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7.3 Recoverable Component of Bias Temperature Instabilities

7.3.1 Modeling of Single Trap Characteristics

The most basic formulation for charge transfer reactions is provided by the two-state model which will be introduced first. As will be pointed out the two-state model can not describe all features observed in the experiments and is therefore extended by two metastable states leading to the well-established four-state NMP model [108, 89]. It has to be noted that the description of the trapping kinetics of single defects using the four-state NMP model is fully consistent with DFT results of likely defect structures.

7.3.2 Two State Modeling

The simplest formulation for charge transfer reactions, i.e. charge capture and emission, is provided by a two state model, see Figure 7.5.

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Figure 7.5: A simple two state Markov process (left) is used to describe RTN signals (right). The transition rates are the reciprocal values of the mean capture and emission time and define the transition probability. It has to be noted that the transition times are exponentially distributed around their mean values, thus in case of equal forward and backward transition rates the RTN is not equal a rectangular signal with a duty cycle of \( \SI {50}{\percent } \).

Initially, RTN signals have been studied to understand the bias and temperature dependent characteristics of the discrete charge capture and emission events [124]. As can be seen, the voltage signal switches between two discrete voltage levels.

The straightforward stochastic description for the two stage process relies on the Markov model which assumes that

  • 1. each state is stable,

  • 2. the transitions are memoryless and

  • 3. only determined by its time independent transition rates.

The charge capture and emission time (math image) and (math image) for a two state defect can be calculated from the transition rates \( \kAB \) and \( \kBA \) using the relations [89]

(7.21–7.22) \{begin}{align} \tauc & = 1/\kAB \quad \mathrm {and}\\ \taue & = 1/\kBA . \{end}{align}

Therein the transition rates (math image) give the probability for the transition from state \( i \) to state \( j \) to occur within a unit time interval.

To provide a solid physical based explanation for charge trapping in oxides, the NMP theory is used [125, 117]. The NMP theory relies on solving the Schrödinger equation for a certain atomic configuration leading to the so called adiabatic energy surfaces. In the case of BTI the change of the charge state of a single defect considerably impacts the position of the surrounding atoms and the atoms. Thus, a system involving \( N \) atoms spans a \( 3N \)-dimensional space as each position of a single atom is given by its three-dimensional coordinates leading to adiabatic energy surfaces which are impossible to implement in device simulators. To simplify the complex adiabatic energy surfaces the atomic positions are reduced to one-dimensional configuration coordinates and the adiabatic energy surface is approximated by a harmonic oscillator. The potential energy surfaces for a two state defect are shown in Figure 7.6

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Figure 7.6:  The potential energy surfaces are shown for the two state process. The energy barriers for the forward and backward transitions are determined by \( \Vi \) and \( \Vj \).

and can be expressed as

(7.23–7.24) \{begin}{align} \Vi (q) & = \Vi + \ci (q - \qi )^2 \\ \Vj (q) & = \Vj + \cj (q - \qj )^2 \{end}{align}

thereby considering the states \( 1 \) and \( 2 \) as the neutral and the charged state, respectively. In the case of donor-like traps, i.e. hole traps, state \( 2 \) represents a positively charged state and in the case of acceptor-like traps, i.e. electron traps, state \( 2 \) is ascribed to a negatively charge state. The barrier \( \VBarrier \) between state \( i \) and \( j \) can get calculated in the classical limit by considering the intersection point of the parabolas and the difference in the energy levels \( \Vij =\Vj -\Vi \) together with \( \qij =\qj -\qi \), which yields

(7.25–7.26) \{begin}{align} \Vi (q) & = \Vi + \ci \dqij ^2 \\ \Vj (q) & = \Vi + \Vij + \cj (\dqij - \qij )^2 \{end}{align}

for the adiabatic potentials. Solving the set of equations for \( \dqij \) for the case of linear electron-phonon coupling (\( \ci =\cj =c \)) leads to the analytic solution

(7.27) \{begin}{align} \dqij =\frac {\frac {\Vij }{c}+\qij ^2}{2\qij } \{end}{align}

for the position of the intersection point and the NMP barrier

(7.28) \{begin}{align} \VBarrier (\qij )=\left (\frac {\Vij +c\qij ^2}{2\sqrt {c}\qij }\right )^2. \{end}{align}

The forward and backward rate, and thus the capture and emission times of the two state process can be calculated by

(7.29–7.30) \{begin}{align} \tauc & = \frac {1}{\kAB }=\frac {1}{\kO } \myexp ^{\beta \VBarrier (\qij )} \\ \taue & = \frac {1}{\kBA }=\frac {1}{\kO } \myexp ^{\beta (\VBarrier (\qji )+\Vi -\Vj )} \{end}{align}

with \( \beta =1/(\kB T) \) and the prefactor \( \kO \). The dependence of the capture and emission times on the difference of the energy levels between state \( 1 \) and state \( 2 \) is shown in Figure 7.7 with corresponding RTN signals.

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Figure 7.7:  The capture and emission times strongly depend on the energy levels \( \Vi \) and \( \Vj \). For \( \Delta V_{\mathrm {12}}=\Vi -\Vj <0 \), the backward rate is larger than the forward rate, i.e \( \tauc <\taue \), whereas \( \Vi >\Vj \) leads to \( \tauc >\taue \). The RTN signals are calculated for the respective regimes.

For \( \Vij <0 \) the forward rate exceeds the backward rate \( \tauc > \taue \) and thus the system is preferably in state \( 2 \). In contrast, \( \Vij >0 \) leads to \( \tauc < \taue \) and then state \( 1 \) is the preferred state. For \( \Vij =0 \) an equal forward and backward rate, \( \tauc = \taue \), is obtained.

So far, the two state charge transfer process has been considered in general terms. To describe the trapping kinetics of oxide defects, state \( 1 \) is assumed to be the neutral state and state \( 2 \) the charged state. Relying on the hole picture the physical process behind the transition \( 1\rightarrow 2 \) is ascribed to hole capture, i.e. electron emission process, and the transition \( 2\rightarrow 1 \) to the hole emission, i.e. electron capture process. In the case of an electron trap, the roles of electrons and holes are exchanged.

7.3.3 Four-State Non-Radiative Multiphonon model

During extensive single-trap studies employing TDDS on pMOSFETs, volatile defects which spontaneously disappeared and reappeared were found. Furthermore, it has been observed that single-trap emission times can be either (i) switching trap or (ii) fixed oxide trap like. For switching traps a strong bias dependent emission time is observed, whereas bias independent emission times are linked to fixed oxide traps.

In order to explain all features observed from single defects the four-state NMP is used, see Figure 7.8.

(image)

Figure 7.8:  The four-state NMP model is used to describe the response of the capture and emission times of single defects to varying biases and temperatures. In general, the model considers two stable states 1 (neutral) and 2 (charged) and two meta-stable states 1’ and 2’. These states are required for an accurate description of the bias dependence of the emission times as well as other features [87].

In principle, the four-state NMP model consists of two stable states 1 and 2 and two meta-stable states 1’ and 2’. The states 1/1’ are considered to be neutral while the states 2/2’ represents a charged defect. The charge capture events are described as a transition from state 1 to state 2 via the meta stable state 2’ and hole emission proceeds from state 2 to state 1 via either state 1’ or state 2’. The model implicitly distinguishes between

  • (i) switching traps with bias dependent emission times, following the pathway \( 2 \rightarrow 1 \rightarrow 1 \) and

  • (ii) fixed oxide traps with bias independent emission times, choosing the pathway via \( 2 \rightarrow 2’ \rightarrow 1 \).

The switching trap characteristics and its corresponding configuration coordinate diagram is visible in Figure 7.9.

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Figure 7.9:  The switching trap behavior is characterized by bias-dependent emission times (left). The four-state NMP model explains the capture and emission time characteristics well. The corresponding potential energy surfaces are shown (right). In order to achieve bias-dependent capture and emission times the transition rates must be dominated by the NMP barriers, thus require \( \EbiasCapture >\EthermalCapture \) and \( \EbiasEmission >\EthermalEmission \) [MWC20].

As can be seen a strong bias and temperature dependence is visible for the capture and the emission time. By using the four-state NMP model, the capture and emission time characteristics and its temperature dependence can be explained. As visible in the configuration coordinate diagram the charge capture event is described by a charge transfer reaction followed by a thermal transition via the pathway \( 1\rightarrow 2’\rightarrow 2 \). It has to be noted that a small thermal transition barrier \( \EthermalCapture \) with respect to a larger bias dependent transition barrier \( \EbiasCapture \) is present, leading to a smaller transition rate \( k_{2’2} \) compared to \( k_{12’} \). Thus the transition from state \( 1 \) to state \( 2 \) is primarily dominated by the latter. The bias dependent charge emission process follows the pathway \( 2\rightarrow 1’\rightarrow 1 \). Again a negligible thermal transition barrier \( \EthermalEmission \) compared to the bias-dependent transition barrier \( \EbiasEmission \) results in a bias-dependent emission time.

The charge capture transition for fixed traps follows the same pathway as obtained for switching traps, i.e. \( 1\rightarrow 2’\rightarrow 2 \), see Figure 7.10.

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Figure 7.10:  The bias-independent emission times are characteristic for fixed oxide traps (left). Using the four-state NMP model the capture and emission times and their temperature dependence can be nicely reproduced. The corresponding potential energy surfaces (right) show the capture pathway defined by the NMP barrier, i.e. \( \EbiasCapture >\EthermalCapture \), however, the emission pathway is primarily determined by the thermal barrier, thus \( \EbiasEmission >\EthermalEmission \) leads to the bias-independent emission time behavior.

In contrast to switching traps, the bias independent charge emission proceeds via the pathway \( 2\rightarrow 2’\rightarrow 1 \). This requires a smaller thermal transition barrier \( \EthermalEmission \) compared to the charge transfer barrier \( \EbiasEmission \).

Trap Position and Bias Dependence

Another important aspect is the determination of the location of the traps in the oxide. For this consider two traps, \( \trapA \) and \( \trapB \), placed at a different depths in the \( \SIO \) gate dielectric of a pMOSFET, see Figure 7.11.

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Figure 7.11:  The two traps \( \trapA \) and \( \trapB \) have different trap levels and different positions inside the gate dielectric. Depending on their depth the trap levels are shifted by a different amount when the gate bias changes. As a consequence, a different impact on the forward transition barriers is obtained (right). In summary, the closer the trap is to the gate the larger the shift of the trap level is and thus the smaller the NMP transition barrier gets [MWC20].

Without any loss of generality both traps are considered to be modeled via a simple two state process. At recovery bias conditions both traps are neutral and their trap levels are below the Fermi-level of the channel. When a stress bias is applied at the gate, the trap level shifts with respect to the Fermi-level of the channel depending on their trap depth due to the electric field in the oxide. As can be seen, \( \trapA \) is located closer to the gate and thus its trap level is shifted further compared to the trap level of \( \trapB \) which is located closer to the channel. The different changes in their trap levels cause different barrier heights \( \EbiasCaptureTrapA \) and \( \EbiasCaptureTrapB \) and thus different transition rates from neutral the state \( 1 \) to the charged state \( 2 \). Overall, the closer the trap is located to the gate, the larger is the observed dependence of the capture time on the stress bias.

Random Telegraph Noise Signals

While performing single trap measurements, defects producing RTN signals have been observed to be rather the rule than the exception. As the capture and emission time of single traps depend on the applied gate bias, the presence of RTN is also strongly gate bias dependent. As previously discussed, typical RTN signals can be described by a two state model. For experimentally detectable RTN signals, the forward and backward transition rates have to be of the same order, otherwise the single charge capture and emission events are not clearly visible in the measured traces.

Quite remarkably, twenty-eight years ago RTN signals have been observed which occasionally disappear and reappear, a phenomenon termed anomalous random telegraph noise (aRTN) [126], see Figure 7.12 (left). Such defects produce an RTN signal for a limited amount of time until the signal disappears. After a random amount of time, the signal reappears. In those days this behavior was observed for approximately \( \SI {4}{\percent } \) of the defects. However, it has been recently found that a significant number of defects show the observed volatility. Considering nanoscale devices, NBTI stress can both decrease and increase the number of traps producing RTN [MWC19]. The occurrence of this phenomena is considerably enhanced when stress pulses with positive gate bias which drive the pMOSFET into accumulation are applied . This corresponds to the observation that in large-area devices NBTI stress increases the noise level in the measurement data, [127, 128].

To explain aRTN signals, at least a three state Markov model has to be considered, see Figure 7.12 (left) [89].

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Figure 7.12:  Several defects have been found which produce RTN. A small number of defects incidentally stop to produce RTN. Such signals are termed aRTN and can be described by a three state Markov chain.

Volatility of Defects

As discussed later in Chapter 8, nanoscale devices allow for individual trap identification by their emission time \( \taue \) and their step height \( d \), that is their contribution to the threshold voltage shift. To identify single traps, the emission events of a series of measured traces at certain bias conditions are collected within the so-called spectral map. As shown in Figure 7.13 clusters are visible each representing a single trap.

After probing a single transistor for several weeks or month, defects have been found to disappear and reappear. For instance, defect #6 which shows an emission time \( \taue \approx \SI {1}{\second } \) and a step height of \( d\approx \SI {3}{\milli \volt } \) suddenly disappeared and remained in his new configuration, see Figure 7.13 (top). The same observation was made for defect \( \#7 \), see Figure 7.13 (bottom). Initially, both observations were considered rare events. However, recent investigations showed that there is a remarkable number of traps showing such a volatile behavior. Quite interestingly, volatile defects have been observed in n-channel and p-channel MOSFETs using (math image) and (math image) gate stacks and are thus not limited to any particular technology. As the phenomenon is stochastic, it is very difficult study it systematically. However, these defects will an essential clue on the chemical nature of oxide traps.

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Figure 7.13:  By analyzing the discrete threshold voltage shifts caused by charge emission events of individual traps in nanoscale devices the so-called spectral maps can be created. Every cluster can be attributed to a single trap. As can be seen, numerable defects have been found in the investigated pMOSFET. Quite noteworthy, defect #6 (top row) and defect #7 (bottom row) suddenly disappear independently from each other. In all subsequent experiments both defects remained invisible [MWC12].

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