonance
state, its energy may be transfer by coupling and lead to
SVE. If single carriers have not enough energy to reach the
resonance state, they can cooperate through EES and then
induce SVE. The third HC mode happens when carrier energy is so low that the resonance is not possible even after
EES. In this case, carriers may induce a step by step vibrational ladder climbing if the current is high enough 共MVE兲.
A modeling of the bending MVE mode, based on a review of
electronic and vibrational mode properties of the Si–H bond,
has been applied to a wide HC stress experimental dataset.
Finally, a complete equation taking into account the three HC
modes 共SVE, EES, and MVE兲 has been proposed, it enables
an accurate description of the CHC to CCC modes including
the temperature effect in actual technology nodes.
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