Ken Myers
The Grasser Model
Abstract
This paper will discuss a current model of the negative bias temperature instability
(NBTI) known as the Grasser Model. This model features the concept that the culprit
behind NBTI is not an interface state, but a combination of interface states and oxygen
vacancies (𝐸′ centers). The basis of Grasser’s argument relies heavily on the HarryDiamond-Laboratories (HDL) model regarding the topic of switching oxide traps.
Nearing the end of the paper, the concept of coupled interface state generation
involving both 𝐸′ and PbH centers will be covered through a modification of the HDL
model.
Introduction
BTI, both positively and negatively biased, is one of the most influential reliability
issues in current metal oxide silicon (MOS) technology. The phenomenon known as
NBTI is caused by applying a negative bias to the gate of a p-channel MOS field effect
transistor (pMOSFET) at high temperatures. The results of this instability are shifts in
the threshold voltage of the device as well as significant losses in the drain current. The
existence of NBTI has been known for many years, but there has not been a significant
amount of effort put into modeling such a phenomenon until more recently. As devices
have become more efficient, the complications of NBTI have been noted with higher
priority, leading to the current search for an accurate model. Currently, two models
stand with the most backers: the Grasser model and the reaction-diffusion model. In this
paper, the concepts behind the Grasser model as well as the drawbacks in other
models will be provided.
Prefacing the Model
Before diving into the meat of the subject, Grasser et al.[1] introduces four
observations that are involved in NBTI experimentation: scalability, bias and
1
temperature dependence of stress, asymmetry between stress and relaxation, and the
bias dependence of recovery.
Using the extended Multi-Stage Multicast (eMSM) scheme, two sets of data
were taken in order to find a scaling factor in both the stress degradation versus time
and the voltage recovery versus time. Fig. 1 depicts data drawn from the eMSM scheme
for stress time while Fig. 2 relates to recovery time. Note that in Fig. 1 the data overlaps
almost completely for the SiON pMOSFET when the appropriate scaling factors are
applied. In Fig. 2 there is extremely similar overlap for the same scaling factors in SiON.
These findings lead Grasser et al. to discard the notion of multiple independent
mechanisms.[1]
Fig. 1: Degradation of the drain current over
multiple stressing conditions. Scaling factors
are given in the box in the top left. Adapted
from Grasser et al.[1]
Fig. 2: Recovery of the voltage shift over
multiple stressing conditions. Adapted from
Grasser et al.[1]
2
In order to test for the bias and temperature dependence of stress in NBTI,
Grasser et al.[1] subjects devices to short stresses of less than -1 V for two seconds.
From this small test, it is noted that the power-law temperature dependence can be
approximated as an Arrhenius process with activation energy (EA) ≈ 70 meV. Another
topic to be brought up through these tests is that elastic tunneling can no longer be
considered due to its lacking in temperature dependence. The final and most pertinent
important observation is that the ability to scale for bias and voltage allows the
conclusion that the single mechanism for the model now contains two stages. [1]
Fig. 3 (left): Shift in threshold voltage versus time for 8 different biases at 125°C
(right): Using the 8 biases, the prefactor ratios Bs/Br were plotted versus temperature.
Adapted from Grasser et al.[1]
As we have discussed in class, there is a huge difference in stress time versus
relaxation time. Plotting these two graphs at a temperature of 125°C results in Fig. 3
(left). After extracting prefactors (Bs for stress and Br for recovery) from their data,
Grasser et al.[1] put together Fig. 3 (right), showing the ratio of Bs/Br to be
approximately 2.5. The error in the value shown in Fig. 3 (right) can be appropriated
from the mobility error in the stressing prefactor. [1]
3
Recovery time of a stressed device has shown a significant dependence on bias
applied during recovery [2]. In NBTI, positive bias improves the recovery time. This
relationship is described by the idea of hydrogen drift and/or the impact of valence band
and interface states on hole trapping [1,2]. This impact is due to both valence band and
interface states during stress. In hole trapping and detrapping, both types of states are
capable of contributing electron pathways, aiding the capture of holes in oxide traps.
When the Fermi level exists in the valence band (negative bias), only valence band
states are able to assist in the recovery of oxide traps. Only after moving the Fermi level
to the conduction band are the interface states able to assist in recovery. This shift is
shown in Fig. 4. [2]
Fig. 4 (right): Improvement of recovery
time with increasing bias. The
densities of recoverable states are
given as a function of the ratio of
stress time to recovery time. As bias is
increased,
the
recovery
time
compared to stress time declines
significantly. Adapted from Huard et
al.[2]
4
Other Models
When reviewing the cases made by other models, two ideas stand out: reactiondiffusion and elastic hole trapping (in some versions, both within the same model) [3].
The main idea behind reaction-diffusion is the back diffusion of H2, a bias independent
process. As shown in Fig. 4, this cannot be true due the the obvious bias dependence
of recovery.
Fig. 5: This graph shows the shift in threshold voltage versus the
recovery time taking place after multiple stressing times. Note the initial
drop, taking place in under 1s. Adapted from Shen et al.[4]
The next downfall to reaction-diffusion theory is the initial recovery response time
As shown in Fig. 5, at least 60% of all stress damage is recovered within the first
second. This holds true up through (and most likely past) 1000 seconds of stressing
time at the given stress of -2.4 V. The drawback in reaction diffusion theory in this case
is the simple fact that the hydrogen that has drifted into the oxide layer from the
interface is not capable of back diffusion at such a high rate. In order to demostrate this,
5
Shen et al.[4] solved the reaction-diffusion equations numerically for the recovery times
(Fig. 6). As shown, when the recovery time equals the time that the device was
stressed, only about 50% of the interface states have been recovered. From these
results, it seems impossible for hydrogen to be recovering in the short amount of time
observed experimentally, creating the necessity for a new model. [4]
Fig. 6: Density of interface states versus recovery
time in multiple stress-time conditions. (a)H0 diffusion
(b) H2 diffusion. Adapted from Shen et al.[4]
Along with reaction-diffusion, we discard the majority of hole-trapping models.
This is due to the fact that many of these models rely on the concept of elastic hole
trapping, which has already been rejected due to its first-order temperature
6
dependence. Secondly, hole-trapping models predict that the stressing prefactor and
the recovery prefactor will be almost equivalent. This is similar to stating that stress and
recovery behaviors will act in a similar fasion, which is shown to be false by Fig. 3. The
lack of convincing data for either of these concepts brings us to the thought of a
combined hole-trapping and interface state process. On-the-fly charge-pumping
measurements give results that allow the belief of fast recovery in interface states. This
opens up the door to a new model. [1]
Hole-Trapping and the 𝑬′ Center
From various 1/f-noise studies and modeling attempts, Grasser puts together the
idea that oxygen vacancies, or 𝐸′ centers, are involved in hole capture. This is due to
the concept that “holes can be captured via a (thermally activated) multiphonon
emission (MPE) process into deep near-interfacial states/border traps.” The reason that
we can consider hole-trapping in this format is due to the temperature activation
required by the tunneling process. [1]
In order to avoid linear field dependence due to MPE, the model must be
changed to an extension of MPE in large electric fields. This extension is known as
multiphonon-field-assisted tunneling (MPFAT). [1]
Moving onto data, Campbell et al.[5] brings to light interesting observations made
through SDR measurements. As shown in Fig. 7, a signal at 𝑔 = 2.0007 ± 0.0003 is
detected in post-stress measurements. This signal is attributed to and oxide vacancy (𝐸′
center) located near the Si/SiO2 interface. This observation enforces the idea that 𝐸′
centers are involved in the NBTI phenomenon.[5]
7
Fig. 7: SDR measurements taken both
pre and post stress of a pMOSFET at
200°C and -5.7 V
Adapted from Campbell et al.[5]
In this model, Grasser et al.[1] claims that the ideal candidate for the 𝐸′ defect in
question is the 𝐸𝛾′ center. The 𝐸𝛾′ center is a defect “thought to be created when a hole
is trapped in the precursor structure”[1]. This precursor structure is also known as a
neutral oxygen vacancy. Once a hole is trapped (a Si/Si bond is broken), the Si atoms
take a new equilibrium position, expanding the distance between the atoms until the
surrounding crystal structure relaxes. An important function of the 𝐸𝛾′ centers generated
by this process is that the center can be repeatedly charged and discharged. This is
possible due to the dipole state in a neutral defect once the 𝐸𝛾′ centered has emmitted
its captured hole(Fig. 8, state 3). Fig. 8, known as the HDL model, shows this process
along with the possibility that instead of recapturing a hole, the neutral defect may return
to its precursor state (Fig. 8, state 1). The neutral 𝐸𝛾′ center is only able to return to state
1 after a long enough time, at which the structure will relax for a second time and the
defect will be healed. [1]
8
Fig. 8: The flow of states shown here is known as the HDL model for switching oxide traps.
State 1 shows the precursor state from which a hole is captured and creates our 𝐸𝛾′ center at
state 2. Our current defect will proceed to emit a hole, creating the neutral defect. At this point
there are two options: the defect can recapture a hole, or, if enough time has passed, will
relax back to the precursor. Adapted from Grasser et al.[1]
Another note-worthy set of observations is brought up by Reddy et al.[6] Through
CV measurements, this group was able to show the relationship between NBTI and
donor-like defects. These measurements assist two of the basic arguments in the
Grasser model. The first is that this model uses the 𝐸𝛾′ center, which is a donor-like
defect and fitting the description. Secondly, reaction-diffusion states that NBTI is mainly
due to interface states (Pb centers) which are not fully donor-like defects. [1,6]
The Current Model
After a hole trapping event occurs in an oxygen vacancy, a positive 𝐸′ is
generated as shown in Fig. 9 at state 2. The left half of the 𝐸′ center is denoted as an
unpassivated silicon dangling bond while the right half contains the captured hole.
Grasser et al.[1] then introduces the concept brought to light by P. M. Lenahan which
assumes that hydrogen passivated silicon dangling bonds previously existed at the
interface. When the device is stressed, the 𝐸 ′ centers near the interface are generated.
9
It is accepted that the hydrogens from the passivated interface states will migrate in
favor of these newly formed dangling bonds. At this point, there are two options for the
hydrogen atom: The atom can stay at the 𝐸′ center and maintain the observation of a
1:1 correlation between fixed oxide charge and interface. On the other hand, the
hydrogen could use the 𝐸′ center as a stepping-stone to migrate further from the
interface. The entirety of the modified HDL model is shown in the figure below, in which
the added state (state 4) depicts the movement of the hydrogen from the P bH to the 𝐸′
center, pushing the silicon dangling bond to the interface. At this point, the hydrogen
may move back to the Pb center, recreating state two from which the complex may
return to the neutral state and relax back into its precursor (state 1). [1]
Fig. 9: (Left) This image shows the flow of the modified HDL model. As shown
before in Fig. 8 we have the three main states with the introduction of the hydrogenpassivated interface state (PbH). Also added in is the possibility of state 4 in which
the hydrogen from the PbH moves to the 𝐸′ center. Adapted from Grasser et al.[1]
10
References
[1] T. Grasser, B. Kaczer, W. Goes, T. Aichinger, P. Hehenberger, and M. Nelhiebel, “A
two stage model for negative bias temperature instability,” inProc. Int. Rel. Phys. Symp.,
2009, pp. 33–44.
[2] V. Huard, C. Parthasarathy, N. Rallet, C. Guerin, M. Mammase, D. Barge, and C.
Ouvrard, “New Characterization and Modeling Approach for NBTI Degradation from
Transistor to Product Level,” in Proc. IEDM, 2007, pp. 797–800.
[3] T. Grasser, B. Kaczer, W. Goes, et al., “The paradigm shift in under-standing the
bias temperature instability: From reaction-diffusion to switching oxide traps,” IEEE
Trans. Electron Devices, vol. 58, no. 11, pp. 3652–3666, Nov. 2011.
[4] C. Shen, M.-F. Li, C. E. Foo, T. Yang, D.M. Huang, A. Yap, G.S. Samudra, and Y.-C.
Yeo, “Characterization and Physical Origin of Fast Vth Transient in NBTI of pMOSFETs
with SiON Dielectric,” in Proc. IEDM, 2006, pp. 333–336.
[5] Campbell J, Lenahan P, Krishnan A, Krishnan S. Observations of NBTI-induced
atomic-scale defects. IEEE Trans Dev Mater Rel 2006;6(2):117–22.
[6] V. Reddy, A.T. Krishnan, A. Marshall, J. Rodriguez, S. Natarajan, T. Rost, and S.
Krishnan, “Impact of Negative Bias Temperature Instability on Digital Circuit Reliability,”
in Proc. IRPS, 2002, pp. 248–254.
11